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Discriminant analysis using SAS macro - 1

Discriminant analysis

Data : Diabetic dataStep 1: Creating SAS- data

Step 2 : Perform the Exploratory scatter plots by group by running the SAS macro DISCRIM and inputting YES to data exploration. Also run stepwise discriminantanalysis to select significant variables.

Step 3: Checking for multi-variate normality and outliers by running the SAS macroDISCRIM and inputting YES to assumption check

Step 4 Run Canonical and PARAMETRIC Discriminant Function analyses by running the SAS macro DISCRIM and leaving data exploration field blank .

Step 5: Examine canonical discriminant analysis results and bi-plot display of discrimination.

Step 6:If Multivariate normality assumption is severely violated, perform non-paramet ric discriminant function analysis by inputting YES to non-parametric option.

Step 7: Check the classification results compare the error rates for cross validation

Step 8:If you have independent validation data set, confirm the discrimination results usingthe validation data

Step 9: Summary and Conclusions


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Creating SAS data set:

If you are using the sample data, open the downloaded discrim-data.SAS file andsubmit.If you are using some other data, create a SAS data set.

proc format;

value gp 3='(3) Overt Diabetic ' 2='(2) Chem. Diabetic' 1='(1) Normal';

run;

data diabetic;

input patient relwt glufast glutest instest sspg group $;

label relwt = 'Relative weight'

glufast = 'Fasting Plasma Glucose'

glutest = 'Test Plasma Glucose'

instest = 'Plasma Insulin during Test'

sspg = 'Steady State Plasma Glucose'

group = 'Clinical Group';

datalines;

1 0.81 80 356 124 55 1

2 0.95 97 289 117 76 1

3 0.94 105 319 143 105 1

4 1.04 90 356 199 108 1

5 1.00 90 323 240 143 1

6 0.76 86 381 157 165 1

7 0.91 100 350 221 119 1

8 1.10 85 301 186 105 1

9 0.99 97 379 142 98 1

10 0.78 97 296 131 94 1

11 0.90 91 353 221 53 1

12 0.73 87 306 178 66 1

13 0.96 78 290 136 142 1

14 0.84 90 371 200 93 1

15 0.74 86 312 208 68 1

16 0.98 80 393 202 102 1

17 1.10 90 364 152 76 1

(complete data not shown)

141 1.05 353 1428 41 480 3

142 0.91 180 923 77 150 3

143 0.90 213 1025 29 209 3144 1.11 328 1246 124 442 3

145 0.74 346 1568 15 253 3

;

PROC PRINT DATA=diabetic(obs=10) label;

TITLE ' Discriminant analysis example';

run;


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Step 2 : Perform the Exploratory scatter plots by running the SAS macro DISCRIM and inputting YES to data exploration

< Open the macro-call window discrim by running the downloaded macro-call filediscrim.sas, enter the appropriate macro variable names, and submit.


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The Method for Selecting Variables is BACKWARD

Observations 145 Variable(s) in the Analysis 5

Class Levels 3 Variable(s) will be Included 0

Significance Level to Stay 0.15

Statistics for Removal, DF = 2, 138

Variable Label

PartialR-

Square F Value Pr > F

relwt Relative weight 0.0749 5.58 0.0047

glufast Fasting Plasma Glucose 0.2781 26.58


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Stepwise Selection Summary

umber

In Entered Removed Label

PartialR-

Square

F

Value Pr > F

Wilks'

Lambda

Pr

ASCC1 glutest Test

PlasmaGlucose

0.7738 242.84


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Forward Selection Summary

epNumber

In Entered Label

PartialR-

SquareF

Value Pr > FWilks'

LambdaPr ASCC

1 1 glutest Test

PlasmaGlucose

0.7738 242.84


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Selected Scatter Plots:


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Step 2 : Perform the canonical and parametric discriminant function analysis by running the SAS macro discrim and keep data exploration field Blank . Also Check for multivariatenormality and influential outliers

< Open the macro-call window discrim by running the downloaded macro-call file discrim.sas, enterthe appropriate macro variable names, and submit.


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Clinical Group=1

Obs s1 s2 s3 s4 s5 r1 r2 r3 r4 r5

1 0.11252 0.029294 -0.099238 1.71461 0.95699 -0.75959 0.14815 -0.71406 5.43254 0.55298

Clinical Group=2

bs s1 s2 s3 s4 s5 r1 r2 r3 r4 r5

2 -0.34755 -0.23682 0.70585 1.24705 -0.65609 -0.72631 -0.39443 0.037414 1.16893 -0.35500

Clinical Group=3

bs s1 s2 s3 s4 s5 r1 r2 r3 r4 r5

3 0.021434 0.37220 0.13407 2.11185 -0.11574 -0.88786 -1.25087 -1.16310 5.82761 -0.44997

Multivariate normality test statistics

M_SKEW

8.433 Multivariate skewness

CHI_SKEW

203.798 skewness chisquare

PVALSKEW

0.000 skewness P-value

M_KURT

52.853 Multivariate kurtosis

Z_KURT

12.847 kurtosis z-value

PVALKURT

0.000 kurtosis P-value


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id rdsq chisq diff

86 46.5157 17.6296 28.8860

144 30.2647 15.0041 15.2606

131 25.7032 13.7552 11.9480

141 24.0769 12.9202 11.1567145 23.2948 12.2891 11.0057

139 22.3210 11.7799 10.5411

134 19.3147 11.3522 7.9625

116 16.3652 10.9827 5.3825

93 15.8799 10.6570 5.2229

82 14.6057 10.3655 4.2402

133 13.9789 10.1014 3.8775

137 13.6007 9.8597 3.7410

69 12.9646 9.6367 3.3279

89 12.1243 9.2364 2.8880

113 12.2535 9.4297 2.8238

136 11.5431 8.8837 2.6593

124 11.3769 8.7218 2.6552

99 11.2212 8.5680 2.6532

114 11.6813 9.0548 2.6264


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Observations 145 DF Total 144

Variables 5 DF Within Classes 142

Classes 3 DF Between Classes 2

Class Level Information

group

VariableName

Frequency Weight Proportion

PriorProbabilit

y

1 _1 76 76.0000 0.524138 0.524138

2 _2 36 36.0000 0.248276 0.248276

3 _3 33 33.0000 0.227586 0.227586

Total-Sample

Variable Label N Sum Mean Variance

StandardDeviatio

n

relwt Relative weight 145 141.71000 0.97731 0.01670 0.1292

glufast Fasting Plasma Glucose 145 17688 121.98621 4087 63.9304

glutest Test Plasma Glucose 145 78824 543.61379 100458 316.9509

instest Plasma Insulin during Test 145 26987 186.11724 14625 120.9352

sspg Steady State Plasma Glucose 145 26710 184.20690 11242 106.0299

II. Canonical discriminant analysis


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group = 1


StandardDeviatio

n


glufast Fasting Plasma Glucose 76 6930 91.18421 67.69895 8.2279

glutest Test Plasma Glucose 76 26598 349.97368 1359 36.8706instest Plasma Insulin during Test 76 13121 172.64474 4741 68.8538


group = 2


StandardDeviatio

n

relwt Relative weight 36 38.01000 1.05583 0.01021 0.1010glufast Fasting Plasma Glucose 36 3575 99.30556 90.04683 9.4893

glutest Test Plasma Glucose 36 17782 493.94444 3070 55.4117



group = 3


StandardDeviatio

n


glufast Fasting Plasma Glucose 33 7183 217.66667 5862 76.5632

glutest Test Plasma Glucose 33 34444 1044 95725 309.3953




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The DISCRIM ProcedureTest of Homogeneity of Within Covariance Matrices

Since the Chi-Square value is significant at the 0.1 level, the within covariance matrices will be used in the discriminant function. Reference: Morrison, D.F. (1976) Multivariate Statistical Methods p252.

Chi-Square DF Pr > ChiSq

396.799635 30


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Eigenvalues of Inv(E)*H= CanRsq/(1-CanRsq)

anonicalrrelation

AdjustedCanonical

Correlation

ApproximateStandard

Error

SquaredCanonical

Correlation Eigenvalue Difference Proportion Cumulative

0.909389 0.906426 0.014418 0.826988 4.7799 4.1355 0.8812 0.8812

0.625998 0.618252 0.050677 0.391874 0.6444 0.1188 1.0000

Test of H0: The canonical correlations in the current row and all that follow are zero

LikelihoodRatio

ApproximateF Value Num DF Den DF Pr > F

1 0.10521315 57.49 10 276


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The DISCRIM ProcedureClassification Summary for Calibration Data: WORK.DIABETIC Resubstitution Summary using Quadratic Discriminant Function

Number of Observations and PercentClassified into group

Fromgroup 1 2 3 Total

1 7598.68

11.32

00.00

76100.00

2 38.33

3391.67

00.00

36100.00

3 00.00

39.09

3090.91

33100.00

Total 7853.79

3725.52

3020.69

145100.00

Priors 0.52414 0.24828 0.22759

Error Count Estimates for group

1 2 3 Total

Rate 0.0132 0.0833 0.0909 0.0483

Priors 0.5241 0.2483 0.2276

Parametric Discriminant Function analysis


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The DISCRIM ProcedureClassification Results for Calibration Data: WORK.DIABETIC

Cross-validation Results using Quadratic Discriminant Function

Posterior Probability of Membership in group

Obs From group Classified into group 1 2 3

26 1 2 * 0.2237 0.5779 0.1984

63 1 2 * 0.3550 0.6380 0.0070

64 1 2 * 0.3708 0.6067 0.0225

75 1 2 * 0.0000 0.9998 0.0002

79 2 1 * 0.6305 0.3624 0.0071

83 2 1* 0.5895 0.4035 0.0070

95 2 3 * 0.0000 0.4623 0.5377

96 2 1 * 0.5005 0.4485 0.0510

107 2 3 * 0.0000 0.1789 0.8211

110 2 1 * 0.8438 0.1498 0.0063

111 2 3 * 0.0000 0.0872 0.9128

131 3 2 * 0.0000 1.0000 0.0000

134 3 2 * 0.0000 0.9777 0.0223

136 3 2* 0.0000 0.9328 0.0672

* Misclassified observation


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The DISCRIM ProcedureClassification Summary for Calibration Data: WORK.DIABETIC

Cross-validation Summary using Quadratic Discriminant Function

Number of Observations and Percent Classified into group

From group 1 2 3 Total

1 7294.74

45.26

00.00

76100.00

2 411.11

2980.56

38.33

36100.00

3 00.00

39.09

3090.91

33100.00

Total 7652.41

3624.83

3322.76

145100.00

Priors 0.52414 0.24828 0.22759


1 2 3 Total

Rate 0.0526 0.1944 0.0909 0.0966

Priors 0.5241 0.2483 0.2276


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Non-parametric Discriminant Function analysis Perform non- parametric discriminant function analysis by running the SAS macro discrim, inputYES to the Non-par field and keep data exploration field Blank .


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Non-parametric Discriminant analysisNearest -neighbour method Mahalanobis distance based on pooled covariance

The DISCRIM Procedure

Classification Summary for Calibration Data: WORK.DIABETIC Resubstitution Summary using 2 Nearest Neighbors


Fromgroup 1 2 3 Other Total

1 7294.74 00.00 00.00 45.26 76100.00

2 00.00

2980.56

00.00

719.44

36100.00

3 00.00

00.00

2884.85

515.15

33100.00

Total 7249.66

2920.00

2819.31

1611.03

145100.00

Priors 0.52414 0.24828 0.22759

Error Count Estimates for group1 2 3 Total

Rate 0.0526 0.1944 0.1515 0.1103

Priors 0.5241 0.2483 0.2276


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Obs From group Classified into group 1 2 364 1 2 * 0.0000 1.0000 0.0000

124 3 2 * 0.0000 1.0000 0.0000

131 3 2 * 0.0000 1.0000 0.0000

134 3 2 * 0.0000 1.0000 0.0000

136 3 2 * 0.0000 1.0000 0.0000




Cross-validation Results using 2 Nearest Neighbors


Cross-validation Summary using 2 Nearest Neighbors



1 7598.68

11.32

00.00

76100.00

2 00.00

36100.00

00.00

36100.00

3 00.00

412.12

2987.88

33100.00

Total 7551.72

4128.28

2920.00

145100.00

Priors 0.52414 0.24828 0.22759


1 2 3 Total

Rate 0.0132 0.0000 0.1212 0.0345

Priors 0.5241 0.2483 0.2276


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Cross-validation Results using 3 Nearest Neighbors

Posterior Probability of Membership in groupObs From group Classified into group 1 2 3

62 1 2 * 0.3363 0.6637 0.0000

63 1 2 * 0.3363 0.6637 0.0000

64 1 2 * 0.0000 1.0000 0.0000

66 1 2 * 0.3363 0.6637 0.0000

69 1 2 * 0.3363 0.6637 0.0000

75 1 2 * 0.3363 0.6637 0.0000

84 2 1 * 0.6604 0.3396 0.000085 2 1 * 0.6604 0.3396 0.0000

96 2 1 * 0.6604 0.3396 0.0000

105 2 1 * 0.6604 0.3396 0.0000

124 3 2 * 0.0000 1.0000 0.0000

131 3 2 * 0.0000 1.0000 0.0000

134 3 2 * 0.0000 1.0000 0.0000

135 3 2 * 0.0000 0.6598 0.3402

136 3 2 * 0.0000 1.0000 0.0000



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Cross-validation Summary using 3 Nearest Neighbors



1 7092.11

67.89

00.00

76100.00

2 4

11.11

32

88.89

0

0.00

36

100.003 0

0.005

15.1528

84.8533

100.00

Total 7451.03

4329.66

2819.31

145100.00

Priors 0.52414 0.24828 0.22759


1 2 3 Total

Rate 0.0789 0.1111 0.1515 0.1034

Priors 0.5241 0.2483 0.2276


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Non-parametric Discriminant analysisUsing Kernal desnsity estimates with unequal bandwidth


Resubstitution Summary using Normal Kernel Density



1 76100.00

00.00

00.00

76100.00

2 00.00

36100.00

00.00

36100.00

3 00.00

00.00

33100.00

33100.00

Total 7652.41

3624.83

3322.76

145100.00

Priors 0.52414 0.24828 0.22759


1 2 3 Total

Rate 0.0000 0.0000 0.0000 0.0000

Priors 0.5241 0.2483 0.2276


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Cross-validation Results using Normal Kernel Density


Obs From group Classified into group 1 2 3

26 1 3 * 0.0286 0.2117 0.7597

62 1 2 * 0.2529 0.7468 0.0003

63 1 2 * 0.2854 0.7017 0.0129

70 1 2 * 0.4493 0.5504 0.0003

75 1 2 * 0.0000 0.6699 0.3301

76 1 2 * 0.3220 0.6244 0.0536

79 2 1 * 0.7210 0.2771 0.0019

96 2 1 * 0.6601 0.3384 0.0015

104 2 3 * 0.0000 0.0245 0.9755

107 2 3 * 0.0000 0.0138 0.9862

111 2 3 * 0.0000 0.0001 0.9999

112 2 3 * 0.0000 0.0840 0.9160

131 3 2 * 0.0000 0.9999 0.0001

134 3 2 * 0.0000 0.9945 0.0055

136 3 2 * 0.0000 0.9949 0.0051



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1 7092.11

56.58

11.32

76100.00

2 25.56

3083.33

411.11

36100.00

3 00.00

39.09

3090.91

33100.00

Total 7249.66

3826.21

3524.14

145100.00

Priors 0.52414 0.24828 0.22759


1 2 3 Total

Rate 0.0789 0.1667 0.0909 0.1034

Priors 0.5241 0.2483 0.2276



Cross-validation Summary using Normal Kernel Density


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Conclusion:

Exploratory scatter plots, stepwise discriminant analysis, checking for multivariate normality and multivariate outliers,canonical, parametric and non-parametric discriminant function analysis were performed using discrim macro and discrim.sasmacro-call file (Fernandez, 2001).The discrim macro was invoked using the parameters specified in the macro-call file discrim..sas (Fernandez, 2001).

References:

Fernandez G.C.J 2001 Discriminant analysis using SAS macro DISCRIM. In Free SAS STAT applications: IV Multivariatemethods. http://www.ag.unr.edu/gf Department of Applied Economics and Statistics, MS 204 , UNR Reno NV 89557

Suggested additional readings:

Statistics:Subhash Sharma Applied Multivariate technique1996 Wiley

SAS publications:

SAS Institute Inc., Technical report P-179 Additional SAS/STAT procedures. Cary, NC SAS Institute Inc., 1988.

SAS Institute Inc., SAS/GRAPH Software: Reference, Version 6 FirstEdition. Volumes 1 and 2, Cary, NC SAS Institute Inc., 1990.

SAS Institute Inc., SAS Guide to macro processing , Version 6 FirstEdition. Cary, NC SAS Institute Inc., 1987.

SAS Institute Inc., SAS/STAT Users Guide Version 6 Fourth Edition.Volumes 1 and 2, Cary, NC SAS Institute Inc., 1989.

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