Two-Sample Multivariate Tests
Mahon & Campbell (1974) recorded data on 200 specimens of Leptograpsus variegatus
crabs on the shore of Western Australia. This occurs in two color forms, blue and orange,
and they collected 50 of each form and sex.
The measurements were: carapace (shell) length CL, carapace width CW, the size the
frontal lobe FL, rear width RW, and body depth BD.
Assess the multivariate normality of the data contained in the data frame Crabs. The
species is contained in the data set. Be sure to take into consideration the fact that there
are two species and two sexes of crabs present in these data.
We will perform tests to contrast the differences between male and female crabs within
species, and also contrast the differences between orange and blue crabs within sex. We
first will form two subsets of the original data. One subset will contain the data on the
blue crabs and theother will contain the data on orange crabs.
> attach(Crabs)
> Blue.Crabs <- Crabs[sp=="B",]
> Orange.Crabs <- Crabs[sp=="O",]
> names(Blue.Crabs)
[1] "cgrp" "sex" "sp" "FL" "RW" "CL" "CW" "BD"
> Blue.Crabs <- Blue.Crabs[,-c(1,3)] # Remove columns for crab group and species
> Orange.Crabs<- Orange.Crabs[,-c(1,3)] # Remove columns for crab group and species
> detach("Crabs")
> attach(Blue.Crabs)
> BC.mat <- Blue.Crabs[,2:6] # Extract the columns containing measurements
> BC.mat[1:4,]
FL RW
CL
CW BD
1 8.1 6.7 16.1 19.0 7.0
2 8.8 7.7 18.1 20.8 7.4
3 9.2 7.8 19.0 22.4 7.7
4 9.6 7.9 20.1 23.1 8.2
> MVttest2(BC.mat,sex,pv=T,box=T,lcp=T)
Perform two-sample T2 test. The required arguments to the function are the data matrix
and the variable used to classify the two groups. In this case BC.mat contains the
measurements and sex contains and indicator for the sex of the crab
(F = female, M = male).
The other options are not required. Setting pv = T indicates that you wish the two sample
variance covariance matrices to be printed (default = F). Setting box = T creates
comparative boxplots for each of the variables (default = F). Setting lcp = T plots the
values of the maximizing linear combination for both groups (default = F). Look for a
great deal of separation between the linear combination scores when the null hypothesis
is rejected.
Output from MVttest2 function (Blue Crabs)
Difference in mean vectors
==================================================================
FL
RW
CL
CW
BD
1.572 -0.42 3.912 4.186 1.534
Sample Variance-Covariance Matrices
===========================================
S1 ( M )
======================================================================
FL
RW
CL
CW
BD
FL 10.26 6.54 23.30 26.61 10.17
RW 6.54 4.46 15.08 17.26 6.55
CL 23.30 15.08 53.42 60.98 23.24
CW 26.61 17.26 60.98 69.78 26.59
BD 10.17 6.55 23.24 26.59 10.24
S2 ( F )
======================================================================
FL
RW
CL
CW
BD
FL 6.91 6.28 15.47 17.80 7.10
RW 6.28 5.95 14.25 16.39 6.56
CL 15.47 14.25 35.04 40.21 16.14
CW 17.80 16.39 40.21 46.29 18.53
BD 7.10 6.56 16.14 18.53 7.58
Test for Equality of Means
=======================================================================
Hotelling's T2 = 241.969 Critical Value = 12.048 p-value= 0
The linear combination most responsible for rejection of the null
hypothesis
lin.comb
FL -2.785
RW -5.205
CL 2.822
CW 0.401
BD -0.552
95% Simultaneous C.I. for Mean Differences (T^2)
===========================================================
LCL
UCL
FL -0.4615290 3.605529
RW -2.0035212 1.163521
CL -0.7048465 8.528846
CW -1.1025639 9.474564
BD -0.5379137 3.605914
95% Simultaneous C.I. for Mean Differences (Bonferroni)
===========================================================
LCL
UCL
FL 0.03299663 3.1110034
RW -1.61843114 0.7784311
CL 0.41790554 7.4060945
CW 0.18354035 8.1884597
BD -0.03405348 3.1020535
Graphs for Selected Options (Blue Crabs)
Comparative Boxplots for Each Variable
Now consider the similar tests for the orange crabs.
>
>
>
>
detach("Blue.Crabs")
attach(Orange.Crabs)
OC.mat <- Orange.crabs[,2:6]
Mvttest2(OC.mat,sex,pv=T,box=T,lcp=T)
Output from MVttest2 function (Orange Crabs)
Difference in mean vectors
==================================================================
FL
RW
CL
CW
BD
-0.968 -2.574 -0.93 -1.848 -0.308
Sample Variance-Covariance Matrices
=======================================
S1 ( M )
===================================================================
FL
RW
CL
CW
BD
FL 12.36 7.60 26.62 29.31 12.32
RW 7.60 4.82 16.53 18.23 7.65
CL 26.62 16.53 57.93 63.77 26.76
CW 29.31 18.23 63.77 70.35 29.49
BD 12.32 7.65 26.76 29.49 12.44
S2 ( F )
===================================================================
FL
RW
CL
CW
BD
FL 8.84 6.73 17.17 19.25 8.01
RW 6.73 5.52 13.40 15.05 6.21
CL 17.17 13.40 34.07 38.08 15.91
CW 19.25 15.05 38.08 42.80 17.76
BD 8.01 6.21 15.91 17.76 7.58
Test for Equality of Means
===============================================================
Hotelling's T2 = 447.033 Critical Value = 12.048 p-value= 0
The linear combination most responsible for rejection of the null
hypothesis
lin.comb
FL -1.121
RW -7.121
CL 3.968
CW -1.258
BD 0.550
95% Simultaneous C.I. for Mean Differences (T^2)
===========================================================
LCL
UCL
FL -3.228160 1.2921598
RW -4.152142 -0.9958583
CL -5.638423 3.7784227
CW -7.069558 3.3735580
BD -2.504263 1.8882631
95% Simultaneous C.I. for Mean Differences (Bonf)
===========================================================
LCL
UCL
FL -2.678521 0.7425208
RW -3.768360 -1.3796401
CL -4.493401 2.6334006
CW -5.799749 2.1037486
BD -1.970163 1.3541629
Graphs from Selected Options (Orange Crabs)
Comparative Boxplots for each Variable
We now consider testing differences between species within a given sex. To do this we
again first form subsets of our data, this time corresponding to sex.
>
>
>
>
>
detach("Orange.Crabs")
attach(Crabs)
Male.Crabs <- Crabs[sex=="M",]
Female.Crabs <- Crabs[sex=="F",]
detach("Crabs")
> attach(Male.Crabs)
> MC.mat <- Male.Crabs[,4:8]
> MVttest2(MC.mat,sp,pv=T,box=T,lcp=T)
Output from MVttest2 (Male Crabs)
Difference in mean vectors
==================================================================
FL
RW
CL
CW
BD
-1.784 -0.544 -1.674 -0.378 -1.974
Sample Variance-Covariance Matrices
===========================================
S1 ( B )
======================================================================
FL
RW
CL
CW
BD
FL 10.26 6.54 23.30 26.61 10.17
RW 6.54 4.46 15.08 17.26 6.55
CL 23.30 15.08 53.42 60.98 23.24
CW 26.61 17.26 60.98 69.78 26.59
BD 10.17 6.55 23.24 26.59 10.24
S2 ( O )
======================================================================
FL
RW
CL
CW
BD
FL 12.36 7.60 26.62 29.31 12.32
RW 7.60 4.82 16.53 18.23
7.65
CL 26.62 16.53 57.93 63.77 26.76
CW 29.31 18.23 63.77 70.35 29.49
BD 12.32 7.65 26.76 29.49 12.44
Test for Equality of Means
===============================================================
Hotelling's T2 = 916.162 Critical Value = 12.048 p-value= 0
The linear combination most responsible for rejection of the null
hypothesis
lin.comb
FL -5.408
RW -3.766
CL -3.762
CW 11.087
BD -11.573
95% Simultaneous C.I. for Mean Differences (T^2)
===========================================================
LCL
UCL
FL -4.118179 0.5501790
RW -2.039355 0.9513555
CL -6.853814 3.5058136
CW -6.188841 5.4328413
BD -4.311756 0.3637559
95% Simultaneous C.I. for Mean Differences (Bonf)
===========================================================
LCL
UCL
FL -3.550540 -0.01746044
RW -1.675706 0.58770610
CL -5.594156 2.24615589
CW -4.775727 4.01972655
BD -3.743247 -0.20475340
Output for Selected Options (Male Crabs)
Repeat for female crabs.
>
>
>
>
detach("Male.Crabs")
attach(Female.Crabs)
FC.mat <- Female.Crabs[,4:8]
MVttest2(FC.mat,spec,pv=T,box=T,lcp=T)
Output and graphs not shown.