ANOVA

Question 1

What does ANOVA stand for, and what is its purpose?

Answer 1

ANalysis Of VAriance (ANOVA) tests for differences in means across three or more groups.

Question 2

Why shouldn’t we use multiple t-tests for comparing more than two groups?

Answer 2

Multiple t-tests increase the Type I error rate, raising the probability of falsely rejecting the null hypothesis.

Question 3

If there are 5 groups, how many pairwise comparisons would there be?

Answer 3

10

Question 4

What is a Type I error?

Answer 4

Rejecting the null hypothesis (𝐻0) when it is actually true.
Probability = 𝛼 (significance level, e.g., 0.05).

Question 5

What is a Type II error?

Answer 5

Failing to reject the null hypothesis (𝐻0) when it is actually false.
Probability = 𝛽.

Question 6

How does performing multiple t-tests increase the Type I error rate?

Answer 6

For 𝑘 comparisons, the probability of at least one Type I error is:1−(1−α)^k

Question 7

How does the Bonferroni correction control the Type I error rate?

Answer 7

Divide the significance level by the number of comparisons:
𝛼𝑘=𝛼/𝑘

Question 8

What are the null and alternative hypotheses for ANOVA?

Answer 8

H0:μA=μB=μC (means are equal)

𝐻1: at least one mean is different.

Question 9

What does the ANOVA test statistic (𝐹) compare?

Answer 9

F= Variabilitybetweengroups / Variabilitywithingroups
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Question 10

What is the formula for the between-group variance (𝑠𝐺2)?

Answer 10

𝑠𝐺2 = sum of squares between SSG / (k-1)

Question 11

What is the formula for the within-group variance (𝑠𝑊2)?

Answer 11

𝑠𝑊2 = sum of squares within (SSW) / (ntot -k)
ntot = total sample size,
𝑘 = number of groups.

Question 12

What does a large F-statistic indicate in ANOVA?

Answer 12

A large F-statistic means between-group variability is large compared to within-group variability, suggesting significant differences between group means.

Question 13

How do you interpret the ANOVA table?

Answer 13

Between groups: Large Mean Square indicates group differences.

Residuals (within groups): Represents random variability.

Large 𝐹-value + small 𝑝-value = reject 𝐻0

Question 14

What degrees of freedom are used in ANOVA?

Answer 14

df1=k−1 (between groups)

𝑑𝑓2=𝑛𝑡𝑜𝑡−𝑘 (within groups)

Question 15

What does a small p-value in ANOVA indicate?

Answer 15

It indicates strong evidence that at least one group mean is significantly different.

Question 16

What is the test statistic in ANOVA?

Answer 16

The F-statistic: SG2 / SW2

Question 17

If the ANOVA test is significant, what should you do next?

Answer 17

Perform a post-hoc test (like Tukey’s test) to identify which groups differ.

Question 18

What assumptions does ANOVA rely on?

Answer 18

Normality of data within groups.

Homogeneity of variances across groups.

Independence of observations.

Question 19

What should you do if ANOVA assumptions are violated?

Answer 19

Use a non-parametric alternative like the Kruskal-Wallis test.

Question 20

Given 𝐹0=5.263, 𝑑𝑓1=2, 𝑑𝑓2=31499df2=31499, and a critical value of 2.996, should you reject 𝐻0?

Answer 20

Yes, reject 𝐻0 because:

𝐹0=5.263 > 2.996 (critical value)

𝑝-value=0.00518 < 0.01.

There is strong evidence that at least one mean is different.

Question 21

What does the 𝑝-value of 0.00518 mean in this context?

Answer 21

It is highly unlikely to obtain an F-statistic this extreme (𝐹0=5.263) if 𝐻0 (all means are equal) were true.

Question 22

After rejecting 𝐻0, what should you do next?

Answer 22

Perform a post-hoc test like Tukey’s HSD to determine which groups are significantly different.

Question 23

Why is the adjusted 𝑝-value (p adj) used in Tukey’s test?

Answer 23

It controls for the family-wise error rate by adjusting the significance level for multiple comparisons.

Question 24

What is the family-wise error rate, and why is it important?

Answer 24

It is the overall probability of making at least one Type I error when conducting multiple comparisons. It is controlled by adjusting 𝛼

Question 25

What are the key assumptions for a valid ANOVA test?

Answer 25

Independence of data within and between groups.

Equal variances across groups (homogeneity of variance).

Normal distribution of data within groups.

Question 26

How can you check the assumption of equal variances?

Answer 26

compare the standard deviation

Rule of thumb: the largest 𝑠 should be no more than twice the smallest.

Alternatively, use Levene’s test in R

Question 27

How do you test for normality in ANOVA?

Answer 27

Plot histograms or Q-Q plots.

Perform a Shapiro-Wilk test in R.

Question 28

What non-parametric test can be used if the ANOVA assumptions are violated?

Answer 28

Kruskal-Wallis test (for non-normal data).

It assumes:
Independent data.

Similar shape of distributions across groups.

Question 29

Why is practical significance different from statistical significance?

Answer 29

Statistical significance means a result is unlikely due to chance, while practical significance considers if the difference is meaningful in real-world terms.

Question 30

What are the steps in hypothesis testing?

Answer 30

State 𝐻0 and 𝐻1

Collect data.

Select a test (e.g., ANOVA, t-test, etc.).

Set 𝛼 (e.g., 0.05).

Find the reference distribution (e.g., F-distribution).

Calculate test-statistic & p-value.

Make a decision (reject/fail to reject 𝐻0).

Interpret the result.

Question 31

Why is the F-distribution used in ANOVA?

Answer 31

It compares the variance between groups to the variance within groups to test if the means differ.

Question 32

What does a large F-statistic imply?

Answer 32

Large differences between group means relative to the within-group variability, suggesting significant group differences.

Question 33

What does failing to reject 𝐻0 mean?

Answer 33

There isn’t enough evidence to conclude that any group means are different.