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Psychology 407
Assignment \#1
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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.
\bigskip
\begin{tabular}{ccc}
Group 1 & Group 2 & Group 3 \\ [2ex] \hline
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 \\
\end{tabular}
\bigskip
Analyze these data using the one-way (single-factor) fixed-effects
analysis-of-variance model, i.e.,
\[ \mathrm{(I)} \ y_{ij} = \mu_{i} + \epsilon_{ij} \ , \]
\noindent where $\epsilon_{ij} \sim N(0,\sigma^{2})$ and are
independent; $Y_{ij}$ is the value for the response variable for
person $j$ within group $i$; $\mu_{i}$ are fixed parameters; $1 \le
i \le r$ (the number of groups); $1 \le j \le n_{i}$ (the number of
observations in group $i$).
\bigskip
a) Construct the appropriate analysis-of-variance (ANOVA) table.
\bigskip
b) What are the estimates of $\mu_{i}$, $1 \le i \le r$; and of
$\sigma^{2}$?
\bigskip
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: $\mathbf{Y} =
\mathbf{X}\mbox{\boldmath $\beta$} + \mbox{\boldmath $\epsilon$}$,
where $\mathbf{X}$ contains 3 indicator variables for its columns.
Indicate the size of the matrices and what they contain.
\bigskip
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 $H_{0}:
\mu_{1} = \mu_{2} = \mu_{3}$? 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))?
\bigskip
Consider the ``effect'' form of the one-way ANOVA model:
\[ \mathrm{(II)} \ y_{ij} = \mu_{\cdot} + \tau_{i} + \epsilon_{ij} \ ,
\]
\noindent where $\mu_{\cdot} = \sum_{i=1}^{r} n_{i} \mu_{i}/n_{T}$,
and $n_{T} = \sum_{i=1}^{r} n_{i} \ .$
\bigskip
e) What are the estimates of $\tau_{i}$, $1 \le i \le r$? In
general, would these be different if we define $\mu_{\cdot} =
\sum_{i=1}^{r} \mu_{i}/r$? In our example?
\bigskip
f) In defining the general linear model for the form in (II), what
would $\mathbf{X}$ look like if \[ \mbox{\boldmath $\beta$} = \left[
\begin{array}{c}
\mu_{\cdot} \\
\tau_{1} \\
\tau_{2} \\
\tau_{3} \\
\end{array} \right] \ ? \]
\noindent Why is it ``impossible'' to use a \mbox{\boldmath $\beta$}
of this form computationally in getting an estimate of
\mbox{\boldmath $\beta$}?
\bigskip
g) If \mbox{\boldmath $\beta$} is given as \[ \left[
\begin{array}{c}
\mu_{\cdot} \\
\tau_{1} \\
\tau_{2} \\
\end{array} \right] \ , \]
\noindent what would $\mathbf{X}$ look like in general if
$\mu_{\cdot} = \sum_{i=1}^{r} n_{i}\mu_{i} / n_{T}$ or $\mu_{\cdot}
= \sum_{i=1}^{r} \mu_{i}/r$.
\noindent In our example, are there any differences? Why?
\bigskip
h) Assuming that the ``true'' means are $\mu_{1} = 4$, $\mu_{2} =
4$, and $\mu_{3} = 7$ and $\sigma^{2} = 6$, what is the power for
rejecting $H_{0}: \mu_{1} = \mu_{2} = \mu_{3}$ given our sample
sizes (and $\alpha = .05$)?
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