1. Term 4, 2006BIO656--Multilevel Models 1 PART 4 Non-linear models Logistic regression Other non-linear models Generalized Estimating Equations (GEE) Examples –Crossover study –British Social Attitudes Survey

2. Term 4, 2006BIO656--Multilevel Models 2 Models for Clustered Data Inferential goals Marginal mean/Population Averaged –Average response across “the population” Mean, conditional on –Other responses in the cluster –Unobserved random effects

3. Term 4, 2006BIO656--Multilevel Models 3 Interpreting Linear Model Coefficients Same interpretation for conditional (cluster-specific) and population-averaged inferences Unit change in dependent variable for a unit change in regressor Multi-level models specify correlations and latent effects: –The random intercept model produces an equal-correlation model (correlation –The latent intercepts can be estimated and used for prediction

4. Term 4, 2006BIO656--Multilevel Models 4 Marginal Models Inferential Target Marginal mean or population-averaged response for different values of predictor variables Examples Difference in mean alcohol consumption for two age groups Rate of alcohol abuse for states with addiction treatment programs compared to those without Public health assessments

5. Term 4, 2006BIO656--Multilevel Models 5 Conditional Models Conditional on other observations in cluster Probability that a person abuses alcohol given family membership or given the number of family members that do Probability that a person will abuse next year, if abuses this year A person’s average alcohol consumption given the average in the neighborhood

6. Term 4, 2006BIO656--Multilevel Models 6 Conditional Models Conditional on random effects Average consumption, conditional on a latent tendency Probability that a person abuses alcohol, conditional on a latent tendency Can be thought of as conditional on unmeasured covariates

7. Term 4, 2006BIO656--Multilevel Models 7 The basic, conditional logistic model Conditional on a random effect, you have the logistic regression: logit(P) = log{P/(1-P)} = u +  +  X u ~ (0,  2 ) Implications Generally, the population averaged (marginal) model will not have the logistic shape In any case, the slope on a covariate will have a different impact in the conditional and marginal models

8. Term 4, 2006BIO656--Multilevel Models 8 Condition on u

9. Term 4, 2006BIO656--Multilevel Models 9 Conditional Logistic and Marginal Shapes Conditional Logistic and Marginal Shapes U  N(0, 4) b =  u=0

10. Term 4, 2006BIO656--Multilevel Models 10 Conditional Logistic and Marginal Shapes U is a two-point mixture at  2 b =  u=0

11. Term 4, 2006BIO656--Multilevel Models 11 Adjust the conditional slope to closely match the marginal curve Assume that there is a population relation that is logistic with term  X How far off is the marginal curve produced from the conditional logistic curve with term  X? Let  * be the slope needed in the conditional logistic so that the marginal curve produced from it comes close to the population relation –“Comes close” means to track the middle part of the population curve

12. Term 4, 2006BIO656--Multilevel Models 12 Non-linear model coefficients Usually, population-averaged (marginal) and conditional models have different shapes –Condition logistic is not population logistic –But, conditional probit is population probit In any case, population-averaged and cluster-specific coefficients have different magnitudes and interpretations because they address different questions For example, when u is a two-point, 50/50 mixture at  2,  = 4 and  * = 8. Need to consider impact on probabilities not just on odds ratios

13. Term 4, 2006BIO656--Multilevel Models 13 SHAPE & SLOPE CHANGES For linear models, regression coefficients in random effects models and marginal models are identical: average of linear model = linear model of average For non-linear models, coefficients have different meanings and values: average of non-linear model  non-linear model of average coefficient value and meaning in average model  coefficient value and meaning in conditional model

14. Term 4, 2006BIO656--Multilevel Models 14 Conditional Logistic and Marginal Shapes Log(odds | u) = u -2.0 + 0.4  X Population prevalences X = 1 X = 0 Cluster-specific probabilities

15. Term 4, 2006BIO656--Multilevel Models 15 Logistic Regression Example Logistic Regression Example Cross-over trial 2 observations per person (before/after) Response 1= not alcohol dependent; 0 = AlcDep (so a high probability is good!) Predictors period (Pd = 0 or 1) treatment group (Trt = 0 or 1) Parameter of interest Treatment vs placebo after/before log(OddsRatio) A positive slope favors the treatment

16. Term 4, 2006BIO656--Multilevel Models 16 Baseline/Follow-up Model i = period, j = person; logit(P) = log(P/[1-P]) Population level (no individual effects) logit(P ij ) =  +  1 PD ij +  2 TR ij +  3 PD ij TR 2ij =  +  1 PD i +  2 TR j +  3 PD i TR j logit(P 2j ) - logit(P 1j ) =  1 +  3 TR j (  3 is the treatment effect) Person-level (individual intercept) logit(P ij ) = u j +  * +  * 1 PD i +  * 2 TR j +  * 3 PD i TR j u j ~ (0,  2 )

17. Term 4, 2006BIO656--Multilevel Models 17 Results for population-level regressions (logistic without multi-level component) Results for population-level regressions (logistic without multi-level component) Marginal Models log(OR) (se) Regressor Standard Logistic (Accounting for correlation) Intercept 0.66 (0.32) 0.67 (0.29) Period -0.27 (0.38) -0.30 (0.23) Treatment  3 0.56 (0.38) 0.57 (0.23) Similar estimates; wrong standard error for Std. Logistic

18. Term 4, 2006BIO656--Multilevel Models 18 The effect of accounting for correlation Treatment effect estimates are the same for marginal logistic and correlation accounted logistic –But, SEs are 0.38 and 0.23 respectively Why is the second smaller than the first? Answer The treatment effect is estimated by contrasting (differencing) period 2 and period 1 The positive, within-person correlation produces a smaller variance of this difference than does assuming independence

19. Term 4, 2006BIO656--Multilevel Models 19 Population-level vs Random Intercept logistic regressions Population-level vs Random Intercept logistic regressions log(OR) (se) MarginalConditional RegressorOrdinary Logistic Regression Logistic (Account for correlation) RE Conditional Logistic Reg. Intercept 0.66 (0.32) 0.67 (0.29) 2.2 (1.00) Period -0.27 (0.38) -0.30 (0.23) -1.0 (0.84) Treatment 0.57 (0.38) 0.57 (0.23) 1.80 (0.93)  = sd(u) 0.03.56 (0.81) 5.00 (2.30)

20. Term 4, 2006BIO656--Multilevel Models 20 Marginal Logistic versus Random Intercept Logistic Unconditional Logistic (Population-level inference): The population AlcnonDep (after/before), treatment/placebo prevalence odds ratio is exp(0.57) = 1.77 Conditional, RE Logistic (Individual-level inference): An individual’s AlcnonDep (after/before), treatment/placebo prevalence odds ratio is exp(1.80) = 6.05 Ratio: (Conditional)/(Marginal) 6.05/1.77 = 3.42 (= e 1.23 ; 1.23 = 1.80-0.57) Different questions; different (but compatible) answers

21. Term 4, 2006BIO656--Multilevel Models 21 Consequence of Conditional/Marginal Slope Differences A population-level analysis that does not build on a multi-level model (that does not include the random effect) can understate the individual- level (cluster level) risk or benefit –Understate environmental risk –Understate benefits of lowering blood pressure –.........

22. Term 4, 2006BIO656--Multilevel Models 22 u = - log(3)u = log(3)Marginal (Population) X = 0P = 0.25P = 0.750.50 X = 1P = 0.50P = 0.900.70 OR(X=1 vs 0)3.00 2.33 (=7/3) logit(pr(Y = 1 | X, u) = u + log(3)X u =  log(3) with probability 1/2 Relation between marginal and conditional ORs 3.00 = (.5/.5)  (.25/.75) = (.9/.1)  (.75/.25)

23. Term 4, 2006BIO656--Multilevel Models 23 u as a missing covariate Without knowing u, a marginal logistic regression predicts 0.50 and 0.70 for X=0 and X=1 respectively –The log(OR) slope on X is 0.847 = log(2.333) If we know u, a logistic regression with it as a covariate (conditional on it) predicts as in the table –The log(OR) slope on X is 1.099 = log(3.00)

24. Term 4, 2006BIO656--Multilevel Models 24 Conditional Logistic and Marginal Shapes Log(odds | u) = u + X X u > 0 u < 0

25. Term 4, 2006BIO656--Multilevel Models 25 Y 2 = 0Y 2 = 1Y 1 marginal Y 1 = 05/163/160.50 Y 1 = 13/165/160.50 Y 2 marginal0.50 1.00 (Y 1, Y 2 ) are in the same cluster The RE model produces the following 2  2 table for X = 0 The RE induces association 5/16 = [(3/4)(3/4) + (1/4)(1/4)]  2 pr(Y 2 =1 | Y 1 = 0) = 3/8 = 3/(3+5) pr(Y 2 =1 | Y 1 = 1) = 5/8 = 5/(3+5)

26. Term 4, 2006BIO656--Multilevel Models 26 Y 2 = 0Y 2 = 1Y 1 marginal Y 1 = 013/10017/1000.30 Y 1 = 117/10053/1000.70 Y 2 marginal0.300.701.00 (Y 1, Y 2 ) are in the same cluster The RE model produces the following 2  2 table for X = 1 The RE induces association 13/100 = [(1/2)(1/2) + (1/10)(1/10)]  2 pr(Y 2 =1 | Y 1 = 0) = 17/30 = 17/(17+13) pr(Y 2 =1 | Y 1 = 1) = 53/70 = 53/(17+53)

27. Term 4, 2006BIO656--Multilevel Models 27 Updating the distribution of u For X = 1 (you can try it for X = 0) pr(u = +log(3) | Y = 0) = pr(u = +log(3), Y = 0)/pr(Y = 0) = (1/2)(1/10)  (3/10) = 1/6 < 0.5 pr(u = +log(3) | Y = 1) = pr(u = +log(3), Y = 1)/pr(Y = 1) = (1/2)(9/10)  (7/10) = 9/14 > 0.5 pr(u = +log(3)) = (1/6)(3/10) + (9/14)(7/10) = 0.5 Can use these to get [Y 2 | Y 1 ]

28. Term 4, 2006BIO656--Multilevel Models 28 Marginal Multi-level, non-linear Models GEE: Marginal mean as a function of covariates Working independence or other working model Followed by Robust SE –“Cluster(id) in Stata –“Robust” Option in SAS Proc Mixed or GenMod –No “robustness” in BUGS Conditional mean, as a function of marginal mean and cluster-specific random effects –Heagerty (1999, Biometrics) –Heagerty and Zeger (2000, Statistical Science)

29. Term 4, 2006BIO656--Multilevel Models 29 Generalized Linear Models (GLMs) Generalized Linear Models (GLMs) g(mean) =  0 +  1 X 1 +... +  p X p (always a marginal model) ModelResponse Y  = E(Y) g(  ) = Distribution Coefficient Interpretation (per unit change in X) LinearContinuous/ Bell-shaped  Gaussian near-Gaussian Change in E(Y) LogisticBinary Log(  /(1-  )) = Logit(  ) Bernoulli Binomial Change in log odds Log- linear Counts Concentration Time to event log(  ) Poisson Log-normal Weibull Change in Log rate

30. Term 4, 2006BIO656--Multilevel Models 30 Baseline/Follow-up Model i = period, j = person; logit(P) = log(P/[1-P]) Population level (no individual effects) logit(P ij ) =  +  1 PD ij +  2 TR ij +  3 PD ij TR 2ij =  +  1 PD i +  2 TR j +  3 PD i TR j logit(P 2j ) - logit(P 1j ) =  1 +  3 TR j (  3 is the treatment effect) Person-level (individual intercept) logit(P ij ) = u j +  * +  * 1 PD i +  * 2 TR j +  * 3 PD i TR j u j ~ (0,  2 )

31. Term 4, 2006BIO656--Multilevel Models 31 Marginal Generalized Linear Models via Generalized Estimating Equations (GEE) Ordinary GLM (linear, logistic, Poisson,..) –Population-average parameters –Logit: O ij = logit(p ij ) =  0 +  1 X ij Then, model association among observations i and i’ in cluster j: corr(log(O ij / O i’j )) = function(G) Solve generalized estimating equation (GEE) –Diggle, Heagerty, Liang and Zeger, 2002) –Gives highly efficient and valid inferences on population-average parameters

32. Term 4, 2006BIO656--Multilevel Models 32 Marginal Models for the Cross-Over Study Marginal Models for the Cross-Over Study log(OR) Estimation method has an effect

33. Term 4, 2006BIO656--Multilevel Models 33 Conditional (RE) Models for the Cross-Over Study Conditional (RE) Models for the Cross-Over Study log(OR)

34. Term 4, 2006BIO656--Multilevel Models 34 Accounting for Clustering via Sample Reuse Standard GEE: “Robust” option in SAS Jackknife Compute  hat Delete a person (in general, a “unit”) Compute  -i i = 1,..., n Compute  i * = n  hat - (n-1)  -i Compute the sampe (co)variance of the  i * Bootstrap Put each person’s data on a token Sample “n” tokens with replacement and compute estimates from the sample Do this “N boot ” times and compute sample (co)variance of the estimates Can get more sophisticated CIs, via BC a

35. Term 4, 2006BIO656--Multilevel Models 35 Estimate Data “Black Box” Procedure FRAMEWORK FOR SAMPLE REUSE

36. Term 4, 2006BIO656--Multilevel Models 36 British Social Attitudes Survey: Conditional and Marginal MLMs S British Social Attitudes Survey: Conditional and Marginal MLMs Note: Subscript order reversed from our usual Response Y ijk = 1 if favor abortion; 0 if not –district i = 1,…264 –person j = 1,…,1056 –year k = 1, 2, 3, 4 Levels 1.Time within person 2.Persons within districts 3.Districts

37. Term 4, 2006BIO656--Multilevel Models 37 Covariates at the three levels Level 1: time Indicators of time Level 2: person Class: upper working; lower working Gender Religion: protestant, catholic, other Level 3: district Percentage protestant (derived)

38. Term 4, 2006BIO656--Multilevel Models 38 Scientific Questions Conditional Model How does a woman’s religion associate with her probability of favoring abortion? How does the predominant religion in a district associate with a woman’s probability of favoring abortion? Marginal Model How does the rate of favoring abortion differ between Protestants and, otherwise similar, Catholics? How does the rate of favoring abortion differ between districts that are predominantly Protestant versus Catholic?

39. Term 4, 2006BIO656--Multilevel Models 39 Schematic of Marginal Random-effects Model

40. Term 4, 2006BIO656--Multilevel Models 40 Conditional Multi-level Model Person and district random effects Modeling the Population Expectation We build a “regression model” for  2

41. Term 4, 2006BIO656--Multilevel Models 41 Conditional Multi-level Model Results All of this is a “regression model” for  2

42. Term 4, 2006BIO656--Multilevel Models 42 Conditional model results How does a woman’s religion associate with her probability of favoring abortion? How does the predominant religion in a district associate with a woman’s probability of favoring abortion?

43. Term 4, 2006BIO656--Multilevel Models 43 Marginal Multi-level Model Person and district random effects If the conditional is logistic, can the marginal be logistic? We simultaneously model the underlying random effects structure, but we are still fitting the marginal model

44. Term 4, 2006BIO656--Multilevel Models 44 Marginal Multi-level Model Results All of this is a “regression model” for  2

45. Term 4, 2006BIO656--Multilevel Models 45 Marginal model results How does the rate of favoring abortion differ between protestants and otherwise similar catholics? How does the predominant religion in a district influence the probability of favoring abortion?

46. Term 4, 2006BIO656--Multilevel Models 46 Refresher: Forests & Trees Multi-Level Models: Explanatory variables from multiple levels –Family –Neighborhood –State Interactions Must take account of correlation among responses from same clusters: Marginal: GEE, MMM Conditional: RE, GLMM

47. Term 4, 2006BIO656--Multilevel Models 47 Key Points “Multi-level” Models: Have covariates from many levels and their interactions Acknowledge correlation among observations from within a level (cluster) Conditional and Marginal Multi-level models have different targets; ask different questions When population-averaged parameters are the focus, use –GEE –Marginal Multi-level Models (Heagerty and Zeger, 2000)

48. Term 4, 2006BIO656--Multilevel Models 48 Key Points (continued) When cluster-specific parameters are the focus, use random effects models that condition on unobserved latent variables that are assumed to be the source of correlation Warning: Model Carefully. Cluster-specific targets often involve extrapolations where there are no actual data for support –e.g. % protestant in neighborhood given a random neighborhood effect

49. Term 4, 2006BIO656--Multilevel Models 49 Recap Population-averaged parameters GEE Marginal multi-level models Cluster-specific parameters and latent effects Random Effects models –built up from latent effects (variance components) Possibly, overlay “Time Series” Models –to induce additional correlation Warning Inferences on latent effects can be very model-dependent

50. Term 4, 2006BIO656--Multilevel Models 50 Working Independence versus modeling correlation Working Independence versus modeling correlation Longitudinal Example Generate data in clusters (i.e., a person) 5 observations per cluster Response is a linear function of time Y it =  0 +  1 t + e it The residuals are first-order autoregressive, AR(1) e it =  e i(t-1) + u it (the u’s are independent) corr(e i(t+s), e it ) =  s Estimate the slope by OLS: assumes independent residuals Maximum likelihood: models the autocorrelation

51. Term 4, 2006BIO656--Multilevel Models 51 Comparisons Compare the following reported Var(  1 ) That reported by OLS (it’s incorrect) That reported by a robustly estimated SE for the OLS slope (It’s correct for the OLS slope) That reported by the MLE model if  It’s correct if the MLE model is correct You can use any working correlation model, but need a robust SE to get valid inferences

52. Term 4, 2006BIO656--Multilevel Models 52 Variance of OLS & MLE Estimates of b versus , the first-lag Correlation OLS reported variance True variance of OLS MLE reported variance

53. Term 4, 2006BIO656--Multilevel Models 53 Analytic Strategy Use a model that fits the observed data well –Directly model observeds or check fit by aggregating a random effects model –“Good” models (candidate models) will give similar observed-data predictions Then, “speculate” on latent effects models by finding several that fit the observed data –See if these give similar messages and produce similar individual-level predictions –Yes  a sturdy finding; No  additional info needed Note:  > 0 indicates that there is unexplained, individual-level heterogeneity

54. Term 4, 2006BIO656--Multilevel Models 54 MLMs Models are multi-level because they Include covariates from many levels (and their interactions) Structure correlation among observations within a cluster Conditional and marginal models Have different goals Ask different questions Can/should get different answers

55. Term 4, 2006BIO656--Multilevel Models 55 Benefits & Drawbacks of working non-independence Benefits Efficient estimates Valid standard errors and sampling distributions Protection from some missing data processes The MLM/RE approach allows estimating conditional-level parameters, estimating latent effects and improving estimates Drawbacks Working non-independence imposes more strict validity requirements on the fixed effects model (the Xs) Can get valid SEs via working independence with robust standard errors –At a sacrifice in efficiency

56. Term 4, 2006BIO656--Multilevel Models 56 There is no free lunch! Working independence models (coupled with robust SEs!!!) are sturdy, but inefficient Fancy models are potentially efficient, but can be fragile