Examples


Example 62.1. Comparison of Robust Estimates

This example illustrates differences in the performance of robust estimates available in the ROBUSTREG procedure.

The following statements generate 1000 random observations. The first 900 observations are from a linear model and the last 100 observations are significantly biased in the y -direction. In other words, ten percent of the observations are contaminated with outliers.

  data a (drop=i);   do i=1 to 1000;   x1=rannor(1234);   x2=rannor(1234);   e=rannor(1234);   if i > 900 then y=100 + e;   else y=10 + 5*x1 + 3*x2 + .5 * e;   output;   end;   run;   proc reg data=a;   model y = x1 x2;   run;   proc robustreg data=a method=m ;   model y = x1 x2;   run;   proc robustreg data=a method=mm;   model y = x1 x2;   run;  

The tables of parameter estimates generated by the ROBUSTREG procedure using M estimation and MM estimation are shown in Output 62.1.2 and Output 62.1.3. For comparison, the ordinary least squares (OLS) estimates produced by the REG procedure are shown in Output 62.1.1. Both the M estimate and the MM estimate correctly estimate the regression coefficients for the underlying model (10, 5, and 3), but the OLS estimate does not.

Output 62.1.2: M Estimates for Data with 10% Contamination
start example
  The ROBUSTREG Procedure   Model Information   Data Set                          WORK.B   Dependent Variable                     y   Number of Covariates                   2   Number of Observations              1000   Method                      M Estimation   Parameter Estimates   Standard   95% Confidence     Chi-   Parameter DF Estimate    Error       Limits       Square Pr > ChiSq   Intercept  1  10.0024   0.0174   9.9683  10.0364  331908     <.0001   x1         1   5.0077   0.0175   4.9735   5.0420 82106.9     <.0001   x2         1   3.0161   0.0167   2.9834   3.0488 32612.5     <.0001   Scale      1   0.5780  
end example
 
Output 62.1.3: MM Estimates for Data with 10% Contamination
start example
  The ROBUSTREG Procedure   Model Information   Data Set                          WORK.B   Dependent Variable                     y   Number of Covariates                   2   Number of Observations              1000   Method                     MM Estimation   Parameter Estimates   Standard   95% Confidence     Chi-   Parameter DF Estimate    Error       Limits       Square Pr > ChiSq   Intercept  1  10.0035   0.0176   9.9690  10.0379  323947     <.0001   x1         1   5.0085   0.0178   4.9737   5.0433 79600.6     <.0001   x2         1   3.0181   0.0168   2.9851   3.0511 32165.0     <.0001   Scale      0   0.6733  
end example
 
Output 62.1.1: OLS Estimates for Data with 10% Contamination
start example
  The REG Procedure   Model: MODEL1   Dependent Variable: y   Parameter Estimates   Parameter       Standard   Variable     DF       Estimate          Error    t Value    Pr > t   Intercept     1       19.06712        0.86322      22.09      <.0001   x1            1        3.55485        0.86892       4.09      <.0001   x2            1        2.12341        0.83039       2.56      0.0107  
end example
 

The next statements demonstrate that if the percentage of contamination is increased to 40%, the M estimates and MM estimates with default options fail to pick up the underlying model. However, by tuning the constant c for the M estimate and the constants INITH and K0 for the MM estimate, you can increase the breakdown values of these estimates and capture the right model. Output 62.1.4 and Output 62.1.5 display these estimates.

Output 62.1.4: M Estimates for Data with 40% Contamination
start example
  The ROBUSTREG Procedure   Model Information   Data Set                          WORK.B   Dependent Variable                     y   Number of Covariates                   2   Number of Observations              1000   Method                      M Estimation   Parameter Estimates   Standard   95% Confidence     Chi-   Parameter DF Estimate    Error       Limits       Square Pr > ChiSq   Intercept  1  10.0137   0.0219   9.9708  10.0565  209688     <.0001   x1         1   4.9905   0.0220   4.9473   5.0336 51399.1     <.0001   x2         1   3.0399   0.0210   2.9987   3.0811 20882.4     <.0001   Scale      1   1.0531  
end example
 
Output 62.1.5: MM Estimates for Data with 40% Contamination
start example
  The ROBUSTREG Procedure   Model Information   Data Set                          WORK.B   Dependent Variable                     y   Number of Covariates                   2   Number of Observations              1000   Method                     MM Estimation   Parameter Estimates   Standard   95% Confidence     Chi-   Parameter DF Estimate    Error       Limits       Square Pr > ChiSq   Intercept  1  10.0103   0.0213   9.9686  10.0520  221639     <.0001   x1         1   4.9890   0.0218   4.9463   5.0316 52535.7     <.0001   x2         1   3.0363   0.0201   2.9970   3.0756 22895.4     <.0001   Scale      0   1.8997  
end example
 
  data b (drop=i);   do i=1 to 1000;   x1=rannor(1234);   x2=rannor(1234);   e=rannor(1234);   if i > 600 then y=100 + e;   else y=10 + 5*x1 + 3*x2 + .5 * e;   output;   end;   run;   proc robustreg data=b method=m(wf=bisquare(c=2));   model y = x1 x2;   run;   proc robustreg data=b method=mm(inith=502 k0=1.8);   model y = x1 x2;   run;  

When there are bad leverage points, the M estimates fail to pick up the underlying model no matter what constant c you use. In this case, other estimates (LTS, S, and MM estimates) in PROC ROBUSTREG, which are robust to bad leverage points, will pick up the underlying model.

The following statements generate 1000 observations with 1% bad high leverage points.

  data b (drop=i);   do i=1 to 1000;   x1=rannor(1234);   x2=rannor(1234);   e=rannor(1234);   if i > 600 then y=100 + e;   else y=10 + 5*x1 + 3*x2 + .5 * e;   if i < 11 then x1=200 * rannor(1234);   if i < 11 then x2=200 * rannor(1234);   if i < 11 then y= 100*e;   output;   end;   run;   proc robustreg data=b method=s(k0=1.8);   model y = x1 x2;   run;   proc robustreg data=b method=mm(inith=502 k0=1.8);   model y = x1 x2;   run;  

Output 62.1.6 displays the S estimates and Output 62.1.7 displays the MM estimates with initial LTS estimates.

Output 62.1.6: S Estimates for Data with 1% Leverage Points
start example
  The ROBUSTREG Procedure   Model Information   Data Set                          WORK.C   Dependent Variable                     y   Number of Covariates                   2   Number of Observations              1000   Method                      S Estimation   Parameter Estimates   Standard   95% Confidence     Chi-   Parameter DF Estimate    Error       Limits       Square Pr > ChiSq   Intercept  1   9.9808   0.0216   9.9383  10.0232  212532     <.0001   x1         1   5.0303   0.0208   4.9896   5.0710 58656.3     <.0001   x2         1   3.0217   0.0222   2.9782   3.0652 18555.7     <.0001   Scale      0   2.2094  
end example
 
Output 62.1.7: MM Estimates for Data with 1% Leverage Points
start example
  The ROBUSTREG Procedure   Model Information   Data Set                          WORK.C   Dependent Variable                     y   Number of Covariates                   2   Number of Observations              1000   Method                     MM Estimation   Parameter Estimates   Standard   95% Confidence     Chi-   Parameter DF Estimate    Error       Limits       Square Pr > ChiSq   Intercept 1   9.9820   0.0215   9.9398  10.0241  215369     <.0001   x1         1   5.0303   0.0206   4.9898   5.0707 59469.1     <.0001   x2         1   3.0222   0.0221   2.9789   3.0655 18744.9     <.0001   Scale      0   2.2134  
end example
 

Example 62.2. Robust ANOVA

The classical analysis of variance (ANOVA) technique based on least squares assumes that the underlying experimental errors are normally distributed. However, data often contain outliers due to recording or other errors. In other cases, extreme responses occurs when control variables in the experiments is set to extremes. It is important to distinguish these extreme points and determine whether they are outliers or important extreme cases. You can use the ROBUSTREG procedure for robust analysis of variance based on M estimation. Typically, there are no high leverage points in a well-designed experiment, so M estimation is appropriate.

The following example shows how to use the ROBUSTREG procedure for robust ANOVA.

An experiment was carried out to study the effects of two successive treatments ( T1 , T2 ) on the recovery time of mice with certain diseases. Sixteen mice were randomly assigned into four groups for the four different combinations of the treatments. The recovery times ( time ) were recorded (in hours).

  data recover;   input id T1 $ T2 $ time;   datalines;   1 0 0 20.2   2 0 0 23.9   3 0 0 21.9   4 0 0 42.4   5 1 0 27.2   6 1 0 34.0   7 1 0 27.4   8 1 0 28.5   9 0 1 25.9   10 0 1 34.5   11 0 1 25.1   12 0 1 34.2   13 1 1 35.0   14 1 1 33.9   15 1 1 38.3   16 1 1 39.9   ;  

The following statements invoke the GLM procedure for a standard ANOVA.

  proc glm data=recover;   class T1 T2;   model time = T1 T2 T1*T2;   run;  

Output 62.2.1 indicates that the overall model effect is not significantatthe10% level and Output 62.2.2 indicates that neither treatment is significantatthe10% level.

Output 62.2.1: Overall ANOVA
start example
  The GLM Procedure   Dependent Variable: time   Sum of   Source                      DF         Squares     Mean Square   F Value    Pr > F   Model                        3     209.9118750      69.9706250      1.86    0.1905   Error                       12     451.9225000      37.6602083   Corrected Total             15     661.8343750   R-Square     Coeff Var      Root MSE    time Mean   0.317167      19.94488      6.136791     30.76875  
end example
 
Output 62.2.2: Model ANOVA
start example
  The GLM Procedure   Dependent Variable: time   Source                      DF       Type I SS     Mean Square   F Value    Pr > F   T1                           1      81.4506250      81.4506250      2.16    0.1671   T2                           1     106.6056250     106.6056250      2.83    0.1183   T1*T2                        1      21.8556250      21.8556250      0.58    0.4609   Source                      DF     Type III SS     Mean Square   F Value    Pr > F   T1                           1      81.4506250      81.4506250      2.16    0.1671   T2                           1     106.6056250     106.6056250      2.83    0.1183   T1*T2                        1      21.8556250      21.8556250      0.58    0.4609  
end example
 

The following statements invoke the ROBUSTREG procedure with the same model.

  proc robustreg data=recover;   class T1 T2;   model time = T1 T2 T1*T2 / diagnostics;   T1_T2: test T1*T2;   output out=robout r=resid sr=stdres;   run;  

Output 62.2.3 shows some basic information about the model and the response variable time .

Output 62.2.3: Model Fitting Information and Summary Statistics
start example
  The ROBUSTREG Procedure   Model Information   Data Set                             WORK.RECOVER   Dependent Variable                           time   Number of Covariates                            2   Number of Continuous Covariates                 0   Number of Discrete Covariates                   2   Number of Observations                         16   Method                               M Estimation   Summary Statistics   Standard   Variable           Q1      Median          Q3        Mean    Deviation         MAD   time          25.5000     31.2000     34.7500     30.7688       6.6425      6.8941  
end example
 

The Parameter Estimates table in Output 62.2.4 indicates that the main effects of both treatments are significantatthe5% level.

Output 62.2.4: Model Parameter Estimates
start example
  The ROBUSTREG Procedure   Parameter Estimates   Standard   95% Confidence     Chi-   Parameter     DF Estimate    Error       Limits       Square Pr > ChiSq   Intercept      1  36.7655   2.0489  32.7497  40.7814  321.98     <.0001   T1        0    1   6.8307   2.8976   12.5100   1.1514    5.56     0.0184   T1        1        0.0000   0.0000   0.0000   0.0000     .        .   T2        0    1   7.6755   2.8976   13.3548   1.9962    7.02     0.0081   T2        1        0.0000   0.0000   0.0000   0.0000     .        .   T1*T2     0 0  1   0.2619   4.0979   8.2936   7.7698    0.00     0.9490   T1*T2     0 1      0.0000   0.0000   0.0000   0.0000     .        .   T1*T2     1 0      0.0000   0.0000   0.0000   0.0000     .        .   T1*T2     1 1      0.0000   0.0000   0.0000   0.0000     .        .   Scale          1   3.5346  
end example
 

The reason for the difference between the traditional ANOVA and the robust ANOVA is explained by Output 62.2.5, which shows that the fourth observation is an outlier. Further investigation shows that the original value of 24.4 for the fourth observation was recorded incorrectly.

Output 62.2.5: Diagnostics
start example
  The ROBUSTREG Procedure   Diagnostics   Standardized   Robust   Obs        Residual      Outlier   4          5.7722         *   Diagnostics Summary   Observation   Type           Proportion      Cutoff   Outlier            0.0625      3.0000  
end example
 

Output 62.2.6 displays the robust test results. The interaction between the two treatments is not significant. Output 62.2.7 displays the robust residuals and standardized robust residuals.

Output 62.2.6: Test of Significance
start example
  The ROBUSTREG Procedure   Robust Linear Tests   T1_T2   Test               Chi-   Test     Statistic   Lambda DF Square Pr > ChiSq   Rho         0.0041   0.7977  1    0.01     0.9431   Rn2         0.0041           1    0.00     0.9490  
end example
 
Output 62.2.7: ROBUSTREG Output
start example
  Obs   T1    T2    time       resid     stdres   1   0     0     20.2   1.7974   0.50851   2   0     0     23.9      1.9026     0.53827   3   0     0     21.9   0.0974   0.02756   4   0     0     42.4     20.4026     5.77222   5   1     0     27.2   1.8900   0.53472   6   1     0     34.0      4.9100     1.38911   7   1     0     27.4   1.6900   0.47813   8   1     0     28.5   0.5900   0.16693   9   0     1     25.9   4.0348   1.14152   10   0     1     34.5      4.5652     1.29156   11   0     1     25.1   4.8348   1.36785   12   0     1     34.2      4.2652     1.20668   13   1     1     35.0   1.7655   0.49950   14   1     1     33.9   2.8655   0.81070   15   1     1     38.3      1.5345     0.43413   16   1     1     39.9      3.1345     0.88679  
end example
 

Example 62.3. Growth Study of De Long and Summers

Robust regression and outlier detection techniques have considerable applications to econometrics. The following example from Zaman, Rousseeuw, and Orhan (2001) shows how these techniques substantially improve the ordinary least squares (OLS) results for the growth study of De Long and Summers.

De Long and Summers (1991) studied the national growth of 61 countries from 1960 to 1985 using OLS.

  data growth;   input country$ GDP LFG EQP NEQ GAP @@;   datalines;   Argentin  0.0089 0.0118 0.0214 0.2286 0.6079   Austria   0.0332 0.0014 0.0991 0.1349 0.5809   Belgium   0.0256 0.0061 0.0684 0.1653 0.4109   Bolivia   0.0124 0.0209 0.0167 0.1133 0.8634   Botswana  0.0676 0.0239 0.1310 0.1490 0.9474   Brazil    0.0437 0.0306 0.0646 0.1588 0.8498   Cameroon  0.0458 0.0169 0.0415 0.0885 0.9333   Canada    0.0169 0.0261 0.0771 0.1529 0.1783   Chile     0.0021 0.0216 0.0154 0.2846 0.5402   Colombia  0.0239 0.0266 0.0229 0.1553 0.7695   CostaRic  0.0121 0.0354 0.0433 0.1067 0.7043   Denmark   0.0187 0.0115 0.0688 0.1834 0.4079   Dominica  0.0199 0.0280 0.0321 0.1379 0.8293   Ecuador   0.0283 0.0274 0.0303 0.2097 0.8205   ElSalvad  0.0046 0.0316 0.0223 0.0577 0.8414   Ethiopia  0.0094 0.0206 0.0212 0.0288 0.9805   Finland   0.0301 0.0083 0.1206 0.2494 0.5589   France    0.0292 0.0089 0.0879 0.1767 0.4708   Germany   0.0259 0.0047 0.0890 0.1885 0.4585   Greece    0.0446 0.0044 0.0655 0.2245 0.7924   Guatemal  0.0149 0.0242 0.0384 0.0516 0.7885   Honduras  0.0148 0.0303 0.0446 0.0954 0.8850   HongKong  0.0484 0.0359 0.0767 0.1233 0.7471   India     0.0115 0.0170 0.0278 0.1448 0.9356   Indonesi  0.0345 0.0213 0.0221 0.1179 0.9243   Ireland   0.0288 0.0081 0.0814 0.1879 0.6457   Israel    0.0452 0.0305 0.1112 0.1788 0.6816   Italy     0.0362 0.0038 0.0683 0.1790 0.5441   IvoryCoa  0.0278 0.0274 0.0243 0.0957 0.9207   Jamaica   0.0055 0.0201 0.0609 0.1455 0.8229   Japan     0.0535 0.0117 0.1223 0.2464 0.7484   Kenya     0.0146 0.0346 0.0462 0.1268 0.9415   Korea     0.0479 0.0282 0.0557 0.1842 0.8807   Luxembou  0.0236 0.0064 0.0711 0.1944 0.2863   Madagasc   0.0102 0.0203 0.0219 0.0481 0.9217   Malawi    0.0153 0.0226 0.0361 0.0935 0.9628   Malaysia  0.0332 0.0316 0.0446 0.1878 0.7853   Mali      0.0044 0.0184 0.0433 0.0267 0.9478   Mexico    0.0198 0.0349 0.0273 0.1687 0.5921   Morocco   0.0243 0.0281 0.0260 0.0540 0.8405   Netherla  0.0231 0.0146 0.0778 0.1781 0.3605   Nigeria   0.0047 0.0283 0.0358 0.0842 0.8579   Norway    0.0260 0.0150 0.0701 0.2199 0.3755   Pakistan  0.0295 0.0258 0.0263 0.0880 0.9180   Panama    0.0295 0.0279 0.0388 0.2212 0.8015   Paraguay  0.0261 0.0299 0.0189 0.1011 0.8458   Peru      0.0107 0.0271 0.0267 0.0933 0.7406   Philippi  0.0179 0.0253 0.0445 0.0974 0.8747   Portugal  0.0318 0.0118 0.0729 0.1571 0.8033   Senegal   0.0011 0.0274 0.0193 0.0807 0.8884   Spain     0.0373 0.0069 0.0397 0.1305 0.6613   SriLanka  0.0137 0.0207 0.0138 0.1352 0.8555   Tanzania  0.0184 0.0276 0.0860 0.0940 0.9762   Thailand  0.0341 0.0278 0.0395 0.1412 0.9174   Tunisia   0.0279 0.0256 0.0428 0.0972 0.7838   U.K.      0.0189 0.0048 0.0694 0.1132 0.4307   U.S.      0.0133 0.0189 0.0762 0.1356 0.0000   Uruguay   0.0041 0.0052 0.0155 0.1154 0.5782   Venezuel  0.0120 0.0378 0.0340 0.0760 0.4974   Zambia   0.0110 0.0275 0.0702 0.2012 0.8695   Zimbabwe  0.0110 0.0309 0.0843 0.1257 0.8875   ;  

The regression equation they used is:

click to expand

where the response variable is the growth in gross domestic product per worker ( GDP ) and the regressors are labor force growth ( LFG ), relative GDP gap ( GAP ), equipment investment ( EQP ), and non-equipment investment ( NEQ ).

The following statements invoke the REG procedure for the OLS analysis:

  proc reg data=growth;   model GDP  = LFG GAP EQP NEQ ;   run;  

The OLS analysis of Output 62.3.1 indicates that GAP and EQP have a significant influence on GDP at the 5% level.

Output 62.3.1: OLS Estimates
start example
  The REG Procedure   Model: MODEL1   Dependent Variable: GDP   Parameter Estimates   Parameter       Standard   Variable     DF       Estimate          Error    t Value    Pr > t   Intercept     1   0.01430        0.01028   1.39      0.1697   LFG           1   0.02981        0.19838   0.15      0.8811   GAP           1        0.02026        0.00917       2.21      0.0313   EQP           1        0.26538        0.06529       4.06      0.0002   NEQ           1        0.06236        0.03482       1.79      0.0787  
end example
 

The following statements invoke the ROBUSTREG procedure with the default M estimation.

  proc robustreg data=growth;   model GDP  = LFG GAP EQP NEQ / diagnostics leverage;   output out=robout r=resid sr=stdres;   run;  

Output 62.3.2 displays model information and summary statistics for variables in the model.

Output 62.3.2: Model Fitting Information and Summary Statistics
start example
  The ROBUSTREG Procedure   Model Information   Data Set                    MYLIB.GROWTH   Dependent Variable                   GDP   Number of Covariates                   4   Number of Observations                61   Method                      M Estimation   Summary Statistics   Standard   Variable           Q1      Median          Q3        Mean    Deviation         MAD   LFG            0.0118      0.0239      0.0281      0.0211      0.00979     0.00949   GAP            0.5796      0.8015      0.8863      0.7258       0.2181      0.1778   EQP            0.0265      0.0433      0.0720      0.0523       0.0296      0.0325   NEQ            0.0956      0.1356      0.1812      0.1399       0.0570      0.0624   GDP            0.0121      0.0231      0.0310      0.0224       0.0155      0.0150  
end example
 

Output 62.3.3 displays the M estimates. Besides GAP and EQP , the robust analysis also indicates that NEQ is significant. This new finding is explained by Output 62.3.4, which shows that Zambia, the sixtieth country in the data, is an outlier. Output 62.3.4 also identifies leverage points based the robust MCD distances; however, there are no serious high leverage points in this data set. Output 62.3.5 displays robust versions of goodness-of-fit statistics for the model.

Output 62.3.3: M estimates
start example
  The ROBUSTREG Procedure   Parameter Estimates   Standard   95% Confidence     Chi-   Parameter DF Estimate    Error       Limits       Square Pr > ChiSq   Intercept  1   0.0247   0.0097   0.0437   0.0058    6.53     0.0106   LFG        1   0.1040   0.1867   0.2619   0.4699    0.31     0.5775   GAP        1   0.0250   0.0086   0.0080   0.0419    8.36     0.0038   EQP        1   0.2968   0.0614   0.1764   0.4172   23.33     <.0001   NEQ        1   0.0885   0.0328   0.0242   0.1527    7.29     0.0069   Scale      1   0.0099  
end example
 
Output 62.3.4: Diagnostics
start example
  The ROBUSTREG Procedure   Diagnostics   Robust                    Standardized   Mahalanobis         MCD                          Robust   Obs      Distance    Distance      Leverage          Residual      Outlier   1        2.6083      4.0639         *   0.9424   5        3.4351      6.7391         *                1.4200   8        3.1876      4.6843         *   0.1972   9        3.6752      5.0599         *   1.8784   17        2.6024      3.8186         *   1.7971   23        2.1225      3.8238         *                1.7161   27        2.6461      5.0336         *                0.0909   31        2.9179      4.7140         *                0.0216   53        2.2600      4.3193         *   1.8082   57        3.8701      5.4874         *                0.1448   58        2.5953      3.9671         *   0.0978   59        2.9239      4.1663         *                0.3573   60        1.8562      2.7135   4.9798         *   61        1.9634      3.9128         *   2.5959   Diagnostics Summary   Observation   Type           Proportion      Cutoff   Outlier            0.0164      3.0000   Leverage           0.2131      3.3382  
end example
 
Output 62.3.5: Goodness-of-Fit
start example
  The ROBUSTREG Procedure   Goodness-of-Fit   Statistic       Value   R-Square       0.3178   AICR          80.2134   BICR          91.5095   Deviance       0.0070  
end example
 

The following statements invoke the ROBUSTREG procedure with LTS estimation, which was used by Zaman, Rousseeuw, and Orhan (2001). The results are consistent with those of M estimation.

  proc robustreg method=lts(h=33) fwls data=growth;   model GDP  = LFG GAP EQP NEQ / diagnostics leverage ;   output out=robout r=resid sr=stdres;   run;  

Output 62.3.6 displays the LTS estimates.

Output 62.3.6: LTS estimates
start example
  The ROBUSTREG Procedure   LTS Profile   Total Number of Observations                  61   Number of Squares Minimized                   33   Number of Coefficients                         5   Highest Possible Breakdown Value          0.4590   LTS Parameter Estimates   Parameter         DF    Estimate   Intercept          1   0.0249   LFG                1      0.1123   GAP                1      0.0214   EQP                1      0.2669   NEQ                1      0.1110   Scale (sLTS)       0      0.0076   Scale (Wscale)     0      0.0109  
end example
 

Output 62.3.7 displays outlier and leverage point diagnostics based on the LTS estimates.

Output 62.3.7: Diagnostics and LTS-Rsquare
start example
  The ROBUSTREG Procedure   Diagnostics   Robust                    Standardized   Mahalanobis         MCD                          Robust   Obs      Distance    Distance      Leverage          Residual      Outlier   1        2.6083      4.0639         *   1.0715   5        3.4351      6.7391         *                1.6574   8        3.1876      4.6843         *   0.2324   9        3.6752      5.0599         *   2.0896   17        2.6024      3.8186         *   1.6367   23        2.1225      3.8238         *                1.7570   27        2.6461      5.0336         *                0.2334   31        2.9179      4.7140         *                0.0971   53        2.2600      4.3193         *   1.2978   57        3.8701      5.4874         *                0.0605   58        2.5953      3.9671         *   0.0857   59        2.9239      4.1663         *                0.4113   60        1.8562      2.7135   4.4984         *   61        1.9634      3.9128         *   2.1201   Diagnostics Summary   Observation   Type           Proportion      Cutoff   Outlier            0.0164      3.0000   Leverage           0.2131      3.3382   R-Square for LTS   Estimation   R-Square      0.7418  
end example
 

Output 62.3.8 displays the final weighted lease squares estimates, which are identical to those reported in Zaman, Rousseeuw, and Orhan (2001).

Output 62.3.8: Final Weighted LS estimates
start example
  The ROBUSTREG Procedure   Parameter Estimates for Final Weighted Least Squares Fit   Standard   95% Confidence     Chi-   Parameter      DF Estimate    Error       Limits       Square Pr > ChiSq   Intercept       1   0.0222   0.0093   0.0405   0.0039    5.65     0.0175   LFG             1   0.0446   0.1771   0.3026   0.3917    0.06     0.8013   GAP             1   0.0245   0.0082   0.0084   0.0406    8.89     0.0029   EQP             1   0.2824   0.0581   0.1685   0.3964   23.60     <.0001   NEQ             1   0.0849   0.0314   0.0233   0.1465    7.30     0.0069   Scale           0   0.0116  
end example
 



SAS.STAT 9.1 Users Guide (Vol. 6)
SAS.STAT 9.1 Users Guide (Vol. 6)
ISBN: N/A
EAN: N/A
Year: 2004
Pages: 127

flylib.com © 2008-2017.
If you may any questions please contact us: flylib@qtcs.net