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Statistical Models for Prognostication
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Predictions: Statistical Models
Insight: Statistical Models
Ingredients: Statistical Models
Theoretical Aspects
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Currently selected section: Regression Models
Problems: Regression
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Example 2

Chapter 8: Statistical Models for Prognostication: Development of Regression Models

A simple measure for calibration is the slope of the regression line for observations against predictions. When predictions are too extreme, the slope will be less than one. We may refer to the slope as "calibration slope." The calibration slope is easily visualized in the plots of predicted against observed values (see graph: Illustration of calibration above).

This form of mis-calibration can statistically be tested by comparison with a model with intercept = 0 and slope = 1 (Miller et al., 1991) (Harrell et al., 1996).

The calibration plot may also provide useful information on the distribution of predicted values, e.g. by plotting symbols for groups of patients. The spread of predictions brings us to the other aspect of model performance: discrimination.

Discrimination refers to the ability to distinguish high risk subjects from low risk subjects, and is commonly quantified by a measure of concordance, the c statistic. For binary outcomes, c is identical to the area under the receiver operating characteristic (ROC) curve (Hanley and McNeil, 1982). C varies between 0.5 and 1.0 for sensible models; the higher the better. For an example of a ROC curve see the graph, "Illustration of 2 ROC curve" below (Steyerberg et al., 2001).

The c statistic is calculated as the fraction of patients with the outcome among pairs of patients where one has the outcome and one not, the patient with the highest prediction being classified as the one with the outcome. Hence, when a model provides no information, c=0.5. The c statistic has been generalized for survival analysis.

Figure 7.2: Calibration
Graphic depiction of ROC curves, described in text
Illustration of 2 ROC curves. The curves refer to a development population
of 544 patients, where the area under the curve was 0.83, and a validation
population of 172 patients, where the area under the curve was 0.80. The
underlying model predicts a benign histology in a residual mass after
chemotherapy for metastatic testicular cancer. More details can be found
elsewhere (Steyerberg et al., 2001).

The c statistic is related in some aspects to R2. Both approach 1 for perfectly discriminating models. An important difference is that c is not dependent on the frequency of the outcome, while R2 is smaller when the outcome is infrequent, and larger when the outcome is more frequent (Ash and Shwartz, 1999).

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