Section 10.8 Model Significance for Simple Linear Regression
You might have noticed another p-value toward the bottom of the output, adjacent to the F-statistic for the linear regression. This p-value corresponds to a test statistic that evaluates the improvement of the regression line’s fit to the data points compared to the mean value of our dependent variable (risky lifestyles). The F-statistic serves as the test statistic for linear regression, assessing how well the regression line fits compared to the mean value of risky lifestyles. The model fit can be tested by following the NHST steps that we used above.
Subsection 10.8.1 Step 1: Formulate the Null and Alternative Hypotheses
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\(H_0\text{:}\) A model including low self-control is not better at explaining risky lifestyles than a baseline model using the mean value of risky lifestyles.
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\(H_A\text{:}\) A model including low self-control is better at explaining risky lifestyles than a baseline model using the mean value of risky lifestyles.
Subsection 10.8.2 Step 2: Calculate the Test Statistic
From the provided output above, you can identify the F-value as \(F(1, 941) = 36.28\text{.}\)
Subsection 10.8.3 Step 3: Determine the Probability (P-Value) of Obtaining a Test Statistic at Least as Extreme as the Observed Value, Assuming no Relationship Exists
The probability of observing an F-value as large as 36.28, or even larger, if the null hypothesis were true, is very low (\(p < 0.05\)).
Subsection 10.8.4 Steps 4 & 5: If the P-Value is Very Small, Typically Less Than 5%, Reject the Null Hypothesis, but if the P-Value is Not Small, Typically 5% or Greater, Retain the Null Hypothesis
Given the small p-value, we can reject the null hypothesis in favor of the alternative hypothesis that a model including low self-control is better at explaining risky lifestyles than a baseline model using the mean value of risky lifestyles.
Subsection 10.8.5 Reporting the Model Significance for the Simple Linear Regression Model
You can add the results regarding the model significance when reporting the results from simple linear regression. Our model significantly outperformed the baseline model (which used the mean of risky lifestyles) in explaining risky lifestyles (\(F(1, 941) = 36.28\text{;}\) \(p < .05\)).
