A 'method' for objects of class phylog.cancor. Shows the original data, and provides p-values and quantiles of the canonical correlations based on the simulated data. There is a print 'method' for this summary.
# S3 method for phylog.cancor
summary(object, ...)an object of class phylog.cancor returned from a previous call to cancor.phylog.
further arguments passed to or from other methods (currently not used)
A list (of class summary.phylog,cancor) with elements
the call to function cancor.phylog.
the likelihood ratio statistic for the test that all canonical correlations are zero
the canonical correlations corresponding to the original (''real'') data set.
the p-value for the test that all canonical correlations are zero
the correlation-wise p-value ---see Details
the multiple correlations p-value; see Details
the quantiles from the simulated canonical correlations; linear interpolation is used. Note that these quantiles are in the spirit of the ''naive p.values''.
the number of simulations used in the analyses
It is necessary to be careful with the null hypothesis you are testing and how the null data set ---the simulations--- are generated. For instance, suppose you want to examine the canonical correlations between sets x and y; you will probably want to generate x and y each with the observed correlations within each set so that the correlations within each set are maintained (but with no correlations among sets). You probably do not want to generate each of the x's as if they were independent of each other x, and ditto for y, since that will destroy the correlations within each set; see some discussion in Manly, 1997.
To test the hypothesis that all population canonical correlations are zero we use the likelihood-ratio statistic from Krzanowski (pp. 447 and ff.); this statistic is computed for the original data set and for each of the simulated data sets, and we obtain the p-value as (number of simulated data sets with LR statistic larger than original (''real'') data + 1) / (number of simulated data sets + 1). Note that a test of this same hypothesis using the Union-Intersection approach is equivalent to the test we implement below for the first canonical correlation.
The p-values for the individual canonical correlations are calculated in two different ways. For the 'component-wise' ones the p-value for a particular correlation is (number of simulated data sets with canonical correlation larger than original (''real'') data + 1) / (number of simulated data sets + 1). With this approach, you can find that the p-value for, say, the second canonical correlation is smaller than the first, which is not sensible. It only makes sense to examine the second canonical correlation if the first one is ''significant'', etc. Thus, when considering the significance of the second canonical correlation we should account for the value of the first. In other words, there is only support against the null hypothesis (of no singificant second canonical correlation) if both the first and the second canonical correlations from the observed data set are larger than most of the simulated data sets. We can account for what happens with the first canonical correlation by computing the p-value of the second canonical correlation as the number of simulations in which the second simulated canonical correlation is larger than the observed, or the first simulated canonical correlation is larger than the observed one, or both (so that the only cases that count agains the null are those where both the first ans second canonical correlations are smaller than the observed ones); these we call 'Multiple' p-values.
Diaz-Uriarte, R., and Garland, T., Jr., in prep. PHYLOGR: an R package for the analysis of comparative data via Monte Carlo simulations and generalized least squares approaches.
Krzanowski, W. J. (1990) Principles of multivariate analysis Oxford University Press.
Manly,B. F. J. (1997) Randomization, bootstraping, and Monte Carlo methods in biology, 2nd ed. Chapman & Hall.
Morrison, D. F. (1990) Multivariate statistcal methods, 3rd ed. McGraw-Hill.
# NOT RUN {
data(SimulExample)
ex1.cancor <- cancor.phylog(SimulExample[,c(1,2,3,4,5,6)],SimulExample[,c(1,2,7,8)])
summary(ex1.cancor)
# }
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