I've been working on the following problem recently: We sent 18 people, 9 each, several times to two different clubs "N" and "O". These people arrived at the club either between 8 and 10 am (10) or between 10 and 12 pm (12). Each club consists of four sectors with ascending price classes. At the end of each test run, the subjects filled out a questionnaire reflecting a score for their satisfaction depending on the different parameters. The aim of the study is to find out how satisfaction can be modelled as a function of the club. You can download the data as csv for one week with this link (without spaces): https: // we.tl/t-I0UXKYclUk
After some try and error, I fitted the following model using the lme4 package in R (the other models were singular, had too strong internal correlations or higher AIC/BIC):
mod <- lmer(Score ~ Club + (1|Sector:Subject) + (1|Subject), data = dl)
Now I wanted to create some diagnostic plots as indicated here.
plot(resid(mod), dl$Score)
plot(mod, col=dl$Club)
library(lattice)
qqmath(mod, id=0.05)
Unfortunately, it turns out that there are still patterns in the residuals that can be attributed to the club but are not captured by the model. I have already tried to incorporate the club into the random effects, but this leads to singularities. Does anyone have a suggestion on how I can deal with these patterns in the residuals? Thank you!
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.
I understand I can use lmer but I would like to undertake a repeated measures anova in order to carry out both a within group and a between group analysis.
So I am trying to compare the difference in metabolite levels between three groups ( control, disease 1 and disease 2) over time ( measurements collected at two timepoints), and to also make a within group comparison, comparing time point 1 with time point 2.
Important to note - these are subjects sending in samples not timed trial visits where samples would have been taken on the same day or thereabouts. For instance time point 1 for one subject could be 1995, time point 1 for another subject 1996, the difference between timepoint 1 and timepoint 2 is also not consistent. There is an average of around 5 years, however max is 15, min is .5 years.
I have 43, 45, and 42 subjects respectively in each group. My response variable would be say metabolite 1, the predictor would be Group. I also have covariates I would like to be accounted for such as age, BMI, and gender. I would also need to account for family ID (which I have as a random effect in my lmer model). My column with Time has a 0 to mark the time point 1 and 1 is timepoint 2). I understand I must segregate the within and between subjects command, however, I am unsure how to do this. From my understanding so far;
If I am using the anova_test, my formula that needs to be specified for between subjects would be;
Metabolite1 ~ Group*Time
Whilst for within subjects ( seeing whether there is any difference within each group at TP1 vs TP2), I am unsure how I would specify this ( the below is not correct).
Metabolite1 ~ Time + Error(ID/Time)
The question is, how do I combine this altogether to specify the between and within subject comparisons I would like and accounting for the covariates such as gender, age and BMI? I am assuming if I specify covariates it will become an ANCOVA not an ANOVA?
Some example code that I found that had both a between and within subject comparison design (termed mixed anova).
aov1 <- aov(Recall~(Task*Valence*Gender*Dosage)+Error(Subject/(Task*Valence))+(Gender*Dosage),ex5)
Where he specifies that the within subject comparison is within the Error term. Also explained here https://rpkgs.datanovia.com/rstatix/reference/anova_test.html
However, mine, which I realise is very wrong currently ( is missing a correct within subject comparison).
repmes<-anova_test(data=mets, Metabolite1~ Group*Time + Error(ID/Time), covariate=c("Age", "BMI",
"Gender", "FamilyID")
I ultimately would like to determine from this with appropriate post hoc tests ( if p < 0.05) whether there are any significant differences in Metabolite 1 expression between groups between the two time points (i.e over time), and whether there are any significant differences between subjects comparing TP1 with TP2. Please can anybody help.
I have a set of data that came from a psychological experiment where subjects were randomly assigned to one of four treatment conditions and their wellbeing w measured on six different occasions. The exact day of measurement on each occasion differs slightly from subject to subject. The first measurement occasion for all subjects is day zero.
I analyse this with lmer :
model.a <- lmer(w ~ day * treatment + (day | subject),
REML=FALSE,
data=exper.data)
Following a simple visual inspection of the change-trajectories of subjects, I'd now like to include (and examine the effect of including) the possibility that the slope of the line for each subject changes at a point mid-way between measurement occasion 3 and 4.
I'm familiar with modeling the alteration in slope by including an additional time-variable in the lmer specification. The approach is described in chapter 6 ('Modeling non-linear change') of the book Applied Longitudinal Data Analysis by Singer and Willett (2005). Following their advice, for each measurement, for each subject, there is now an additional variable called latter.day. For measurements up to measurement 3, the value of latter.day is zero; for later measurements, latter.day encodes the number of days after day 40 (which is the point at which I'd like to include the possible slope-change).
What I cannot see is how to adjust the lmer coding of the examples in the Singer and Willett cases to suit my own problem ... which includes the same point-of-slope-change for all subjects as well as a between-subjects factor (treatment). I'd appreciate help on how to write the specification for lmer.
I am a biologist working in aquaculture nutrition research and until recently I haven't paid much attention to the power of statistics. The usual method of analysis had been to run ANOVA on final weights of animals given various treatments and boom, you have a result. I have tried to improve my results by designing an experiment that could track individuals growth over time but I am having a really hard time trying to understand which model to use for the data I have.
For simplified explanation of my experiment: I have 900 abalone/snails which were sourced from a single cohort (spawned/born at the same time). I have individually marked each abalone (id) and recorded a length and weight at Time 0. The animals were then randomly assigned 1 of 6 treatment diets (n=30 abalone per treatment) each replicated n=5 times (n=150 abalone / replicate). Each replicate looks like a randomized block design where each treatment is only replicate once within each block and each is assigned to independent tank with n=30 abalone/tank (n treatment). Abalone were fed a known amount of feed for 90 days before being weighed and measured again (Time 1). They are back in their homes for another 90 days before the concluding the experiment.
From my understanding:
fixed effects - Time, Treatment
nested random effects - replicate, id
My raw data entered is in Long format with each row being a unique animal and columns for Time (0 or 1), Replicate (1-5), Treatment (1-6), Sex (M or F) Animal ID (1-900), Length (mm), Weight (g), Condition Factor (Weight/Length^2.99*5655)
I have used columns from my raw data and converted them to factors and vectors before using the new variables to create a data frame.
id<-as.factor(data.long[,5])
time<-as.factor(data.long[,1])
replicate<-as.factor(data.long[,2])
treatment<-data.long[,3]
weight<-as.vector(data.long[,7])
length<-as.vector(data.long[,6])
cf<-as.vector(data.long[,10])
My data frame is currently in the following structure:
df1<-data.frame(time,replicate,treatment,id,weight,length,cf)
I am struggling to understand how to nest my individual abalone within replicates. I can convert the weight data to change from initial but I think the package nlme already accounts this change when coded correctly. I could also create another measure of Specific Growth Rate for each animal at Time 1 but this would not allow the Time factor to be used.
lme(weight ~ time*treatment, random=~1 | id, method="ML", data=df1))
I would like to structure a mixed effects model so that my code takes into account the individual animal variability to detect statistical differences in their weight at Time 1 between treatments.