The ROBUSTREG Procedure

When the model has a single independent continuous variable (with or without the intercept), the ROBUSTREG procedure automatically creates a plot of robust fit against this independent variable.

The following simple example shows the fit plot. The data, from Rousseeuw and Leroy (1987, Table 3), include the logarithm of surface temperature and the logarithm of light intensity for 47 stars in the direction of the constellation Cygnus.

data star;
   input index x y @@;
   label x = 'Log Temperature'
         y = 'Log Light Intensity';
   datalines;
 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
;

The following statements use the MM method to plot the robust fit of the logarithm of light intensity against the logarithm of the surface temperature:

ods graphics on;
proc robustreg data=star method=mm;
   model y = x;
run;

Figure 98.22 shows the fit plot. Confidence limits are added to the plot by default.

Figure 98.22: Robust Fit

You can suppress the confidence limits by specifying the NOLIMITS option, as shown in the following statements:

proc robustreg data=star method=mm plot=fitplot(nolimits);
   model y = x;
run;

The distance-distance (DD) plot is mainly used for leverage-point diagnostics. It is a scatter plot of the robust distances (or projected robust distances) against the classical Mahalanobis distances (or projected classical Mahalanobis distances) for the independent variables. For more information about the robust distance, see the section Leverage-Point and Outlier Detection.

You can use the PLOT=DDPLOT option to request this plot. The following statements use the Stack data set in the section M Estimation to create the single plot shown in Figure 98.5:

proc robustreg data=Stack plot=ddplot;
   model y = x1 x2 x3;
run;

The reference lines represent the cutoff values. The diagonal line is also drawn to show the distribution of the distances. By default, all outliers and leverage points are labeled with observation numbers. To change the default, you can use the LABEL= option as described in Table 98.2.

If you specify ID variables in the ID statement, the values of the first ID variable instead of observation numbers are used as labels.

The residual-distance (RD) plot is used for both outlier and leverage-point diagnostics. It is a scatter plot of the standardized robust residuals against the robust distances. For more information about the robust distance, see the section Leverage-Point and Outlier Detection.

You can use the PLOT=RDPLOT option to request the RD plot. The following statements use the Stack data set in the section M Estimation to create the plot shown in Figure 98.4:

proc robustreg data=Stack plot=rdplot;
   model y = x1 x2 x3;
run;

The reference lines represent the cutoff values. By default, all outliers and leverage points are labeled with observation numbers. To change the default, you can use the LABEL= option as described in Table 98.2.

If you specify ID variables in the ID statement, the values of the first ID variable instead of observation numbers are used as labels.

PROC ROBUSTREG produces a histogram and a Q-Q plot for the standardized robust residuals. The histogram is superimposed with a normal density curve and a kernel density curve. The following statements use the Stack data set from the section M Estimation to create the plots in Figure 98.6 and Figure 98.7:

proc robustreg data=Stack plots=(histogram qqplot);
   model y = x1 x2 x3;
run;