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Example 52.1: Crime Rates

The following data provide crime rates per 100,000 people in seven categories for each of the fifty states in 1977. Since there are seven numeric variables, it is impossible to plot all the variables simultaneously. Principal components can be used to summarize the data in two or three dimensions, and they help to visualize the data. The following statements produce Output 52.1.1:

   data Crime;
      title 'Crime Rates per 100,000 Population by State';
      input State $1-15 Murder Rape Robbery Assault 
            Burglary Larceny Auto_Theft;
      datalines;
   Alabama        14.2 25.2  96.8 278.3 1135.5 1881.9 280.7
   Alaska         10.8 51.6  96.8 284.0 1331.7 3369.8 753.3
   Arizona         9.5 34.2 138.2 312.3 2346.1 4467.4 439.5
   Arkansas        8.8 27.6  83.2 203.4  972.6 1862.1 183.4
   California     11.5 49.4 287.0 358.0 2139.4 3499.8 663.5
   Colorado        6.3 42.0 170.7 292.9 1935.2 3903.2 477.1
   Connecticut     4.2 16.8 129.5 131.8 1346.0 2620.7 593.2
   Delaware        6.0 24.9 157.0 194.2 1682.6 3678.4 467.0
   Florida        10.2 39.6 187.9 449.1 1859.9 3840.5 351.4
   Georgia        11.7 31.1 140.5 256.5 1351.1 2170.2 297.9
   Hawaii          7.2 25.5 128.0  64.1 1911.5 3920.4 489.4
   Idaho           5.5 19.4  39.6 172.5 1050.8 2599.6 237.6
   Illinois        9.9 21.8 211.3 209.0 1085.0 2828.5 528.6
   Indiana         7.4 26.5 123.2 153.5 1086.2 2498.7 377.4
   Iowa            2.3 10.6  41.2  89.8  812.5 2685.1 219.9
   Kansas          6.6 22.0 100.7 180.5 1270.4 2739.3 244.3
   Kentucky       10.1 19.1  81.1 123.3  872.2 1662.1 245.4
   Louisiana      15.5 30.9 142.9 335.5 1165.5 2469.9 337.7
   Maine           2.4 13.5  38.7 170.0 1253.1 2350.7 246.9
   Maryland        8.0 34.8 292.1 358.9 1400.0 3177.7 428.5
   Massachusetts   3.1 20.8 169.1 231.6 1532.2 2311.3 1140.1
   Michigan        9.3 38.9 261.9 274.6 1522.7 3159.0 545.5
   Minnesota       2.7 19.5  85.9  85.8 1134.7 2559.3 343.1
   Mississippi    14.3 19.6  65.7 189.1  915.6 1239.9 144.4
   Missouri        9.6 28.3 189.0 233.5 1318.3 2424.2 378.4
   Montana         5.4 16.7  39.2 156.8  804.9 2773.2 309.2
   Nebraska        3.9 18.1  64.7 112.7  760.0 2316.1 249.1
   Nevada         15.8 49.1 323.1 355.0 2453.1 4212.6 559.2
   New Hampshire   3.2 10.7  23.2  76.0 1041.7 2343.9 293.4
   New Jersey      5.6 21.0 180.4 185.1 1435.8 2774.5 511.5
   New Mexico      8.8 39.1 109.6 343.4 1418.7 3008.6 259.5
   New York       10.7 29.4 472.6 319.1 1728.0 2782.0 745.8
   North Carolina 10.6 17.0  61.3 318.3 1154.1 2037.8 192.1
   North Dakota    0.9  9.0  13.3  43.8  446.1 1843.0 144.7
   Ohio            7.8 27.3 190.5 181.1 1216.0 2696.8 400.4
   Oklahoma        8.6 29.2  73.8 205.0 1288.2 2228.1 326.8
   Oregon          4.9 39.9 124.1 286.9 1636.4 3506.1 388.9
   Pennsylvania    5.6 19.0 130.3 128.0  877.5 1624.1 333.2
   Rhode Island    3.6 10.5  86.5 201.0 1489.5 2844.1 791.4
   South Carolina 11.9 33.0 105.9 485.3 1613.6 2342.4 245.1
   South Dakota    2.0 13.5  17.9 155.7  570.5 1704.4 147.5
   Tennessee      10.1 29.7 145.8 203.9 1259.7 1776.5 314.0
   Texas          13.3 33.8 152.4 208.2 1603.1 2988.7 397.6
   Utah            3.5 20.3  68.8 147.3 1171.6 3004.6 334.5
   Vermont         1.4 15.9  30.8 101.2 1348.2 2201.0 265.2
   Virginia        9.0 23.3  92.1 165.7  986.2 2521.2 226.7
   Washington      4.3 39.6 106.2 224.8 1605.6 3386.9 360.3
   West Virginia   6.0 13.2  42.2  90.9  597.4 1341.7 163.3
   Wisconsin       2.8 12.9  52.2  63.7  846.9 2614.2 220.7
   Wyoming         5.4 21.9  39.7 173.9  811.6 2772.2 282.0
   ;

   proc princomp out=Crime_Components;
   run;

The first component is a measure of overall crime rate since the first eigenvector shows approximately equal loadings on all variables. The second eigenvector has high positive loadings on the variables Auto_Theft and Larceny and high negative loadings on the variables Murder and Assault. There is also a small positive loading on Burglary and a small negative loading on Rape. This component seems to measure the preponderance of property crime over violent crime. The interpretation of the third component is not obvious.

A simple way to examine the principal components in more detail is to display the output data set sorted by each of the large components. The following statements produce Output 52.1.2 through Output 52.1.3:

   proc sort;
      by Prin1;
   run;

   proc print;
      id State;
      var Prin1 Prin2 Murder Rape Robbery 
          Assault Burglary Larceny Auto_Theft;
      title2 'States Listed in Order of Overall Crime Rate';
      title3 'As Determined by the First Principal Component';
   run;

   proc sort;
      by Prin2;
   run;

   proc print;
      id State;
      var Prin1 Prin2 Murder Rape Robbery 
          Assault Burglary Larceny Auto_Theft;
      title2 'States Listed in Order of Property Vs. 
              Violent Crime';
      title3 'As Determined by the Second Principal Component';
   run;

Another recommended procedure is to make scatter plots of the first few components. The sorted listings help to identify observations on the plots. The following statements produce Output 52.1.4 through Output 52.1.5:

   title2 'Plot of the First Two Principal Components';
   %plotit(data=Crime_Components, labelvar=State, 
           plotvars=Prin2 Prin1, color=black, colors=blue);
   run;

   title2 'Plot of the First and Third Principal Components';
   %plotit(data=Crime_Components, labelvar=State, 
           plotvars=Prin3 Prin1, color=black, colors=blue);
   run;

The most striking feature of the plot of the first and third principal components is that Massachusetts and New York are outliers on the third component.

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