Generalized Linear Mixed Models to analyse data from R espondent- D riven S ampling

1. Generalized Linear Mixed Modelsto 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

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 Design

24. Possible correlation structure for a RDS Design

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:

29. Some results II IDU Study:

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 0,313 76,0% 0,046 Heckatorn 0,141 96,5% 0,003 GLMM 0,143 96,0% -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

43. HIV risk factors - IDU

44. IDU – final model for HIV status log(OR) OR 95%-CI p-value Network size (Ref: 100+) less than 100-0.75 0.47 0.30...0.75 0.002 Duration of injection career (Ref: 0-2 years) 3-5 years0.94 2.55 1.16...5.62 0.020 6-9 years 1.61 5.00 2.19..11.38 0.001 10 years or more 1.31 3.69 1.42...9.61 0.008 Place of residence (Ref: Tallinn) Kohtla-Järve 1.79 5.99 2.53..14.19 0.005 Number of sexual partners during last year (Ref: 0) one-1.37 0.25 0.09...0.69 0.007 more than one-1.07 0.34 0.13...0.89 0.028 Age group (Ref: <20) 20-24-0.43 0.65 0.33...1.29 0.218 25-29-0.90 0.41 0.18...0.89 0.026 30 or more-1.26 0.28 0.11...0.75 0.012

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