Solved – Repeated measures over time with small $n$

multiple-comparisonsrepeated measureswilcoxon-signed-rank

I was given data to analyze for a study looking at the effects of a treatment on iron levels at four different time points (before treatment, the day treatment ended, 4 weeks after treatment, and 2-4 months after treatment). There is no control group. They are looking to see if there are significant increases in iron levels at each of the 3 post-treatment time points to the before treatment (baseline) level. Eleven patients had baseline levels but only 8 patients had complete data for all 4 time points ($n$=11, 10, 9, and 8 for each time point). Not only were iron levels measured, but two other laboratory measures were taken at each time point to be compared to baseline.

I have a few questions as to how to analyze this. I first thought an RM ANOVA would be appropriate to analyze this data, but I was concerned about the small sample size, the loss of data, and the non-normal distribution of the data. I then considered comparing each post-treatment measure to baseline using Wilcoxon signed-rank tests, but then I run into the issue of multiple comparisons. However, I have read some literature that downplays needing to run multiple comparisons. So overall, I'm dealing with small sample sizes, incomplete data, and multiple comparisons (and whether or not its necessary).

I hope this all made sense. I'm new to CrossValidated and was directed here by a colleague as a place to learn from experienced statisticians, so I'd appreciate any advice! Thanks!

Edited to add raw data from comment:

There are four total time points and the outcome variable is continuous. For example, the results at each time point look similar to this: 

 Baseline (n=11): [2, 7, 7, 3, 6, 3, 2, 4, 4, 3, 14] 
 1st Post (n=10): [167, 200, 45, 132, ., 245, 199, 177, 134, 298, 111]
 2nd Post (n=9):  [75, 43, 23, 98, 87, ., 300, ., 118, 202, 156]
 3rd Post (n=8):  [23, 34, 98, 112, ., 200, ., 156, 54, 18, .]

Best Answer

I have re-think your problem and found Friedman's test which is a non-parametric version of a one way ANOVA with repeated measures.

I hope you have some basic skills with R.

# Creating a source data.frame
my.data<-data.frame(value=c(2,7,7,3,6,3,2,4,4,3,14,167,200,45,132,NA,
245,199,177,134,298,111,75,43,23,98,87,NA,300,NA,118,202,156,23,34,98,
112,NA,200,NA,156,54,18,NA),
post.no=rep(c("baseline","post1","post2","post3"), each=11),
ID=rep(c(1:11), times=4))

# you must install this library
library(pgirmess)

Perform test Friedman's test...

friedman.test(my.data$value,my.data$post.no,my.data$ID)

    Friedman rank sum test

data:  my.data$value, my.data$post.no and my.data$ID
Friedman chi-squared = 14.6, df = 3, p-value = 0.002192

and then find between which groups the difference exist by non-parametric post-hoc test. Here you have all possible comparisons.

friedmanmc(my.data$value,my.data$post.no,my.data$ID)
Multiple comparisons between groups after Friedman test 
p.value: 0.05 
Comparisons
               obs.dif critical.dif difference
baseline-post1      25     15.97544       TRUE
baseline-post2      21     15.97544       TRUE
baseline-post3      20     15.97544       TRUE
post1-post2          4     15.97544      FALSE
post1-post3          5     15.97544      FALSE
post2-post3          1     15.97544      FALSE

As you can see only baseline (first time point) is statistically different from others.

I hope this will help you.

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