Supplementary exercise 9.20 of IPS7e
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Data on coronary heart disease (CHD) patients cross-classified with
respect to survival for one year and ownership of a pet.

The study design is not quite clear here, but it seems that the 92 patients
are a SRS from some population (perhaps patients with CHD that have been
treated at a particular hospital over a period of time). Thus, both
classification criteria in the table are response variables, and the
model is a single multinomial distribution on 4 cells.

Minitab commands and output:

MTB > WOpen "R:\Chapter 9\ex09_020.mtw".
Retrieving worksheet from file: ‘R:\Chapter 9\ex09_020.mtw’
Worksheet was saved on 24/10/2014

MTB > XTabs 'Status' 'Ownership';
SUBC>   Layout 1 1;
SUBC>   Frequencies 'Survivors';
SUBC>   Counts;
SUBC>   RowPercents;
SUBC>   ColPercents;
SUBC>   TotPercents;
SUBC>   ChiSquare;
SUBC>   Expected;
SUBC>   Fisher;
SUBC>   DMissing 'Status' 'Ownership'.
Tabulated Statistics: Status, Ownership 
Using frequencies in Survivors

Rows: Status   Columns: Ownership

          nopet     pet     All

alive        28      50      78
          35.90   64.10  100.00
          71.79   94.34   84.78
          30.43   54.35   84.78
         33.065  44.935

dead         11       3      14
          78.57   21.43  100.00
          28.21    5.66   15.22
          11.96    3.26   15.22
          5.935   8.065

All          39      53      92
          42.39   57.61  100.00
         100.00  100.00  100.00
          42.39   57.61  100.00

Cell Contents:      Count
                    % of Row
                    % of Column
                    % of Total
                    Expected count


Pearson Chi-Square = 8.851, DF = 1, P-Value = 0.003
(Likelihood Ratio Chi-Square = 9.011, DF = 1, P-Value = 0.003)

Fisher’s exact test: P-Value =  0.0063747


Answers to questions:
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(a) Not an experiment, because no action/treatment was imposed (to
either give or not give the patient a pet).

(b) What the researchers think about causal effects, has no bearing on
whether pet ownership is an explanatory variable or not. If the study
design was as described above, both variables are response variables.

To examine the probability of survival for patients with or without
a pet, we compute conditional probabilities. They are in each column (we
condition on column values). The sample survival probability for
pet-owners was 50/53=94% and for non pet-owners 28/39=72%. It seems
that pet-owners tend to survive more frequently than non pet-owners.

(c) Null hypothesis H0: equal survival probability among pet-owners and
non pet-owners (based on assuming two independent binomial distributions
in the two groups), or H0: no association between survival and
pet-ownership (based on assuming a multinomial model on 4 categories
corresponding to the interpretation of both variables as response variables). 
The two-sided alternative hypothesis Ha is unequal survival probabilities, or 
an association between survival and ownership, respectively.

(d) X^2 = 8.85, P = 0.003.
Note that all expected values are > 5, so the use of the X^2 statistic
and the chi-square distribution should be ok.

Fisher's exact test (with a two-sided alternative) gives P=0.006, but
even if the value is slightly different (as it will typically be), the
conclusion remains the same.

(e) Conclusion: strong indication of an association between survival and
pet-ownership. However, we should not conclude that this is a causal
relation (pet-ownership affecting survival), because there may be
lurking variables, in particular the condition of the patients (severely
ill patients may not be able to have pets).