The design would suggest either repeated measures ANCOVA or a multilevel linear model (with occasions of measurement clustered within "subjects", or "cells" in your case), both using the baseline (pre-treatment) measurement as a covariate.

However, judging from the graph you uploaded, the DV appears to be positively skewed in all cells of the design, and there appear to be outliers in some cells (and for the baseline measure). Given that the degree of skew looks similar across the board, I think the presence of outliers is likely more problematic. I'd be looking at some measures of influence.

What software do you have access to?

I suggest panel regression.

Area --> dependent var

Treatment (TOFA =1, DMSO =0) --> independent var

Time --> time series var as a covariate

Your distribution is right skewed and therefore the family of regression can be Gamma or Poisson (of course Poisson is for count, but your area has decimals).

Given that your observations are percentages, you might want to consider an appropriately scaled (i.e. to 0 to 1) inverse sine transformation, which would affect the skewness mentioned by others.

The suggestion by David Eugene Booth (below) would be good for formal testing, while the inverse sine transform may help if you want a visual impression comparing within-group and between-group variations.

I suggest beta regression which is designed for such things. See:

https://rcompanion.org/handbook/J_02.html

Best, D. Booth

Bruce Weaver I have access to Minitab, Matlab and Origin atm

Thank you for all your suggestions!

David Eugene Booth David A. Jones Seyyed Amir Yasin Ahmadi Bruce Weaver

A first note: It would be good scientific practice to fix the analysis plan before you do the experiment.

You have a 2*4 factorial design with repeated measaures on the time factor.

The test of interest is the interaction, probably interaction contrasts comparing the difference between groups of each timepoint with the baseline difference. The conditional distribution of the response variable is skew.

One option you have to adress this is by mixed quantuile regression models. These models offer a way to model the dependency of some quantile, e.g. the median, on the design factors, instead of the avarage. Outliers are the irrelevant. if you wish to model some other aspect of the distribution, e.g. the 75% quantile, this is possible in that framework, which is interesting when your signal might be reflected in changes in skewness rather then changes in the central tendency.

Software is available for R, e.g. package lqmm. Have fun :)

Hello Ewan Hislop. I failed to spot the % sign on the Y-axis of your graph earlier. In light of that, the beta regression (suggested by David Eugene Booth) makes sense. But ordinary beta regression will not handle the repeated measures data properly. You'll need to use generalized estimating equations (GEE) or a mixed modeling approach to do that. I am not very familiar with Minitab or Matlab (and have never heard of the other package you mentioned), so do not know if they have routines for estimating such models. HTH.

Bruce Weaver Hi Bruce. What user friendly programs do you suggest for this type of modelling? Thanks

One of the packages I use is Stata. I suspect that you could take the GEE approach using its -betareg- command with the vce(clustvar) option, using a unique ID code as the cluster variable.

• https://www.stata.com/manuals/rbetareg.pdf

For the mixed effects (multilevel) approach, I would investigate the -meglm- command:

• https://www.stata.com/manuals/memeglm.pdf

Other readers of this thread will no doubt have other suggestions. I believe there are some useRs in the crowd, for example. ;-)

It appears that Matlab can fit generalized linear mixed effects models:

• https://www.mathworks.com/help/stats/fit-a-generalized-linear-mixed-effects-model.html

Thanks Bruce Weaver This is really helpful.

Thanks Bruce Weaver

I strongly agree with GEE (or random effect model of panel regression instead). Stata is the best and easiest choice for repeated measure models.

The problem is to choose a suitable family (Gaussian, Gamma, Poisson, or beta that of course it does not support repeated measures as you mentioned). Now we have two problems for the DV.

1- Skewness

2- Being as percentage!

I think Poisson family can solve both of them. Each percentage can be broken as "event per exposure". I have previously worked with color diagnosis softwares of tissue slides (e.g. ImageJ, motic image plus). They report both percentage and net number of pixels. Ewan Hislop can choose fat drop pixels as event and total pixels as exposure.

Hereby, the command will be:

xtset id time

xtpoisson fat_pixel drug ib(0).time, exposure(total_pixel) re

However, Ewan Hislop does not have access to Stata. I am not familiar with Matlab, but the author can use R as free source software.

I do have enough experience in statistics. I will be happy to know whether you are agree with braking rate to event per exposure. Because it was my personal experience!!

Best!

@Bruce Weaver. If you look at my link there is no problem with R. Notice you have to write the regression RHS yourself. This is a simple ANCOVA regression design with DV as Post and Pre as an IV for example. I note that the author of that review is @Salvatore Magnifico a well known contributor to these posts. Examples of such an ANCOVA model can be found here: https://b-ok.cc/book/3591248/4df2ac

Best wishes to all, David Booth PS R is a free download so we all have access.

David Eugene Booth , Ewan Hislop wrote:

"I was testing prostate cells treated with a drug and evaluating the %area of lipid droplet size. I have acquired a % area of the lipid droplet for approximately 60 cells at 0hr (no treatment), 15hr (drug vs control), 24hr (drug vs control), 48hr (drug vs control) and calculated mean change in lipid droplet size at each timepoint."

I took him to mean that the same cell was measured at multiple time-points--i.e., repeated measures. You appear be treating it as if there are independent groups of cells at each time point. Perhaps Ewan needs to clarify which it is.

Cheers,

Bruce

Bruce Weaver Thanks for clarifying that our questioner was doing "repeated measures" which since the days of the development of "repeated measures" models for dealing with the rapidly mutating HIV virus, newer models developed by biostatisticians are called longitudinal models. One of the leaders in this development, Prof. Marie Davidian at North Carolina State University has I think the best summary of the subject in the notes for her doctoral course on the subject. Prof. Davidian's Notes can be found here. :

https://www4.stat.ncsu.edu/~davidian/st732/notes.html

I note that these models were developed to handle much larger repeated measures than the 5-10 or so repeats in repeated measures ANOVA which would not be suitable in any case for a rapidly mutating biological entity.. Thus, at this point, I would think that Ewan Hislop would want to consider Prof Magnifico's notes on Beta Regression with R code included [see my earlier post ] and consider a longitudinal RHS to the model.. Prof. Davidian's notes include R code as well as SAS. R is a free download for everyone. My favorite introductory R book is at this link https://b-ok.cc/book/5825760/ed313f

With much usable research grade code. I hope Ewan finds this discussion helpful.

Best, David Booth

Bruce Weaver David Eugene Booth Seyyed Amir Yasin Ahmadi To clarify...

Cells from the same culture were plated at a concentration of 5x10^5 cells/ml and imaged at various timepoints (0hr) cells in DMSO and at (15, 24, 48hr) cells withm(DRUG) and with (CONTROL - DMSO) using SRS microscopy. The cells whether it be with DMSO or DRUG were imaged 3 times in the concentration of (5x10^5 cells/ml ). Each image equated to approximately 20 cells. From this I used Image J to measure the %area of lipid droplets compared to the whole cell area for approximately 60 cells at each timepoint (0hr, 15hr, 24hr, 48hr). Ultimately to observe if the lipid-altering drug changed the distribution of intracellular lipid. In this case the size/area of lipid droplets over time.

I originally was told to use paired T-tests to compared the mean differences in %area between timepoints 0hr-15hr 15hr-24hr etc. But, since starting this thread I believe this is not the way to measure/show significant changes in %area.

I have little experience with R and stata. I will be sure to check out the introductory R book suggested by David Eugene Booth

Again, thank you for taking time out your day to offer your wealth of statistical knowledge.

Regards,

Ewan

Ewan Hislop , bear in mind that Matlab, which you have and are already somewhat familiar with, may be able to estimate the same model you would get via R.

• https://www.mathworks.com/help/stats/fit-a-generalized-linear-mixed-effects-model.html

Ewan Hislop Please note note that if you need R code much can be adapted from the links to Prof. Davidian and Prof Magnifico as well as the book. Asking questions is never a bad approach. Best wishes, David Booth