Assignment 3

1. The question concerns data from a case-control study of esophageal cancer in Ile-et-Vilaine, France. The data is distributed with R and maybe obtained along with a description of the variables by :

   > data(esoph)

   > esoph

      agegp     alcgp    tobgp ncases ncontrols

   1  25-34 0-39g/day 0-9g/day      0        40

   2  25-34 0-39g/day    10-19      0        10

   3  25-34 0-39g/day    20-29      0         6

      …deleted…

   88   75+      120+    10-19      1         1

   > help(esoph)

   1. Fit a binomial GLM with interaction effects between all three predictors. Start from the saturated model and use backward elimination to simplify the model as far as is reasonable.

   2. All three factors are ordered and so special contrasts have been used appropriate for ordered factors involving linear term, quadratic term, cubic term, ..., etc. Further simplification of the model is possible by eliminating some of these terms. Use the unclass() command in R to convert some or all factors to a numerical representation and show how the model may be simplified.

   3. Does your final model fit the data? Is the test you make accurate for this data?

   4. What is the predicted effect of moving to a category one higher in alcohol consumption? Compute a 95% confidence interval for this predicted effect.

   5. Bearing in mind that this is a case-control study, what can be said about the predicted probability that a 25 year old who does not smoke or drink will get esophageal cancer?

2. The data come from a study of breast cancer in Wisconsin (Bennet and Mangasarian, 1992). There are 681 cases of potentially cancerous tumors of which 238 are actually malignant. Determining whether a tumor is really malignant is traditionally determined by an invasive surgical procedure. The purpose of this study was to determine whether a new procedure, called fine needle aspiration, which draws only a small sample of tissue could be effective in determining tumor status. The data has the following response and nine predictors:

   • Class - 0 if malignant, 1 if benign

   • Adhes - marginal adhesion

   • BNucl - bare nuclei
   • Chrom - bland chromatin
   • Epith - epithelial cell size
   • Mitos - mitoses
   • Nnucl - normal nucleoli
   • Thick - clump thickness
   • UShap - cell shape uniformity
   • USize - cell size uniformity

   The predictor values are determined by a doctor observing the cells and rating them on a scale from 1 (normal) to 10 (most abnormal) with respect to the particular characteristic.

   • Fit a binomial regression with Class as the response and the other nine variables as predictors. Report the residual deviance and associated degrees of freedom. Can this information be used to determine if this model fits the data? Explain.
   • Use AIC as criterion to determine the best subset of variables. (Use the step() function).
   • Use the reduced model to predict the outcome for a new patient with predictor variables 1,1,3,2,1,1,4,1,1 (same order as above). Give a confidence interval for your prediction.
   • Suppose that a cancer is classified as benign if p>0.5 and malignant if p<0.5. Compute the number of errors of both types that will be made if this method is applied to the current data with the reduced model.
   • Suppose we change the cutoff to 0.9 so that p<0.9 is classified as malignant and p>0.9 as benign. Computer the number of errors in this case. Discuss the issues in determining the cutoff.
   • It is usually misleading to use the same data to fit a model and test its predictive ability. To investigate this, split the data into two parts - assign every third observation to a test set and the remaining 2/3 of the data to a training set. Use the training set to determine the model and the test set to assess its predictive performance. Compare the outcome to the previously obtained results.