# Multiple possible models for predicting a dependent variable: Which one should we choose?# This is a problem for linear regression. Here are three models for Hinton's study time data.study.time = c(40,43,18,10,25,33,27,17,30,47),exam.score=c(58,73,56,47,58,54,45,32,68,69),iq = c(118,128,110,114,138,120,106,124,132,130))ml1 = lm(exam.score ~ study.time, data = studytimedata)# We can inspect the models using "summary". But what do we conclude from this inspection?# We have the same problem for logistic regression, demonstrated here for the problem of # predicting the dative alternation -- what influences whether we will see "John gave Mary the book"# or "John gave the book to Mary"?library(rms)library(languageR)md1 = lrm(RealizationOfRecipient ~ AnimacyOfRec, data = dative)md2 = lrm(RealizationOfRecipient ~ DefinOfRec, data = dative)md3 = lrm(RealizationOfRecipient ~ AnimacyOfRec + DefinOfRec, data = dative)md4 = lrm(RealizationOfRecipient ~ SemanticClass, data = dative)md5 = lrm(RealizationOfRecipient ~ SemanticClass + AnimacyOfRec + DefinOfRec, data = dative)md6 = lrm(RealizationOfRecipient ~ LengthOfTheme, data = dative)# Model comparison: linear regression, nested models. Use F-test (ANOVA)anova(ml1, ml3)# Model comparison: logistic regression, nested models. Here, we can use likelihood ratio.# lrm() returns the model deviance in the "deviance" entry.# This is a vector with two members: deviance for the model with only the intercept,# and deviance for the models with all its parameters. We are interested in the latter.dev.1 = md1$deviance[2]dev.5 = md5$deviance[2]# log likelihood ratio: deviance of simpler model - deviance of more complex model.# Approximately chi-square distributed.# Null hypothesis: no difference between the two models.# lrtest() is a function of the library rlm that does likelihood ratio tests. lrtest(md1, md3)lrtest(md1, md5)# In non-nested cases, we can use the Akaike Information Criterion, AIC. # # >>> Lower AIC is better <<<# # Here it is for linear regressionAIC(ml1)AIC(ml2)AIC(ml3)# And here it is for logistic regressionAIC(md1)AIC(md2)AIC(md3)AIC(md4)AIC(md5)AIC(md6) |
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