Graphical 'Test' of Multivariate Normality
------------------------------------------

AUTHOR:   Richard Goldstein, Qualitas
SUPPORT:  Written communication only, EMAIL goldst@@harvarda.bitnet
	  or 37 Kirkwood Road, Brighton MA 02135.

               ^multnorm^ varlist [^in^ range] [^if^ exp]

^multnorm^ is a graphical procedure for examining multivariate normality, via
diagnostics from a standard linear regression.  Thus, you must enter at least 
2 variables; do not enter any options other than ^in^ and/or ^if^.  Cases with 
missing values on any variable in varlist are dropped prior to producing the 
new variables and the graph.

'Test' of multivariate normality (actually a graph); taken from Stevens, J
(1986) ^Applied Multivariate Statistics for the Social Sciences^, Hillsdale:
L Erlbaum Assoc., Publishers, pp. 207-212, and from Thompson, B (1990),
"MULTINOR:  A Fortran Program that Assists in Evaluating Multi-variate
Normality", ^Educational and Psychological Measurement^, 50:845-8.  Note that
Stevens contains some typo's and approximations; although Thompson does not
mention any problem with Stevens' calculation of Mahalanobis Distance,
Thompson's graph agrees with that calculated here for Stevens' data and NOT
with Stevens' calculations or his graph.  Note also that the formula for
Mahalanobis Distance used here is considered 'inappropriate' for use as a
measure of leverage by Velleman PF and Welsch RE (1981), "Efficient Computing
of Regression Diagnostics", ^The American Statistician^, 35:  234-242; the
variable MD2 used and reported here, and used by Thompson and, apparently,
Stevens, appears as equation 29 in Velleman & Welsch.

If the variables are multivariate normal, then the graph will approximate
a 45-degree line.

Note that the calculations are relatively slow.

In the two examples below, the graphs are not shown; the first example is
the Stevens data, and the data are shown first.  The second example is the
Stata auto.dta provided with Stata to all users.

Example One-Stevens Data:
-------------------------

	. ^use stevens^

	. ^list^

	            wi         wc         pc
	  1.       5.8        9.7        8.9
	  2.      10.6       10.9         11
	  3.       8.6        7.2        8.7
	  4.       4.8        4.6        6.2
	  5.       8.3       10.6        7.8
	  6.       4.6        3.3        4.7
	  7.       4.8        3.7        6.4
	  8.       6.7          6        7.2
	  9.       7.1        8.4        8.4
	 10.       6.2          3        4.3
	 11.       4.2        5.3        4.2
	 12.       6.9        9.7        7.2
	 13.       5.6        4.1        4.3
	 14.       4.8        3.8        5.3
	 15.       2.9        3.7        4.2
	 16.       6.1        7.1        8.1
	 17.      12.5       11.2        8.9
	 18.       5.2        9.3        6.2
	 19.       5.7       10.3        5.5
	 20.         6        5.7        5.4
	 21.       5.2        7.7        6.9
	 22.       7.2        5.8        6.7
	 23.       8.1        7.1        8.1
	 24.       3.3          3        4.9
	 25.       7.6        7.7        6.2
	 26.       7.7        9.7        8.9

	. ^multnorm wi wc pc^

	           MD2       chi2
	  1.  .6940832   .1798865
	  2.  .8292803    .389966
	  3.  .9431213   .5674309
	  4.  1.194305   .7331322
	  5.  1.280711   .8938028
	  6.  1.306861   1.052871
	  7.  1.406288   1.212533
	  8.  1.423692   1.374434
	  9.  1.727728   1.539968
	 10.  1.770294   1.710439
	 11.  1.787973   1.887161
	 12.  1.795986   2.071538
	 13.   2.00243   2.265146
	 14.  2.381555   2.469824
	 15.  2.413529   2.687782
	 16.  2.671461    2.92176
	 17.  2.716661   3.175261
	 18.  2.751626   3.452895
	 19.  2.758187   3.760947
	 20.  2.808616   4.108345
	 21.  3.925842   4.508442
	 22.   4.90097   4.982584
	 23.  5.404338    5.56821
	 24.  5.893517   6.340872
	 25.  7.680526   7.494797
	 26.  10.53042   9.923156

Example Two--STATA auto data:
-----------------------------

	. ^use auto^
	(1978 Automobile Data)

	. ^multnorm price mpg weight^

	           MD2       chi2
	  1.  .4823955   .0879764
	  2.  .4881752   .1865578
	  3.  .4927304   .2664138
	  4.  .5064312   .3381441
	  5.  .5876299   .4050903
	  6.   .589836   .4688687 
	  7.  .6252201   .5304068 
	  8.  .6869202    .590303 
	  9.  .7496714   .6489731
	 10.  .8043185   .7067225
	 11.  .8366966   .7637849
	 12.  .8725201   .8203464
	 13.  .9046813   .8765573
	 14.   .991994   .9325455
	 15.  1.024452   .9884196
	 16.  1.033603   1.044274
	 17.  1.064712   1.100195
	 18.  1.132212   1.156258
	 19.  1.140015   1.212533
	 20.  1.183668   1.269087
	 21.  1.190048   1.325981
	 22.  1.232133   1.383276
	 23.  1.240343    1.44103
	 24.  1.281489   1.499298
	 25.  1.301269   1.558136
	 26.  1.314231   1.617602
	 27.  1.328227   1.677749
	 28.   1.38066   1.738637
	 29.  1.387493   1.800322
	 30.  1.401482   1.862865
	 31.  1.430728   1.926327
	 32.  1.465629   1.990772
	 33.  1.569834   2.056267
	 34.  1.571902   2.122884
	 35.  1.620626   2.190695
	 36.  1.674173   2.259777
	 37.  1.785907   2.330218
	 38.  1.898156   2.402103
	 39.  1.903607   2.475529
	 40.  1.929491   2.550598
	 41.  2.197038    2.62742
	 42.  2.277208   2.706115
	 43.   2.30725   2.786811
	 44.  2.367661   2.869651
	 45.  2.514124   2.954788
	 46.  2.571822   3.042392
	 47.  2.699493   3.132652
	 48.  2.720322   3.225772
	 49.  2.812051   3.321984
	 50.  2.836437   3.421544
	 51.  2.854816    3.52474
	 52.  3.111239   3.631898
	 53.  3.316182   3.743385
	 54.  3.316417   3.859622
	 55.  3.436041   3.981088
	 56.  3.439405   4.108345
	 57.  3.561527    4.24204
	 58.  3.946049   4.382935
	 59.  4.056426   4.531934
	 60.  4.751401   4.690122
	 61.  4.832843   4.858812
	 62.  4.941287   5.039619
	 63.  5.107936   5.234561
	 64.  5.687628   5.446204
	 65.  6.183751   5.677877
	 66.  6.207568   5.934019
	 67.  6.556265   6.220731
	 68.  6.850253   6.546726
	 69.  6.876239   6.925084
	 70.  7.460886   7.376768
	 71.  7.720632   7.938627
	 72.  8.366157   8.685135
 	 73.  16.45519    9.80805
	 74.  18.55515   12.19098