I'm hoping the informed can kindly help me with some suggestions on the appropriate statistical techniques to use for panel data collected from 60 biofilter columns over 6 months.
1. The data. There are several different designs. Each design represents a unique combination of vegetation, filter media and saturation zone type. There are ~5 replicates for each design. In some months, they were all dosed with greywater, and with stormwater in other months. We have collected outflow concentration of several pollutants including P, N, TSS and metals, each month.
2. My plan A. This was my original plan. The idea is to treat the data set as panel data and run a regression using the appropriate method. Since we've been tracking the performance of each biofilter through time, I surmise that the 'Pooled Cross-Section' method probably isn't good. So, I should use 'Fixed/Random Effect.' This plan has been put in doubt due to further review of literature.
3. Plan B, your suggestion. When I looked at how other did their analysis on measurements collected over time, I found that none of them used the methods I stated above. They just used k-way ANOVA most of the time. At best, they included Kruskal-Wallis and Principle Component Analysis. What also seem strange to me is that, even though they collected multiple pollutant measurements at different times, they presented only 1 single value in the their papers. Did they some how take an average? If so, what's the averaging method?
I would appreciate any suggestions.
Dear Yuan,
There are many approaches to deal with the combination of longitudinal and multiple response variables. This is an very interesting paper http://smm.sagepub.com/content/20/4/299.abstract.
Some alternatives are
1 - Analyse each outcome using orthodox approaches for longitudinal and repeated measures. You can use for example linear mixed models as implemented in the lme4 https://cran.r-project.org/web/packages/lme4/index.html or GEE https://www.jstatsoft.org/article/view/v015i02 R packages. Of course, you will ignore the possible correlation between outcomes and you have the inconvenient in your example to interpret 60 models.
2 - You can use a statistical model framework that was designed explicitly to deal with multiple response variables and longitudinal and/or repeated measures structures, as for example the class of multivariate covariance generalized linear models http://onlinelibrary.wiley.com/doi/10.1111/rssc.12145/abstract. It will easily deal with multiple responses and take into account the longitudinal structure in your data. However, in your example it can be hard since you have a 60 x 60 correlation matrix to estimate along with regression and covariance structures of each response variable. It is a really large model and it can be hard to fit and interpret.
3 - You can use a two stage approach. In the first step, you use some technique for dimension reduction as for example principal components (here an example in R https://www.r-bloggers.com/computing-and-visualizing-pca-in-r/) or factor analysis. Based on one of these techniques you construct let say k < 60 new outcomes that will be a combination of the original 60 outcomes. In the second step, you use the techniques in 2 to analyse the new outcomes. This approach has the disadvantage that your interpretation will corresponds to the new outcomes and not to the original ones.
The best approach will depend on your data and goals.
However, I am not sure that you really have multiple response variables. If for all filters you measure the same attribute then you have just one response variable and the filters are just a factor with 60 levels. Thus, you can use the approaches in 1.
Good luck!
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