Study Questions, Chapter 13, part 1

1. In the CRSF design in the ANOVA, what have we added to our conceptualization ("clouds" diagram) of the underlying process for the random variable(s)?

The design, i.e. the treatment variable, the independent variable

2. What are the assumptions for the CRSF ANOVA?

underlying distribution of the error is normal

random sampling and random assignment

for our formulas, equality of variances

3. What is it that we say is normally distributed in the General Linear Model for the CRSF ANOVA?

the error, or noise

4. What addition changes the analytical model into the statistical model?

the error random variable

5. What hypotheses do we test in the ANOVA?

hypotheses about means

6. Why, then, is it called the Analysis of Variance?

Traditionally, we use two estimates of error variance, one of which has an expected value the same as the other only if the null hypothesis is true, in an F test of the hypothesis

7. For the CRSF design, do we have to assume equal variances?

no, but then we'd have to use a different formula

8. Write out the specification for a CRSF design that includes 3 observations in each of 3 groups. (Your specification will contain 9 lines, where each line is an equation.) Use means coding. The first and fourth lines are provided to get you started.

y₁₁ = mu₁ + epsilon₁₁

y₁₂ = mu₁ + epsilon₁₂

y₁₃ = mu₁ + epsilon₁₃

y₂₁ = mu₂ + epsilon₂₁

y₂₂ = mu₂ + epsilon₂₂

y₂₃ = mu₂ + epsilon₂₃

y₃₁ = mu₃ + epsilon₃₁

y₃₂ = mu₃ + epsilon₃₂

y₃₃ = mu₃ + epsilon₃₃

9. What is the expected value (y-hat) for each observation in the CRSF design?

the mean of its group

10. What method do we use to estimate mu and sigma squared?

least squares

11. What is the hypothesis for the main effect of treatments, in terms of group means?

H₀: mu₁ = mu₂ = ... = mu_k

12. What is the hypothesis for the main effect of treatments, in terms of effects?

H₀: alpha_i = 0

13. Name two estimates of the error variance that can be derived from the data. Which one is always unbiased? Which one is unbiased if the null hypothesis is true?

MSA, MSE

MSE

MSA

14. If the null hypothesis is false, in what way is the treatment mean square (MSA) biased, as an estimator for sigma squared?

it is too large

15. Is the F test in the ANOVA inherently one-tailed or two-tailed?

one-tailed