Question 1

Which of the following is NOT an advantage of using a two-way ANOVA over running two separate one-way ANOVAs when analyzing a factorial design with two categorical factors?

Accepted Answer

It guarantees that the main effects will always be significant if the interaction is significant.

Question 2

In a balanced two-way ANOVA with $$r$$ levels of factor A, $$c$$ levels of factor B, and $$n$$ observations per cell, what is the correct formula for the total sample size $$N$$?

Accepted Answer

$$N = r 	imes c 	imes n$$

Question 3

Consider the two-way fixed effects ANOVA model: $$Y_{ijk} = \mu.. + \alpha_i + \beta_j + \gamma_{ij} + E_{ijk}. $$What does the parameter $$\gamma_{ij}$$ represent?

Accepted Answer

The interaction effect between level $$i of A $$and level $$j of B$$

Question 4

Which of the following statements about the interpretation of main effects in the presence of a significant interaction in two-way ANOVA is correct?

Accepted Answer

Main effects may be misleading because the effect of one factor depends on the level of the other; simple effects should be reported instead.

Question 5

In the decomposition of variation for a balanced two-way ANOVA, which of the following correctly expresses the total sum of squares ($$SSY$$)?

Accepted Answer

$$SSY = SSR + SSC + SSRC + SSE$$

Question 6

A clinical trial compares two drugs (A and B) at two dose levels (Low and High) in a 2×2 factorial design. The ANOVA table shows a significant interaction (p \u003c 0.01) and a significant main effect for Dose (p \u003c 0.01), but a non-significant main effect for Drug (p = 0.47). What is the most appropriate interpretation?

Accepted Answer

The significant interaction means the effect of Dose differs by Drug; main effects should not be interpreted without considering the interaction.

Question 7

Which of the following patterns in an interaction plot indicates the presence of a disordinal (qualitative) interaction?

Accepted Answer

The lines cross each other.

Question 8

In a two-way ANOVA, which null hypothesis is tested by the F-ratio $$F = MSRC / MSE in a $$fixed effects model?

Accepted Answer

No interaction between factors A and B

Question 9

Suppose you are analyzing a 3×2 factorial RCT with equal cell sizes using two-way ANOVA. The interaction term is significant. What is the recommended next step for interpreting the results?

Accepted Answer

Report and interpret the simple effects (the effect of one factor at each level of the other).

Question 10

Which of the following is a key assumption of two-way ANOVA that CANNOT be checked using residual plots?

Accepted Answer

Independence of observations

Question 11

In the context of two-way ANOVA, what is the main practical consequence of having unequal cell sizes (an unbalanced design)?

Accepted Answer

The sums of squares are no longer orthogonal, and the order of terms in the model affects the results.

Question 12

Which of the following best describes the difference between fixed and random effects in two-way ANOVA?

Accepted Answer

Fixed effects are estimated as constants; random effects are modeled as random variables drawn from a distribution.

Question 13

In a mixed effects two-way ANOVA where factor A is fixed and factor B is random, which mean square should be used as the denominator for the F-test of the main effect of factor A?

Accepted Answer

$$MSRC$$

Question 14

A multi-center clinical trial uses a two-way ANOVA with treatment (Active vs Placebo) as a fixed effect and site (6 hospitals) as a random effect. What is the main reason for treating site as a random effect?

Accepted Answer

Inference is intended to generalize to all possible sites, not just those included in the study.

Question 15

Which of the following is the correct formula for the cell mean $$\mu_{ij} in a $$two-way ANOVA model with effect (sum-to-zero) coding for a 2×2 design?

Accepted Answer

$$\mu_{ij} = \mu.. + \alpha_i + \beta_j + \gamma_{ij}$$

Question 16

In the regression formulation of two-way ANOVA with effect coding, what is the interpretation of the coefficient for the interaction term (e.g., $$\gamma_{11} in a 2×2 $$design)?

Accepted Answer

It represents the difference in the effect of factor A between the two levels of factor B.

Question 17

A researcher fits a two-way ANOVA model in R using both aov() and lm() on a balanced design. Which of the following is true regarding the F-statistics and p-values for the main effects and interaction?

Accepted Answer

They will be identical for both aov() and lm() models.

Question 18

Which of the following post-hoc procedures is most appropriate for comparing all possible pairs of treatment cells in a two-way ANOVA with a significant interaction?

Accepted Answer

TukeyHSD()

Question 19

In a two-way ANOVA, the partial eta-squared ($$\$$eta^2$$) statistic is used to:

Accepted Answer

Measure the proportion of total variance explained by each effect.

Question 20

A 3×2 factorial RCT for depression treatments finds a significant treatment × severity interaction. The simple effects analysis shows that combined therapy is much more effective than either CBT or medication alone, but only in the severe group. What does this imply about the main effect of treatment?

Accepted Answer

The effect of treatment depends on severity; main effects alone are misleading.

Two-Way Analysis of Variance (ANOVA) with Equal Cell Sizes: Concepts, Applications, and Mixed Models

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