I'm trying to figure out how to analyze data for a set of experiments in which subjects were each assigned to a treatment group on day 0, and then the dependent variable was measured at various days throughout the three-week experiment. I think that means I need to do a two-factor ANOVA with repeated measures on one factor.
"Subject-by-trial" design "Day" = "within subject" or "repeated measures" factor "Treatment" = "between subject" factor "Obs" = measured (dependent) variable
If that weren't complicated enough, I don't have an equal number of subjects in each treatment group, and observations are missing for some days for some subjects (hence the "unbalanced" part).
Here's an example of what the data look like:
mydata <- data.frame(
Subject = c(13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65),
Day = c(rep(c("Day1", "Day3", "Day6"), each=28)),
Treatment = c(rep(c("B", "A", "C", "B", "C", "A", "A", "B", "A", "C", "B", "C", "A", "A", "B", "A", "C", "B", "C", "A", "A"), each = 4)),
Obs = c(6.472687, 7.017110, 6.200715, 6.613928, 6.829968, 7.387583, 7.367293, 8.018853, 7.527408, 6.746739, 7.296910, 6.983360, 6.816621, 6.571689, 5.911261, 6.954988, 7.624122, 7.669865, 7.676225, 7.263593, 7.704737, 7.328716, 7.295610, 5.964180, 6.880814, 6.926342, 6.926342, 7.562293, 6.677607, 7.023526, 6.441864, 7.020875, 7.478931, 7.495336, 7.427709, 7.633020, 7.382091, 7.359731, 7.285889, 7.496863, 6.632403, 6.171196, 6.306012, 7.253833, 7.594852, 6.915225, 7.220147, 7.298227, 7.573612, 7.366550, 7.560513, 7.289078, 7.287802, 7.155336, 7.394452, 7.465383, 6.976048, 7.222966, 6.584153, 7.013223, 7.569905, 7.459185, 7.504068, 7.801867, 7.598728, 7.475841, 7.511873, 7.518384, 6.618589, 5.854754, 6.125749, 6.962720, 7.540600, 7.379861, 7.344189, 7.362815, 7.805802, 7.764172, 7.789844, 7.616437, NA, NA, NA, NA))
I know I'm supposed to test for sphericity, and I'm pretty sure I need to use a Type II or III SS because of the unbalanced design. (Even if the example data I gave are not unbalanced, the full dataset is.) Perhaps the cars package is called for?
I've also read that I must appropriately specify the nature of the correlations between measurements in the same subject. The data are approximately evenly spaced in time (the full dataset includes days 1, 3, 6, 9, 12, 15, 18 and 21), so I should probably use AR(1), though perhaps I need to use the Akaike information criterion to determine which structure, as compound symmetry might be appropriate. (And no, I don't fully understand any of that, let alone how to use the Akaike whatnot to decide which structure, or how to specify a structure somewhere once I figure out which one is appropriate.)
I am by now hopelessly confused about exactly what analysis I need to do and how to make it happen in R. I just want to know whether or not the treatment means are significantly different, and if so, on which days. I'm not sure if there's an interaction with time, or how to calculate standard error of the mean and 95% confidence intervals for each time point so I can graph the data. (If the ANOVA results say the treatment effect is significant, can I then run pairwise t-tests as usual? Do I need to do some kind of Bonferroni adjustment because there are several different treatments? Is there a fancy correction I can do to deal with the repeated measures?)
Any help figuring out either precisely what sequence of statistical tests I need to run or how to run them in R would be immensely appreciated.
~A Baffled Biologist