# F and t statistics in a regression

In a multiple linear regression, why is it possible to have a highly significant F statistic (p<.001) but have very high p-values on all the regressor's t tests?

In my model, there are 10 regressors. One has a p-value of 0.1 and the rest are above 0.9

For dealing with this problem see the follow-up question.

-
See also here: how can a regression be significant yet all predictors be non-significant, & for a discussion of the opposite case, see here: significant t-test vs non-significant F-statistic. –  gung Sep 13 '12 at 15:15

As Rob mentions, this occurs when you have highly correlated variables. The standard example I use is predicting weight from shoe size. You can predict weight equally well with the right or left shoe size. But together it doesn't work out.

Brief simulation example

RSS = 3:10 #Right shoe size

##Fit a joint model
m = lm(weights ~ LSS + RSS)

##F-value is very small, but neither LSS or RSS are significant
summary(m)

##Fitting RSS or LSS separately gives a significant result.
summary(lm(weights ~ LSS))

-
+1:> faily good way to make it intuitive. –  user603 Oct 13 '10 at 13:17
(+1) Very nice example! –  chl Oct 13 '10 at 13:25
It is interesting and important to note that both of your models predict equally well, in this case. High correlations among predictors are not necessarily a problem for prediction. Multicolinearity is only a problem when 1) analysts try to inappropriately interpret multiple regression coefficients; 2) the model is not estimable; and 3) SEs are inflated and coefficients are unstable. –  Brett Magill Jun 9 '11 at 14:27