ENH: multivariate (multiple endpoint) stats

[josef-pkt]

We should get extension for one, 2 or k-sample tests to multivariate samples, i.e. the analogue of some of the t_test, ... to the multivariate case.

one reference (that I didn't read): equivalence tests for multivariate means

Hoffelder, Thomas, Rüdiger Gössl, and Stefan Wellek. 2015. “Multivariate Equivalence Tests for Use in Pharmaceutical Development.” Journal of Biopharmaceutical Statistics 25 (3): 417–37. doi:10.1080/10543406.2014.920344.

missing descriptive statistics
multivariate skewness and kurtosis - Mardia see #3280 for usage in correcting chi2 statistic

[josef-pkt]

I just saw by chance that Stata has a nice collection of multivariate tests mv_test for correlation, covariance and means.
Should be easy enough to implement based on their formula collection.

see also #3280 using robust methods that should be appropriate for elliptically symmetric distributions even with heavier tails than normal.

[josef-pkt]

one reference that looks interesting or is recent enough to provide a starting point for the relevant literature.
multivariate analog of t-test with unequal variances with improved small sample properties

Tamae Kawasaki & Takashi Seo (2015) A Two Sample Test for Mean
Vectors with Unequal Covariance Matrices, Communications in Statistics - Simulation and
Computation, 44:7, 1850-1866, DOI: 10.1080/03610918.2013.824587
http://dx.doi.org/10.1080/03610918.2013.824587

[josef-pkt]

another semi-random idea: unequal covariances but with shrinkage

Suppose the second sample is not large enough to get a reliable covariance estimate in the case of covariance heterogeneity. In that case we could get an intermediate solution between pooled and separate covariance matrices by shrinking the second covariance to the first.
The underlying model is something like a random covariance model or a multivariate GARCH, where our "prior" is that the two covariances are close to each other but not necessarily identical.
Essentially, we would be using a weighted average pooled covariance where the weights need not correspond to the relative sample sizes.

see #3199 #2847 for outlier test (nuisance parameters taken from the first sample) and
#3197 covariance shrinkage and regularization.

a bit related (I have not yet looked closely at those): exponentially weighted moving average, EWMA, control charts for covariance or correlation.

[josef-pkt]

Thinking about multivariate equivalence testing again, a quick search finds also these (not read)
(connection: matching or covariate balancing tests in randomized trials (and IPW matching?): the burden of proof in low powered experiments should be to show equivalence not no difference)

Chervoneva, Inna, Terry Hyslop, and Walter W. Hauck. 2007. “A Multivariate Test for Population Bioequivalence.” Statistics in Medicine 26 (6): 1208–23. doi:10.1002/sim.2605.

Tsukada, Shin-ichi. 2014. “Equivalence Testing of Mean Vector and Covariance Matrix for Multi-Populations under a Two-Step Monotone Incomplete Sample.” Journal of Multivariate Analysis 132 (November): 183–96. doi:10.1016/j.jmva.2014.08.005.

BTW: I have no idea about how to do this. We have avoided multivariate/multi-parameter one sided tests and inequality restrictions so far.

[josef-pkt]

multivariate equivalence

Wang, W., J. T. Gene Hwang, and A. Dasgupta. 1999. “Statistical Tests for Multivariate Bioequivalence.” Biometrika 86 (2): 395–402. https://doi.org/10.1093/biomet/86.2.395.
I found this in a pile of printed articles, Looks at pointwise rejection regions (max_i).

Pallmann, Philip, and Thomas Jaki. 2017. “Simultaneous Confidence Regions for Multivariate Bioequivalence.” Statistics in Medicine 36 (29): 4585–4603. https://doi.org/10.1002/sim.7446.
I don't remember this one, added to zotero at the same time as wang et al.

I added equivalence testing for oneway anova based on Wellek which is equivalence in terms of a joint F-statistic or effect size and not for individual, pointwise bounds.
The equivalent for multivariate mean would be in terms of Mahalanobis distance, i.e. hotelling T statistic. If that is converted to pointwise, then it is very conservative according to Wang et al 1999

There related issues on simultaneous confidence intervals.