Comprehensive ANCOVA, Repeated Measures, and MANOVA Key Concepts

What is the purpose of ANCOVA?

To test for differences between group means while adjusting for known extraneous variables (covariates).

What are the primary advantages of using ANCOVA?

It reduces error variance by explaining unexplained variance and provides greater experimental control.

What type of variables are typically used as covariates in ANCOVA?

Continuous variables.

What is the assumption of independence of the covariate and treatment effect in ANCOVA?

It assumes the covariate is independent of the treatment effect; if they overlap, the covariate may reduce the experimental effect, complicating interpretation.

What is the homogeneity of regression slopes assumption in ANCOVA?

It assumes the relationship between the outcome variable and the covariate is the same across all treatment groups.

What does a significant interaction effect between a covariate and a treatment factor indicate?

It indicates a violation of the homogeneity of regression slopes assumption.

What does partial eta squared (partial η2) quantify in ANCOVA?

The proportion of variance that a variable explains that is not explained by other variables in the model.

What is a repeated-measures design?

A design where the same entities participate in all conditions of an experiment or provide data at multiple time points.

What are the main benefits of a repeated-measures design?

Increased sensitivity to experimental effects by reducing unsystematic variance and improved economy by requiring fewer participants.

What are some non-statistical limitations of repeated-measures designs?

Practice and fatigue effects, carryover effects, contrast effects, and context effects.

What is the assumption of sphericity in repeated-measures ANOVA?

It assumes that the variances of the differences between all possible pairs of conditions are equal.

What test is used to check for the violation of sphericity?

Mauchly's test.

What happens if the assumption of sphericity is violated?

It leads to a loss of power and an F-statistic that does not follow the expected distribution.

Which estimates are used to correct for the violation of sphericity?

Greenhouse-Geisser estimate and Huynh-Feldt estimate.

When should the Huynh-Feldt estimate be used to correct for sphericity?

When the estimate of sphericity is greater than 0.75.

When should the Greenhouse-Geisser estimate be used to correct for sphericity?

When the Greenhouse-Geisser estimate is less than 0.75 or when nothing is known about sphericity.

What is the difference between a random intercept model and a random slope model?

A random intercept model allows the baseline (intercept) to vary between participants, while a random slope model also allows the effect of the predictor (slope) to vary between participants.

What is the effect of a covariate on the unexplained variance in ANCOVA?

It reduces the unexplained variance (SSr) by accounting for some of it.

Why is independence of the covariate and treatment effect preferred in ANCOVA?

It makes the interpretation of the results more straightforward.

What is the consequence of heterogeneity of slopes in ANCOVA?

It suggests that the relationship between the covariate and the outcome depends on the treatment group, which can be modeled using multilevel linear models.

What does a p-value of less than .05 in Mauchly's test indicate?

That the assumption of sphericity has been violated.

What is the primary statistical limitation of repeated-measures designs?

Observations are not independent because the same participants are measured multiple times.

What is compound symmetry?

A condition where both the variances across conditions are equal and the covariances between pairs of conditions are equal.

What is the purpose of adjusting means in ANCOVA?

To estimate what the group means would be if all participants had the same score on the covariate.

What are the degrees of freedom for the total sum of squares (SST)?

N - 1

What does the within-participant sum of squares (SSW) measure?

Individual differences within an entity rather than within a condition.

What is the relationship between the residual sum of squares (SSR), SSW, and SSM?

SSR = SSW - SSM

How is the F statistic calculated in the variance approach?

F = Mean Square Model / Mean Square Residual

What is the between-participants sum of squares (SSB)?

SSB = SST - SSW

What is a mixed design in statistics?

A design that includes both independent variables measured using different entities and variables measured using repeated measures.

What are the two primary assumptions for mixed designs?

Homogeneity of variance and Sphericity.

Why is MANOVA preferred over multiple ANOVAs?

It controls the familywise error rate (Type I error) and accounts for relationships between dependent variables.

What are four common test statistics used in MANOVA?

Pillai's Trace, Wilk's Lambda, Roy's Largest Root, and Hotelling-Lawley.

What is a matrix in the context of multivariate statistics?

A grid of numbers arranged in columns and rows.

What is an identity matrix?

A square matrix where the diagonals are 1 and the off-diagonals are 0.

What does the letter H represent in MANOVA matrices?

The hypothesis sum of squares and cross-products matrix.

What does the letter E represent in MANOVA matrices?

The error sum of squares and cross-products matrix.

What does the letter T represent in MANOVA matrices?

The total sum of squares and cross-products (SSCP) matrix.

What do cross-products represent in multivariate models?

The total combined error between two variables, representing an unstandardized estimate of their correlation.

What are discriminant variates in MANOVA?

Underlying linear dimensions of outcome variables used to predict group membership.

How is the number of discriminant variates determined?

It is the smaller of p (number of outcome variables) or k - 1 (number of groups).

What do eigenvalues (λi) represent in discriminant analysis?

The variance explained by each discriminant variate.

What does Pillai's Trace measure?

The sum of the proportion of explained variance on the discriminant functions.

What is the formula for the model cross-product?

CP_M = CP_T - CP_R

What does HE^-1 represent in the context of MANOVA?

The ratio of the systematic variance in the model to the unsystematic variance in the model.

What is a square matrix?

A matrix with an equal number of columns and rows.

What is the Hotelling-Lawley trace?

The sum of the eigenvalues for each variate, representing the sum of SSM/SSR for each variate.

What does Wilk's Lambda represent?

The product of the unexplained variance on each of the variates, representing the ratio of error variance to total variance (SSR/SST).

What is Roy's Largest Root?

The eigenvalue for the first variate only, representing the proportion of explained to unexplained variance for the first discriminant function.

What is the purpose of Box's Test in MANOVA?

It tests the assumption of homogeneity of covariance matrices; a significant result (p < 0.05) indicates the assumption has been violated.

What is the primary advantage of MANOVA over conducting several separate ANOVAs?

It prevents the inflation of the Type I error rate that occurs when conducting multiple ANOVAs.

What is a requirement for the number of outcome variables in a MANOVA?

MANOVA requires more than one outcome variable.

What is the consequence of high correlation between dependent variables in MANOVA?

The loss of degrees of freedom.

What does Discriminant Function Analysis predict?

A categorical dependent variable using one or more predictor variables.

What is a two-way repeated-measures ANOVA?

A test comparing several means when there are two independent variables, and the same entities are used in all conditions.

How many effects are there in a two-way repeated-measures ANOVA?

Three.

Which row should be interpreted in a repeated-measures ANOVA if sphericity is not assumed?

The Greenhouse-Geisser row.

What is the consequence of violating the assumption of sphericity?

It reduces statistical power, affects the F-statistic distribution, and increases Type I error rates in post hoc tests.

What is a Mixed ANOVA?

A test comparing means with two or more independent variables, where at least one is measured using the same entities and at least one is measured using different entities.

What does the presence of parallel lines in a graph indicate?

There is no interaction between the variables.

What are practice effects in repeated-measures studies?

Changes in participant performance due to the nature of repeated testing rather than the independent variable.

What are potential sources of bias in a repeated-measures factorial design?

Outliers and non-normality.

Which assumption is not relevant when conducting a repeated-measures ANOVA?

Homogeneity of variance.

What is the definition of a repeated-measures factorial design?

An experiment comparing the means of two or more categorical predictors, each measured using the same entities in all conditions.

What is the purpose of contrasts in research?

To test specific hypotheses related to particular research questions.

What does a three-way ANOVA analyze?

Differences between means across three independent variables.

What are cross-products in statistical analysis?

They represent a total value for the combined error between two variables.

Why are repeated-measures studies often more efficient regarding sample size?

Because each participant is measured multiple times, allowing for smaller groups.

What is the assumption of multivariate normality in MANOVA?

The assumption that the residuals follow a multivariate normal distribution.

What is the assumption of independence in MANOVA?

The assumption that residuals should be statistically independent.

Is it possible to calculate underlying linear dimensions of dependent variables for a MANOVA?

Yes.

What is the primary factor that strengthens the power of a MANOVA?

A large effect size combined with one outcome variable showing a large group difference.