Using the lme4 package for mixed effect models in R, I am trying to figure out what is the difference in modelling a one-way ANOVA within subject and a one-way ANOVA between subject.

Suppose first, that each subject see all three treatments (A, B and C). In each treatment, each subject gives me one measure (DV). I can model this within-subject design as follows:

`lmer(DV ~ treatment + (1|subject), data = My_Data) `

Now, suppose each subject sees only one treatment. What is the corresponding model? Would it be the same? If yes, how will lmer() know that it is a between subject design?

**Contents**hide

#### Best Answer

The model formula will be the same in both cases. lmer knows whether the factor is within or between.

We can see this with a simple simulation. First, simulate a within-subject design:

`> set.seed(15) > dt.se <- expand.grid(subject = 1:10, treatmentWithin = as.factor(c("A", "B", "C"))) > > dt.se$DV <- dt.se$subject * as.numeric(dt.se$treatmentWithin) + rnorm(nrow(dt.se), 0, 1) > > xtabs(~ treatmentWithin + subject, dt.se) subject treatmentWithin 1 2 3 4 5 6 7 8 9 10 A 1 1 1 1 1 1 1 1 1 1 B 1 1 1 1 1 1 1 1 1 1 C 1 1 1 1 1 1 1 1 1 1 `

So we can see that each subject receives each treatment once. Then we fit the model:

`> summary(lmm1 <- lmer(DV ~ treatmentWithin + (1|subject), data = dt.se)) Linear mixed model fit by REML ['lmerMod'] Formula: DV ~ treatmentWithin + (1 | subject) Data: dt.se REML criterion at convergence: 167.3 Scaled residuals: Min 1Q Median 3Q Max -1.5772 -0.4341 0.0261 0.4912 1.9308 Random effects: Groups Name Variance Std.Dev. subject (Intercept) 29.8 5.46 Residual 10.5 3.24 Number of obs: 30, groups: subject, 10 Fixed effects: Estimate Std. Error t value (Intercept) 5.68 2.01 2.83 treatmentWithinB 5.24 1.45 3.61 treatmentWithinC 11.29 1.45 7.78 `

Now we create a new treatment variable for a between-subject design:

`> dt.se$treatmentBetween <- as.factor(rep(c(rep(c("A", "B", "C"), 3), "A"), 3)) > xtabs(~ treatmentBetween + subject, dt.se) subject treatmentBetween 1 2 3 4 5 6 7 8 9 10 A 3 0 0 3 0 0 3 0 0 3 B 0 3 0 0 3 0 0 3 0 0 C 0 0 3 0 0 3 0 0 3 0 `

Now we see that each subject receives only one treatment – 3 times each. So now we can fit the model with the same formula:

`> summary(lmm2 <- lmer(DV ~ treatmentBetween + (1|subject), data = dt.se)) Linear mixed model fit by REML ['lmerMod'] Formula: DV ~ treatmentBetween + (1 | subject) Data: dt.se REML criterion at convergence: 191.9 Scaled residuals: Min 1Q Median 3Q Max -1.2516 -0.6507 -0.0686 0.3562 2.1658 Random effects: Groups Name Variance Std.Dev. subject (Intercept) 28.8 5.36 Residual 41.4 6.43 Number of obs: 30, groups: subject, 10 Fixed effects: Estimate Std. Error t value (Intercept) 11.146 3.262 3.42 treatmentBetweenB -0.448 4.982 -0.09 treatmentBetweenC 0.590 4.982 0.12 `

### Similar Posts:

- Solved – Using mixed models for multiple observations per subject and per period
- Solved – Does the blocking factor confound the fixed effect in the model
- Solved – How is correlation between fixed intercept and fixed effect calculated in lmer of R, lme4
- Solved – R lme4 interpret MLM confidence intervals
- Solved – Can a variable be included in a mixed model as a fixed effect and as a random effect at the same time