Survey-Weighted OLS¶
Overview¶
Survey weights correct for differential selection probabilities and nonresponse in complex survey data. Ignoring weights can produce biased coefficient estimates and underestimated standard errors.
The original Zelig package (Imai, King, and Lau 2007, 2008) required separate model types for survey-weighted estimation (e.g., "normal.survey" instead of "ls"). zelig2 simplifies this: pass a numeric weight vector to weights and the model is automatically estimated via survey::svyglm (Lumley 2004). No separate model type is needed.
Note on the original Zelig's survey implementation
The original Zelig's survey models (normal.survey, logit.survey, etc.) contain a known double-weighting bug (IQSS/Zelig#332): weights are passed to both svydesign() and svyglm(), producing incorrect coefficient estimates and standard errors. zelig2 does not have this bug --- its estimates match survey::svyglm() exactly. See Comparison with Zelig for a detailed analysis.
This vignette uses data from the U.S. Census Bureau's Household Pulse Survey (Week 62, N = 58,202), a nationally representative survey with person-level probability weights (PWEIGHT).
Model Estimation with Survey Weights¶
library(zelig2)
z_weighted <- zelig2(
mh_score ~ age + college + income_k,
model = "ls",
data = pulse,
weights = pulse$pweight,
num = 1000L
)
summary(z_weighted)
zelig2: Least Squares (Linear Regression)
Formula: mh_score ~ age + college + income_k
N: 58202
Survey-weighted: yes
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 7.08844445 0.12424980 57.0499 < 2.2e-16 ***
age -0.05369919 0.00194374 -27.6267 < 2.2e-16 ***
college -0.40248340 0.05548744 -7.2536 4.059e-13 ***
income_k -0.01016999 0.00038834 -26.1883 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Standard errors reflect the survey design and are generally larger than unweighted estimates.
Comparing Weighted vs. Unweighted Estimates¶
z_unweighted <- zelig2(mh_score ~ age + college + income_k,
model = "ls", data = pulse)
coef(z_unweighted)
Substantial Differences
The college effect is -0.540 unweighted but -0.402 weighted --- a 34% difference. The unweighted sample overrepresents college-educated respondents, who differ systematically in mental health outcomes.
Survey-weighted standard errors are also larger (college SE: 0.029 unweighted vs. 0.055 weighted). This reflects the effective sample size reduction from weighting and properly accounts for the survey design.
Simulating Quantities of Interest¶
Set covariates and simulate expected values from the weighted model:
z_weighted <- setx(z_weighted, age = 40, college = 1, income_k = 75)
z_weighted <- sim(z_weighted)
summary(z_weighted)
A 40-year-old college graduate earning $75,000 has an expected mental health score of 3.78 (95% CI: 3.69, 3.87). The wider confidence interval compared to the unweighted model (SD 0.045 vs. 0.023) reflects the additional uncertainty from survey weighting.
First Differences¶
Compare college graduates to non-graduates under the weighted model:
z_weighted <- zelig2(mh_score ~ age + college + income_k,
model = "ls", data = pulse,
weights = pulse$pweight, num = 1000L)
z_weighted <- setx(z_weighted, age = 40, college = 0, income_k = 75)
z_weighted <- setx1(z_weighted, age = 40, college = 1, income_k = 75)
z_weighted <- sim(z_weighted)
summary(z_weighted)
--- Simulation Summary ( 1000 draws) ---
Expected Values:
Mean: 4.1784 SD: 0.0499 [4.0852, 4.2774]
First Differences:
Mean: -0.4016 SD: 0.0539 [-0.5137, -0.3021]
College graduates have mental health scores 0.40 points lower (better) than non-graduates (95% CI: -0.51, -0.30). Note that the weighted first difference (-0.40) is attenuated relative to the unweighted estimate (-0.54), reflecting the correction for sample composition.
Alternative Ways to Specify Weights¶
Pre-built Survey Design¶
Pass an existing survey::svydesign object:
library(survey)
pulse_design <- svydesign(ids = ~1, weights = ~pweight, data = pulse)
z <- zelig2(mh_score ~ age + college + income_k,
model = "ls", data = pulse,
survey_design = pulse_design)
Complex Designs¶
For surveys with clustering, stratification, or finite population corrections:
z <- zelig2(mh_score ~ age + college + income_k,
model = "ls", data = pulse,
weights = pulse$pweight,
ids = ~cluster_id,
strata = ~stratum)
These arguments are passed to survey::svydesign internally.
When to Use Survey Weights
Use weights whenever your data come from a complex survey design: national surveys (NHANES, CPS, ACS), multi-stage cluster samples, stratified samples, or surveys with oversampling or differential nonresponse.
References¶
- King, G., Tomz, M., and Wittenberg, J. (2000). "Making the Most of Statistical Analyses: Improving Interpretation and Presentation." American Journal of Political Science, 44(2), 347--361.
- Imai, K., King, G., and Lau, O. (2008). "Toward a Common Framework for Statistical Analysis and Development." Journal of Computational and Graphical Statistics, 17(4), 892--913.
- Lumley, T. (2004). "Analysis of Complex Survey Samples." Journal of Statistical Software, 9(8), 1--19.