Repeated measures analysis: why nest experimental factors within subject factor?

Question

Consider a pure repeated measures design, with (let's say) 3 experimental within-subject factors A, B, and C, and (for simplicity) 2 levels per factor. So we have 2*2*2 = 8 measurements per subject.

Now I would like to test the fixed effects with a linear mixed effects model. I have read in several sources (for example Andy Field's Book "Discovering Statistics using R", and this site: http://www.jason-french.com/tutorials/repeatedmeasures.html ) that with lme, one should use the following syntax:

model <- lme(dv ~ A*B*C, random = ~1|id/A/B/C)

However, I do not understand why you would "nest" the factors within the subject in the random part of the model, and not just use (1|id). What is the point of this, and what does it do?

Conceptually, I don't understand why one would nest the experimental fixed factors within the random subject factor. The way I understood nesting until now, you would only use it to account for the fact that certain lower factor levels only exist within certain higher factor levels - like pupils within classes within schools within cities, etc. How does this apply to a repeated measures design with fully crossed within-subject factors?

Mathematically, the way I understood this is that such a model would first estimate a random intercept for each subject, capturing random differences in the average values of the dependent variable between subjects. So in the case of, (let's say) 20 subjects, we get 20 different random intercepts. Then, apparently, the model estimates random intercepts for each combination of subject and level of factor A (resulting in 40 random intercepts), then for each combination of subject, factor A and factor B (80 random intercepts), all the way down to the most specific level, where we get as many estimated random intercepts as we have measurements (160). What is the point of this, and why would we not only estimate a random intercept per subject (1|subject)? Also, wouldn't all of these random intercepts together explain the dependent variable nearly perfectly, and leave little to nothing to be explained by the fixed effects?

Lastly, my intuition tells me that these random intercepts should at least partially explain the same information as would be captured by entering random slopes of the experimental factors into the model. Is that correct?

Answer 1

This is a very interesting question. I have been thinking this and related things for a long time.

For me the key to understanding this is to realise that: Random intercepts for a grouping factor are not always sufficient to capture the random variation in the data that is in excess of the residual variation. Because of this, we sometimes see models with random intecepts for interactions between a fixed factor and a grouping variable (and even sometimes random intercepts just for a fixed factor). Generally we advise that a factor can be fixed or random (intercepts) but not both - however there are important exceptions of which your example here is one.

Another hindrence to understanding this is for people like me who come from a multilevel modelling / mixed model background in observational social and medical sciences, we are often caught up in thinking about repeated measures and nesting vs crossed random effects without an understanding that things are a bit different in experimental analysis. More on this a bit later.

Judging by the comments we have both discovered the same thing. In the context of a repeated measures ANOVA, if you want to obtain the same results with lmer then you fit:

 y ~ A + B + (1|id) + (1|id:A) + (1|id:B)

where I have discarded factor C without loss of generality.

and the reason why some people specify 1|id/A/B is that they are using nmle:lme and not lme4:lmer. I am not sure why this is needed in lme() but I am fairly sure that to replicate a repeated measures anova - where there is variation for each combination of id and the factors - then you fit the model above in lmer(). Note that (1|id/A/B) seems similar, however it is wrong because it would also fit (1|id:A:B) which is indistinguishable from the residual variance (as noted in your comment also).

It is important to note (and therefore worth repeating) that we only fit this type of model where we have reason to believe that there is variation for each combination of id and the factors. Typically with mixed models we would not do this. We need to understand experimental design. One type of experiment where this is common is the so-called split-plot design where blocking has also been used. This type of experimental design employs randomisation at different "levels" - or rather different combinations of factors, and this is why analysis of such experiemnts often includes random intercept terms that at first glance seem odd. However, the random structure is a property of the experimental design and without knowledge of this, it is virtually impossible to select the correct structure.

So, with regards to the your actual question where the experiment has a repeated factorial design, we can use our friend, simulation, to investigate further.

We will simulate data for the models:

 y ~ A + B + (1|id)

and

 y ~ A + B + (1|id) + (1|id:A) + (1|id:B)

and look at what happens when we use both models to analyse both datasets.

set.seed(15)
n <- 100 # number of subjects
K <- 4 # number of measurements per subject

# set up covariates
df <- data.frame(id = rep(seq_len(n), each = K),
                 A = rep(c(0,0,1,1), times = n),
                 B = rep(c(0,1), times = n * 2)
)

# 
df$y <- df$A + 2 * df$B + 3 * df$id + rnorm(n * K)

m0 <- lmer(y ~ A + B + (1|id) , data = df)
m1 <- lmer(y ~ A + B + (1|id) + (1|id:A) + (1|id:B), data = df)
summary(m0)

m0 is the "right" model for these data because we simply created y with fixed effects for id (which we capture with random intercepts) and unit variance. This is a bit of an abuse but it is convenient and does what we want:

 Groups   Name        Variance Std.Dev.
 id       (Intercept) 842.1869 29.0205 
 Residual               0.9946  0.9973 
Number of obs: 400, groups:  id, 100

Fixed effects:
            Estimate Std. Error t value
(Intercept) 50.47508    2.90333   17.39
A            1.01277    0.09973   10.15
B            2.06675    0.09973   20.72

as we can see, we recover unit variance in the residual and good estimates for the fixed effects. However:

> summary(m1)

Random effects:
 Groups   Name        Variance  Std.Dev.
 id:B     (Intercept) 0.000e+00  0.0000 
 id:A     (Intercept) 8.724e-03  0.0934 
 id       (Intercept) 8.422e+02 29.0204 
 Residual             9.888e-01  0.9944 
Number of obs: 400, groups:  id:B, 200; id:A, 200; id, 100

Fixed effects:
            Estimate Std. Error t value
(Intercept) 50.47508    2.90334   17.39
A            1.01277    0.10031   10.10
B            2.06675    0.09944   20.78

This is a singular fit - zero estimates for the variance of the id:B term and close to zero for id:A - which we happen to know is correct here because we didn't simulate any variance for those "levels". Also we find:

> anova(m0, m1)

m0: y ~ A + B + (1 | id)
m1: y ~ A + B + (1 | id) + (1 | id:A) + (1 | id:B)
   npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)
m0    5 1952.8 1972.7 -971.39   1942.8                     
m1    7 1956.8 1984.7 -971.39   1942.8 0.0052  2     0.9974

meaning that we strongly prefer the (correct) model m0

So now we simulate data with variation at these "levels" too. Since m1 is the model we want to simlulate for, we an use it's design matrix for the random effects:

# design matrix for the random effects
Z <- as.matrix(getME(m1, "Z"))

# design matrix for the fixed effects
X <-  model.matrix(~ A + B, data = df)

betas <- c(10, 2, 3) # fixed effects coefficients
D1 <- 1 # SD of random intercepts for id
D2 <- 2 # SD of random intercepts for id:A
D3 <- 3 # SD of random intercepts for id:B

# we simulate random effects
b <- c(rnorm(n*2, sd = D3), rnorm(n*2, sd = D2), rnorm(n, sd = D1))
# the order here is goverened by the order that lme4 creates the Z matrix

# linear predictor
lp <- X %*% betas + Z %*% b

# add residual variance of 1
df$y <- lp + rnorm(n * K)

m2 <- lmer(y ~ A + B + (1|id) + (1|id:A) + (1|id:B), data = df)
m3 <- lmer(y ~ A + B + (1|id), data = df)
summary(m2)

'm2` is the corect model here and we obtain:

Random effects:
 Groups   Name        Variance Std.Dev.
 id:B     (Intercept) 6.9061   2.6279  
 id:A     (Intercept) 4.4766   2.1158  
 id       (Intercept) 2.9117   1.7064  
 Residual             0.8704   0.9329  
Number of obs: 400, groups:  id:B, 200; id:A, 200; id, 100

Fixed effects:
            Estimate Std. Error t value
(Intercept)  10.3870     0.3866  26.867
A             1.8123     0.3134   5.782
B             3.0242     0.3832   7.892

the SD for the id intercept is a little high, but otherwise we have good estimates for the random and fixed effects. On the other hand:

> summary(m3)

Random effects:
 Groups   Name        Variance Std.Dev.
 id       (Intercept) 6.712    2.591   
 Residual             8.433    2.904   
Number of obs: 400, groups:  id, 100

Fixed effects:
            Estimate Std. Error t value
(Intercept)  10.3870     0.3611  28.767
A             1.8123     0.2904   6.241
B             3.0242     0.2904  10.414

although the fixed effects point estimates are OK, their standard errors are larger. The random structure is, of course, completely wrong. And finally:

> anova(m2, m3)

Models:
m3: y ~ A + B + (1 | id)
m2: y ~ A + B + (1 | id) + (1 | id:A) + (1 | id:B)
   npar    AIC    BIC   logLik deviance Chisq Df Pr(>Chisq)    
m3    5 2138.1 2158.1 -1064.07   2128.1                        
m2    7 1985.7 2013.7  -985.87   1971.7 156.4  2  < 2.2e-16 ***

showing that we strongly prefer m2