Online Resource 1
Refusal Bias in the Estimation of HIV Prevalence
Section 1
The fieldwork protocol was developed such that nurses were randomly assigned to PSUs, covering one or a few blocks of houses, and to households within PSUs, as follows. All nurses jointly went to each of the neighborhoods that encompass several PSUs. Nurses worked in changing teams of two or three. Teams were randomly assigned to PSUs within neighborhoods such that the number of respondents was divided approximately evenly over the nurses. The number of respondents per PSU was known from the socioeconomic questionnaire in the first stage of the survey. When all households within a neighborhood had been visited, the entire group of nurses moved on to the next neighborhood. There are no indications that nurses bypassed the assignment schedule and systematically swapped assigned households among each other dependent on certain household characteristics.
Table S1 Regressions of PSU-level household characteristics on nurse IDcodes
Average Age / Average Adult Education (highest grade completed) / Average Log of per Capita Consumption / Average Wealth / Average Water Quality / Average Roof Quality / Average Toilet Quality / Average Access to Electricity / Average Homeownership / Average EmployedNurse A / –0.428 / 0.000 / –0.030 / –0.117 / –0.022 / –0.011 / –0.014 / 0.002 / –0.075 / 0.032
(0.746) / (0.409) / (0.166) / (0.169) / (0.075) / (0.033) / (0.067) / (0.076) / (0.054) / (0.023)
Nurse B / –0.540 / –0.041 / –0.061 / –0.145 / –0.026 / –0.014 / –0.018 / 0.007 / –0.086 / 0.028
(0.728) / (0.402) / (0.160) / (0.165) / (0.074) / (0.033) / (0.065) / (0.076) / (0.053) / (0.021)
Nurse C / –0.015 / 0.163 / –0.001 / –0.066 / –0.023 / –0.001 / –0.034 / –0.004 / –0.022 / 0.016
(0.349) / (0.222) / (0.080) / (0.083) / (0.045) / (0.028) / (0.034) / (0.038) / (0.037) / (0.016)
Nurse D / 0.388 / –0.125 / –0.025 / 0.024 / 0.002 / –0.029* / 0.007 / 0.028 / 0.045 / –0.012
(1.108) / (0.326) / (0.154) / (0.138) / (0.053) / (0.014) / (0.050) / (0.050) / (0.040) / (0.012)
Nurse E / 0.564 / –0.028 / 0.001 / 0.070 / 0.040 / –0.035* / 0.035 / 0.062 / 0.039 / –0.011
(1.126) / (0.341) / (0.158) / (0.144) / (0.057) / (0.016) / (0.052) / (0.054) / (0.041) / (0.013)
Nurse F / –1.059 / –0.245 / –0.233 / –0.048 / 0.004 / –0.050 / –0.016 / 0.141* / –0.054 / –0.002
(0.719) / (0.380) / (0.156) / (0.156) / (0.079) / (0.028) / (0.066) / (0.070) / (0.056) / (0.023)
Notes:Values are coefficients of linear regressions on nurse dummy variables and a constant. Nurse code G is the omitted category. One observation is used per nurse per PSU (N=292). Standard errors, shown in parentheses, are corrected for clustering at PSU level.
*p.05; **p.01; ***p.001
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Section 2: Nurse Heterogeneity
For the instruments to be valid,they should not be directly correlated with HIV status other than through the participation effect. A more subtle implication is that there should be no unobservedvariables that are both correlated with the ability of the nurses to motivate individuals to participate in the test, and that are also correlated with the probability of being HIV-positive. If nurse effects are heterogeneous by HIV status—that is, if Zi is correlated with unobserved variables in εi1—the exogeneity requirement will be violated (Manski 1989).
A potential unobserved factor that may induce differences in the capacity of nurses to convince HIV-positive versus HIV-negative individuals to participate is related to respondents’ fear of exposure. This may be attenuated for example by the (unobserved) HIV status of the interviewer herself or her attitudes toward HIV-infected individuals. By definition, unobserved factors cannot be examined. However, the interaction effects of nurse performance with the biomedical variables in the dataset may yield some insights into the presence of heterogeneity. The six biomarkers, which are highly correlated with HIV infection, can serve as a proxy for fear of exposure. We again estimate test participation and include the interaction terms of nurse codes with the biomarker factor score (column 1 in Table S2A). Heterogeneous nurse effects by HIV status on participation appear limited to two nurses, one of whom seems to be more skilled than others in convincing individuals with a high biomarker score to participate, while the other nurse seems less skilled. The interaction terms of heterogeneous nurse effects are jointly significant.
Alternative interactions of nurse IDcodes with age or consumption (in Table S2A, columns 2 and 3, respectively) are also suggestive of heterogeneity. However, the pattern of significant interaction effects is not consistent over nurses but differs by characteristic.
Table S2AInteraction effects of nurse IDcodes with biomarkers/age/consumption on test participation
Dependent Variable: HIV Test ParticipationInteraction Variable / Biomarker Score / Age / Log p.c. Consumption
(1) / (2) / (3)
Interaction Variable Nurse A / –0.189 / 0.033* / 0.508**
(0.129) / (0.014) / (0.165)
Interaction Variable Nurse B / –0.231* / 0.005 / 0.524**
(0.104) / (0.011) / (0.170)
Interaction Variable Nurse C / –0.118 / 0.013 / 0.237
(0.118) / (0.012) / (0.165)
Interaction Variable Nurse D / 0.684* / 0.043*** / 0.194
(0.279) / (0.011) / (0.188)
Interaction Variable Nurse E / –0.245 / 0.027 / 0.067
(0.155) / (0.014) / (0.171)
Interaction Variable Nurse F / –0.032 / –0.007 / 0.417
(0.244) / (0.018) / (0.261)
Joint Significance of the Interaction Terms (pvalue) / .011* / .000*** / .001**
Notes: The coefficients are from probit estimations of the likelihood of testparticipation, with standard errors corrected for clustering at the PSUlevel in parentheses. Other explanatory variables included in the estimation are the socioeconomic characteristics, neighborhood dummy variables, nurse codes, AIDS knowledge score, biomarker dummy variables, and stigma dummy variables. Nurse code G is the omitted category.
*p .05;**p.01;***p.001
To test the sensitivity of the main findings in Table 4 to potential heterogeneity across nurses, all results were also estimated including the interaction terms of biomarkers, age, and consumption with nurse ID codes in the first stage (TEST equation). This does not qualitatively affect the findings as is clear from a comparison of Table S2B with the main results in Table 4.
Table S2B Overview of the results (Heckman predictions include interaction terms of biomarkers, age and consumption with nurse ID in TEST equation)%HIV / 95% Confidence Interval (delta method)
(1) / (2) / (3)
A. Prevalence and Predictions for Full Sample
All individuals
HIV prevalence / 10.53
Probit prediction / 10.54 / 8.64 / 12.44
Heckman prediction / 11.86 / 8.43 / 14.61
Males
HIV prevalence / 8.95
Probit prediction / 9.01 / 6.77 / 11.25
Heckman prediction / 10.35 / 6.28 / 14.43
Females
HIV prevalence / 11.76
Probit prediction / 11.78 / 9.23 / 14.34
Heckman prediction / 13.87 / 9.52 / 18.23
Predicted
%HIV / 95% Confidence Interval (delta method)
B. Heckman Predictions for Participants/Nonparticipants
All individuals
Participants (85.8%a) / 10.56 / 8.65 / 12.48
Nonparticipants (14.2%) / 17.37 / 0.27 / 34.47
Males
Participants (83.7%) / 8.95 / 6.67 / 11.23
Nonparticipants (16.3%) / 17.67 / –5.01 / 40.35
Females
Participants (87.6%) / 11.8 / 9.25 / 14.36
Nonparticipants (12.4%) / 28.2 / 3.19 / 53.22
Notes: The predicted means are calculated by averaging the predictions for all persons with complete information on all relevant socioeconomic and biomedical variables (excluding HIV). The predicted means for participants and nonparticipants are calculated by averaging the Heckman predictions of the conditional probabilities: that is,P(HIV=1|TEST=1) and P(HIV=1|TEST=0), respectively, for participants and nonparticipants. We obtain the predictions for males and females by running separate regressions by gender.
aProportion of participants to nonparticipants in HIV test.
Next, we investigated the robustness of the results by excluding respectively nurses A and B, A and D, and B and D. The results do not change when heterogeneity is concentrated on nurse A or D. However, they are sensitive to the exclusion of nurse B from the estimation. This may be related to the fact that nurse B is one of the lowest performing nurses. Excluding her from the estimation weakens the instruments (the likelihood ratio drops from 297.89 to 209.38).
In theory, any heterogeneity will introduce a bias, but in practice, the bias may be very small in size and significance.Bias would be upward for a skilled nurse who is relatively more likely to convince HIV-positive versus HIV-negative respondents, and downward for a low-performing nurse who is worse in eliciting participation from HIV-positive compared withHIV-negative individuals. Without additional information on the unobserved factors, we cannot draw firm conclusions with respect to the consequences of potential heterogeneity for the results. This limitation affectsany analysesusing a Heckman selection model, although it is rarely recognized in applied work.
Section 3
Table S3Probit regressions of the likelihood of HIV infection
HIVInfectionAll / Males / Females
Female / –1.620
(1.482)
Age / 0.199*** / 0.210*** / 0.340***
(0.051) / (0.055) / (0.064)
Age,Squared / –0.002*** / –0.002*** / –0.005***
(0.001) / (0.001) / (0.001)
AgeFemale / 0.145
(0.077)
Age,Squared Female / –0.003**
(0.001)
Has Partner / 0.059 / 0.067 / –0.269
(0.206) / (0.211) / (0.153)
Has Partner Female / –0.290
(0.231)
Highest Grade Completed / 0.064 / –0.003 / 0.135*
(0.035) / (0.042) / (0.065)
Highest Grade Completed,Squared / –0.006* / –0.000 / –0.012**
(0.003) / (0.003) / (0.004)
Household Size / –0.008 / 0.011 / –0.022
(0.015) / (0.021) / (0.020)
Number ofChildren / –0.424*** / –0.422*** / –0.212**
(0.081) / (0.082) / (0.065)
Number ofChildren Female / 0.208*
(0.097)
Wealth and Income
Log of (pc consumption / 1,000) / –0.214* / –0.140 / 0.006
(0.096) / (0.098) / (0.089)
Log of (pc consumption / 1,000) female / 0.256*
(0.115)
Wealth index / –0.111 / –0.257 / –0.177
(0.131) / (0.153) / (0.132)
Wealth index female / –0.150
(0.140)
Unemployed / –0.242 / –0.234 / 0.209
(0.225) / (0.219) / (0.158)
Unemployed female / 0.439
(0.290)
Neighborhood (South-Central omitted)
Tobias Hainyeko / 0.181 / 0.252 / 0.173
(0.297) / (0.385) / (0.376)
Moses Garoëb / –0.033 / 0.007 / –0.025
(0.238) / (0.373) / (0.283)
Samora Machel / 0.344 / 0.646* / 0.176
(0.209) / (0.303) / (0.270)
Katutura / 0.070 / 0.573* / –0.241
(0.184) / (0.290) / (0.230)
AIDS Knowledge
AIDS knowledge score / 0.127 / 0.157 / –0.119
(0.088) / (0.088) / (0.084)
AIDS knowledge score x female / –0.238
(0.128)
Stigma
HIV+ food / 0.313 / 0.328 / –0.390*
(0.187) / (0.195) / (0.179)
HIV+ kiss / –0.132 / –0.127 / 0.172
(0.163) / (0.166) / (0.163)
HIV+ care / –0.130 / –0.253 / 0.086
(0.297) / (0.314) / (0.306)
HIV+ food female / –0.700**
(0.220)
HIV+ kiss female / 0.330
(0.252)
HIV+ care female / 0.167
(0.417)
Biomarkers
Cough / 0.305 / 0.271 / –0.154
(0.260) / (0.260) / (0.282)
X-ray / 0.248 / 0.226 / 0.185
(0.221) / (0.223) / (0.172)
AFBtest / 0.329 / 0.365 / 0.767**
(0.325) / (0.319) / (0.264)
Tuberculosis / –0.212 / –0.215 / 0.709*
(0.280) / (0.277) / (0.278)
Weight loss / 0.125 / 0.100 / 0.227
(0.198) / (0.196) / (0.156)
Diarrhea / –0.304 / –0.317 / 0.610*
(0.337) / (0.315) / (0.241)
Cough female / –0.450
(0.402)
X-ray female / –0.083
(0.263)
AFBtest female / 0.350
(0.414)
Tuberculosis female / 1.004*
(0.411)
Weight loss female / 0.130
(0.241)
Diarrhea female / 0.844
(0.457)
Constant / –5.251*** / –5.981*** / –6.613***
(1.119) / (1.282) / (1.277)
N / 1,710 / 749 / 961
Pseudo-R2 / .230 / .232 / .241
Note:Standard errors, shown in parentheses, are corrected for clustering at the PSU level.
*p.05; **p.01; ***p.001
Section 4
Table S4Heckman selection model
HIV-Infection Equation / Test-Participation EquationAll / Males / Females / All / Males / Females
Female / –1.463 / –0.453
(1.463) / (1.038)
Age / 0.202*** / 0.214*** / 0.315*** / –0.057 / –0.056 / 0.000
(0.050) / (0.054) / (0.067) / (0.031) / (0.033) / (0.027)
Age, Squared / –0.002*** / –0.002*** / –0.004*** / 0.001 / 0.001 / –0.000
(0.001) / (0.001) / (0.001) / (0.000) / (0.000) / (0.000)
Age Female / 0.134 / 0.057
(0.076) / (0.039)
Age, Squared Female / –0.002* / –0.001
(0.001) / (0.000)
Has Partner / 0.073 / 0.073 / –0.248 / –0.035 / –0.011 / 0.124
(0.198) / (0.206) / (0.152) / (0.176) / (0.184) / (0.122)
Has Partner Female / –0.301 / 0.158
(0.225) / (0.200)
Highest Grade Completed / 0.060 / –0.007 / 0.131* / 0.026 / 0.072 / –0.033
(0.034) / (0.042) / (0.062) / (0.035) / (0.039) / (0.060)
Highest Grade Completed, Squared / –0.006* / –0.000 / –0.011** / 0.000 / –0.002 / 0.004
(0.003) / (0.003) / (0.004) / (0.003) / (0.003) / (0.004)
Household Size / –0.009 / 0.011 / –0.021 / 0.006 / 0.013 / –0.001
(0.015) / (0.022) / (0.018) / (0.020) / (0.025) / (0.023)
Number of Children / –0.435*** / –0.429*** / –0.214*** / 0.114 / 0.115 / –0.012
(0.081) / (0.082) / (0.063) / (0.072) / (0.077) / (0.049)
Number of Children Female / 0.217* / –0.117
(0.095) / (0.078)
Wealth and Income
Log of (pc consumption / 1,000) / –0.198* / –0.135 / 0.037 / –0.074 / –0.022 / –0.217*
(0.096) / (0.100) / (0.086) / (0.091) / (0.096) / (0.096)
Log of (pc consumption / 1,000) female / 0.255* / –0.098
(0.114) / (0.109)
Wealth index / –0.100 / –0.249 / –0.146 / –0.123 / –0.187 / –0.141
(0.128) / (0.160) / (0.122) / (0.113) / (0.119) / (0.098)
Wealth index female / –0.143 / –0.066
(0.135) / (0.110)
Unemployed / –0.249 / –0.239 / 0.159 / 0.107 / 0.096 / 0.092
(0.224) / (0.216) / (0.156) / (0.133) / (0.134) / (0.131)
Unemployed female / 0.427 / –0.009
(0.290) / (0.185)
Neighborhood (South-Central Windhoek omitted)
Tobias Hainyeko / 0.140 / 0.229 / 0.063 / 0.332 / 0.331 / 0.443
(0.299) / (0.384) / (0.351) / (0.224) / (0.284) / (0.275)
Moses Garoëb / –0.062 / –0.006 / –0.120 / 0.261 / 0.105 / 0.490*
(0.245) / (0.372) / (0.274) / (0.238) / (0.308) / (0.249)
Samora Machel / 0.284 / 0.608 / 0.041 / 0.678** / 0.717* / 0.683*
(0.223) / (0.313) / (0.271) / (0.252) / (0.287) / (0.303)
Katutura / 0.047 / 0.547 / –0.257 / 0.106 / 0.278 / –0.009
(0.186) / (0.309) / (0.210) / (0.193) / (0.218) / (0.222)
AIDS Knowledge
AIDS knowledge score / 0.113 / 0.150 / –0.143 / –0.052 / 0.009 / –0.101
(0.088) / (0.090) / (0.079) / (0.079) / (0.090) / (0.101)
AIDS knowledge score female / –0.239 / 0.010
(0.127) / (0.115)
Stigma
HIV+ food / 0.340 / 0.346 / –0.338* / –0.115 / –0.115 / 0.036
(0.179) / (0.189) / (0.170) / (0.145) / (0.144) / (0.155)
HIV+ kiss / –0.096 / –0.108 / 0.239 / –0.214 / –0.217 / –0.364**
(0.168) / (0.178) / (0.155) / (0.126) / (0.143) / (0.130)
HIV+ care / –0.049 / –0.214 / 0.144 / –0.859** / –0.969*** / –0.227
(0.324) / (0.328) / (0.299) / (0.276) / (0.278) / (0.301)
HIV+ food female / –0.714*** / 0.158
(0.217) / (0.184)
HIV+ kiss female / 0.328 / –0.099
(0.252) / (0.158)
HIV+ care female / 0.109 / 0.589*
(0.436) / (0.282)
Biomarkers
Cough / 0.277 / 0.258 / –0.112 / 0.269 / 0.243 / –0.153
(0.254) / (0.255) / (0.263) / (0.226) / (0.236) / (0.161)
X-ray / 0.228 / 0.217 / 0.178 / 0.222 / 0.207 / –0.096
(0.216) / (0.223) / (0.165) / (0.150) / (0.149) / (0.154)
AFBtest / 0.340 / 0.375 / 0.759** / –0.043 / –0.136 / 0.229
(0.324) / (0.317) / (0.247) / (0.284) / (0.294) / (0.205)
Tuberculosis / –0.202 / –0.214 / 0.660* / –0.203 / –0.141 / –0.094
(0.284) / (0.279) / (0.286) / (0.294) / (0.291) / (0.321)
Weight loss / 0.113 / 0.093 / 0.314* / 0.053 / 0.085 / –0.452***
(0.196) / (0.195) / (0.155) / (0.181) / (0.181) / (0.128)
Diarrhea / –0.221 / –0.266 / 0.586* / –0.222 / –0.311 / 0.262
(0.328) / (0.300) / (0.230) / (0.251) / (0.273) / (0.189)
Cough female / –0.402 / –0.427
(0.394) / (0.273)
X-ray female / –0.060 / –0.340
(0.262) / (0.183)
AFBtest female / 0.352 / 0.195
(0.411) / (0.333)
Tuberculosis female / 0.980* / 0.171
(0.415) / (0.402)
Weight loss female / 0.181 / –0.478*
(0.239) / (0.217)
Diarrhea female / 0.767 / 0.392
(0.437) / (0.292)
Nurses (nurse G omitted)
Nurse A / –0.013 / 0.405 / –0.366
(0.202) / (0.284) / (0.233)
Nurse B / –1.183*** / –0.931*** / –1.474***
(0.193) / (0.249) / (0.227)
Nurse C / –0.852*** / –0.754*** / –0.978***
(0.190) / (0.228) / (0.238)
Nurse D / –0.162 / –0.130 / –0.217
(0.179) / (0.208) / (0.217)
Nurse E / 0.078 / 0.200 / –0.091
(0.206) / (0.262) / (0.232)
Nurse F / 0.005 / 0.439 / –0.389
(0.290) / (0.426) / (0.289)
Constant / –5.176*** / –5.943*** / –6.012*** / 2.342** / 1.718 / 2.572**
(1.118) / (1.321) / (1.315) / (0.858) / (0.896) / (0.890)
ρ (rho)a / –0.329 / –0.198 / –0.622
(pvalue corresponding to H0: ρ=0) / (.191) / (.526) / (.061)
N / 1,992 / 895 / 1,097
Note:Standard errors, shown in parentheses, are corrected for clustering at the PSU level.
*p.05; **p.01; ***p.001
Section 5
Table S5 HIV prevalence prediction by stratum (percentages)Stratum of ProbitPredicted HIV Probability*
(1) / Refusals
(2) / Min. Probit Estimate
(stratum)
(3) / Max. Probit Estimate
(stratum)
(4) / Mean Probit Estimate
(stratum)
(5) / Observed HIV Prevalence
(6) / Mean
Heckman Estimate
(7)
1 / 20.55 / 0.00 / 1.15 / 0.45 / 0.95 / 0.64
2 / 12.06 / 1.15 / 3.54 / 2.18 / 2.29 / 2.79
3 / 11.53 / 3.54 / 7.81 / 5.48 / 5.38 / 6.68
4 / 12.06 / 7.86 / 17.86 / 12.57 / 11.14 / 14.56
5 / 14.57 / 17.89 / 93.87 / 32.06 / 32.65 / 34.69
aEach stratum represents one-fifth of the sample ranked from lowest probit predicted probability of HIV infection (stratum 1) to highest predicted probability (stratum 5).
Section6: Effects of Sample Size on Validation of the Model
To gain more insight into the effect of sample size on the width of the confidence intervals (CIs), we have simulated the validation estimations increasing our sample twofold, fivefold, and tenfold by appending the dataset to itself. Note that when multiplying the dataset, it is imperative to assume that new PSUs are added every time that the dataset is multiplied. Otherwise, multiplying the dataset has no effect on the standard errors: the reduction in the standard errors is exactly countered by the increase in within-cluster correlations.
As expected, larger sample sizes increase precision but do not affect point estimates. When sample size is increased tenfold, the probit and the conditional Heckman CI no longer encompass the estimated prevalence rate of 14.8%, but the unconditional Heckman CI still does.
Table S6. Simulated sample size increases
HIV (%) / Sample Increase / 95% CI(delta method)
Probit Estimate / 13.55 / Sample 1 / 10.60 / 16.50
Sample 2 / 11.47 / 15.63
Sample 5 / 12.23 / 14.86
Sample 10 / 12.62 / 14.48
Unconditional Heckman Estimate / 15.65 / Sample 1 / 11.36 / 19.94
Sample 2 / 12.62 / 18.67
Sample 5 / 13.74 / 17.56
Sample 10 / 14.30 / 17.00
Heckman Estimate, Conditional on Test Refusal / 23.68 / Sample 1 / 7.15 / 40.21
Sample 2 / 12.02 / 35.34
Sample 5 / 16.32 / 31.04
Sample 10 / 18.48 / 28.89
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Section 7
Fig. S7Kernel density of the subsamples, probit vs. Heckman
1