Dear friends,
I am trying to find a solution to statistically compare the overall correlation strength between variables in two datasets. The two datasets included measures of the same variables. I want to see whether the overall correlation strength between the variables in one dataset is stronger than the other. To do so, I conducted PCA on the two dataset separately, and calculated the variance explained by the first principle component. If the variance explained by the first principle component is high, in theory the variables are overall more linearly corrected. So now I have two PC1, each for one dataset. My question is how I can perform a statistical test on the difference, so I can say the PC1 for the first dataset is significantly higher or lower than the PC1 for the second dataset.
Many thanks,
Qiang
Since these will be based on different weights of the variables (i.e., the first eigenvector for each), I'm curious what you are trying to conclude. Also, you don't say if it is the same units for each (e.g., the people in the first dataset matched one-to-one with those in the second. Here is some R code for the percentile bootstrap for both of these (and you can do more things like the boot library).
x1
Your purpose is not dimensionallity-reduction, PCA does that mainly, based on correlation or covariance matrices.
Your purpose is merely finding and comparing correlation strenght between a set of variables. Then you better calculate a correlation matrix for each data set and compare them using the p-values associated with each correlation coefficient in the matrix (Pearson, Spearman or other). I think this approach is more straightforward for your purpose.
Ette Etuk Thank you for the suggestion. However, as I am actually comparing two covariance matrices, I am not sure F test can be used. F-test seems more used for comparing the variance of two samples that measure the same variable.
Iván Felipe Benavides Thank you for the suggestion. But here I am actually doing some kind of dimensionality reduction. My question is similar as which sample has a lower/higher dimensionality? I would say if the first or the first few principal components of a sample explain considerable variance than the other sample, then the first sample has a lower dimensionality. But I also want to have some stats for the comparison.
Ok
Then, from my experience you would have 2 options.
1) Just compare the correlation values by themselves and according to your knowledge of your study system, stabish a threshold to conclude if the difference is big or small.
2) Run the PCA N times, each time with a different random subset of cases (with replacement). Thus, you willl have a distribution of N correlation values for each data set. Then you can do different things: 1) use a t-test to compare between the two distributions, 2) compare their histograms, 3) Calculate confidence intervals to conclude if the difference is statistically significant.
Feel free to contact (inbox) if you want further help.
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