Hi, Frank,

There are many ways, but I'll only mention optimal data analysis (ODA), a statistically-motivated non-parametric algorithm that, by formulation, explicitly maximizes model predictive accuracy in training analysis (1), provides exact p values (2, 3), and for which efficacy is established in validity analysis (4).

In ODA, group is the class ("dependent") measure in your study, and the eight counts occurring at the different time points are the attributes ("independent" variables).

As stated your hypothesis is non-directional ("exploratory"), you merely want to determine if two groups differ in terms of bacterial count across time (test A), or at each of the eight time points (test B).

Statistical analysis (exploratory and confirmatory) is used to reveal a model which enables the investigator understand the relationship between class membership status and score on an attribute. If this understanding is in reality correct, then the model allows the investigator to make valid (accurate) predictions for the sample that was used to derive the model, as well as for independent random samples.

Using ODA one may compare all-possible pairings (e.g., method 1 vs. 2, 1 vs. 3, and 2 vs. 3), or (to minimize the number of tests of statistical hypotheses which are needed to complete analysis--and thereby to maximize statistical power) one may conduct an optimal range test that compares all three groups simultaneously (5).

The pair-wise and range test methods may be used across all time points considered simultaneously (test A), or for each of the eight time points considered individually (test B).

Regardless of the type of hypothesis tested, and the type of analysis conducted, the additional sample which you have can be treated as a hold-out ("independent sample") validity sample--in which the training model is applied to classify the hold-out sample, and the resulting accuracy is determined (6). Using the "holdout" command in ODA software, every model identified in every analysis conducted for the training sample (1a, 2a, 3a) may be applied to classify the observations in the validity sample (1b, 2b, 3c, respectively) to assess model cross-generalizability across samples.

All models can be directly compared in terms of accuracy adjusted for chance (3, 7), and the adjusted accuracy measure may be further adjusted for model complexity (8). ODA methodology is powerful, requiring relatively small samples to achieve strong statistical power (9). And yet, in contrast to legacy methods, ODA models don't find effects in random data (10-12).

References

(1) https://odajournal.com/2014/04/06/a-statistical-guide-for-the-ethically-perplexed-chapter-4-panter-sterba-handbook-of-ethics-in-quantitative-methodology-routledge-2011-clarifying-disorientation-regarding/

(2) Carmony, L., Yarnold, P.R., & Naeymi-Rad, F. (1998). One-tailed Type I error rates for balanced two-category UniODA with a random ordered attribute. Annals of Operations Research, 74, 223-238.

(3) https://odajournal.com/2020/06/11/generating-novometric-confidence-intervals-in-r-bootstrap-analyses-to-compare-model-and-chance-ess/

(4) https://odajournal.com/2020/06/28/what-is-novometric-data-analysis/

(5) https://odajournal.com/2013/11/22/oda-range-test-vs-one-way-analysis-of-variance-patient-race-and-lab-results/

(6) Yarnold, P.R., & Soltysik, R.C. Optimal data analysis: Guidebook with software for Windows. Washington, D.C.: APA Books, 2005.

(7) https://odajournal.com/2017/04/18/what-is-optimal-data-analysis/

(8) https://odajournal.com/2016/11/04/theoretical-aspects-of-the-d-statistic/

(9) https://odajournal.com/2020/02/08/statistical-power-analysis-in-oda-cta-and-novometrics-invited/

(10) https://odajournal.com/2019/03/21/some-machine-learning-algorithms-find-relationships-between-variables-when-none-exist-cta-doesnt/

(11) https://odajournal.com/2019/04/11/effect-of-sample-size-on-discovery-of-relationships-in-random-data-by-classification-algorithms/

(12) https://odajournal.com/2019/04/22/multi-layer-perceptron-neural-net-model-identifies-effect-in-random-data/

Dear Frank, some of the best statisticians have worked on this problem And there are many excellent references. If you will google the following topics you will get extremely good answers.

1. Comparing growth curve models

2. Growth curve (statistics) Wikipedia

probably the best software for these models is either R or SAS. You will find references in your google output. Please contact me if you have further questions.

David Booth

There are 2 fundamental ways to the comparison of two samples of exotic (non-Gaussian) distributed random variables.

Step 1:

Assuming that there are samples from the same distribution, the amount can be constructed, for example of squared difference, normalized by the product of the standard deviations to be distributed (also presumably) a chi-square or the ratio of the standard deviations (it will be Fischer distribution with corresponding degrees of freedom). Your real-world data will show you whether this hypothesis must be REJECTED or not.

Another way is to use the criteria for comparison of samples that do not depend on the distribution such as Kolmogorov-Smirnov test.

This is not a problem about what is usually called "exponentially distributed data". Possibly it is reasonable to assume that the counts in each cell are Poisson distributed.

In that case you can now estimate A and B by maximum likelihood. The statistical package "R" contains several functions for solving the resulting optimization problem. The log likelihood (if the counts are independent and Poisson distributed) is sum_i N_i log(lambda_i) - lambda_i where lambda_i = A x_i^B.

You may want to maximize this quantity through choice of A and B