1. Generalized Linear Mixed Models to analyse data from Respondent-Driven Sampling
Märt Möls, Krista Fischer,
Anneli Uusküla, Helle Kilgi
Tartu University, Estonia

2. Hard-to-reach populations
Intravenous Drug Users
Example: Estonian Intravenous Drug Users Survey (Data collection 2005)
Commercial Sex Workers
Example: Estonian Commercial Sex Workers Survey (Data collection 2006)
gay men; street youth; homeless; bayesian etc.

3. Respondent-Driven Sampling (RDS)

4. Respondent-Driven Sampling (RDS)

5. Respondent-Driven Sampling (RDS)

6. Respondent-Driven Sampling (RDS)

7. Respondent-Driven Sampling (RDS)

8. Respondent-Driven Sampling (RDS)

9. Respondent-Driven Sampling (RDS)

10. Respondent-Driven Sampling (RDS)

11. Estonian IDU Study

12. Basic description of the Studies
Intravenous Drug Users
Commercial Sex Workers
Sample size
450
Tallinn - 350; Kohtla-Järve-100
227
Number of seeds 9 43
Median network size 50 6
Number of “waves”
HIV+
62%
7,5%

13. How to analyse a RDS study?
Naive Analysis - Treat the data as from usual random sample. Convinient, often used, but... ;
Heckathorn’s method – use Markov Chains to derive asymptotically unbiased (under plausible assumptions) estimates of proportions;
Use Linear Mixed Models/Generalized Linear Mixed Models with suitable covariance structure.

14. Naive Analysis - problems
Undersampling of respondents with few friends (small network size);
Bias due to non-random selection of seeds;
Friends are “more similar” to each other than two randomly choosen persons from the target population
Association between recruiter and recruited (from IDU study):
HIV status p-value=0,0009
age p-value=0,03
type of drug used p-value=0,00002

15. Heckathorn’s method
"Respondent-Driven Sampling: A New Approach to the Study of Hidden Populations." By Douglas D. Heckathorn.  Social Problems, 1997.
"Respondent-Driven Sampling II: Deriving Valid Population Estimates from Chain-Referral Samples of Hidden Populations." By Douglas D. Heckathorn. Social Problems, 2002.
+ (asymptoticaly) unbiased results under plausible assumptions (connected population; individuals sample randomly among one’s friends; if A knows B then B knows A; + a few technical ones).
- A method to just estimate proportions?

16. Linear Mixed Models / GLMM
AIM:
To provide classical modelling possibilities to analyse samples collected based on a RDS design.

17. Remedy 1 - Undersampling
Undersampling can be corrected using weighted averages or related techniques (estimation of certain linear functions of parameters). Corrections are regularly applied if stratified random sampling has been used, for example.

18. Distribution of network sizes

19. Correction for undersampling
1. Include network size to the model of interest, eg.:
logit(P(HIV+|NS)) = c+f(NS)
2. Integrate NS out from the final result based on the estimated proportions of network sizes, eg:
P(HIV+) = ∑ P(HIV+|NS=i) P(NS=i)
3. Use delta method to calculate se for logistic regression; exact calculations for linear models.

20. There is nothing new in analysing correlated observations.
For example, one can use

21. Time Series Analysis

22. Repeated Measures / Multilevel Analysis

23. A possible correlation structure for a RDS Design1 r
r 2
r 3

24. Possible correlation structure for a RDS DesignH I 1 r
r 2
r 3 s sr
sr2

25. Some possible modeling approaches using such correlation structure:
Linear Mixed Models
Generalized Linear Mixed Models
Generalized Latent Variable Models / Generalized Structural Equation Models
....

26. Correction for seed selection bias?
May-be one can consider the seed selection bias as a random “effect” added to the seeds (seeds are correlated to each other due to the random “selection bias” effect);
This added random “effect” is propagating (in a decreasing way) along the recruitment path – so the “offsprings” of different seeds start to be less correlated after each new recruitment wave.
- Suggested approach is really computer intensive. Estimating linear/generalized linear models by using this kind of correlation structure can really be too slow.

27. Does it work? Comparison with other methods Simulations
Does the proposed model fit to the real data?

28. Some results I CSW Study: 7,5% 4,7% 0,01747 - 0,0169 4%..12% 3%..9%
Naive
Heckathorn
GLMM
HIV+ (%)
7,5%
4,7% se
0,01747 -
0,0169
95%-CI
4%..12%
3%..9%
1%..8%
Average Age
29,3 NA
29,0
0,6
1,2
28,1...30,5
26,6..31,3

29. Some results II IDU Study: 54% 47% 89% 91% 90% Average Age 24,2 NA
Naive
Heckathorn
GLMM
Tallinn, HIV+ (%)
54%
47%
Kohtla-Järve, HIV+ (%)
89%
91%
90%
Average Age
24,2 NA
23,7

30. Simulation – creating a population

31. Simulation – creating a population

32. Simulation – creating a population

33. Simulation – creating a population

34. Simulation – creating a population

35. Simulation – sampling

36. Simulation – sampling

37. Simulation – sampling

38. Simulation – sampling

39. Simulation results (True population prevalence about 0,45)
95%-CI
Method MSE coverage bias
Naive 0, ,0% ,046
Heckatorn 0, ,5% ,003
GLMM 0, ,0% ,006

40. Does the model fit? Example model (for age) from the IDU study
Correlation parameter estimates:
r = 0,084
s = 0,152
AIC = 2700 (RDS correlation structure)
AIC = 2704 (Indipendence)

41. Problems with the CSW-study

42. HIV risk factors - CSW Category of SW Brothel Street Other
Naive Analysis
OR (95% CI)
(unadjusted)
GLMM
Category of SW
Brothel
Street
Other
4,0 (1,2...15,3)
12,6 (3,0...56,8) 1
3,6 (1,2...10,4)
3,1 (0,7...14,1)
Drug use: No
Yes
8,3 (2,3...27,8)
2,3 (0,6...8,5)
Years of CSW
0,84 (0,70...0,95)
0,76 (0,67..0,87)

43. HIV risk factors - IDU Gender: male female 1 1.2 (0.7...2.1)
Naive Analysis
OR (95% CI)
(unadjusted)
GLMM
Gender: male
female 1
1.2 ( )
1.3 ( )
Years of drug use:
2 years or less
3..5 years
6..9 years
10 years or more
2.8 ( )
4.8 ( )
3.5 ( )
2.4 ( )
3.8 ( )
2.5 ( )

44. IDU – final model for HIV status
log(OR) OR %-CI p-value
Network size (Ref: 100+)
less than
Duration of injection career (Ref: 0-2 years)
3-5 years
6-9 years
10 years or more
Place of residence (Ref: Tallinn)
Kohtla-Järve
Number of sexual partners during last year (Ref: 0)
one
more than one
Age group (Ref: <20)
30 or more

45. Conclusions
To draw an RDS sample is often temptingly convienient – however, naive analysis of such samples may result in misleading inferences.
GLMM analysis of RDS samples requires assumptions that are realistic in many practical settings.
One can analyse an RDS study by using standard software (for example R)
Not all hidden populations are networked; not all questions concerning RDS have been answered