Biostatistics Study NotesFrom concepts and derivations to interpretation

CHAPTER 06 · TOPIC 04

Two-Way ANOVA and MANOVA

Separate main effects from interactions in two-way ANOVA and test multiple related dependent variables jointly with MANOVA.

On this page
  1. Two-way ANOVA: Examining two factors simultaneously
  2. Two-way ANOVA model
  3. Why Examine the Interaction First?
  4. Two-way ANOVA conditions
  5. MANOVA: Examining Multiple Dependent Variables Simultaneously
  6. Constructing a MANOVA Test
  7. MANOVA Assumptions and Limitations
  8. Follow-Up Analyses after a Significant MANOVA
  9. Distinguishing the Three Methods

Two-way ANOVA and multivariate analysis of variance (MANOVA) are extensions in different directions. Two-way ANOVA deals with two categorical factors for one quantitative dependent variable, while MANOVA deals with multiple related quantitative dependent variables simultaneously.

Two-way ANOVA: Examining two factors simultaneously#

If treatment is factor A, gender is factor B, and blood pressure is the dependent variable, the main effect of A, the main effect of B, and the A×B interaction can be tested in the same model.

effectWhat to ask
Main effect of factor AWhen averaged at the level of B, does the mean differ between the levels of A?
Main effect of factor BWhen averaged at the level of A, does the average differ between the levels of B?
A×B interactionDoes the effect of A change depending on the level of B?

Two-way ANOVA model#

Yijk=μ+αi+βj+(αβ)ij+εijkY_{ijk}=\mu+\alpha_i+\beta_j+(\alpha\beta)_{ij}+\varepsilon_{ijk}
symbolmeaning
μGrand average of all conditions
αᵢi-th level effect of factor A
βⱼj-th level effect of factor B
(αβ)ᵢⱼInteraction of level combinations
εᵢⱼₖError not explained by model
A main effectH0,A:α1=⋯=αa=0H_{0,A}:\alpha_1=\cdots=\alpha_a=0
B main effectH0,B:β1=⋯=βb=0H_{0,B}:\beta_1=\cdots=\beta_b=0
interactionH0,AB:(αβ)ij=0for all i,jH_{0,AB}:(\alpha\beta)_{ij}=0\quad\text{for all }i,j

Why Examine the Interaction First?#

If the interaction is significant, the effect of one factor depends on the level of the other. For example, if a drug is effective in only one sex, a sex-averaged drug main effect alone hides the data structure. Typically, you look at interactions first and then interpret the main effects.

Significant interactions are followed up with simple main effects, such as comparing treatments within each gender. These comparisons also require multiplicity adjustment.

Two-way ANOVA conditions#

  • Observations are independent. For repeated measurements, use repeated measures or mixed models
  • Errors within each factor combination are approximately normal
  • The error variance is equal for each combination
  • There are enough observations in each cell to estimate the interaction

MANOVA: Examining Multiple Dependent Variables Simultaneously#

When measuring multiple related quantitative outcomes from the same subject, such as systolic blood pressure, diastolic blood pressure, and heart rate, MANOVA compares a vector of population means rather than a single mean for each group.

mean vector of jth groupμj=(μj1,μj2,…,μjp)T\boldsymbol{\mu}_j=(\mu_{j1},\mu_{j2},\ldots,\mu_{jp})^{\mathsf T}
null hypothesisH0:μ1=μ2=⋯=μkH_0:\boldsymbol{\mu}_1=\boldsymbol{\mu}_2=\cdots=\boldsymbol{\mu}_k

Performing separate ANOVAs for each dependent variable increases type I error due to multiple testing, and differences in combinations of results cannot be directly tested. MANOVA takes advantage of the covariance between the dependent variables.

Constructing a MANOVA Test#

Expand the between-group and within-group sums of squares of univariate ANOVA to sum of squares and cross-product matrices H (effect) and E (error), and create multivariate statistics from their eigenvalues.

statisticsinterpretation
Pillai's traceRelatively robust to assumption violations. The larger the difference, the greater the group difference.
Wilks' lambdaThe smaller the value, the larger the group difference in the mean vector.
Hotelling–Lawley traceAccumulate intergroup information for multiple discriminant directions
Roy's largest rootFocus on the one discriminant direction with the largest difference

MANOVA Assumptions and Limitations#

  • observations are independent
  • The dependent variable vector for each group is approximately multivariate normal.
  • The covariance matrices of each group are equal
  • The dependent variables are related but not collinear enough to nearly overlap.
  • There is a sufficient sample size to estimate the covariance of multiple dependent variables.

If the variables of interest are largely unrelated, there is little benefit to integration, and if the correlations are extremely high, the same concept may be being measured redundantly. The variables of interest are selected in advance from the research question rather than after looking at the data.

Follow-Up Analyses after a Significant MANOVA#

  1. See which multivariate tests for factors or interactions are significant
  2. Adjust multiplicity by performing ANOVA and planned contrast for each dependent variable according to the research question.
  3. Evaluate direction and substantive meaning from group mean vector, confidence interval, and effect size
  4. If necessary, examine the discriminant function or canonical variate and return to the original scale dependent variable for interpretation.

Distinguishing the Three Methods#

methodcategorical factorquantitative dependent variablemain question
One-way ANOVA11Are the group means of one factor different?
Two-way ANOVA21Are there two main effects and an interaction?
MANOVA1 or more2 or moreAre the group mean vectors different?