Difference between F-test and separate t tests on each variable? Why can't we  separately test each variable(using t-test), whether or not it is equal to 0? Why should we use the F-test for checking if all coefficients combined are equal to 0?
 A: To elaborate on the answer of @Erik, there may be an issue of multiplicity control with t-tests. Suppose (for simplicity) that none of the variables are related to the outcome, and that the variables are independent.
Then, when testing each variable with, e.g., a t-test at level $\alpha$, you will reject that null with probability $\alpha$. Hence, the number of $k$ rejections in $N$ tests is a Binomial random variable with
probability mass function
$$\Pr\nolimits_k=\left(%
\begin{array}{c}
  N \\
  k \\
\end{array}%
\right)\alpha^k(1-\alpha)^{N-k}.$$
Take $\alpha=0.05$ and $N=20$. Then, the probability of at least one false rejection, the Familywise error rate, is given by 
$$FWER=\sum_{j=1}^{20}\left(
\begin{array}{c}
  20 \\
  j \\
\end{array}%
\right)0.05^j(1-0.05)^{20-j}=0.6415$$
Obviously, the problem only gets worse with $N$, so if you test many hypotheses at some fixed $\alpha$ each, you are almost bound to get at least one false positive.
A: Consider the case of one factor with many levels. For example, different cell line. The F-test can show you whether the dependant variable is influenced by cell line and the t-test can show whether two specific cell lines differ.
There are some issues with using only the t-values:


*

*If you have coefficients, you will need to correct for false positives

*Your regression coefficients depend on the reference level of your factor. If you set one of the extremes as the references you will get more significances.

*It is possible that you have low power for each individual t-test, but high power for the F-Test. This is the case when you have many groups with small sample sizes.

