# What is the best statistical analysis to analyse significant differences in different metabolites over time (time 0,24 and 48 hours)

I have 40 different metabolites where I measured their concentration at time 0, 24 and 48 hours. I would like to know if there are significant differences over time in each metabolite. I am not interested if there are or not differences between metabolites. I was thinking in running one way Anova repeated measures + Tukey tests per metabolite. so at the end I would have 40 different analysis (Anova + Tukey).

• Can you clarify how many data points you have for a single metabolite at a single time point? Just 1, or 10, or ? Then maybe I can help you. Commented Aug 1 at 17:36
• Are the timepoints assessed because the metabolites grow over time? Commented Aug 1 at 17:42
• I have 3 data points per each time per metabolite: for example for metabolite A in time 0 I have (12,16,17), time 24 (20,22,24) and time 48 (30,33,34) and son on. Commented Aug 1 at 18:05
• the metabolites are assessed over time due their concentrations varied a lot. this is an experiment where I put X plant sample rich in metabolites combined with faecal sample (rich in bacteria). the bacteria hydrolyses metabolites, therefore their concentrations change over time Commented Aug 1 at 18:16
• You do not have enough data for this to be anything more than a preliminary characterisation of the system. Graph the data points and assess them by eye while considering what you know about the particular metabolites. Then design an experiment to assess the behaviour of the interesting ones. Commented Aug 1 at 22:02

You certainly can conduct 40 separate 1-way ANOVA's, with the time point as your factor (3 levels). Or you could conduct 40 regressions, with the time point as your regressor. Since ANOVA is basically a special case of regression (see e.g. here, here, here, or here), chose which ever you are more comfortable with. But...

At each time point, for any given metabolite, you only have 3 values. That sample is quite small. It may even beg the question; should you really run formal statistical tests (no matter what they are)? A statistical significance on a sample size of 3 may not be very convincing to your audience?

Furthermore, if you perform 40 ANOVA's, your chance to get at least 1 (or more) false significant one, just by chance (due to random sampling variations) is $$87%$$... So you need to adjust your $$\alpha$$ level for your 40 ANOVA's. Which will reduce your power, which is already low due the sample size. Having said this, the example data you gave in the comments is significant (both for for 1-way ANOVA and OLS regression) at the "traditional" .05 significance level and at the Bonferroni corrected level of .00125.

Note also that you also should check the ANOVA (or regression) assumptions for all 40 tests. The chance that they are "reasonably" met for all 40 tests may not be very high. And with only 9 residuals, you can truly only eyeball these assumptions...

With all the above caveats, you may indeed be better off just plotting your results, for each metabolite, with a linear fit, and eyeball the graphs; it should be clear which metabolites increase, which stay stable, etc...

A small note: if you decide to use ANOVA, I would not use Tukey, because it runs all comparisons, and thus further reduces your power, which is already challenged. I would use Dunnett instead, comparing to time 0 (which is basically a control). Yes, it is only 2 vs. 3, but, any little bit helps.

• +1 However, I would not recommend a linear model for the time-course of metabolites in a mixed system like this. It is perfectly possible for the actual pattern to be a rise and then a fall. And with such tiny samples any statistical analysis that is not guided by detailed prior knowledge is going to be a random crapshoot. Commented Aug 1 at 22:06
• @MichaelLew, I share your scepticism re. linear model. But with only 3 time points, fitting a higher order model is an even more futile endeavor. So linear is about all one can do... Commented Aug 1 at 22:37
• @MichaelLew . I see is very interesting what you mention. In all the years of experience I have in biological experiments I have never seen any laboratories of analytical chemistry (where is my area of study) running more than a triplicate for a particular sample. this because these types of experiments are expensive and requite tons of work to only run one triplicate for analysis. also if this is the case, why would you consider journal in sciences allow the publications of manuscripts providing only triplicates for statistical analysis? Commented Aug 1 at 22:45
• @jginestet . taking your advice of running Anova and Dunnet instead of Tukey to compare the control (time 0) against time 24 and tim 48 do you think this could be consider as a multiple t test comparison. If so, should I correct for P value with FDR. I know I only have triplicates for each time in each metabolite, this type of experiment is not like preliminary due there is not a different way to assess metabolization of chemical compounds by bacteria (known as microbiota sometimes). Commented Aug 1 at 23:02
• @ricardor, for every ANOVA you run, there is a 5% chance of a false positive (FP) result (significant, even when $H_0$ is true), assuming you use $\alpha=.05$). When you run 40 ANOVA's (or 40 significance tests), you can expect -on average- about 2 FP's, and the probability of getting at least 1 is .87 (\$1-(0.95)^40). If this is a preliminary study (you will run confirmatory tests on the positives), then you may not care too much for FP's. But if this is a final study, or it will be published, you may need to protect against FP's. Commented Aug 2 at 16:23

I think your idea makes total sense here. Just make sure to check for the assumptions of ANOVA, as metabolites might not display all the necessary properties in terms of homoscedasticity etc. Mild violations could also be tolerated, it depends on the specific case.

In addition, remember to correct your p-values by multiple hypothesis testing: by chance, you are expected to get a few positive hits, and you want to be more stringent (use FDR).

Furthermore, as your variable is ordered, you might consider using a linear model (or a generalized one) to search for monotonic trends.