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Setting Up a Test for the Slope of a Regression Model
When we believe a linear relationship exists between two quantitative variables, we often use a Linear Regression Models to model this relationship. However, the observed slope from a sample ( $ \hat{b} $ ) might just be due to random chance, even if there’s no true linear relationship in the population. A hypothesis test for the slope of a regression model allows us to determine if there is statistically significant evidence of a linear relationship between the two variables in the population.
Purpose of the Test
The primary goal of this test is to assess whether the explanatory variable ( $ x $ ) has a statistically significant linear association with the response variable ( $ y $ ) in the population. If the true population slope ( $ \beta $ ) is zero, it implies that there is no linear relationship between $ x $ and $ y $ .
Hypotheses
The hypotheses for testing the slope of a regression model are stated in terms of the population regression slope, $ \beta $ .
- Null Hypothesis ( $ H_0 $ ): There is no linear relationship between the explanatory variable ( $ x $ ) and the response variable ( $ y $ ) in the population. $$ H_0: \beta = 0 $$ This means that the true slope of the population regression line is zero.
- Alternative Hypothesis ( $ H_a $ ): There is a linear relationship between the explanatory variable ( $ x $ ) and the response variable ( $ y $ ) in the population. The alternative hypothesis can be one-sided or two-sided, depending on the research question.
- Two-sided: The slope is not zero. $$ H_a: \beta \neq 0 $$ * One-sided (positive relationship): The slope is greater than zero. $$ H_a: \beta > 0 $$ * One-sided (negative relationship): The slope is less than zero. $$ H_a: \beta < 0 $$
Conditions for Inference
Before carrying out a test for the slope of a regression model, several conditions must be met to ensure the validity of the results. These are often remembered by the acronym LINER:
| Condition | Description | How to Check |
|---|---|---|
| Linear | The true relationship between $ x $ and $ y $ is linear. | Examine the [Representing the Relationship Between Two Quantitative Variables |
| Independent | Individual observations are independent of each other. | This is usually ensured by [Random Sampling and a Collection |
| Normal | For any fixed value of $ x $ , the response $ y $ varies normally around the true regression line. | Examine a histogram or normal probability plot of the [Residuals |
| Equal Variance | The standard deviation of the response $ y $ is the same for all values of $ x $ . | Examine the [Residuals |
| Random | The data come from a well-designed [Introduction to Planning a Study | random sample](./../introduction-to-planning-a-study |
If these conditions are not reasonably met, the inference procedure for the slope may not be appropriate, and the conclusions drawn could be unreliable.
Test Statistic
The test statistic for the slope of a regression model is a $ t $ -statistic. It measures how many standard errors the observed sample slope ( $ \hat{b} $ ) is away from the hypothesized population slope (typically 0).
The formula for the test statistic is:
$$ t = \frac{\hat{b} - \beta_0}{SE_{\hat{b}}} $$
Where:
- $ \hat{b} $ is the sample slope (estimate of $ \beta $ ).
- $ \beta_0 $ is the hypothesized population slope (usually 0 under $ H_0 $ ).
- $ SE_{\hat{b}} $ is the standard error of the slope. This value is typically provided in the regression output from statistical software.
The degrees of freedom for this $ t $ -distribution are $ df = n - 2 $ , where $ n $ is the number of observations in the sample.
This test is often referred to as a $ t $ -test for the slope. Once the test statistic is calculated, it can be used to find the Interpreting p-Values|p-value to make a decision regarding the null hypothesis, as outlined in Carrying Out a Test for the Slope of a Regression Model.