Deviance R^{2} is usually considered the proportion of the total deviance in the response variable that the model explains.

The higher the deviance R^{2}, the better the model fits your data. Deviance R^{2} is always between 0% and 100%.

Deviance R^{2} always increases when you add additional terms to a model. For example, the best 5-term model will always have an R^{2} that is at least as high as the best 4-term model. Therefore, deviance R^{2} is most useful when you compare models of the same size.

Goodness-of-fit statistics are just one measure of how well the model fits the data. Even when a model has a desirable value, you should check the residual plots and goodness-of-fit tests to assess how well a model fits the data.

You can use a fitted line plot to graphically illustrate different deviance R^{2} 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.

The data arrangement affects the deviance R^{2} value. The deviance R^{2} is usually higher for data with multiple trials per row than for data with a single trial per row. Deviance R^{2} values are comparable only between models that use the same data format. For more information, go to How data formats affect goodness-of-fit in binary logistic regression.

Adjusted deviance R^{2} 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.

Use adjusted deviance R^{2} to compare models that have different numbers of terms. Deviance R^{2} always increases when you add a term to the model. The adjusted deviance R^{2} value incorporates the number of terms 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 R^{2} |
Adjusted Deviance R^{2} |
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 R^{2}, which indicates that cooling rate improves the model. The third step, which adds cooking temperature to the model, increases the deviance R^{2} but not the adjusted deviance R^{2}. These results indicate that cooking temperature does not improve the model. Based on these results, you consider removing cooking temperature from the model.

The data arrangement affects the adjusted deviance R^{2} value. For the same data, the adjusted deviance R^{2} is usually higher for data with multiple trials per row than for data with a single trial per row. Use the adjusted deviance R^{2} only to compare the fit of models that have the same data format. For more information, go to How data formats affect goodness-of-fit in binary logistic regression.

The Akaike's Information Criterion (AIC), corrected Akaike’s Information Criterion (AICc), and the Bayesian Information Criterion (BIC) are measures of the relative quality of a model that account for fit and the number of terms in the model.

Use AIC, AICc and BIC to compare different models. Smaller values are desirable. However, the model with the least value for a set of predictors does not necessarily fit the data well. Also use tests and plots to assess how well the model fits the data.

- AICc and AIC
- When the sample size is small relative to the parameters in the model, AICc performs better than AIC. AICc performs better because with relatively small sample sizes, AIC tends to be small for models with too many parameters. Usually, the two statistics give similar results when the sample size is large enough relative to the parameters in the model.
- AICc and BIC
- Both AICc and BIC assess the likelihood of the model and then apply a penalty for adding terms to the model. The penalty reduces the tendency to overfit the model to the sample data. This reduction can yield a model that performs better in general.