# Chapter 4 Statistical models for prediction

In chapter 4, we consider different statistical models for different types of outcomes.

## Competing risks

Clinical decision-making often relies on a subject’s absolute risk of a disease event of interest. However, in a frail population, competing risk events may preclude the occurrence of the event of interest. Competing-risk regression models can hence be useful, specifically the Fine and Gray model. We recently applied competing risks methods to coronary risk prediction (Epidemiology, July 2009). An excellent tutorial was published by Putter et al in Statistics in Medicine in 2007.

Fine and Gray regression models can be fitted with the contributed R package cmprsk and the function FGR in the R package riskRegression provides a convenient formula interface to fit these models. An IPCW-estimator for the concordance probability has recently been proposed and implemented in the R-function cindex in the R package pec. Some additional R functions associated with our paper on coronary risk prediction (Epidemiology, July 2009) and programmed by Marcel Wolbers are available but are now largely superseded by the above-mentioned packages.

## Categorical outcomes

For categorical outcomes, we can use a polytomous logistic regression model. The use of this model was illustrated with a case study on prediction of the histology of residual masses in testicular cancer patients (section 4.3.2).
For ordinal outcomes, we can consider a straightforward extension of the standard logistic model: the proportional odds logistic model. The use of this model was illustrated with the prediction of outcome at 6 months after traumatic brain injury (section 4.4.1). We can also try to predict the histology of residual masses in testicular cancer patients with a proportional odds logistic model. We illustrate below that the proportional odds assumption is violated.

### Example revisited: histology in testicular cancer

We developed polytomous regression models for the histology of residual masses in testicular cancer patients in the previous section. We could also have used a proportional odds model. However, Figure 4.extra shows that the proportional odds assumption does not hold in this case for many predictors. This is also noted from Table 4.extra, where the proportional odds provides an approximate average estimate of widely varying odds ratios. This is in contrast to the example of predicting the GOS in traumatic brain injury. Figure 4.extra Assessment of the proportional odds assumption for each of 6 predictors (univariate analysis) to predict the histology at resection for testicular cancer patients (n=821). The circle, and triangle symbols correspond to the categorizations necrosis/teratoma vs cancer, and necrosis vs teratoma/cancer. For example, the overall logit for necrosis = -0.3, or a probability of 43% (349/821 patients), and the logit for necrosis/teratoma = 2 (88%, 722/821). The proportional odds assumption is not satisfied for most predictors, since the horizontal distance between the points is not constant within each category, nor is the vertical pattern for the same symbols. 