Model summary statistics for Simple Binary Logistic Regression

Find definitions and interpretation guidance for the model summary statistics.

Deviance R-Sq

Deviance R2 is usually considered the proportion of the deviance in the response variable that the model explains.

Interpretation

The higher the deviance R2, the better the model fits your data. Deviance R2 is always between 0% and 100%.

Deviance R2 always increases when you add additional predictors to a model. For example, the best 5-predictor model will always have an R2 that is at least as high as the best 4-predictor model. Therefore, deviance R2 is most useful when you compare models of the same size.

For binary logistic regression, the format of the data affects the deviance R2 value. The deviance R2 is usually higher for data in Event/Trial format. Deviance R2 values are comparable only between models that use the same data format.

Deviance R2 is just one measure of how well the model fits the data. Even when a model has a high R2, you should check the residual plots to assess how well the model fits the data.

You can use a fitted line plot to graphically illustrate different deviance R2 values. The first plot illustrates a model that explains approximately 96% of the deviance in the response. The second plot illustrates a model that explains about 60% of the deviance in the response. The more deviance that a model explains, the closer the data points fall to the curve. Theoretically, if a model could explain 100% of the deviance, the fitted values would always equal the observed values and all of the data points would fall on the curve.  Adjusted deviance R2 is the proportion of deviance in the response that is explained by the model, adjusted for the number of predictors in the model relative to the number of observations.

Interpretation

Use adjusted deviance R2 to compare models that have different numbers of predictors. Deviance R2 always increases when you add a predictor to the model. The adjusted deviance R2 value incorporates the number of predictors in the model to help you choose the correct model.

For example, you work for a potato chip company that examines factors that affect crumbled potato chips. You receive the following results as you add predictors:
Step % Potato Cooling rate Cooking temp Deviance R2 Adjusted Deviance R2 P-value
1 X     52% 51% 0.000
2 X X   63% 62% 0.000
3 X X X 65% 62% 0.000

The first step yields a statistically significant regression model. The second step, which adds cooling rate to the model, increases the adjusted deviance R2, which indicates that cooling rate improves the model. The third step, which adds cooking temperature to the model, increases the deviance R2 but not the adjusted deviance R2. These results indicate that cooking temperature does not improve the model. Based on these results, you consider removing cooking temperature from the model.

For binary logistic regression, the format of the data affects the adjusted deviance R2 value. For the same data, the adjusted deviance R2 is usually higher when the data are in Event/Trial format than when the data are in Binary Response/Frequency format. Use the adjusted deviance R2 only to compare the fit of models that have the same data format.

AIC

Akaike Information Criterion is a measure of the relative quality of a model that accounts for fit and the number of terms in the model. The statistic has no interpretation without a comparison value.

Interpretation

Use AIC to compare different models. The smaller the AIC, the better the model fits the data. However, the model with the smallest AIC for a set of predictors does not necessarily fit the data well. Also use goodness-of-fit tests and residual plots to assess how well a model fits the data.

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