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The Importance of Variance Analysis

This critical topic is too often taught to only a handful of students—or neglected in the b-school curriculum altogether.

Variance analysis is an essential tool for business graduates to have in their toolkits as they enter the workforce. Over our decades of experience in executive education, we’ve observed that managers across all industries and functions use variance analysis to measure the ability of their organizations to meet their commitments.

Because variance analysis is such a powerful risk management tool, there is a strong case for including it in the finance portion of any MBA curriculum. Yet fewer than half of finance professors believe they should be teaching this subject; they view it as a topic more typically taught in accounting classes. At the same time, in practice, variance analysis is such a cross-functional tool that it could be taught throughout the business school curriculum—but it’s not. We perceive a worrisome disconnect between the way variance analysis is taught and the way it is used in real life.

Variance Analysis and Its Applications

There are three periods in the life of a business plan: prior period to plan, plan to actual, and prior period to actual. For instance, if a business plan is being formulated for 2019, the “prior period” would be 2018, the “plan to actual” would be the budget for 2019, and the “prior period to actual” would be what really happens in 2019. These three stages are also referred to as planning, meeting commitments, and growth.

For each of these periods, variance analysis looks at the deviations between the targeted objective and the actual outcome. The most common variances are found in price, volume, cost, and productivity. When executives conduct an operational review, they will need to explain why there were positive or negative variances in any of these areas. For instance, did the company miss a target because it lost an anticipated national account or failed to lock in a price contract due to competitive pressure?

Executives who understand variances will improve their risk management, make better decisions, and be more likely to meet commitments. In the process, they’ll produce outcomes that can give an organization a real competitive advantage and, ultimately, create shareholder value.

Most businesses apply variance analysis at the operating income level to determine what they projected and what they achieved. The variances usually are displayed in the form of floating bar charts—also known as walk, bridge, or waterfall charts. These graphics are often used in internal corporate documents as well as in investor-facing documents such as quarterly earnings presentations.

While variance analysis can be applied in many functional areas, it is used most often in finance-related fields. Yet, the majority of finance programs at both the graduate and undergraduate levels don’t cover it at all. We surveyed finance faculty in 2013 and accounting faculty in 2017 to determine how they teach and use variance analysis. Among other things, we learned that:

• More than 80 percent of accounting faculty believe that variance analysis is important to a finance career, and they are far more likely to teach it than their finance faculty colleagues.
• Only 59 percent of finance faculty and 48 percent of accounting faculty are familiar with examples of walk charts from real-world companies. Yet these visual portrayals of operating margin variances are commonplace in quarterly earnings presentations and readily found on investor relations websites.

Because universities mostly fail to teach this important topic, corporate educators have been left to fill the learning gap. Many global organizations, in fact, make variance analysis a key subject in their development programs for entry-level financial professionals.

The University Response

We believe it’s critical for universities to better align their curricula with the skills that today’s employers seek in the graduates they hire. Not only do we think variance analysis should be included in the business curriculum, but we could even make an argument for running it as a capstone business course. We offer these suggestions for ways that faculty could integrate this powerful tool across the business school program:

• Both accounting and finance faculty should, as much as is practical, incorporate variance analysis into their classes, particularly focusing on financial planning and analysis. We acknowledge that a dearth of corporate finance texts on the topic will make this a challenge for finance professors. The two of us employ teaching materials in our graduate business and undergraduate finance classes based on experience in the corporate world, and we would be glad to share them with others.
• Faculty who use case studies should always include a case specific to variance analysis tools. Students who pursue careers in corporate finance will almost certainly be required to use such tools, particularly as data and predictive analytics applications are enhanced to improve forecasting accuracy. Two sources of such case studies are TRI Corporation and Harvard Business Publishing.
• Professors can introduce students to real-world applications of variance analysis by showing how it is used in investor relations (IR) pitches. As instructors, the two of us routinely search IR sites for applications of variance analysis. We specifically look for operating margin variance walks (floating bars, brick charts) for visual applications that can make the topic come to life for students. Here’s an example from Ingersoll Rand:

• Faculty from accounting and finance programs should collaborate on when, where, and how to teach variance analysis. At the very least, this will ensure that students gain an understanding of the topic from either a finance or an accounting perspective, but the ideal would be for them to benefit from both perspectives for a holistic understanding. At Fairfield University, accounting programs introduce students to the theory of variance analysis. Then finance programs take an operational and cross-functional approach that addresses planning, meeting commitments, and growth.
• Both accounting and finance faculty should help finance majors understand variance analysis from a practitioner’s standpoint. Discussions about pricing, supply chain, manufacturing costs, risk management, and inflation and deflation around cost inputs can help students grasp the necessity of making trade-offs and balancing short-term and long-term business goals. To make sure students understand the practitioner’s viewpoint, we use corporate business simulations that are more operationally focused, as opposed to being academic in tone.
• To extend the topic to all majors, not just finance and accounting students, faculty from disciplines such as strategy and operations could also incorporate variance analysis into their classes. For instance, if they use business simulations for their capstone courses, they could add a component that covers variance analysis. At Fairfield, we use a variety of competitive business simulations from the corporate world.
• Finally, professors can bring in guest speakers from almost any business functional area and ask them to explain, as part of their presentations, how variance analysis is relevant in their fields. As an example, we often have senior finance executives from Stanley Black & Decker—a company known well-known for its ability to grow and meet its commitments via variance analysis—present to our graduate program. We tap other companies from Fairfield County as well.

In the graduate classes we teach at Fairfield University, we have always tried to connect theory with practice. And we’ve long believed that creating a culture of meeting and exceeding commitments requires aligning interaction across functions in the workplace. With this article, we hope that, at the very least, we can start a larger discussion about the need for cross-disciplinary teaching of variance analysis.

  For more about variance analysis materials, contact us at  [email protected]  or  [email protected] .

Estimation and statistical inferences of variance components in the analysis of single-case experimental design using multilevel modeling

• Published: 10 September 2021
• Volume 54 , pages 1559–1579, ( 2022 )

Cite this article

• Haoran Li   ORCID: orcid.org/0000-0003-0886-4172 1 ,
• Wen Luo 1 ,
• Eunkyeng Baek 1 ,
• Christopher G. Thompson 1 &
• Kwok Hap Lam 1  

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Multilevel models (MLMs) can be used to examine treatment heterogeneity in single-case experimental designs (SCEDs). With small sample sizes, common issues for estimating between-case variance components in MLMs include nonpositive definite matrix, biased estimates, misspecification of covariance structures, and invalid Wald tests for variance components with bounded distributions. To address these issues, unconstrained optimization, model selection procedure based on parametric bootstrap, and restricted likelihood ratio test (RLRT)-based procedure are introduced. Using simulation studies, we compared the performance of two types of optimization methods (constrained vs. unconstrained) when the covariance structures are correctly specified or misspecified. We also examined the performance of a model selection procedure to obtain the optimal covariance structure. The results showed that the unconstrained optimization can avoid nonpositive definite issues to a great extent without a compromise in model convergence. The misspecification of covariance structures would cause biased estimates, especially with small between case variance components. However, the model selection procedure was found to attenuate the magnitude of bias. A practical guideline was generated for empirical researchers in SCEDs, providing conditions under which trustworthy point and interval estimates can be obtained for between-case variance components in MLMs, as well as the conditions under which the RLRT-based procedure can produce acceptable empirical type I error rate and power.

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Single-case experimental designs (SCEDs) are commonly used in research to evaluate the effectiveness of an intervention by repeatedly measuring an outcome of a small number of cases over time. The SCEDs utilize various designs including, but not limited to, AB, ABAB, multiple baseline across cases, multiple baseline across behaviors or settings, alternating treatments, changing criterion, multiple probe across cases, and multiple probe across behaviors or settings (Pustejovsky et al., 2019 ; Shadish & Sullivan, 2011 ). Among them, multiple baseline design (MBD) is the mostly adopted type of SCED design (Pustejovsky et al., 2019 ). MBD is comprised of interrupted time series data from multiple cases, settings, or behaviors where an intervention is introduced sequentially within different time series (Baek & Ferron, 2013 ; Ferron et al., 2010 ; Moeyaert et al., 2017 ; Pustejovsky et al., 2019 ; Shadish & Sullivan, 2011 ). This design enables researchers to attribute changes in level and trend to the intervention instead of external events, history, or maturation, especially when the randomization is introduced in the design (Kratochwill & Levin, 2010 ; Levin et al., 2018 ), which makes it possible to use a randomization test for inferential purposes (Michiels et al., 2020 )

Combining research findings from multiple cases within an SCED study can provide strong evidence for the average treatment effect across cases (Moeyaert et al., 2017 ; Shadish & Rindskopf, 2007 ). In addition to the average treatment effect, it is informative to estimate the variation of an intervention effect across cases. It is likely that a treatment effect is positive on average but has a large variation to the extent that the treatment is not effective or has adverse effects for some cases. In other words, a large variation in the treatment effect indicates that the treatment is not equally effective for all cases. Further analyses may be needed to identify individual characteristics to explain variation in the treatment effect. Thus, the interpretation of the average effect should be accompanied by the associated variation in effect across cases (Barlow et al., 2009 ; Kratochwill & Levin, 2014 ; Moeyaert et al., 2017 ).

There has been a great amount of single-case data analytical techniques including visual analysis, nonoverlap indices, descriptive indices quantifying changes in level and slope, standardized mean difference indices, procedures based on regression analysis, simulation-based procedures, and randomization test (Manolov & Moeyaert, 2017 ). As a very flexible statistical approach, multilevel modeling (MLM) has been proposed to analyze SCED data (Ferron et al., 2009 ; Moeyaert et al., 2014a ; Shadish et al., 2013a ; Van den Noortgate & Onghena, 2003a , 2003b ). In a single study, a two-level model can capture both the average treatment effect (through the estimated fixed effect) and the between-case variability in treatment effects (through estimated variance-covariance components). Extant studies show that the treatment effect can be estimated accurately, and its statistical inferences based on adjusted degrees of freedom have good statistical properties (i.e., type I error rates and coverage rates close to the nominal level) in the analysis of SCED using MLMs (Baek et al., 2020 ; Baek & Ferron, 2013 ; Ferron et al., 2009 ; Moeyaert et al., 2017 ).

For the estimation and statistical inferences of the between-case variation in treatment effects, however, there are several challenges facing researchers when using MLMs. First, researchers may encounter the issue of nonpositive definiteness when using restricted maximum likelihood (REML) estimators in the constrained optimization, leading to inadmissible estimates of variance components. Second, estimates of between-case variance components tend to be biased using either REML or Bayesian approaches (Baek et al., 2020 ; Baek & Ferron, 2020 ; Ferron et al., 2009 ; Hembry et al., 2015 ; Moeyaert et al., 2017 ; Moeyaert et al., 2013a , 2013c ). Though the bias is expected to be mitigated with more cases and/or larger between-case variance components, it is unknown what number of cases is needed and/or what values of the between-case variance components should be to achieve accurate estimates. Third, although the misspecification of covariance structures (i.e., constraining a non-zero covariance to be zero) may have little impact on the accuracy of the estimation of large between-case variance components (Moeyaert et al., 2017 ), it is unknown whether the estimation is accurate when the between-case variance is small or moderate. Last, the commonly used Wald test is not appropriate for testing the between-case variance components in SCEDs because it requires large sample sizes. It also has unfavorable properties when the null value of tested variance component is zero or close to zero (West et al., 2014 ). A sophisticated procedure based on the restricted likelihood ratio test (RLRT) is needed to conduct significance tests and the performance of such tests (i.e., type I error rate and power) has not been systematically examined (Shadish et al., 2013a ).

To fill the gap in the literature, our study serves three purposes. First, we addressed the issue of non-positive definiteness by using the unconstrained optimization through matrix parameterization and compared its performance with the traditional constrained optimization method. Second, we examined the impact of misspecifications of covariance structures and explored the feasibility of using a “post hoc” model selection procedure (test for the covariance component to determine whether it should be kept in the fitted model) to obtain the optimal covariance structure. Third, we introduced and evaluated the performance of the RLRT-based procedure to test the between-case variance components. A colloquial (non-technical) illustration was provided and the empirical type I error rate and power of the RLRT-based procedure for testing the between-case variance components was evaluated. Last, we provided a guideline about the conditions for obtaining accurate estimation and statistical inferences for between-case variance components with MLMs in SCEDs.

The remaining of the paper is organized as follows. First, we review the basic MLM for MBD, the issue of nonpositive definiteness, and an unconstrained optimization approach as a positive solution. Second, we review the performance of MLMs to estimate between-case variance components, the impact of misspecification of covariance structures, and model selection procedure to obtain the optimal covariance structures. Next, we review the RLRT-based procedures for testing between-case variance components. Following this portion of the literature, we present methods for the current simulation study, followed by the results. Last, we discuss the findings, limitations, and future directions.

Multilevel modeling for MBD data

The basic expression of a multilevel model for MBD is

where i and j are the index of measurements and cases, respectively, Phase is a binary variable (0 = baseline and 1 = treatment), γ 00 is the average baseline level, γ 10 is the average treatment effect (i.e., change in level), μ 0 j and μ 1 j are random effects associated with the baseline level and treatment effect for individual j , and e ij is the within-case error term. We assume that μ 0 j and μ 1 j follow a multivariate normal distribution with a mean vector of zeros, variances indicated by \( {\sigma}_{u0}^2 \) and \( {\sigma}_{u1}^2 \) , respectively, and a single covariance term indicated by σ u 0 u 1 (i.e., the G matrix). Although in the basic multilevel model the level-1 error ( e ij ) is assumed to follow an independent and identically normal distribution (i.i.d.), more complex error structures where autocorrelation is assumed, such as the first-order autoregressive, can be applied (Baek & Ferron, 2013 ; Ferron et al., 2009 ). We provide the matrix form of this model in Appendix A.

Non-positive definiteness and unconstrained parameterization

For an estimated variance-covariance G matrix, we refer to G as positive semidefinite or nonnegative definite when all its eigenvalues ≥0, or equivalently \( {\sigma}_{u0}^2\ge 0 \) , \( {\sigma}_{u1}^2\ge 0 \) and \( {\sigma}_{u0}^2{\sigma}_{u1}^2\ge {\sigma}_{u0u1}^2 \) . Positive semidefinite G matrices include positive definite and singular matrices. When some of its eigenvalues =0, a G matrix is referred to as a singular or degenerate matrix with statistically dependent vectors. When all eigenvalues in G are greater than 0, the nonredundant G matrix (i.e., random effects are not perfectly correlated) is positive definite, indicating that \( {\sigma}_{u0}^2>0 \) , \( {\sigma}_{u1}^2>0 \) and \( {\sigma}_{u0}^2{\sigma}_{u1}^2>{\sigma}_{u0u1}^2 \) .

Some optimization algorithms to implement REML estimation impose constraints on the parameters to ensure that estimated G matrices are positive definite during the iteration process (i.e., constrained optimization; West et al., 2014 ). However, such constraints are not imposed at the last iteration when the estimation converges. Hence, it is possible to obtain nonpositive definite G matrices as the final estimates, especially when the cluster-level sample size is small (e.g., a small number of cases in SCEDs) because little information is available for estimating variance components (Chung et al., 2015 ).

Two primary scenarios that would lead to nonpositive definiteness are as follows. First, a variance component estimate might be close to 0 (accounting for the finite numeric precision of a computer) or outside the parameter space (i.e., estimated to be negative). This arises given the lack of variation in the random effects, after controlling for the fixed effects. When this situation is encountered, statistical software, such as Proc MIXED in SAS and MIXED in SPSS, would provide a warning (e.g., ‘the estimated G matrix is not positive definite’ in SAS or ‘this covariance estimate is redundant’ in SPSS) and by default set the variance estimate equal to 0 (West et al., 2014 ). Second, there is also a possibility that estimates of between-case variance components are significantly larger than 0, while the G matrix remains nonpositive definite. This occurs when the condition \( {\sigma}_{u0}^2{\sigma}_{u1}^2>{\sigma}_{u0u1}^2 \) is not satisfied. Similar warning messages are provided in this scenario (e.g., ‘the estimated G matrix is not positive definite’ in SAS). Under both scenarios, applied researchers often choose to either ignore warning messages and report variance components estimates regardless or not report the estimates with concerns about nonpositive definiteness.

The issue of nonpositive definiteness has an impact on methodological investigations, which rely on Monte Carlo simulation. If there is a large proportion of nonpositive definite cases, results should be interpreted with extreme caution and studies that evaluate estimators would risk severe bias if conclusions are only based on cases with positive definite matrices (Gill & King, 2004 ). The issue of nonpositive definiteness also has an impact on empirical studies. As Kiernan ( 2018 ) indicated, when the G matrix is nonpositive definite, we may not compare parameter estimates across different statistical programs. Moreover, the nonpositive definiteness encountered in the last iteration in statistical packages could lead to inadmissible solutions (Demidenko, 2013 ; Stram & Lee, 1994 ).

Several approaches have been suggested to resolve the problem of nonpositive definiteness. The most intuitive practice is to remove random effects associated with zero variance estimates and re-conduct analyses (Kiernan, 2018 ). This is acceptable in the first scenario where there is not much variation. Under the second scenario, however, it is difficult to determine whether the issue is caused by variance components or covariance. Alternatively, if the interest is only on the fixed effects, one can remove the positive definiteness constraints for the G matrix via use of the ‘nobound’ option in SAS (West et al., 2014 ). Nonetheless, it is common to encounter negative variance estimates (i.e., G matrix is not positive semidefinite), which are difficult to interpret. Lastly, a more promising approach to deal with nonpositive definiteness is to employ different types of matrix parameterization in the iteration process (Pinheiro & Bates, 1996 ). Matrix parameterization can reduce a constrained optimization problem (i.e., G matrices are forced to be positive definite in the iteration process) to an unconstrained problem and ensures that G matrices are at least positive semidefinite throughout the entire estimation process (Demidenko, 2013 ; West et al., 2014 ). More specifically, the upper-triangular elements in the G matrix can be reparametrized in the way that the resulting estimate must be positive semidefinite (McNeish & Bauer, 2020 ).

Next, we briefly describe a commonly used parameterization, namely, log-Cholesky parameterization, for unstructured G matrices and for diagonal G matrices where we assume that the random effects are independent. Let G denote a q  ×  q positive semidefinite variance-covariance matrix. As G is symmetric, there are only q ( q  + 1)/2 (i.e., G is unstructured) or q (i.e., G is diagonal) unconstrained parameters. We denote a minimum set of unconstrained parameters by a vector θ . The G matrix is represented by G  =  L T L , where L  =  L ( θ ) is a q  ×  q matrix. Any G matrix defined as the above decomposition is positive semidefinite (Pinheiro & Bates, 1996 ). In log-Cholesky parameterization, L is an upper triangular matrix and θ includes the logarithms of the diagonal elements of L and the upper off-diagonal elements. For example, if \( G=\left[\begin{array}{cc}2& 1\\ {}1& 2\end{array}\right] \) , the Cholesky decomposition of \( G=\left[\begin{array}{cc}1.414& 0\\ {}0.707& 1.225\end{array}\right]\left[\begin{array}{cc}1.414& 0.707\\ {}0& 1.225\end{array}\right] \) . The log-Cholesky parameterization of G is θ  = (log (1.414), 0.707, log (1.225)) T . In models where the covariance is constrained to be 0, G becomes a diagonal matrix that can be parametrized by fewer parameters in θ . Under this situation, the matrix can be simply parameterized by the logarithm of the diagonal elements of L . For example, if \( G=\left[\begin{array}{cc}9& 0\\ {}0& 4\end{array}\right] \) , the Cholesky decomposition of \( G=\left[\begin{array}{cc}3& 0\\ {}0& 2\end{array}\right]\left[\begin{array}{cc}3& 0\\ {}0& 2\end{array}\right] \) . The parameterization of G is θ  = (log (3), log (2)) T .

It should be noted that matrix parameterization with a diagonal G matrix in some software ensures positive definiteness (e.g., nlme package in R), which is a stronger assumption compared to the positive semi-definiteness. Thus, maximum likelihood estimates may not exist when the true variance component is 0. Demidenko ( 2013 ) showed that the estimated model was more likely to fail to converge when matrices were forced to be positive definite using matrix parameterization and concluded that the best way to cope with this issue is not to cope at all. Put another way, they follow the principle that it is better to get any solution (even negative variance estimates are informative as it may shed light on which random effect is problematic and thus can be omitted) than no solution at all, as in the case of failing to converge.

In summary, there is a controversy on whether and how to deal with nonpositive definiteness. The impact of using constrained optimization versus using the unconstrained optimization on the performance of MLMs for SCED should be systematically examined. We expect that the impact of nonpositive definiteness on the estimates of variance components is trivial in the first scenario (i.e., when there is a lack of variation in the random effects). In this case, the final estimates of the variance components will be set to 0 when the G matrix is not positive definite in most programs that implement the constrained optimization method. In the second scenario (i.e., when the condition \( {\sigma}_{u0}^2{\sigma}_{u1}^2>{\sigma}_{u0u1}^2 \) is not satisfied), we expect that the constrained optimization would yield inadmissible correlation estimates between random effects (i.e., r < −1.0 or r > 1.0).

Estimation of between-case variance components

Several simulation studies examined the performance of MLMs with REML to estimate the variance components in the MBD. In general, the estimates of between-level variance components were biased, however, the directions of the biases were not consistent. Some studies found moderate to large positive biases in the estimates of variance in the treatment effect (e.g., Ferron et al. 2009 , Moeyaert et al., 2013a , 2013c ), especially when the true variance component was small (e.g., \( {\sigma}_{u1}^2<.\mathrm{0.5} \) ). Some studies found small to moderate negative

biases (Baek et al., 2020 ; Joo & Ferron, 2019 ; Moeyaert et al., 2017 ) when the variance component was moderate ( \( {\sigma}_{u1}^2=.\mathrm{0.5}\ \mathrm{or}\ 2.0 \) ). For estimates of level-1 error variance, readers are referred to simulation studies with emphasize on the level-1 error structures (see Baek & Ferron, 2013 , 2020 ; Joo et al., 2019 ). In summary, with different magnitude of between-case variance, the estimates of between-case variance components are biased in general, which made it difficult to decide the degree of heterogeneity in treatment effects among cases.

As previous findings are inconsistent, it is necessary to replicate the simulation studies and improve the generalizability of the findings by considering a wider range for the magnitude of the variance components. Besides, previous studies based on REML with constrained optimization (e.g., default optimization method with SAS Proc MIXED) may eliminate nonpositive definite cases and thus the observed bias could be partly due to high rates of non-positive G matrix, especially when variance components are small. Hence, it is important to re-examine the estimates using constrained and unconstrained optimization methods. Last, previous studies adopted a diagonal covariance structure, but this practice can lead to underfitted covariance structures in reality (Moeyaert et al., 2016 ), which we will elaborate in the following section. Therefore, there is a need to revisit the estimation of variance components with consideration of a wider range of between-case variance components, different optimization methods, and different covariance structures through the basic model with REML. We hope that this can serve as the starting point to clarify the direction of the bias, and in which conditions the estimates of between-case variance components are trustworthy with REML.

Specification of covariance structures

In addition to the issues of non-positive definiteness and the general issue of biased estimation of variance components due to small sample sizes, the misspecification of covariance structures could also impact the estimates of between-case variance components. A common type of misspecification is to ignore the covariance when the true covariance is nonzero, which leads to an underfitted model. Previous studies in contexts of general MLM showed that underfitting covariance structures could result in biased variance estimates and inflated type I error rate for the test of fixed effects (Barr et al., 2013 ; Hoffman, 2015 ). In contrast, another type of misspecification is to assume a full covariance structure when there is no correlation between random effects, leading to an overfitted model. This could result in loss of power to detect significant fixed effects especially when the sample size is small (Matuschek et al., 2017 ). In SCEDs, a nonzero covariance indicates that the treatment effect is correlated with the baseline level, which is a common phenomenon (Moeyaert et al., 2016 ). However, previous studies to combine SCED data across cases and studies using MLMs typically assumed a diagonal covariance structure (Denis et al., 2011 ; Moeyaert et al., 2013b , 2014b ; Van den Noortgate & Onghena, 2003a , 2007 ), which could lead to an underfitted covariance structure. On the other hand, overfitting is possible when treatment effects are unrelated to baseline levels. Moeyaert et al. ( 2016 ) evaluated consequences of the misspecification of covariance structures (i.e., overfitting and underfitting) and found that it had no effect on the estimation and inferences for the treatment effect, and the between-case variance estimates were also unbiased in both correctly specified and misspecified covariance structures. However, the authors only considered conditions where the true variance components are large (i.e., \( {\sigma}_{u0}^2={\sigma}_{u1}^2=2\ \mathrm{or}\ 8 \) ). It is unknown whether the findings can be generalized to conditions where the true between-case variance is small in size.

Without theories or prior knowledge about the relationship between baseline levels and treatment effects among cases, empirical researchers may seek a “post hoc” model selection procedure based on model fit to determine an optimal covariance structure. Due to the small sample sizes in SCEDs, the commonly used RLRT statistic to compare model fit does not follow a Chi-square distribution. Likewise, the information criterion (e.g., AIC and BIC) is not appropriate because they become equivalent to RLRTs at different alpha levels (e.g., AIC is equivalent to RLRTs at alpha = .157) and thus inherit the issue of the regular RLRT when the sample size is small. Hence, a parametric bootstrap (PB) approach (see Davison & Hinkley, 1997 ) based on the RLRT was proposed to determine whether the covariance is statistically significantly different from 0. PB simulates the null distribution the RLRT statistic, upon which an approximated p value can be calculated. Specifically, the first step is to simulate B (e.g., B = 500) bootstrap samples from the null model (i.e., a reduced model without the tested covariance). Then, the differences between the deviance of the reduced and the full model for each simulated date set are computed (e.g., T ∗  = { t 1 , …,  t B }, where  t is the RLRT statistic). Last, the observed deviance difference ( t obs ) from the empirical data is compared against the simulated null distribution provided by T ∗ and the approximated p value is calculated as follows:

where n extreme is the number of values in T ∗ that are equal or greater than the t obs .

The significant level for test based on PB should be chosen with caution. Within the context of model selection, we should not simply assume alpha =.05. Indeed, the significance level should be considered as the relative weight of model complexity and goodness of fit (Matuschek et al., 2017 ). When alpha = 0, an infinity penalty on the model complexity is implied and we always choose a reduced model, irrespective the evidence provided by the data. However, when alpha = 1, an infinite penalty on the lack of fit is implied and we always choose a more complex model. From this perspective, choosing alpha = .05 implies an overly strong penalty for model complexity. Following the same practice in Matuschek et al. ( 2017 ), we choose alpha = .20 for the PB test to select an optimal covariance structure.

Significance tests for variance components

There are two scenarios in which the significance tests for variance components are conducted. In the first scenario, the diagonal G matrix is adopted (either assumed or based on the model selection procedure). In this case, the null and alternative hypotheses are:

We obtain the difference in the -2*log(restricted likelihood) between two nested models, that is, the constrained model in which the tested parameter is constrained to 0, and the reference model in which the tested parameter is freely estimated. Using asymptotic theory, regular restricted likelihood ratio test assumes that the difference follows a χ 2 distribution with degrees of freedom equal to the difference in the number of covariance parameters. However, there is a case where the test of a variance component is not commonplace because the tested parameter value in the null hypothesis is at the boundary of the parameter space. Stram and Lee ( 1994 ) found that the regular likelihood ratio test is overconservative when the null value of between-case variance component is 0. In this case, inference based on the regular restricted likelihood ratio test is not accurate.

More accurate sampling distributions of the RLRT statistic have been proposed to deal with this boundary issue in testing variance components in MLM. Self and Liang ( 1987 ) and Stram and Lee ( 1994 ) showed that the RLRT for the presence of a single variance component has an asymptotic 0.5 \( {\chi}_0^2 \) : 0.5 \( {\chi}_1^2 \) mixture distribution under i.i.d. assumptions. We denote this test as RLRT SL where \( {\chi}_0^2 \) implies a probability mass at 0 and \( {\chi}_1^2 \) is the Chi-square distribution with df  = 1. The calculation of p value for the RLRT SL is

where χ 2 is the difference in the -2*log(restricted likelihood) between the nested model and the reference model.

The second scenario for testing variance components is when the unstructured G matrix is adopted (either assumed or based on the model selection procedure). In this case to formally obtain the p value, we need to test the variance and covariance simultaneously (i.e., both the variance and covariance are equal to 0 in the null hypothesis) because it is impossible to specify a reduced model in which the covariance is nonzero but the variance being tested is 0. Similar to the case when testing only one parameter, the RLRT SL has an asymptotic 0.5 \( {\chi}_1^2 \) : 0.5 \( {\chi}_2^2 \) mixture distribution under i.i.d. assumptions, while \( {\chi}_1^2 \) and \( {\chi}_2^2 \) are the chi-square distributions with df  = 1 and 2, respectively. We calculate p value for the RLRT SL as

where χ 2 is the difference in the -2*log(restricted likelihood) between the nested model and the reference model. In addition, when the i.i.d. assumption is violated, the generalized least squares (GLS) transformation (Wiencierz et al., 2011 ) can be applied to RLRT SL to account for autocorrelated errors in the context of SCED. Details of the transformation are presented in Appendix B .

Confidence intervals for variance components

It is well known that the Wald-type confidence interval (CI) performs poorly for small data sets and for parameters like variance components, which are known to have a skewed or bounded sampling distribution (SAS Institute, 2017 ). One alternative approach that can be applied in the MLM framework is the Satterthwaite approximation (Satterthwaite, 1946 ), which constructs confidence intervals for parameters with a lower boundary of 0. To calculate the bounds of the CI, we use

where the degrees of freedom v  = 2 Z 2 , Z is the Wald statistic \( {\hat{\sigma}}^2/ SE\left({\hat{\sigma}}^2\right) \) , and the denominators are 1 −  α /2 and α /2 ( α = .05 by default) quantiles of the χ 2 distribution with v degrees of freedom. The small sample size properties of this approximation in the context of SCED with presence of autocorrelation have not been examined so far.

Simulation conditions

To examine the performance of MLMs with different optimization methods and covariance structures, and the performance of the RLRT-based procedure for the between-case variance component, two Monte Carlo simulation studies were conducted. In Simulation Study 1, we generated data using Eq. ( 1 ) with zero covariance between the baseline level and treatment effect, and varied following design factors: series length I , number of cases J , treatment effect ( γ 10 ), between-case variance ( \( {\sigma}_{u0}^2 \) and \( {\sigma}_{u1}^2 \) ) and autocorrelation ( ρ ). In Simulation Study 2, an additional design factor was varied, namely, the correlation between the baseline level and treatment effect ( r ).

The number of cases J simulated in each study took on a value of 4 or 8, and the simulated series length took on a value of 10 or 20. This represents characteristics of SCED research based on the reviews of single case analysis (Pustejovsky et al., 2019 ; Shadish & Sullivan, 2011 ) and are consistent with previous simulation studies (Baek et al., 2020 ; Ferron et al., 2009 ; Moeyaert et al., 2017 ). Because we focused on the MBD across cases design, the start of the treatment is staggered across cases as shown in Table 1 . These staring points were chosen so that both the baseline and treatment phase contained a sufficient number of measurements. For all conditions, the intercept γ 00 was set to 0. To identify values for treatment effects and variance-covariance components that are authentic in SCED data, we reviewed five meta-analyses of single-case studies (Alen et al., 2009 ; Denis et al., 2011 ; Kokina & Kern, 2010 ; Shogren et al., 2004 ; Wang et al., 2011 ) and a reanalysis with MLMs that quantified the treatment effects and the variance components based on the data from above five meta-analyses (Moeyaert et al., 2017 ). According to the reanalysis results, a median standardized treatment effect size was 2.0. Hence, we set the treatment effect ( γ 10 ) to be either 2.0 or 0. In terms of variance components, the results of the reanalysis for the baseline level ( \( {\sigma}_{u0}^2 \) ), treatment effect ( \( {\sigma}_{u1}^2 \) ), within-case residuals ( \( {\sigma}_e^2 \) ) is provided in Table 2 (for full results, see Moeyaert et al., 2017 ). For between-case variance components, the estimates covered a wide range of .27 to 7.96 from five meta-analyses with similar number of reviewed studies. The relatively large standard errors also indicate uncertainty of the point estimation in each meta-analysis. Thus, to reflect the large range and uncertainty of the between-case variance component estimates, \( {\sigma}_{u0}^2 \) and \( {\sigma}_{u1}^2 \) were given a value of 0, .3, .5, 1.0, 2.0, 4.0, or 8.0 in Simulation Study 1 (zero covariance) and .3, .5, 1.0, 2.0, 4.0, or 8.0 in Simulation Study 2 (nonzero covariance). This setting also covers the range of adopted values for between-case variance components in previous simulation studies (Baek et al., 2020 ; Baek & Ferron, 2020 ; Ferron et al., 2009 ; Hembry et al., 2015 ; Moeyaert et al., 2017 ; Moeyaert et al., 2013a , 2013c , 2016 ).

In Simulation Study 1, covariance σ u 0 u 1 was set to 0. In Simulation Study 2, covariances σ u 0 u 1 were generated based on negative correlations between the baseline level and treatment effect found in the above meta-analyses (Alen et al., 2009 ; Denis et al., 2011 ; Kokina & Kern, 2010 ; Shogren et al., 2004 ; Wang et al., 2011 ) and the reanalysis using MLM (Moeyaert et al., 2016 ). Specifically, we used the lower and upper bound of the correlation (i.e., r  =  − .3 or − .7) following Moeyaert et al. ( 2016 ).

For within-case variance, the results based on the reanalysis were quite similar each other with estimates in the range of 1.000 to 1.146 and small standard errors. Thus, the within-case error was generated with variance of 1.0 and assumed homogeneous across phases in both simulation studies. In terms of autocorrelations among repeated measures in SCEDs, Shadish and Sullivan ( 2011 ) found that the average estimate for single-cases studies with MBD was .32. Considering the typical autocorrelated values found in behavior data and used in previous simulation studies (Ferron et al., 2009 ; Shadish & Sullivan, 2011 ; Sideridis & Greenwood, 1997 ), the autocorrelation ρ was set to .2, .4 or .6 in Simulation Study 1 and 2. The summarized design factors and conditions are presented in Table 3 . Combining all the design factors, there were a total of 168 conditions and 288 conditions in Simulation Study 1 and 2, respectively. In each condition, 2000 independent data sets (i.e., replications) were simulated.

In both simulation studies, we analyzed data by Eq. ( 1 ) with various G matrices and REML estimation with either constrained or unconstrained optimization. Specifically, we considered the following five different combinations: unconstrained optimization with a diagonal matrix (UD), constrained optimization with a diagonal matrix (CD), unconstrained optimization with an unstructured matrix (UU), constrained optimization with an unstructured matrix (CU), and unconstrained optimization with the model selection procedure (UM). The UM was implemented through the parametric bootstrap with the GLS transformation. The constrained optimization with model selection procedure was not considered due to the low convergence rate with a small number of cases ( J  = 4). The unconstrained optimization was implemented by using the lme () function in the nlme package in R 4.0.3 (Pinherio et al. 2019 ) and the constrained optimization was implemented in SAS 9.4 using Proc MIXED. Kenward-Roger method was used to estimate degree of freedom and standard error for the treatment effect (Kenward & Roger, 1997 ). R code for the estimation with the model selection procedure, GLS transformation, parametric bootstrap, and RLRT SL are provided in Appendix C . Both the R and the SAS code for the simulation is available from the Open Science Framework project (see the link in the open practices statement).

Performance measures

Outcomes examined included the convergence rate and positive definiteness rate, as well as relative bias, mean squared error (MSE) and coverage rate of the 95% confidence intervals for the estimated parameters. Outcomes were calculated based on converged replications except for the bias of correlation estimates (inadmissible estimates caused by nonpositive definiteness were excluded). The relative bias is the absolute bias divided by the population true value Footnote 1 . Bias in a parameter estimate was considered to be acceptable if the relative bias was less than 5% (Carsey & Harden, 2013 ). The MSE was calculated as the expected value for the squared differences between the estimates and population parameters. The coverage rate of the 95% confidence interval was calculated as the proportion of the 95% confidence intervals that contains the true population values. To account for simulation error caused by a finite number of repetitions in our simulation, we computed the bounds around the coverage rate which follows a binomial distribution: \( p\pm 1.96\times \sqrt{p\left(1-p\right)/r} \) , where p is the expected probability (0.95) and r is the number of repetitions (2000). Therefore, we consider coverage rates between 94.0% and 96.0% ( \( 0.95\pm 1.96\times \sqrt{0.95\left(1-0.95\right)/2000} \) ) to be acceptable.

To evaluate the performance of RLRT SL , empirical type I error rate and power were calculated. The empirical type I error rate was computed as the proportion of repetitions where the true variance component was 0, but the p value of the test is less than .05 (i.e., the nominal α level). Empirical power was computed as the proportion of repetitions where the true variance component is non-zero and the p -value of the test is less than .05. With 2000 simulated datasets, we expect that the empirical type I error rate ranges from .04 to .06 ( \( .05\pm 1.96\times \sqrt{0.95\left(1-0.95\right)/2000} \) ).

We expect that the positive definiteness rate for the unconstrained optimization is significantly higher than that of the constrained optimization in both studies. When diagonal matrix is correctly specified (i.e., models specified with diagonal matrix in Study 1), only the first scenario of nonpositive definiteness will occur, and thus we expect that differences in variance components estimates are small between constrained and unconstrained optimization. With unstructured matrix, however, estimates of the correlation between random effects are likely to be inadmissible (| r | > 1) for constrained optimization, thus leading to more nonpositive definiteness than unconstrained optimization in both studies. We also expect that the misspecification of covariance structures would cause biased estimates of between-case variance components and the model selection procedure would mitigate the biases.

In each simulation study, we first compared combinations of covariance structures and optimization methods (i.e., five model specifications: UD, CD, UU, CU, and UM) in terms of the model convergence rate and positive definiteness rate of the estimated G matrix. We then analyzed relative bias, coverage rate, MSE for the estimates of between-case variance components. We further investigated the bias/relative bias for the estimates of the covariance and correlation between random effects, and the impact of nonpositive definiteness on their estimates. Last, we illustrated the type I error rate and power for the RLRT-based procedure to test between-case variance components. The results for the average baseline level, average treatment effect, level 1 residual variance and autocorrelation are not primary focus of this study and thus are provided in the supplemental material (see Table S1 for Study1 and Table S2 for Study 2).

To study the impact of the design factors on the above performance outcomes, we conducted a series of ANOVA analyses. For the positive definiteness rate, and performance outcomes for the estimates of between-case variance and covariance, we conducted a mixed ANOVA with the between-subject design factors including series length, number of cases, treatment effect, between-case variance, correlation (only included in Study 2), autocorrelation, and the within-subject factor of model specification. For type I error rate and power in Study 1, we investigated whether they were dependent on the design factors. As the nonzero covariance is generated and thus the variance is nonzero in Study 2, we only investigated whether the power was dependent on the design factors in each procedure. Main effects and two-way interaction effects were estimated. We also calculated the η 2 to determine whether effect sizes were small (.01), medium (.06) or large (.14) based on the benchmarks given by Cohen ( 1988 ). The overview of η 2 for the between-case variance components is shown in supplemental Tables S3 to S6 . The overview of η 2 for the covariance and correlation is shown in supplemental Table S7 . We focused on the most meaningful and important findings in the following sections, that is, the impact of model specification on the relative bias for between-case variance components, and the type I error rate and power of statistical inferences. To avoid discussing trivial effects, we will primarily look at the design factors whose main effects and/or associated interaction effects have large effect sizes ( η 2 ≥ .14). Results presented in following tables and figures varied according to these high-impact factors. All results highlighted in the results section were statistically significant at .05 alpha (actually, p < .001).

Convergence rates

For the convergence rate, most of the variability was associated with the model specification (Study 1: F (4, 624) = 646.71, η 2  = .36; Study 2: F (4, 1104) = 1126.55, η 2  = .36) and its interaction with the number of cases (Study 1: F (4, 624) = 489.84, η 2  = .27; Study 2: F (4, 1104) = 936.44, η 2  = .30). As shown in Table 4 , the convergence rates were high under all conditions (above 98%) except when the unstructured matrix was specified, and the constrained optimization was used (CU) for condition of four cases (convergence rate = 81.12% and 83.30% for Study 1 and 2, respectively).

Positive definiteness

For the positive definiteness rate, most of the variability was associated with the model specification (Study 1: F (4, 624) = 5246.23, η 2  = .46; Study 2: F (4, 1104) = 10330.90, η 2  = .46), between-case variance (Study 1: F (6, 117) = 4429.57, η 2  = .18; Study 2: F (5, 232) = 2755.53, η 2  = .16), and their interaction (Study 1: F (24, 624) = 488.21, η 2  = .26; Study 2: F (20, 1104) = 935.01, η 2  = .21). As shown in Table 5 , the unconstrained optimization with diagonal matrix had 100% positive definiteness rates across all conditions. For unconstrained optimization with unstructured matrix (UU) or model selection procedure (UM), positive definiteness rates were consistently high across levels of the between-case variance (above 95%). On the other hand, as between-case variance became small, positive definiteness rates decreased dramatically with constrained optimization (i.e., CD and CU). Overall, the positive definiteness rate was less than 40% on average when the variance was small (<.3).

Between-case variance components

Relative bias.

In Study 1 with zero covariance, the relative bias of the variance in the baseline level was mostly associated with the size of the variance in the baseline level ( F (5, 99) = 120.94, η 2  = .32), model specification ( F (4, 532) = 359.86, η 2  = .13), and their interaction ( F (20, 532) = 69.77, η 2  = .13). As illustrated in Fig. 1 , overfitting the G matrix (i.e., UU and CU) would overestimate the variance of the baseline level when the variance component is small (i.e., < 0.5). Although UD and CD were better than the others when the variance of the treatment effect was small, the differences became smaller as the variance component increased. As expected, the model selection procedure (UM) slightly mitigated the bias caused by the misspecification.

Relative bias in the baseline level variation for different model specifications as a function of variance in simulation Study 1(zero covariance). Note . UD = unconstrained and diagonal; CD = constrained and diagonal; UU = unconstrained and unstructured; CU = constrained and unstructured; UM = unconstrained and model selection

Similar results were found for the variance of the treatment effect. The size of the variance in the treatment effect ( F (5, 99) = 572.82, η 2  = .46) had statistically significant and large impact on the relative bias. The model specification ( F (4, 532) = 216.58, η 2  = .07) and its interaction with the variance ( F (20, 532) = 51.21, η 2  = .08) only had statistically significant and medium effects. Although the pattern was similar to the variance of the baseline level, one main difference was that when the size of the variance of the treatment effect was small (< 0.5), regardless of model specification methods, the variance was overestimated Fig. 2 .

Relative bias in the treatment effect variation for different model specifications as a function of variance in simulation Study 1 (zero covariance). Note . UD = unconstrained and diagonal; CD = constrained and diagonal; UU = unconstrained and unstructured; CU = constrained and unstructured; UM = unconstrained and model selection

In Study 2 with nonzero covariance, the relative bias of the variance in the baseline level was largely associated with the model specification ( F (4, 1104) = 4211.35, η 2  = .45) and its interaction with the variance in the baseline level ( F (20, 1104) = 424.35, η 2  = .23). As illustrated in Fig. 3 , ignoring the covariance would cause negatively biased variance estimate of the intercept when the variance was 4 or less whereas specifying an unstructured covariance matrix (i.e., UU and CU) could result in positively biased variance estimate of the intercept when the variance was 0.5 or less. As expected, the model selection procedure (UM) had the best performance with acceptable relative biases for all sizes of variance except when the variance was small (0.3).

Relative bias in the baseline level variation for different model specifications as a function of variance in simulation Study 2 (nonzero covariance). Note . UD = unconstrained and diagonal; CD = constrained and diagonal; UU = unconstrained and unstructured; CU = constrained and unstructured; UM = unconstrained and model selection

Similar patterns were found for the variance of the treatment effect in Study 2. The model specification ( F (4, 1104) = 2248.08, η 2  = .33) and its interaction with the variance in the treatment effect ( F (20, 1104) = 283.58, η 2  = .21) had statistically significant and large effects on the relative bias. However, it seems to be more challenging to obtain accurate estimates of the variance of the treatment effect than the baseline level. As illustrated in Fig. 4 , the variance of the treatment should be at least 1.0 in order for the relative biases to fall within the acceptable level even using the best performing method (i.e., model selection procedure (UM)). Table 6 showed the relative biases under various conditions based on the model selection procedure Footnote 2 .

Relative bias in the treatment effect variation for different model specifications as a function of variance in simulation Study 2 (nonzero covariance). Note . UD = unconstrained and diagonal; CD = constrained and diagonal; UU = unconstrained and unstructured; CU = constrained and unstructured; UM = unconstrained and model selection

Coverage rates and MSE

The coverage rate for the variance of the baseline level in both studies was largely associated with the amount of variance (Study 1: F (5, 99) = 496.51, η 2  = .60; Study 2: F (5, 232) = 701.78, η 2  = .42). As the variance increased, the average coverage rates approached the nominal level of .95. The effect of model specification had small to moderate effect (Study 1: F (4, 532) = 263.43, η 2  = .02; Study 2: F (4, 1104) = 1319.49, η 2  = .12). The average coverage rate for all model specifications were close to the nominal level (Study 1: range of .943 to .954; Study 2: range of .945 to .9). For model selection procedure, the coverage rate was close to the nominal level when variance in the baseline level was larger than 1.0.

Similar patterns were found for the coverage rate of the variance of the treatment effect in both studies. The most influential factor was the size of the variance (Study 1: F (5, 99) = 1183.88, η 2  = .74; Study 2: F (5, 232) = 2086.93, η 2  = .56). As the variance increased, coverage rates also approached the nominal level of .95. The effect of model specification was small to moderate (Study 1: F (4, 532) = 177.16, η 2  = .01; Study 2: F (4, 1104) = 1115.49, η 2  = .09). The average coverage rates for different model specifications were slightly below or close to the nominal level (Study 1: range of .926 to .941; Study 2: range of .926 to .959). For model selection procedure (UM), the coverage rate was close to the nominal level when variance in the treatment effect was larger than 1.0.

MSE can be very useful to compare different methods in terms of efficiency of estimators. In both studies, MSE was mostly associated with the between-case variance ( η 2 in the range of .7931 to .8117) as MSE inherently depended on the nominal value of the between-case variance. Model specification had statistically significant but trivial effects on the MSE ( η 2 in the range of .0002 to .0005), but it showed slight disadvantage for the constrained optimization with unstructured matrix (CU). We provided the average MSE for each specification in Table 7 .

Covariance and correlation

The bias of covariance estimates was largely associated with the number of cases (Study 1: F (1, 117) = 100.60, η 2  = .24; Study 2: F (1, 232) = 99.47, η 2  = .19). In addition, the between-case variance had a medium to large effect on the biases (Study 1: F (6, 117) = 11.89, η 2  = .17; Study 2: F (5, 232) = 100.60, η 2  = .09). Overall, the bias/relative bias of the covariance estimates tended to be underestimated in Study 1 (average bias = −.052) and in Study 2 (average bias = −.032; average relative bias = −.151). As illustrated in Table 8 , the magnitude of the biases/relative biases decreased with larger number of case and size of between-case variance.

As expected, correlation estimates in constrained optimization were out of boundary (i.e., inadmissible) when nonpositive definiteness was encountered in both studies. Therefore, large proportions (same as the nonpositive definiteness rate) of inadmissible correlation estimates were obtained in the constrained optimization, especially with small amount of the between-case variance. On the other hand, all correlation estimates were admissible in the unconstrained optimization. Therefore, we only evaluated the bias/relative bias of the correlation estimates in the unconstrained optimization and found that they tended to be underestimated in Study 1 (average bias = −.031) and overestimated in Study 2 (average bias = .099; average relative bias = .216). As illustrated in Table 8 , ANOVA results showed that the magnitude of bias in correlation estimates reduced with larger between-case variance (Study 1: F (6, 117) = 1781.92, η 2  = .90; Study 2: F (5, 232) = 166.29, η 2  = .35). In addition, the interaction effect between the variance and correlation is significant and large in Study 2 ( F (5, 232) = 73.27, p  < .001, η 2  = .15). Thus, with larger magnitude of the correlation, the bias would decrease more rapidly when the between-case variance increased.

Empirical type I error rates

The empirical type I error rates of the RLRT SL using unconstrained optimization Footnote 3 were examined based on Study 1. For both variance components, the empirical type I error rates were mostly associated with series length (baseline level: F (1, 9) = 468.62, η 2  = .39; treatment effect: F (1, 9) = 145.30, η 2  = .40), autocorrelation (baseline level: F (2, 9) = 209.46, η 2  = .35; treatment effect: F (2, 9) = 25.13, η 2  = .14), and model specification (baseline level: F (2, 36) = 511.71, η 2  = .15; treatment effect: F (2, 36) = 380.75, η 2  = .37). The empirical type I error rates of RLRT SL based on the series length, autocorrelation and model specification were shown in Table 9 . For model selection procedure with typical autocorrelation ( ρ  = .32) found in MBD (see Shadish & Sullivan, 2011 ), the type I error rates for variance in the baseline level and treatment effect were close to the nominal level when the series length was 20 and were inflated when the series length was 10, regardless of the number of cases. For the model specification with diagonal or unstructured matrix, the type I error rates were even closer to the nominal level.

Empirical power

In Study 1, the empirical power of the RLRT SL for the variance components was mostly associated with the amount of the variance (baseline level: F (5, 99) = 945.64, η 2  = .71; treatment effect: (5, 99) = 3898.30, η 2  = .85). In addition, there were medium effect of the series length (baseline level: F (1, 99) = 477.65, η 2  = .06; treatment effect: F (1, 99) = 592.36, η 2  = .03) and number of cases (baseline level: F (1, 99) = 945.64, η 2  = .13; treatment effect: F (1, 99) = 2014.59, η 2  = .09). Similar results were found for the empirical power in Study 2. Tables 10 and 11 showed the empirical power under various combinations of variance size, number of cases, and series length for Study 1 and 2, respectively. For the variance of the baseline level, the power reached the commonly accepted .80 threshold without inflated type I error rate when the series length = 20 in conditions where the amount of variance ≥ 2.0 and the number of cases = 4, or when the amount of variance ≥ 1.0 and the number of cases = 8 in both studies. For the variance of the treatment effect, the power reached .80 without inflated type I error rate when the series length = 20 in conditions where the amount of variance ≥ 4.0 and the number of cases = 4, or when the amount of variance ≥ 2.0 and the number of cases = 8 in both studies. As model specification had little impact on the empirical power of the test (all η 2  < .01), the power for model specification with diagonal and unstructured matrix was provided in supplemental Table S10 and S11 .

The purpose of this study was threefold. The first was to evaluate the impact of optimization methods (i.e., constrained vs. unconstrained) and specifications of covariance structures (diagonal vs. unstructured vs model selection procedure) on between-case variance components using MLMs in the analysis of SCEDs. The second was to evaluate an RLRT-based procedure to make statistical inferences for between-case variance components. Lastly, based on the findings from the simulation studies, we aimed to provide guidelines to show empirical researchers the conditions under which the estimates of between-case variance components were reliable, and the test procedure had acceptable type I error rate and power.

The convergence rates were very high (≥ 98%) in all conditions except for the estimation of an unstructured matrix using the constrained optimization with a small number of cases ( J  = 4). The unconstrained optimization had advantages over the constrained optimization because the positive definiteness rates were over 90% and consistent across levels of the between-case variance when using unconstrained optimization. On the other hand, the constrained optimization frequently encountered non-positive definiteness (proportion ≥ 60%) when the variance was small (< .3).

Although the unconstrained optimization is effective in reducing non-positive definite estimates, it did not affect the estimates of the between-case variance components. Regardless of the optimization methods, positive biases in the estimated variance of treatment effects were present when the true variance size was small (i.e., 1.0 or below), even when the covariance structure was correctly specified. This is partly because that the between-case variance is close to the boundary of its parameter space and the asymptotic assumptions of REML are seriously violated with small samples. Another possible reason is the biased estimate of the autocorrelation of level 1 errors. Though the estimates of autocorrelation are not of our interest in this study, we found negative biases in the estimates, which are consistent with previous findings (Ferron et al., 2009 ).

Overall, the bias of between-case variance estimates would decrease when the size of between-case variance components became larger, which is consistent with previous work (Ferron et al. 2009 , Moeyaert et al., 2013a , 2013c ). In general, overfitting the covariance structure tended to overestimate the between-case variance components, whereas underfitting the covariance structure tended to underestimate the between-case variance components, regardless of the choice of optimization methods. However, the positive biases caused by overfitting reduced to an acceptable level when the variance was 1.0 or above, whereas the negative biases caused by underfitting were not acceptable until the variance was as large as 8.0. Hence our findings partially supported the conclusion in Moeyaert et al. ( 2016 ) that correctly specified model did not significantly overperform than the mis-specified model when the between-case variance was large.

In practice, applied researchers seldom know the true covariance structure in the population or the true size of the variance, therefore it is difficult to determine whether the covariance structure is correctly specified or the impact of misspecifications. Our findings showed that the post hoc model selection procedure based on the parameter bootstrap could mitigate the bias caused by misspecifications. Specifically, point estimates for the variance in the baseline level were accurate when the variance ≥ 1.0 with four cases or variance ≥ 0.5 with eight cases. This was also true for the variance in the treatment effect when the variance ≥ 2.0 with four cases or variance ≥ 1.0 with eight cases. Despite the differences in point estimates, the coverage rates of the confidence intervals were similar across the various model specifications. For the model selection procedure, the interval estimates for between case variance components was close to the nominal level when the variance ≥ 1.0. The average MSE of between-case variance estimates for five model specifications were also very alike, with sight disadvantages in terms of efficiency for the estimator in the constrained optimization with unstructured matrix.

Although the choice of optimization methods did not affect the estimates of variance components, it did have an impact on the estimates of the correlation between random effects. The constrained optimization resulted in a high rate of nonpositive definiteness with inadmissible correlation estimates. On the other hand, although the unconstrained optimization seldom encountered non-positive definiteness issues, the estimated correlation coefficient was likely to be biased. Hence, we caution researchers interpreting the estimated correlation between the random effects. For covariance estimates, there were no significant differences between constrained and unconstrained optimization methods. However, caution is also needed in certain conditions because biased estimates are obtained.

Last but not least, through the examinations of the empirical type I error rate and power of the RLRT SL for testing between-case variance components, we found that at least 20 measurements are needed to ensure that the empirical type I error rate is close to the nominal level. Because the GLS transformation applied to RLRT SL is based on the estimates of autocorrelations, a sufficient series length can prevent inflated type I error rate by providing more accurate estimations of autocorrelations. For single case studies, Ferron et al. ( 2010 ) found a median of 24 measurements and Shadish and Sullivan ( 2011 ) found a median of 20 measurements. Thus, empirical type I error rate is expected be close to the nominal level for at least half of single case studies. On the other hand, although the estimates of between-case variance components have little bias when the variance is sufficiently large, studies with few cases might be under-powered. In the next section, we provided specific guidelines for the minimally detectable effect size for variance component under commonly encountered conditions in SCED.

Recommendations

A practical guideline was summarized for empirical researchers about the estimation and statistical inference for between-case variance in SCEDs. First, we recommend using the unconstrained optimization if nonpositive definite issues are encountered, especially when the estimate of correlation/covariance between random effects is of interests. Second, we recommend using the model selection procedure (see Appendix C for the R code) to determine the optimal covariance structure (i.e., either diagonal or unstructured) if there is no strong theoretical grounding or prior knowledge regarding the structure of the covariance matrix. Third, researchers should be aware that when the estimated variance components are small (0.5 or less), they are likely to be overestimated, indicating that the true parameter values are even smaller, especially when the number of cases is small (i.e., n = 4). Fourth, when conducting null hypothesis significance test for variance components, researcher should use the RLRT based on the asymptotic mixture distribution (i.e., RLRT SL ) and apply the GLS transformation to account for autocorrelated errors (see Appendix C for the R code). Fifth, researchers should make sure that the power for testing variance components is sufficient. For the power of .80 without an inflated type I error rate (i.e., given series length = 20), the minimally detectable variance of the baseline level is 2.0 when there are four cases, or 1.0 when there are eight cases. For the variance of the treatment effect, the minimally detectable variance is 4.0 when the number of cases = 4, or 2.0 when the number of cases = 8. Last but not least, researchers should be cautious when interpreting the covariance between random effects as the covariance estimates can be interpreted with confidence only when there are four cases and the estimated variance components are larger than 4.0 or when there are eight cases and the estimated variance components are larger than the 1.0. More caution is warranted for the correlation estimates as they are biased across all conditions.

Limitations and future research

Results of this study are limited to the chosen conditions. Although we chose typical conditions in SCEDs, conclusions from our study should be further validated before they can be generalized to other conditions. Specifically, we only simulated data based on MBD across cases, which is most common design in SCEDs. However, other designs such as ABAB, MBD across behaviors and alternating treatments are not rare in single case studies. Under those designs, conclusions in our study should be carefully reexamined.

Due to the limitation of REML, future studies should continue to explore the Bayesian approaches. Bayesian approaches are promising because they are not based on the asymptotic assumptions. However, previous studies using Bayesian approach did not generate consistent results with different choice of priors for the between-case variance components (Baek et al., 2020 ; Hembry et al., 2015 ; Joo & Ferron, 2019 ; Moeyaert et al., 2017 ). Future study should consider the construction of reasonable informative priors to improve the variance estimation. The Bayesian approach may also provide better estimates of autocorrelations (Shadish et al., 2013b ), which can be used in the GLS transformation for the RLRT. In addition, with small series length ( I  = 10) and small number of cases ( J  = 4) most of the conditions are under-powered with RLRT-based procedures. With reasonable informative priors for the between-case variance component (see Tsai & Hsiao, 2008 ), Bayesian inference based on the posterior distribution can be an alternative to the RLRT-based procedures to improve the power for detecting the significance of between-case variance components.

Absolute bias is calculated when the population value of a parameter is zero.

Relative biases based on diagonal and unstructured covariance structure specification can be found in supplemental Table S8 and S9 , respectively).

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(DOCX 60 kb)

Matrix form: Y  =  Xβ  +  Zμ  +  ϵ

Y : n x 1 column vector ( n = total observations = i * j) that contains all data on the dependent variable.

X : n x 2 matrix that contains data on all the independent variables.

β : 2 x 1 column vector that contains the fixed effects

Z : design matrix with dimension of n x (2 j ) ( j = the number of cases).

μ : 2 j x 1 column vector that contains all the random effect.

ϵ : n x 1 column vector that contains all the level 1 residuals.

GLS transformation

Consider a simple regression model Y  =  Xβ  +  ϵ where the i.i.d. assumption is not fulfilled, and the covariance matrix of the errors is specified by \( \boldsymbol{V}={\sigma}_e^2\boldsymbol{R} \) . We can obtain the best linear unbiased estimator of β as the generalized least square (GLS) estimator. The GLS estimate of β can also be obtained as the ordinary least square estimator (OLS) estimate of a transformed model \( \overset{\sim }{\boldsymbol{Y}}=\overset{\sim }{\boldsymbol{X}}\boldsymbol{\beta} +\overset{\sim }{\boldsymbol{\epsilon}} \) where \( \overset{\sim }{\boldsymbol{Y}}={\boldsymbol{V}}^{-1/2}\boldsymbol{Y} \) , \( \overset{\sim }{\boldsymbol{X}}={\boldsymbol{V}}^{-1/2}\boldsymbol{X} \) , and \( \overset{\sim }{\boldsymbol{\epsilon}}={\boldsymbol{V}}^{-1/2}\boldsymbol{\epsilon} \) . In the transformed model, the error covariance is equal to \( {\sigma}_e^2I \) .

We use the GLS transformation to derive the distribution of the RLRT statistic under the first-order autoregressive error structure. The matrix form of the reduced model with only one random effect is

where  μ is a j  × 1 column vector that contains only one random effect and \( \boldsymbol{\epsilon} \sim N\left(\mathbf{0},{\sigma}_e^2\boldsymbol{R}\right) \) . R is a block diagonal matrix and has the same first-order autoregressive structure within each block. The RLRT for testing \( {\sigma}_{u0}^2 \) and \( {\sigma}_{u1}^2 \) is equivalent to test them in the GLS-transformed model

where \( \overset{\sim }{\boldsymbol{Y}}={\boldsymbol{R}}^{-1/2}\boldsymbol{Y} \) , \( \overset{\sim }{X}={\boldsymbol{R}}^{-1/2}\boldsymbol{X} \) , \( \overset{\sim }{\boldsymbol{Z}}={\boldsymbol{R}}^{-1/2}\boldsymbol{Z} \) , and \( \overset{\sim }{\boldsymbol{\epsilon}}={\boldsymbol{R}}^{-1/2}\boldsymbol{\epsilon} \) ~ \( N\left(\mathbf{0},{\sigma}_e^2I\right) \) . In the AR (1) case, the transformation matrix R −1/2 is block diagonal with j blocks of \( {\boldsymbol{R}}_i^{-1/2} \) with dimension of i  ×  i :

where we use estimate \( \hat{\rho} \) from the full model as autocorrelation parameter ρ .

Consider the condition with series length I = 10 and the number of cases J = 4. The transformation matrix R is then a 40 by 40 sparse matrix containing four diagonal blocks of \( {R}_i^{-1/2} \) with dimension of 10 by 10 as defined in Appendix B . An example R code for the transformation and the significance tests under this condition are provided below.

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Li, H., Luo, W., Baek, E. et al. Estimation and statistical inferences of variance components in the analysis of single-case experimental design using multilevel modeling. Behav Res 54 , 1559–1579 (2022). https://doi.org/10.3758/s13428-021-01691-6

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DOI : https://doi.org/10.3758/s13428-021-01691-6

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What is Variance Analysis: Types, Examples and Formula

Table of Content

Key takeaways.

• Variance analysis compares the actual vs expected cash flows and keeps track of the financial metrics of your businesses. 
• Different variance analysis formula measures specific financial metrics, providing insights into specific aspects of performance. 
• Leveraging AI capabilities to analyze differences helps stakeholders achieve a better understanding of the finances and make well-informed decisions.

Introduction

In any business, having a grasp of projected cashflows, and available cash is crucial for daily financial operations. Enterprises utilize variance to measure the disparity between expected and actual cash flow.

Variance analysis involves assessing the reasons for the variances and understanding their impact on financial performance. In this ever-changing global economy, variance is an important metric for enterprises to track more than ever as it helps understand how accurate your cash forecasts are and whether you need to adjust your financial plans or take corrective actions to survive in the ever-changing volatile business environment. 

By the end of this blog, you will be able to understand variance analysis, its importance, and how to calculate it so you can leverage the cash properly and make strategic and informed business decisions.

What is Variance Analysis?

Variance analysis measures the difference between the forecasted cash position and the actual cash position. A positive variance occurs when actual cash flow surpasses the forecasted amount, while a negative variance indicates the opposite. Variance analysis helps you understand where you went over or under budget and why. 

This analysis provides insights into budget deviations and their underlying causes. It holds significance by enabling financial performance monitoring, trend identification, and informed decision-making for future planning. Through variance analysis, you can stay aligned with financial objectives and progressively enhance your profitability.

Types of Variance Analysis

Different types of variances can occur in the cash forecasting process due to reasons such as changes in market scenarios, customer behavior, and timing issues, among other factors. These variances can impact both sales revenue and expenses. By understanding the core impacts of these variances, companies can make necessary adjustments to their budgets, mitigate risks, and improve their overall financial performance.

Broadly, variances can be classified into two major categories:

• Materials, Labor, and Variable Overhead Variances
• Fixed Overhead Variances

Materials, labor, and variable overhead variances

These include price/rate variances and efficiency and quantity variances. Price/rate variances show the differences between industry-standard costs and actual pricing for materials, while efficiency variances and quantity variances refer to the differences between actual input values and the expected input values specified. This analysis plays a crucial role in managing procurement costs, making informed decisions, optimizing cost structures, and maintaining positive cash flow.

Fixed overhead variances 

Fixed overhead variances include volume variances and budget variances. Volume variances measure the difference between the actual revenue and budgeted revenue that is derived solely from changes in sales volume. Meanwhile, budget variances indicate the differences between actual and budgeted amounts. These variances help businesses understand the influence of sales volume fluctuations on financial performance, provide insights into the effectiveness of financial planning , and identify areas of overperformance or underperformance. 

Budget variances 

Budget variances can be divided into two subgroups: expense variances and revenue variances. Expense variance measures actual costs compared to the budgeted costs while revenue variances measure actual revenue with the budgeted revenue. Positive revenue variances represent revenue that exceeds the expected revenue, while negative revenue variances represent lower expected revenue.

Budget variance analysis are important to understand the reasons behind the deviations from the budgeted amounts. It enables the identification of avenues for enhancing business processes, boosting revenue, and cutting costs. By examining revenue variances, you can uncover possibilities for long-term efficiency improvements and increased business value.

Let’s take a look at an example of variance in budgeting 

Let’s say that your enterprise sells gadgets, and you’ve projected that you’ll sell $1 million worth of gadgets in the next quarter. However, at the end of the quarter, you find that you’ve only sold $800,000 worth of gadgets. That’s a variance of $200,000, or 20% of your original plan. By analyzing this variance, you can figure out what went wrong and take steps to improve your sales performance in the next quarter. Here, variance analysis becomes the vital tool that enables you to quickly identify such changes and adjust your strategies accordingly to manage your financial performance and optimize cash forecasting .

Role of Variance Analysis 

In periods of market instability, your business could face unforeseen fluctuations in revenue, costs, or other financial indicators. In such cases, one of the most crucial tools in your financial management system is variance analysis.

Variance analysis allows you to track the financial performance of your organization and implement proactive measures to decrease risks and enhance financial health. It enables businesses to compare their expected cash flow with their actual cash flow and to identify the root reasons for any discrepancies. Businesses can acquire an important understanding of their cash flow performance and decide on appropriate actions in response to fluctuating market conditions.

For instance, If a company realizes its cash inflows are lower, it can cut costs or alter its pricing strategy to stay profitable. Likewise, if its real cash outflows exceed because of unforeseen costs, it can modify its financial plan or explore other funding choices.

Variance Analysis Formula 

The key components of variance are relatively straightforward; actuals vs. expected. Let’s look into the key variance analysis formula that focuses on specific financial metrics. These formulas unveil gaps between expected and actual results, providing insights into specific aspects of performance. 

Cost variance formula

Cost variance measures the difference between actual costs and budgeted costs. The cost variance formula is:

Cost Variance = Actual Costs – Budgeted Costs

This formula helps identify cost control issues, inefficiencies, and opportunities for improvement.

Efficiency variance formula

Efficiency variance measures the difference between actual input values (e.g., labor hours, machine hours) and budgeted or standard input values. The efficiency variance formula is:

Efficiency Variance = (Actual Input – Budgeted Input) × Standard Rate

This formula helps organizations identify variations in productivity and pinpoint areas for improvement.

Volume variance formula

Volume variance, also known as sales volume variance, measures the impact of changes in sales volume on revenue compared to the budgeted volume. The variance volume formula is: 

Volume Variance = (Actual Sales Volume – Budgeted Sales Volume) × Budgeted Selling Price

This formula helps organizations to understand the contribution of sales volume to revenue performance.

Budget variance formula

Budget variance measures the actual revenue with the budgeted revenue. The budget variance formula is:

Budget Variance = Actual Revenue – Budgeted Revenue

This formula aids in evaluating pricing strategies, market demand, and sales effectiveness.

Examples of Variance Analysis 

For instance, let’s consider, a company that plans to create a new mobile app with a projected cost of $50,000. The expected timeline for completion is 4 months, with a budgeted labor cost of $10,000 per month. The target is to release the application with 10 key features. Here are the examples that demonstrate different types of variances under this scenario:

Cost variance:

During the development process, the company implements cost-saving measures and efficient resource allocation, resulting in lower actual costs. The actual cost of the project at completion is $45,000. The cost variance can be calculated as follows:

Cost Variance = $45,000 – $50,000 = -$5,000

Here, the negative cost variance of -$5,000 indicates that the company has achieved cost savings of $5,000 compared to the budgeted cost for the project.

Efficiency variance:

The project is efficiently managed, and the team completes the development in 3.5 months instead of the budgeted 4 months. Assuming a budgeted labor cost of $10,000 per month, the efficiency variance can be calculated as follows:

Efficiency Variance = (3.5 months – 4 months) × $10,000 = -$5,000

The negative efficiency variance of -$5,000 indicates that the project was completed ahead of schedule, resulting in labor cost savings of $5,000.

Volume variance:

The final version of the mobile application is released with 12 key features instead of the budgeted 10 features. Assuming a budgeted revenue of $2,000 per feature, the volume variance can be calculated as follows:

Volume Variance = (12 features – 10 features) × $2,000 per feature = $4,000

The positive volume variance of $4,000 indicates that the company delivered additional features, resulting in increased revenue of $4,000 compared to the budgeted amount.

Budget variance:

The company spent $8,000 on marketing and promotional activities for the mobile application launch, while the budgeted amount was $10,000. The budget variance can be calculated as follows:

Budget Variance = $8,000 – $10,000 = -$2,000

The negative budget variance of -$2,000 indicates that the company spent $2,000 less than the budgeted amount for marketing and promotional activities.

In these scenarios, the company achieved cost savings, enhanced efficiency, delivered additional features, and spent less than the budgeted amount on marketing expenses. These variances provide insights into cost management, efficiency, revenue generation, and budget adherence within the given software development project scenario.

Benefits of Conducting Variance Analysis

Let’s take a look at the top 4 benefits enterprises can reap by conducting variance analysis for cash forecasting:

Identify discrepancies

Variance analysis helps identify discrepancies between the actual cash inflows and outflows and the forecasted amounts. By comparing the forecasted cash flow with the actual cash flow, it is easier to identify any discrepancies, enabling the stakeholders to take corrective measures. 

Refine cash forecasting techniques

Conducting variance analysis allows for a review of past forecasts to identify any errors or biases that may have impacted accuracy. This information can be used to refine forecasting techniques, improve future forecasts make adjustments to existing forecast templates, or build new ones.

Improve financial decision-making

Understanding the reasons for variances can provide valuable insights that can help improve financial decision-making, which is critical in a volatile market. For example, if a variance is caused by unexpected expenses, management may decide to reduce expenses or explore cost-saving measures.

Better cash management

By analyzing variances, companies can identify areas where cash management can be improved. This can include better management of accounts receivable or accounts payable, more effective inventory management, or renegotiating payment terms with the suppliers.

Role of AI in Variance Analysis for Cash Forecasting

Amid turbulent market conditions, as companies prepare for 2024 and beyond, enterprises’ finance chiefs professionals are recommending various enhancements to improve decision-making. The most commonly mentioned improvements are the adoption of digital technologies, AI, and automation, and the enhancement of forecasting, scenario planning, and consistency in measuring key performance indicators, as per the Deloitte CFO Signals Survey .

This goes to show the significance of the adoption of advanced technologies, such as AI, for companies preparing for uncertain markets. The challenge with traditional variance analysis is that it is difficult for treasurers to create low-variance cash flow forecasts for enterprises as they utilize manual methods and spreadsheets while dealing with large volumes of data. Moreover, relying on manual variance reduction approaches leads to high variance and can be time-consuming, labor-intensive, and expensive, thus, delaying the decision-making process.

Here’s how AI addresses this challenge and enables it to take variance analysis to the next level: 

• AI-based cash forecasting software helps in variance analysis by taking additional steps to improve the accuracy of the cash forecast by 90-95%. 
• It enables organizations to continuously improve the forecast by understanding the key drivers of variance. 
• It compares cash forecasts to actual results to check for variances, aligning the forecast with other aspects such as monthly, quarterly, and yearly forecasts, thus, ensuring that the forecast is accurate across various scenarios.
• AI also analyzes the accuracy of cash forecasts through a line item analysis across multiple horizons and makes tweaks to the algorithm through an AI-assisted review process. 
• Finally, AI fine-tunes the forecast model and enhances the data as needed to achieve the desired level of forecast accuracy. 

Benefits of Leveraging AI in Variance Analysis

Here are some of the key benefits you can achieve by conducting variance analysis with AI in cash forecasting:

Automated reporting

AI can streamline the process of reporting discrepancies in cash flow by delivering consistent reports that emphasize developments and regularities. Automating the process of reporting allows organizations to save time and resources that would otherwise have been spent on manual reporting. This also guarantees consistent and accurate reporting, removing the chance of human mistakes. By receiving frequent updates on discrepancies in cash flow as they occur, you can effectively monitor your business’s cash flow and pinpoint opportunities for enhancement to optimize your financial results.

Faster, data-driven decision-making

AI can assist in making quicker, better-informed decisions about managing cash flow by providing in-depth insights on cash forecasts in real time. This can assist companies in promptly addressing fluctuations in cash flow and implementing necessary measures. This is especially crucial in periods of market volatility when cash flow trends can quickly fluctuate and unforeseen circumstances may arise.

Real-time cash analysis & better liquidity management 

With AI at its core, cash flow forecasting software can learn from industry-wide seasonal fluctuations to improve forecasting accuracy. AI-powered cash forecasting software that enables variance analysis can also create snapshots of different forecasts and variances to compare them for detailed, category-level analysis. Offering such comprehensive visibility, helps you respond quickly to changes in cash flow, take corrective action as needed, and manage your enterprise’s liquidity better. 

Improved cash forecasting accuracy with real-time cash analysis

AI streamlines your examination of cash flow by delving deeply into and analyzing a large volume of data from various sources, such as past cash flow information, market trends, and economic indicators, in real time. Therefore, it allows for an immediate understanding of discrepancies in cash flow. This can offer a more in-depth assessment of cash flow discrepancies, enabling the recognition of trends and patterns that may not be visible through manual review.

How HighRadius Can Help Automate Variance Analysis in Cash Forecasting? 

In a rapidly evolving business landscape, market uncertainties and disruptions can have a significant impact on an enterprise’s financial stability. That’s why having a robust cash forecasting system with AI at its core is essential for businesses to conduct automated variance analysis. HighRadius’ cash forecasting software enables more advanced and sophisticated variance analysis that helps you achieve up to 95% global cash flow forecast accuracy. 

By leveraging its AI capabilities in data analysis, pattern recognition, real-time integration, and predictive modeling, it empowers finance teams to gain deeper insights, improve accuracy, and make more informed decisions to manage cash flow effectively. Furthermore, our solution helps continuously improve the forecast by understanding the key drivers of variance. The AI algorithm learns from historical data and feedback, continuously improving their accuracy and effectiveness over time. This iterative learning process enhances the quality of variance analysis results. 

Our AI-based cash forecasting solution supports drilling down into variances across various cash flow categories, geographies, and entity-level variances performing a root cause analysis, and helps achieve up to 98% automated cash flow category tagging. 

1) What is the difference between standard costing and variance analysis?

Standard costing is setting an estimated (standard) cost on metrics such as input values, materials, cost of labor, and overhead based on industrial trends and historical data. Variance analysis focuses on analyzing and interpreting differences (variances) between actual costs and standard costs.

2) What are the three main sources of variance in an analysis?

In variance analysis, the three main sources of variance are material variances (differences in material usage or cost), labor variances (variations in labor productivity or wage rates), and overhead variances (deviations in overhead costs).

3) What is P&L variance analysis?

P&L (profit & loss) variance analysis is the process of comparing actual financial results to expected results in order to identify differences or variances. This type of variance analysis is typically performed on a company’s income statement, which shows its revenues, expenses, and net profit or loss over a specific period of time. 

4) Why is the analysis of variance important?

The analysis of variance is important to keep track of as it tells about the financial health of your business. With proper variance analysis, you can measure the financial performance of your business, keep track of over and under-performing financial metrics, and identify areas for improvement.

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Variance Analysis and Flexible Budgeting

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Home » ANOVA (Analysis of variance) – Formulas, Types, and Examples

ANOVA (Analysis of variance) – Formulas, Types, and Examples

Table of Contents

Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a statistical method used to test differences between two or more means. It is similar to the t-test, but the t-test is generally used for comparing two means, while ANOVA is used when you have more than two means to compare.

ANOVA is based on comparing the variance (or variation) between the data samples to the variation within each particular sample. If the between-group variance is high and the within-group variance is low, this provides evidence that the means of the groups are significantly different.

ANOVA Terminology

When discussing ANOVA, there are several key terms to understand:

• Factor : This is another term for the independent variable in your analysis. In a one-way ANOVA, there is one factor, while in a two-way ANOVA, there are two factors.
• Levels : These are the different groups or categories within a factor. For example, if the factor is ‘diet’ the levels might be ‘low fat’, ‘medium fat’, and ‘high fat’.
• Response Variable : This is the dependent variable or the outcome that you are measuring.
• Within-group Variance : This is the variance or spread of scores within each level of your factor.
• Between-group Variance : This is the variance or spread of scores between the different levels of your factor.
• Grand Mean : This is the overall mean when you consider all the data together, regardless of the factor level.
• Treatment Sums of Squares (SS) : This represents the between-group variability. It is the sum of the squared differences between the group means and the grand mean.
• Error Sums of Squares (SS) : This represents the within-group variability. It’s the sum of the squared differences between each observation and its group mean.
• Total Sums of Squares (SS) : This is the sum of the Treatment SS and the Error SS. It represents the total variability in the data.
• Degrees of Freedom (df) : The degrees of freedom are the number of values that have the freedom to vary when computing a statistic. For example, if you have ‘n’ observations in one group, then the degrees of freedom for that group is ‘n-1’.
• Mean Square (MS) : Mean Square is the average squared deviation and is calculated by dividing the sum of squares by the corresponding degrees of freedom.
• F-Ratio : This is the test statistic for ANOVAs, and it’s the ratio of the between-group variance to the within-group variance. If the between-group variance is significantly larger than the within-group variance, the F-ratio will be large and likely significant.
• Null Hypothesis (H0) : This is the hypothesis that there is no difference between the group means.
• Alternative Hypothesis (H1) : This is the hypothesis that there is a difference between at least two of the group means.
• p-value : This is the probability of obtaining a test statistic as extreme as the one that was actually observed, assuming that the null hypothesis is true. If the p-value is less than the significance level (usually 0.05), then the null hypothesis is rejected in favor of the alternative hypothesis.
• Post-hoc tests : These are follow-up tests conducted after an ANOVA when the null hypothesis is rejected, to determine which specific groups’ means (levels) are different from each other. Examples include Tukey’s HSD, Scheffe, Bonferroni, among others.

Types of ANOVA

Types of ANOVA are as follows:

One-way (or one-factor) ANOVA

This is the simplest type of ANOVA, which involves one independent variable . For example, comparing the effect of different types of diet (vegetarian, pescatarian, omnivore) on cholesterol level.

Two-way (or two-factor) ANOVA

This involves two independent variables. This allows for testing the effect of each independent variable on the dependent variable , as well as testing if there’s an interaction effect between the independent variables on the dependent variable.

Repeated Measures ANOVA

This is used when the same subjects are measured multiple times under different conditions, or at different points in time. This type of ANOVA is often used in longitudinal studies.

Mixed Design ANOVA

This combines features of both between-subjects (independent groups) and within-subjects (repeated measures) designs. In this model, one factor is a between-subjects variable and the other is a within-subjects variable.

Multivariate Analysis of Variance (MANOVA)

This is used when there are two or more dependent variables. It tests whether changes in the independent variable(s) correspond to changes in the dependent variables.

Analysis of Covariance (ANCOVA)

This combines ANOVA and regression. ANCOVA tests whether certain factors have an effect on the outcome variable after removing the variance for which quantitative covariates (interval variables) account. This allows the comparison of one variable outcome between groups, while statistically controlling for the effect of other continuous variables that are not of primary interest.

Nested ANOVA

This model is used when the groups can be clustered into categories. For example, if you were comparing students’ performance from different classrooms and different schools, “classroom” could be nested within “school.”

ANOVA Formulas

ANOVA Formulas are as follows:

Sum of Squares Total (SST)

This represents the total variability in the data. It is the sum of the squared differences between each observation and the overall mean.

• yi represents each individual data point
• y_mean represents the grand mean (mean of all observations)

Sum of Squares Within (SSW)

This represents the variability within each group or factor level. It is the sum of the squared differences between each observation and its group mean.

• yij represents each individual data point within a group
• y_meani represents the mean of the ith group

Sum of Squares Between (SSB)

This represents the variability between the groups. It is the sum of the squared differences between the group means and the grand mean, multiplied by the number of observations in each group.

• ni represents the number of observations in each group
• y_mean represents the grand mean

Degrees of Freedom

The degrees of freedom are the number of values that have the freedom to vary when calculating a statistic.

For within groups (dfW):

For between groups (dfB):

For total (dfT):

• N represents the total number of observations
• k represents the number of groups

Mean Squares

Mean squares are the sum of squares divided by the respective degrees of freedom.

Mean Squares Between (MSB):

Mean Squares Within (MSW):

F-Statistic

The F-statistic is used to test whether the variability between the groups is significantly greater than the variability within the groups.

If the F-statistic is significantly higher than what would be expected by chance, we reject the null hypothesis that all group means are equal.

Examples of ANOVA

Examples 1:

Suppose a psychologist wants to test the effect of three different types of exercise (yoga, aerobic exercise, and weight training) on stress reduction. The dependent variable is the stress level, which can be measured using a stress rating scale.

Here are hypothetical stress ratings for a group of participants after they followed each of the exercise regimes for a period:

• Yoga: [3, 2, 2, 1, 2, 2, 3, 2, 1, 2]
• Aerobic Exercise: [2, 3, 3, 2, 3, 2, 3, 3, 2, 2]
• Weight Training: [4, 4, 5, 5, 4, 5, 4, 5, 4, 5]

The psychologist wants to determine if there is a statistically significant difference in stress levels between these different types of exercise.

To conduct the ANOVA:

1. State the hypotheses:

• Null Hypothesis (H0): There is no difference in mean stress levels between the three types of exercise.
• Alternative Hypothesis (H1): There is a difference in mean stress levels between at least two of the types of exercise.

2. Calculate the ANOVA statistics:

• Compute the Sum of Squares Between (SSB), Sum of Squares Within (SSW), and Sum of Squares Total (SST).
• Calculate the Degrees of Freedom (dfB, dfW, dfT).
• Calculate the Mean Squares Between (MSB) and Mean Squares Within (MSW).
• Compute the F-statistic (F = MSB / MSW).

3. Check the p-value associated with the calculated F-statistic.

• If the p-value is less than the chosen significance level (often 0.05), then we reject the null hypothesis in favor of the alternative hypothesis. This suggests there is a statistically significant difference in mean stress levels between the three exercise types.

4. Post-hoc tests

• If we reject the null hypothesis, we conduct a post-hoc test to determine which specific groups’ means (exercise types) are different from each other.

Examples 2:

Suppose an agricultural scientist wants to compare the yield of three varieties of wheat. The scientist randomly selects four fields for each variety and plants them. After harvest, the yield from each field is measured in bushels. Here are the hypothetical yields:

The scientist wants to know if the differences in yields are due to the different varieties or just random variation.

Here’s how to apply the one-way ANOVA to this situation:

• Null Hypothesis (H0): The means of the three populations are equal.
• Alternative Hypothesis (H1): At least one population mean is different.
• Calculate the Degrees of Freedom (dfB for between groups, dfW for within groups, dfT for total).
• If the p-value is less than the chosen significance level (often 0.05), then we reject the null hypothesis in favor of the alternative hypothesis. This would suggest there is a statistically significant difference in mean yields among the three varieties.
• If we reject the null hypothesis, we conduct a post-hoc test to determine which specific groups’ means (wheat varieties) are different from each other.

How to Conduct ANOVA

Conducting an Analysis of Variance (ANOVA) involves several steps. Here’s a general guideline on how to perform it:

• Null Hypothesis (H0): The means of all groups are equal.
• Alternative Hypothesis (H1): At least one group mean is different from the others.
• The significance level (often denoted as α) is usually set at 0.05. This implies that you are willing to accept a 5% chance that you are wrong in rejecting the null hypothesis.
• Data should be collected for each group under study. Make sure that the data meet the assumptions of an ANOVA: normality, independence, and homogeneity of variances.
• Calculate the Degrees of Freedom (df) for each sum of squares (dfB, dfW, dfT).
• Compute the Mean Squares Between (MSB) and Mean Squares Within (MSW) by dividing the sum of squares by the corresponding degrees of freedom.
• Compute the F-statistic as the ratio of MSB to MSW.
• Determine the critical F-value from the F-distribution table using dfB and dfW.
• If the calculated F-statistic is greater than the critical F-value, reject the null hypothesis.
• If the p-value associated with the calculated F-statistic is smaller than the significance level (0.05 typically), you reject the null hypothesis.
• If you rejected the null hypothesis, you can conduct post-hoc tests (like Tukey’s HSD) to determine which specific groups’ means (if you have more than two groups) are different from each other.
• Regardless of the result, report your findings in a clear, understandable manner. This typically includes reporting the test statistic, p-value, and whether the null hypothesis was rejected.

When to use ANOVA

ANOVA (Analysis of Variance) is used when you have three or more groups and you want to compare their means to see if they are significantly different from each other. It is a statistical method that is used in a variety of research scenarios. Here are some examples of when you might use ANOVA:

• Comparing Groups : If you want to compare the performance of more than two groups, for example, testing the effectiveness of different teaching methods on student performance.
• Evaluating Interactions : In a two-way or factorial ANOVA, you can test for an interaction effect. This means you are not only interested in the effect of each individual factor, but also whether the effect of one factor depends on the level of another factor.
• Repeated Measures : If you have measured the same subjects under different conditions or at different time points, you can use repeated measures ANOVA to compare the means of these repeated measures while accounting for the correlation between measures from the same subject.
• Experimental Designs : ANOVA is often used in experimental research designs when subjects are randomly assigned to different conditions and the goal is to compare the means of the conditions.

Here are the assumptions that must be met to use ANOVA:

• Normality : The data should be approximately normally distributed.
• Homogeneity of Variances : The variances of the groups you are comparing should be roughly equal. This assumption can be tested using Levene’s test or Bartlett’s test.
• Independence : The observations should be independent of each other. This assumption is met if the data is collected appropriately with no related groups (e.g., twins, matched pairs, repeated measures).

Applications of ANOVA

The Analysis of Variance (ANOVA) is a powerful statistical technique that is used widely across various fields and industries. Here are some of its key applications:

Agriculture

ANOVA is commonly used in agricultural research to compare the effectiveness of different types of fertilizers, crop varieties, or farming methods. For example, an agricultural researcher could use ANOVA to determine if there are significant differences in the yields of several varieties of wheat under the same conditions.

Manufacturing and Quality Control

ANOVA is used to determine if different manufacturing processes or machines produce different levels of product quality. For instance, an engineer might use it to test whether there are differences in the strength of a product based on the machine that produced it.

Marketing Research

Marketers often use ANOVA to test the effectiveness of different advertising strategies. For example, a marketer could use ANOVA to determine whether different marketing messages have a significant impact on consumer purchase intentions.

Healthcare and Medicine

In medical research, ANOVA can be used to compare the effectiveness of different treatments or drugs. For example, a medical researcher could use ANOVA to test whether there are significant differences in recovery times for patients who receive different types of therapy.

ANOVA is used in educational research to compare the effectiveness of different teaching methods or educational interventions. For example, an educator could use it to test whether students perform significantly differently when taught with different teaching methods.

Psychology and Social Sciences

Psychologists and social scientists use ANOVA to compare group means on various psychological and social variables. For example, a psychologist could use it to determine if there are significant differences in stress levels among individuals in different occupations.

Biology and Environmental Sciences

Biologists and environmental scientists use ANOVA to compare different biological and environmental conditions. For example, an environmental scientist could use it to determine if there are significant differences in the levels of a pollutant in different bodies of water.

Advantages of ANOVA

Here are some advantages of using ANOVA:

Comparing Multiple Groups: One of the key advantages of ANOVA is the ability to compare the means of three or more groups. This makes it more powerful and flexible than the t-test, which is limited to comparing only two groups.

Control of Type I Error: When comparing multiple groups, the chances of making a Type I error (false positive) increases. One of the strengths of ANOVA is that it controls the Type I error rate across all comparisons. This is in contrast to performing multiple pairwise t-tests which can inflate the Type I error rate.

Testing Interactions: In factorial ANOVA, you can test not only the main effect of each factor, but also the interaction effect between factors. This can provide valuable insights into how different factors or variables interact with each other.

Handling Continuous and Categorical Variables: ANOVA can handle both continuous and categorical variables . The dependent variable is continuous and the independent variables are categorical.

Robustness: ANOVA is considered robust to violations of normality assumption when group sizes are equal. This means that even if your data do not perfectly meet the normality assumption, you might still get valid results.

Provides Detailed Analysis: ANOVA provides a detailed breakdown of variances and interactions between variables which can be useful in understanding the underlying factors affecting the outcome.

Capability to Handle Complex Experimental Designs: Advanced types of ANOVA (like repeated measures ANOVA, MANOVA, etc.) can handle more complex experimental designs, including those where measurements are taken on the same subjects over time, or when you want to analyze multiple dependent variables at once.

Disadvantages of ANOVA

Some limitations or disadvantages that are important to consider:

Assumptions: ANOVA relies on several assumptions including normality (the data follows a normal distribution), independence (the observations are independent of each other), and homogeneity of variances (the variances of the groups are roughly equal). If these assumptions are violated, the results of the ANOVA may not be valid.

Sensitivity to Outliers: ANOVA can be sensitive to outliers. A single extreme value in one group can affect the sum of squares and consequently influence the F-statistic and the overall result of the test.

Dichotomous Variables: ANOVA is not suitable for dichotomous variables (variables that can take only two values, like yes/no or male/female). It is used to compare the means of groups for a continuous dependent variable.

Lack of Specificity: Although ANOVA can tell you that there is a significant difference between groups, it doesn’t tell you which specific groups are significantly different from each other. You need to carry out further post-hoc tests (like Tukey’s HSD or Bonferroni) for these pairwise comparisons.

Complexity with Multiple Factors: When dealing with multiple factors and interactions in factorial ANOVA, interpretation can become complex. The presence of interaction effects can make main effects difficult to interpret.

Requires Larger Sample Sizes: To detect an effect of a certain size, ANOVA generally requires larger sample sizes than a t-test.

Equal Group Sizes: While not always a strict requirement, ANOVA is most powerful and its assumptions are most likely to be met when groups are of equal or similar sizes.

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Budget Variance Analysis: A Detailed Overview on Financial Performance Measurement

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Budget Variance Analysis Definition

Budget variance analysis is a financial analysis method that compares the actual financial outcomes with the budgeted figures to identify deviations, called variances. This process helps organizations understand their operational performance, identify potential problems, and take corrective actions to improve financial management.

Components of Budget Variance Analysis

Actual results.

Actual results represent the real, quantifiable outcomes from a given period, which may span from a quarter to a fiscal year. These are the actual expenses and revenues your company realized during this time frame. This is an essential part of budget variance analysis as it provides the real data you will compare with your predicted outcomes or budgeted results.

Budgeted Results

Simply put, budgeted results are the anticipated revenues and expenses of a business for a particular time frame. Before the beginning of a period, companies plan an estimated budget based on multiple factors such as past data, market analysis, predictions, etc. This approximation of revenues and expenses provides a goal for the teams to strive to meet.

The Variance

Variance is the difference between the actual results and the budgeted results. It reflects the numerical differences in individual budget items. It can be favorable if the actual revenues are higher than anticipated, or if the real expenses are less than expected. Conversely, a variance is unfavorable when revenues are lower than anticipated, or expenses are higher than expected.

Percentage Variance

Though the variance provides a numerical difference, the percentage variance is also critical because it offers more context to the difference. To calculate the percentage variance, divide the variance by the budgeted amount, then multiply by 100 to get the percentage. This component of the budget variance analysis helps businesses comprehend the magnitude of the variance and also allows for an easier and more meaningful comparison across different budget items or time frames.

Together, these components comprise a useful tool for businesses to monitor and control their financial operations, as well as to measure the effectiveness of their planning process.

Role of Budget Variance Analysis in Financial Management

Budget Variance Analysis plays a significant role in financial management by serving as a tool to conduct a periodical inspection of an organization's financial health. In essence, it evaluates the gap between an organization's planned financial outcomes and the actual results.

Role in Tracking Financial Performance

One of the primary roles of Budget Variance Analysis in financial management is tracking the performance of an organization based on its finances. This process is achieved by regularly comparing the projected budget against the actual expenses incurred. More often than not, the projected budget will vary from the actual expenses, and this variance can be either positive or negative.

A positive variance indicates that an organization's expenses are less than the projected budget, suggesting a favorable financial performance. Conversely, a negative variance, wherein the actual expenses exceeded the projected budget, signals a potential financial issue that needs immediate attention.

By reviewing these variances routinely, financial managers can promptly address financial discrepancies and implement necessary controls to mitigate further financial deviations. Essentially, Budget Variance Analysis serves as a diagnostic tool, offering an opportunity to analyze the discrepancies between the planned financial pathway and the actual financial state of an organization.

Role in Decision Making

More so, Budget Variance Analysis provides a solid basis for decision-making processes, especially for stakeholders. This analytical tool offers an insight into whether the existing financial strategy is working or if it requires adjustments. For instance, a recurring negative variance might imply the need for an overhaul in the financial strategy.

Budget Variance Analysis offers real-time, actionable intelligence to the stakeholders. It provides them insights about the areas where budget estimations are inaccurate and helps identify the factors causing the variance. Understanding these factors could assist stakeholders in deciding whether to change the strategy or continue with the current one.

Likewise, by evidencing the financial performance of an organization, the Budget Variance Analysis equips stakeholders with financial foresight, enabling them to make informed investment decisions, ensuring the company's growth and profitability.

In summary, Budget Variance Analysis is a crucial aspect of financial management, acting as a guidance tool for future business judgments and investment decisions.

Understanding Favorable and Unfavorable Variances

In the realm of budget variance analysis, variances are divided into two categories: favorable and unfavorable. The nature of these variances and their impact on a company's finances differ significantly.

Favorable Variance

Favorable variances paint a positive picture for a company's financial health. They occur when the actual revenues exceed the budgeted ones, or if the actual costs are lower than the budgeted costs. Essentially, a favorable variance suggests a company's financial performance is better than what was anticipated.

For example, if a company budgets $10,000 for its monthly marketing expenses and ends up spending only $8,000, this creates a favorable variance of $2,000. Here, the company's health is improved because it saved a sizable amount of money.

However, it's worth noting that not all favorable variances are a sign of positive financial health. There may be instances where a lower than planned expense could mean a missed growth opportunity or reduction in quality.

Unfavorable Variance

Unfavorable variances, on the other hand, present potential problems in a company's finances. These occur when the actual revenues are less than the budgeted amounts, or if the actual costs exceed the budgeted costs. Simply put, an unfavorable variance indicates a company's performance is not on par with its predictions.

For instance, if a company forecasts sales of $50,000 but actual sales come in at $40,000, they're looking at an unfavorable variance of $10,000. This could imply the company's financial health is not as strong as projected as its revenues have fallen short.

Similarly to favorable variances, context is important. An unfavorable variance isn't always an indicator of poorly performing financials. It may be caused by external factors that are outside of the company's control, such as a sudden increase in material costs or market downturns.

In summary, both favorable and unfavorable variances serve as tools for financial insight. They help in unraveling the performance of a company by comparing actual results with planned expectations. However, they should not be viewed in isolation but contextually, understanding the nuanced factors that contributed to the variances.

Types of Budget Variance Analysis

In the realm of budget variance analysis, four primary types are usually considered. These include sales variance, cost variance, material variance, and labor variance.

Sales Variance

Sales Variance is the difference between the actual sales and the budgeted or forecasted sales. In other words, if your organization predicted a particular level of sales for a given period and the actual sales either fell short or surpassed that forecast, you have a sales variance.

This variance is further categorized into two types:

• Sales Price Variance: This is calculated by comparing the actual price at which goods or services were sold to the budgeted price.
• Sales Volume Variance: Alternatively, this measures the difference between the actual quantity of an item sold and the budgeted amount.

Cost Variance

Cost Variance, as the name implies, is concerned with the difference between the actual cost and the budgeted cost of production or operations. A positive cost variance occurs when the actual cost is less than the budgeted cost. A negative cost variance reveals that the actual cost exceeds the budget.

Much like sales variance, cost variance can also be broken down into the following:

• Direct Cost Variance: The difference between the standard cost of direct materials and labor, and the actual cost incurred.
• Indirect Cost Variance: The variance between the standard cost of indirect materials, labor and expenses, and the actual cost incurred.

Material Variance

Material Variance measures the difference between the budgeted cost of materials and the actual cost of materials used in production. This type of variance helps organizations identify inefficiencies in their procurement process and wastage in material utilization.

Furthermore, material variance is divided into:

• Material Price Variance: The difference between the standard cost and the actual cost for a given quantity of materials.
• Material Usage Variance: Which relates to the difference between the standard quantity of materials expected to be used for the number of units produced, and the actual quantity of material used.

Labor Variance

Finally, Labor Variance represents the difference between the budgeted labor costs (amount of work hours at a certain pay rate) and the actual labor costs. This helps identify overtime issues, labor efficiency, and other potential labor-related concerns within a business.

Just like other variances, labor variance is split into:

• Labor Rate Variance: Which is the difference between the budgeted hourly labor rate and actual rate paid.
• Labor Efficiency Variance: Which measures the variance between the budgeted labor hours for actual output and the actual hours worked.

Computation of Budget Variance

The formula for budget variance.

The formula for calculating budget variance is quite simple:

Basically, you subtract the actual, really occurred value from the budgeted, planned cost or revenue. The result will either be a positive value, a negative value, or zero.

The Calculation Process

The calculation process of budget variance involves the following steps:

Step 1: Gathering the Data

The initial step involves collecting data on the actual income and expenses that occurred over the period you want to analyze. Alongside the actual figures, you also need to gather the forecasted or budgeted figures for the same time frame.

Step 3: Subtracting to Find the Variance

Once you have obtained both sets of data, subtract the budgeted number from the actual one. Follow the budget variance formula mentioned earlier to do so.

Step 4: Analyzing the Results

If the result of the calculation is zero, it means your actuals matched your budgeted amount perfectly.

A positive budget variance indicates that the actual income was higher than expected, or the actual expenses were less than anticipated. It's generally a good sign, but it might also state that you've set your targets too low.

A negative budget variance reveals that the actual income was lower than projected, or the expenses were higher. This result often points to a problem area, but it could also mean that your forecasts were overly optimistic.

Remember that budget variance analysis isn't about assigning blame for any discrepancies, but rather about identifying where operations might need to be altered to improve financial performance in the future.

Factors Affecting Budget Variance

There are several external and internal factors that can significantly affect budget variance.

Economic Fluctuations

Economic conditions play a significant role in budget variance. Inflation, recession, changes in interest rates and exchange rates can all impact an organization's financial performance, leading to differences between budgeted and actual figures. For example, during an economic downturn, companies might experience lower sales than budgeted. Conversely, during a period of economic growth, sales might exceed budget expectations, leading to a positive variance.

Changes in Market Trends

Market trends can greatly affect a company's revenue and expenses, and therefore its budget variance. New trends may increase demand for a company's products or services, leading to higher than expected sales and a favorable budget variance. On the other hand, a shift in consumer preferences away from a company's product could result in a negative budget variance. Similarly, changes in the competitive landscape, including new entrants or pricing strategies can also lead to variances.

Internal Organizational Changes

Internal changes within an organization can also lead to budget variances. This might include changes in company policy, introduction of new products or services, operational efficiency improvements, etc. For instance, if a company decides to use a new production technique that reduces costs, this could lead to lower than projected expenses and a favorable budget variance. Similarly, loss of key staff or less productive work hours could effectively lower outputs and lead to an unfavorable variance. Therefore, it is essential for companies to consider and anticipate these internal factors during the budgeting process.

Keep in mind that these factors, whether external or internal, are not isolated. They can intersect and interact in complex ways. For example, an economic downturn (an external factor) might spur a company to streamline its operations (an internal change), which could then affect the actual figures compared to the budgeted ones. Thus, a holistic approach is needed when performing budget variance analysis.

Implications of Budget Variance on Corporate Social Responsibility and Sustainability

Budget variance, by tipping the scales of planned expenditures, can have multiple implications on a company's Corporate Social Responsibility (CSR) and sustainability initiatives. These effects largely circulate around the unpredictability of expenses or savings and their potential impact on environmentally friendly activities and community services.

Unexpected Costs

When a budget variance results in unforeseen expenses, the allocation to CSR and sustainability may be jeopardized. Companies might be inclined to divert resources to areas that are closely linked with immediate business operations and profitability.

For example, funds intended for installing solar panels or other renewable energy sources might be reallocated to cover the unexpected costs. Similarly, new eco-friendly product development may be halted due to budgetary constraints.

Unanticipated Savings

Conversely, an unexpected savings through budget variance can provide a windfall for CSR and sustainability initiatives. This surplus could accelerate the transition to sustainable practices, fund new research for environmental-friendly processes, or amplify support for community projects.

For instance, unexpected savings could be channeled towards upgrading company facilities with energy-efficient equipment, enhancing waste management systems, or scaling up the scope of community development programs such as digital literacy classes or health clinics.

Balance between CSR and Sustainability

Navigating budget variance requires a balanced approach that does not undermine the commitment towards CSR and sustainability. Even when unexpected costs arise, companies should endeavor to maintain their social and environmental initiatives, as these long-term investments can significantly contribute to their reputation, stakeholder satisfaction, and ultimately, their business success.

Regular Analysis and Adjustment

Consequently, companies need to regularly perform budget variance analysis and adjust their financial plans accordingly. It involves taking proactive measures, like securing an emergency fund, to ensure that CSR and sustainability initiatives are never compromised. Furthermore, benefits arising from unexpected savings can be strategically invested to enhance their positive social and ecological impact.

In essence, dealing with the impact of budget variance on CSR and sustainability is all about maintaining a long-term perspective and fostering adaptability. Companies who manage this successfully not only ensure their own growth and survival, but also their relevance and respect in the societies they operate in.

Caveats and Limitations of Budget Variance Analysis

The drawbacks of budget variance analysis.

While budget variance analysis is a key weapon in your financial management arsenal, it's far from flawless. Perhaps the most obvious limitation is the potential for inaccuracy in your initial budget. If your original budget estimates were incorrect, the resulting variance analysis will, of course, be flawed.

Additionally, the static nature of budget variance analysis can be seen as a shortcoming. This tool doesn't adjust for changes in business or economic circumstances, as it reflects a point in time and doesn't take into consideration your company's dynamic environment. Thus, some unexpected business occurrences will typically remain unaddressed.

Variance Analysis is Not Prognostic

For all its benefits, it's crucial to recognize that budget variance analysis isn't a predictive tool. It doesn't provide insights into future markets or customer behaviors. It looks at financial deviations historically, without the capacity to forecast future financial situations.

Not Suited for All Types of Budgets

Another limitation lies in the fact that budget variance analysis may not be ideally suited to all types of budgets. It's most effective when applied to fixed budgets, where expenses are planned to remain stable. However, when it comes to flexible budgets that incorporate operational changes, variance analysis might create confusion and distortion.

Use In Conjunction with Other Tools

Finally, remember that budget variance analysis shouldn't be used in isolation. Relying solely on it could lead to a myopic view of your financial status. It's just one metric that gives you a snapshot of your financial performance—it doesn't offer a comprehensive view. Use variance analysis as part of a greater suite of tools, including cash flow analysis, balance sheet analysis, and profitability ratios among others, to get a well-rounded perspective of your financial status.

Remember, optimal financial management requires a multi-tool approach—don't rely solely on budget variance analysis to steer your financial decision-making.

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Case Study on Variance Analysis

Variance analysis case study:.

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