Solved – Dataset of repeated measures with unbalanced sample size and unequal variace – What to do

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I have a dataset of three groups of cells treated with 10 different compounds and am not sure how to check for significant differences between those treatments. Within each group the data is also paired.

I tried doing an two-way ANOVA; however, that doesn't work because of the unbalanced sample size. I then used GLM in SPSS, which at least gives me the correct means and standard deviations for each treatment, but it doesn't calculate correct p-values since it assumes equal variances.

Could somebody suggest the correct test in this case or knows how to modify the GLM accordingly?

Edit:
So here is how my dataset looks like in priciple, I only have 8 more treatments:
enter image description here

I now would like to compare the treatment conditions to each other, so for example 'is there a significant difference between the number of puncta in control and starvation conditions?'.

So far I used univariate GLM with the Puncta as dependent variable and group and treatment as independent factors.

Best Answer

In the past I had a similar problem with the analysis of clinical data using 2-way ANOVA. What I did was to use the Anova function of the R package car and set the test type to III (type="III"). If you had posted more information, maybe I could provide also a code example. The results I got seemed reasonable.

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Solved – Might be an unbalanced within subjects repeated measures

It's not imbalanced because your repeated measures should be averaged across such subgroups within subject beforehand. The only thing imbalanced is the quality of the estimates of your means.

Just as you aggregated your accuracies to get a percentage correct and do your ANOVA in the first place you average your latencies as well. Each participant provides 6 values, therefore it is not imbalanced.

Most likely though... the ANOVA was not the best analysis in the first place. You should probably be using mixed-effect modelling. For the initial test of the accuracies you'd use mixed effects logistic regression. For the second one you propose it would be a 3-levels x 2-correctnesses analysis of the latencies. Both would have subjects as a random effect.

In addition it's often best to do some sort of normality correction on the times like a log or -1/T correction. This is less of a concern in ANOVA because you aggregate across a number of means first and that often ameliorates the skew of latencies through the central limit theorem. You could check with a boxcox analysis to see what fits best.

On a more important note though... what are you expecting to find? Is this just exploratory? What would it mean to have different latencies in the correct and incorrect groups and what would it mean for them to interact? Unless you are fully modelling the relationship between accuracy and speed in your experiment, or you have a full model that you are testing, then you are probably wasting your time. A latency with an incorrect response means that someone did something other than what you wanted them to... and it could be anything. That's why people almost always only work with the latencies to the correct responses.

(these two types of responses also often have very different distributions with incorrect much flatter because they disproportionately make up both the short and long latencies)

Mixed Model – Unbalanced Mixed Effect ANOVA for Repeated Measures

The lme/lmer functions from the nlme/lme4 packages are able to deal with unbalanced designs. You should make sure that time is a numeric variable. You would also probably want to test for different types of curves as well. The code will look something like this:

library(lme4)
#plot data with a plot per person including a regression line for each
xyplot(heart.rate ~ time|id, groups=treatment, type= c("p", "r"), data=heart)

#Mixed effects modelling
#variation in intercept by participant
lmera.1 <- lmer(heart.rate ~ treatment * time + (1|id), data=heart)
#variation in intercept and slope without correlation between the two
lmera.2 <- lmer(heart.rate ~ treatment * time + (1|id) + (0+time|id), data=heart)
#As lmera.1 but with correlation between slope and intercept
lmera.3 <- lmer(heart.rate ~ treatment * time + (1+time|id), data=heart)

#Determine which random effects structure fits the data best
anova(lmera.1, lmera.2, lmera.3)

To get quadratic models use the formula "heart.rate ~ treatment * time * I(time^2) + (random effects)".

Update:
In this case where treatment is a between-subjects factor, I would stick with the model specifications above. I don't think the term (0+treatment|time) is one that you want included in the model, to me it doesn't make sense in this instance to treat time as a random-effects grouping variable.

But to answer your question of "what does the correlation -0.504 mean between treat0 and treat1" this is the correlation coefficient between the two treatments where each time grouping is one pair of values. This makes more sense if id is the grouping factor and treatment is a within-subjects variable. Then you have an estimate of the correlation between the intercepts of the two conditions.

Before making any conclusions about the model, refit it with lmera.2 and include REML=F. Then load the "languageR" package and run:

plmera.2<-pvals.fnc(lmera.2)
plmera.2

Then you can get p-values, but by the looks of it, there is probably a significant effect of time and a significant effect of treatment.

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Solved – Might be an unbalanced within subjects repeated measures

Mixed Model – Unbalanced Mixed Effect ANOVA for Repeated Measures

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