Analysis of one group of subjects (data provided)

Question

I have one group ($n = 39$) of subjects pre- and post-tested on a continuous variable. I also have a gender variable coded $0$ and $1$.

I was wondering how I can analyze this data so to detect any changes from pre- to post-test controlling for the gender variable?

Here is my data in R:

set.seed(62)
pre = rnorm(39)
post = rnorm(39, 3)
gender = rbinom(39, 1, .6)

Answer 1

I think the most straight forward approach (keep in mind there are several ways to skin this cat) is to perform a simple regression analysis on the difference in pre- and post-test scores with a single independent dummy variable for gender.

So for example, you'd want to preprocess your data and obtain $y_{diff_i}=y_{pre_i}-y_{post_i}$ for the $i$th subject $i=1$ to $39$. Then simply fit the model:

$y_{diff_i}=\beta_0+\beta_{gender}X_{gender_i}+\epsilon_i$

If you are interested in testing for a gender effect, you'd test whether:

$H_0: \hat{\beta}_{gender}=0$ vs. $H_1: \hat{\beta}_{gender}\ne0$.

Then, assuming there's no gender effect, to determine if there was a statistically significant difference between the pre and post tests, you simply test whether:

$H_0: \hat{\beta}_0=0$ vs. $H_1: \hat{\beta}_0\ne0$.

If you do find an important gender effect, you'd want to go about determining the average increase in score for each gender separately with the use of a contrast. Assuming men are coded 1 and women are coded 0, then estimated pre-post change for men becomes:

$\hat{\beta}_0+\hat{\beta}_{gender}$

and the average pre-post test change for women is estimated by just $\hat{\beta}_0$.

So continuing with your toy example, in R this could be fit as follows:

> y_diff=pre-post
> myfit<-lm(y_diff~gender)
> summary(myfit)

Call:
lm(formula = y_diff ~ gender)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.1334 -0.9911  0.4681  1.1861  2.4989 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -3.1081     0.4209  -7.385 8.75e-09 ***
gender       -0.3158     0.5480  -0.576    0.568    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.683 on 37 degrees of freedom
Multiple R-squared:  0.008892,  Adjusted R-squared:  -0.01789 
F-statistic: 0.332 on 1 and 37 DF,  p-value: 0.568

Of course, you'll need to carry out the usual model diagnostics and such, but I think you are familiar with those details.

Other possible methods include fitting a model and adjusting for the pre-test score as well as gender:

$y_{post_i}=\beta_0+\beta_{gender}X_{gender_i}+\beta_{pre}Y_{pre_i}+\epsilon_i$. See the first comment by rolando2 for more info on a model like this.