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        Psychology 407

        Assignment \#8


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I. Goodwin and Michael (1981) performed a study of newborn infants,
recording birth position (left or right) and head position
preference (whether the child consistently looked right, had a bias
toward looking right, had a bias toward looking left, or
consistently looked left).  The observed frequencies are shown
below:

\bigskip

\normalsize

\begin{tabular}{ccccc}

  &  &  Head position preference & & \\

  & Consistent right &  Biased right  &  Biased left  &  Consistent
  left \\

  Birth position &  &   &  &  \\

  Left  &  29  &  8  &  8  &  4  \\

  Right &  18  &  7  &  12  &  13  \\

  \end{tabular}

  \bigskip

  \Large

  a)  Test the association between the two attributes using
  Pearson's chi-square test of association.  Obtain the expected
  frequencies under attribute independence and interpret the
  discrepancies between ``obtained'' and ``expected''.

  \bigskip

  b)  Test the goodness-of-fit of the column marginal sums against a
  uniform distribution.

  \bigskip

  c)  Obtain and interpret Goodman-Kruskals's index of predictive
  association for predicting head \emph{from}  birth position.

  \bigskip

  II.  In a review of the evidence regarding the therapeutic value
  of ascorbic acid (Vitamin C) for treating the common cold, Pauling
  (1971)  describes a 1961 French study involving 279 skiers during
  two periods of 5--7 days.  The study was double-blind with one
  group of 140 subjects receiving a placebo while a second group of
  139 received 1 gram of ascorbic acid per day.  Of interest is the
  relative occurrence of colds for the two groups where the data are
  as follows.

  \bigskip

  \begin{tabular}{ccc}

   & Cold  &  No Cold \\

   Placebo  &  31  &  109  \\

   Ascorbic Acid  &  17  &  122  \\

   \end{tabular}

   \bigskip

   a) Test the association using Pearson's chi-square test; obtain
   the expected values of the cell frequencies under independence.

   \bigskip

   b)  What is the four-fold point correlation?

   \bigskip

   III.  For the present purposes, assume mental patients can be
   classified as psychotics or neurotics.  A  psychiatrist, while
   studying the symptoms of a sample of 20 from each of these
   populations found that six patients in the neurotic group had
   suicidal feelings and two in the psychotic group:

   \bigskip

   \begin{tabular}{ccc}

     &  Psychotics  &  Neurotics \\

     Suicidal feelings &  2  &  6  \\

     No suicidal feelings  &  18  &  14  \\

     \end{tabular}

     \bigskip

     a)  Carry out a test of association using Fisher's exact test.
     How does the p-value compare to the chi-square approximation?

     \bigskip

     IV.  A psychiatrist wished to assess the effect of the symptom
     ``depersonalization'' of the prognosis of depressed patients.
     Twenty-three depressed patients, who were diagnosed as being
     ``depersonalized'', were matched one-to-one for age, sex,
     duration of illness, and on certain personality variables, with
     23 depressed patients who were diagnosed as not being
     ``depersonalized''.  The number of pairs of patients from the
     two samples who, on discharge after a course of ECT, were
     diagnosed as ``recovered'' or ``not recovered'' are given
     below:

     \bigskip

     \begin{tabular}{cccc}

       &  &  Depersonalized  &  \\
         &  &  Not recovered  &  Recovered \\

           &  Recovered  &  5  &  14  \\
           Not depersonalized  &  &  &  \\

             &  Not Recovered  &  2  &  2  \\

             \end{tabular}

             \bigskip

             a)  Using McNemar's test of correlated proportions,
             test the equality of the ``recovered'' proportions for
             the two types of patients.  Carry out the test using
             the chi-square approximation \emph{and} the exact
             version based on the binomial distribution.

             \bigskip

             V.  In viewing a set of video tapes of 50 couples
             involved in marital counseling, two raters assigned the
             couples to one of three behavioral categories based on
             the interactions observed.  The data turned out as
             follows ($C_{1}$, $C_{2}$, and $C_{3}$ denote the
             behavioral categories):

             \bigskip

             \begin{tabular}{ccccc}

               &  &  & rater b &  \\
               
  & &$C_{1}$  & $C_{2}$ & $C_{3}$ \\

  &  $C_{1}$  &  14  &  2  &  3  \\

rater a &  $C_{2}$  &  0  &  10  &  1  \\

  &   $C_{3}$  &  6  &  2  &  12  \\

  \end{tabular}

  \bigskip

  a)  Obtain Cohen's Kappa statistic for these data and test its
  significance against zero using the hypothesis of independence
  between the raters.




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