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Psychology 406
Assignment \#13
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Suppose that an experimenter is interested in ``level of
aspiration'' as the dependent variable in an experiment. An
experimental task has been developed consisting of a difficult game
apparently involving motor skill, yielding a numerical score that
can be attached to a person's performance. But this appearance is
deceptive: unknown to the subject, the game is actually under the
control of the experimenter, so that each subject is made to obtain
exactly the same score. After a fixed number of trials, during
which the subject unknowingly receives the preassigned score, the
individual is asked to predict what the score will be on the next
group of trials. However, before this prediction, the subject is
given ``information'' about how the score compares with some
fictitious norm group. In one experimental condition, the subject
is told that the first performance is above average for the norm
group; in the second that it is average; and in the third that it is
below average. There are thus three possible experimental
``standings'' that might be given to any subject. (Of course, after
the experiment, each subject is full informed of this little ruse by
the experimenter.)
\smallskip
The dependent score value Y is based on the report the subject makes
about anticipated performance in the next group of trials. Because
each subject has obtained the same score, this anticipated score on
the next set of trials is treated as equivalent to a level of
aspiration that the subject has set. Each subject is tested
privately, and no communication is allowed between subjects until
the entire experiment is completed. Each of the three groups
contains 20 randomly assigned subjects.
\smallskip
In addition to the dependent measure, Y, prior to the experiment
each subject had been tested on a game very similar to that used in
the experiment proper, and a ``skill score'', $\mathrm{X}_{1}$,
obtained for each. The data that resulted from this experiment can
be represented in the following form:
\bigskip
\begin{tabular}{cc|cc|cc}
\multicolumn{2}{c}{above average} & \multicolumn{2}{c}{average} & \multicolumn{2}{c}{below average}
\\ \hline
Y & $\mathrm{X}_{1}$ & Y & $\mathrm{X}_{1}$ & Y & $\mathrm{X}_{1}$
\\ \hline
52 & 44 & 28 & 38 & 15 & 23 \\
48 & 47 & 35 & 26 & 14 & 17 \\
43 & 30 & 34 & 36 & 23 & 31 \\
50 & 38 & 32 & 30 & 21 & 25 \\
43 & 40 & 34 & 36 & 14 & 27 \\
44 & 45 & 27 & 23 & 20 & 35 \\
46 & 36 & 31 & 45 & 21 & 25 \\
46 & 41 & 27 & 28 & 16 & 28 \\
43 & 40 & 29 & 34 & 20 & 30 \\
49 & 43 & 25 & 37 & 14 & 37 \\
38 & 48 & 43 & 40 & 23 & 32 \\
42 & 24 & 34 & 36 & 25 & 32 \\
42 & 39 & 33 & 41 & 18 & 34 \\
35 & 36 & 42 & 29 & 26 & 48 \\
33 & 46 & 41 & 39 & 18 & 39 \\
38 & 33 & 37 & 37 & 26 & 38 \\
39 & 38 & 37 & 47 & 20 & 30 \\
34 & 26 & 40 & 34 & 19 & 24 \\
33 & 41 & 36 & 47 & 22 & 31 \\
34 & 36 & 35 & 31 & 17 & 19 \\
\end{tabular}
\bigskip
In addition to Y and $\mathrm{X}_{1}$, define three ``dummy''
variables, $\mathrm{X}_{2}$, $\mathrm{X}_{3}$, and $\mathrm{X}_{4}$:
$\mathrm{X}_{j}$ = 1, if the subject belongs to group $j - 1$; = 0,
otherwise. The SYSTAT output that is attached gives the basic
statistics in addition to information on fitting a variety of
models. (In giving the basic statistics, Groups 1, 2, and 3 are
above average, average, and below average, respectively.) The
models that are fitted are given as (a) through (f) below:
\bigskip
Model (a):
\[ \mathrm{Y} = \beta_{0} + \beta_{1}(\mathrm{X}_{1}\mathrm{X}_{2})
+ \beta_{2}(\mathrm{X}_{1}\mathrm{X}_{3}) +
\beta_{3}(\mathrm{X}_{1}\mathrm{X}_{4}) + \epsilon \
\]
Model (b):
\[ \mathrm{Y} = \beta_{0} + \beta_{1}\mathrm{X}_{1} + \epsilon \
\]
Model (c):
\[ \mathrm{Y} = \beta_{0} + \beta_{1}\mathrm{X}_{2}
+ \beta_{2}\mathrm{X}_{3} + \epsilon \
\]
Model (d):
\[ \mathrm{Y} = \beta_{0} + \beta_{1}(\mathrm{X}_{1}\mathrm{X}_{2})
+ \beta_{2}(\mathrm{X}_{1}\mathrm{X}_{3}) +
\beta_{3}(\mathrm{X}_{1}\mathrm{X}_{4}) + \beta_{4}\mathrm{X}_{2} +
\beta_{5}\mathrm{X}_{3} + \epsilon \
\]
Model (e):
\[ \mathrm{Y} = \beta_{0} + \beta_{1}\mathrm{X}_{1}
+ \beta_{2}\mathrm{X}_{2} + \beta_{3}\mathrm{X}_{3} + \epsilon \
\]
Model (f):
\[ \mathrm{X}_{1} = \beta_{0} + \beta_{1}\mathrm{X}_{2}
+ \beta_{2}\mathrm{X}_{3} + \epsilon \
\]
\bigskip
Questions:
\bigskip
i) Model (c) performs a ``one-way analysis of variance'' on the
dependent measure Y in relation to the 3 groups. Show explicitly
the relation between the means on Y within each of the three groups
(given in the basic statistics) and the estimated means on Y for all
the various combinations of values that $\mathrm{X}_{2}$ and
$\mathrm{X}_{3}$ can take on. What does the analysis-of-variance
table say about the ``effectiveness'' of the 3 treatments? Why
isn't an $\mathrm{X}_{4}$ term included in model (c)?
\smallskip
ii) Looking at model (f), carry out a similar interpretation as in
(c). Are the results surprising? Why?
\smallskip
iii) Plot the regression lines of Y on $\mathrm{X}_{1}$ implied by
model (d) for the three separate groups. Superimpose on this plot
the regressions of Y on $\mathrm{X}_{1}$ implied by model (e) for
the three separate groups. Carry out a test of model (d) versus
model (e) and interpret.
\smallskip
iv) Plot the regression lines of Y on $\mathrm{X}_{1}$ implied by
model (e) and model (b).
\smallskip
\noindent Carry out a test of model (e) versus model (b) and
interpret. This is called ``analysis-of-covariance'', and
supposedly is a way of assessing the effectiveness of the three
treatments. How is it different than what was done in (i)? (It may
help to interpret what was done in (i) as a comparison of model (c)
against a restricted model, $\mathrm{Y} = \beta_{0} + \epsilon$.)
\smallskip
\noindent Analysis of covariance is based on an assumption that
model (e) is the ``Full Model''. How does this relate to what was
done in (iii)?
\smallskip
v) Suppose I have some given value on $\mathrm{X}_{1}$, say P.
Using model (e), what are the expected values on Y for the three
separate groups. Suppose I have a second given value on
$\mathrm{X}_{1}$, say Q. What are the expected values on Y for the
three separate groups, again using model (e), and what are the
relationships between the two sets of expected values.
\smallskip
\noindent Now, do the same for model (d) and comment on the
differences from using model (e).
\smallskip
\noindent Using these interpretations, why is it argued that one
cannot compare the effectiveness of treatments merely by comparing
model (d) and (a) (when model (e) cannot be assumed correct)?
\smallskip
\noindent Also, why is it argued that we can actually ``control''
for the effect of $\mathrm{X}_{1}$ in assessing treatment
effectiveness when model (e) is ``true'' but not if model (d) is
``true''?
\smallskip
vi) Carry out a test of model (d) versus (b). What is this a test
of any how does it differ from a comparison of model (d) versus
model (a) and of model (e) versus model (b)?
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