B1.3 Univariable Linear Regression

Linear regression with continuous exposure

In section B1.1 we examined the association between two continuous variables (SBP and BMI [continuous]) using scatterplots and correlations. We can also assess the relationship between two continuous variables by using linear regression. In comparison with the regression using a categorical exposure, if we have a continuous variable we interpret the beta coefficient as the “average increase in [outcome] for a 1 unit change in [exposure]’.

We can fit a simple linear regression model to the data using the lm() function as lm(formula, data)and the fitted model can be saved to an object for further analysis. For example my.fit <- lm(Y ~ X, data = my.data) performs a linear regression of Y (response variable) versus X (explanatory variable) by taking them from the dataset my.data. The result is stored on an object called my.fit.

Other different functional forms for the relationship can be specified via formula. The data argument is used to tell R where to look for the variables used in the formula.

When the model is saved in an object we can use the summary() function to extract information about the linear model (estimates of parameters, standard errors, residuals, etc). The function confint() can be used to compute confidence intervals of the estimates in the fitted model.

Let us now fit a linear regression model with SBP as our response variable, and continuous BMI as our explanatory variable.

> fit1 <- lm(sbp ~ bmi, data=white.data)

> summary(fit1)

Question 1.3a: Fit a linear model of SBP versus BMI (continuous) in the Whitehall data and interpret your findings.

Linear regression with categorical exposure

Instead of conducting multiple pairwise comparisons in an ANOVA, we could run a linear regression to compare the mean difference in each age group compared to a reference category.

First, we need to tell R to treat our exposure as a categorical variable:

white.data$bmi_fact<-factor(white.data$bmi_grp4)

We can use is.factor() to check that the new created variable is indeed a factor variable.

is.factor(white.data$bmi_fact)

## [1] TRUE

Then, we use table() to count the number of participants in each BMI category.

table(white.data$bmi_fact)

##

##    1    2    3    4

##   50 1793 2091  376

We can use aggregate() to compute summary statistics for SBP by BMI.

aggregate(white.data$sbp, list(white.data$bmi_fact), FUN=mean, na.rm=TRUE)
##   Group.1        x
## 1       1 126.6600
## 2       2 129.5695
## 3       3 131.3661
## 4       4 133.2394

The results indicate that there are mean SBP increases across the four categories of BMI.

Now we fit a linear regression model called fit2 with SBP as our response and BMI groups as our explanatory variable.

fit2 <- lm(sbp ~ bmi_fact, data=white.data)
summary(fit2)

Call:
lm(formula = sbp ~ bmi_fact, data = white.data)

Residuals:
    Min      1Q  Median      3Q     Max 
-43.366 -12.239  -1.570   9.634 100.430 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  126.660      2.474  51.187   <2e-16 ***
bmi_fact2      2.910      2.509   1.160   0.2462    
bmi_fact3      4.706      2.504   1.879   0.0602 .  
bmi_fact4      6.579      2.634   2.498   0.0125 *  

If we want confidence intervals, we could then type:

confint(fit2, level=0.95)

By default R will use the lowest category of bmi_fact as the reference in a regression. But this category is the smallest and hence it is not a good reference category as all comparisons with it will have large confidence intervals. We can change the reference category using the ‘relevel’ function:
white.data<-within(white.data,bmi_fact<-relevel(bmi_fact,ref=2))

Then we fit the same linear regression again:
> fit2 <- lm(sbp ~ bmi_fact, data=white.data)
> summary(fit2)
> confint(fit2, level=0.95)

Question 1.3b: Using the new reference category of BMI group=2, what conclusions do you make about the relationship between SBP and BMI groups based on the regression output given by summary(fit2)?