Treatment vs. effect contrasts

regression
statistics
contrasts
Author

Jürgen Wilbert

Published

May 17, 2023

Abstract
Here is a simple example to show the differences between treatment and effect contrasts.

Example dataset

Create a random dataset with criteria y, predictors x1, x2 and gender.

  • y and gender are correlated
  • y and x1 are correlated only if gender is 1
  • y and x2 are correlated only if gender is 1
  • x1 and x2 are correlated
  • x1 and x2 have an interaction effect on y only if gender is 1
set.seed(1234)
n <- 2000
gender <- rep(0:1, each = n/2)
y <- sample(0:10, n, replace = TRUE) + gender * sample(0:10, n, replace = TRUE)
x1 <- sample(0:10, n, replace = TRUE) + gender * y
x2 <- x1 + sample(0:10, n, replace = TRUE) + gender * y
y <- y + (x1 > median(x1) & x2 > median(x2) & gender == 1) * sample(0:10, n, replace = TRUE) * 2

dat <- data.frame(y = y, x1 = x1, x2 = x2, gender = gender)

Descriptives

wmisc::nice_corrmatrix(dat, type = "html", numbered_columns = FALSE)
Table
Correlation matrix
Variable n M SD y x1 x2 gender
y 2000 11.67 9.83 －
x1 2000 10.09 6.80 .79*** －
x2 2000 20.30 12.88 .82*** .95*** －
gender 2000 0.50 0.50 .68*** .74*** .79*** －
Note. ✞p < .10; p < .05; p < .01; p < .001. .

Contrasts

The left part of the table is with gender as treatment contrast (0 vs. 1) and the right part with gender as effect contrast (-1 vs. 1)

# Gender has values 0 vs. 1 (treatment contrast)
fit1 <- lm(y ~ gender * x1 * x2, data = dat)

# Gender hast -1 vs. 1 (effect contrast)
dat$gender <- car::recode(dat$gender, "0 = -1; 1 = 1")

fit2 <- lm(y ~ gender * x1 * x2, data = dat)

sjPlot::tab_model(fit1, fit2, show.se = TRUE, show.ci = FALSE, col.order = c("est", "se", "std.est", "p"), digits = 4, dv.labels = c("Treatment contrast<br> for gender", "Effect contrast<br> for gender"))
  Treatment contrast
for gender		 Effect contrast
for gender
Predictors Estimates std. Error p Estimates std. Error p
(Intercept) 5.2577 0.6189 <0.001 -4.6274 0.6427 <0.001
gender -19.7703 1.2853 <0.001 -9.8852 0.6427 <0.001
x1 -0.0695 0.1358 0.609 0.6640 0.0851 <0.001
x2 -0.0270 0.0795 0.734 0.4363 0.0482 <0.001
gender × x1 1.4671 0.1702 <0.001 0.7335 0.0851 <0.001
gender × x2 0.9267 0.0963 <0.001 0.4633 0.0482 <0.001
x1 × x2 0.0056 0.0116 0.629 -0.0124 0.0059 0.036
(gender × x1) × x2 -0.0360 0.0118 0.002 -0.0180 0.0059 0.002
Observations 2000 2000
R2 / R2 adjusted 0.738 / 0.737 0.738 / 0.737
  • The intercept in model1 (treatment contrast) is the mean of y for gender 0
  • The intercept in model2 (effect contrast) is the mean of y for all data
  • All predictors in model1 without a gender term are effects for gender 0 while the interactions with a gender term are effects for gender 1
  • All predictors in model2 without a gender term are effects across gender while the interactions with a gender term are the effects of gender (subtracted for gender 0 and added for gender 1).
  • gender has a significant effect on y in both models
  • x1, x2 and x1*x2 only have a significant effect on y in model 2
  • All gender interactions are significant in both models where the effect sizes and the standard errors of model2 are half of the corresponding values in model1, so p is identical.

For those who love Anovas ;-)

fit1  |> car::Anova(type = "III") |> wmisc::nice_table()
Sum Sq Df F value Pr(>F)
1834.702440 1 72.1670347 3.810176e-17
6014.888596 1 236.5924110 1.552720e-50
6.660056 1 0.2619697 6.088269e-01
2.943411 1 0.1157775 7.336959e-01
1889.686051 1 74.3297855 1.324176e-17
2352.201419 1 92.5225789 1.930416e-21
5.937356 1 0.2335427 6.289624e-01
235.982758 1 9.2822550 2.344213e-03
50642.613735 1992 NA NA 
fit2  |> car::Anova(type = "III") |> wmisc::nice_table()
Sum Sq Df F value Pr(>F)
1318.0715 1 51.845636 8.472011e-13
6014.8886 1 236.592411 1.552720e-50
1548.5771 1 60.912449 9.559850e-15
2085.6571 1 82.038201 3.098825e-19
1889.6861 1 74.329785 1.324176e-17
2352.2014 1 92.522579 1.930416e-21
112.2512 1 4.415342 3.574248e-02
235.9828 1 9.282255 2.344213e-03
50642.6137 1992 NA NA