About Backward Stepwise and Backward Removal

In the backward stepwise model selection procedure, variables are sequentially removed from a full (all regression terms included) model.  In contrast, forward procedures start with an empty (intercept only) model, and proceed by adding terms.  To simplify the description, this overview describes the process for stepwise linear regression, with modifications that apply to logistic and Poisson regression presented below.

The backward stepwise procedure

In the first round of backward stepwise iterations, the regression terms (i.e., datasets that you selected in the model definition step) are each removed from the "full" starting model, and the regression calculation is performed to find the improvement in the residual sum of squares for each of these resulting models relative to the starting model.  For each new model, SpaceStat calculates a p-value for the change in the sum of squares; this calculation is based on an F-distribution and incorporates the degrees of freedom in the regression term and the error variance.  When all of the models missing one term have been created, the backward stepwise procedure selects the term associated with the highest p-value as the first round candidate for removal from the model.  This variable’s p-value is then compared to the "p to stay" cut-off value you have specified in the stepwise procedure dialog box, and if it is higher than the cut-off, the candidate term will be removed from the regression model.  This provides a new starting model for the next round. If in this or in any of the subsequent rounds the highest candidate p-value is not higher than the "p to stay" value you have specified in the regression dialog, then the backward stepwise procedure will stop.  

The subsequent rounds start with the same procedure; the terms still in the model are examined in turn to see what their p-values are for removal from the model. The highest p-value candidate is again compared against the "p to stay" to determine if it is to be removed from the model. Once more than one term has been removed from the model, SpaceStat will loop through each of the terms that have been removed and, on the basis of an F-distribution, calculate the p-value associated with restoring the term into the model.  Again, this p-value is calculated by comparing the resulting decrease in residual sum of squares associated with the model that now includes the term that had previously been removed to the error variance of the larger model.  The term with the lowest p-value becomes the candidate for restoration into the model, and if this p-value is smaller than the "p to enter" value you have entered in the dialog box, then the term will be restored to the model.

SpaceStat allows as many rounds of this procedure as there are possible regression terms in your proposed model (i.e., as many rounds as the number of terms you defined in the model definition step). If a term "returns (is later added after having been removed), SpaceStat augments the number of remaining rounds to allow all the remaining terms to have a chance to be removed.  If, however, during a particular round, the same term is restored and then removed, this event will trigger an end to the backward stepwise regression procedure.  The integrity of the results obtained in backward stepwise regression depends on the choice of "p to enter" and "p to stay"; the "p to stay" value should not be made smaller than the "p to enter" value.

Backward removal

Setting the "p to enter" value to 0 ensures that no term removed from the model will be allowed to return. This reduces the backward stepwise procedure to the "Backward Removal" procedure used in other software.

Logistic and Poisson backward stepwise regression

Rather than an F-test, to determine whether variables enter or leave logistic regression and Poisson models, SpaceStat calculates the exact log-likelihood difference attributable to the focal variable. The log-likehood is evaluated with a chi-squared test to determine significance relative to the "p to enter" and "p to stay" cut offs you set in the regression settings.   

Click here to see how to perform stepwise regression.