8.5 Binary predictors.
Does gender also play a roll in FEV? Let’s look at the separate effects of height and age on FEV1, and visualize how gender plays a roll.
ht.plot <- ggplot(fev_long, aes(x=ht, y=fev1)) +
geom_point(aes(col=gender)) +
geom_smooth(se=FALSE, aes(col=gender), method="lm") +
geom_smooth(se=FALSE, col="red", method="lm") +
scale_color_viridis_d() +
theme(legend.position = c(0.15,0.85))
age.plot <- ggplot(fev_long, aes(x=age, y=fev1)) +
geom_point(aes(col=gender)) +
geom_smooth(se=FALSE, aes(col=gender), method="lm") +
geom_smooth(se=FALSE, col="red", method="lm") +
scale_color_viridis_d(guide=FALSE)
grid.arrange(ht.plot, age.plot, ncol=2)
- The points are colored by gender
- Each gender has it’s own best fit line in the same color as the points
- The red line is the best fit line overall - ignoring gender
Is gender a moderator for either height or age?
Let’s compare the models with, and without gender
| Dependent variable: | ||
| fev1 | ||
| W/o gender | w/ gender | |
| age | -0.02*** (-0.03, -0.01) | -0.02*** (-0.03, -0.02) |
| ht | 0.16*** (0.15, 0.18) | 0.11*** (0.08, 0.13) |
| genderF | -0.64*** (-0.79, -0.48) | |
| Constant | -6.74*** (-7.84, -5.63) | -2.24*** (-3.71, -0.77) |
| Observations | 300 | 300 |
| Adjusted R2 | 0.57 | 0.65 |
| Residual Std. Error | 0.53 (df = 297) | 0.48 (df = 296) |
| F Statistic | 197.57*** (df = 2; 297) | 182.77*** (df = 3; 296) |
| Note: | p<0.1; p<0.05; p<0.01 | |
- Gender is a binary categorical variable, with reference group “Male”.
- This is detected because the variable that shows up in the regression model output is
genderF. So the estimate shown is for males, compared to females. - More details on how categorical variables are included in multivariable models is covered in section 8.6.
- This is detected because the variable that shows up in the regression model output is
Interpretation of Coefficients
The regression equation for the model without gender is
\[ y = -6.74 - 0.02 age + 0.16 height \]
- \(b_{0}:\) For someone who is 0 years old and 0 cm tall, their FEV is -6.74L.
- \(b_{1}:\) For every additional year older an individual is, their FEV1 decreases by 0.02L.
- \(b_{2}:\) For every additional cm taller an individual is, their FEV1 increases by 0.16L.
The regression equation for the model with gender is
\[ y = -2.24 - 0.02 age + 0.11 height - 0.64genderF \]
- \(b_{0}:\) For a male who is 0 years old and 0 cm tall, their FEV is -2.24L.
- \(b_{1}:\) For every additional year older an individual is, their FEV1 decreases by 0.02L.
- \(b_{2}:\) For every additional cm taller an individual is, their FEV1 increases by 0.16L.
- \(b_{3}:\) Females have 0.64L lower FEV compared to males.
Note: The interpretation of categorical variables still falls under the template language of “for every one unit increase in \(X_{p}\), \(Y\) changes by \(b_{p}\)”. Here, \(X_{3}=0\) for males, and 1 for females. So a 1 “unit” change is females compared to males.
Which model fits better? What measure are you using to quanitify “fit”?