# 95% confidence interval for the difference between two values from a categorical variable given some r output

The r output I would get is just the summary of a fitted model. For example:

fitmodel = lm(formula = response ~ categorical + predictor)
summary(fitmodel)


I'm also given a $t_{0.25}$ value.

The categorical variable has 2 possible values ("yes" and "no"). So, from the output, how do I find (by hand) a 95% confidence interval for the difference between "yes" and "no"? In particular, how would I find the stand error for the difference?

What I know:

• I know how to find the coefficients from the output
• I know the general formula for the confidence interval

Let's take a toy example in R:

set.seed(135)
fEffect <- sample(c(0,1), size = 15, replace = TRUE)
rEffect <- rnorm(15, mean = 3)
y <- 2 * fEffect + 0.4 * rEffect + rnorm(15)

summary(lm(y ~ fEffect + rEffect))


(I set the seed for replicability.) The output from this linear model is:

Call:
lm(formula = y ~ fEffect + rEffect)

Residuals:
Min       1Q   Median       3Q      Max
-0.98166 -0.83153 -0.08039  0.75780  1.27464

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.2176     1.1684  -0.186  0.85540
fEffect       2.0093     0.4751   4.229  0.00117 **
rEffect       0.5156     0.3496   1.475  0.16605
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.8564 on 12 degrees of freedom
Multiple R-squared:  0.6533,    Adjusted R-squared:  0.5956
F-statistic: 11.31 on 2 and 12 DF,  p-value: 0.001736


As you can see from this output, the estimate for the fixed effect is 2.0093, and the standard error of the fixed effect is 0.4751, the Student's $t$ statistic for $\alpha = 0.25$ on 12 degrees of freedom is 2.179. Thus, the confidence interval for this estimate is

$$\beta_1 \in 2.0093 \pm 2.179 \times 0.4751 = (0.9741,3.0445).$$

As we can see, this interval does not contain 0, so we reject the null claim that $\beta_1 = 0$. Furthermore, while we know that the response, $y$, is in fact related to the random effect (because we created it to be so), the confidence interval for that estimate does contain 0:

$$\beta_2 \in 0.5156 \pm 2.179 \times 0.3496 = (-0.2462,1.2774).$$

We therefore do not have sufficient evidence to reject the claim that $\beta_2 = 0$.

If you are interested in finding out how that standard error value is calculated, that question was answered previously in this Cross Validated question.

• Is fEffect supposed to be the categorical variable? – sucksatmath Dec 7 '17 at 21:02
• All right, I'm a little confused. Why do you need to measure the difference between two $\beta$ estimates? – Gabriel J. Odom Dec 7 '17 at 22:10
• Yes, fEffect is the fixed effect. It is a 0 - 1 indicator. – Gabriel J. Odom Dec 7 '17 at 22:11
• I just need to know how to do it for a written test in school. – sucksatmath Dec 7 '17 at 22:49
• A confidence interval for the difference of two regression slopes doesn't make any sense. Here's why: the estimate of a regression coefficient has a Student's $t$ distribution (see this). There is not, to my knowledge, a closed form of the difference of two random variables from a Student's $t$ distribution (see this). – Gabriel J. Odom Dec 7 '17 at 23:24