GLM (family=binomial) is foucusd on when the response is dichotomous(yes/no, male/female, etc..). I'm wondering how to judge if the model we built is good eough? As we know, in OLS regression some criterion like R^2 and adjusted R^2 can tell us how much variations are explained but not for GLM. See example I performed:
> summary(fit.full)
Call:
glm(formula = ynaffair ~ gender + age + yearsmarried + children +
+religiousness + education + occupation + rating, family = binomial(),
data = Affairs)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.6575 -0.7459 -0.5714 -0.2552 2.5099
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.71792 0.96165 0.747 0.455336
gendermale 0.28665 0.23973 1.196 0.231811
age -0.04494 0.01831 -2.454 0.014142 *
yearsmarried 0.09686 0.03236 2.993 0.002758 **
childrenyes 0.37088 0.29466 1.259 0.208147
religiousness -0.32230 0.09003 -3.580 0.000344 ***
education 0.01795 0.05088 0.353 0.724329
occupation 0.03210 0.07194 0.446 0.655444
rating2 -0.02312 0.58177 -0.040 0.968303
rating3 -0.84532 0.57619 -1.467 0.142354
rating4 -1.13916 0.55740 -2.044 0.040981 *
rating5 -1.61050 0.56649 -2.843 0.004470 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 675.38 on 600 degrees of freedom
Residual deviance: 608.22 on 589 degrees of freedom
AIC: 632.22
After removed the insignificant variables, the reduced model look like below,although the AIC decreasd, we still do not know if this is the model with the lowest AIC we can achieved:
> summary(fit.reduced)
Call:
glm(formula = ynaffair ~ age + yearsmarried + religiousness +
+rating, family = binomial(), data = Affairs)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.5117 -0.7541 -0.5722 -0.2592 2.4123
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.10220 0.71849 1.534 0.125014
age -0.03588 0.01740 -2.062 0.039224 *
yearsmarried 0.10113 0.02933 3.448 0.000565 ***
religiousness -0.32571 0.08971 -3.631 0.000282 ***
rating2 0.11848 0.57258 0.207 0.836068
rating3 -0.70168 0.56671 -1.238 0.215658
rating4 -0.96190 0.54230 -1.774 0.076109 .
rating5 -1.49502 0.55550 -2.691 0.007118 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 675.38 on 600 degrees of freedom
Residual deviance: 613.63 on 593 degrees of freedom
AIC: 629.63
And we perform the ANOVA, suggesting that the reduced model with
four predictors fits as well as the full model:
> anova(fit.reduced, fit.full, test="Chisq")
Analysis of Deviance Table
Model 1: ynaffair ~ age + yearsmarried + religiousness + +rating
Model 2: ynaffair ~ gender + age + yearsmarried + children +
+religiousness + education + occupation + rating
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
1 593 613.63
2 589 608.22 4 5.4124 0.2475
