MULTIVARIATE HOMEWORK #5
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1 MULTIVARIATE HOMEWORK #5 Fisher s dataset on differentiating species of Iris based on measurements on four morphological characters (i.e. sepal length, sepal width, petal length, and petal width) was subjected to Factor Analysis, Cluster Analysis, and Discriminant Analysis. From the SAS output that follows, please answer the following questions. 1. Do the results from the Factor Analysis agree with Fisher s results indicate that the four measurements of flower morphology can each be considered a single factor? 2. How many factors were identified and what criterion was used to determine the total number of factors? 3. Why wasn t the dependent variable Species used in the Factor Analysis? 4. In the results from the Cluster Analysis, what criteria did you use for determining the number of clusters? 5. What percentage of the variation in Species was explained by the number of clusters you determined to be meaningful. 6. In the Discriminant Analysis, did the four morphological variables significantly contribute to assigning the iris plants into one of three species? Explain your answer. 7. In the Discriminant Analysis, what results were provided that were not provided in the Factor Analysis?
2 Simple Linear Correlation Results 04:19 Sunday, June 24, The CORR Procedure 5 Variables: sepallength sepalwidth petallength petalwidth species Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum Label sepallength sepal length sepalwidth sepal width petallength petal length petalwidth petal width species sepallength sepal length sepalwidth sepal width petallength petal length petalwidth petal width Pearson Correlation Coefficients, N = 150 Prob > r under H0: Rho=0 sepallength sepalwidth petallength petalwidth species species
3 Factor Analysis of Fishers Iris Data 04:19 Sunday, June 24, The FACTOR Procedure Input Data Type Raw Data Number of Records Read 150 Number of Records Used 150 N for Significance Tests 150
4 Factor Analysis of Fishers Iris Data 04:19 Sunday, June 24, The FACTOR Procedure Initial Factor Method: Principal Components Prior Communality Estimates: ONE Eigenvalues of the Correlation Matrix: Total = 4 Average = 1 Eigenvalue Difference Proportion Cumulative factor will be retained by the MINEIGEN criterion. Factor Pattern Factor1 sepallength sepal length sepalwidth sepal width petallength petal length petalwidth petal width
5 Factor Analysis of Fishers Iris Data 04:19 Sunday, June 24, The FACTOR Procedure Initial Factor Method: Principal Components Variance Explained by Each Factor Factor Final Communality Estimates: Total = sepallength sepalwidth petallength petalwidth
6 Factor Analysis of Fishers Iris Data 04:19 Sunday, June 24, The FACTOR Procedure Rotation Method: Varimax Note: Rotation not possible with 1 factor.
7 Cluster analysis Using Wards Method 04:19 Sunday, June 24, The CLUSTER Procedure Ward's Minimum Variance Cluster Analysis Eigenvalues of the Covariance Matrix Eigenvalue Difference Proportion Cumulative Root-Mean-Square Total-Sample Standard Deviation Root-Mean-Square Distance Between Observations Number of Clusters Clusters Joined Freq Semipartial R-Square R-Square Cluster History Approximate Expected R-Square Cubic Clustering Criterion Pseudo F Statistic Pseudo t-squared Tie 15 CL24 CL CL21 CL CL18 CL CL16 CL CL14 CL CL20 CL CL27 CL CL15 CL CL10 CL CL13 CL CL9 CL CL12 CL CL6 CL CL3 CL CL5 CL
8 Cluster analysis Using Wards Method 04:19 Sunday, June 24, The CLUSTER Procedure Ward's Minimum Variance Cluster Analysis
9 Cluster analysis Using Wards Method 04:19 Sunday, June 24, The FREQ Procedure Table of CLUSTER by species CLUSTER species Frequency Total Total
10 04:19 Sunday, June 24,
11 Discriminant Analysis of the Iris Dataset 04:19 Sunday, June 24, The DISCRIM Procedure Total Sample Size 150 DF Total 149 Variables 4 DF Within Classes 147 Classes 3 DF Between Classes 2 Number of Observations Read 150 Number of Observations Used 150 species Variable Name Class Level Information Frequency Weight Proportion Prior Probability 1 _ _ _ Pooled Covariance Matrix Information Covariance Matrix Rank Natural Log of the Determinant of the Covariance Matrix
12 Discriminant Analysis of the Iris Dataset 04:19 Sunday, June 24, The DISCRIM Procedure Squared Distance to species From species F Statistics, NDF=4, DDF=144 for Squared Distance to species From species Prob > Mahalanobis Distance for Squared Distance to species From species Generalized Squared Distance to species From species
13 Discriminant Analysis of the Iris Dataset 04:19 Sunday, June 24, The DISCRIM Procedure Linear Discriminant Function for species Variable Label Constant petallength petal length petalwidth petal width sepallength sepal length sepalwidth sepal width
14 Discriminant Analysis of the Iris Dataset 04:19 Sunday, June 24, The DISCRIM Procedure Classification Summary for Calibration Data: WORK.CAN Resubstitution Summary using Linear Discriminant Function Number of Observations and Percent Classified into species From species Total Total Priors Error Count Estimates for species Total Rate Priors
15 Discriminant Analysis of the Iris Dataset 04:19 Sunday, June 24, The DISCRIM Procedure Classification Results for Calibration Data: WORK.CAN Cross-validation Results using Linear Discriminant Function Posterior Probability of Membership in species Obs From species Classified into species * * * * Misclassified observation
16 Discriminant Analysis of the Iris Dataset 04:19 Sunday, June 24, The DISCRIM Procedure Classification Summary for Calibration Data: WORK.CAN Cross-validation Summary using Linear Discriminant Function Number of Observations and Percent Classified into species From species Total Total Priors Error Count Estimates for species Total Rate Priors
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