Psychology 407 Assignment #1

Suppose we randomly assign 60 people to three groups and ask each member in the group to rate a specific geometric figure on a 1 to 10 point scale based on its “complexity”. A large rating will imply a more complex figure, and all members in a group rate the same figure independently. The data and figures rated by each group are as follows.

Group 1 Group 2 Group 3
6  10  5
2  8  10
2  9  10
6  4  3
3  5  10
1  3  10
4  3  10
4  4  10
2  5  10
4  2  7
4  10  10
3  3  10
6  8  4
1  7  10
2  9  9
5  7  1
2  8  10
2  9  10
2  6  10
2  9  9

Analyze these data using the one-way (single-factor) fixed-effects analysisof-variance model, i.e.,

(I) y_{ij} = μ_{i} + ϵ_{ij} ,

where ϵ_{ij} ∼ N(0, σ^{2}) and are independent; Y_{ij} is the value for the response variable for person j within group i; μ_{i} are fixed parameters; 1 ≤ i ≤ r (the number of groups); 1 ≤ j ≤ n_{i} (the number of observations in group i).

a) Construct the appropriate analysis-of-variance (ANOVA) table.

b) What are the estimates of μ_{i}, 1 ≤ i ≤ r; and of σ^{2}?

c) Show (no need to compute anything) schematically how the analysis-of-variance model given in (I) can be recast in matrix terms as a linear model having the usual form: Y = Xβ + ϵ, where X contains 3 indicator variables for its columns. Indicate the size of the matrices and what they contain.

Assuming that the model given in (c) if the Full model, what is the form of the Reduced model that would be used in testing H_{0} : μ_{1} = μ_{2} = μ_{3}? Considering the general test statistic in Kutner, et al. (p. 75, equation 2.70), what are the numerical values for its constituent terms (Hint: use the numerical results in (a))?

Consider the “effect” form of the one-way ANOVA model:

(II) y_{ij} = μ_{·} + τ_{i} + ϵ_{ij} ,

where μ_{·} = ∑_{i=1}^{r} n_{i}μ_{i}/n_{T}, and n_{T} = ∑_{i=1}^{r} n_{i} .

e) What are the estimates of τ_{i}, 1 ≤ i ≤ r? In general, would these be different if we define μ_{·} = ∑_{i=1}^{r} μ_{i}/r? In our example?

f) In defining the general linear model for the form in (II), what would X look like if

β = [ \begin{matrix}μ_{·} \\ τ_{1} \\ τ_{2} \\ τ_{3} \end{matrix} ] ?

Why is it “impossible” to use a β of this form computationally in getting an estimate of β? g) If β is given as

[ \begin{matrix}μ_{·} \\ τ_{1} \\ τ_{2} \end{matrix} ] ,

what would X look like in general if μ_{·} = ∑_{i=1}^{r} n_{i}μ_{i}/n_{T} or μ_{·} = ∑_{i=1}^{r} μ_{i}/r. In our example, are there any differences? Why?

h) Assuming that the “true” means are μ_{1} = 4, μ_{2} = 4, and μ_{3} = 7 and σ^{2} = 6, what is the power for rejecting H_{0} : μ_{1} = μ_{2} = μ_{3} given our sample sizes (and α = .05)?