I have some issues trying to conduct an Anova for a repeated survey.
This is my data basis: 40 participants from 3 school classes and I wanted to know how motivated the pupils were for learning a certain subject at two different points of time.
Now I wanted to conduct an ANOVA in order to find out whether there are significant differences in the motivation of these three groups (i.e. classes). However, I cannot seem to figure it out.
I've tried the following:
> anova.rm <- ezANOVA(daten.long, dv=.(motivation),
+ wid=.(id), within=.(time) )
Warnung: "time" will be treated as numeric.
Warnung: There is at least one numeric within variable, therefore aov() will be used for computation and no assumption checks will be obtained.
> anova.rm
$ANOVA
Effect DFn DFd F p p<.05 ges
1 time 1 2 3.887892 0.1873993 0.6603199
--> In this one-factored analysis the motivation is already related to the class, but it doesn't give me any information whether there are any differences between the groups if I do a post hoc-test (such as Fishers Least Significance Difference or the pairwise.t.test)
Therefore, I've tried to use the multifactorial variance analysis:
> anova.mixed <- ezANOVA(daten.long, dv=.(motivation), wid=.(id), within=.(time), between=.(class))
Warnung: "time" will be treated as numeric.
Warnung: "class" will be treated as numeric.
Warnung: There is at least one numeric within variable, therefore aov() will be used for computation and no assumption checks will be obtained.
> anova.mixed
$ANOVA
Effect DFn DFd F p p<.05 ges
1 class 1 1 0.05304433 0.8558908 0.04881692
2 time 1 1 16.05556143 0.1556988 0.34263494
3 class:time 1 1 7.25926249 0.2262510 0.19071808
Same problem for this try. I have no idea whether I'm on the right track to get to the point if there are any significant differences between my groups at the two points of time.
I'm thankful for your help and advice! :)
Related
Could someone help me to determine the correct random variable structure in my binomial GLMM in lme4?
I will first try to explain my data as best as I can. I have binomial data of seedlings that were eaten (1) or not eaten (0), together with data of vegetation cover. I try to figure out if there is a relationship between vegetation cover and the probability of a tree being eaten, as the other vegetation is a food source that could attract herbivores to a certain forest patch.
The data is collected in ~90 plots scattered over a National Park for 9 years now. Some were measured all years, some were measured only a few years (destroyed/newly added plots). The original datasets is split in 2 (deciduous vs coniferous), both containing ~55.000 entries. Per plot about 100 saplings were measured every time, so the two separate datasets probably contain about 50 trees per plot (though this will not always be the case, since the decid:conif ratio is not always equal). Each plot consists of 4 subplots.
I am aware that there might be spatial autocorrelation due to plot placement, but we will not correct for this, yet.
Every year the vegetation is surveyed in the same period. Vegetation cover is estimated at plot-level, individual trees (binary) are measured at a subplot-level.
All trees are measured, so the amount of responses per subplot will differ between subplots and years, as the forest naturally regenerates.
Unfortunately, I cannot share my original data, but I tried to create an example that captures the essentials:
#set seed for whole procedure
addTaskCallback(function(...) {set.seed(453);TRUE})
# Generate vector containing individual vegetation covers (in %)
cover1vec <- c(sample(0:100,10, replace = TRUE)) #the ',number' is amount of covers generated
# Create dataset
DT <- data.frame(
eaten = sample(c(0,1), 80, replace = TRUE),
plot = as.factor(rep(c(1:5), each = 16)),
subplot = as.factor(rep(c(1:4), each = 2)),
year = as.factor(rep(c(2012,2013), each = 8)),
cover1 = rep(cover1vec, each = 8)
)
Which will generate this dataset:
>DT
eaten plot subplot year cover1
1 0 1 1 2012 4
2 0 1 1 2012 4
3 1 1 2 2012 4
4 1 1 2 2012 4
5 0 1 3 2012 4
6 1 1 3 2012 4
7 0 1 4 2012 4
8 1 1 4 2012 4
9 1 1 1 2013 77
10 0 1 1 2013 77
11 0 1 2 2013 77
12 1 1 2 2013 77
13 1 1 3 2013 77
14 0 1 3 2013 77
15 1 1 4 2013 77
16 0 1 4 2013 77
17 0 2 1 2012 46
18 0 2 1 2012 46
19 0 2 2 2012 46
20 1 2 2 2012 46
....etc....
80 0 5 4 2013 82
Note1: to clarify again, in this example the number of responses is the same for every subplot:year combination, making the data balanced, which is not the case in the original dataset.
Note2: this example can not be run in a GLMM, as I get a singularity warning and all my random effect measurements are zero. Apparently my example is not appropriate to actually use (because using sample() caused the 0 and 1 to be in too even amounts to have large enough effects?).
As you can see from the example, cover data is the same for every plot:year combination.
Plots are measured multiple years (only 2012 and 2013 in the example), so there are repeated measures.
Additionally, a year effect is likely, given the fact that we have e.g. drier/wetter years.
First I thought about the following model structure:
library(lme4)
mod1 <- glmer(eaten ~ cover1 + (1 | year) + (1 | plot), data = DT, family = binomial)
summary(mod1)
Where (1 | year) should correct for differences between years and (1 | plot) should correct for the repeated measures.
But then I started thinking: all trees measured in plot 1, during year 2012 will be more similar to each other than when they are compared with (partially the same) trees from plot 1, during year 2013.
So, I doubt that this random model structure will correct for this within plot temporal effect.
So my best guess is to add another random variable, where this "interaction" is accounted for.
I know of two ways to possibly achieve this:
Method 1.
Adding the random variable " + (1 | year:plot)"
Method 2.
Adding the random variable " + (1 | year/plot)"
From what other people told me, I still do not know the difference between the two.
I saw that Method 2 added an extra random variable (year.1) compared to Method 1, but I do not know how to interpret that extra random variable.
As an example, I added the Random effects summary using Method 2 (zeros due to singularity issues with my example data):
Random effects:
Groups Name Variance Std.Dev.
plot.year (Intercept) 0 0
plot (Intercept) 0 0
year (Intercept) 0 0
year.1 (Intercept) 0 0
Number of obs: 80, groups: plot:year, 10; plot, 5; year, 2
Can someone explain me the actual difference between Method 1 and Method 2?
I am trying to understand what is happening, but cannot grasp it.
I already tried to get advice from a colleague and he mentioned that it is likely more appropriate to use cbind(success, failure) per plot:year combination.
Via this site I found that cbind is used in binomial models when Ntrails > 1, which I think is indeed the case given our sampling procedure.
I wonder, if cbind is already used on a plot:year combination, whether I need to add a plot:year random variable?
When using cbind, the example data would look like this:
>DT3
plot year cover1 Eaten_suc Eaten_fail
8 1 2012 4 4 4
16 1 2013 77 4 4
24 2 2012 46 2 6
32 2 2013 26 6 2
40 3 2012 91 2 6
48 3 2013 40 3 5
56 4 2012 61 5 3
64 4 2013 19 2 6
72 5 2012 19 5 3
80 5 2013 82 2 6
What would be the correct random model structure and why?
I was thinking about:
Possibility A
mod4 <- glmer(cbind(Eaten_suc, Eaten_fail) ~ cover1 + (1 | year) + (1 | plot),
data = DT3, family = binomial)
Possibility B
mod5 <- glmer(cbind(Eaten_suc, Eaten_fail) ~ cover1 + (1 | year) + (1 | plot) + (1 | year:plot),
data = DT3, family = binomial)
But doesn't cbind(success, failure) already correct for the year:plot dependence?
Possibility C
mod6 <- glmer(cbind(Eaten_suc, Eaten_fail) ~ cover1 + (1 | year) + (1 | plot) + (1 | year/plot),
data = DT3, family = binomial)
As I do not yet understand the difference between year:plot and year/plot
Thus: Is it indeed more appropriate to use the cbind-method than the raw binary data? And what random model structure would be necessary to prevent pseudoreplication and other dependencies?
Thank you in advance for your time and input!
EDIT 7/12/20: I added some extra information about the original data
You are asking quite a few questions in your question. I'll try to cover them all, but I do suggest reading the documentation and vignette from lme4 and the glmmFAQ page for more information. Also I'd highly recommend searching for these topics on google scholar, as they are fairly well covered.
I'll start somewhere simple
Note 2 (why is my model singular?)
Your model is highly singular, because the way you are simulating your data does not indicate any dependency between the data itself. If you wanted to simulate a binomial model you would use g(eta) = X %*% beta to simulate your linear predictor and thus the probability for success. One can then use this probability for simulating the your binary outcome. This would thus be a 2 step process, first using some known X or randomly simulated X given some prior distribution of our choosing. In the second step we would then use rbinom to simulate binary outcome while keeping it dependent on our predictor X.
In your example you are simulating independent X and a y where the probability is independent of X as well. Thus, when we look at the outcome y the probability of success is equal to p=c for all subgroup for some constant c.
Can someone explain me the actual difference between Method 1 and Method 2? ((1| year:plot) vs (1|year/plot))
This is explained in the package vignette fitting linear mixed effects models with lme4 in the table on page 7.
(1|year/plot) indicates that we have 2 mixed intercept effects, year and plot and plot is nested within year.
(1|year:plot) indicates a single mixed intercept effect, plot nested within year. Eg. we do not include the main effect of year. It would be somewhat similar to having a model without intercept (although less drastic, and interpretation is not destroyed).
It is more common to see the first rather than the second, but we could write the first as a function of the second (1|year) + (1|year:plot).
Thus: Is it indeed more appropriate to use the cbind-method than the raw binary data?
cbind in a formula is used for binomial data (or multivariate analysis), while for binary data we use the raw vector or 0/1 indicating success/failure, eg. aggregate binary data (similar to how we'd use glm). If you are uninterested in the random/fixed effect of subplot, you might be able to aggregate your data across plots, and then it would likely make sense. Otherwise stay with you 0/1 outcome vector indicating either success or failures.
What would be the correct random model structure and why?
This is a topic that is extremely hard to give a definitive answer to, and one that is still actively researched. Depending on your statistical paradigm opinions differ greatly.
Method 1: The classic approach
Classic mixed modelling is based upon knowledge of the data you are working with. In general there are several "rules of thumb" for choosing these parameters. I've gone through a few in my answer here. In general if you are "not interested" in the systematic effect and it can be thought of as a random sample of some population, then it could be a random effect. If it is the population, eg. samples do not change if the process is repeated, then it likely shouldn't.
This approach often yields "decent" choices for those who are new to mixed effect models, but is highly criticized by authors who tend towards methods similar to those we'd use in non-mixed models (eg. visualizing to base our choice and testing for significance).
Method 2: Using visualization
If you are able to split your data into independent subgroups and keeping the fixed effect structure a reasonable approach for checking potential random effects is the estimate marginal models (eg. using glm) across these subgroups and seeing if the fixed effects are "normally distributed" between these observations. The function lmList (in lme4) is designed for this specific approach. In linear models we would indeed expect these to be normally distributed, and thus we can get an indication whether a specific grouping "might" be a valid random effect structure. I believe the same is approximately true in the case of generalized linear models, but I lack references. I know that Ben Bolker have advocated for this approach in a prior article of his (the first reference below) that I used during my thesis. However this is only a valid approach for strictly separable data, and the implementation is not robust in the case where factor levels are not shared across all groups.
So in short: If you have the right data, this approach is simple, fast and seemingly highly reliable.
Method 3: Fitting maximal/minimal models and decreasing/expanding model based on AIC or AICc (or p-value tests or alternative metrics)
Finally an alternative to use a "step-wise"-like procedure. There are advocates of both starting with maximal and minimal models (I'm certain at least one of my references below talk about problems with both, otherwise check glmmFAQ) and then testing your random effects for their validity. Just like classic regression this is somewhat of a double-edged sword. The reason is both extremely simple to understand and amazingly complex to comprehend.
For this method to be successful you'd have to perform cross-validation or out-of-sample validation to avoid selection bias just like standard models, but unlike standard models sampling becomes complicated because:
The fixed effects are conditional on the random structure.
You will need your training and testing samples to be independent
As this is dependent on your random structure, and this is chosen in a step-wise approach it is hard to avoid information leakage in some of your models.
The only certain way to avoid problems here is to define the space
that you will be testing and selecting samples based on the most
restrictive model definition.
Next we also have problems with choice of metrics for evaluation. If one is interested in the random effects it makes sense to use AICc (AIC estimate of the conditional model) while for fixed effects it might make more sense to optimize AIC (AIC estimate of the marginal model). I'd suggest checking references to AIC and AICc on glmmFAQ, and be wary since the large-sample results for these may be uncertain outside a very reestrictive set of mixed models (namely "enough independent samples over random effects").
Another approach here is to use p-values instead of some metric for the procedure. But one should likely be even more wary of test on random effects. Even using a Bayesian approach or bootstrapping with incredibly high number of resamples sometimes these are just not very good. Again we need "enough independent samples over random effects" to ensure the accuracy.
The DHARMA provides some very interesting testing methods for mixed effects that might be better suited. While I was working in the area the author was still (seemingly) developing an article documenting the validity of their chosen method. Even if one does not use it for initial selection I can only recommend checking it out and deciding upon whether one believes in their methods. It is by far the most simple approach for a visual test with simple interpretation (eg. almost no prior knowledge is needed to interpret the plots).
A final note on this method would thus be: It is indeed an approach, but one I would personally not recommend. It requires either extreme care or the author accepting ignorance of model assumptions.
Conclusion
Mixed effect parameter selection is something that is difficult. My experience tells me that mostly a combination of method 1 and 2 are used, while method 3 seems to be used mostly by newer authors and these tend to ignore either out-of-sample error (measure model metrics based on the data used for training), ignore independence of samples problems when fitting random effects or restrict themselves to only using this method for testing fixed effect parameters. All 3 do however have some validity. I myself tend to be in the first group, and base my decision upon my "experience" within the field, rule-of-thumbs and the restrictions of my data.
Your specific problem.
Given your specific problem I would assume a mixed effect structure of (1|year/plot/subplot) would be the correct structure. If you add autoregressive (time-spatial) effects likely year disappears. The reason for this structure is that in geo-analysis and analysis of land plots the classic approach is to include an effect for each plot. If each plot can then further be indexed into subplot it is natural to think of "subplot" to be nested in "plot". Assuming you do not model autoregressive effects I would think of time as random for reasons that you already stated. Some years we'll have more dry and hotter weather than others. As the plots measured will have to be present in a given year, these would be nested in year.
This is what I'd call the maximal model and it might not be feasible depending on your amount of data. In this case I would try using (1|time) + (1|plot/subplot). If both are feasible I would compare these models, either using bootstrapping methods or approximate LRT tests.
Note: It seems not unlikely that (1|time/plot/subplot) would result in "individual level effects". Eg 1 random effect per row in your data. For reasons that I have long since forgotten (but once read) it is not plausible to have individual (also called subject-level) effects in binary mixed models. In this case It might also make sense to use the alternative approach or test whether your model assumptions are kept when withholding subplot from your random effects.
Below I've added some useful references, some of which are directly relevant to the question. In addition check out the glmmFAQ site by Ben Bolker and more.
References
Bolker, B. et al. (2009). „Generalized linear mixed models: a practical guide for ecology and evolution“. In: Trends in ecology & evolution 24.3, p. 127–135.
Bolker, B. et al. (2011). „GLMMs in action: gene-by-environment interaction in total fruit production of wild populations of Arabidopsis thaliana“. In: Revised version, part 1 1, p. 127–135.
Eager, C. og J. Roy (2017). „Mixed effects models are sometimes terrible“. In: arXiv preprint arXiv:1701.04858. url: https://arxiv.org/abs/1701.04858 (last seen 19.09.2019).
Feng, Cindy et al. (2017). „Randomized quantile residuals: an omnibus model diagnostic tool with unified reference distribution“. In: arXiv preprint arXiv:1708.08527. (last seen 19.09.2019).
Gelman, A. og Jennifer Hill (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.
Hartig, F. (2019). DHARMa: Residual Diagnostics for Hierarchical (Multi-Level / Mixed) Regression Models. R package version 0.2.4. url: http://florianhartig.github.io/DHARMa/ (last seen 19.09.2019).
Lee, Y. og J. A. Nelder (2004). „Conditional and Marginal Models: Another View“. In: Statistical Science 19.2, p. 219–238.
doi: 10.1214/088342304000000305. url: https://doi.org/10.1214/088342304000000305
Lin, D. Y. et al. (2002). „Model-checking techniques based on cumulative residuals“. In: Biometrics 58.1, p. 1–12. (last seen 19.09.2019).
Lin, X. (1997). „Variance Component Testing in Generalised Linear Models with Random Effects“. In: Biometrika 84.2, p. 309–326. issn: 00063444. url: http://www.jstor.org/stable/2337459
(last seen 19.09.2019).
Stiratelli, R. et al. (1984). „Random-effects models for serial observations with binary response“. In:
Biometrics, p. 961–971.
What is the assign attribute of a linear model fit? It's supposed to somehow provide the position of the response term, but in practice it seems to enumerate all coefficients in the model. It's my understanding that assign is a carryover from S and it's not supported by glm(). I need to extract the equivalent information for glm, but I don't understand what the implementation does for lm and can't seem to find the source code either. The help file for lm.fit says, unhelpfully:
non-null fits will have components assign, effects and (unless not requested) qr relating to the linear fit, for use by extractor functions such as summary and effects
You find this in help("model.matrix"), which creates these values:
There is an attribute "assign", an integer vector with an entry for
each column in the matrix giving the term in the formula which gave
rise to the column. Value 0 corresponds to the intercept (if any), and
positive values to terms in the order given by the term.labels
attribute of the terms structure corresponding to object.
So, it maps the design matrix to the formula.
The numbers from $assign represent the corresponding predictor variable. If your predictor is categorical with 3 levels, you will see the corresponding number (3-1) times in your $assign call. Example:
data(mpg, package = "ggplot2")
m = lm(cty ~ hwy + class,data = mpg)
m$assign
[1] 0 1 2 2 2 2 2 2
# Note how there is six 2's to represent the indicator variables
# for the various 'class' levels. (class has 7 levels)
You will see the quantitative predictors will only have one value (hwy in the example above), since they are represented by one term in the design formula.
I'm having a huge problem with a nested model I am trying to fit in R.
I have response time experiment with 2 conditions with 46 people each and 32 measures each. I would like measures to be nested within people and people nested within conditions, but I can't get it to work.
The code I thought should make sense was:
nestedmodel <- lmer(responsetime ~ 1 + condition +
(1|condition:person) + (1|person:measure), data=dat)
However, all I get is an error:
Error in checkNlevels(reTrms$flist, n = n, control) :
number of levels of each grouping factor must be < number of observations
Unfortunately, I do not even know where to start looking what the problem is here.
Any ideas? Please, please, please? =)
Cheers!
This might be more appropriate on CrossValidated, but: lme4 is trying to tell you that one or more of your random effects is confounded with the residual variance. As you've described your data, I don't quite see why: you should have 2*46*32=2944 total observations, 2*46=92 combinations of condition and person, and 46*32=1472 combinations of measure and person.
If you do
lf <- lFormula(responsetime ~ 1 + condition +
(1|condition:person) + (1|person:measure), data=dat)
and then
lapply(lf$reTrms$Ztlist,dim)
to look at the transposed random-effect design matrices for each term, what do you get? You should (based on your description of your data) see that these matrices are 1472 by 2944 and 92 by 2944, respectively.
As #MrFlick says, a reproducible example would be nice. Other things you could show us are:
fit the model anyway, using lmerControl(check.nobs.vs.nRE="ignore") to ignore the test, and show us the results (especially the random effects variances and the statement of the numbers of groups)
show us the results of with(dat,table(table(interaction(condition,person))) to give information on the number of replicates per combination (and similarly for measure)
I have a three factor contigency table that explores the association between committed crimes, Shoplifting or other theft acts here, gender and prior convictions on the one hand and lenient setences on the other. Lenient senteces is the response variable here and is binary ,1 for receiving a lenient sentence, 0 otherwise.
Crime Gender Priorconv Yes No
1 Shoplifting Men N 24 1
2 Other Theft Acts Men N 52 9
3 Shoplifting Women N 48 3
4 Other Theft Acts Women N 22 2
5 Shoplifting Men P 17 6
6 Other Theft Acts Men P 60 34
7 Shoplifting Women P 15 6
8 Other Theft Acts Women P 4 3
You can recreate the table using these commands
table1<-expand.grid(Crime=factor(c("Shoplifting","Other Theft Acts")),Gender=factor(c("Men","Women")),
Priorconv=factor(c("N","P")))
table1<-data.frame(table1,Yes=c(24,52,48,22,17,60,15,4),No=c(1,9,3,2,6,34,6,3))
I have been trying to run a logistic regression but quickly ran into trouble when I tried to include interactions between my variables. The glm works perfectly without the interactions. The code I have been using is
fit<-glm(cbind(Yes,No)~Crime+Gender+Priorconv+I(Crime*Priorconv),data=table1,family=binomial)
and the error I have been getting
Error in `contrasts<-`(`*tmp*`, value = contr.funs[1 + isOF[nn]]) :
contrasts can be applied only to factors with 2 or more levels
In addition: Warning message:
In Ops.factor(Crime, Priorconv) : * not meaningful for factors
Could you please tell how I could deal with this error?
Thank you
By specifying I(Crime*Priorconv) you are asking R to compute the value Crime*Priorconv, which it refuses to do (because it doesn't make sense to multiply factors). If Crime and Priorconv were already numeric dummy variables (e.g. 0/1 coding with 0=shoplifting, 1=other and 0=N, 1=P) then it would make sense to multiply them, and you would use the I() notation to indicate that you wanted to multiply them.
Otherwise (if you don't use I()), R will interpret * as "interaction plus all lower-order effects", i.e. Crime*Priorconv corresponds to 1+Crime+Priorconv+Crime:Priorconv (where : denotes the interaction). R would automatically handle the redundancies (i.e. the fact that you have already specified main effects of Crime and Priorconv): in a formula context, including redundant main effects and explicitly including the intercept (1) or not are all equivalent. These formulae will all specify the same model:
1+Crime+Priorconv+Crime:Priorconv
Crime+Priorconv+Crime*Priorconv
Crime+Priorconv+Crime:Priorconv
Crime*Priorconv
but I prefer the last one: as #J.R. points out in his answer you can take advantage of the * notation to express your model more compactly.
You can use x:y in the formula to specify interactions between x and y, eg.:
fit<-glm(cbind(Yes,No)~Crime+Gender+Priorconv+Crime:Priorconv,data=table1,family=binomial)
or a little shorter:
fit<-glm(cbind(Yes,No)~Gender+Crime*Priorconv,data=table1,family=binomial)
I have a question concerning multi level regression models in R, specifically how to add predictors for my level 2 "measure".
Please consider the following example (this is not a real dataset, so the values might not make much sense in reality):
date id count bmi poll
2012-08-05 1 3 20.5 1500
2012-08-06 1 2 20.5 1400
2012-08-05 2 0 23 1500
2012-08-06 2 3 23 1400
The data contains
different persons ("id"...so it's two persons)
the body mass index of each person ("bmi", so it doesn't vary within an id)
the number of heart problems each person has on a specific day ("count). So person 1 had three problems on August the 5th, whereas person 2 had no difficulties/problems on that day
the amount of pollutants (like Ozon or sulfit dioxide) which have been measured on that given day
My general research question is, if the amount of pollutants effects the numer of heart problems in the population.
In a first step, this could be a simple linear regression:
lm(count ~ poll)
However, my data for each day is so to say clustered within persons. I have two measures from person 1 and two measures from person 2.
So my basic idea was to set up a multilevel model with persons (id) as my level 2 variable.
I used the nlme package for this analysis:
lme(fixed=count ~ poll, random = ~poll|id, ...)
No problems so far.
However, the true influence on level 2 might not only come from the fact that I have different persons. Rather it would be much more likely that the effect WITHIN a person might come from his or her bmi (and many other person related variables, like age, amount of smoking and so on).
To make a longstory short:
How can I specify such level two predictors in the lme function?
Or in other words: How can I setup a model, where the relationship between heart problems and pollution is different/clustered/moderated by the body mass index of a person (and as I said maybe additionally by this person's amount of smoking or age)
Unfortunately, I don't have a clue, how to tell R, what I want. I know oif other software (one of them called HLM), which is capable of doing waht I want, but I'm quite sure that R can this as well...
So, many thanks for any help!
deschen
Short answer: you do not have to, as long as you correctly specify random effects. The lme function automatically detects which variables are level 1 or 2. Consider this example using Oxboys where each subject was measured 9 times. For the time being, let me use lmer in the lme4 package.
library(nlme)
library(dplyr)
library(lme4)
library(lmerTest)
Oxboys %>% #1
filter(as.numeric(Subject)<25) %>% #2
mutate(Group=rep(LETTERS[1:3], each=72)) %>% #3
lmer(height ~ Occasion*Group + (1|Subject), data=.) %>% #4
anova() #5
Here I am picking 24 subjects (#2) and arranging them into 3 groups (#3) to make this data balanced. Now the design of this study is a split-plot design with a repeated-measures factor (Occasion) with q=9 levels and a between-subject factor (Group) with p=3 levels. Each group has n=8 subjects. Occasion is a level-1 variable while Group is level 2.
In #4, I did not specify which variable is level 1 or 2, but lmer gives you correct output. How do I know it is correct? Let us check the multi-level model's degrees of freedom for the fixed effects. If your data is balanced, the Kenward–Roger approximation used in the lmerTest will give you exact dfs and F/t-ratios according to this article. That is, in this example dfs for the test of Group, Occasion, and their interaction should be p-1=2, q-1=8, and (p-1)*(q-1)=16, respectively. The df for the Subject error term is (n-1)p = 21 and the df for the Subject:Occasion error term is p(n-1)(q-1)=168. In fact, these are the "exact" values we get from the anova output (#5).
I do not know what algorithm lme uses for approximating dfs, but lme does give you the same dfs. So I am assuming that it is accurate.