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Chapter 12. Analysis of variance (ANOVA)

Unit 12.5. Questions for repetition

1. State Fisher's leading idea behind one-way ANOVA: how does the comparison of "variability between samples" to "variability within samples" lead to a decision about the equality of population means?

2. Define the four sums of squares used in one-way ANOVA $latex -$ $latex SS_g, SSW, SSB, SST$ $latex -$ and prove that $latex SST = SSW + SSB$.

3. State Theorem 12.1 on the distribution of $latex \sum (X_i - \bar X)^2 / \sigma^2$ and Theorem 12.2 on chi square addition and subtraction. Where does the independence of samples enter the derivation?

4. Derive the distribution of $latex SSW / \sigma^2$ (no assumption on the means) and the distribution of $latex SSB / \sigma^2$ under the null hypothesis. What is then the distribution of $latex MSG / MSW$ under the null?

5. State the decision rule for one-way ANOVA and write out the format of the one-way ANOVA table (sources of variation, sums of squares, degrees of freedom, mean squares, F ratio).

6. Explain the difference between one-way and two-way ANOVA in terms of the experiment. What is a blocking factor and why is it useful?

7. Write down the two-way ANOVA population model $latex x_{gb} = \mu + \mu_g + \mu_b + \varepsilon_{gb}$. Derive the population models for $latex \bar x_{g \cdot}, \bar x_{\cdot b}, \bar x$ and the variances of the group and block sample means.

8. Define the four sums of squares used in two-way ANOVA $latex -$ $latex SST, SSG, SSB, SSE$ $latex -$ and outline the proof that $latex SST = SSG + SSB + SSE$.

9. State the three null hypotheses considered in two-way ANOVA. For each one, give the chi-square distribution of the corresponding sum of squares (and of $latex SSE$ under $latex H_0$). What F-statistic tests each hypothesis, and what are its degrees of freedom?

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