Assignment 3

1. 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.

2. The dataset infert presents data from a study of secondary infertility (failure to conceive after at least one previous conception). The variables in the dataset are:

   • education - 0=0-5 years; 1=6-11 years; 2=12+ years

   • age - age in years of case

   • parity - number of prior pregnancies
   • induced - number of prior induced abortions, 0=0; 1=1; 2=2 or more
   • case - case status, 1=case; 0=control
   • spontaneous - number of prior spontaneous abortions, 0=0; 1=1; 2=2 or more
   • stratum - matched set number, 1-83
   • pooled.stratum - stratum number, 1-63

   The factors of interest are induced abortions and spontaneous abortions (e.g., miscarriages). The study matched each case of infertility with two controls who were not infertile, matching on age, education, and parity.

   • Construct cross-classified tables by number of spontaneous and induced abortions separately for cases and controls. Comment on the differences between the two tables.
   • Fit a binary response model with only spontaneous and induced as predictors. Determine the statistical significance of these predictors. Express the effects of the predictors in terms of odds.
   • Fit a binary response model with only education, age, and parity as predictors. Explain how the significance (or lack thereof) of these predictors should be interpreted.
   • Now put all five predictors in a binary response model. Interpret the results in terms of odds.
   • Fit a matched case control model appropriate to the data. Interpret the output and compare the odds to those found in the previous model.
   • The spontaneous and induced predictors could be viewed as ordinal due to the grouping in the highest level. Refit the model using ordinal factors rather than numerical variables for these two predictors. Is there evidence that the ordinal representation is necessary?