Spearman correlation coefficient with repeated measures

Question

I have a data set where 2 continuous variables(bio markers) are being measured a variable amount of times for each subject. Some subjects have 1 measurement while others have towards 100. The data is from an observational study of subjects under no intervention and began out as a single measurement however it has changed setup and is now a longitudinal study with drop outs. It is expected that there is a nonlinear correlation between the markers which at the same time are highly influenced by the lifestyle of the subject which is why it is necessary to use a repeated measurement setup.

I would like to calculate something resembling a repeated measures Spearman correlation coefficient. Is this possible in a reasonable way? Something which approaches the ordinary Spearman coefficient when number of measures per subject approaches 1.

(If a solution can be posted in R I will be happy but I can implement a general outline if necessary). Right now my only idea is to collapse each subject by taking the mean or calculate a Spearman correlation coefficient for each subject and then collapse with a mean.

Here is a synthetic data example where I have added some correlation coefficients. spearman_mean is calculating a Spearman for each subject an calculating the mean. mean_spearman is calculating a subject mean and then Spearman. rmcorr is a linear mixed model based approach which resembles a Pearson coefficient (as far as I understand). If a "real" repeated measurement Spearman is not available then if you can give me some pointers towards a best practice it would be really helpful.

library(dplyr)
library(ggplot2)
N = 100
set.seed(1234)
df <- data.frame(id_num = sample(1:3, N, replace = T, prob = c(0.1,0.5,0.4)),
                 var1 = rnorm(N, mean = 5, sd = 0.5)) %>%
  mutate(id = factor(LETTERS[id_num]),
         var2 = exp(1/sqrt(id_num)*var1)) %>% 
  arrange(id)
df$var2 <- df$var2 + rnorm(N, sd = 3)

df %>% 
  ggplot(aes(x = var1, y = var2, color = id)) + geom_point(size = 2)

df_cor <- data.frame(pearson = cor(df$var1, df$var2, method = "pearson"),
                     spearman = cor(df$var1, df$var2, method = "spearman"),
                     rmcorr = rmcorr::rmcorr(participant = id, measure1 = var1, measure2 = var2, dataset = df)$r,
                     df %>% group_by(id) %>% summarise(spear = cor(var1, var2, method = "spearman")) %>% summarise(spearman_mean = mean(spear)),
                     df %>% group_by(id) %>% 
                       summarise(var1 = mean(var1),
                                 var2 = mean(var2)) %>% 
                       summarise(mean_spearman = cor(var1, var2, method = "spearman")))

df_cor %>% knitr::kable()

pearson	spearman	rmcorr	spearman_mean	mean_spearman
0.3331734	0.5253885	0.622991	0.9114043	0.5

Answer 1

Mohr & Marcon (2005) seems to describe what you are trying to do. They suggest using a weighted mean of the within-subject Spearman correlations (weighted by number of observations minus one), and then using permutations or a normal approximation to calculate a P value. I'm not aware of any R code for this, unfortunately.

Donna L. Mohr & Rebecca A. Marcon (2005) Testing for a ‘within-subjects’ association in repeated measures data, Journal of Nonparametric Statistics, 17:3, 347-363, DOI: 10.1080/10485250500038694