Using R and the `anova`

function we can easily compare **nested **models. Where we are dealing with regression models, then we apply the `F-Test`

and where we are dealing with logistic regression models, then we apply the `Chi-Square Test`

. By nested, we mean that the independent variables of the **simple **model will be a **subset **of the more **complex **model. In essence, we try to find the best parsimonious fit of the data. Note that we should fit the models on the same dataset.

The Null Hypothesis is that the **simple **model is better and we reject the null hypothesis if the p-value is less than 5% inferring that the **complex **model is is significantly better than the simple one.

## Example of Comparing Nested Models

Let’s work with the `LifeCycleSavings`

dataset by considering as dependent variable the `sr`

and the rest as independent variables (IV).

Let’s say that we want to compare the following two models:

`fit0`

which is the \(sr = \alpha\)`fit1`

which is the \(sr = \alpha +\beta \times pop15\)

fit0 <- lm(sr ~ 1, data = LifeCycleSavings) fit1 <- lm(sr ~ pop15, data = LifeCycleSavings) summary(fit0) summary(fit1)

Notice the P-value of the F-Test of the fit1 model is 0.0008866 which actually tests the Null Hypothesis that “all the beta coefficients are zero” versus the alternative hypothesis that “at least one beta coefficient is not zero”. Since we have only one beta coefficient, the pop15 the p-value of the F-Test is the same as the p-value of the T-Test as we can see above.

Now, if we compare the `fit0`

vs the `fit1`

, in essence, we test if we should include the `pop15`

coefficient or not, thus we expect to get the same p-value. Let’s compare the nested models using `anova`

:

anova(fit0, fit1, test='F')

As expected we got the same p-value, and we can say that we should prefer the `fit1`

compared to `fit0`

model.

Let’s make another comparison by comparing the `fit1`

compared to the `fit4`

which contains all the IVs.

fit4<-lm(sr~pop15+pop75+dpi+ddpi, data = LifeCycleSavings) summary(fit4)

Let’s compare the two models:

anova(fit1, fit4, test='F')

The p-value is** 0.04177** forcing us to reject the null hypothesis that the `fit1`

models is better. Finally, let’s compare the `fit1`

model versus the `fit3`

which contains the first 3 IV of the dataset.

fit3<-lm(sr~pop15+pop75+dpi, data = LifeCycleSavings) anova(fit1, fit3, test='F')

In this case, the p-value is** 0.1318** which means that we should accept the null hypothesis that the `fit1`

is better than the `fit3`

.