Fixed Effects Versus Random Effects Models for Multilevel and ...

Fixed Effects Versus Random Effects Models for Multilevel and ...

Fixed Versus Random Effects Models for Multilevel and Longitudinal Data Analysis Ashley H. Schempf, PhD MCH Epidemiology Training Course June 1, 2012 Outline Clustered Data Fixed Effects Models Random Effects Models GEE Models Hybrid Models Applied Examples Homework Assignment Clustered Data Involves nesting/clustering of observations or data points Multilevelclustering over space A B C 1 2 3 D 4 5 Neighborhood j 6 7 8 9 10 11 12 Individual i Panel/Longitudinalclustering over time Time t A

Individual/ Unit i 1 B C D 2 1 1 1 3 2 2 2 3 3 3 repeated measurements Unique features Correlation of data within clusters Violation of independence; as a modeling assumption errors must be independent Complexity/redundancy must be accounted for Variation at multiple levels allows richer examination and distinction of effects Neighborhood/family versus individual effects Cross-sectional (selection) versus longitudinal (causation) A lot of bias can be introduced with single-level data (omitted variables, selection) Racial disparities when contextual differences arent examined (neighborhood level data omitted) Association between dieting and weight (longitudinal data omitted) Between versus Within Cluster Effects Factors that only vary between clusters are cluster level effects Multilevel: walkability, crime level Longitudinal: race/ethnicity, sex However, any factor that varies within cluster can also vary between cluster Multilevel: income/poverty, race

Individual-level and neighborhood aggregated (e.g. % poverty, % black) Longitudinal: smoking, activity, diet At each time point but also averaged for an individual (e.g. average activity level over time) Between versus Within Cluster Effects In multilevel cases, we may care about both between and within-cluster effects Contextual effect of living in more versus less segregated neighborhoods (% Black) Individual effect of race/ethnicity In longitudinal cases, the between-cluster effects of within-cluster variables tend to represent confounded cross-sectional inference Comparing a person who smokes to one who doesnt Comparing outcomes within a person when they smoke and after they quit Handling Clustered Data Many ways of accounting for complex errors and violation of non-independence Robust SEs, random effects, GEE, survey analysis Many ways of disentangling between and within-cluster effects Fixed effects, hybrid models The correct choice lies in your purpose Applied Data Example To demonstrate these options, Ill use a dataset of birth certificate information from two counties in North Carolina Multilevel data structure: births nested within neighborhoods (Census block groups) Covariate of interest: race (Black-White) Continuous and dichotomous outcome: gestational age and preterm birth (<37 weeks) Schempf AH, Kaufman JS. Accounting for context in studies of health inequalities: a review and comparison of approaches. Ann Epidemiol. forthcoming Schempf AH, Kaufman JS, Messer LC, Mendola P. The neighborhood contribution to black-white perinatal disparities: an example from two north Carolina counties, 1999-2001. Am J Epidemiol. 2011;174(6):744-52. Fixed Effects Account for all cluster-level variation by holding cluster constant

All inference is therefore within-cluster Can be implemented either by entering dummy variables for n-1 clusters conditional approach Continuous outcome: de-meaning or subtracting the cluster means from all variables before running model Binary outcome: conditional logistic regression reg ga_clean race i.b_group i.b_group _Ib_group_1-392 (_Ib_group_1 for b_g~p==370630001011 omitted) note: _Ib_group_51 omitted because of collinearity note: _Ib_group_155 omitted because of collinearity Source | SS df MS -------------+-----------------------------Model | 5064.7562 390 12.9865543 Residual | 141130.634 31098 4.53825437 -------------+-----------------------------Total | 146195.391 31488 4.64289223 Number of obs F(390, 31098) Prob > F R-squared Adj R-squared Root MSE = = = = = = 31489 2.86 0.0000 0.0346 0.0225 2.1303 -----------------------------------------------------------------------------ga_clean | Coef. Std. Err. t P>|t|

[95% Conf. Interval] -------------+---------------------------------------------------------------race | -.4692281 .032978 -14.23 0.000 -.5338663 -.40459 _Ib_group_2 | .0241545 .3980305 0.06 0.952 -.7560012 .8043102 _Ib_group_3 | .5450826 .3702998 1.47 0.141 -.1807199 1.270885 _Ib_group_4 | -.1346887 .4903915 -0.27 0.784 -1.095876 .8264983 _Ib_group_5 | .435551 .4606856 0.95 0.344 -.4674114 1.338513 _Ib_group_6 | -.2657666 .5106346 -0.52 0.603 -1.266631 .7350977 _Ib_group_7 | .3649276 .4704894 0.78 0.438 -.5572506 1.287106 _Ib_group_8 | -.6098332 .5706181 -1.07

0.285 -1.728268 .5086011 _Ib_group_9 | .30786 .6502678 0.47 0.636 -.9666911 1.582411 _Ib_group_10 | .1086299 .5707573 0.19 0.849 -1.010077 1.227337 . Accounting for neighborhood differences (within-neighborhood inference), Black infants are delivered -.47 weeks earlier than White infants proc glm data=nc.data_final; class b_group ; model ga_clean= race b_group /clparm solution; quit; R-Square 0.034644 Coeff Var 5.476830 Root MSE 2.130318 ga_clean Mean 38.89692 Source race b_group DF 1 389 Type I SS 2585.192623 2479.563572

Mean Square 2585.192623 6.374199 F Value 569.64 1.40 Pr > F <.0001 <.0001 Source race b_group DF 1 389 Type III SS 918.775440 2479.563572 Mean Square 918.775440 6.374199 F Value 202.45 1.40 Pr > F <.0001 <.0001 Parameter Estimate Intercept race b_group b_group b_group b_group b_group 39.45939128

-0.46922813 -0.67109834 -0.64694387 -0.12601570 -0.80578708 -0.23554737 370630001011 370630001012 370630001021 370630002001 370630002002 Parameter Intercept race b_group b_group b_group b_group B B B B B B Standard Error t Value Pr > |t| 0.32492837 0.03297796 0.45201887 0.40694049 0.37957976 0.49753595 0.46848328 121.44 -14.23 -1.48 -1.59 -0.33 -1.62 -0.50

<.0001 <.0001 0.1376 0.1119 0.7399 0.1053 0.6151 95% Confidence Limits 370630001011 370630001012 370630001021 370630002001 38.82251860 -0.53386626 -1.55707353 -1.44456361 -0.87000731 -1.78097758 40.09626396 -0.40459000 0.21487684 0.15067587 0.61797592 0.16940342 Accounting for neighborhood differences (within-neighborhood inference), Black infants are delivered -.47 weeks earlier than White infants xtreg ga_clean race, i(b_group) fe Fixed-effects (within) regression Group variable: b_group Number of obs Number of groups = = 31489 390 R-sq: Obs per group: min = avg = max =

1 80.7 652 within = 0.0065 between = 0.3141 overall = 0.0177 corr(u_i, Xb) = 0.2308 F(1,31098) Prob > F = = 202.45 0.0000 -----------------------------------------------------------------------------ga_clean | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------race | -.4692281 .032978 -14.23 0.000 -.5338663 -.40459 _cons | 39.04992 .0161171 2422.89 0.000 39.01833 39.08151 -------------+---------------------------------------------------------------sigma_u | .39512117 sigma_e | 2.1303179 rho | .03325698 (fraction of variance due to u_i) -----------------------------------------------------------------------------F test that all u_i=0: F(389, 31098) = 1.40 Prob > F = 0.0000 Same result without the fixed coefficient output for all the clusters

proc glm data=nc.data_final; absorb b_group; model ga_clean = race; quit; The GLM Procedure Dependent Variable: ga_clean Sum of Source DF Squares Mean Square Model 390 5064.7562 12.9866 Error 31098 141130.6344 4.5383 Corrected Total 31488 146195.3906 R-Square 0.034644 Coeff Var 5.476830 F Value Pr > F 2.86 <.0001 Root MSE ga_clean Mean 2.130318 38.89692 Source DF b_group race 389 4145.980755 10.658048 2.35 <.0001 1 918.775440 918.775440

202.45 <.0001 Source DF race 1 Parameter race Type I SS Type III SS 918.7754401 Mean Square Mean Square F Value 918.7754401 Standard Estimate Error -.4692281302 F Value t Value 0.03297796 Pr > F Pr > F 202.45 Pr > |t| -14.23 <.0001 <.0001

Comparison to Conventional Regression Use cluster-robust SEs to account for complex error (individual and cluster) reg ga_clean race, vce(cl b_group) Linear regression Number of obs F( 1, 389) Prob > F R-squared Root MSE = = = = = 31489 351.42 0.0000 0.0177 2.1356 (Std. Err. adjusted for 390 clusters in b_group) -----------------------------------------------------------------------------| Robust ga_clean | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------race | -.6112235 .0326053 -18.75 0.000 -.6753281 -.5471189 _cons | 39.09623 .014152 2762.59 0.000 39.0684 39.12405 ------------------------------------------------------------------------------ Crude effect: -0.61 weeks Adjusting for neighborhood: -0.47 weeks

Neighborhood explained 23% of the racial disparity (assuming there are no confounders of neighborhood) Can use surveyreg for cluster-robust SEs in SAS proc surveyreg data=nc.data_final; cluster b_group; class b_group ; model ga_clean= race /clparm solution; run; The SURVEYREG Procedure Regression Analysis for Dependent Variable ga_clean Estimated Regression Coefficients Parameter Estimate Standard Error Intercept race 39.0962254 -0.6112235 0.01415202 0.03260526 t Value Pr > |t| 2762.59 -18.75 <.0001 <.0001 95% Confidence Interval 39.0684014 39.1240495 -0.6753281 -0.5471189 NOTE: The denominator degrees of freedom for the t tests is 389. Logistic Model for Binary Outcome logit ptb_total i.race i.b_group, or Logistic regression Log likelihood = -9091.5835

Number of obs LR chi2(367) Prob > chi2 Pseudo R2 = = = = 31157 687.08 0.0000 0.0364 -----------------------------------------------------------------------------ptb_total | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | 1.614217 .0849715 9.10 0.000 1.455979 1.789652 _Ib_group_2 | .7499313 .4644189 -0.46 0.642 .2227859 2.524383 _Ib_group_3 | .7817543 .4527046 -0.43 0.671 .2512752 2.432154 _Ib_group_4 | 1.837389 1.208127 0.93 0.355 .506426 6.66632 _Ib_group_5 | 1.012978

.6832426 0.02 0.985 .2700685 3.799497 _Ib_group_6 | 1.871619 1.284711 0.91 0.361 .4874589 7.186159 _Ib_group_7 | .6577316 .504969 -0.55 0.585 .1460644 2.961781 _Ib_group_8 | 4.113144 2.752025 2.11 0.035 1.108285 15.26498 _Ib_group_9 | .6818469 .7794273 -0.34 0.738 .0725551 6.407749 _Ib_group_10 | 1.130029 1.000033 0.14 0.890 .1994381 6.40281 Accounting for neighborhood differences (within-neighborhood inference), the odds of PTB are 1.61 times greater for Black than White infants 23 clusters with 322 observations were dropped because of non-varying outcomesall 0 or 1division by 0 for an OR margins race, vce(unconditional) post Predictive margins Expression

Number of obs = 31157 : Pr(ptb_total), predict() (Std. Err. adjusted for 367 clusters in b_group2) -----------------------------------------------------------------------------| Unconditional | Margin Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | 0 | .0753146 .0021845 34.48 0.000 .071033 .0795962 1 | .1154796 .0039339 29.35 0.000 .1077693 .12319 -----------------------------------------------------------------------------. lincom _b[1.race] - _b[0.race] ( 1) Risk Difference - 0bn.race + 1.race = 0 -----------------------------------------------------------------------------| Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------(1) | .040165 .0046569 8.62 0.000

.0310377 .0492924 -----------------------------------------------------------------------------. nlcom _b[1.race] / _b[0.race] _nl_1: Risk Ratio _b[1.race] / _b[0.race] -----------------------------------------------------------------------------| Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------_nl_1 | 1.533297 .0713801 21.48 0.000 1.393394 1.673199 ------------------------------------------------------------------------------ proc logistic data=nc.data_final; class b_group ; model ptb_total (desc)= race b_group; run; Odds Ratio Estimates Point Estimate Effect race b_group b_group b_group b_group b_group b_group b_group b_group b_group b_group b_group b_group b_group b_group 370630001011

370630001012 370630001021 370630002001 370630002002 370630002003 370630003011 370630003012 370630003013 370630003021 370630003022 370630003023 370630004011 370630004012 vs vs vs vs vs vs vs vs vs vs vs vs vs vs 371830544023 371830544023 371830544023 371830544023 371830544023 371830544023 371830544023 371830544023 371830544023 371830544023 371830544023 371830544023 371830544023 371830544023 1.614 2.118 1.588 1.656 3.891 2.145

3.964 1.393 8.710 1.444 2.393 1.250 2.515 1.409 2.206 95% Wald Confidence Limits 1.456 0.387 0.314 0.347 0.727 0.390 0.709 0.219 1.599 0.120 0.311 0.198 0.431 0.187 0.347 1.790 11.580 8.034 7.897 20.828 11.787 22.160 8.850 47.453 17.316 18.388 7.881 14.684 10.615 14.033 Would need to use SUDAAN or binomial/poisson models for RD or RR in SAS xtlogit ptb_total race, i(b_group) fe or clogit ptb_total race, group(b_group) vce(cl b_group) or

note: multiple positive outcomes within groups encountered. note: 23 groups (332 obs) dropped because of all positive or all negative outcomes. Iteration 0: Iteration 1: Iteration 2: log pseudolikelihood = -8466.3107 log pseudolikelihood = -8456.9979 log pseudolikelihood = -8456.9975 Conditional (fixed-effects) logistic regression Log pseudolikelihood = -8456.9975 Number of obs Wald chi2(1) Prob > chi2 Pseudo R2 = = = = 31157 84.25 0.0000 0.0047 (Std. Err. adjusted for 367 clusters in b_group) -----------------------------------------------------------------------------| Robust ptb_total | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------1.race | 1.604089 .0825837 9.18 0.000 1.450126 1.774398 ------------------------------------------------------------------------------ The conditional approach is recommended for non-linear models because of the incidental parameters problem with dummy variables, leading to upward bias. Mainly a problem for small clusters so not too different in this sample (avg cluster size ~80) 1.61 versus 1.60

proc logistic data=nc.data_final; strata b_group; model ptb_total (desc) = race; run; The LOGISTIC Procedure Conditional Analysis Model Fit Statistics Criterion AIC SC -2 Log L Without Covariates 16994.399 16994.399 16994.399 With Covariates 16915.995 16924.352 16913.995 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 80.4035 1 <.0001 Score 82.2342 1 <.0001 Wald 81.7008 1 <.0001 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq race 1

0.4726 0.0523 81.7008 <.0001 Odds Ratio Estimates Effect race Point Estimate 1.604 95% Wald Confidence Limits 1.448 1.777 Comparison to Conventional Regression logit ptb_total race, vce(cl b_group) or Logistic regression Log pseudolikelihood = -9312.2722 Number of obs Wald chi2(1) Prob > chi2 Pseudo R2 = = = = 31489 278.47 0.0000 0.0163 (Std. Err. adjusted for 390 clusters in b_group) -----------------------------------------------------------------------------| Robust ptb_total | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | 2.02922 .0860517

16.69 0.000 1.86738 2.205085 ------------------------------------------------------------------------------ Crude OR: 2.03 Adjusting for neighborhood: 1.60 Neighborhood explained ~40% of the racial disparity in PTB (assuming there are no confounders of neighborhood) N.B. For percent change in OR, you always need to subtract the null (1.0) first (0.6-1.03)/1.03 = -.41 or a drop of 41% after controlling for contextual differences proc surveylogistic data=nc.data_final; cluster b_group; model ptb_total (desc)= race; run; Testing Global Null Hypothesis: BETA=0 Test Likelihood Ratio Score Wald Chi-Square DF Pr > ChiSq 308.0644 324.9556 278.4631 1 1 1 <.0001 <.0001 <.0001 Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error

Wald Chi-Square Pr > ChiSq Intercept race 1 1 -2.6016 0.7077 0.0285 0.0424 8352.0534 278.4631 <.0001 <.0001 Odds Ratio Estimates Effect Point Estimate race 95% Wald Confidence Limits 2.029 1.867 2.205 Association of Predicted Probabilities and Observed Responses Percent Concordant Percent Discordant Percent Tied Pairs 32.9 16.2 50.8 80536248

Somers' D Gamma Tau-a c 0.167 0.340 0.027 0.584 Fixed Effects: Benefits & Disadvantages Benefits: Provides within-cluster effects that are not confounded by cluster-level factors because all cluster variation is removed (accounts for unobservable confounding) No minimum number of clusters Disadvantages: Does not allow estimation of observable cluster-level effects so often seen more in longitudinal analyses where betweencluster effects may not be of interest Can be inefficient/less precise due to less degrees of freedom (each cluster counts as parameter) and it only exploits one level of variation Random Effects Alternative to fixed effects that models only one additional parameter (instead of k-1) by making greater assumptions More efficient but vulnerable to bias Average cluster-specific intercept with the cluster-level variance estimated (2) Accounts for variability in the outcome across neighborhoods but not for covariates (corr oj, xi = 0) Allows estimates of variance at both levels and of cluster-level covariates because the cluster-level variance isnt completely removed from model xtreg ga_clean race, i(b_group) re mle Random-effects ML regression Group variable: b_group Number of obs Number of groups = = 31489

390 Random effects u_i ~ Gaussian Obs per group: min = avg = max = 1 80.7 652 Log likelihood = -68563.638 LR chi2(1) Prob > chi2 = = 389.27 0.0000 -----------------------------------------------------------------------------ga_clean | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | -.5883143 .027936 -21.06 0.000 -.6430679 -.5335606 _cons | 39.07689 .0180206 2168.46 0.000 39.04157 39.11221 -------------+---------------------------------------------------------------/sigma_u | .1364134 .021792 .0997419 .1865676 /sigma_e | 2.131475

.0085433 2.114796 2.148285 rho | .0040792 .0013011 .0021387 .0074679 -----------------------------------------------------------------------------Likelihood-ratio test of sigma_u=0: chibar2(01)= 18.07 Prob>=chibar2 = 0.000 Cluster-specific, within-neighborhood interpretation but significantly higher than FE estimate of -0.47 Intracluster Correlation = 0.004 (proportion of variance that occurs at neighborhood level) 0.1362/(0.1362+2.132) = 0.004 - Significant neighborhood variation but a small fraction of overall variability (0.4%) proc glimmix data=nc.data_final method=quad; class b_group; model ga_clean = race /solution ; random intercept/ subject=b_group; run; The GLIMMIX Procedure Optimization Information Optimization Technique Dual Quasi-Newton Parameters in Optimization 4 Lower Boundaries 2 Upper Boundaries 0 Fixed Effects Not Profiled Starting From GLM estimates Quadrature Points 1 Covariance Parameter Estimates Standard Cov Parm Subject Estimate Intercept Residual b_group Error 0.01861 0.005946

4.5432 0.03642 Solutions for Fixed Effects Effect Standard Estimate Error DF t Value Intercept 39.0769 0.01802 389 race -0.5883 0.02793 31098 Pr > |t| 2168.69 <.0001 -21.06 <.0001 Type III Tests of Fixed Effects Effect race Num DF 1 Den DF 31098 F Value 443.54 Pr > F <.0001 xtlogit ptb_total race, i(b_group) re or Random-effects logistic regression Group variable: b_group

Number of obs Number of groups = = 31489 390 Random effects u_i ~ Gaussian Obs per group: min = avg = max = 1 80.7 652 Log likelihood = -9310.5391 Wald chi2(1) Prob > chi2 = = 265.00 0.0000 -----------------------------------------------------------------------------ptb_total | OR Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | 1.992814 .0844133 16.28 0.000 1.834049 2.165323 -------------+---------------------------------------------------------------/lnsig2u | -4.059262 .6231052 -5.280526 -2.837999

-------------+---------------------------------------------------------------sigma_u | .131384 .040933 .0713425 .241956 rho | .0052196 .0032354 .0015447 .0174837 -----------------------------------------------------------------------------Likelihood-ratio test of rho=0: chibar2(01) = 3.47 Prob >= chibar2 = 0.031 Cluster-specific, within-neighborhood interpretation but significantly higher than FE estimate of 1.60 Intracluster Correlation = 0.005 (most of variance occurs within neighborhood at the individual level, 99.5%, rather than between neighborhoods at neighborhood level, 0.5%) 0.1314/(0.1314 + 2/3) = 0.005 proc glimmix data=nc.data_final method=quad; class b_group; model ptb_total (descending) = race /solution dist=bin link=logit oddsratio; random intercept / subject=b_group; run; Covariance Parameter Estimates Standard Cov Parm Subject Estimate Error Intercept b_group 0.01746 0.01081 Solutions for Fixed Effects Standard Estimate Error Effect Intercept race race DF

t Value -2.5945 0.02907 389 0.6894 0.04238 31098 Pr > |t| -89.26 16.27 <.0001 <.0001 Odds Ratio Estimates 95% Confidence _race Estimate DF Limits 1.3261 0.3261 1.992 31098 1.834 Type III Tests of Fixed Effects Effect race Num DF 1 Den DF 31098 F Value 264.60 Pr > F <.0001

2.165 Random Intercept + Slope Also possible to allow random normal variation in covariate effect across neighborhood, e.g. allowing racial disparity to vary by neighborhood xtmixed ga_clean race || b_group: race, mle Mixed-effects ML regression Group variable: b_group Number of obs = 31489 Number of groups = 390 Obs per group: min = 1 avg = 80.7 max = 652 Wald chi2(1) = 304.72 Log likelihood = -68537.949 Prob > chi2 = 0.0000 -----------------------------------------------------------------------------ga_clean | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | -.6060449 .0347181 -17.46 0.000 -.6740911 -.5379988 _cons | 39.09577 .0147177 2656.37 0.000 39.06693 39.12462 ----------------------------------------------------------------------------------------------------------------------------------------------------------Random-effects Parameters | Estimate Std. Err.

[95% Conf. Interval] -----------------------------+-----------------------------------------------b_group: Independent | sd(race) | .3447168 .0374273 .2786404 .4264626 sd(_cons) | .0221739 .0600709 .0001096 4.485535 -----------------------------+-----------------------------------------------sd(Residual) | 2.127355 .0085317 2.110699 2.144143 -----------------------------------------------------------------------------LR test vs. linear regression: chi2(2) = 69.45 Prob > chi2 = 0.0000 Note: LR test is conservative and provided only for reference. Appears to be significant variation across neighborhoods but the point estimate or average within-neighborhood disparity is not correct based on comparisons to FE models (-0.47) proc glimmix data=nc.data_final method=quad; class b_group; model ga_clean = race /solution ; random intercept race/ subject=b_group; Run; The GLIMMIX Procedure Covariance Parameter Estimates Cov Parm Subject Standard Estimate Error Intercept b_group 0.000494 0.003077 race b_group 0.1188 0.02584 Residual

4.5256 0.03631 Solutions for Fixed Effects Standard Estimate Error Effect DF t Value Pr > |t| Intercept 39.0958 0.01498 389 2609.64 <.0001 race -0.6060 0.03488 352 -17.38 <.0001 Type III Tests of Fixed Effects Effect race Num DF 1 Den DF F Value Pr > F 352 301.93 <.0001 xtmelogit ptb_total race || b_group: race, or Mixed-effects logistic regression Group variable: b_group

Integration points = 7 Log likelihood = -9310.2914 Number of obs Number of groups = = 31489 390 Obs per group: min = avg = max = 1 80.7 652 Wald chi2(1) Prob > chi2 = = 255.50 0.0000 -----------------------------------------------------------------------------ptb_total | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | 1.991382 .0858163 15.98 0.000 1.830093 2.166887 ----------------------------------------------------------------------------------------------------------------------------------------------------------Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+-----------------------------------------------b_group: Independent | sd(race) |

.1218997 .0899226 .0287138 .5175045 sd(_cons) | .1101542 .0562791 .040468 .2998403 -----------------------------------------------------------------------------LR test vs. logistic regression: chi2(2) = 3.96 Prob > chi2 = 0.1380 Note: LR test is conservative and provided only for reference. No indication of significant neighborhood variation in the PTB racial disparity; average neighborhood-specific disparity is biased relative to FE (1.60) proc glimmix data=nc.data_final method=quad; class b_group; model ptb_total (descending) = race /solution dist=bin link=logit oddsratio; random intercept race/ subject=b_group; run; Covariance Parameter Estimates Cov Parm Subject Standard Estimate Error Intercept b_group 0.01230 0.01248 race b_group 0.01495 0.02201 Solutions for Fixed Effects Standard Estimate Error Effect Intercept race

race DF -2.5967 0.02875 0.6886 0.04312 t Value 389 352 Pr > |t| -90.31 15.97 <.0001 <.0001 Odds Ratio Estimates 95% Confidence _race Estimate DF Limits 1.3261 0.3261 1.991 352 1.829 Type III Tests of Fixed Effects Effect race Num DF 1 Den DF

F Value Pr > F 352 255.04 <.0001 2.167 When will RE approximate FE? When there is no between-cluster confounding No clustering of X No variation in outcome by cluster Even when confounding is present, RE can still approximate FE under certain conditions Normally, a composite of within and between effects but weighted toward the within-effect when it is more precise Large cluster size High ICC 50 100 150 200 Example 15 20 25 x 30 35 Most of variation is between rather than within cluster (ICC=0.99)

so within-effect is going to be very precise (little variability) ---------------------------------------------------------------Variable | ols olsc gee fe re -------------+-------------------------------------------------x | -2.2152 -2.2152 2.0552 2.0563 2.0503 | 0.799 1.189 0.044 0.045 0.054 _cons | 166.0880 166.0880 67.3646 67.3404 67.4793 | 18.868 28.958 13.615 1.049 7.058 ---------------------------------------------------------------legend: b/se For most multilevel neighborhood studies, ICC is quite low (<10%) so only a huge cluster size could compensate to get valid within-cluster estimates in the presence of neighborhood confounding We can also control for observable cluster-level factors but there are many factors that may not be measured or imperfectly measured e.g. built environment, air quality/toxins, health care access/quality, fresh foods, social cohesion So it can be rare to get the FE estimate, which controls for all factors--observed and unobserved, by controlling only for a few observed factors in a RE model xtreg ga_clean race poverty, i(b_group) re mle Random-effects ML regression

Group variable: b_group Number of obs Number of groups = = 31489 390 Random effects u_i ~ Gaussian Obs per group: min = avg = max = 1 80.7 652 Log likelihood = -68554.58 LR chi2(2) Prob > chi2 = = 407.38 0.0000 -----------------------------------------------------------------------------ga_clean | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | -.5368362 .030145 -17.81 0.000 -.5959193 -.477753 poverty | -.0064216 .0015111

-4.25 0.000 -.0093832 -.0034599 _cons | 39.1246 .0205574 1903.19 0.000 39.08431 39.16489 -------------+---------------------------------------------------------------/sigma_u | .1304851 .0218083 .0940368 .1810607 /sigma_e | 2.131108 .0085401 2.114436 2.147913 rho | .003735 .0012466 .0018992 .007028 -----------------------------------------------------------------------------Likelihood-ratio test of sigma_u=0: chibar2(01)= 15.89 Prob>=chibar2 = 0.000 So controlling for poverty moves us closer to the within-cluster effect but doesnt control for all important neighborhood factors Crude: -0.61 FE: -0.47 Controlling for neighborhood poverty: -0.54 Explains about half of the neighborhood contribution (0.07/0.14) And 11.5% of overall disparity (0.07/-0.61) Hausman test for consistency in estimates from FE and RE models hausman ga_clean_fe ga_clean_re_poverty ---- Coefficients ---| (b) (B) (b-B) sqrt(diag(V_b-V_B)) | ga_clean_fe ga_clean_r~y Difference S.E. -------------+---------------------------------------------------------------race | -.4692281 -.5500752 .080847

.0156404 -----------------------------------------------------------------------------b = consistent under Ho and Ha; obtained from xtreg B = inconsistent under Ha, efficient under Ho; obtained from xtreg Test: Ho: difference in coefficients not systematic chi2(1) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 26.72 Prob>chi2 = 0.0000 Rejects the null of equivalence between the FE and RE estimator **From a RE model that is based on generalized least squares, not maximum likelihood xtlogit ptb_total race poverty, i(b_group) re or Random-effects logistic regression Group variable: b_group Number of obs Number of groups = = 31489 390 Random effects u_i ~ Gaussian Obs per group: min = avg = max = 1 80.7 652 Log likelihood = -9297.2331 Wald chi2(2) Prob > chi2 = =

316.05 0.0000 -----------------------------------------------------------------------------ptb_total | OR Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | 1.804966 .0834998 12.77 0.000 1.64851 1.976272 poverty | 1.010437 .0020192 5.20 0.000 1.006487 1.014403 -------------+---------------------------------------------------------------/lnsig2u | -4.32276 .7583429 -5.809085 -2.836435 -------------+---------------------------------------------------------------sigma_u | .1151661 .0436677 .0547739 .2421452 rho | .0040153 .0030328 .0009111 .0175106 -----------------------------------------------------------------------------Likelihood-ratio test of rho=0: chibar2(01) = 2.21 Prob >= chibar2 = 0.068 So controlling for poverty moves us closer to the within-cluster effect but doesnt control for all important neighborhood factors Crude: 2.03 FE: 1.60 Controlling for neighborhood poverty: 1.80 Explains about half of the neighborhood contribution (0.2/0.43) And 20% of overall disparity (0.2/1.03) Hausman test for consistency in estimates from FE and RE models hausman ptb_fe ptb_re_poverty

---- Coefficients ---| (b) (B) (b-B) sqrt(diag(V_b-V_B)) | ptb_fe ptb_re_pov~y Difference S.E. -------------+---------------------------------------------------------------race | .4725559 .590542 -.1179861 .0243549 -----------------------------------------------------------------------------b = consistent under Ho and Ha; obtained from xtlogit B = inconsistent under Ha, efficient under Ho; obtained from xtlogit Test: Ho: difference in coefficients not systematic chi2(1) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 23.47 Prob>chi2 = 0.0000 Rejects the null of equivalence between the FE and RE estimator Random Effects: Benefits & Disadvantages Benefits: Ability to estimate covariates both within and between cluster (level 1 and 2 effects) Ability to partition variance at multiple levels Examine variation in effects across cluster Efficient/parsimonious Disadvantages: Within-cluster effects can be significantly biased Requires ~30 clusters for estimation of cluster variance with random normal assumption GEE Handles clustered data with complex error treated as a nuisance rather than explicitly controlled (FE) or modeled as an interest (RE) Within-cluster correlation specified as Independent (robust SEs, point estimates unchanged) Exchangeable (similar to RE point estimates)

Unstructured (allows variation in correlation) Inference is population-averaged rather than cluster-specific (only difference is for odds ratio since the average of each cluster-specific OR overall OR; not collapsible) xtreg ga_clean race, i(b_group) pa corr(ind) vce(robust) xtgee ga_clean race, corr(ind) vce(robust) Iteration 1: tolerance = 6.133e-15 GEE population-averaged model Group variable: Link: Family: Correlation: Scale parameter: Pearson chi2(31489): Dispersion (Pearson): b_group identity Gaussian independent 4.560647 143610.20 4.560647 Number of obs Number of groups Obs per group: min avg max Wald chi2(1) Prob > chi2 = = = = = = = 31489 390 1 80.7 652 351.43 0.0000

Deviance Dispersion = 143610.20 = 4.560647 (Std. Err. adjusted for clustering on b_group) -----------------------------------------------------------------------------| Semirobust ga_clean | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | -.6112235 .0326047 -18.75 0.000 -.6751276 -.5473194 _cons | 39.09623 .0141518 2762.63 0.000 39.06849 39.12396 ------------------------------------------------------------------------------ Results are very similar to OLS regression with cluster-robust SEs xtreg ga_clean race, i(b_group) pa corr(exc) xtgee ga_clean race, corr(exc) vce(robust) Iteration Iteration Iteration Iteration Iteration 1: 2: 3: 4: 5: tolerance tolerance tolerance

tolerance tolerance = = = = = .00997269 .00071555 .00004882 3.319e-06 2.256e-07 GEE population-averaged model Group variable: b_group Link: identity Family: Gaussian Correlation: exchangeable Scale parameter: 4.560803 Number of obs Number of groups Obs per group: min avg max Wald chi2(1) Prob > chi2 = = = = = = = 31489 390 1 80.7 652 350.49

0.0000 (Std. Err. adjusted for clustering on b_group) -----------------------------------------------------------------------------| Semirobust ga_clean | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | -.5939304 .0317245 -18.72 0.000 -.6561093 -.5317514 _cons | 39.08105 .013988 2793.91 0.000 39.05364 39.10847 ------------------------------------------------------------------------------ Results are very similar to the random intercept model with cluster-robust SEs proc genmod data=nc.data_final; class b_group; model ga_clean = race; repeated subject=b_group /corr=ind; run; Analysis Of GEE Parameter Estimates Empirical Standard Error Estimates Standard 95% Confidence Parameter Estimate Error Limits Z Pr > |Z| Intercept 39.0962 0.0141 39.0685 39.1239 2766.18 <.0001 race -0.6112 0.0326 -0.6750 -0.5474 -18.77 <.0001 proc genmod data=nc.data_final; class b_group; model ga_clean = race; repeated subject=b_group /corr=exc;

run; Exchangeable Working Correlation Correlation 0.0028899011 Analysis Of GEE Parameter Estimates Empirical Standard Error Estimates Standard 95% Confidence Parameter Estimate Error Limits Z Pr > |Z| Intercept 39.0811 0.0140 39.0537 39.1084 2797.50 <.0001 race -0.5939 0.0317 -0.6560 -0.5318 -18.75 <.0001 xtlogit ptb_total race, i(b_group) pa corr(ind) vce(robust) or xtgee ptb_total race, fam(bin) link(logit) corr(ind) vce(robust) eform Iteration 1: tolerance = 1.545e-07 GEE population-averaged model Group variable: Link: Family: Correlation: Scale parameter: Pearson chi2(31489): Dispersion (Pearson): b_group logit binomial independent 1 31489.00 1 Number of obs Number of groups Obs per group: min avg max Wald chi2(1) Prob > chi2 = = = = =

= = 31489 390 1 80.7 652 278.47 0.0000 Deviance Dispersion = = 18624.54 .5914619 (Std. Err. adjusted for clustering on b_group) -----------------------------------------------------------------------------| Semirobust ptb_total | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | 2.02922 .0860517 16.69 0.000 1.867381 2.205086 ------------------------------------------------------------------------------ Results are very similar to logistic regression with clusterrobust SEs xtlogit ptb_total race, i(b_group) pa corr(exc) vce(robust) or xtgee ptb_total race, fam(bin) link(logit) corr(exc) vce(robust) eform Iteration Iteration Iteration Iteration Iteration 1: 2: 3:

4: 5: tolerance tolerance tolerance tolerance tolerance = = = = = .00895091 .00071972 .00006077 5.223e-06 4.495e-07 GEE population-averaged model Group variable: b_group Link: logit Family: binomial Correlation: exchangeable Scale parameter: 1 Number of obs Number of groups Obs per group: min avg max Wald chi2(1) Prob > chi2 = = = = = = =

31489 390 1 80.7 652 269.83 0.0000 (Std. Err. adjusted for clustering on b_group) -----------------------------------------------------------------------------| Semirobust ptb_total | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | 1.995782 .0839587 16.43 0.000 1.837828 2.167313 ------------------------------------------------------------------------------ Results are very similar to RE logistic regression with cluster-robust SEs ICC is so low that marginal OR cluster-specific OR proc genmod data=nc.data_final; class b_group; model ptb_total = race /dist=bin link=logit; repeated subject=b_group/corr=ind; estimate 'race or' race -1 1 /exp; run; Contrast Estimate Results Label Mean Mean L'Beta Standard L'Beta Estimate Confidence Limits Estimate Error Alpha Confidence Limits race or 0.6699 Exp(race or)

0.6513 0.6880 0.7077 0.0424 0.05 0.6246 0.7907 2.0292 0.0859 0.05 1.8676 2.2049 proc genmod data=nc.data_final; class b_group; model ptb_total = race /dist=bin link=logit; repeated subject=b_group/corr=exc; estimate 'race or' race -1 1 /exp; run; Contrast Estimate Results Label Mean Mean L'Beta Standard L'Beta Estimate Confidence Limits Estimate Error Alpha Confidence Limits race or 0.6662 Exp(race or) 0.6476 0.6843 0.6910 0.0420 0.05 0.6087 0.7734 1.9958 0.0839 0.05 1.8380 2.1671 GEE: Benefits & Disadvantages Benefits: Ability to estimate both within and betweencluster effects and adjust SEs for clustering Examine cross-level interaction Disadvantages: Within-cluster effects can be significantly biased Marginal inference leads to effect estimates closer to the null in logistic models (depending on ICC) No variance components Hybrid Models Obtain the appropriate within-neighborhood effect in random effects, GEE, or general cluster-robust models Contain the advantages of RE or GEE models without the bias in the within-cluster effects Incorporate the cluster-mean of the covariate to account for all between-cluster variation related to the covariate (aggregated variable, % Black)

Centering (subtracting cluster-mean) Centering + cluster-mean covariate adjustment Cluster-mean covariate adjustment egen race_bgc=mean(race), by(b_group) gen race_c=race-race_bgc xtreg ga_clean race_c, i(b_group) re mle Random-effects ML regression Group variable: b_group Random effects u_i ~ Gaussian Number of obs Number of groups Obs per group: min avg max = = = = = 31489 390 1 80.7 652 LR chi2(1) = 201.73 Log likelihood = -68657.406 Prob > chi2 = 0.0000 -----------------------------------------------------------------------------ga_clean | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race_c | -.4692281 .0329833 -14.23 0.000 -.5338742 -.404582 _cons | 38.85962

.0206466 1882.13 0.000 38.81915 38.90008 -------------+---------------------------------------------------------------/sigma_u | .291513 .0203577 .2542229 .3342729 /sigma_e | 2.130663 .0085431 2.113985 2.147473 rho | .0183752 .0025312 .0139516 .0239413 -----------------------------------------------------------------------------Likelihood-ratio test of sigma_u=0: chibar2(01)= 193.82 Prob>=chibar2 = 0.000 Estimate now corresponds to within-neighborhood effect of race obtained in FE model (-0.47) -Note that the ICC is now much larger than in the RE model (0.004 0.018) -Neighborhood variance increased because centering removed the association between race and neighborhood before estimation so this refers to the ICC from a null model -No other variables in the model (including random intercept) will be adjusted for PROC SUMMARY NWAY DATA=nc.data; VAR race; CLASS b_group; OUTPUT OUT=cluster MEAN=race_bgc; RUN; DATA nc.data_final; MERGE nc.data cluster; BY b_group; race_c=race-race_bgc; RUN; proc glimmix data=nc.data_final method=quad; class b_group; model ga_clean = race_c /solution ; random intercept / subject=b_group; run; Covariance Parameter Estimates Cov Parm Subject Intercept Residual

b_group Effect Intercept race_c Standard Estimate Error 0.08497 0.01187 4.5397 0.03641 Solutions for Fixed Effects Standard Estimate Error DF t Value 38.8596 -0.4692 0.02065 389 0.03298 31098 Pr > |t| 1882.23 -14.23 <.0001 <.0001 xtlogit ptb_total race_c, i(b_group) re or Random-effects logistic regression Group variable: b_group Number of obs Number of groups = = 31489 390 Random effects u_i ~ Gaussian

Obs per group: min = avg = max = 1 80.7 652 Log likelihood = -9384.2229 Wald chi2(1) Prob > chi2 = = 83.13 0.0000 -----------------------------------------------------------------------------ptb_total | OR Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race_c | 1.610407 .0841611 9.12 0.000 1.453621 1.784104 -------------+---------------------------------------------------------------/lnsig2u | -2.192009 .1819926 -2.548708 -1.83531 -------------+---------------------------------------------------------------sigma_u | .3342037 .0304113 .2796115 .3994546 rho | .0328355 .0057796 .023213 .046258 -----------------------------------------------------------------------------Likelihood-ratio test of rho=0: chibar2(01) =

83.11 Prob >= chibar2 = 0.000 OR now corresponds to within-neighborhood effect of race obtained in FE model (1.61) ICC is now much larger than in the RE model (0.005 0.032) because it is not adjusted for racial clustering/segregation across neighborhoods PROC SUMMARY NWAY DATA=nc.data; VAR race; CLASS b_group; OUTPUT OUT=cluster MEAN=race_bgc; RUN; DATA nc.data_final; MERGE nc.data cluster; BY b_group; race_c=race-race_bgc; RUN; proc glimmix data=nc.data_final method=quad; class b_group; model ptb_total (descending) = race_c /solution dist=bin link=logit oddsratio; random intercept / subject=b_group; run; Covariance Parameter Estimates Standard Cov Parm Subject Estimate Error Intercept b_group 0.1113 0.02016 Effect Intercept race_c race_c 1 Solutions for Fixed Effects Standard Estimate Error DF t Value -2.3408 0.4764 0.02861 389 0.05226 31098 Pr > |t|

-81.82 9.12 <.0001 <.0001 Odds Ratio Estimates 95% Confidence _race_c Estimate DF Limits -1E-9 1.610 31098 1.454 1.784 egen race_bgc=mean(race), by(b_group) gen race_c=race-race_bgc xtreg ga_clean race_c race_bgc, i(b_group) re mle Random-effects ML regression Group variable: b_group Random effects u_i ~ Gaussian Number of obs = 31489 Number of groups = 390 Obs per group: min = 1 avg = 80.7 max = 652 LR chi2(2) = 431.28 Log likelihood = -68542.63 Prob > chi2 = 0.0000 -----------------------------------------------------------------------------ga_clean | Coef.

Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race_c | -.4692281 .0329838 -14.23 0.000 -.5338752 -.4045811 race_bgc | -.8450466 .0474493 -17.81 0.000 -.9380455 -.7520477 _cons | 38.89084 .0143968 2701.36 0.000 38.86263 38.91906 -------------+---------------------------------------------------------------/sigma_u | .1207709 .021501 .0851965 .1711996 /sigma_e | 2.130694 .008534 2.114033 2.147486 rho | .0032025 .0011389 .0015566 .0062758 -----------------------------------------------------------------------------Likelihood-ratio test of sigma_u=0: chibar2(01)= 13.58 Prob>=chibar2 = 0.000 Within-neighborhood effect of race corresponds to that obtained in FE model (-0.47) Between-neighborhood effect is not adjusted for the individual level effect so it reflects the ecological effect (contextual and individual effect) comparing outcomes between neighborhoods with a higher versus lower % Black b = w + c if c=0 then b = w and no RE biasb = b = w + c if c=0 then b = w and no RE biasw + b = w + c if c=0 then b = w and no RE biasc if b = w + c if c=0 then b = w and no RE biasc=0 then b = w + c if c=0 then b = w and no RE biasb = b = w + c if c=0 then b = w and no RE biasw and no RE bias -0.85 = -0.47 + b = w + c if c=0 then b = w and no RE biasc so b = w + c if c=0 then b = w and no RE biasc = -0.38 -Note that the ICC is now small again because its adjusted for cluster mean of race (composition/segregation) egen race_bgc=mean(race), by(b_group)

xtreg ga_clean race race_bgc, i(b_group) re mle Random-effects ML regression Group variable: b_group Random effects u_i ~ Gaussian Log likelihood = -68542.63 Number of obs Number of groups Obs per group: min avg max LR chi2(2) Prob > chi2 = = = = = = = 31489 390 1 80.7 652 431.28 0.0000 -----------------------------------------------------------------------------ga_clean | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------race | -.4692281 .0329838 -14.23 0.000 -.5338752 -.4045811 race_bgc | -.3758185 .0577872 -6.50

0.000 -.4890794 -.2625576 _cons | 39.04385 .0179707 2172.64 0.000 39.00863 39.07907 -------------+---------------------------------------------------------------/sigma_u | .1207709 .021501 .0851965 .1711996 /sigma_e | 2.130694 .008534 2.114033 2.147486 rho | .0032025 .0011389 .0015566 .0062758 -----------------------------------------------------------------------------Likelihood-ratio test of sigma_u=0: chibar2(01)= 13.58 Prob>=chibar2 = 0.000 Without cluster mean centering , the cluster mean variable now refers to the contextual effect of racial segregation because it is now correlated with and can be adjusted for the individual effect Refers to the effect of living in a more segregated neighborhood regardless of whether you are black or white b = w + c if c=0 then b = w and no RE biasc = b = w + c if c=0 then b = w and no RE biasb - b = w + c if c=0 then b = w and no RE biasw = -0.38 Because it is significantly different from zero, there is evidence of a contextual effect , b = w + c if c=0 then b = w and no RE biasw b = w + c if c=0 then b = w and no RE biasb corr oj, xi 0 Caveats Cluster-mean adjustment is a convenient way to account for cluster-level confounding but there are some important considerations Cluster mean must always come from sample (not external population, e.g. census % Black) Other covariates may also require cluster-mean adjustment if theyre confounders of the principal covariate and also vary by cluster Should always compare to FE estimates For complex survey data, weights may need to be scaled for multilevel models and the hybrid approach may only work if there is minimal sampling bias Examples

Multilevel data structure Neighborhood models Sibling studies Longitudinal data structure Policy evaluation (state panels) Neighborhood Contribution to Black White Perinatal Disparities Used hybrid models (cluster-mean adjustment) to evaluate the impact of neighborhood inequalities to racial disparities Adjusted for individual-level factors that could influence neighborhood selection (age, education, marital status, gravidity) Neighborhoods made a significant contribution to PTB (~15% disparity reduction) Effect of neighborhood segregation wasnt fully explained by neighborhood poverty May need to consider context across the life course Schempf AH, Kaufman JS, Messer LC, Mendola P. The neighborhood contribution to black-white perinatal disparities: an example from two north Carolina counties, 1999-2001. Am J Epidemiol. 2011;174(6):744-52. Neighborhood Contribution to Hypertension Disparities Used cluster-mean centering for all covariates compared to RE model with neighborhood-level covariates Neighborhood differences helped to entirely explain the residual black-white hypertension disparity after adjustment for individual-level factors OR 1.5 1.0 Measured neighborhood variables (poverty/affluence, Hispanic, age composition) accounted for entire neighborhood contribution OR still 1.0 without cluster-mean centering Morenoff JD, House JS, Hansen BB, et al. Understanding social disparities in hypertension prevalence, awareness, treatment, and control: the role of neighborhood context. Soc Sci Med. 2007;65(9):1853-1866. Gestational Weight Gain and Child BMI Siblings nested within mothers (CPP) Used FE models to account for unobserved family level factors (environmental/genetic) that may be associated with both GWG and child BMI GEE (independent) models showed associations between GWG and child BMI FE models, comparing only changes in GWG between pregnancies within the same mother, showed no effect on child BMI May be nothing deterministic about in-utero exposures

Branum AM, Parker JD, Keim SA, Schempf AH. Prepregnancy body mass index and gestational weight gain in relation to child body mass index among siblings. Am J Epidemiol. 2011 Nov 15;174(10):1159-65. Breastfeeding and Child BMI Breastfeeding is often associated with many positive health outcomes but its unclear whether this is due to confounding by selection (unobserved characteristics) Siblings nested within mother (add-health) Used FE models to contrast outcomes for discordant siblings (one breastfed, one formula fed) No effect of different breastfeeding exposure on adolescent overweight within siblings to the same mother Changing a womans breastfeeding behavior is not likely to change child BMI; many benefits of breastfeeding but reducing obesity may not be one Nelson MC, Gordon-Larsen P, Adair LS. Are adolescents who were breast-fed less likely to be overweight? Analyses of sibling pairs to reduce confounding. Epidemiology. 2005 Mar;16(2):247-53. Policy Evaluation Cross sectional analyses comparing jurisdictions with a policy to those without can be severely biased (between-cluster) States/counties/hospitals may institute policies in response to a problem (reverse causality) States/counties/hospitals may institute policies if they are more progressive; already healthier (selection, omitted variable bias) Need longitudinal inference, comparing outcomes before and after policy change within a given unit to contemporaneous controls for time trend irrespective of policy change Difference in Difference 5 4 3 Treatment Control 2 1 0 Before After T=Treatment C=Control B=Before A=After DiD1) (YT YC, A) (YT YC, B) = 1 2) (YA YB, T) (YA YB, C) = 1 Cross-sectional: (YT YC, A) = 3

Pre-Post: (YA YB, T) = 3 Could be an example of selection for a positive health indicator e.g. breastfeeding laws Difference in Difference 5 4 3 Treatment Control 2 1 0 Before After T=Treatment C=Control B=Before A=After DiD1) (YT YC, A) (YT YC, B) = -1 2) (YA YB, T) (YA YB, C) = -1 Cross-sectional: (YT YC, A) = 1 Pre-Post: (YA YB, T) = 1 Could be an example of reverse causality for a negative indicator e.g. school nutrition/activity policies and obesity Fixed Effects Panel Analysis Design - Same as Difference in Difference but extends beyond two periods and requires multiple units Yjt = b = w + c if c=0 then b = w and no RE biaso + b = w + c if c=0 then b = w and no RE biasPjt + b = w + c if c=0 then b = w and no RE biasj + b = w + c if c=0 then b = w and no RE biast + b = w + c if c=0 then b = w and no RE biasXjt + ejt - Within-unit contrasts with between-unit contemporaneous controls are achieved by entering unit level dummy variables (b = w + c if c=0 then b = w and no RE biasj) along with time indicator (b = w + c if c=0 then b = w and no RE biast) - 3+ time points can control for unit-specific trends that may be simultaneous to policy changes Sample Software Code SAS STATA Proc reg; *Proc logistic; Class state year; Model Y= x state year; Run; xi: reg y x i.year i.state

xi: logit y x i.year i.state Proc logistic; Class state year; Model Y= x year; Strata state; Run; xi: xtreg y x i.year, i(state) fe xi: xtlogit y x i.year, i(state) fe xi: clogit y x i.year, group(state) Applied Example Objective To evaluate and compare the impact of statespecific changes in smoking-related policies on childhood asthma prevalence & severity Data Individual-level outcome data come from available waves of NSCH (2003 & 2007) Parent-reported current asthma Severity of current asthma (mild v. moderate/severe) Chronic ear infection (3+ in past year) Control factors: child age, sex, race/ethnicity, primary language, family structure, insurance status/type, household poverty, and parental education Longitudinal state policy data from CDC Cigarette Taxes Clean air legislation Medicaid coverage of cessation services Data Steps Append both years of survey data Set statement in SAS Append in STATA Need to assign unique ID numbers Merge policy level detail by state and year If exploring lags, create in policy database prior to merging (manually or with lag function) State Year Tax Tax_Lag1 Tax_Lag2 CT

2003 0.59 0.34 0.34 CT 2007 2.00 1.51 1.51 Model Specification Used OLS instead of logistic because output is interpretable as probabilities Y = b = w + c if c=0 then b = w and no RE bias0 + b = w + c if c=0 then b = w and no RE biassex + b = w + c if c=0 then b = w and no RE biasage + b = w + c if c=0 then b = w and no RE biasrace + b = w + c if c=0 then b = w and no RE biaslanguage + b = w + c if c=0 then b = w and no RE biasfam_structure + b = w + c if c=0 then b = w and no RE biaseducation + b = w + c if c=0 then b = w and no RE biaspoverty + b = w + c if c=0 then b = w and no RE biasinsurance + b = w + c if c=0 then b = w and no RE biasyear + b = w + c if c=0 then b = w and no RE biasstate + b = w + c if c=0 then b = w and no RE biastax Results Cigarette taxes Asthma prevalence: 16% per $1 increase, p=0.09 Moderate/severe asthma: 29% per $1 increase, p=0.04 Clean air legislation No significant effects Medicaid coverage for cessation therapy Chronic ear infection: 60% with expansion, p<0.01 Comparison of Model Inference Effects for a $1 increase in cigarette taxes, relative decrease from 2003 outcome level 3.1% Outcome Moderate/Severe Asthma Prevalence Fixed Effects Panel Analysis -29.8% (-57.7% , -1.9%) Cross-Sectional

3.8% (-4.4%, 12%) Interpretation Fixed effects: On average, states that increase their tax by $1 see a significant ~30% drop in moderate/severe asthma Cross-sectional: On average, tax differences between states are not associated with moderate/severe asthma State-Specific Inference Kentucky 2003 2007 Moderate/Severe Asthma 4.3% 3.2% -1.1% 3 30 27 Cigarette Taxes Predicted effect = 0.27 * -0.9% pts = -0.24% pts so the tax hike was responsible for 22% of the decline in moderate/severe asthma (-.24%/-1.1%) If state had increased taxes by $1 (73 more than actual), decline would have been greater by 0.73 * -0.9% = -0.66% -- translates to a relative drop of 41% instead of 26% (absolute change of -1.76% instead of -1.1% relative to baseline of 4.3%) Smoking Legislation and Childrens Secondhand Smoke Exposure Cross-sectional models showed that clean air legislation was associated with reduced exposure Fixed effect (longitudinal models) showed no association so it appears that more progressive states enacted legislation However, there was an impact of cigarette taxes that served to reduce racial/ethnic and income

disparities Greater effects for White and low-income children Hawkins SS, Chandra A, Berkman L. The Impact of Tobacco Control Policies on Disparities in Children's Secondhand Smoke Exposure: A Comparison of Methods. Matern Child Health J. 2012 Mar 29. [Epub ahead of print] Medical Marijuana Laws and Adolescent Use RE models showed marijuana laws to be associated with greater use FE models showed no association between medical marijuana laws and use Likely a reverse causal association that states with greater use among the constituency are more likely to pass legislation Harper S, Strumpf EC, Kaufman JS. Do medical marijuana laws increase marijuana use? Replication study and extension. Ann Epidemiol. 2012 Mar;22(3):207-12. Summary For multilevel analyses, you typically want to examine effects at both levels so you need models that allow variation at multiple levels Generally want to use RE or GEE models Use hybrid models incorporating cluster means to account for cluster-level confounding and achieve true withincluster effects For panel/longitudinal analyses, you typically only care about the within-unit inference obtained from FE models Generally want to use conventional FE models Homework Assignment Dataset of births matched to same mother (momid) Use this to compare inference and interpret the effects of smoking on birthweight (birwt is continuous and LBW, <2500 g) from these models Conventional regression with cluster robust SEs Random intercept GEE (exchangeable) Fixed effects Hybrid fixed effects (cluster mean adjusting) Include smoke and the covariates of mage, male, married, hsgrad, somecoll, collgrad, black, pretri2, pretri3, novisit Describe the benefits and disadvantages of the above models Conventional regression with cluster robust SEs proc surveyreg data=new; cluster momid; model birwt = smoke mage male black pretri2 pretri3 novisit run;

Number of Observations Mean of birwt Sum of birwt married hsgrad somecoll collgrad; 8604 3469.9 29855287 Design Summary Number of Clusters 3978 Fit Statistics R-square Root MSE Denominator DF 0.09307 502.33 3977 Estimated Regression Coefficients Parameter black pretri2 pretri3 novisit married hsgrad somecoll collgrad Estimate Standard Error t Value Pr > |t| -222.62920 15.16062 48.01495 -192.22265 50.13834

58.16388 87.25244 102.54592 28.9780315 17.9602368 38.4015782 85.1568122 26.7366384 25.6906615 28.3130890 29.1243183 -7.68 0.84 1.25 -2.26 1.88 2.26 3.08 3.52 <.0001 0.3987 0.2112 0.0240 0.0608 0.0236 0.0021 0.0004 NOTE: The denominator degrees of freedom for the t tests is 3977. Random Intercept proc glimmix data=new method=quad(initpl=5) noinitglm; class momid; model birwt = smoke mage male black pretri2 pretri3 novisit collgrad /solution ; random intercept / subject=momid; run; married hsgrad somecoll Covariance Parameter Estimates Cov Parm Subject Intercept

Residual momid Estimate Standard Error 114042 138531 4252.84 2904.36 Solutions for Fixed Effects Effect Intercept smoke mage male black pretri2 pretri3 novisit married hsgrad somecoll collgrad Estimate Standard Error DF t Value Pr > |t| 3106.48 -219.73 7.8386 120.67 -216.99 8.1155 43.7332 -158.13

53.1767 62.9070 88.8345 100.40 40.9177 18.2327 1.3469 9.5870 28.2373 15.5535 34.5642 54.7582 25.4802 24.9831 27.2365 27.9142 3972 4620 4620 4620 4620 4620 4620 4620 4620 4620 4620 4620 75.92 -12.05 5.82 12.59 -7.68 0.52 1.27 -2.89 2.09 2.52 3.26 3.60 <.0001 <.0001 <.0001 <.0001 <.0001 0.6018

0.2058 0.0039 0.0369 0.0118 0.0011 0.0003 GEE (Exchangeable) proc genmod data=new; class momid; model birwt=smoke mage male black pretri2 pretri3 novisit repeated subject=momid /corr=exc; run; married hsgrad somecoll collgrad ; Exchangeable Working Correlation Correlation 0.4428629745 GEE Fit Criteria QIC QICu 8624.9632 8616.0000 Analysis Of GEE Parameter Estimates Empirical Standard Error Estimates Parameter Estimate Intercept smoke mage male black pretri2 pretri3 novisit married hsgrad somecoll collgrad 3108.079 -220.554 7.7884 120.5962 -217.102

8.2117 43.7602 -158.761 53.1095 62.8929 88.8847 100.5458 Standard Error 42.6106 19.1370 1.4092 9.6715 29.2310 15.7123 36.4548 73.4419 26.6567 25.7679 28.3292 29.0306 95% Confidence Limits 3024.564 -258.062 5.0264 101.6403 -274.394 -22.5838 -27.6899 -302.705 0.8632 12.3887 33.3604 43.6469 3191.594 -183.046 10.5504 139.5521 -159.810 39.0073 115.2103 -14.8177 105.3557 113.3971 144.4089 157.4447

Z Pr > |Z| 72.94 -11.53 5.53 12.47 -7.43 0.52 1.20 -2.16 1.99 2.44 3.14 3.46 <.0001 <.0001 <.0001 <.0001 <.0001 0.6012 0.2300 0.0306 0.0463 0.0147 0.0017 0.0005 Fixed Effects proc glm data=new; absorb momid; model birwt=smoke mage male black pretri2 pretri3 novisit collgrad ; run; quit; married hsgrad somecoll Dependent Variable: birwt Parameter smoke mage male black pretri2 pretri3 novisit married hsgrad

somecoll collgrad Estimate Standard Error t Value Pr > |t| -104.2849992 22.7041539 125.3693243 0.0000000 2.0485058 45.4074853 -99.1457340 0.0000000 0.0000000 0.0000000 0.0000000 29.14818572 3.00947000 10.93838681 . 18.62282276 41.48998267 67.59271486 . . . . -3.58 7.54 11.46 . 0.11 1.09 -1.47 . . . . 0.0004 <.0001

<.0001 . 0.9124 0.2738 0.1425 . . . . B B B B B NOTE: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable. Hybrid Fixed Effects in RE Model PROC SUMMARY NWAY DATA=mla.smoking2; VAR smoke mage male pretri2 pretri3 novisit; CLASS momid; OUTPUT OUT=cluster MEAN=m_smoke m_age m_male m_pretri2 m_pretri3 m_novisit; RUN; DATA new; MERGE mla.smoking2 cluster; BY momid; RUN; proc glimmix data=new method=quad(initpl=5) noinitglm; class momid; model birwt = smoke m_smoke mage male black pretri2 pretri3 novisit m_pretri3 m_novisit/solution ; random intercept / subject=momid; run; Covariance Parameter Estimates Cov Parm Subject Intercept Residual momid Estimate Standard

Error 114019 137212 4219.03 2867.98 Solutions for Fixed Effects Effect Estimate Standard Error DF t Value Pr > |t| Intercept smoke m_smoke mage male black pretri2 pretri3 novisit married hsgrad somecoll collgrad m_age m_male m_pretri2 m_pretri3 m_novisit 3227.63 -104.28 -184.57 22.7041 125.37 -224.76 2.0485 45.4075

-99.1457 44.0668 63.3957 91.8638 108.02 -18.3473 -20.9414 22.7864 2.4144 -154.42 45.8709 29.2192 37.2826 3.0168 10.9651 28.3522 18.6682 41.5911 67.7575 26.0170 25.2540 27.8175 28.9992 3.3681 22.3241 33.7079 74.8960 114.85 3966 4620 4620 4620 4620 4620 4620 4620 4620 4620 4620 4620 4620 4620 4620 4620 4620 4620 70.36

-3.57 -4.95 7.53 11.43 -7.93 0.11 1.09 -1.46 1.69 2.51 3.30 3.73 -5.45 -0.94 0.68 0.03 -1.34 <.0001 0.0004 <.0001 <.0001 <.0001 <.0001 0.9126 0.2750 0.1435 0.0904 0.0121 0.0010 0.0002 <.0001 0.3483 0.4991 0.9743 0.1788 married hsgrad somecoll collgrad m_age m_male m_pretri2

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