Lesson 107 — One-way ANOVA

An experiment compares 3 groups with 10 observations each. Determine $df_B$ and $df_W$.

A researcher uses 5 groups with 10 participants each. Determine $df_B$ and $df_W$.

In an experiment with 3 groups, $SSB = 40$ ($df_B = 2$) and $SSW = 150$ ($df_W = 30$). Calculate $MS_B$ and $MS_W$.

From the values in exercise 107.3, calculate the $F$ statistic.

The value $F = 4.0$ with $df = (2, 30)$ and $\alpha = 0.05$ (critical $\approx 3.32$). What is the correct conclusion?

$SST = 200$ and $SSB = 80$. Calculate $\eta^2$ and classify the effect size.

Using the data from exercise 107.6 ($SST = 200$, $SSB = 80$), determine $SSW$.

Why, under $H_0$, is it expected that $F \approx 1$? Explain in terms of what $MS_B$ and $MS_W$ estimate.

Three groups with $n = 15$ each. Group means: 9, 11, and 13. Calculate the grand mean $\bar{Y}$.

Using the data from exercise 107.9, calculate $SSB$.

Why not perform multiple $t$-tests to compare 4 groups? Calculate the probability of at least one false positive with $\alpha = 0.05$.

List the three assumptions of one-way ANOVA. With $n = 8$ per group and standard deviations $s_1 = 3$, $s_2 = 3$, $s_3 = 9$, which assumption is most suspect?

A study has $k = 3$ groups with $n = 8$. Shapiro-Wilk rejects normality in one group ($p = 0.008$). Variance ratio: 9:1. What test to use?

What is Levene's test for before ANOVA? What conclusion to draw from $p = 0.38$?

ANOVA rejects $H_0$ in an experiment with 5 groups. What does this mean? What to do next?

Compare Tukey HSD and Bonferroni: which is more conservative? When to use each?

For $k = 2$ groups, how do ANOVA $F$ and two-sample $t$ relate? Do p-values coincide?

Describe the shape of the $F$ distribution with small degrees of freedom. Why is $F$ never negative?

Convert $\eta^2 = 0.09$ to Cohen's $f$ and classify the effect size.

Why does $E[MS_W] = \sigma^2$ even under $H_1$, but $E[MS_B] > \sigma^2$ under $H_1$?

Three groups with $n = 8$ each. Means: 12, 15, and 18. Calculate $SSB$ and $MS_B$.

Continuation of exercise 107.21: $SSW = 336$ and $df_W = 21$. Calculate $F$ and decide at $\alpha = 0.05$ (critical $\approx 3.47$).

Using the data from exercises 107.21–107.22 ($SSB = 144$, $SSW = 336$), calculate $\eta^2$.

4 groups with $n = 12$ each. $SSB = 120$ and $SSW = 440$. Conduct the complete ANOVA at $\alpha = 0.05$ (critical $F_{3,44} \approx 2.82$) and calculate $\eta^2$.

Complete the ANOVA table: $MS_B = 12$, $MS_W = 2$. Calculate $F$.

A teacher wants to compare three teaching methods (A, B, C) with 20 students each, evaluated by test. Formalize the ANOVA model, hypotheses, and necessary assumptions.

A clinical study compares 4 weight loss diets with 40 participants each. Describe how to verify ANOVA assumptions before conducting the test.

A researcher compares 3 ML algorithms tested on the same 30 datasets. Is it appropriate to use one-way ANOVA? Justify.

Five stores have weekly sales monitored for 30 weeks. You want to use ANOVA. Outline: $df_B$, $df_W$, and if $n = 30$ is sufficient to detect medium effect (Cohen's $f = 0.25$, 80% power).

A chemistry lab compares four catalyst concentrations (0, 5, 10, 20 g/L) on reaction yield, with 10 replications each. Justify the use of one-way ANOVA and list assumptions to verify.

4 diets, 25 people each. Weight loss (kg) — means per diet: 3, 4, 5, and 4.5. Calculate $SSB$.

$SST = 300$ and $\eta^2 = 0.5$. Determine $SSB$ and $SSW$.

Algebraically derive the decomposition $SST = SSB + SSW$. Show explicitly why the cross terms cancel when summing over $j$ for each fixed $i$.

Show that, for $k = 2$ balanced groups, the ANOVA $F$ statistic is equal to the square of the two-sample $t$ statistic with pooled variance.

Argue (without full proof) why, under $H_0$, $SSB/\sigma^2 \sim \chi^2_{k-1}$ and $SSW/\sigma^2 \sim \chi^2_{N-k}$ are independent. How does this imply $F \sim F_{k-1, N-k}$?

What happens to ANOVA when groups have very different sizes (extreme imbalance)? Is the test still valid?

To detect medium effect ($f = 0.25$) between 4 groups at $\alpha = 0.05$ with 80% power, how many subjects per group are needed (approximately)? With $n = 25$ per group, is the study adequately powered?

What is the Bayes factor $BF_{10}$ in Bayesian ANOVA? How to interpret $BF_{10} = 15$ versus $BF_{10} = 0.08$?

Demonstrate that $SST = SSB + SSW$, showing that cross terms cancel when summing over $j$ for each fixed $i$.

Show that $E[MS_B] = \sigma^2$ under $H_0$ (for balanced groups, $n_i = n$).

Derive the Kruskal-Wallis test statistic and explain why it is the non-parametric analog of one-way ANOVA.

Derive the $F^*$ statistic of Welch's ANOVA for unequal variances. Explain how denominator degrees of freedom are adjusted.