We demonstrated how to draw a probability plot of the generalized distances against a Chi-Square distribution using the bone density data below. We discussed tests of equal variances and covariances using the lizard data below.
An example of a macro for determining generalized distances for plotting a Chi-Square probability plot for assessing multivariate normality follows:
GMacro
GeneralizedDistances
# First, make a column of means of the variables
# by putting them into separate rows and then transposing the rows into a column
Statistics '<>Dradius'-'<>ulna';
Mean c20-c25.
Transpose c20-c25;
STORE c27.
#Clean up the worksheet
Erase c20-c25.
Name c20 "means".
Let c20 = c27
Erase c26-c27.
# Second, make a column of data for each individual
Transpose '<>Dradius'-'<>ulna';
STORE c21-c44.
# Third, store the variance-covariance matrix
Name m1 "COVARIANCE"
Covariance '<>Dradius'-'<>ulna' 'COVARIANCE'.
# Fourth, Invert the variance-covariance matrix
name m2 "Inverse"
Inverse 'COVARIANCE' 'INVERSE'.
#Fifth, copy each column of data for each individual into a matrix
#subtract the mean and calculate its generalized distance:
Name m3 "data"
Name m4 "dataT"
Name k2 'answer'
Name c45 "distances"
DO k1 = 21:44
copy ck1 'data'
subtract 'means' 'data' 'data'
Transpose 'data' 'dataT'
Multiply 'dataT' 'Inverse' M6
Multiply M6 'data' 'answer'
let k3 = k1 - 20
let 'distances'(k3) = 'answer'
ENDDO
#sort distances in place
Sort 'distances' 'distances';
By 'distances'.
# form a column of probability ranks
Let k1 = N('distances')
Name c46 "Ranks"
Set 'Ranks'
1( 1 : k1 / 1 )1
End.
Let 'Ranks' = ('Ranks' - .5)/k1
# Find the chisquare for each probability.
# Since there are 6 responses, we have 6 d.f. for the distribution
Name C47 "ChiSq"
InvCDF 'Ranks' 'ChiSq';
ChiSquare 6.
#plot
Fitline 'ChiSq' 'distances';
Confidence 95.0.
EndMacro
These examples are taken from Johnson and Wichern: Applied Multivariate Statistical Analysis. The first set of data consists of measurements of body size in male and female Cophosaurus texanus lizards. The data were provided by Kevin E, Bonine. The second set of data consists of bone mineral content before and after treatment for bone loss and were provided by Everett Smith.
Sex weight(grams) Snout-Vent-Length(mm) Hind-Limb-Span(mm) f 5.526 59.0 111.5 m 10.401 75.0 142.0 f 9.213 69.0 124.0 f 8.953 67.5 125.0 m 7.063 62.0 129.5 f 6.610 62.0 123.0 m 11.273 74.0 140.0 f 2.445 47.0 97.0 m 15.493 86.5 162.0 f 9.004 69.0 126.5 m 8.199 70.5 136.0 f 6.601 64.5 116.0 m 7.622 67.5 135.0 m 10.067 73.0 136.5 m 10.091 73.0 135.5 m 10.888 77.0 139.0 f 7.610 61.5 118.0 m 7.733 66.5 133.5 m 12.015 79.5 150.0 m 10.049 74.0 137.0 f 5.149 59.5 116.0 f 9.158 68.0 123.0 m 12.132 75.0 141.0 f 6.978 66.5 117.0 f 6.890 63.0 117.0
subject Dradius radius Dhumerus humerus Dulna ulna 1yrDradius 1yrradius 1yrDhumerus 1yrhumerus 1yrDulna 1yrulna DR r Dh h DU u 1 1.103 1.052 2.139 2.238 0.873 0.872 1.027 1.051 2.268 2.246 0.869 0.964 -0.076 -0.001 0.129 0.008 -0.004 0.092 2 0.842 0.859 1.873 1.741 0.590 0.744 0.857 0.817 1.718 1.710 0.602 0.689 0.015 -0.042 -0.155 -0.031 0.012 -0.055 3 0.925 0.873 1.887 1.809 0.767 0.713 0.875 0.880 1.953 1.756 0.765 0.738 -0.050 0.007 0.066 -0.053 -0.002 0.025 4 0.857 0.744 1.739 1.547 0.706 0.674 0.873 0.698 1.668 1.443 0.761 0.698 0.016 -0.046 -0.071 -0.104 0.055 0.024 5 0.795 0.809 1.734 1.715 0.549 0.654 0.811 0.813 1.643 1.661 0.551 0.619 0.016 0.004 -0.091 -0.054 0.002 -0.035 6 0.787 0.779 1.509 1.474 0.782 0.571 0.640 0.734 1.396 1.378 0.753 0.515 -0.147 -0.045 -0.113 -0.096 -0.029 -0.056 7 0.933 0.880 1.695 1.656 0.737 0.803 0.947 0.865 1.851 1.686 0.708 0.787 0.014 -0.015 0.156 0.030 -0.029 -0.016 8 0.799 0.851 1.740 1.777 0.618 0.682 0.886 0.806 1.742 1.815 0.687 0.715 0.087 -0.045 0.002 0.038 0.069 0.033 9 0.945 0.876 1.811 1.759 0.853 0.777 0.991 0.923 1.931 1.776 0.844 0.656 0.046 0.047 0.120 0.017 -0.009 -0.121 10 0.921 0.906 1.954 2.009 0.823 0.765 0.977 0.925 1.933 2.106 0.869 0.789 0.056 0.019 -0.021 0.097 0.046 0.024 11 0.792 0.825 1.624 1.657 0.686 0.668 0.825 0.826 1.609 1.651 0.654 0.726 0.033 0.001 -0.015 -0.006 -0.032 0.058 12 0.815 0.751 2.204 1.846 0.678 0.546 0.851 0.765 2.352 1.980 0.692 0.526 0.036 0.014 0.148 0.134 0.014 -0.020 13 0.755 0.724 1.508 1.458 0.662 0.595 0.770 0.730 1.470 1.420 0.670 0.580 0.015 0.006 -0.038 -0.038 0.008 -0.015 14 0.880 0.866 1.786 1.811 0.810 0.819 0.912 0.875 1.846 1.809 0.823 0.773 0.032 0.009 0.060 -0.002 0.013 -0.046 15 0.900 0.838 1.902 1.606 0.723 0.677 0.905 0.826 1.842 1.579 0.746 0.729 0.005 -0.012 -0.060 -0.027 0.023 0.052 16 0.764 0.757 1.743 1.794 0.586 0.541 0.756 0.727 1.747 1.860 0.656 0.506 -0.008 -0.030 0.004 0.066 0.070 -0.035 17 0.733 0.748 1.863 1.869 0.672 0.752 0.765 0.764 1.923 1.941 0.693 0.740 0.032 0.016 0.060 0.072 0.021 -0.012 18 0.932 0.898 2.028 2.032 0.836 0.805 0.932 0.914 2.190 1.997 0.883 0.785 0.000 0.016 0.162 -0.035 0.047 -0.020 19 0.856 0.786 1.390 1.324 0.578 0.610 0.843 0.782 1.242 1.228 0.577 0.627 -0.013 -0.004 -0.148 -0.096 -0.001 0.017 20 0.890 0.950 2.187 2.087 0.758 0.718 0.879 0.906 2.164 1.999 0.802 0.769 -0.011 -0.044 -0.023 -0.088 0.044 0.051 21 0.688 0.532 1.650 1.378 0.533 0.482 0.673 0.537 1.573 1.330 0.540 0.498 -0.015 0.005 -0.077 -0.048 0.007 0.016 22 0.940 0.850 2.334 2.225 0.757 0.731 0.949 0.900 2.130 2.159 0.804 0.779 0.009 0.050 -0.204 -0.066 0.047 0.048 23 0.493 0.616 1.037 1.268 0.546 0.615 0.463 0.637 1.041 1.265 0.570 0.634 -0.030 0.021 0.004 -0.003 0.024 0.019 24 0.835 0.752 1.509 1.422 0.618 0.664 0.776 0.743 1.442 1.411 0.585 0.640 -0.059 -0.009 -0.067 -0.011 -0.033 -0.024 25 0.915 0.936 1.971 1.869 0.869 0.868 * * * * * * * * * * * *