Understanding, Testing, and Relaxing Sphericity of Repeated Measures ANOVA with Manifest and Latent Variables Using SEM

Colloquial Definition of Sphericity

In the psychology literature, the colloquial definition of sphericity states that the variances of all pairwise differences between repeated measures are equal (see for example, Field, 1998; Field et al., 2012, pp. 550–552; Lane, 2016; Nimon, 2012). For example, in a one-way U-RM-ANOVA with three levels (i.e., A1, A2, and A3; not the same as this article’s guiding empirical example), the common definition of sphericity prescribes

The colloquial definition is sufficient for sphericity for designs with one within-subjects factor, but does not clearly generalize to designs with more than one within-subjects factor.

For example, researchers may infer that sphericity for the 2 × 3 design from the empirical example implies that the 13 unique pairwise differences across the 6 repeated measures must have equal variances, such that

Unfortunately, this constraint is neither required nor implied by sphericity.

A second confusing aspect of the colloquial definition of sphericity concerns the separate tests of sphericity provided by statistical software for each main or interaction effect (i.e., each main and interaction effect receive their own test of sphericity unless the effect has one degree of freedom; see the section Sphericity for Main and Interaction Effects of U-RM-ANOVA for details). In particular, if the colloquial definition of sphericity were true, then only one test should be needed regardless of the within-subjects design (i.e., the definition refers to pairwise differences rather than main/interaction effects). Therefore, the outputs from statistical software implementing U-RM-ANOVA do not corroborate the colloquial definition.

We want to briefly mention that, in fact, an omnibus test can be constructed to test for sphericity in multiple effects simultaneously. This omnibus test can decrease the Type I error rate of incorrectly rejecting the assumption that sphericity holds. This test will be described in the section Testing Sphericity Using L-RM-ANOVA.

Separate from the colloquial definition, psychology texts often describe compound symmetry as a special case of sphericity, and then describe assumptions in terms of compound symmetry (Field, 1998; Haverkamp & Beauducel, 2017; Maxwell & Delaney, 2004). Even though these texts do not claim equality across compound symmetry and sphericity, the compound symmetry simplification also does not easily generalize to higher-order within-subjects designs.

Original Definition of Sphericity

Huynh and Feldt (1970, p. 1587, Theorem 3) provide the original definition of sphericity . In general (i.e., not specific to the guiding empirical example), a P × P matrix $\mathbf{\Sigma }$ (e.g., a variance covariance matrix), conforms to sphericity if, and only if,

wherein C is a (P − 1) × P orthogonal contrast matrix, ${\sigma }^2$ is some positive constant, and I is a (P − 1) × (P − 1) identity matrix. Therefore, a given matrix (e.g., $\mathbf{\Sigma }$) adheres to sphericity if (after pre- and post-multiplying by an orthogonal contrast matrix), the diagonal elements are equal and the off-diagonal elements are zero.

However, the definition of sphericity requires more specificity for higher-order U-RM-ANOVAs (i.e., Huynh & Feldt, 1970, definition was for a single repeated measures factor, not for within-subjects designs with more than one repeated measures factor). Similarly, multivariate statistics textbooks describing U-RM-ANOVA often provide the original definition by Huynh and Feldt (1970), without explaining how to extend the definition to obtain separate tests for each main/interaction effect (for example, see Stevens, 2002, p. 421). Hence, we explain the original (1970) definition using the variance covariance matrix of the contrast data (i.e., ${\mathbf{\text{V}}}_{\mathbf{\pi }}$) from the empirical example to show how to generalize the definition to multi-factorial designs.

Sphericity for Main and Interaction Effects of U-RM-ANOVA

Following the empirical example, let ${\mathbf{\text{V}}}_{\mathit{y}}$ refer to the R × R variance covariance matrix of y, and ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ refer to the (R − 1) × (R − 1) variance covariance matrix of $\mathbf{\pi }$. Then, adhering to the definitions for C given in Equation 1, it follows that

and sphericity for each main/interaction effect will correspond to specific constraints within ${\mathbf{\text{V}}}_{\mathbf{\pi }}$. In particular, for a given main or interaction effect, sphericity will hold if both (1) the variances in ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ created from the effect’s contrasts are all equal, and (2) the covariances in ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ created from the effect’s contrasts are all zero. To clarify that definition, we explicitly define ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ from the empirical example as and note that the variances and covariances in ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ that respectively reflect the main effect of A, the main effect of B, and the interaction are $\{{\sigma }_{{\pi }_1}^2\}$, $\{{\sigma }_{{\pi }_2}^2,{\sigma }_{{\pi }_3}^2,{\sigma }_{{\pi }_2{\pi }_3}\}$, and $\{{\sigma }_{{\pi }_4}^2,{\sigma }_{{\pi }_5}^2,{\sigma }_{{\pi }_4{\pi }_5}\}$. Then, sphericity holds for the main effect of B if both ${\sigma }_{{\pi }_2}^2={\sigma }_{{\pi }_3}^2$ and ${\sigma }_{{\pi }_2{\pi }_3}=0$, regardless of the remaining elements in ${\mathbf{\text{V}}}_{\mathbf{\pi }}$; sphericity holds for the interaction if both ${\sigma }_{{\pi }_4}^2={\sigma }_{{\pi }_5}^2$ and ${\sigma }_{{\pi }_4{\pi }_5}=0$, regardless of the remaining elements in ${\mathbf{\text{V}}}_{\mathbf{\pi }}$; and sphericity is not relevant for the main effect of A because there is only one variance and no covariances (i.e., at least two variances are needed to form an equality constraint between them, and at least one covariance is needed to be set equal to zero).

Thus, the general definition for sphericity in U-RM-ANOVA refers to ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ (as opposed to variances of all pairwise differences). Once the set of orthogonal contrasts for the main/interaction effects of a specific U-RM-ANOVA design are identified and organized into an (R − 1) × R matrix C, sphericity holds for each effects if, and only if, the variances in ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ that reflect the effect are equal, and the covariances in ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ that reflect the effect equal zero.

Latent Repeated Measures ANOVA (L-RM-ANOVA)

The SEM framework can be used to test sphericity because SEMs can (1) estimate ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ from ${\mathbf{\text{V}}}_{\mathit{y}}$, (2) place constraints on the estimated elements in ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ that conform to sphericity, and (3) perform likelihood ratio tests to quantify statistical significance (analogous to Mauchly’s test). This section characterizes estimation of ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ from ${\mathbf{\text{V}}}_{\mathit{y}}$ (because the latter two points are commonplace), as a special case of latent repeated measures analysis of variance (L-RM-ANOVA, Langenberg et al., 2020; an extension of the growth components approach given by Mayer et al., 2012). L-RM-ANOVA estimates individual contrasts (i.e., $\mathbf{\pi }$) as a set of latent variables in SEM. Stated differently, L-RM-ANOVA identifies how to rewrite Equation 2 as an SEM.

To clarify, recall that the SEM measurement model for P manifest variables and Q latent variables may be written as

wherein y is P dimensional vector of manifest variables, $\mathbf{\nu }$ is a P dimensional column vector of manifest intercepts, $\mathbf{\Lambda }$ is a P × Q factor loading matrix, $\mathbf{\eta }$ is a Q dimensional column vector of latent variables; and $\mathbf{\text{ε}}$ is P dimensional column vector of residuals. L-RM-ANOVA rewrites Equation 2 as a special case of Equation 8, wherein $\mathbf{\pi }$ will be estimated as the set of latent variables.

In particular, L-RM-ANOVA sets P = Q, $\mathbf{\nu }=\mathbf{\text{0}}$, and $\mathbf{\epsilon }=\mathbf{\text{0}}$ (i.e., there are as many latent variables as manifest variables, and the latent variables fully account for the means, variances, and covariances of the manifest variables), such that

Equation 9 is similar to Equation 2 (i.e., $\mathbf{\pi }=\mathbf{\text{C}}\mathit{y}$), when conceptualizing $\mathbf{\pi }$ as a set of latent variables (i.e., $\mathbf{\pi }=\mathbf{\eta }$), and C as a matrix of loadings ( $\mathbf{\text{C}}=\mathbf{\text{Λ}}$). Conceptually, multiplying both sides of Equation 2 by the inverse of C should yield an identical form as the SEM measurement model in Equation 9 (i.e., $\mathit{y}={\mathbf{\text{C}}}^{-1}\mathbf{\pi }$). However, C cannot be inverted because it is a not a square matrix (currently R × (R − 1)). L-RM-ANOVA augments C by concatenating a row that contains the constant $1/\sqrt{R}$ to the top of the matrix; creating an invertible matrix ${\mathbf{\text{C}}}^{\ast }$ that maintains orthogonality. Following the guiding example,

In general, the concatenated row can be any vector that is pairwise linearly independent to the other rows. However, the way we chose the this row, it can be interpreted as an intercept that contributes to each of the manifest variables. The mean of this contrast equals the mean across all of the dependent variables multiplied by the constant $R/\sqrt{R}$. Replacing C with ${\mathbf{\text{C}}}^{\ast }$ into Equation 2 and solving for y yields which follows the structure of the SEM measurement model in Equation 9. Equation 11 demonstrates how L-RM-ANOVA identifies latent contrasts: augmenting the orthogonal contrast matrix C to create ${\mathbf{\text{C}}}^{\ast }$, and using the inverted version of ${\mathbf{\text{C}}}^{\ast }$ as a factor loading matrix in SEM. Estimating Equation 11 as an SEM estimates ${\mathbf{\text{V}}}_{\mathbf{\pi }}$, because the means, variances, and covariances of latent variables may be freely estimated. Figure 2 depicts the model prescribed in Equation 11 as applied to the guiding example. Rectangles represent the manifest variables y and the circles represent the latent contrast variables $\mathbf{\pi }$. The weights of the arrows going from $\mathbf{\pi }$ to y can be found in the ${\mathbf{\text{C}}}^{{\ast }^{-1}}$ matrix. Importantly, the manifest intercepts and residual variances are forced to equal zero; the means/intercepts, variances, and covariances of $\mathbf{\pi }$ are freely estimated; and an additional latent variable (i.e., ${\pi }_0$) emerges because ${\mathbf{\text{C}}}^{\ast }$ contains an extra row relative to C (i.e., the intercept). The latent variables ${\pi }_1$– ${\pi }_5$ have an identical interpretation as their original interpretation in the section Review of Orthogonal Contrast Matrices, and the new latent variable ${\pi }_0$ indicates a mean across the R repeated measures. Next, this article demonstrates how to test sphericity using L-RM-ANOVA in the SEM framework.

Testing Sphericity Using L-RM-ANOVA

Sphericity for a given main/interaction effect may be tested by performing a ${\chi }^2$-difference test across a model that assumes sphericity for the effect versus a model that does not. Following the guiding example, sphericity for the main effect of B and the interaction may be tested (sphericity is not relevant for the main effect of A because it only has one degree of freedom; see the section Sphericity for Main and Interaction Effects of U-RM-ANOVA). Sphericity for the main effect of B prescribes that both ${\sigma }_{{\pi }_2}^2={\sigma }_{{\pi }_3}^2$ and ${\sigma }_{{\pi }_2,{\pi }_3}=0$; whereas sphericity for the interaction effect prescribes ${\sigma }_{{\pi }_4}^2={\sigma }_{{\pi }_5}^2$, and ${\sigma }_{{\pi }_4,{\pi }_5}=0$ (see the section Sphericity for Main and Interaction Effects of U-RM-ANOVA for details). The constrained covariance matrices ${\mathbf{\text{V}}}_{{\mathbf{\pi }}_{\text{B}}}$ and ${\mathbf{\text{V}}}_{{\mathbf{\pi }}_{\text{A}\times \text{B}}}$ are given by:

where ${\sigma }_{\text{B}}^2$ is the equal variance across ${\pi }_2$ and ${\pi }_3$, and ${\sigma }_{\text{A}\times \text{B}}^2$ is the equal variance across ${\pi }_4$ and ${\pi }_5$, and omitted cells are freely estimated. In SEM, models with these constraints will be compared against models that do not place any constraints at all. Table 1 presents the results of both Mauchly’s test of sphericity and the L-RM-ANOVA based ${\chi }^2$-difference tests. The results are virtually identical across the two approaches; providing evidence that sphericity does not hold for either effect.

As mentioned earlier, it is also possible to test both sphericity assumptions simultaneously. That is, we can perform an omnibus test that tests for sphericity of in the effect of B and the interaction effect of A and B. This omnibus test can decrease the Type I error rate that can arise due to multiple testing. The results for the omnibus test can also be found in Table 1.

Simulation 1: Comparing Mauchly’s Sphericity Test to the SEM Based Test

In the previous sections, we revisited the original definition of sphericity and showed that SEM can be used to test for sphericity in multi-factorial experimental designs. We further compared Mauchly’s Sphericity Test to the SEM based test using an empirical example, where both tests showed very similar results. It is, however, important to know the statistical properties across different settings, for instance, sample sizes and degrees of departure from sphericity. In this section, we will conduct a small-scale simulation study to compare both tests. The aim of this study is to guide applied researchers to decide in which situation which test is to be preferred.

Testing Main and Interaction Effects of U-RM-ANOVA Using L-RM-ANOVA

A given main or interaction effect may be tested by performing a ${\chi }^2$-difference test across two models: a constrained model in which the means belonging to a particular effect are fixed to zero (i.e., conforming to the null hypothesis), and a second unconstrained model in which the means are freely estimated (i.e., confirming to the alternative hypothesis). But, to adhere to the assumptions of U-RM-ANOVA, both models used in ${\chi }^2$-difference test must impose sphericity.

Following the guiding example, the null hypothesis for the main effect of A prescribes ${\mu }_{{\pi }_1}=0$; the null hypothesis for the main effect of B prescribes both ${\mu }_{{\pi }_2}=0$ and ${\mu }_{{\pi }_3}=0$; and the null hypothesis for interaction effect prescribes both ${\mu }_{{\pi }_4}=0$ and ${\mu }_{{\pi }_5}=0$ (see the section Review of Orthogonal Contrast Matrices for details). Therefore, testing the main effect of A compares a model that constrains ${\mu }_{{\pi }_1}=0$, versus a model that freely estimates ${\mu }_{{\pi }_1}$ (without any extra constraints for the sphericity assumption). Testing the main effect of B compares a model that constrains ${\mu }_{{\pi }_2}=0$ and ${\mu }_{{\pi }_3}=0$, versus a model that freely estimates ${\mu }_{{\pi }_2}$ and ${\mu }_{{\pi }_3}$ (while both models impose sphericity for the main effect of B). Testing the interaction effect compares a model that constrains ${\mu }_{{\pi }_4}=0$ and ${\mu }_{{\pi }_5}=0$, versus a model that freely estimates ${\mu }_{{\pi }_4}$ and ${\mu }_{{\pi }_5}$ (while both models impose sphericity for the interaction effect).

Table 2 displays the relevant test statistics, and their p-values, for the main and interaction effects from U-RM-ANOVA and L-RM-ANOVA. Importantly, the product of an F-value and its numerator degrees of freedom (i.e., ${\mathit{\text{df}}}_{\text{num}}$) is asymptotically equivalent to the ${\chi }^2$-difference value from L-RM-ANOVA (i.e., $F_{\text{value}}\times {\mathit{\text{df}}}_{\text{num}}\approx {\chi }^2\iff F_{\text{value}}\approx {\chi }^2/{\mathit{\text{df}}}_{\text{num}}$, e.g., Fahrmeir et al., 2013; Kohler, 1982; Lu & Zhang, 2010); and the approximate F-values are given from L-RM-ANOVA using that transformation.

The test statistics and p-values differ across U-RM-ANOVA and L-RM-ANOVA (as opposed to the tests of sphericity from the previous sub-section). The difference across the approaches arises because U-RM-ANOVA bases its test statistics (and p-values) on F-ratios (and F-distributions), whereas L-RM-ANOVA uses ${\chi }^2$-differences (and compares those values to ${\chi }^2$-distributions). Therefore, discrepancies may arise across the approaches, even though they are designed to test the same hypotheses.

To narrow the gap, the following sub-section (after the excursus) shows how to reproduce both the sums of squares and the F-values from U-RM-ANOVA using the parameter estimates of L-RM-ANOVA. Furthermore, p-values from U-RM-ANOVA may then be more closely reproduced using L-RM-ANOVA by comparing the reproduced F-values to an F distribution.

Calculating Sums of Squares, Mean Squares, and F-Ratios Using L-RM-ANOVA

As noted earlier in this article (section Review of Orthogonal Contrast Matrices), orthonormal contrast matrices provide the opportunity to reproduce the sums of squares, mean squares, and F-ratios for each effect from U-RM-ANOVA (Voelkle, 2007). This sub-section shows how to reproduce each of these components using L-RM-ANOVA, and Table 3 shows the estimates across U-RM-ANOVA and L-RM-ANOVA. Importantly, all sums of squares, mean squares, and F-ratios are formed from an L-RM-ANOVA that both creates latent contrasts using an orthonormal matrix, and imposes sphericity. The general formulas to calculate the sums of squares are:

The sums of squares for a given effect from U-RM-ANOVA (written as, e.g., ${\mathit{\text{SS}}}_{\text{A}}$, ${\mathit{\text{SS}}}_{\text{B}}$, or ${\mathit{\text{SS}}}_{\text{A}\times \text{B}}$) may be reproduced in L-RM-ANOVA by summing across the squared means of the effect’s latent contrasts, and multiplying the sum by the sample size (Equation 13). Following the guiding example, ${\mathit{\text{SS}}}_{\text{A}}={\mu }_{{\pi }_1}^2\times N$; ${\mathit{\text{SS}}}_{\text{B}}=({\mu }_{{\pi }_2}^2+{\mu }_{{\pi }_3}^2)\times N$; and ${\mathit{\text{SS}}}_{\text{A}\times \text{B}}=({\mu }_{{\pi }_4}^2+{\mu }_{{\pi }_5}^2)\times N$ (N = 169 in the guiding example).

The residual sums of squares for a given effect from U-RM-ANOVA (written as, e.g., ${\mathit{\text{RSS}}}_{\text{A}}$, ${\mathit{\text{RSS}}}_{\text{B}}$, or ${\mathit{\text{RSS}}}_{\text{A}\times \text{B}}$) may be reproduced in L-RM-ANOVA by summing across the variances of the effect’s latent contrasts, and multiplying the sum by the sample size (Equation 14). Following the guiding example, ${\mathit{\text{RSS}}}_{\text{A}}={\sigma }_{{\pi }_1}^2\times N$, ${\mathit{\text{RSS}}}_{\text{B}}=({\sigma }_{{\pi }_2}^2+{\sigma }_{{\pi }_3}^2)\times N$, and ${\mathit{\text{RSS}}}_{\text{A}\times \text{B}}=({\sigma }_{{\pi }_4}^2+{\sigma }_{{\pi }_5}^2)\times N$ (again, N = 169 in the guiding example).

The mean squares for a given effect from U-RM-ANOVA (written as, e.g., ${\mathit{\text{MS}}}_{\text{A}}$, ${\mathit{\text{MS}}}_{\text{B}}$, or ${\mathit{\text{MS}}}_{\text{A}\times \text{B}}$ ) may be reproduced in L-RM-ANOVA by dividing the effect’s sums of squares by its numerator degrees of freedom (i.e., the number of contrasts that underlies the effect, Equation 15), such that ${\mathit{\text{MS}}}_{\text{A}}={\mathit{\text{SS}}}_{\text{A}}/{\mathit{\text{df}}}_{{\text{A}}_{\text{num}}}$; ${\mathit{\text{MS}}}_{\text{B}}={\mathit{\text{SS}}}_{\text{B}}/{\mathit{\text{df}}}_{{\text{B}}_{\text{num}}}$; and ${\mathit{\text{MS}}}_{\text{A}\times \text{B}}={\mathit{\text{SS}}}_{\text{A}\times \text{B}}/{\mathit{\text{df}}}_{\text{A}\times {\text{B}}_{\text{num}}}$. In the guiding example, ${\mathit{\text{df}}}_{{\text{A}}_{\text{num}}}=1$, ${\mathit{\text{df}}}_{{\text{B}}_{\text{num}}}=2$, and ${\mathit{\text{df}}}_{\text{A}\times {\text{B}}_{\text{num}}}=2$.

The mean squares of the residuals for a given effect from U-RM-ANOVA (written as, e.g., ${\mathit{\text{MSR}}}_{\text{A}}$, ${\mathit{\text{MSR}}}_{\text{B}}$, or ${\mathit{\text{MSR}}}_{\text{A}\times \text{B}}$ ) may be reproduced in L-RM-ANOVA by dividing each effect’s residual sums of squares by the product of N − 1 and the effect’s numerator degrees of freedom (Equation 16). Following the guiding example, we have ${\mathit{\text{MSR}}}_{\text{A}}={\mathit{\text{RSS}}}_{\text{A}}/((N-1)\times {\mathit{\text{df}}}_{{\text{A}}_{\text{num}}})$; ${\mathit{\text{MSR}}}_{\text{B}}={\mathit{\text{RSS}}}_{\text{B}}/((N-1)\times {\mathit{\text{df}}}_{{\text{B}}_{\text{num}}})$; and ${\mathit{\text{MSR}}}_{\text{A}\times \text{B}}={\mathit{\text{RSS}}}_{\text{A}\times \text{B}}/((N-1)\times {\mathit{\text{df}}}_{\text{A}\times {\text{B}}_{\text{num}}})$.

F-values from U-RM-ANOVA may be reproduced in L-RM-ANOVA by dividing the effect’s mean squares by its mean squared residuals (Equation 17). Following the guiding example leads to $F_{\text{A}}={\mathit{\text{MS}}}_{\text{A}}/{\mathit{\text{MSR}}}_{\text{A}}$, $F_{\text{B}}={\mathit{\text{MS}}}_{\text{B}}/{\mathit{\text{MSR}}}_{\text{B}}$, and $F_{\text{A}\times \text{B}}={\mathit{\text{MS}}}_{\text{A}\times \text{B}}/{\mathit{\text{MSR}}}_{\text{A}\times \text{B}}$. As shown in Table 3, all values are virtually identical across U-RM-ANOVA and L-RM-ANOVA; and therefore researchers may compute F and p-values that closely match those from U-RM-ANOVA using L-RM-ANOVA.

Finally, we would like to note that a major advantage of SEM is that sums of squares can also be calculated in the presence of missing values. Parameter estimates can be obtained through full information maximum likelihood, which are then used to calculate sums of squares as shown above. The above calculations can further be extended to mixed within- and between-subjects designs with any number of factors. We refer the interested reader to Langenberg et al. (2022), which includes instructions in the appendix to calculate F-values and the effect size measure ${\eta }_p^2$ for larger within- and between-subjects designs.

Testing Main and Interaction Effects Without the Assumption of Sphericity Using L-RM-ANOVA

In contrast to U-RM-ANOVA, L-RM-ANOVA may test main/interaction effects without the assumption of sphericity. The model comparison procedure described above (i.e., ${\chi }^2$-difference tests described in the section Testing Main and Interaction Effects of U-RM-ANOVA Using L-RM-ANOVA) can relax sphericity by estimating all elements of ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ in both models (i.e., removing the constraints that describe sphericity). The right part of Table 2 provides estimates of main and interaction effects from L-RM-ANOVA when estimated without the assumption of sphericity.

Currently, U-RM-ANOVA relies on post-hoc corrections to relax sphericity (e.g., Greenhouse-Geisser and Huynh-Feldt corrections), which computationally (and conceptually) differ from direct estimation of ${\mathbf{\text{V}}}_{\mathbf{\pi }}$ as performed via L-RM-ANOVA. Table 2 also provides p-values associated with these common post-hoc corrections.

Imposing the assumption of sphericity can increase power of hypothesis tests in U-RM-ANOVA. This is true for any type of assumption imposed to statistical models as fewer parameters need to be estimated if the assumptions is true. However, if the assumption, in fact, does not hold, an increased Type I error rate may arise (e.g., Haverkamp & Beauducel, 2017).

Simulation 2: Comparing U-RM-ANOVA and L-RM-ANOVA With and Without Sphericity

In the previous two sections, we compared hypothesis tests of U-RM-ANOVA and L-RM-ANOVA with and without the assumption of sphericity. Both approaches yield very similar F-values and p-values. It remains to be answered whether any of the approaches outperforms the others in terms of statistical properties. For instance, L-RM-ANOVA is estimated through maximum likelihood which is known to have and inflated Type 1 error in small samples (e.g., Green & Babyak, 1997; Hu et al., 1992; Muthén & Kaplan, 1985; Raykov & Widaman, 1995). U-RM-ANOVA, in contrast, is said to have a power advantage but should also have and inflated Type 1 error when sphericity does not hold. In this section, we will examine the statistical properties of the aforementioned approaches across several settings. We have two main hypotheses: (1) We expect L-RM-ANOVA to have an inflated Type 1 error when testing main and interaction effects in small samples as compared to RM-ANOVA because it is estimated through maximum likelihood which relies on asymptotic theory, and (2) we also expect the models that assume sphericity to have an inflated Type 1 error when testing main and interaction effects and larger bias of effect size estimates when sphericity does not hold.