Discriminant and Classification Analysis
Fisher Iris Data Examples
In this example, the three populations of Iris (Iris setosa,
Iris versicolor, and Iris virginica) are to be discriminated
based upon their flower "shapes." To get a measure of shape,
two new variables are constructed from the original length
and width variables
- y1=log(sepallen/sepalwid)
- y2=log(petallen/petalwid)
where sepallen and sepalwid are the flower sepal lengths
and widths, respectively, and petallen and petalwid are
the flower petal lengths and widths, respectively.
By using only 2 response variables (y1 and y2), the results
of the various discriminant analyses can be easily illustrated
graphically (2-dimensional plots).
Discriminant analysis approaches used in this example are
(1) linear discriminant analysis, (2) quadratic discriminant
analysis, and (3) canonical discriminant analysis. You might
also wish to explore these data using non-parametric discriminant
analysis techniques.
Data Set
SAS Data Step
Data Listing
Raw Data Scatter Plot
Linear Discriminant Analysis
Program
Output
Estimated Densities
Iris setosa (Contour) (Surface)
Iris versicolor (Contour) (Surface)
Iris virginica (Contour) (Surface)
Posterior Probabilities
Iris setosa (Contour) (Surface)
Iris versicolor (Contour) (Surface)
Iris virginica (Contour) (Surface)
Classification Regions
Quadratic Discriminant Analysis
Program
Output
Estimated Densities
Iris setosa (Contour) (Surface)
Iris versicolor (Contour) (Surface)
Iris virginica (Contour) (Surface)
Posterior Probabilities
Iris setosa (Contour) (Surface)
Iris versicolor (Contour) (Surface)
Iris virginica (Contour) (Surface)
Classification Regions
Canonical Discriminant Analysis
Program
Output
Canonical Variates Scatter Plot
Another view of canonical structure
Another view of standardized coefficients
*************************************************************;
* DISCRIM.SAS -- Discriminant Analysis with Interpretation *;
* Fisher Iris Data -- Discriminant Analysis Examples. *;
* Part 1. Three population linear discriminant analysis. *;
* Part 2. Three population quadratic discriminant analysis. *;
* Part 3. Three population canonical discriminant analysis. *;
* See Table 11.5 of Johnson and Wichern, 3rd Ed., p. 566. *;
*************************************************************;
Title1 "Discriminant Analysis of the Fisher Iris Data";
GOptions FText=SwissX CText=Black HText=1.0
FTitle=SwissX CTitle=Black HTitle=1.0
HSize=6.0 in VSize=4.5 in NoPrompt;
Options PS=58 LS=79 PageNo=1 NoDate
FORMCHAR='|----|+|---+=|-/\<>*';
Proc Format;
Value Specname
1='SETOSA '
2='VERSICOLOR'
3='VIRGINICA ';
Run;
Data Iris;
Input sepallen sepalwid petallen petalwid species @@;
y1=log(sepallen/sepalwid);
y2=log(petallen/petalwid);
Format species specname.;
Label sepallen='Sepal Length in mm.'
sepalwid='Sepal Width in mm.'
petallen='Petal Length in mm.'
petalwid='Petal Width in mm.'
y1="Log Sepal Shape"
y2="Log Petal Shape";
Datalines;
50 33 14 02 1 64 28 56 22 3 65 28 46 15 2 67 31 56 24 3
63 28 51 15 3 46 34 14 03 1 69 31 51 23 3 62 22 45 15 2
59 32 48 18 2 46 36 10 02 1 61 30 46 14 2 60 27 51 16 2
65 30 52 20 3 56 25 39 11 2 65 30 55 18 3 58 27 51 19 3
68 32 59 23 3 51 33 17 05 1 57 28 45 13 2 62 34 54 23 3
77 38 67 22 3 63 33 47 16 2 67 33 57 25 3 76 30 66 21 3
49 25 45 17 3 55 35 13 02 1 67 30 52 23 3 70 32 47 14 2
64 32 45 15 2 61 28 40 13 2 48 31 16 02 1 59 30 51 18 3
55 24 38 11 2 63 25 50 19 3 64 32 53 23 3 52 34 14 02 1
49 36 14 01 1 54 30 45 15 2 79 38 64 20 3 44 32 13 02 1
67 33 57 21 3 50 35 16 06 1 58 26 40 12 2 44 30 13 02 1
77 28 67 20 3 63 27 49 18 3 47 32 16 02 1 55 26 44 12 2
50 23 33 10 2 72 32 60 18 3 48 30 14 03 1 51 38 16 02 1
61 30 49 18 3 48 34 19 02 1 50 30 16 02 1 50 32 12 02 1
61 26 56 14 3 64 28 56 21 3 43 30 11 01 1 58 40 12 02 1
51 38 19 04 1 67 31 44 14 2 62 28 48 18 3 49 30 14 02 1
51 35 14 02 1 56 30 45 15 2 58 27 41 10 2 50 34 16 04 1
46 32 14 02 1 60 29 45 15 2 57 26 35 10 2 57 44 15 04 1
50 36 14 02 1 77 30 61 23 3 63 34 56 24 3 58 27 51 19 3
57 29 42 13 2 72 30 58 16 3 54 34 15 04 1 52 41 15 01 1
71 30 59 21 3 64 31 55 18 3 60 30 48 18 3 63 29 56 18 3
49 24 33 10 2 56 27 42 13 2 57 30 42 12 2 55 42 14 02 1
49 31 15 02 1 77 26 69 23 3 60 22 50 15 3 54 39 17 04 1
66 29 46 13 2 52 27 39 14 2 60 34 45 16 2 50 34 15 02 1
44 29 14 02 1 50 20 35 10 2 55 24 37 10 2 58 27 39 12 2
47 32 13 02 1 46 31 15 02 1 69 32 57 23 3 62 29 43 13 2
74 28 61 19 3 59 30 42 15 2 51 34 15 02 1 50 35 13 03 1
56 28 49 20 3 60 22 40 10 2 73 29 63 18 3 67 25 58 18 3
49 31 15 01 1 67 31 47 15 2 63 23 44 13 2 54 37 15 02 1
56 30 41 13 2 63 25 49 15 2 61 28 47 12 2 64 29 43 13 2
51 25 30 11 2 57 28 41 13 2 65 30 58 22 3 69 31 54 21 3
54 39 13 04 1 51 35 14 03 1 72 36 61 25 3 65 32 51 20 3
61 29 47 14 2 56 29 36 13 2 69 31 49 15 2 64 27 53 19 3
68 30 55 21 3 55 25 40 13 2 48 34 16 02 1 48 30 14 01 1
45 23 13 03 1 57 25 50 20 3 57 38 17 03 1 51 38 15 03 1
55 23 40 13 2 66 30 44 14 2 68 28 48 14 2 54 34 17 02 1
51 37 15 04 1 52 35 15 02 1 58 28 51 24 3 67 30 50 17 2
63 33 60 25 3 53 37 15 02 1
;
Proc Sort Data=Iris;
By Species;
Run;
Proc Print Data=Iris Split=" ";
Run;
Title3 'Plot of Original Shape Measurements';
Proc GPlot Data=Iris;
Plot y1*y2=Species;
Symbol1 V=Dot H=0.7 I=None C=Black;
Symbol2 V=Star H=0.7 I=None C=Black;
Symbol3 V=Square H=0.7 I=None C=Black;
Run; Quit;
*************************************************************;
* Construct some artificial Y1 and Y2 data to be classified.*;
* These observations can be used to plot and explore the *;
* classification regions and to construct a classification *;
* chart for classifying future observations. *;
*************************************************************;
/* find the min and max values of y1 and y2 in the data set
so we'll have an idea as to the potential range of y1 and y2
values */
Proc Means Data=Iris NoPrint;
Var Y1 Y2;
Output Out=Stats Min=MinY1 MinY2 Max=MaxY1 MaxY2;
Run;
/* Now generate the test values. These data will be used
and reported using the testout=, testdata=, and testoutd=
options on the Proc Discrim statement. */
Data PlotData;
If _N_=1 Then Set Stats;
Y1Inc=(MaxY1-MinY1)/50;
Y2Inc=(MaxY2-MinY2)/50;
Do Y1 = (MinY1-Y1Inc) To (MaxY1+Y1Inc) By Y1Inc;
Do Y2 = (MinY2-Y2Inc) To (MaxY2+Y2Inc) By Y2Inc;
Output;
Keep Y1 Y2;
End;
End;
Stop;
Run;
*************************************************************;
* Part 1. Linear Discriminant Analysis *;
*************************************************************;
Title2 'Linear Discriminant Function Analysis';
Proc Discrim Data=Iris
Testdata=Plotdata TestOut=PlotP TestOutD=PlotD
Method=Normal Pool=Yes
WCov Distance MANOVA Simple;
Class Species;
Var y1 y2;
Run;
%Macro Graphs;
%* Produce various plots of densities and probabilities *;
%* for the discriminant analysis. *;
Title3 'Plot of Estimated Densities';
Proc GContour Data=PlotD;
Title4 "Iris setosa";
Plot y1*y2=Setosa;
Run;
Proc G3D Data=PlotD;
Plot y1*y2=Setosa;
Run;
Proc GContour Data=PlotD;
Title4 "Iris versicolor";
Plot y1*y2=Versicol;
Run;
Proc G3D Data=PlotD;
Plot y1*y2=Versicol;
Run;
Proc GContour Data=PlotD;
Title4 "Iris virginica";
Plot y1*y2=Virginic;
Run;
Proc G3D Data=PlotD;
Plot y1*y2=Virginic;
Run;
Title3 'Plot of Posterior Probabilities';
Proc GContour Data=PlotP;
Title4 "Iris setosa";
Plot y1*y2=Setosa;
Run;
Proc G3D Data=PlotP;
Plot y1*y2=Setosa;
Run;
Proc GContour Data=PlotP;
Title4 "Iris versicolor";
Plot y1*y2=Versicol;
Run;
Proc G3D Data=PlotP;
Plot y1*y2=Versicol;
Run;
Proc GContour Data=PlotP;
Title4 "Iris virginica";
Plot y1*y2=Virginic;
Run;
Proc G3D Data=PlotP;
Plot y1*y2=Virginic;
Run;
Title3 'Plot of Classification Results';
Proc GPlot Data=PlotP;
Plot y1*y2=_Into_;
Symbol1 V=Dot H=0.7 I=None C=Black;
Symbol2 V=Star H=0.7 I=None C=Black;
Symbol3 V=Square H=0.7 I=None C=Black;
Run; Quit;
%Mend;
%Graphs;
*************************************************************;
* Part 2. Quadratic Discriminant Analysis. *;
*************************************************************;
Title2 'Quadratic Discriminant Function Analysis';
Proc Discrim Data=Iris
TestData=PlotData TestOut=PlotP TestOutD=PlotD
Method=Normal Pool=No Short NoClassify
WCov Distance MANOVA Simple;
Class Species;
Var y1 y2;
Run;
%Graphs;
*************************************************************;
* Part 3. Canonical Discriminant Analysis. *;
*************************************************************;
Title2 "Canonical Discriminant Function Analysis";
Proc CanDisc Data=Iris All Out=OIris;
Class Species;
Var y1 y2;
Run;
Title3 "Show What Canonical Structure Is";
Proc Corr Data=OIris;
Var Can1 Can2;
With y1 y2;
Run;
Title3 "Show What Standardized Coefficients Are";
Proc Reg Data=OIris;
Model Can1 Can2 = y1 y2;
Run; Quit;
Title3 "Plot of Observations In Space of Canonical Variates";
Proc GPlot Data=OIris;
Plot Can1*Can2=Species;
Symbol1 V=Dot H=0.7 I=None C=Black;
Symbol2 V=Star H=0.7 I=None C=Black;
Symbol3 V=Square H=0.7 I=None C=Black;
Run;
Discriminant Analysis of the Fisher Iris Data 1
Sepal Sepal Petal Petal
Length Width Length Width Log Log
in in in in Sepal Petal
OBS mm. mm. mm. mm. SPECIES Shape Shape
1 50 33 14 2 SETOSA 0.41552 1.94591
2 46 34 14 3 SETOSA 0.30228 1.54045
3 46 36 10 2 SETOSA 0.24512 1.60944
4 51 33 17 5 SETOSA 0.43532 1.22378
5 55 35 13 2 SETOSA 0.45199 1.87180
6 48 31 16 2 SETOSA 0.43721 2.07944
7 52 34 14 2 SETOSA 0.42488 1.94591
8 49 36 14 1 SETOSA 0.30830 2.63906
9 44 32 13 2 SETOSA 0.31845 1.87180
10 50 35 16 6 SETOSA 0.35667 0.98083
11 44 30 13 2 SETOSA 0.38299 1.87180
12 47 32 16 2 SETOSA 0.38441 2.07944
13 48 30 14 3 SETOSA 0.47000 1.54045
14 51 38 16 2 SETOSA 0.29424 2.07944
15 48 34 19 2 SETOSA 0.34484 2.25129
16 50 30 16 2 SETOSA 0.51083 2.07944
17 50 32 12 2 SETOSA 0.44629 1.79176
18 43 30 11 1 SETOSA 0.36000 2.39790
19 58 40 12 2 SETOSA 0.37156 1.79176
20 51 38 19 4 SETOSA 0.29424 1.55814
21 49 30 14 2 SETOSA 0.49062 1.94591
22 51 35 14 2 SETOSA 0.37648 1.94591
23 50 34 16 4 SETOSA 0.38566 1.38629
24 46 32 14 2 SETOSA 0.36291 1.94591
25 57 44 15 4 SETOSA 0.25886 1.32176
26 50 36 14 2 SETOSA 0.32850 1.94591
27 54 34 15 4 SETOSA 0.46262 1.32176
28 52 41 15 1 SETOSA 0.23767 2.70805
29 55 42 14 2 SETOSA 0.26966 1.94591
30 49 31 15 2 SETOSA 0.45783 2.01490
31 54 39 17 4 SETOSA 0.32542 1.44692
32 50 34 15 2 SETOSA 0.38566 2.01490
33 44 29 14 2 SETOSA 0.41689 1.94591
34 47 32 13 2 SETOSA 0.38441 1.87180
35 46 31 15 2 SETOSA 0.39465 2.01490
36 51 34 15 2 SETOSA 0.40547 2.01490
37 50 35 13 3 SETOSA 0.35667 1.46634
38 49 31 15 1 SETOSA 0.45783 2.70805
39 54 37 15 2 SETOSA 0.37807 2.01490
40 54 39 13 4 SETOSA 0.32542 1.17865
41 51 35 14 3 SETOSA 0.37648 1.54045
42 48 34 16 2 SETOSA 0.34484 2.07944
43 48 30 14 1 SETOSA 0.47000 2.63906
44 45 23 13 3 SETOSA 0.67117 1.46634
45 57 38 17 3 SETOSA 0.40547 1.73460
46 51 38 15 3 SETOSA 0.29424 1.60944
47 54 34 17 2 SETOSA 0.46262 2.14007
48 51 37 15 4 SETOSA 0.32091 1.32176
49 52 35 15 2 SETOSA 0.39590 2.01490
50 53 37 15 2 SETOSA 0.35937 2.01490
51 65 28 46 15 VERSICOLOR 0.84218 1.12059
Discriminant Analysis of the Fisher Iris Data 2
Sepal Sepal Petal Petal
Length Width Length Width Log Log
in in in in Sepal Petal
OBS mm. mm. mm. mm. SPECIES Shape Shape
52 62 22 45 15 VERSICOLOR 1.03609 1.09861
53 59 32 48 18 VERSICOLOR 0.61180 0.98083
54 61 30 46 14 VERSICOLOR 0.70968 1.18958
55 60 27 51 16 VERSICOLOR 0.79851 1.15924
56 56 25 39 11 VERSICOLOR 0.80648 1.26567
57 57 28 45 13 VERSICOLOR 0.71085 1.24171
58 63 33 47 16 VERSICOLOR 0.64663 1.07756
59 70 32 47 14 VERSICOLOR 0.78276 1.21109
60 64 32 45 15 VERSICOLOR 0.69315 1.09861
61 61 28 40 13 VERSICOLOR 0.77867 1.12393
62 55 24 38 11 VERSICOLOR 0.82928 1.23969
63 54 30 45 15 VERSICOLOR 0.58779 1.09861
64 58 26 40 12 VERSICOLOR 0.80235 1.20397
65 55 26 44 12 VERSICOLOR 0.74924 1.29928
66 50 23 33 10 VERSICOLOR 0.77653 1.19392
67 67 31 44 14 VERSICOLOR 0.77071 1.14513
68 56 30 45 15 VERSICOLOR 0.62415 1.09861
69 58 27 41 10 VERSICOLOR 0.76461 1.41099
70 60 29 45 15 VERSICOLOR 0.72705 1.09861
71 57 26 35 10 VERSICOLOR 0.78495 1.25276
72 57 29 42 13 VERSICOLOR 0.67576 1.17272
73 49 24 33 10 VERSICOLOR 0.71377 1.19392
74 56 27 42 13 VERSICOLOR 0.72951 1.17272
75 57 30 42 12 VERSICOLOR 0.64185 1.25276
76 66 29 46 13 VERSICOLOR 0.82236 1.26369
77 52 27 39 14 VERSICOLOR 0.65541 1.02450
78 60 34 45 16 VERSICOLOR 0.56798 1.03407
79 50 20 35 10 VERSICOLOR 0.91629 1.25276
80 55 24 37 10 VERSICOLOR 0.82928 1.30833
81 58 27 39 12 VERSICOLOR 0.76461 1.17865
82 62 29 43 13 VERSICOLOR 0.75984 1.19625
83 59 30 42 15 VERSICOLOR 0.67634 1.02962
84 60 22 40 10 VERSICOLOR 1.00330 1.38629
85 67 31 47 15 VERSICOLOR 0.77071 1.14210
86 63 23 44 13 VERSICOLOR 1.00764 1.21924
87 56 30 41 13 VERSICOLOR 0.62415 1.14862
88 63 25 49 15 VERSICOLOR 0.92426 1.18377
89 61 28 47 12 VERSICOLOR 0.77867 1.36524
90 64 29 43 13 VERSICOLOR 0.79159 1.19625
91 51 25 30 11 VERSICOLOR 0.71295 1.00330
92 57 28 41 13 VERSICOLOR 0.71085 1.14862
93 61 29 47 14 VERSICOLOR 0.74358 1.21109
94 56 29 36 13 VERSICOLOR 0.65806 1.01857
95 69 31 49 15 VERSICOLOR 0.80012 1.18377
96 55 25 40 13 VERSICOLOR 0.78846 1.12393
97 55 23 40 13 VERSICOLOR 0.87184 1.12393
98 66 30 44 14 VERSICOLOR 0.78846 1.14513
99 68 28 48 14 VERSICOLOR 0.88730 1.23214
100 67 30 50 17 VERSICOLOR 0.80350 1.07881
101 64 28 56 22 VIRGINICA 0.82668 0.93431
102 67 31 56 24 VIRGINICA 0.77071 0.84730
Discriminant Analysis of the Fisher Iris Data 3
Sepal Sepal Petal Petal
Length Width Length Width Log Log
in in in in Sepal Petal
OBS mm. mm. mm. mm. SPECIES Shape Shape
103 63 28 51 15 VIRGINICA 0.81093 1.22378
104 69 31 51 23 VIRGINICA 0.80012 0.79633
105 65 30 52 20 VIRGINICA 0.77319 0.95551
106 65 30 55 18 VIRGINICA 0.77319 1.11696
107 58 27 51 19 VIRGINICA 0.76461 0.98739
108 68 32 59 23 VIRGINICA 0.75377 0.94204
109 62 34 54 23 VIRGINICA 0.60077 0.85349
110 77 38 67 22 VIRGINICA 0.70622 1.11365
111 67 33 57 25 VIRGINICA 0.70819 0.82418
112 76 30 66 21 VIRGINICA 0.92954 1.14513
113 49 25 45 17 VIRGINICA 0.67294 0.97345
114 67 30 52 23 VIRGINICA 0.80350 0.81575
115 59 30 51 18 VIRGINICA 0.67634 1.04145
116 63 25 50 19 VIRGINICA 0.92426 0.96758
117 64 32 53 23 VIRGINICA 0.69315 0.83480
118 79 38 64 20 VIRGINICA 0.73186 1.16315
119 67 33 57 21 VIRGINICA 0.70819 0.99853
120 77 28 67 20 VIRGINICA 1.01160 1.20896
121 63 27 49 18 VIRGINICA 0.84730 1.00145
122 72 32 60 18 VIRGINICA 0.81093 1.20397
123 61 30 49 18 VIRGINICA 0.70968 1.00145
124 61 26 56 14 VIRGINICA 0.85278 1.38629
125 64 28 56 21 VIRGINICA 0.82668 0.98083
126 62 28 48 18 VIRGINICA 0.79493 0.98083
127 77 30 61 23 VIRGINICA 0.94261 0.97538
128 63 34 56 24 VIRGINICA 0.61677 0.84730
129 58 27 51 19 VIRGINICA 0.76461 0.98739
130 72 30 58 16 VIRGINICA 0.87547 1.28785
131 71 30 59 21 VIRGINICA 0.86148 1.03302
132 64 31 55 18 VIRGINICA 0.72490 1.11696
133 60 30 48 18 VIRGINICA 0.69315 0.98083
134 63 29 56 18 VIRGINICA 0.77584 1.13498
135 77 26 69 23 VIRGINICA 1.08571 1.09861
136 60 22 50 15 VIRGINICA 1.00330 1.20397
137 69 32 57 23 VIRGINICA 0.76837 0.90756
138 74 28 61 19 VIRGINICA 0.97186 1.16643
139 56 28 49 20 VIRGINICA 0.69315 0.89609
140 73 29 63 18 VIRGINICA 0.92316 1.25276
141 67 25 58 18 VIRGINICA 0.98582 1.17007
142 65 30 58 22 VIRGINICA 0.77319 0.96940
143 69 31 54 21 VIRGINICA 0.80012 0.94446
144 72 36 61 25 VIRGINICA 0.69315 0.89200
145 65 32 51 20 VIRGINICA 0.70865 0.93609
146 64 27 53 19 VIRGINICA 0.86305 1.02585
147 68 30 55 21 VIRGINICA 0.81831 0.96281
148 57 25 50 20 VIRGINICA 0.82418 0.91629
149 58 28 51 24 VIRGINICA 0.72824 0.75377
150 63 33 60 25 VIRGINICA 0.64663 0.87547
Discriminant Analysis of the Fisher Iris Data 4
Linear Discriminant Function Analysis
Discriminant Analysis
150 Observations 149 DF Total
2 Variables 147 DF Within Classes
3 Classes 2 DF Between Classes
Class Level Information
Output Prior
SPECIES SAS Name Frequency Weight Proportion Probability
SETOSA SETOSA 50 50.0000 0.333333 0.333333
VERSICOLOR VERSICOL 50 50.0000 0.333333 0.333333
VIRGINICA VIRGINIC 50 50.0000 0.333333 0.333333
Discriminant Analysis of the Fisher Iris Data 5
Linear Discriminant Function Analysis
Discriminant Analysis Within-Class Covariance Matrices
SPECIES = SETOSA DF = 49
Variable Y1 Y2
Y1 0.0060705993 0.0011655624 Log Sepal Shape
Y2 0.0011655624 0.1511271809 Log Petal Shape
-----------------------------------------------------------------
SPECIES = VERSICOLOR DF = 49
Variable Y1 Y2
Y1 0.0105307308 0.0044191864 Log Sepal Shape
Y2 0.0044191864 0.0090071768 Log Petal Shape
-----------------------------------------------------------------
SPECIES = VIRGINICA DF = 49
Variable Y1 Y2
Y1 0.0114432865 0.0080857213 Log Sepal Shape
Y2 0.0080857213 0.0199940192 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 6
Linear Discriminant Function Analysis
Discriminant Analysis Simple Statistics
Total-Sample
Variable N Sum Mean
Y1 150 97.19306 0.64795
Y2 150 202.10414 1.34736
Total-Sample
Variable Variance Std Dev Label
Y1 0.04490 0.21189 Log Sepal Shape
Y2 0.19444 0.44096 Log Petal Shape
----------------------------------------------------------
SPECIES = SETOSA
Variable N Sum Mean
Y1 50 19.11748 0.38235
Y2 50 92.87038 1.85741
SPECIES = SETOSA
Variable Variance Std Dev Label
Y1 0.00607 0.07791 Log Sepal Shape
Y2 0.15113 0.38875 Log Petal Shape
----------------------------------------------------------
SPECIES = VERSICOLOR
Variable N Sum Mean
Y1 50 38.25185 0.76504
Y2 50 58.59985 1.17200
SPECIES = VERSICOLOR
Variable Variance Std Dev Label
Y1 0.01053 0.10262 Log Sepal Shape
Y2 0.00901 0.09491 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 7
Linear Discriminant Function Analysis
Discriminant Analysis Simple Statistics
SPECIES = VIRGINICA
Variable N Sum Mean
Y1 50 39.82373 0.79647
Y2 50 50.63391 1.01268
SPECIES = VIRGINICA
Variable Variance Std Dev Label
Y1 0.01144 0.10697 Log Sepal Shape
Y2 0.01999 0.14140 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 8
Linear Discriminant Function Analysis
Discriminant Analysis Pooled Covariance Matrix Information
Covariance Natural Log of the Determinant
Matrix Rank of the Covariance Matrix
2 -7.5229646
Discriminant Analysis of the Fisher Iris Data 9
Linear Discriminant Function Analysis
Discriminant Analysis Pairwise Squared Distances Between Groups
2 _ _ -1 _ _
D (i|j) = (X - X )' COV (X - X )
i j i j
Squared Distance to SPECIES
From
SPECIES SETOSA VERSICOLOR VIRGINICA
SETOSA 0 28.81518 37.28957
VERSICOLOR 28.81518 0 0.63321
VIRGINICA 37.28957 0.63321 0
F Statistics, NDF=2, DDF=146 for
Squared Distance to SPECIES
From
SPECIES SETOSA VERSICOLOR VIRGINICA
SETOSA 0 357.73950 462.94874
VERSICOLOR 357.73950 0 7.86130
VIRGINICA 462.94874 7.86130 0
Prob Mahalanobis Distance for
Squared Distance to SPECIES
From
SPECIES SETOSA VERSICOLOR VIRGINICA
SETOSA 1.0000 0.0001 0.0001
VERSICOLOR 0.0001 1.0000 0.0006
VIRGINICA 0.0001 0.0006 1.0000
Discriminant Analysis of the Fisher Iris Data 10
Linear Discriminant Function Analysis
Discriminant Analysis Pairwise Generalized Squared Distances Between Groups
2 _ _ -1 _ _
D (i|j) = (X - X )' COV (X - X )
i j i j
Generalized Squared Distance to SPECIES
From
SPECIES SETOSA VERSICOLOR VIRGINICA
SETOSA 0 28.81518 37.28957
VERSICOLOR 28.81518 0 0.63321
VIRGINICA 37.28957 0.63321 0
Discriminant Analysis of the Fisher Iris Data 11
Linear Discriminant Function Analysis
Discriminant Analysis
Multivariate Statistics and F Approximations
S=2 M=-0.5 N=72
Statistic Value F Num DF Den DF Pr F
Wilks' Lambda 0.11524006 142.0409 4 292 0.0001
Pillai's Trace 0.89753721 59.8378 4 294 0.0001
Hotelling-Lawley Trace 7.56666236 274.2915 4 290 0.0001
Roy's Greatest Root 7.55198075 555.0706 2 147 0.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.
Discriminant Analysis of the Fisher Iris Data 12
Linear Discriminant Function Analysis
Discriminant Analysis Linear Discriminant Function
_ -1 _ -1 _
Constant = -.5 X' COV X Coefficient Vector = COV X
j j j
SPECIES
SETOSA VERSICOLOR VIRGINICA Label
CONSTANT -31.96540 -36.82598 -37.30186
Y1 26.81354 75.10137 79.93662 Log Sepal Shape
Y2 28.89977 13.81970 10.79932 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 13
Linear Discriminant Function Analysis
Discriminant Analysis
Classification Summary for Calibration Data: WORK.IRIS
Resubstitution Summary using Linear Discriminant Function
Generalized Squared Distance Function:
2 _ -1 _
D (X) = (X-X )' COV (X-X )
j j j
Posterior Probability of Membership in each SPECIES:
2 2
Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
j k k
Number of Observations and Percent Classified into SPECIES:
From SPECIES SETOSA VERSICOLOR VIRGINICA Total
SETOSA 49 1 0 50
98.00 2.00 0.00 100.00
VERSICOLOR 0 41 9 50
0.00 82.00 18.00 100.00
VIRGINICA 0 16 34 50
0.00 32.00 68.00 100.00
Total 49 58 43 150
Percent 32.67 38.67 28.67 100.00
Priors 0.3333 0.3333 0.3333
Error Count Estimates for SPECIES:
SETOSA VERSICOLOR VIRGINICA Total
Rate 0.0200 0.1800 0.3200 0.1733
Priors 0.3333 0.3333 0.3333
Discriminant Analysis of the Fisher Iris Data 14
Linear Discriminant Function Analysis
Discriminant Analysis Classification Summary for Test Data: WORK.PLOTDATA
Classification Summary using Linear Discriminant Function
Generalized Squared Distance Function:
2 _ -1 _
D (X) = (X-X )' COV (X-X )
j j j
Posterior Probability of Membership in each SPECIES:
2 2
Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
j k k
Number of Observations and Percent Classified into SPECIES:
SETOSA VERSICOLOR VIRGINICA Total
Total 1340 1065 404 2809
Percent 47.70 37.91 14.38 100.00
Priors 0.3333 0.3333 0.3333
Discriminant Analysis of the Fisher Iris Data 15
Quadratic Discriminant Function Analysis
Discriminant Analysis
150 Observations 149 DF Total
2 Variables 147 DF Within Classes
3 Classes 2 DF Between Classes
Class Level Information
Output Prior
SPECIES SAS Name Frequency Weight Proportion Probability
SETOSA SETOSA 50 50.0000 0.333333 0.333333
VERSICOLOR VERSICOL 50 50.0000 0.333333 0.333333
VIRGINICA VIRGINIC 50 50.0000 0.333333 0.333333
Discriminant Analysis of the Fisher Iris Data 16
Quadratic Discriminant Function Analysis
Discriminant Analysis Within-Class Covariance Matrices
SPECIES = SETOSA DF = 49
Variable Y1 Y2
Y1 0.0060705993 0.0011655624 Log Sepal Shape
Y2 0.0011655624 0.1511271809 Log Petal Shape
-----------------------------------------------------------------
SPECIES = VERSICOLOR DF = 49
Variable Y1 Y2
Y1 0.0105307308 0.0044191864 Log Sepal Shape
Y2 0.0044191864 0.0090071768 Log Petal Shape
-----------------------------------------------------------------
SPECIES = VIRGINICA DF = 49
Variable Y1 Y2
Y1 0.0114432865 0.0080857213 Log Sepal Shape
Y2 0.0080857213 0.0199940192 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 17
Quadratic Discriminant Function Analysis
Discriminant Analysis Simple Statistics
Total-Sample
Variable N Sum Mean
Y1 150 97.19306 0.64795
Y2 150 202.10414 1.34736
Total-Sample
Variable Variance Std Dev Label
Y1 0.04490 0.21189 Log Sepal Shape
Y2 0.19444 0.44096 Log Petal Shape
----------------------------------------------------------
SPECIES = SETOSA
Variable N Sum Mean
Y1 50 19.11748 0.38235
Y2 50 92.87038 1.85741
SPECIES = SETOSA
Variable Variance Std Dev Label
Y1 0.00607 0.07791 Log Sepal Shape
Y2 0.15113 0.38875 Log Petal Shape
----------------------------------------------------------
SPECIES = VERSICOLOR
Variable N Sum Mean
Y1 50 38.25185 0.76504
Y2 50 58.59985 1.17200
SPECIES = VERSICOLOR
Variable Variance Std Dev Label
Y1 0.01053 0.10262 Log Sepal Shape
Y2 0.00901 0.09491 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 18
Quadratic Discriminant Function Analysis
Discriminant Analysis Simple Statistics
SPECIES = VIRGINICA
Variable N Sum Mean
Y1 50 39.82373 0.79647
Y2 50 50.63391 1.01268
SPECIES = VIRGINICA
Variable Variance Std Dev Label
Y1 0.01144 0.10697 Log Sepal Shape
Y2 0.01999 0.14140 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 19
Quadratic Discriminant Function Analysis
Discriminant Analysis Pairwise Squared Distances Between Groups
2 _ _ -1 _ _
D (i|j) = (X - X )' COV (X - X )
i j j i j
Squared Distance to SPECIES
From
SPECIES SETOSA VERSICOLOR VIRGINICA
SETOSA 0 113.97047 105.56748
VERSICOLOR 27.94081 0 2.39395
VIRGINICA 33.91155 4.25455 0
Discriminant Analysis of the Fisher Iris Data 20
Quadratic Discriminant Function Analysis
Discriminant Analysis
Multivariate Statistics and F Approximations
S=2 M=-0.5 N=72
Statistic Value F Num DF Den DF Pr F
Wilks' Lambda 0.11524006 142.0409 4 292 0.0001
Pillai's Trace 0.89753721 59.8378 4 294 0.0001
Hotelling-Lawley Trace 7.56666236 274.2915 4 290 0.0001
Roy's Greatest Root 7.55198075 555.0706 2 147 0.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.
Discriminant Analysis of the Fisher Iris Data 21
Quadratic Discriminant Function Analysis
Discriminant Analysis Classification Summary for Test Data: WORK.PLOTDATA
Classification Summary using Quadratic Discriminant Function
Generalized Squared Distance Function:
2 _ -1 _
D (X) = (X-X )' COV (X-X ) + ln |COV |
j j j j j
Posterior Probability of Membership in each SPECIES:
2 2
Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
j k k
Number of Observations and Percent Classified into SPECIES:
SETOSA VERSICOLOR VIRGINICA Total
Total 1839 336 634 2809
Percent 65.47 11.96 22.57 100.00
Priors 0.3333 0.3333 0.3333
Discriminant Analysis of the Fisher Iris Data 22
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis
150 Observations 149 DF Total
2 Variables 147 DF Within Classes
3 Classes 2 DF Between Classes
Class Level Information
SPECIES Frequency Weight Proportion
SETOSA 50 50.0000 0.333333
VERSICOLOR 50 50.0000 0.333333
VIRGINICA 50 50.0000 0.333333
Discriminant Analysis of the Fisher Iris Data 23
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis Within-Class SSCP Matrices
SPECIES = SETOSA
Variable Y1 Y2
Y1 0.297459366 0.057112559 Log Sepal Shape
Y2 0.057112559 7.405231864 Log Petal Shape
-----------------------------------------------------------------
SPECIES = VERSICOLOR
Variable Y1 Y2
Y1 0.5160058088 0.2165401319 Log Sepal Shape
Y2 0.2165401319 0.4413516644 Log Petal Shape
-----------------------------------------------------------------
SPECIES = VIRGINICA
Variable Y1 Y2
Y1 0.5607210370 0.3962003441 Log Sepal Shape
Y2 0.3962003441 0.9797069390 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 24
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis
Pooled Within-Class SSCP Matrix
Variable Y1 Y2
Y1 1.374186212 0.669853035 Log Sepal Shape
Y2 0.669853035 8.826290467 Log Petal Shape
Between-Class SSCP Matrix
Variable Y1 Y2
Y1 5.31562170 -10.28549881 Log Sepal Shape
Y2 -10.28549881 20.14562817 Log Petal Shape
Total-Sample SSCP Matrix
Variable Y1 Y2
Y1 6.68980791 -9.61564577 Log Sepal Shape
Y2 -9.61564577 28.97191864 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 25
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis Within-Class Covariance Matrices
SPECIES = SETOSA DF = 49
Variable Y1 Y2
Y1 0.0060705993 0.0011655624 Log Sepal Shape
Y2 0.0011655624 0.1511271809 Log Petal Shape
-----------------------------------------------------------------
SPECIES = VERSICOLOR DF = 49
Variable Y1 Y2
Y1 0.0105307308 0.0044191864 Log Sepal Shape
Y2 0.0044191864 0.0090071768 Log Petal Shape
-----------------------------------------------------------------
SPECIES = VIRGINICA DF = 49
Variable Y1 Y2
Y1 0.0114432865 0.0080857213 Log Sepal Shape
Y2 0.0080857213 0.0199940192 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 26
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis
Pooled Within-Class Covariance Matrix DF = 147
Variable Y1 Y2
Y1 0.0093482055 0.0045568234 Log Sepal Shape
Y2 0.0045568234 0.0600427923 Log Petal Shape
Between-Class Covariance Matrix DF = 2
Variable Y1 Y2
Y1 0.0531562170 -.1028549881 Log Sepal Shape
Y2 -.1028549881 0.2014562817 Log Petal Shape
Total-Sample Covariance Matrix DF = 149
Variable Y1 Y2
Y1 0.0448980396 -.0645345354 Log Sepal Shape
Y2 -.0645345354 0.1944424070 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 27
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis
Within-Class Correlation Coefficients / Prob |R|
SPECIES = SETOSA
Variable Y1 Y2
Y1 1.00000 0.03848
Log Sepal Shape 0.0 0.7908
Y2 0.03848 1.00000
Log Petal Shape 0.7908 0.0
---------------------------------------------------
SPECIES = VERSICOLOR
Variable Y1 Y2
Y1 1.00000 0.45375
Log Sepal Shape 0.0 0.0009
Y2 0.45375 1.00000
Log Petal Shape 0.0009 0.0
---------------------------------------------------
SPECIES = VIRGINICA
Variable Y1 Y2
Y1 1.00000 0.53456
Log Sepal Shape 0.0 0.0001
Y2 0.53456 1.00000
Log Petal Shape 0.0001 0.0
Discriminant Analysis of the Fisher Iris Data 28
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis
Pooled Within-Class Correlation Coefficients / Prob |R|
Variable Y1 Y2
Y1 1.00000 0.19234
Log Sepal Shape 0.0 0.0192
Y2 0.19234 1.00000
Log Petal Shape 0.0192 0.0
Between-Class Correlation Coefficients / Prob |R|
Variable Y1 Y2
Y1 1.00000 -0.99393
Log Sepal Shape 0.0 0.0702
Y2 -0.99393 1.00000
Log Petal Shape 0.0702 0.0
Total-Sample Correlation Coefficients / Prob |R|
Variable Y1 Y2
Y1 1.00000 -0.69069
Log Sepal Shape 0.0 0.0001
Y2 -0.69069 1.00000
Log Petal Shape 0.0001 0.0
Discriminant Analysis of the Fisher Iris Data 29
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis Simple Statistics
Total-Sample
Variable N Sum Mean
Y1 150 97.19306 0.64795
Y2 150 202.10414 1.34736
Total-Sample
Variable Variance Std Dev Label
Y1 0.04490 0.21189 Log Sepal Shape
Y2 0.19444 0.44096 Log Petal Shape
----------------------------------------------------------
SPECIES = SETOSA
Variable N Sum Mean
Y1 50 19.11748 0.38235
Y2 50 92.87038 1.85741
SPECIES = SETOSA
Variable Variance Std Dev Label
Y1 0.00607 0.07791 Log Sepal Shape
Y2 0.15113 0.38875 Log Petal Shape
----------------------------------------------------------
SPECIES = VERSICOLOR
Variable N Sum Mean
Y1 50 38.25185 0.76504
Y2 50 58.59985 1.17200
SPECIES = VERSICOLOR
Variable Variance Std Dev Label
Y1 0.01053 0.10262 Log Sepal Shape
Y2 0.00901 0.09491 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 30
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis Simple Statistics
SPECIES = VIRGINICA
Variable N Sum Mean
Y1 50 39.82373 0.79647
Y2 50 50.63391 1.01268
SPECIES = VIRGINICA
Variable Variance Std Dev Label
Y1 0.01144 0.10697 Log Sepal Shape
Y2 0.01999 0.14140 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 31
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis
Total-Sample Standardized Class Means
Variable SETOSA VERSICOLOR VIRGINICA
Y1 -1.253490416 0.552562098 0.700928318 Log Sepal Shape
Y2 1.156683088 -0.397690196 -0.758992892 Log Petal Shape
Pooled Within-Class Standardized Class Means
Variable SETOSA VERSICOLOR VIRGINICA
Y1 -2.747075410 1.210962391 1.536113019 Log Sepal Shape
Y2 2.081514495 -0.715665265 -1.365849230 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 32
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis Pairwise Squared Distances Between Groups
2 _ _ -1 _ _
D (i|j) = (X - X )' COV (X - X )
i j i j
Squared Distance to SPECIES
From
SPECIES SETOSA VERSICOLOR VIRGINICA
SETOSA 0 28.81518 37.28957
VERSICOLOR 28.81518 0 0.63321
VIRGINICA 37.28957 0.63321 0
F Statistics, NDF=2, DDF=146 for
Squared Distance to SPECIES
From
SPECIES SETOSA VERSICOLOR VIRGINICA
SETOSA 0 357.73950 462.94874
VERSICOLOR 357.73950 0 7.86130
VIRGINICA 462.94874 7.86130 0
Prob Mahalanobis Distance for
Squared Distance to SPECIES
From
SPECIES SETOSA VERSICOLOR VIRGINICA
SETOSA 1.0000 0.0001 0.0001
VERSICOLOR 0.0001 1.0000 0.0006
VIRGINICA 0.0001 0.0006 1.0000
Discriminant Analysis of the Fisher Iris Data 33
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis
Univariate Test Statistics
F Statistics, Num DF= 2 Den DF= 147
Total Pooled Between RSQ/
Variable STD STD STD R-Squared (1-RSQ)
Y1 0.2119 0.0967 0.2306 0.794585 3.8682
Y2 0.4410 0.2450 0.4488 0.695350 2.2825
Univariate Test Statistics
Variable F Pr F Label
Y1 284.3124 0.0001 Log Sepal Shape
Y2 167.7606 0.0001 Log Petal Shape
Average R-Squared: Unweighted = 0.7449676
Weighted by Variance = 0.7139657
Multivariate Statistics and F Approximations
S=2 M=-0.5 N=72
Statistic Value F Num DF Den DF Pr F
Wilks' Lambda 0.11524006 142.0409 4 292 0.0001
Pillai's Trace 0.89753721 59.8378 4 294 0.0001
Hotelling-Lawley Trace 7.56666236 274.2915 4 290 0.0001
Roy's Greatest Root 7.55198075 555.0706 2 147 0.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.
Discriminant Analysis of the Fisher Iris Data 34
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis
Adjusted Approx Squared
Canonical Canonical Standard Canonical
Correlation Correlation Error Correlation
1 0.939717 0.939249 0.009579 0.883068
2 0.120288 . 0.080738 0.014469
Eigenvalues of INV(E)*H
= CanRsq/(1-CanRsq)
Eigenvalue Difference Proportion Cumulative
1 7.5520 7.5373 0.9981 0.9981
2 0.0147 . 0.0019 1.0000
Test of H0: The canonical correlations in the
current row and all that follow are zero
Likelihood
Ratio Approx F Num DF Den DF Pr F
1 0.11524006 142.0409 4 292 0.0001
2 0.98553082 2.1582 1 147 0.1439
Total Canonical Structure
CAN1 CAN2
Y1 0.947698 0.319169 Log Sepal Shape
Y2 -0.885372 0.464882 Log Petal Shape
Between Canonical Structure
CAN1 CAN2
Y1 0.999072 0.043070 Log Sepal Shape
Y2 -0.997749 0.067060 Log Petal Shape
Pooled Within Canonical Structure
CAN1 CAN2
Y1 0.715024 0.699100 Log Sepal Shape
Y2 -0.548520 0.836138 Log Petal Shape
Discriminant Analysis of the Fisher Iris Data 35
Canonical Discriminant Function Analysis
Canonical Discriminant Analysis
Total-Sample Standardized Canonical Coefficients
CAN1 CAN2
Y1 1.867295076 1.224975162 Log Sepal Shape
Y2 -1.282005465 1.311206984 Log Petal Shape
Pooled Within-Class Standardized Canonical Coefficients
CAN1 CAN2
Y1 0.8520466795 0.5589561248 Log Sepal Shape
Y2 -.7124014960 0.7286285761 Log Petal Shape
Raw Canonical Coefficients
CAN1 CAN2
Y1 8.812502686 5.781141421 Log Sepal Shape
Y2 -2.907330352 2.973553518 Log Petal Shape
Class Means on Canonical Variables
SPECIES CAN1 CAN2
SETOSA -3.823510522 -0.018843683
VERSICOLOR 1.541637490 0.155420683
VIRGINICA 2.281873032 -0.136577000
Discriminant Analysis of the Fisher Iris Data 36
Canonical Discriminant Function Analysis
Show What Canonical Structure Is
Correlation Analysis
2 'WITH' Variables: Y1 Y2
2 'VAR' Variables: CAN1 CAN2
Simple Statistics
Variable N Mean Std Dev Sum
Y1 150 0.647954 0.211892 97.193057
Y2 150 1.347361 0.440956 202.104138
CAN1 150 0 2.904684 0
CAN2 150 0 1.000531 0
Simple Statistics
Variable Minimum Maximum Label
Y1 0.237672 1.085709 Log Sepal Shape
Y2 0.753772 2.708050 Log Petal Shape
CAN1 -7.571585 4.580913
CAN2 -2.773825 2.946968
Pearson Correlation Coefficients / Prob |R| under Ho: Rho=0 / N = 150
CAN1 CAN2
Y1 0.94770 0.31917
Log Sepal Shape 0.0001 0.0001
Y2 -0.88537 0.46488
Log Petal Shape 0.0001 0.0001
Discriminant Analysis of the Fisher Iris Data 37
Canonical Discriminant Function Analysis
Show What Standardized Coefficients Are
Model: MODEL1
Dependent Variable: CAN1
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value ProbF
Model 2 1257.14117 628.57058 . .
Error 147 0 0
C Total 149 1257.14117
Root MSE 0.00000 R-square 1.0000
Dep Mean -0.00000 Adj R-sq 1.0000
C.V. 0.00000
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob |T|
INTERCEP 1 -1.792871 0.00000000 . .
Y1 1 8.812503 0.00000000 . .
Y2 1 -2.907330 0.00000000 . .
Variable
Variable DF Label
INTERCEP 1 Intercept
Y1 1 Log Sepal Shape
Y2 1 Log Petal Shape
Dependent Variable: CAN2
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value ProbF
Model 2 149.15820 74.57910 2.4108614E15 0.0001
Error 147 4.54739E-12 3.093463E-14
C Total 149 149.15820
Root MSE 0.00000 R-square 1.0000
Dep Mean 0.00000 Adj R-sq 1.0000
C.V. 61562488961
Discriminant Analysis of the Fisher Iris Data 38
Canonical Discriminant Function Analysis
Show What Standardized Coefficients Are
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob |T|
INTERCEP 1 -7.752362 0.00000011 -68657232.17 0.0001
Y1 1 5.781141 0.00000009 61479042.374 0.0001
Y2 1 2.973554 0.00000005 65806844.246 0.0001
Variable
Variable DF Label
INTERCEP 1 Intercept
Y1 1 Log Sepal Shape
Y2 1 Log Petal Shape