Prostate specific antigen is useful as a preoperative marker for patients with prostate cancer [Stamey]. We use the dataset from [Hastie] (and [Stamey]) to model the dependence of the logarithm of prostate specific antigen (lpsa) on the given (8) predictors. We use multilinear regression, and choose what predictors to keep using backwards stepwise selection.
The univariate plots suggest a linear relationship between many of the predictors and the response.
Unsurprisingly, a multilinear model (using all the predictors) fits the data well, improving the base error (of the model which simply predicts the average) by 50.5%.
(As explained in the IPYnotebook, the intercept is irrelevant in this graph: one must only observe the slope).
However, some of the predictors turn out to be superfluous.
Backwards selection tests the null-hypothesis using the Z-scores (which follow t-student distributions with 67-(p+1) d.o.f where p is the number of predictors) and provides a way to select the significant predictors.
Using backwards selection with a threshold of 5% on the p-value of the Z-scores actually improves our error on the test set: the base error is improved by 53.2%.
Analyzing the Z-scores more carefully, one sees that it may actually be better to cap the p-values at 6% instead.
Indeed the resulting model (which drops the predictors gleason, age, lcp and pgg45) behaves the best, with an improvement of 56.7% w.r.t. the base error.
Here is a table with the errors for the different models:
- [Hastie]: 'The Elements of Statistical Learning' by Trevor Hastie et al.
- [Gareth]: 'Introduction to Statistical Learning' by Gareth James et al.
- [Casella]: 'Statistical Inference' by George Casella and Roger Berger.
- [Downey]: 'Think stats' by Allen Downey.
- [Stamey]: Stamey, Thomas A., et al. "Prostate specific antigen in the diagnosis and treatment of adenocarcinoma of the prostate. II. Radical prostatectomy treated patients." The Journal of urology 141.5 (1989): 1076-1083.