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The Linear Predictor

In the regression models that we consider, the regression formula is a linear combination of regression coefficients and predictors. The values that result from the regression formula may also be referred to as the "linear predictor" or "prognostic index."

The linear predictor (LP) summarizes the prognosis for every patient. The LP can be calculated for the patients in the data set under study as: LP = a + b₁x₁ + b₂x₂ + … + b_ix_i, with a denoting the intercept, and b₁ to b_i regression coefficients for i covariables x₁ to x_i. In the data set under study, the LP by definition has a perfect fit when we include the LP in a regression model:

Y ~ a + b LP,

Where the slope b is unity and the intercept a is zero.

The LP can also be calculated for patients not included in the modeling process:

LP_model = a_model + b_{1, model} x_{1, new} + b_{2, model} x_{2, new} + … + b_{I, model} x_{I, new,} with a denoting the intercept from the model, b_{1, model} to b_{i, model} the regression coefficients from the model, and x_{i, new} to x_{i, new} the covariables for the new patients. We can include this LP in a regression model:

Y_new ~ a_new + b_new LP_model,

Where, if the model provides perfectly valid predictions, the slope b_new is unity and the intercept a_new zero.

In practice, however, the slope of the LP is often less than unity in new patients (Spiegelhalter, 1986) (Van Houwelingen and Le Cessie, 1990) (Harrell et al., 1996): high predictions are too high and low predictions too low. Hence, the linear predictor is an important concept to assess model performance (see: Development of Regression Models, Evaluation of Performance) and, more specifically, calibration of predictions (see: Development of Regression Models, Evaluation of Performance).