I'd like to know whether the solution proposed below is valid/acceptable and any justification available.
We have two biological conditions, and for each condition we measured 3 time series, so at each time point we have up to 3 data points per condition. For a priori reasons we believe the time series follow a Gaussian mixture model. We'd like to test for a significant difference in means between conditions at a small number of time points (e.g. t = 14, 28, and 38). The plot below is an example, but we have thousands of similar comparisons, and most are not so clear. What's the best way to do this?
A first approach was to compute a t-test at the time points of interest (after transforming the data into something approximately normal). However this gave us at most 6 data points for each comparison, threw out a ton of data, and it didn't use our assumption about the Gaussian mixture model.
Here is the proposed solution. Fit a Gaussian mixture model to both conditions, calculate confidence intervals around the fitted curve, and use these confidence intervals to assess significance at given times. That is, we would evaluate whether the fitted models differ at the time points of interest. (See below.) I could see a problem with this approach if the Gaussian mixture models don't describe the raw data, but in general they do very well (mean R^2 is around 0.9). So, given that Gaussian mixture models seem to be a good description of our data, is this a valid/acceptable solution? If so, how would we carry it out?
Edit: I asked a similar question here and linked to a different, also similar question where Rob Hyndman suggested using a parametric bootstrap. Unfortunately I'm really not familiar with parametric bootstrapping, so I'm not sure whether it would be appropriate here. This explanation of parametric boostrap says it involves "positing a model on the statistic you want to estimate" (in our case the means?), and estimating those parameters "by repeated sampling of the ecdf". I believe the ecdf is just our data.
However I'm not clear how to go from there (bootstrapped distributions of parameters for the Gaussian mixture models) to answering whether there's a significant difference at a specific time point. My understanding is that parametric bootstrapping directly compares the model parameters, i.e. Gaussian parameters in our case. But we're not interested in comparing model parameters. Instead we want to compare the model value, e.g. y_condition1(t=38) vs y_condition2(t=38). So I'm not sure parametric bootstrapping is the way to go.