Section 10.9 Conducting Multiple Linear Regression
Multiple linear regression involves incorporating multiple independent variables to predict the dependent variable. In the context of predicting risky lifestyles among inmates, it’s clear that factors beyond just low self-control may play a role. For instance, age could be a significant demographic factor, as younger individuals might be more inclined to engage in risky behaviors compared to older inmates. Therefore, multiple linear regression is better suited for real-life scenarios where multiple factors influence dependent variables. All you need to do is to tweak the R codes that we used to perform a simple linear regression.
rl_by_lsc_age <- lm(formula = RL ~ LSC + AGE, data = Inmate_Survey, na.action = na.exclude)
summary(object = rl_by_lsc_age)
As you can see, even after including age in our regression model, low self-control remained statistically significant. Low self-control was positively and significantly associated with risky lifestyles (\(b = 0.09\text{;}\) \(t = 5.51\text{;}\) \(p < .05\)). Age was also a significant predictor of risky lifestyles (\(b = -0.01\text{;}\) \(t = -2.30\text{;}\) \(p < .05\)). Age was negatively and significantly associated with risky lifestyles.
Subsection 10.9.1 Model Fit for Linear Regression
In the outputs for both simple linear regression and multiple linear regression, you may have observed multiple R-squared and adjusted R-squared values located just above the F-statistic. These statistics serve to evaluate the overall goodness of fit of the regression model. R-squared (aka the coefficient of determination) is calculated by determining the proportion of variance in the dependent variable that can be explained by the independent variables incorporated in the model. It ranges from 0 to 1, with 0 signifying that the independent variables account for none of the variance in the dependent variable, and 1 indicating that they explain all of the variance. In our multiple linear regression, for instance, the R-squared value is 0.04252. To determine the percentage of variance explained by the model, multiply this value by 100. Therefore, 4.25% of the variance in risky lifestyles is explained by both low self-control and age.
Now, what is adjusted R-squared? As additional variables are added to the model, the R-squared value tends to increase. Adjusted R-squared serves to counteract this tendency by slightly penalizing the R-squared value for each additional variable introduced into the model. This adjustment ensures that the measure appropriately accounts for the complexity of the model and prevents overestimation of its explanatory power.
