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EDIT 2: I originally thought I needed to run a two-factor ANOVA with repeated measures on one factor, but I now think a linear mixed-effect model will work better for my data. I think I nearly know what needs to happen, but am still confused by few points.

The experiments I need to analyze look like this:

  • Subjects were assigned to one of several treatment groups
  • Measurements of each subject were taken on multiple days
  • So:
    • Subject is nested within treatment
    • Treatment is crossed with day

(each subject is assigned to only one treatment, and measurements are taken on each subject on each day)

My dataset contains the following information:

  • Subject = blocking factor (random factor)
  • Day = within subject or repeated measures factor (fixed factor)
  • Treatment = between subject factor (fixed factor)
  • Obs = measured (dependent) variable

UPDATE OK, so I went and talked to a statistician, but he's an SAS user. He thinks that the model should be:

Treatment + Day + Subject(Treatment) + Day*Subject(Treatment)

Obviously his notation is different from the R syntax, but this model is supposed to account for:

  • Treatment (fixed)
  • Day (fixed)
  • the Treatment*Day interaction
  • Subject nested within Treatment (random)
  • Day crossed with "Subject within Treatment" (random)

So, is this the correct syntax to use?

m4 <- lmer(Obs~Treatment*Day + (1+Treatment/Subject) + (1+Day*Treatment/Subject), mydata)

I'm particularly concerned about whether the Day crossed with "Subject within Treatment" part is right. Is anyone familiar with SAS, or confident that they understand what's going on in his model, able to comment on whether my sad attempt at R syntax matches?

Here are my previous attempts at building a model and writing syntax (discussed in answers & comments):

m1 <- lmer(Obs ~ Treatment * Day + (1 | Subject), mydata)

How do I deal with the fact that subject is nested within treatment? How does m1 differ from:

m2 <- lmer(Obs ~ Treatment * Day + (Treatment|Subject), mydata)
m3 <- lmer(Obs ~ Treatment * Day + (Treatment:Subject), mydata)

and are m2 and m3 equivalent (and if not, why)?

Also, do I need to be using nlme instead of lme4 if I want to specify the correlation structure (like correlation = corAR1)? According to Repeated Measures, for a repeated-measures analysis with repeated measures on one factor, the covariance structure (the nature of the correlations between measurements of the same subject) is important.

When I was trying to do a repeated-measures ANOVA, I'd decided to use a Type II SS; is this still relevant, and if so, how do I go about specifying that?

Here's an example of what the data look like:

mydata <- data.frame(
  Subject  = c(13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 
               34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65, 13, 14, 15, 16, 17, 18, 
               19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 
               40, 62, 63, 64, 65, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 
               29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65), 
  Day       = c(rep(c("Day1", "Day3", "Day6"), each=28)), 
  Treatment = c(rep(c("B", "A", "C", "B", "C", "A", "A", "B", "A", "C", "B", "C", 
                      "A", "A", "B", "A", "C", "B", "C", "A", "A"), each = 4)), 
  Obs       = c(6.472687, 7.017110, 6.200715, 6.613928, 6.829968, 7.387583, 7.367293, 
                8.018853, 7.527408, 6.746739, 7.296910, 6.983360, 6.816621, 6.571689, 
                5.911261, 6.954988, 7.624122, 7.669865, 7.676225, 7.263593, 7.704737, 
                7.328716, 7.295610, 5.964180, 6.880814, 6.926342, 6.926342, 7.562293, 
                6.677607, 7.023526, 6.441864, 7.020875, 7.478931, 7.495336, 7.427709, 
                7.633020, 7.382091, 7.359731, 7.285889, 7.496863, 6.632403, 6.171196, 
                6.306012, 7.253833, 7.594852, 6.915225, 7.220147, 7.298227, 7.573612, 
                7.366550, 7.560513, 7.289078, 7.287802, 7.155336, 7.394452, 7.465383, 
                6.976048, 7.222966, 6.584153, 7.013223, 7.569905, 7.459185, 7.504068, 
                7.801867, 7.598728, 7.475841, 7.511873, 7.518384, 6.618589, 5.854754, 
                6.125749, 6.962720, 7.540600, 7.379861, 7.344189, 7.362815, 7.805802, 
                7.764172, 7.789844, 7.616437, NA, NA, NA, NA))
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I think that your approach is correct. Model m1 specifies a separate intercept for each subject. Model m2 adds a separate slope for each subject. Your slope is across days as subjects only participate in one treatment group. If you write model m2 as follows it's more obvious that you model a separate intercept and slope for each subject

m2 <- lmer(Obs ~ Treatment * Day + (1+Day|Subject), mydata)

This is equivalent to:

m2 <- lmer(Obs ~ Treatment + Day + Treatment:Day + (1+Day|Subject), mydata)

I.e. the main effects of treatment, day and the interaction between the two.

I think that you don't need to worry about nesting as long as you don't repeat subject ID's within treatment groups. Which model is correct, really depends on your research question. Is there reason to believe that subjects' slopes vary in addition to the treatment effect? You could run both models and compare them with anova(m1,m2) to see if the data supports either one.

I'm not sure what you want to express with model m3? The nesting syntax uses a /, e.g. (1|group/subgroup).

I don't think that you need to worry about autocorrelation with such a small number of time points.

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  • $\begingroup$ This is not correct. Treatment is a level-2 variable, it cannot be nested within Subjects. $\endgroup$ – Patrick Coulombe May 15 '13 at 2:41
  • $\begingroup$ About autocorrelation and the number of time points: I only show three in this example data, but my real data contain observations on 8 different days, so I think it probably will be an issue. Any ideas how to put that in? $\endgroup$ – phosphorelated May 15 '13 at 16:37
  • 1
    $\begingroup$ Also, I'm now quite confused about nesting; is (1+Treatment|Subject) different from (1+Treatment/Subject)? What does the "|" mean, in plain English? I don't understand the explanations I've read. $\endgroup$ – phosphorelated May 15 '13 at 16:41
  • $\begingroup$ Hi. What's here a "separate slope for each subject"? because subject is a factor variable, not a continuous variable. $\endgroup$ – skan Feb 22 '16 at 20:25
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I don't feel comfortable enough to comment on your autocorrelated errors issue (nor about the different implementations in lme4 vs. nlme), but I can speak to the rest.

Your model m1 is a random-intercept model, where you have included the cross-level interaction between Treatment and Day (the effect of Day is allowed to vary between Treatment groups). In order to allow for the change over time to differ across participants (i.e. to explicitly model individual differences in change over time), you also need to allow for the effect of Day to be random. To do this, you would specify:

m2 <- lmer(Obs ~ Day + Treatment + Day:Treatment + (Day | Subject), mydata)

In this model:

  • The intercept if the predicted score for the treatment reference category at Day=0
  • The coefficient for Day is the predicted change over time for each 1-unit increase in days for the treatment reference category
  • The coefficients for the two dummy codes for the treatment groups (automatically created by R) are the predicted difference between each remaining treatment group and the reference category at Day=0
  • The coefficients for the two interaction terms are the difference in the effect of time (Day) on predicted scores between the reference category and the remaining treatment groups.

Both the intercepts and the effect of Day on score are random (each subject is allowed to have a different predicted score at Day=0 and a different linear change over time). The covariance between intercepts and slopes is also being modeled (they are allowed to covary).

As you can see, the interpretation of the coefficients for the two dummy variables are conditional on Day=0. They will tell you if the predicted score at Day=0 for the reference category is significantly different from the two remaining treatment groups. Therefore, where you decide to center your Day variable is important. If you center at Day 1, then the coefficients tell you whether the predicted score for the reference category at Day 1 is significantly different from the predicted score of the two remaining groups. This way, you could see if there are pre-existing differences between the groups. If you center at Day 3, then the coefficients tell you whether the predicted score for the reference category at Day 3 is significantly different from the predicted score of the two remaining groups. This way, you could see if there are differences between the groups at the end of the intervention.

Finally, note that Subjects are not nested within Treatment. Your three treatments are not random levels of a population of levels to which you want to generalize your results--rather, as you mentioned, your levels are fixed, and you want to generalize your results to these levels only. (Not to mention, you shouldn't use multilevel modeling if you have only 3 upper-level units; see Maas & Hox, 2005.) Instead, treatment is a level-2 predictor, i.e. a predictor which takes a single value across Days (level-1 units) for each subject. Therefore, it is merely included as a predictor in your model.

Reference:
Maas, C. J. M., & Hox, J. J. (2005). Sufficient sample sizes for multilevel modeling. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences, 1, 86-92.

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  • 1
    $\begingroup$ It is not estimable by lmer because the number of obs <= number random effects and residual variance are probably unidentifiable. $\endgroup$ – Shuguang Mar 13 '15 at 22:09
  • $\begingroup$ The formula structure in the answer is correct. In order to override the error mentioned by @Shuguang, you would need to add ...,control=lmerControl(check.nobs.vs.nRE="ignore"). see this link for further explanation by Ben Bolker. $\endgroup$ – NiuBiBang Sep 21 '15 at 2:05
  • $\begingroup$ Nice explanations. Could you please explain a little bit more why "Subjects are not nested within Treatment" and why you don't create a + (Treatment | Subject) error term and why not just (1 | Subject) or even (1|Treatment*Day)? $\endgroup$ – skan Feb 22 '16 at 20:35
  • $\begingroup$ Technically you could nest subjects within treatment, however if the predictor is one that would be the same no matter how many times you ran the experiment, it should be a fixed (not a random) effect. Factors that would be different every time you ran the experiment, such as individual characteristics of the subject - e.g. their starting value or their idiosyncratic response to changes in the treatment over time - are random effects.(1 + Day|Subject) means a random slopes model, which allows each subject's initial value (Intercept) and rate of change on the outcome to be different. $\endgroup$ – llewmills Feb 22 at 3:05

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