The algorithms of PROC ROBUSTREG assume that a response variable is linearly dependent on the regressors. However, in practice, a response often depends on some factors in a nonlinear manner. This example demonstrates how a nonlinear response-factor relationship can be modeled by using constructed effects. (See the section EFFECT Statement of Chapter 19: Shared Concepts and Topics, for details.)
The following data set contains 526 female observations and 474 male observations sampled from 2003 National Health and Nutrition
Examination Survey (NHANES). Each observation is composed of three values: BMI
(body mass index), Age
, and Gender
, measured for subjects whose ages are between 20 and 60.
data BMI; input BMI Age Gender $ @@; datalines; 46.16 30.33 F 20.67 31.83 F 30.98 51.33 F 30.71 31.42 F 29.81 30.50 M 19.94 25.08 F 29.97 41.67 F 24.48 26.92 F 34.34 51.25 F 20.24 53.67 F 27.72 60.25 F 32.85 41.67 M 22.75 47.50 F 32.78 22.42 F 43.07 29.50 F 38.34 58.50 F 40.03 39.92 F 21.78 56.42 M 28.77 39.83 F 28.77 28.75 F 29.73 54.25 M 33.75 35.67 M 28.48 35.83 M 22.12 29.58 F ... more lines ... 26.98 42.50 F 29.44 39.75 M 25.60 52.67 F 19.30 22.00 F 26.53 27.92 F 23.77 29.00 F 29.86 60.58 M 25.41 44.08 M 26.53 24.83 M 33.33 42.08 F 30.52 32.50 F 31.89 38.17 F 32.20 35.92 F 21.73 26.67 M 32.10 39.33 M 25.13 51.75 M ;
The goal of this analysis is to evaluate whether the BMI
by Age
curves are different between women and men at a 5% significance level. In order to provide sufficient flexibility to model
the effect of Age
on BMI
, you can use regression splines that you define with an EFFECT statement. In this example, a regression spline of degree
2 with three knots is used for variable Age
. The knots are placed at the 25, 50, and 75 percentiles of Age
. This analysis assume that there is no interaction between Gender
and Age
, so that the BMI
by Age
curves for women and men are the same up to a constant. The following statements produce the BMI
by Age
scatter plot shown in Output 80.4.1:
proc sort data=bmi; by age; run; ods graphics on; proc sgplot data=bmi; scatter x=age y=bmi/group=gender; run;
The observations with large BMI
values (for example, BMI
> 40) are outliers that can substantially influence an ordinary least squares (OLS) analysis. Output 80.4.1 shows that the distributions of BMI
conditional on Age
are skewed toward the side of large BMI
, and there are more observations with large BMI
values (outliers) in the female group. Hence you can expect a significant Gender
difference in the BMI
by Age
OLS regression analysis. This expectation is confirmed by the OLS Gender
p-value = 0.0059 in Output 80.4.2, which is produced by the following statements:
proc glmselect data=bmi; class gender; effect Age_Sp=spl(age/degree=2 knotmethod=percentiles(3)); model bmi = gender age_sp /selection=none showpvalues; output out=out_ols P=pred R=res; run;
Output 80.4.2: OLS Estimates
Parameter Estimates | |||||
---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error | t Value | Pr > |t| |
Intercept | 1 | 29.890089 | 1.022825 | 29.22 | <.0001 |
Gender F | 1 | 1.167332 | 0.422565 | 2.76 | 0.0058 |
Gender M | 0 | 0 | . | . | . |
Age_Sp 1 | 1 | -4.404487 | 1.473761 | -2.99 | 0.0029 |
Age_Sp 2 | 1 | -3.329537 | 1.374096 | -2.42 | 0.0156 |
Age_Sp 3 | 1 | -0.966875 | 1.314964 | -0.74 | 0.4623 |
Age_Sp 4 | 1 | -1.611621 | 1.123854 | -1.43 | 0.1519 |
Age_Sp 5 | 1 | -0.484787 | 1.701281 | -0.28 | 0.7757 |
Age_Sp 6 | 0 | 0 | . | . | . |
A robust regression method can reduce the outlier influence by automatically assigning smaller or even zero weights to outliers.
For the BMI data, a robust regression method is likely to set less weight on observations with large BMI, so more female observations
would receive smaller weights than male observations. The following statements invoke PROC ROBUSTREG with the BMI
data set:
proc robustreg data=bmi method=s seed=100; class gender; effect Age_Sp=spl(age/degree=2 knotmethod=percentiles(3)); model bmi = gender age_sp; output out=out_s P=pred R=res; run;
Output 80.4.3 shows the parameter estimates and the diagnostics summary produced by PROC ROBUSTREG with the S method. In contrast to OLS,
the robust p-value=0.5573 of the Gender
coefficient indicates that the Gender
effect is not significant. The outlier diagnostics based on the S estimates find 19 outliers that are assigned lower weights
by the S method than by the OLS method.
Output 80.4.3: S Estimates and S Diagnostics Summary
Parameter Estimates | ||||||||
---|---|---|---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error | 95% Confidence Limits | Chi-Square | Pr > ChiSq | ||
Intercept | 1 | 28.2858 | 1.0081 | 26.3100 | 30.2616 | 787.33 | <.0001 | |
Gender | F | 1 | 0.2409 | 0.4114 | -0.5654 | 1.0473 | 0.34 | 0.5581 |
Gender | M | 0 | 0.0000 | . | . | . | . | . |
Age_Sp | 1 | 1 | -3.8956 | 1.4376 | -6.7133 | -1.0779 | 7.34 | 0.0067 |
Age_Sp | 2 | 1 | -1.8692 | 1.3430 | -4.5014 | 0.7630 | 1.94 | 0.1640 |
Age_Sp | 3 | 1 | -0.8336 | 1.2877 | -3.3574 | 1.6903 | 0.42 | 0.5174 |
Age_Sp | 4 | 1 | -0.2329 | 1.1055 | -2.3997 | 1.9338 | 0.04 | 0.8331 |
Age_Sp | 5 | 1 | 0.0055 | 1.6632 | -3.2543 | 3.2652 | 0.00 | 0.9974 |
Age_Sp | 6 | 0 | 0.0000 | . | . | . | . | . |
Scale | 0 | 6.1715 |
Diagnostics Summary | ||
---|---|---|
Observation Type | Proportion | Cutoff |
Outlier | 0.0190 | 3.0000 |
To further compare the OLS and S outputs, the following statements plot the BMI
predictions in variable Age
for both methods in the same graph, which is shown in Output 80.4.4:
data out2_s; set out_s; if gender="F" then label="F, S "; if gender="M" then label="M, S "; run; data out2_ols; merge bmi out_ols; if gender='F' then label='F, OLS'; if gender='M' then label='M, OLS'; keep pred bmi gender age label; run; data out2; set out2_ols out2_s; run; proc sgplot data=out2; series x=age y=pred/group=label; run;
You can observe the following differences between the OLS and S predictions:
The OLS prediction is larger
The OLS curves have a local maximum near Age
= 35
Then, a question remains: is the significance of the Gender
effect for the OLS regression due solely to the outlying observations? To tentatively answer this question, the following
statements drop the observations with the top 10% of BMI
values from the original data set and reapply OLS and S methods on the reduced data set:
data three; set bmi; where bmi<38.315; run; proc robustreg data=three method=s seed=100; class gender; effect Age_Sp=spl(age/degree=2 knotmethod=percentiles(3)); model bmi = gender age_sp; output out=out_s P=pred R=res; run;
data out2_s; set out_s; if gender="F" then label="F, S "; if gender="M" then label="M, S "; run; proc glmselect data=three outdesign=four; class gender; effect Age_Sp=spl(age/degree=2 knotmethod=percentiles(3)); model bmi = gender age_sp /selection=none showpvalues; output out=out_ols P=pred R=res; run;
data out2_ols; merge three out_ols; if gender='F' then label='F, OLS'; if gender='M' then label='M, OLS'; keep pred bmi gender age label; run; data out2; set out2_ols out2_s; run; proc sgplot data=out2; series x=age y=pred/group=label; run; ods graphics off;
Output 80.4.5: S Estimates
Parameter Estimates | ||||||||
---|---|---|---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error | 95% Confidence Limits | Chi-Square | Pr > ChiSq | ||
Intercept | 1 | 27.6427 | 0.9741 | 25.7334 | 29.5520 | 805.23 | <.0001 | |
Gender | F | 1 | -0.2650 | 0.4023 | -1.0535 | 0.5234 | 0.43 | 0.5100 |
Gender | M | 0 | 0.0000 | . | . | . | . | . |
Age_Sp | 1 | 1 | -3.1859 | 1.4032 | -5.9361 | -0.4356 | 5.15 | 0.0232 |
Age_Sp | 2 | 1 | -1.5354 | 1.3051 | -4.0934 | 1.0226 | 1.38 | 0.2394 |
Age_Sp | 3 | 1 | -0.3776 | 1.2499 | -2.8273 | 2.0721 | 0.09 | 0.7626 |
Age_Sp | 4 | 1 | 0.3299 | 1.0668 | -1.7610 | 2.4208 | 0.10 | 0.7572 |
Age_Sp | 5 | 1 | 0.0949 | 1.6221 | -3.0845 | 3.2742 | 0.00 | 0.9534 |
Age_Sp | 6 | 0 | 0.0000 | . | . | . | . | . |
Scale | 0 | 4.9440 |
Output 80.4.6: OLS Estimates
Parameter Estimates | |||||
---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error | t Value | Pr > |t| |
Intercept | 1 | 27.841568 | 0.780817 | 35.66 | <.0001 |
Gender F | 1 | -0.040924 | 0.317749 | -0.13 | 0.8976 |
Gender M | 0 | 0 | . | . | . |
Age_Sp 1 | 1 | -3.253964 | 1.121292 | -2.90 | 0.0038 |
Age_Sp 2 | 1 | -0.975273 | 1.034172 | -0.94 | 0.3459 |
Age_Sp 3 | 1 | -0.508979 | 0.999609 | -0.51 | 0.6108 |
Age_Sp 4 | 1 | 0.089393 | 0.852774 | 0.10 | 0.9165 |
Age_Sp 5 | 1 | -0.113706 | 1.298157 | -0.09 | 0.9302 |
Age_Sp 6 | 0 | 0 | . | . | . |
In the reduced data set, 71 female observations and 29 male observations are dropped. Output 80.4.5 and Output 80.4.6 respectively show the refitted S and OLS parameter estimates, and Output 80.4.7 displays the fitted curves on the reduced data set. You can see that Gender
is no longer significant for the OLS model, and the OLS turning pattern has also disappeared, but the new S curves do not
change much from the previous ones. The OLS BMI
by Age
curves in Output 80.4.7 are closer to the S curves than to the OLS curves in Output 80.4.4. This suggests that indeed the difference between the OLS and S estimate results are due solely to the influence of the outlying
observations.