Handbook_of_statistical_analysis_using_SAS

Final Communality Estimates and Variable Weights

Total Communality: Weighted = 9.971975 Unweighted = 4.401648

Display 13.10

Here, the scree plot suggests perhaps three factors, and the formal significance test for number of factors given in Display 13.10 confirms that more than two factors are needed to adequately describe the observed correlations. Consequently, the analysis is now extended to three factors, with a request for a varimax rotation of the solution.

proc factor data=pain method=ml n=3 rotate=varimax; var p1-p9;

run;

The output is shown in Display 13.11. First, the test for number factors indicates that a three-factor solution provides an adequate description of the observed correlations. We can try to identify the three common factors by examining the rotated loading in Display 13.11. The first factor loads highly on statements 1, 3, 4, and 8. These statements attribute pain relief to the control of doctors, and thus we might label the factor doctors’ control of pain. The second factor has its highest loadings on statements 6 and 7. These statements associated the cause of pain as one’s own actions, and the factor might be labelled individual’s responsibility for pain. The third factor has high loadings on statements 2 and 5. Again, both involve an individual’s own responsibility for their pain but now specifically because of things they have not done; the factor might be labelled lifestyle responsibility for pain.

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The FACTOR Procedure

Initial Factor Method: Maximum Likelihood

Prior Communality Estimates: SMC

0.46369858 0.37626982 0.54528471 0.51155233 0.39616724

0.55718109 0.48259656 0.56935053 0.25371373

Preliminary Eigenvalues: Total = 8.2234784 Average = 0.91371982

3 factors will be retained by the NFACTOR criterion.

Convergence criterion satisfied.

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Significance Tests Based on 123 Observations

The FACTOR Procedure

Initial Factor Method: Maximum Likelihood

Squared Canonical Correlations

0.90182207 0.83618918 0.60884385

Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 15.8467138 Average = 1.76074598

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Variance Explained by Each Factor

Final Communality Estimates and Variable Weights

Total Communality: Weighted = 15.846716 Unweighted = 5.128747

The FACTOR Procedure

Rotation Method: Varimax

Orthogonal Transformation Matrix

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Variance Explained by Each Factor

Final Communality Estimates and Variable Weights

Total Communality: Weighted = 15.846716 Unweighted = 5.128747

Display 13.11

Exercises

13.1Repeat the principal components analysis of the Olympic decathlon data without removing the athlete who finished last in the competition. How do the results compare with those reported in this chapter (Display 13.5)?

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13.2Run a principal components analysis on the pain data and compare the results with those from the maximum likelihood factor analysis.

13.3Run principal factor analysis and maximum likelihood factor analysis on the Olympic decathlon data. Investigate the use of other methods of rotation than varimax.

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Chapter 14

Cluster Analysis: Air

Pollution in the U.S.A.

14.1Description of Data

The data to be analysed in this chapter relate to air pollution in 41 U.S. cities. The data are given in Display 14.1 (they also appear in SDS as Table 26). Seven variables are recorded for each of the cities:

1.SO2 content of air, in micrograms per cubic metre

2.Average annual temperature, in °F

3.Number of manufacturing enterprises employing 20 or mor e workers

4.Population size (1970 census), in thousands

5.Average annual wind speed, in miles per hour

6.Average annual precipitation, in inches

7.Average number of days per year with precipitation

In this chapter we use variables 2 to 7 in a cluster analysis of the data to investigate whether there is any evidence of distinct groups of cities. The resulting clusters are then assessed in terms of their air pollution levels as measured by SO2 content.

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Display 14.1

14.2Cluster Analysis

Cluster analysis is a generic term for a large number of techniques that have the common aim of determining whether a (usually) multivariate data set contains distinct groups or clusters of observations and, if so, find which of the observations belong in the same cluster. A detailed account of what is now a very large area is given in Everitt, Landau, and Leese (2001).

The most commonly used classes of clustering methods are those that lead to a series of nested or hierarchical classifications of the observations, beginning at the stage where each observation is regarded as forming a single-member “cluster” and ending at the stage where all the observations are in a single group. The complete hierarchy of solutions can be displayed as a tree diagram known as a dendrogram. In practice, most users are interested in choosing a particular partition of the data, that is, a particular number of groups that is optimal in some sense. This entails “cutting” the dendrogram at some particular level.

Most hierarchical methods operate not on the raw data, but on an inter-individual distance matrix calculated from the raw data. The most commonly used distance measure is Euclidean and is defined as:

where xik and xjk are the values of the kth variable for observations i and j. The different members of the class of hierarchical clustering techniques arise because of the variety of ways in which the distance between a cluster containing several observations and a single observation, or between two clusters, can be defined. The inter-cluster distances used

by three commonly applied hierarchical clustering techniques are

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Single linkage clustering: distance between their closest observations

Complete linkage clustering: distance between the most remote observations

Average linkage clustering: average of distances between all pairs of observations, where members of a pair are in different groups

Important issues that often need to be considered when using clustering in practice include how to scale the variables before calculating the distance matrix, which particular method of cluster analysis to use, and how to decide on the appropriate number of groups in the data. These and many other practical problems of clustering are discussed in Everitt et al. (2001).

14.3Analysis Using SAS

The data set for Table 26 in SDS does not contain the city names shown in Display 14.1; thus, we have edited the data set so that they occupy the first 16 columns. The resulting data set can be read in as follows:

data usair;

infile 'n:\handbook2\datasets\usair.dat' expandtabs; input city $16. so2 temperature factories population wind-

speed rain rainydays; run;

The names of the cities are read into the variable city with a $16. format because several of them contain spaces and are longer than the default length of eight characters. The numeric data are read in with list input.

We begin by examining the distributions of the six variables to be used in the cluster analysis.

proc univariate data=usair plots; var temperature--rainydays; id city;

run;

The univariate procedure was described in Chapter 2. Here, we use the plots option, which has the effect of including stem and leaf plots, box plots, and normal probability plots in the printed output. The id statement

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