Using R for lack-of-fit F-test -


i learnt how use r perform f-test lack of fit of regression model, $h_0$: "there no lack of fit in regression model".

$$f_{lof} = \frac{mslf}{mspe} = \frac{sslf(\text{model}) / df_1}{sspe/df_2}$$

where df_1 degrees of freedom sslf (lack-of-fit sum of squares) , df_2 degrees of freedom sspe (sum of squares due pure error).

in r, f-test (say model 2 predictors) can calculated

anova(lm(y~x1+x2), lm(y~factor(x1)*factor(x2)))

example output:

model 1: y ~ x1 + x2 model 2: y ~ factor(x1) * factor(x2)   res.df    rss df sum of sq      f pr(>f) 1     19 18.122                            2     11 12.456  8    5.6658 0.6254 0.7419

f-statistic: 0.6254 p-value of 0.7419.

since p-value greater 0.05, not reject $h_0$ there no lack of fit. therefore model adequate.

what want know why use 2 models , why use command factor(x1)*factor(x2)? apparently, 12.456 model 2, magically sspe model 1.

why?

you testing whether model interaction improves model fit.

model 1 corresponds additive effect of x1 , x2.

one way "check" if complexity of model adequate (in case whether multiple regression additive effects make sense data) compare proposed model more flexible/complex model.

your model 2 has role of more flexible model. first predictors made categorical (by using factor(x1) , factor(x2)) , interaction between them constructed factor(x1)*factor(x2). interaction model includes additive model special case (i.e., model 1 nested in model 2) , has several parameters provide potentially better fit data.

you can see difference in number of parameters between 2 models in output anova. model 2 has 8 parameters allow better fit because p-value non-significant conclude model 2 (with flexibility based on additional 8 parameters) not provide better fit data. thus, additive model provides decent enough fit data when compared model 2.

note trick above making categories (factors) of x1 , x2 works when number of unique values x1 , x2 low. if x1 , x2 numeric , each individual has own value model 2 not useful end same number of parameters hav observations. in situations more ad hoc modifications such binning variables used.


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