The task: I need to calculate a one-sample t-test to see if the mean of a vector of values differs significantly from zero.
Data: The dataset consist of about 25 such vectors, each with a length of 19 (counts collected at different points in time). So there's the small sample problem, but additionally, some data from these vectors (up to 10-20%) are missing. However, since the vectors are essentially transformed trendlines, it seems it would be possible to linearly interpolate the missing values prior to the transformation.
Edit to answer comments about data: each of the 25 vectors (or rows of a table, or samples, if you will) are conceived the following way: counts of certain occurrences of X and Y were observed on 19 consequtive occasions, and normalized so that each value of a vector reflects the propertion of X (vs Y) at that time point when the observation was made (so they are in essence trends of X). The values in these trend vectors were then further transformed (the function itself is irrelevant here, and a bit too involved). But this yields those vectors; the null hypothesis is that their mean is 0, the alternative is that it is not 0.
Proportions of X over time: x = 0.05 0.09 0.11 0.13 0.14 0.16 0.22 0.23 0.23 0.24 0.34 0.26 0.29 0.37 0.40 0.34 0.48 0.53 0.46 Transformed into: xtr = 0.36 0.14 0.11 0.03 0.08 0.24 0.03 -0.01 0.04 0.27 -0.17 0.07 0.21 0.05 -0.12 0.31 0.09 -0.13 (element at first position gets removed in the process)
One sample t-test of xtr yields p-value of 0.01993; if vector x is interpolated to length 50, the p-value of the t-test on the transformed vector is 0.00005278.
Problem: I am not sure how to intepret the p-values after that though: the more interpolation I add (used the
approx function in R), the more significant results I get - starting with 3 significant (alpha=0.05) out of 25 at no interpolation and only interpolating missing; but going up to 14-17 out of 25 with p<0.05 results, if adding more interpolated datapoints; but interestingly with not much increases when getting up to 180-200 interpolated points. The plot below (going in increments of 10 interpolated points) illustrates this.
How to make sense of that? On the one hand, the data collection was done in a sparse manner across time, so it is reasonable to assume there could be more data in there (if the data were collected more frequently and by doing a broader sweep out there) - this makes intepolation tempting, particularly in cases where there seems to be a qualitatively pretty straightforward upward or downward trend (just with gaps in some places, or just being too small samples in general). Then again, if I add too much, almost each test becomes "significant".
Question: Is there some (justifiable) middleground here - would it be valid to use (some justifiable amount of) interpolation prior to a t-test, and if so, should I (somehow) reconsider the alpha threshold for significance?**