ADVANCED INFERENTIAL STATISTICS: ANOVA, MULTIPLE GROUP COMPARISONS &
POST-HOC ANALYSIS
SECTION A: ANALYSIS OF VARIANCE (ANOVA) FUNDAMENTALS
Questions 1-40
1. When comparing three or more population means, why is performing multiple t-
tests problematic?
A) It increases the probability of Type I error beyond the nominal alpha level
B) It reduces statistical power substantially
C) It requires larger sample sizes than ANOVA
D) It cannot be used with categorical independent variables
Answer: A) It increases the probability of Type I error beyond the nominal alpha level
Rationale: Conducting multiple pairwise t-tests inflates the family-wise error rate. For k
groups, the probability of at least one Type I error becomes 1 - (1-α)^m, where m is the
number of comparisons, far exceeding the nominal α level.
2. In a one-way ANOVA, the total variability in the data is partitioned into which two
components?
A) Between-group variance and within-group variance
B) Systematic variance and error variance
C) Explained variance and unexplained variance
D) All of the above are correct
Answer: D) All of the above are correct
Rationale: These are different ways of describing the same partition. Between-group
variance represents systematic/explained variance due to group differences, while within-
group variance represents error/unexplained variance.
,3. What is the F-ratio in ANOVA?
A) MS_between / MS_error
B) MS_error / MS_between
C) SS_between / SS_error
D) SS_total / SS_between
Answer: A) MS_between / MS_error
Rationale: The F-ratio is the ratio of mean square between groups to mean square error.
Larger values indicate that group differences exceed what would be expected by chance.
4. The degrees of freedom for the between-groups sum of squares in a one-way ANOVA
with k groups is:
A) k - 1
B) N - k
C) N - 1
D) k(n - 1)
Answer: A) k - 1
Rationale: The between-groups df equals the number of groups minus one, representing
the number of independent comparisons possible among group means.
5. Which assumption is NOT required for a one-way ANOVA?
A) Normality of residuals within each group
B) Homogeneity of variances across groups
C) Independence of observations
D) Normality of the independent variable
Answer: D) Normality of the independent variable
Rationale: ANOVA assumes normality of the dependent variable residuals, not the
independent variable, which is categorical. The independent variable can be any nominal
or ordinal categorical variable.
6. Levene's test is used to check which ANOVA assumption?
A) Normality of residuals
B) Homogeneity of variance
,C) Independence of observations
D) Linearity
Answer: B) Homogeneity of variance
Rationale: Levene's test assesses the null hypothesis that variances are equal across
groups. A significant result (p < .05) indicates violation of the homogeneity assumption.
7. When the F-test in ANOVA is significant, what can we conclude?
A) All group means are different from each other
B) At least one group mean is different from the others
C) The groups are all equivalent
D) The effect size is large
Answer: B) At least one group mean is different from the others
Rationale: A significant F-test indicates that not all group means are equal. However, it
does not specify which groups differ, necessitating post-hoc tests.
8. What is the relationship between the t-test and ANOVA?
A) ANOVA is a generalization of the t-test to more than two groups
B) t-test is a special case of ANOVA for two groups
C) Both A and B are correct
D) They are unrelated statistical tests
Answer: C) Both A and B are correct
Rationale: For two groups, the F-statistic from ANOVA equals the square of the t-statistic,
and the p-values are identical. ANOVA generalizes the t-test to handle multiple groups.
9. In ANOVA, the sum of squares total (SS_total) equals:
A) SS_between + SS_within
B) SS_between × SS_within
C) SS_between - SS_within
D) SS_between / SS_within
Answer: A) SS_between + SS_within
Rationale: The total sum of squares is partitioned into between-group variation and within-
group variation. This fundamental identity underlies the ANOVA decomposition.
, 10. Which measure of effect size is most commonly reported with ANOVA?
A) Cohen's d
B) Eta-squared (η²)
C) Omega-squared (ω²)
D) R²
Answer: B) Eta-squared (η²)
Rationale: Eta-squared represents the proportion of total variance explained by the
grouping variable, calculated as SS_between / SS_total. It's the most frequently reported
effect size for ANOVA.
11. A researcher obtains F(3, 60) = 4.50, p = .006. How many groups were compared?
A) 3
B) 4
C) 60
D) 63
Answer: B) 4
Rationale: The first degrees of freedom (df1 = 3) equals k - 1, where k is the number of
groups. Therefore, k = df1 + 1 = 4 groups.
12. What does the within-groups mean square (MS_within) estimate?
A) Population variance assuming the null hypothesis is true
B) Variance of group means
C) Population variance regardless of the null hypothesis
D) Variance due to the independent variable
Answer: A) Population variance assuming the null hypothesis is true
Rationale: MS_within is the pooled variance estimate that does not depend on group
differences. Under the null hypothesis, it serves as an unbiased estimate of the common
population variance.
13. Which of the following is an appropriate method for testing normality in ANOVA?
A) Shapiro-Wilk test
B) Kolmogorov-Smirnov test
C) Q-Q plots
D) All of the above
POST-HOC ANALYSIS
SECTION A: ANALYSIS OF VARIANCE (ANOVA) FUNDAMENTALS
Questions 1-40
1. When comparing three or more population means, why is performing multiple t-
tests problematic?
A) It increases the probability of Type I error beyond the nominal alpha level
B) It reduces statistical power substantially
C) It requires larger sample sizes than ANOVA
D) It cannot be used with categorical independent variables
Answer: A) It increases the probability of Type I error beyond the nominal alpha level
Rationale: Conducting multiple pairwise t-tests inflates the family-wise error rate. For k
groups, the probability of at least one Type I error becomes 1 - (1-α)^m, where m is the
number of comparisons, far exceeding the nominal α level.
2. In a one-way ANOVA, the total variability in the data is partitioned into which two
components?
A) Between-group variance and within-group variance
B) Systematic variance and error variance
C) Explained variance and unexplained variance
D) All of the above are correct
Answer: D) All of the above are correct
Rationale: These are different ways of describing the same partition. Between-group
variance represents systematic/explained variance due to group differences, while within-
group variance represents error/unexplained variance.
,3. What is the F-ratio in ANOVA?
A) MS_between / MS_error
B) MS_error / MS_between
C) SS_between / SS_error
D) SS_total / SS_between
Answer: A) MS_between / MS_error
Rationale: The F-ratio is the ratio of mean square between groups to mean square error.
Larger values indicate that group differences exceed what would be expected by chance.
4. The degrees of freedom for the between-groups sum of squares in a one-way ANOVA
with k groups is:
A) k - 1
B) N - k
C) N - 1
D) k(n - 1)
Answer: A) k - 1
Rationale: The between-groups df equals the number of groups minus one, representing
the number of independent comparisons possible among group means.
5. Which assumption is NOT required for a one-way ANOVA?
A) Normality of residuals within each group
B) Homogeneity of variances across groups
C) Independence of observations
D) Normality of the independent variable
Answer: D) Normality of the independent variable
Rationale: ANOVA assumes normality of the dependent variable residuals, not the
independent variable, which is categorical. The independent variable can be any nominal
or ordinal categorical variable.
6. Levene's test is used to check which ANOVA assumption?
A) Normality of residuals
B) Homogeneity of variance
,C) Independence of observations
D) Linearity
Answer: B) Homogeneity of variance
Rationale: Levene's test assesses the null hypothesis that variances are equal across
groups. A significant result (p < .05) indicates violation of the homogeneity assumption.
7. When the F-test in ANOVA is significant, what can we conclude?
A) All group means are different from each other
B) At least one group mean is different from the others
C) The groups are all equivalent
D) The effect size is large
Answer: B) At least one group mean is different from the others
Rationale: A significant F-test indicates that not all group means are equal. However, it
does not specify which groups differ, necessitating post-hoc tests.
8. What is the relationship between the t-test and ANOVA?
A) ANOVA is a generalization of the t-test to more than two groups
B) t-test is a special case of ANOVA for two groups
C) Both A and B are correct
D) They are unrelated statistical tests
Answer: C) Both A and B are correct
Rationale: For two groups, the F-statistic from ANOVA equals the square of the t-statistic,
and the p-values are identical. ANOVA generalizes the t-test to handle multiple groups.
9. In ANOVA, the sum of squares total (SS_total) equals:
A) SS_between + SS_within
B) SS_between × SS_within
C) SS_between - SS_within
D) SS_between / SS_within
Answer: A) SS_between + SS_within
Rationale: The total sum of squares is partitioned into between-group variation and within-
group variation. This fundamental identity underlies the ANOVA decomposition.
, 10. Which measure of effect size is most commonly reported with ANOVA?
A) Cohen's d
B) Eta-squared (η²)
C) Omega-squared (ω²)
D) R²
Answer: B) Eta-squared (η²)
Rationale: Eta-squared represents the proportion of total variance explained by the
grouping variable, calculated as SS_between / SS_total. It's the most frequently reported
effect size for ANOVA.
11. A researcher obtains F(3, 60) = 4.50, p = .006. How many groups were compared?
A) 3
B) 4
C) 60
D) 63
Answer: B) 4
Rationale: The first degrees of freedom (df1 = 3) equals k - 1, where k is the number of
groups. Therefore, k = df1 + 1 = 4 groups.
12. What does the within-groups mean square (MS_within) estimate?
A) Population variance assuming the null hypothesis is true
B) Variance of group means
C) Population variance regardless of the null hypothesis
D) Variance due to the independent variable
Answer: A) Population variance assuming the null hypothesis is true
Rationale: MS_within is the pooled variance estimate that does not depend on group
differences. Under the null hypothesis, it serves as an unbiased estimate of the common
population variance.
13. Which of the following is an appropriate method for testing normality in ANOVA?
A) Shapiro-Wilk test
B) Kolmogorov-Smirnov test
C) Q-Q plots
D) All of the above
