Testing the Individual Model Parameters

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

Assumptions

the-multiple-regression-model_202509091642#assumptions-for-the-error-component

Test statistic

$$t=\frac{{\hat{\beta }}_i}{s_{{\hat{\beta }}_t}}\sim t_v$$

Hypotheses

	Two-tailed	Lower-tailed	Upper-tailed
Null hypothesis	$H_0:{\beta }_i=0$	$H_0:{\beta }_i=0$	$H_0:{\beta }_i=0$
Alternative hypothesis	$H_a:{\beta }_i\ne 0$	$H_a:{\beta }_i<0$	$H_a:{\beta }_i>0$
Rejection region	$	t	> t_{\alpha/2,v}$

P-value

$$\begin{array}{rl}{H}_a& :{\beta }_i>0,p\text{-value}=\{\begin{array}{ll}P/2& t>0\\ 1-P/2& t<0\end{array}\\ & \\ H_a& :{\beta }_i<0,p\text{-value}=\{\begin{array}{ll}1-P/2& t>0\\ P/2& t<0\end{array}\end{array}$$

Confidence interval

A $100(1-\alpha )\%$ confidence interval for a $\beta$ parameter is found by:

$${\hat{\beta }}_i\pm t_v^{-1}(\alpha /2)\times s_{{\hat{\beta }}_t}$$