Question

In linear regression , why can we see the lack of fit test as the comparison between the model we choose and the saturated model?

Answer 1

Let the two alternative models are under consideration, one model is simpler or more parsimonious than the other,one of the models is the saturated model. Another common situation is to consider ‘nested’ models, where one model is obtained from the other one by putting some of the parameters to be zero. Suppose now we test

The difference in the null and alternative hypothesis from the section above. Here to test the null hypothesis that an arbitrary group of k coefficients from the model is set equal to zero (e.g. no relationship with the response), we need to fit two models:

• the reduced model which omits the k predictors in question, and
• the current model which includes them

The likelihood-ratio statistic is

the degrees of freedom is k (the number of coefficients in question).

To perform the test, we must look at the "Model Fit Statistics" section and examine the value of "−2 Log L" for "Intercept and Covariates." Here, the reduced model is the "intercept-only" model (e.g. no predictors) and "intercept and covariates" is the current model we fitted. For our running example, this would be equivalent to testing "intercept-only" model vs. full (saturated) model (since we have only one covariate).

For our example, ΔG² = 5176.510 − 5147.390 = 29.1207 with df = 2 − 1 = 1. Notice that this matches Deviance we got in the earlier text above.

Another way to calculate the test statistic is

ΔG² = G² from reduced model
            −G² from current model,

where the G²'s are the overall goodness-of-fit statistics.

This value of -2 Log L is useful to compare two nested models which differ by an arbitrary set of coefficients.

Also notice that the ΔG² we calculated for this example equals to

Likelihood Ratio     29.1207    1    <.0001

from "Testing Global Hypothesis: BETA=0" section

Testing the Joint Significance of All Predictors.

Testing the null hypothesis that the set of coefficients is simultaneously zero.

H₀ : β₁ = β₂ = ... = 0 versus the alternative that at least one of the coefficients β₁, . . . , β_k is not zero.

the alternative that the current model (in this case saturated model) is correct

Using SAS:-

the SAS output, three different chisquare statistics for this test are displayed in the section

"Testing Global Null Hypothesis: Beta=0," corresponding to the likelihood ratio, score and Wald tests. Recall their definitions from the very first lessons.

The Homer-Lemeshow Statistic

An alternative statistic for measuring overall goodness-of-fit is Hosmer-Lemeshow statistic.

This is a Pearson-like χ² that is computed after data are grouped by having similar predicted probabilities. It is more useful when there is more than one predictor and/or continuous predictors in the model too.