rmANOVA is old-school, and it won't adress your scientific questions adequately. And you can't use incomplete time series (if in one run one value is missing, the whole run will be waisted). I suggest you use a more modern approach!

You should first develop some idea what kind of time-course or trend you want to model. You would ideally be able to describe this by some function with meaningful coefficients. The values of coefficients will then be estimated using the data you have, and you can perform significance tests for these meaningful coefficients. For instance, one of the simplest models is a lineat change over time, in which case your function is that of a straight line, with a coefficient for the intercept and another for the slope. Here, the slope coefficient is the interesting one, and it is meaningful, as it gives you the (expected) change over time (which you assumed is constant). The model can include coefficients for the differenc in slopes between different groups (that's called a "time:group interaction"), and these coefficients are often the scientifically most interesting thing ("is the change over time different between the groups?").

If your data is (more or less) unlimited growth, then the change will not be constant but proportional to the current amount. Here one would typically use an exponential function that has an (interesting and meaningful) coefficient for the growth rate. And again, the time:group interaction allows you to specifically estimate and test differences in the rates between groups. The growth could be limited, in which case one would chose a logistic function where one could test initial rates, or limiting values, or EC50 values - all of which are meaningful and can be interesting, depending on the research question.

If you have reapeated measures, you can model the variance between the individuals (in your case: the starting materials) w.r.t. any of the coefficient in the model. For instance, consider again the constant-change model. You may here usually assume that the intercepts vary between subjects, but it is also possible to think of the slope varying between subjects, or both. This can be considered in the model. Instead of taking the coefficient(s) being a constant value, the model takes them as a "random variable" with an expected value and a variance. The respective coefficients are termed "random effects" in contrst to the "fixed effects" in the model, like the difference between groups. This allows the model to "see" more precisely the fixed effects. A model containing both, fixed and random effects, is called a mixed effects model, somtimes a hierarchical model, because it models a hierarchy of "sources of variation" in your data. A good read on that topic is this book: http://www.stat.columbia.edu/~gelman/arm/

rmANOVA is old-school, and it won't adress your scientific questions adequately. And you can't use incomplete time series (if in one run one value is missing, the whole run will be waisted). I suggest you use a more modern approach!

You should first develop some idea what kind of time-course or trend you want to model. You would ideally be able to describe this by some function with meaningful coefficients. The values of coefficients will then be estimated using the data you have, and you can perform significance tests for these meaningful coefficients. For instance, one of the simplest models is a lineat change over time, in which case your function is that of a straight line, with a coefficient for the intercept and another for the slope. Here, the slope coefficient is the interesting one, and it is meaningful, as it gives you the (expected) change over time (which you assumed is constant). The model can include coefficients for the differenc in slopes between different groups (that's called a "time:group interaction"), and these coefficients are often the scientifically most interesting thing ("is the change over time different between the groups?").

If your data is (more or less) unlimited growth, then the change will not be constant but proportional to the current amount. Here one would typically use an exponential function that has an (interesting and meaningful) coefficient for the growth rate. And again, the time:group interaction allows you to specifically estimate and test differences in the rates between groups. The growth could be limited, in which case one would chose a logistic function where one could test initial rates, or limiting values, or EC50 values - all of which are meaningful and can be interesting, depending on the research question.

If you have reapeated measures, you can model the variance between the individuals (in your case: the starting materials) w.r.t. any of the coefficient in the model. For instance, consider again the constant-change model. You may here usually assume that the intercepts vary between subjects, but it is also possible to think of the slope varying between subjects, or both. This can be considered in the model. Instead of taking the coefficient(s) being a constant value, the model takes them as a "random variable" with an expected value and a variance. The respective coefficients are termed "random effects" in contrst to the "fixed effects" in the model, like the difference between groups. This allows the model to "see" more precisely the fixed effects. A model containing both, fixed and random effects, is called a mixed effects model, somtimes a hierarchical model, because it models a hierarchy of "sources of variation" in your data. A good read on that topic is this book: http://www.stat.columbia.edu/~gelman/arm/

You have a repeated measures protocol. So look into methods for repeated measures testing. The statistical problem is that the groups in each time increment are not independent. Thus sample size is not the total number of observations, but must be adjusted. Good luck.

General (linear) mixed model as mentioned Jochen Wilhelm will be appropriate tto Your data.

rmANOVA is a case of that kind of models when at every timepoint we have full set of measures. If not - GLMM helps to estimate effects and difference in effects of materials, timepoints and so on.

Thanks for the answers. It helped me to scale down the options however I have still one question:

Does these models (general mixed model or rMANOVA) fit the dependency of the categorical data. Let's explain my experiment. We get a sample from a donor and measure immediately the amount of cells (T-cells for example). After that, we distribute the sample and we incubate one sample in the refrigerator and one in the freezer. We measure at interval of a week for four consecutive weeks. Thus, we have dependency of the conditions and within each condition we have dependency at different time points.

If I take a look at the model, I expect exponential decay of cells.

To analyze repeated measure data, there are two major ways as mentioned; one is a mixed model and another is a multivariate model.

Both of them aims to take the dependency of time into account.

But since you have a specific hypothesis for the pattern of decay (in this case, the exponential decay of cell as time goes), the mixed model approach is easier I think.

What you need is just writing link formula of regression model as you do for non-linear regression.

Of course, you can use multivariate approach. But constructing the covariance structure and its likelihood function requires a bit of technique.

Yes.

P.S.

If you use R, lme4 package are useful.

See the instruction for "nlme" function of the following pdf.

https://cran.r-project.org/web/packages/lme4/lme4.pdf

Dear Jesse,

after Your refinement I think that instead use data directly You can transform data at first.

Exponential distibution has parameter lambda.

For each sample from donor and each condition You can estimate parameter lambda. Then You can succesfully use rmAnova or GLMM where dependent variable is lamblda (without timepoint, it decrease size of model.)

lamblda is interpretable and links to half-life

Anothers apporoach to transform Your data described above by Jochen Wilhelm.

What parameters of cells trends Your need - it is Your choice.

Here you have one of our papers with repeated measures: