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\Large
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Psychology 407
Computer Assignment \#1
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An experiment was carried out to study the effect of a small lesion
introduced into a particular structure in a rat's brain on its
ability to perform in a discrimination problem. The particular
structure studied is bilaterally symmetric, so that the lesion could
be introduced into the structure on the right side of the brain, the
left side, both sides, or neither side (a control group). Four
groups of randomly selected rats were formed, and given the various
treatments. Originally, the control group contained 7 rats and each
of the experimental groups had 14 rats, but due either to death or
postoperative incapacity, only the following numbers were actually
observed in the discrimination situation. The experimenter was
prepared to assume that loss of rats during the study was a purely
random event, which had no bearing on the results. The final data
were as shown below.
\bigskip
\begin{tabular}{ccccc}
Treatment & I & II & III & IV \\
& 20& 24 & 20 & 27 \\
& 18 & 22 & 22 & 35 \\
& 26 & 25 & 30 & 18 \\
& 19 & 25 & 27 & 24 \\
& 26 & 20 & 22 & 28 \\
& 24 & 21 & 24 & 32 \\
& 26 & 34 & 28 & 16 \\
& & 18 & 21 & 18 \\
& & 32 & 23 & 25 \\
& & 23 & 25 & \\
& & 22 & 18 & \\
& & & 30 & \\
& & & 32 & \\
& & & & \\
\end{tabular}
\bigskip
a) Obtain the basic descriptive summary statistics for the four
treatment groups (Use the STATS module and the BY command). Also,
obtain the basic descriptive summary statistics aggregated over all
four treatment groups.
\bigskip
b) Carry out the usual single-factor analysis-of-variance in
different ways, and compare the consistency of the results.
\smallskip
i) Use the automatic coding of the dummy variables using the
CATEGORY command in ANOVA.
\smallskip
ii) Using multiple regression in GLM, construct zero-one indicator
variables and fit the model (without an additive constant).
\smallskip
iii) Using multiple regression in GLM, construct dummy variables
that reflect the use of a \emph{weighted} mean in constructing the
additive constant in the ANOVA model.
\bigskip
c) Interpret the results substantively and discuss what further
analyses might be warranted and how they might be conducted, e.g.,
post-hoc tests.
\newpage
A sociologist selected a random sample of 45 adjunct professors who
teach in the evening division of a large metropolitan university for
a study of special problems associated with teaching in the evening
division. The data collected include the amount of payment received
by the faculty member for teaching a course during the past
semester. The sociologist classified the faculty members by subject
matter of course (Factor A) and highest degree earned (Factor B).
The earnings per course (in thousand dollars) follow:
\bigskip
\begin{tabular}{cccc}
Factor B: & Bachelor's & Master's & Doctorate \\ [2ex] \hline
Factor A & & & \\ [2ex]
Humanities & 1.7 & 1.8 & 2.5 \\
& 1.9 & 2.1 & 2.7 \\
& & & 2.9 \\
& & & 2.5 \\
& & & 2.6 \\
& & & 2.8 \\
& & & 2.7 \\
& & & 2.9 \\ [2ex]
Social sciences & 2.5 & 2.7 & 3.5 \\
& 2.3 & 2.4 & 3.3 \\
& 2.6 & 2.6 & 3.6 \\
& 2.4 & 2.4 & 3.4 \\
& & 2.5 & \\ [2ex]
Engineering & 2.7 & 2.9 & 3.7 \\
& 2.8 & 3.0 & 3.6 \\
& & 2.8 & 3.7 \\
& & 2.7 & 3.8 \\
& & & 3.9 \\ [2ex]
Management & 2.5 & 2.3 & 3.3 \\
& 2.6 & 2.8 & 3.4 \\
& & & 3.3 \\
& & & 3.5 \\
& & & 3.6 \\ \hline
\end{tabular}
\bigskip
a) Using STATS and the BY command (several times), obtain the basic
descriptive summary statistics for Factor A alone, Factor B alone,
and for the combination of Factor A \emph{and} Factor B. Also,
obtain the basic descriptive summary statistics aggregated over all
cells in the design.
\bigskip
b) Carry out a two-way ANOVA using ANOVA and the CATEGORY command,
and interpret substantively. Graph the means and interpret the
results in terms of the graph.
\bigskip
c) The analysis carried out in (a) is based on a particular dummy
coding of the categories, which you can save and look at (see ANOVA
and SAVE/MODEL). Do so, and show numerically that the tests used in
(a) could be obtained alternatively by testing sets of regression
coefficients against zero (using the simultaneous tests as discussed
in GLM).
\newpage
Several different methods of teaching the FORTRAN programming
language are to be compared. However, the experimenter had reason
to believe that general ability of the students might also affect
their achievement under any given method. It was considered
desirable to remove the possible linear effects of general ability
on programming achievement though use of the analysis of covariance.
The subjects were twenty female college freshmen who were assigned
at random to five groups, each representing a different method of
teaching this material. Prior to the experimental sessions, each
subject was given a general ability test (score x) and, after the
training, achievement was measured (score y). No student had prior
experience with programming languages. The results are shown below
(each x and y value has been reduced by 100 for computational
convenience; this has no effect whatsoever on the final analysis).
The desire of the experimenter was to treat the general ability
score as the concomitant variable, and to ask if differences due to
method exist even after adjustment for general ability.
\bigskip
\begin{tabular}{ccc|cc|cc|cc|cc}
Groups: & 1 & & 2 & & 3 & & 4 & & 5 & \\ \hline
& x & y & x & y & x & y & x & y & x & y \\
& 10 & 18 & 22 & 40 & 30 & 38 & 35 & 25 & 11 & 15 \\
& 20 & 17 & 31 & 22 & 31 & 40 & 37 & 45 & 16 & 17 \\
& 15 & 23 & 16 & 28 & 18 & 41 & 41 & 50 & 19 & 20 \\
& 12 & 19 & 17 & 31 & 22 & 40 & 30 & 51 & 25 & 23 \\
\end{tabular}
\bigskip
a) Obtain the basic descriptive summary statistics for the five
treatment groups for both the covariate and the dependent measure
(use the STATS module and the BY command). Also, obtain the basic
descriptive summary statistics aggregated over all five groups.
\bigskip
b) Carry out an analysis-of-covariance on these data, testing first
for the homogeneity of regression slopes. Provide the adjusted
group means and show how they relate to Kutner, et al. on page 931
(22.22). Are they the same? Comment. Interpret the results
substantively.
\bigskip
c) Redo the analyses in (a) from first principles using the
general linear model with 0-1 indicator variables. In particular,
show how the comparison of Full and Reduced models allows one to
test: (i) the homogeneity of regression slopes; (ii) assuming
homogeneity of regression slopes, that the adjusted group means are
or are not significantly different from each other.
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