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Apologies in advance for my lack of statistical knowledge/insight!

I am trying to calculate the sample size for a clinical trial of treatment A versus placebo with primary outcome of prolongation of treatment time ('the longer the better'). We know that the primary outcome is skewed. The minimal clinically relevant treatment effect would be a difference of 5 days between the medians.

As suggested by prof. Harrell, I have tried calculating the sample size using a proportional odds model based on data from a similar (already published) trial. I based my calculations on this example: https://www.fharrell.com/post/pop/

However, I am unsure whether I have provided the right input for the functions popower and pomodm in R.

This is my code:

pwr.t.test(n=180/2, d=5/sd(p), type = 'two.sample') The previous trial had 180 patients in total (90 treatment A, 90 placebo); this yield a power of 44% to detect a difference of 5 days (in means) (two-sample t-test), which I think can be improved by using a proportional odds model instead.

p <- df_placebo$`prolongation in days, where p is a list of prolongation in days (non-integer, unique values) for each patient in the placebo group from the previous trial. Am I correct in only using placebo data for the sample size calculation?

kp <- table (p); kp <- kp /sum(kp), this is based on the example blog-post but I am not sure whether this is correct for my case or whether I should input a table of probabilities based on frequencies instead (which would be 1/90 for every 'prolongation level', as there are 90 patients in the placebo group and each 'prolongation' is a unique value.

kx <- as.numeric(names(kp))

library(Hmisc)

w <- pomodm(kx, kp, odds.ratio=1.0), this yields the original mean and median for the placebo group, which to me seems correct.

Next step to find out which OR corresponds to a difference in medians of 5 days between placebo and treatment group:

z <- data.table(or=ors)
u <- z[, as.list(pomodm(kx, kp, odds.ratio=or) - w), by=or]
m <- meltData(or ~ mean + median, data=u)   
ggplot(m, aes(x=or, y=value, color=variable)) + geom_line() +
  geom_vline(xintercept=1.29, alpha=0.3) +
  geom_hline(yintercept=5) + 
  guides(color=guide_legend(title='')) +
  xlab('Odds Ratio') + ylab('Difference Between Groups') 

enter image description here

A five-day increase seems to correspond to an OR of 1.29. But if I input this into popower this yields a power of only 16% with 180 patients in total. popower(kp, odds.ratio=1.29, 180)

I am not sure what I am doing wrong. So sorry if this is not a very smart question; R and statistics are very new to me and I have just started learning more a few weeks ago by reading articles and blog posts.

Thank you all very much in advance for your help. Catherina

Edited to show distribution of prolongation data (ln-transformed, blue = placebo, red = treatment A, from previous trial): Red = treatment A, blue = placebo (data from previous trial)

Prolongation data on the original scale (placebo and treatment A, respectively): enter image description here

enter image description here

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1 Answer 1

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You've done a good job Catherina. Yes, the external data from a placebo group forms the reference data. table(p) / sum(table(p)) is the right vector; it is the relative frequencies. A minor issue is that popower wants you to provide relative frequencies that are averaged over control and treatment subjects. If you let $a$ be square root of the odds ratio, you can apply an odds ratio of $a$ to the placebo proportions to get the proportions for the between-treatment average. I wish I had used the control group for the proportions in popower and posamsize. To check the calculations you can apply an odds ratio of $\frac{1}{a}$ to the modified proportions to make sure you get the original placebo proportions.

The bigger questions are

  • which effect size to use in computing power (mean vs. median)
  • what is the real MCID, i.e., were the experts right in say a difference in medians of 5? Did they assume the mean and median were the same when selecting 5?

The only way I can think of to explore this is to find a transformation of p that yields a symmetric distribution, and solve for the difference in means on that transformed scale that corresponds to a difference in medians of 5 on the original scale. Compute the SD on that transformed scale and repeat pwr.t.test on the just-solved-for difference in means. That should be more comparable to popower.

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  • $\begingroup$ Thank you! Using the square root of the odds ratio to obtain relative frequencies averaged over control and treatment subjects indeed did not impact the power very much; still 15% when aiming for a difference of 5 days in medians (part I). $\endgroup$ Commented Apr 28 at 19:01
  • $\begingroup$ The closest I could get to a symmetric distribution was by ln-transforming the data, althought the distribution was not log-normal. Then I did this: mean_placebo <- 2.25039457 # mean of log normal transformed data in previous trial mean_treatment <- log(exp(mean_placebo) + 5) #MCRD of 5 days on the untransformed scale (not sure if this is right) MCRD <- mean_treatment - mean_placebo #log-scale MCRD sd_log <- 1.23184 # Standard deviation on the log scale pwr.t.test(n=180/2, d=MCRD/sd_log, type = 'two.sample') This yields a power of 63%. $\endgroup$ Commented Apr 28 at 19:04
  • $\begingroup$ I also tried doing the following: applied the OR matching a 5 day difference in medians on the data on the original scale, ln-transformed the data, calculated the difference in means on the transformed scale and the SD on the transformed scale and then repeated pwr.t.test with this difference in means. This yields a power of 11% (more comparable to popower). $\endgroup$ Commented Apr 29 at 6:55
  • $\begingroup$ Hard to figure out. In the graph above you show negative values of prolongation of treatment. How can that be? How did you take logs then? $\endgroup$ Commented Apr 29 at 11:10
  • $\begingroup$ The graph with the red and blue lines are the ln-transformed data (sorry that wasn't clear; some prolongation times were really short, leading to negative values when ln-transformed). I have added descriptions of the untransformed data as well (for placebo and treatment A). $\endgroup$ Commented Apr 29 at 12:37

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