Well, if you had a model you could fit it and discuss the parameter estimates. But I suppose you don't have a model. So you can just describe what you see in the figure. It's straight forward and not a big deal.

Sidenote: What are the response and the error bars? Is it possible to get negative response values? If not, you should reconsider how to indicate uncertainty.

Given you use the correct error distribution (see sidenote above) you can estimate an interaction of time and group in an appropriate (generalized?, mixed?) linear model. This interaction would give you the "difference" beween the curves, i.e. it would be zero if the curves are parallel and non-zero if they were not parallel. You can also derive a p-value for observing such "non-parallelity" given a zero population interaction.

Dear Jochen,

many thanks for your comments. Actually, the graph is just an example, but the case is about several curves like those (see attached), that need statistics to be compared apart the view impact.

The response variable is a 'gene expression', that is continuous and normally distributed after a log-transofrmation. Bars are 95% confidence intervals. Do not care about the negative 95%CI, because it may be realistic.

I already analyzed data by repeated measure ANOVA (GLM proc) in order to obtain final univariate tests per each time point, and by repeated measure ANOVA (Mixed proc) to obtain tests comparing each treatment. However, perhaps I could do better. In fact, these procedures cannot differentiate properly the curves, or statistical difference did not meet biological means. For example, the two curves shown in the first attached graph resulted statistically non different, but it is clear that their progress is opposite. Thus I realized that repeated measure analysis may not be appropriate in this case.

Any suggestion?

Hey Giovani,

As Dr. Wilhelm pointed out, maybe an -- more costly, but also more meaningful -- option is to set a model for your curves, fit the model(s) for your curves and then go on to discuss parameter estimates. Nothing wrong with repeated measures (rp) ANOVA in my opinion, but maybe a more structured model could do you more good. Besides, depending on your experimental/sampling design, rp-ANOVA may lack the power to detect differences in means.

I'm actually facing this problem myself, and decided to fit a more structured growth model and then devise a likelihood ratio test to compare parameter estimates for my groups. Maybe this could be suitable for you as well.

Hope this helps,

Luiz

Fine, I was just wondering why the vertical axis starts at 0 (does not show negative values). In the figure shown in your first post even the error bar of the time point "5" is partly hidden. If these are log-expressions, then you are right; negative values are well possible (so why is the axis starting at 0 then, although there are confidence intervals going into the negative region?) and a symmetric/normal error distribution is reasonable.

rp-ANOVA is also fine if there is a repeated structure in your data, e.g. a sample from the same individual was analyzed at each time point. And still, I am of the same opinion as Luiz above and I also can't give you a perfect ready-to-use solution.

One not-so-good-solution: You have only few time points. Most interesting seem the points at "1" and "5". You might trash the data inbetween (oh, a terrible idea, I know!) and particularily investigate the group x time interaction in the remaining data.

Thanks Luiz and Jochen. Probably I would fail to find a good model fitting my curves, because they are very different in shape, while the way of testing interactions sounds good. Once tested hypothesis of interactions, could I test significance of interaction Treatment*Time per each couple of curves (e.g.: contrast) in order to yield a kind of multiple comparison test?

Sure.

Great!

May you help me to perform contrasts? I have five treatments (A to E) and five Time points (0,1,2,3,5).

Should I impose intercept=0? Should I impose Time as a continuous variable, or as a classification one?

Given parameters estimates like the attached, and, for example, a B vs. D contrast written in SAS as follows: contrast 'B vs D' DNA 0 -1 0 1 0;

How do I write contrast for interactions D vs. D? I mean, how to write the matrix to perform contrasts?

Please wait. I found this web-page http://www.ats.ucla.edu/stat/sas/faq/estimate_statement.htm that may be appropriate for me, do you agree?

Agree.

Infinite thanks for your support!

Interaction contrast analysis satisfied me very much. I'm wondering how to present data of such a interaction contrast in a easy-to-understand manner. Can you indicate me any papers dealing with application of this analysis?

History shows that following older publications is an easy way to go, often successful, but also often leads to the establishment of errors, suboptimal solutions, and stupid ideas. I would encourage you to think what you want to show the reader and then to find a way how this message can be transprorted in a way the reader will most easily and correctly grasp.

Apart from this, I actually have no paper in mind that deals with this. So I could't give you an example even if I wanted to. There are probably others who might know such papers.

One example from my field of work is the presentation of (log) "fold-changes" of gene expression from real-time PCRs. The expression values are normalized per sample to an internal reference (loading control), and these normalized expressions are compared between groups (fold-changes). Actually, these fold-changes are nothing but the interactions of gene(target,reference) x group (treated,control). However, it is usually not recognized as such. So it is common to present interactions without having in mind that these are interactions - and still the reader (believe) to understand what the authors have written.

Thanks Jochen.

The post is actually old, but you can use this web server: http://bioinf.wehi.edu.au/software/compareCurves/

There is also an R package of the same name.

Yo may have a look at growthcurver, an R package specifically designed to analyze microbial growth curves. It is particularly well suited for the statistical and graphical analysis of growth curves performed in a plate reader. A standard form of the logistic equation is fitted to the data, estimating parameters like doubling time, carrying capacity, and growth rate. The paper describing the R package is freely available here:

Article Growthcurver: An R package for obtaining interpretable metri...

The package vignette can be found here: https://cran.r-project.org/web/packages/growthcurver/vignettes/Growthcurver-vignette.html

Additional usage examples can be found here:

https://rpubs.com/angelov/growthcurver

Thank for very interesting and useful discussion

for the comparaison of grow cuves by statistical procédure...