8.5 Testing one coefficient
Does the machinery of Unit 7 still work?
It does, unchanged apart from the degrees of freedom. Before using it, it is worth recalling why it works at all.
Every coefficient is an estimate
The four numbers in the first column of a regression output are not facts about the population. They were computed from one sample of 680 students.
A different sample of 680 students would have produced four different numbers.
Each \(\hat{\beta}_j\) is a statistic — a quantity computed from a sample — and so each has a sampling distribution, exactly as the sample mean did in Unit 2.
Section 7.3 drew that distribution for the simple regression slope by taking five thousand samples and plotting the result. The same is true of every coefficient here, and of all of them jointly.
That is what makes the rest of this unit possible. Unbiasedness says the distribution of \(\hat{\beta}_j\) is centred on \(\beta_j\). The standard error estimates its width. And the Central Limit Theorem of Unit 1 says that in reasonably large samples the distribution is approximately normal, whatever the distribution of the errors — which is why CLRM6 matters so little in practice.
So the problem is the one this book has been solving since Unit 2. We have a single draw from a distribution centred on the quantity we want, and we know roughly how wide that distribution is. Everything that followed from that — standard errors, confidence intervals, \(t\) statistics, \(p\)-values — applies here without modification.
The test
To test whether a single population coefficient equals zero,
\[H_0 : \beta_j = 0 \qquad\qquad t = \frac{\hat{\beta}_j}{\mathrm{se}(\hat{\beta}_j)} \;\sim\; t_{\,n-k-1}\]
and confidence intervals follow as before.
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -7.6603 1.2944 -5.918 0
#> attendance 0.1360 0.0104 13.040 0
#> cum_gpa 0.5748 0.0441 13.026 0
#> admission_score 0.0835 0.0158 5.299 0#> 2.5 % 97.5 %
#> (Intercept) -10.2019 -5.1188
#> attendance 0.1155 0.1565
#> cum_gpa 0.4881 0.6614
#> admission_score 0.0526 0.1144Every coefficient is significant at any conventional level, and the interval for attendance runs from 0.116 to 0.157.
The null tested is \(\beta_j = 0\) in the model as specified.
Rejecting it says attendance is related to grades among students alike in prior GPA and admission score. It says nothing about a model with different controls, where the coefficient, its standard error and its significance may all differ.
A \(t\) statistic is a statement about one column of one regression.