Handbook_of_statistical_analysis_using_SAS

©2002 CRC Press LLC

Display 13.4

We can use the first two principal component scores to produce a useful plot of the data, particularly if we label the points in an informative manner. This can be achieved using an annotate data set on the plot statement within proc gplot. As an example, we label the plot of the principal component scores with the athlete's overall position in the event.

proc rank data=pcout out=pcout descending; var score;

ranks posn;

data labels;

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set pcout;

retain xsys ysys '2'; y=prin1;

x=prin2; text=put(posn,2.);

keep xsys ysys x y text;

proc gplot data=pcout;

plot prin1*prin2 / annotate=labels; symbol v=none;

run;

proc rank is used to calculate the finishing position in the event. The variable score is ranked in descending order and the ranks stored in the variable posn.

The annotate data set labels has variables x and y which hold the horizontal and vertical coordinates of the text to be plotted, plus the variable text which contains the label text. The two further variables that are needed, xsys and ysys, define the type of coordinate system used. A value of '2'means that the coordinate system for the annotate data set is the same as that used for the data being plotted, and this is usually what is required. As xsys and ysys are character variables, the quotes around

'2'are necessary. The assignment statement text=put(posn,2.); uses the put function to convert the numeric variable posn to a character variable text that is two characters in length.

In the gplot step, the plotting symbols are suppressed by the v=none option in the symbol statement, as the aim is to plot the text defined in the annotate data set in their stead. The resulting plot is shown in Display 13.5. We comment on this plot later.

Next, we can plot the total score achieved by each athlete in the competition against each of the first two principal component scores and also find the corresponding correlations. Plots of the overall score against the first two principal components are shown in Displays 13.6 and 13.7 and the correlations in Display 13.8.

goptions reset=symbol; proc gplot data=pcout;

plot score*(prin1 prin2); run;

proc corr data=pcout; var score prin1 prin2;

run;

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Display 13.5

Display 13.6

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Display 13.7

Display 13.6 shows the very strong relationship between total score and first principal component score — the correlation of the two variables is found from Display 13.8 to be 0.96 which is, of course, highly significant. But the total score does not appear to be related to the second principal component score (see Display 13.7 and r = 0.16).

And returning to Display 13.5, the first principal component score is seen to largely rank the athletes in finishing position, confirming its interpretation as an overall measure of performance.

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Pearson Correlation Coefficients, N = 33

Prob > |r| under H0: Rho=0

Display 13.8

13.3.2 Statements about Pain

The SAS procedure proc factor can accept data in the form of a correlation, or covariance matrix, as well as in the normal rectangular data matrix. To analyse a correlation or covariance matrix, the data need to be read into a special SAS data set with type=corr or type=cov. The correlation matrix shown in Display 13.2 was edited into the form shown in Display 13.9 and read in as follows:

data pain (type = corr);

infile 'n:\handbook2\datasets\pain.dat' expandtabs missover; input _type_ $ _name_ $ p1 - p9;

run;

The type=corr option on the data statement specifies the type of SAS data set being created. The value of the _type_ variable indicates what type of information the observation holds. When _type_=CORR, the values of the variables are correlation coefficients. When _type_=N, the values are the sample sizes. Only the correlations are necessary but the sample sizes have been entered because they will be used by the maximum likelihood method for the test of the number of factors. The _name_ variable identifies the variable whose correlations are in that row of the matrix. The missover option in the infilestatement obviates the need to enter the data for the upper triangle of the correlation matrix.

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Display 13.9

Both principal components analysis and maximum likelihood factor analysis might be applied to the pain statement data using proc factor. The following, however, specifies a maximum likelihood factor analysis extracting two factors and requesting a scree plot, often useful in selecting the appropriate number of components. The output is shown in Display 13.10.

proc factor data=pain method=ml n=2 scree; var p1-p9;

run;

The FACTOR Procedure

Initial Factor Method: Maximum Likelihood

Prior Communality Estimates: SMC

p1 p2 p3 p4 p5

0.46369858 0.37626982 0.54528471 0.51155233 0.39616724

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Preliminary Eigenvalues: Total = 8.2234784 Average = 0.91371982

2 factors will be retained by the NFACTOR criterion.

The FACTOR Procedure

Initial Factor Method: Maximum Likelihood

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Number

The FACTOR Procedure

Initial Factor Method: Maximum Likelihood

Convergence criterion satisfied.

Significance Tests Based on 123 Observations

0.86999919 0.76633961

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Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 9.97197304 Average = 1.107997

The FACTOR Procedure

Initial Factor Method: Maximum Likelihood

Variance Explained by Each Factor

Factor1 6.69225960 2.95023735

Factor2 3.27971562 1.45141080

©2002 CRC Press LLC