ASSIGNMENT 2: BINARY RESPONSE MODELS
In the following contingency table we have 79 patients cross-classified by three factors: A= the condition of the patient, B = whether the patient received antitoxin or not and C = whether the patient survived or not. Prime interest is given on the effect of antitoxin to deaths from tetanus.
| * | * |
SURVIVAL |
* |
| (A) CONDITION | (B) ANTITOXIN | No (deaths) | Yes (survivors) |
| More Severe | Yes | 15 | 06 |
| More Severe | No | 22 | 04 |
| Less Severe | Yes | 05 | 15 |
| Less Severe | No | 07 | 05 |
There are two ways of analysing these sets of data. We may either use binary response models (logistic, probit or clog-log regression ) or log-linear models for contingency tables. In each kind of analysis different relationships are under investigation.
A...Fit the following model on BUGS:
g(pij)= ́ + ái + âj , rij~ Binomial(pij , nij) , i,j=1,2,
where pij is the probability of survival, rij is the number of survivals and nij is the total number of patients with i condition and j antitoxin. The function g(pij) is called the link function because is used to link the response variable/factor (survival) with the explanatory factors (condition & antitoxin). With this model we ‘measure’ the effect of the explanatory variables on the response. Fit all the following three links using both sum to zero and corner parametrazation:
B...Give the statistics and histograms for the model coefficients with a brief comment. Check Convergence for each coefficient. Give brief interpretation for each parameter. Compare the Bayesian estimates with the classical MLE estimates (fitted on any statistical program). Produce the posterior distribution of the ODDS (plots & statistics).
C...Give the Residuals, st.residuals and probability of more extreme values for each model. Are there any outliers? Summarise the residual results in a diagram of the 95% confidence intervals (for residuals and st. residuals).
D... Which of the three models seems to be better according to deviance and approximate Bayes Factor? Also try to calculate the Bayes Factor via MCMC estimation.
E... Give the posterior distributions of the fitted probabilities (plots, stats). Also summarise the 95% CI of the fitted probabilities in CI plot (CI = confidence interval).
