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Psychology 406
Assignment \#12
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In an experiment designed to assess a possible curvilinear
relationship between level of background noise on task performance,
an experimenter assigned (at random) 10 subjects to each of 6 noise
levels (assumed to be equally-spaced values of 1, 2, ..., 6) and
obtained a ``number correct'' score on a heavily speeded performance
measure. The data for this study turned out as follows (the columns
are labeled by Noise Level):
\bigskip
\begin{tabular}{cccccc}
one & two & three & four & five & six \\ \hline
18 & 34 & 39 & 37 & 15 & 14 \\
24 & 36 & 41 & 32 & 18 & 19 \\
20 & 39 & 35 & 25 & 27 & 5 \\
26 & 43 & 48 & 28 & 28 & 25 \\
23 & 48 & 44 & 29 & 22 & 7 \\
29 & 28 & 38 & 31 & 24 & 13 \\
27 & 30 & 42 & 34 & 21 & 10 \\
33 & 33 & 47 & 38 & 19 & 16 \\
32 & 37 & 53 & 43 & 13 & 20 \\
38 & 42 & 33 & 23 & 33 & 11 \\
\end{tabular}
\bigskip
Because of possible computational issue in fitting polynomial models
(question: what are they?), the 6 noise levels will be coded as
deviations from the mean noise level (a value of 3.5); thus, the 6
noise levels are actually -2.5, -1.5, -.5, +.5, +1.5, +2.5. These
latter deviation values should be assumed in \emph{everything} that
follows.
\bigskip
Summary Information on Performance (the standard deviation is based
on an unbiased variance estimate):
\bigskip
\begin{tabular}{cccc}
Noise level & Sample Size & Mean & Standard Deviation \\
-2.5 & 10 & 27.0 & 6.164 \\
-1.5 & 10 & 37.0 & 6.164 \\
-.5 & 10 & 42.0 & 6.164 \\
+.5 & 10 & 32.0 & 6.164 \\
+1.5 & 10 & 22.0 & 6.164 \\
+2.5 & 10 & 14.0 & 6.164 \\
Overall & 60 & 29.0 & 11.087 \\
\end{tabular}
\bigskip
(\emph{Any} indication that these data are ``made up''?)
\bigskip
The end two pages give SYSTAT results on fitting a variety of
polynomial models. Here, PER stands for performance and NOISED
stands for noise deviated from the mean.
\bigskip
Questions:
\bigskip
a) Plot the data: noise against performance. Indicate on the plot
the mean performance level within each noise level.
\bigskip
b) Replot just the mean performance levels within each noise level
and on this graph represent all \emph{five} linear/curvilenear
functions given by the SYSTAT output. Comment on what appears to
provide a ``reasonable'' fit.
\bigskip
c) Calculate a ``pure error'' sum-of-squares from the summary
information provided for performance. What would a plot in (b) look
like if a polynomial of order 5 were fitted? And what would be the
residual sum-of-squares? Provide the analysis-of-variance table for
fitting the order 5 polynomial. (If in a previous life you studied
one-way analysis-of-variance, comment on the correspondence between
the last table you gave and what would be usually provided in the
one-way analysis of variance context.)
\bigskip
d) Obtain the ``extra'' sums-of-squares indicated (here, X is the
noise level):
\bigskip
SSR(X); SSR($\mathrm{X}^{2}$ $\mid$ X);
SSR($\mathrm{X}^{3}$ $\mid$ $\mathrm{X}$, $\mathrm{X}^{2})$;
SSR($\mathrm{X}^{4}$ $\mid$ $\mathrm{X}$, $\mathrm{X}^{2}$,
$\mathrm{X}^{3})$;
SSR($\mathrm{X}^{5}$ $\mid$ $\mathrm{X}$, $\mathrm{X}^{2}$,
$\mathrm{X}^{3}$, $\mathrm{X}^{4})$; and
SSR($\mathrm{X}^{3}$, $\mathrm{X}^{4}$, $\mathrm{X}^{5}$ $\mid$
$\mathrm{X}$, $\mathrm{X}^{2})$
SSR($\mathrm{X}^{4}$, $\mathrm{X}^{5}$ $\mid$ $\mathrm{X}$,
$\mathrm{X}^{2}$, $\mathrm{X}^{3})$.
\bigskip
Test whether there is a significant lack-of-fit for a second order
and for a third order model using the ``pure error'' term --- give
the two corresponding analysis-of-variance tables. Comment on how
these tests relate to the intuition you provided in (b).
\bigskip
e) What is the relation between all of the residual mean squares
generated in the SYSTAT analyses and the mean square for pure error?
Are they all estimates of error? In what sense and under what
conditions?
\bigskip
f) Look at the SYSTAT analysis for the third order model. Show
numerically how the test for the coefficient on $\mathrm{X}^{3}$ can
be generated using the extra sum of squares principle. In carrying
out this test, what assumption is being made about the residual
mean-squares for the third order model.
\bigskip
g) Look at the SYSTAT analysis for the second order model. What do
the given tolerances tell you about the relation between X and
$\mathrm{X}^{2}$? Why should this relation hold here for our data?
Comment on the change or lack of change in the regression
coefficients as the order of the model increases. How general would
you expect such a result to be when other data sets are considered?
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