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Example 9.1: LMS and LTS with Substantial Leverage Points: Hertzsprung-Russell Star Data

The following data are reported in Rousseeuw and Leroy (1987, p. 27) and are based on Humphrey (1978) and Vansina and De Greve (1982). The 47 observations correspond to the 47 stars of the CYG OB1 cluster in the direction of Cygnus. The regressor variable (column 2) x is the logarithm of the effective temperature at the surface of the star (T_e), and the response variable (column 3) y is the logarithm of its light intensity (L / L₀). The results for LS and LMS on page 28 of Rousseeuw and Leroy (1987) are based on a more precise (five decimal places) version of the data set. This data set is remarkable in that it contains four substantial leverage points (giant stars) corresponding to observations 11, 20, 30, and 34 that greatly affect the results of L₂ and even L₁ regression.

   ab =  { 1  4.37  5.23,   2  4.56  5.74,   3  4.26  4.93,
           4  4.56  5.74,   5  4.30  5.19,   6  4.46  5.46,
           7  3.84  4.65,   8  4.57  5.27,   9  4.26  5.57,
          10  4.37  5.12,  11  3.49  5.73,  12  4.43  5.45,
          13  4.48  5.42,  14  4.01  4.05,  15  4.29  4.26,
          16  4.42  4.58,  17  4.23  3.94,  18  4.42  4.18,
          19  4.23  4.18,  20  3.49  5.89,  21  4.29  4.38,
          22  4.29  4.22,  23  4.42  4.42,  24  4.49  4.85,
          25  4.38  5.02,  26  4.42  4.66,  27  4.29  4.66,
          28  4.38  4.90,  29  4.22  4.39,  30  3.48  6.05,
          31  4.38  4.42,  32  4.56  5.10,  33  4.45  5.22,
          34  3.49  6.29,  35  4.23  4.34,  36  4.62  5.62,
          37  4.53  5.10,  38  4.45  5.22,  39  4.53  5.18,
          40  4.43  5.57,  41  4.38  4.62,  42  4.45  5.06,
          43  4.50  5.34,  44  4.45  5.34,  45  4.55  5.54,
          46  4.45  4.98,  47  4.42  4.50 } ;

   a = ab[,2]; b = ab[,3];

   print "*** Hertzsprung-Russell Star Data: Do LMS ***";
      optn = j(8,1,.);
      optn[2]= 3;    /* ipri */
      optn[3]= 3;    /* ilsq */
      optn[8]= 3;    /* icov */
   call lms(sc,coef,wgt,optn,b,a);

Partial output for LS regression is shown in Output 9.1.2.

Looking at the column Best Crit in the iteration history table, you see that, with complete enumeration, the optimal solution is found very early.

The output for WLS regression follows. Due to the size of the scaled residuals, six observations (with numbers 7, 9, 11, 20, 30, 34) were assigned zero weights in the following WLS analysis.

The LTS regression leads to similar results:

   print "*** Hertzsprung-Russell Star Data: Do LTS ***";
      optn = j(8,1,.);
      optn[2]= 3;    /* ipri */
      optn[3]= 3;    /* ilsq */
      optn[8]= 3;    /* icov */
   call lts(sc,coef,wgt,optn,b,a);

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