Preliminary

Variable definitions

Some variables that will be used throughout this page:

• $n :$ Number of observations
• $p + 1 :$ Number of coefficients ($\beta$) in the model
• $t_{q,v} :$ Point percent function of a Student’s-t distribution at the $q-$th quantile
• $v = n - (k+1) :$ The degrees of freedom in $t_{q,v}$

Warning

Failing to reject $H_0$ does not mean that the independent variable does not explain the dependent variable.

Instead, several conclusions are possible:

• There is no relationship
• A relationship exists, but a Type II error occurred
• A relationship exists, but is different than the hypothesized model

The most you can say after testing is:

• If $H_0$ is rejected: There is a sufficient evidence for the hypothesized relationship
• Else: There is insufficient evidence for the hypothesized relationship

Recommendations

1. First, test the overall model adequacy.

   If $H_0$ is rejected, continue to step 2

   Else, consider hypothesizing a different model

2. Conduct t-tests on the most “important” $\beta$ coefficients. Usually only involves $\beta$s involved with higher-order terms

   Conducting a series of t-tests leads to an overall high Type I error rate