Regression/ANCOVA models as described above are most useful when
- they contain a few clinically relevant and interpretable prognostic variables
- the parameters or coefficients are estimated with relatively high precision
- the prognostic factors each carry independent information about the outcome variable
- the model is consistent with other clinical and biological data
For a given situation, however, it may not be easy to construct a model that satisfies these criteria.
With available statistical software in modern computers, portions of the model-building process are automatic. Caution must be exercised, however, for the following reasons.
- The criteria employed by the software may be inappropriate, e.g., relying solely on p-values.
- There are poor statistical properties when performing a large number of tests and refitting models.
- It is not possible to incorporate outside information into the model-building process.
- The software may not handle the problem of missing data very well.
The model-building process requires thought and an understanding of the clinical situation. Some statisticians only use prognostic variables in the model for which there exist plausible biological reasons for their inclusion.
Computer software to assist in the construction and evaluation of a model follows several approaches.
One approach is called a step-up or forward selection process, in which the initial model contains no regressors but they enter the model one at a time. In this situation, a regressor enters the model if its p-value is less than a critical value, say 0.05.
Another approach is called the step-down or backward selection process, in which the initial model contains all of the regressors. In this situation, a regressor is eliminated from the model if its p-value is not less than the critical value.
A third approach, called stepwise selection, is a modification of forward selection. In this situation, after a new variable enters the model, all the variables that had entered the model previously are reexamined to see if their p-values have changed. If any of the revised p-values exceed the critical value, then the corresponding variables are eliminated from the model.
A fourth approach involves finding the best one-variable-model, the best two-variable model, etc. with the help of software, and then using judgment as to which is the best overall model, i.e., if the (c+1)-variable model is only slightly better than the c-variable model, the latter is selected. It is prudent to attempt a variety of models and approaches to determine if the results are consistent.
Some statisticians favor the backward selection or step-down process, although there is no universal agreement among statisticians. It is not unusual for a particular data set to discover that step-up and step-down selection algorithms lead to different models. The main reason for this is that the regressors/covariates are not completely independent of each other.
When a variable is entered into or removed from a model, the p-values of the other variables will change. Consider a linear model with two potential regressors, \(X_1\) and \(X_2\), and suppose that they are strongly correlated (“independent variables” is a misnomer). Suppose that in a model with \(X_1\) only, \(X_1\) is significant, and in a model with \(X_2\) only, \(X_2\) is significant. When a model is constructed with both \(X_1\) and \(X_2\), however, the contribution by \(X_2\) to the model is no longer statistically significant. Because \(X_1\) and \(X_2\) are strongly correlated, \(X_2\) has very little predictive power when \(X_1\) already is in the model.
Initial screening of the entire set of candidate regressors/covariates is advised. Many statisticians recommend that each potential regressor be examined individually in a simple model. This can help identify potential regressors for which there is not a strong biological justification.
Usually, the critical significance level in this first-stage approach is more lenient, say 0.10 or 0.15. Then all of the regressors that meet this first-stage criterion and/or that have biological/clinical justification comprise the set of regressors that are subjected to the model-building process. Clinical input always should augment this first-stage process.