Discriminant Analysis Output
Using School Referrals
In the School Referrals file, the variable MDT is actually a grouping variable created using a cluster analysis routine. The other variables we will use are:
- VIQ = Verbal IQ from the Wechsler Intelligence Test
- PIQ = Performance IQ from the Wechsler Intelligence Test
- READ = the Reading score from an individual achievement test
- ARITH = an Arithmetic scor from the achievement test
- UNX = a score that measures under-reaction (shyness) in children
- OVX = a score that measures over-reaction (agression) in children
First you should choose MGLH/Fully factorial (M)ANOVA from the menus.
Next fill out the following dialog box like that shown below. Be sure to click More and Extended Output.
LEVELS ENCOUNTERED DURING PROCESSING ARE:
MDT
1.000 2.000 3.000 4.000 5.000 6.000
7.000
NUMBER OF CASES PROCESSED: 200
DEPENDENT VARIABLE MEANS
VIQ PIQ READ ARITH UNX
85.855 90.430 87.750 81.525 4.205
OVX
8.080
-1
ESTIMATES OF EFFECTS B = (X'X) X'Y
VIQ PIQ READ ARITH
CONSTANT 81.683 87.701 98.277 83.130
MDT 1.000 -16.983 -18.251 -30.527 -17.730
MDT 2.000 16.317 6.999 -6.777 1.170
MDT 3.000 3.180 2.806 -11.531 -2.137
MDT 4.000 20.442 14.549 3.848 6.995
MDT 5.000 -15.683 0.299 10.723 17.870
MDT 6.000 28.408 27.299 14.541 21.961
UNX OVX
CONSTANT 6.097 8.316
MDT 1.000 -4.297 -0.566
MDT 2.000 -3.997 18.434
MDT 3.000 -1.923 -2.845
MDT 4.000 6.778 -4.441
MDT 5.000 8.903 -6.316
MDT 6.000 -1.369 4.048
STANDARDIZED ESTIMATES OF EFFECTS
VIQ PIQ READ ARITH
CONSTANT 0.000 0.000 0.000 0.000
MDT 1.000 -0.353 -0.365 -0.648 -0.467
MDT 2.000 0.339 0.140 -0.144 0.031
MDT 3.000 0.101 0.086 -0.376 -0.086
MDT 4.000 0.287 0.196 0.055 0.124
MDT 5.000 -0.129 0.002 0.090 0.186
MDT 6.000 0.456 0.422 0.239 0.448
UNX OVX
CONSTANT 0.000 0.000
MDT 1.000 -0.294 -0.020
MDT 2.000 -0.274 0.642
MDT 3.000 -0.202 -0.152
MDT 4.000 0.314 -0.104
MDT 5.000 0.241 -0.087
MDT 6.000 -0.073 0.109
TOTAL SUM OF PRODUCT MATRIX
VIQ PIQ READ ARITH UNX
VIQ 44536.795
PIQ 30346.470 48057.020
READ 23596.750 15546.500 42543.500
ARITH 21420.225 19051.850 21519.250 27693.875
UNX -120.055 -406.630 1822.250 262.475 4088.595
OVX 6933.320 4690.120 2560.000 2070.600 -1523.280
OVX
OVX 15828.720
RESIDUAL SUM OF PRODUCT MATRIX E'E = Y'Y-Y'XB
VIQ PIQ READ ARITH UNX
VIQ 21544.368
PIQ 10806.088 29791.143
READ 7581.738 -84.246 23739.385
ARITH 6165.046 4000.457 7274.803 14137.777
UNX -1557.498 -2224.524 -519.333 -1609.428 3041.883
OVX 743.211 1901.029 471.588 -62.268 -383.838
OVX
OVX 7433.305
RESIDUAL COVARIANCE MATRIX S
Y.X
VIQ PIQ READ ARITH UNX
VIQ 111.629
PIQ 55.990 154.358
READ 39.284 -0.437 123.002
ARITH 31.943 20.728 37.693 73.253
UNX -8.070 -11.526 -2.691 -8.339 15.761
OVX 3.851 9.850 2.443 -0.323 -1.989
OVX
OVX 38.515
RESIDUAL CORRELATION MATRIX R
Y.X
VIQ PIQ READ ARITH UNX
VIQ 1.000
PIQ 0.427 1.000
READ 0.335 -0.003 1.000
ARITH 0.353 0.195 0.397 1.000
UNX -0.192 -0.234 -0.061 -0.245 1.000
OVX 0.059 0.128 0.036 -0.006 -0.081
OVX
OVX 1.000
SQUARED MULTIPLE CORRELATIONS
VIQ PIQ READ ARITH UNX
0.516 0.380 0.442 0.489 0.256
OVX
0.530
LEAST SQUARES MEANS.
MDT = 1.000 N OF CASES = 20.000
VIQ PIQ READ ARITH UNX
LS. MEAN 64.700 69.450 67.750 65.400 1.800
SE 2.363 2.778 2.480 1.914 0.888
OVX
LS. MEAN 7.750
SE 1.388
MDT = 2.000 N OF CASES = 20.000
VIQ PIQ READ ARITH UNX
LS. MEAN 98.000 94.700 91.500 84.300 2.100
SE 2.363 2.778 2.480 1.914 0.888
OVX
LS. MEAN 26.750
SE 1.388
MDT = 3.000 N OF CASES = 138.000
VIQ PIQ READ ARITH UNX
LS. MEAN 84.862 90.507 86.746 80.993 4.174
SE 0.899 1.058 0.944 0.729 0.338
OVX
LS. MEAN 5.471
SE 0.528
MDT = 4.000 N OF CASES = 8.000
VIQ PIQ READ ARITH UNX
LS. MEAN 102.125 102.250 102.125 90.125 12.875
SE 3.735 4.393 3.921 3.026 1.404
OVX
LS. MEAN 3.875
SE 2.194
MDT = 5.000 N OF CASES = 2.000
VIQ PIQ READ ARITH UNX
LS. MEAN 66.000 88.000 109.000 101.000 15.000
SE 7.471 8.785 7.842 6.052 2.807
OVX
LS. MEAN 2.000
SE 4.388
MDT = 6.000 N OF CASES = 11.000
VIQ PIQ READ ARITH UNX
LS. MEAN 110.091 115.000 112.818 105.091 4.727
SE 3.186 3.746 3.344 2.581 1.197
OVX
LS. MEAN 12.364
SE 1.871
MDT = 7.000 N OF CASES = 1.000
VIQ PIQ READ ARITH UNX
LS. MEAN 46.000 54.000 118.000 55.000 2.000
SE 10.565 12.424 11.091 8.559 3.970
OVX
LS. MEAN 0.000
SE 6.206
Next select MGLH/Test of effects from the Stats menu
Fill out the next two dialogs as shown. Don't forget th click More, Extended View, and Save Scores and Results as a file -- Named CANON here.
TEST FOR EFFECT CALLED: MDT
NULL HYPOTHESIS CONTRAST AB
VIQ PIQ READ ARITH UNX
1 -16.983 -18.251 -30.527 -17.730 -4.297
2 16.317 6.999 -6.777 1.170 -3.997
3 3.180 2.806 -11.531 -2.137 -1.923
4 20.442 14.549 3.848 6.995 6.778
5 -15.683 0.299 10.723 17.870 8.903
6 28.408 27.299 14.541 21.961 -1.369
OVX
1 -0.566
2 18.434
3 -2.845
4 -4.441
5 -6.316
6 4.048
-1
INVERSE CONTRAST A(X'X) A'
1 2 3 4 5
1 0.073
2 0.023 0.073
3 0.029 0.029 0.042
4 0.012 0.012 0.018 0.126
5 -0.041 -0.041 -0.035 -0.052 0.394
6 0.017 0.017 0.023 0.006 -0.047
6
6 0.102
-1 -1
HYPOTHESIS SUM OF PRODUCT MATRIX H = B'A'(A(X'X) A') AB
VIQ PIQ READ ARITH UNX
VIQ 22992.427
PIQ 19540.382 18265.877
READ 16015.012 15630.746 18804.115
ARITH 15255.179 15051.393 14244.447 13556.098
UNX 1437.443 1817.894 2341.583 1871.903 1046.712
OVX 6190.109 2789.091 2088.412 2132.868 -1139.442
OVX
OVX 8395.415
ERROR SUM OF PRODUCT MATRIX G = E'E
VIQ PIQ READ ARITH UNX
VIQ 21544.368
PIQ 10806.088 29791.143
READ 7581.738 -84.246 23739.385
ARITH 6165.046 4000.457 7274.803 14137.777
UNX -1557.498 -2224.524 -519.333 -1609.428 3041.883
OVX 743.211 1901.029 471.588 -62.268 -383.838
OVX
OVX 7433.305
UNIVARIATE F TESTS
VARIABLE SS DF MS F P
VIQ 22992.427 6 3832.071 34.329 0.000
ERROR 21544.368 193 111.629
PIQ 18265.877 6 3044.313 19.722 0.000
ERROR 29791.143 193 154.358
READ 18804.115 6 3134.019 25.479 0.000
ERROR 23739.385 193 123.002
ARITH 13556.098 6 2259.350 30.843 0.000
ERROR 14137.777 193 73.253
UNX 1046.712 6 174.452 11.069 0.000
ERROR 3041.883 193 15.761
OVX 8395.415 6 1399.236 36.330 0.000
ERROR 7433.305 193 38.515
MULTIVARIATE TEST STATISTICS
WILKS' LAMBDA = 0.087
F-STATISTIC = 17.109 DF = 36, 828 PROB = 0.000
PILLAI TRACE = 1.669
F-STATISTIC = 12.392 DF = 36,1158 PROB = 0.000
HOTELLING-LAWLEY TRACE = 3.969
F-STATISTIC = 20.545 DF = 36,1118 PROB = 0.000
THETA = 0.687 S = 6, M =-0.5, N = 93.0 PROB = 0.000
TEST OF RESIDUAL ROOTS
ROOTS 1 THROUGH 6
CHI-SQUARE STATISTIC = 469.947 DF = 36 PROB = 0.000
ROOTS 2 THROUGH 6
CHI-SQUARE STATISTIC = 246.536 DF = 25 PROB = 0.000
ROOTS 3 THROUGH 6
CHI-SQUARE STATISTIC = 87.775 DF = 16 PROB = 0.000
ROOTS 4 THROUGH 6
CHI-SQUARE STATISTIC = 46.009 DF = 9 PROB = 0.000
ROOTS 5 THROUGH 6
CHI-SQUARE STATISTIC = 20.808 DF = 4 PROB = 0.000
ROOTS 6 THROUGH 6
CHI-SQUARE STATISTIC = 0.216 DF = 1 PROB = 0.642
CANONICAL CORRELATIONS
1 2 3 4 5
0.829 0.749 0.442 0.350 0.319
6
0.033
DEPENDENT VARIABLE CANONICAL COEFFICIENTS
STANDARDIZED BY CONDITIONAL (WITHIN GROUPS) STANDARD DEVIATIONS
1 2 3 4 5
VIQ 0.304 -0.368 0.813 -0.104 0.561
PIQ 0.389 0.318 -0.012 -0.225 -0.133
READ 0.288 0.225 -0.864 -0.459 0.467
ARITH 0.472 0.098 0.019 0.306 -0.929
UNX 0.516 0.278 0.008 0.830 0.267
OVX 0.159 -0.899 -0.291 0.286 -0.063
6
VIQ -0.507
PIQ 1.001
READ 0.021
ARITH -0.386
UNX 0.110
OVX 0.145
CANONICAL LOADINGS (CORRELATIONS BETWEEN CONDITIONAL
DEPENDENT VARIABLES AND DEPENDENT CANONICAL FACTORS)
1 2 3 4 5
VIQ 0.644 -0.229 0.506 -0.388 0.278
PIQ 0.510 -0.000 0.302 -0.365 -0.147
READ 0.550 0.090 -0.594 -0.412 0.268
ARITH 0.643 0.056 -0.039 -0.162 -0.636
UNX 0.220 0.309 -0.074 0.832 0.394
OVX 0.192 -0.895 -0.276 0.166 -0.046
6
VIQ -0.221
PIQ 0.703
READ -0.306
ARITH -0.389
UNX 0.055
OVX 0.237
GROUP CLASSIFICATION FUNCTION COEFFICIENTS
1 2 3 4 5
VIQ 0.156 0.358 0.230 0.345 -0.118
PIQ 0.363 0.428 0.487 0.566 0.611
READ 0.317 0.398 0.416 0.492 0.610
ARITH 0.659 0.808 0.794 0.890 1.178
UNX 0.878 1.200 1.235 1.962 2.064
OVX 0.123 0.593 0.038 -0.001 -0.015
6 7
VIQ 0.307 -0.167
PIQ 0.600 0.430
READ 0.535 0.942
ARITH 1.034 0.292
UNX 1.559 0.656
OVX 0.192 -0.117
GROUP CLASSIFICATION CONSTANTS
1 2 3 4 5
-53.183 -101.212 -86.653 -126.354 -133.182
6 7
-142.739 -73.951
CANONICAL SCORES HAVE BEEN SAVED
Open the CANON file and select Tables/Tabulate from the Stats menu.
Fill out the Dialog as follows.
--------------------------------------------------------------------------------
TABLE OF GROUP (ROWS) BY PREDICT (COLUMNS)
FREQUENCIES
1.000 2.000 3.000 4.000 5.000 6.000
-------------------------------------------------------------
1.000 20 0 0 0 0 0
2.000 0 19 1 0 0 0
3.000 8 6 111 9 1 2
4.000 0 0 0 8 0 0
5.000 0 0 0 0 2 0
6.000 0 1 0 0 0 10
7.000 0 0 0 0 0 0
-------------------------------------------------------------
TOTAL 28 26 112 17 3 12
7.000 TOTAL
-----------
1.000 0 20
2.000 0 20
3.000 1 138
4.000 0 8
5.000 0 2
6.000 0 11
7.000 1 1
-----------
TOTAL 2 200
WARNING: MORE THAN ONE-FIFTH OF FITTED CELLS ARE SPARSE (FREQUENCY < 5)
SIGNIFICANCE TESTS ARE SUSPECT
TEST STATISTIC VALUE DF PROB
PEARSON CHI-SQUARE 734.807 36 0.000
LIKELIHOOD RATIO CHI-SQUARE 309.073 36 0.000
MCNEMAR SYMMETRY CHI-SQUARE 25.571 21 0.223
COEFFICIENT VALUE ASYMPTOTIC STD ERROR
PHI 1.917
CRAMER V 0.783
CONTINGENCY 0.887
GOODMAN-KRUSKAL GAMMA 0.926 0.031
KENDALL TAU-B 0.777 0.041
STUART TAU-C 0.512 0.047
COHEN KAPPA 0.750 0.043
SPEARMAN RHO 0.803 0.039
SOMERS D (COLUMN DEPENDENT) 0.879 0.035
LAMBDA (COLUMN DEPENDENT) 0.670 0.051
UNCERTAINTY (COLUMN DEPENDENT) 0.571 0.051
In this case, you can plot these groups using the three best discriminant scores using the 3D-Spinning Plot.
