Treatment vs. effect contrasts

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)

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)
(Intercept)	1834.702440	1	72.1670347	0.0000000
gender	6014.888596	1	236.5924110	0.0000000
x1	6.660056	1	0.2619697	0.6088269
x2	2.943411	1	0.1157775	0.7336959
gender:x1	1889.686051	1	74.3297855	0.0000000
gender:x2	2352.201418	1	92.5225789	0.0000000
x1:x2	5.937356	1	0.2335427	0.6289624
gender:x1:x2	235.982758	1	9.2822550	0.0023442
Residuals	50642.613736	1992	NA	NA

fit2  |> car::Anova(type = "III") |> wmisc::nice_table()

	Sum Sq	Df	F value	Pr(>F)
(Intercept)	1318.0715	1	51.845636	0.0000000
gender	6014.8886	1	236.592411	0.0000000
x1	1548.5771	1	60.912449	0.0000000
x2	2085.6571	1	82.038201	0.0000000
gender:x1	1889.6861	1	74.329785	0.0000000
gender:x2	2352.2014	1	92.522579	0.0000000
x1:x2	112.2512	1	4.415342	0.0357425
gender:x1:x2	235.9828	1	9.282255	0.0023442
Residuals	50642.6137	1992	NA	NA