Performing hypothesis test in R to assess reduced model I have a multiple linear regression model with 10 covariates, and I want to perform the hypothesis test that 5 of the covariates are equal to 0, versus the alternative hypothesis that at least one of them is not equal to 0. More precisely: $$ H_0: \beta_{age} = \beta_{tc} = \beta_{ldl} = \beta_{tch} = \beta_{glu} = 0 \ \text{versus} \ H_1 : \text{at least one} \neq 0$$
So I want to perform this test in the full model using R to discover whether the reduced model which does not include these covariates is preferable to the full model. How do I go about doing this? Below is a summary of the full model in R.

 A: This is exactly what an F-test does. What follows depends on the regression models being nested.
Under your null hypothesis, you have a 6-parameter model (five variables and the intercept). You calculate its sum of squared errors, $SSE_0$, via sum((L_reduced$residuals)^2), where L_reduced is the lm for this model.
Under the alternative hypothesis, you have an 11-parameter model (ten variables and the intercept). You calculate its sum of squared errors, $SSE_1$, via sum((L_full$residuals)^2), where L_full is the lm for this model. 
Let $\sigma^2$ be the error variance and $n$ the sample size. Then under the null hypothesis:
$$ \dfrac{SSE_0 - SSE_1}{\sigma^2} \sim \chi^2_{11-6} \text{ }\text{ }\text{  and  }\text{ }\text{ } \dfrac{SSE_1}{\sigma^2} \sim \chi^2_{n-11}$$
Consequently:
$$ \dfrac{(SSE_0 - SSE_1)/(11-6)}{SSE_1/(n-11)}= F\sim F_{11-6, n -11}$$
You now have a test statistic to compare to $F_{11-6, n -11}$ via 1-pf(F,11-6,n-11). This will say if the alternative model holds;$^{\dagger}$ if that is the case, it must be that at least one of those $\beta$ coefficients is not zero. $\square$
All of this comes from chapter 3 of Agresti:
Agresti, Alan. Foundations of linear and generalized linear models. John Wiley & Sons, 2015.
$^{\dagger}$ There are the usual caveats about what hypothesis testing means.
