## 8.4 Example

The analysis in example ?? concluded that FEV1 in fathers significantly increases by 0.12 (95% CI:0.09, 0.15) liters per additional inch in height (p<.0001). Looking at the multiple $$R^{2}$$ (correlation of determination), this simple model explains 25% of the variance seen in the outcome $$y$$.

However, FEV tends to decrease with age for adults, so we should be able to predict it better if we use both height and age as independent variables in a multiple regression equation.

• What direction do you expect the slope coefficient for age to be? For height?

Fitting a regression model in R with more than 1 predictor is done by adding each variable to the right hand side of the model notation connected with a +.

mv_model <- lm(FFEV1 ~ FAGE + FHEIGHT, data=fev)
summary(mv_model)
##
## Call:
## lm(formula = FFEV1 ~ FAGE + FHEIGHT, data = fev)
##
## Residuals:
##      Min       1Q   Median       3Q      Max
## -1.34708 -0.34142  0.00917  0.37174  1.41853
##
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.760747   1.137746  -2.427   0.0165 *
## FAGE        -0.026639   0.006369  -4.183 4.93e-05 ***
## FHEIGHT      0.114397   0.015789   7.245 2.25e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5348 on 147 degrees of freedom
## Multiple R-squared:  0.3337, Adjusted R-squared:  0.3247
## F-statistic: 36.81 on 2 and 147 DF,  p-value: 1.094e-13
confint(mv_model)
##                   2.5 %      97.5 %
## (Intercept) -5.00919751 -0.51229620
## FAGE        -0.03922545 -0.01405323
## FHEIGHT      0.08319434  0.14559974

Holding height constant, a father who is one year older is expected to have a FEV value 0.03 (0.01, 0.04) liters less than another man (p<.0001).

Holding age constant, a father who is 1cm taller than another man is expected to have a FEV value of 0.11 (.08, 0.15) liter greater than the other man (p<.0001).

For the model that includes age, the coefficient for height is now 0.11, which is interpreted as the rate of change of FEV1 as a function of height after adjusting for age. This is also called the partial regression coefficient of FEV1 on height after adjusting for age.

Both height and age are significantly associated with FEV in fathers (p<.0001 each).