Longtudnal Logstc Regresson: Breastfeedng of Nepalese Chldren PART II GEE models (margnal, populaton average) covered last lab Random Intercept models (subject specfc) Transton models Scentfc Queston Determne whether the breastfeedng of Nepalese chldren vares wth chld age and/or sex of chld. Data: Nepal Data (nepal.dta) as modfed n Lab 10 Outcome: Y j =I(breastfeedng j ) for ndvdual at vst number j We use vst number as our tme. We prepare our dataset lke we dd n Lab 10.. use "nepal.dta", clear ************************************** ** Dataset prep work from lab 10 **** ************************************** ** drop extra varables **. drop age age3 age4 t ** drop other varables we're not usng n ths analyss. drop wt ht arm day month year ded alve mage lt ** gen vst number varable to use as our tme varable **. sort d age. by d: gen vst=_n. tab vst vst Freq. Percent Cum. ------------+----------------------------------- 1 00 0.00 0.00 00 0.00 40.00 3 00 0.00 60.00 4 00 0.00 80.00 5 00 0.00 100.00 ------------+----------------------------------- Total 1,000 100.00. xtset d vst panel varable: d (strongly balanced) tme varable: vst, 1 to 5 delta: 1 unt Drop observatons wth mssng values on our outcome. drop f bf==. (53 observatons deleted). xtdes 1
d: 1,,..., 00 n = 199 vst: 1,,..., 5 T = 5 Delta(vst) = 1 unt Span(vst) = 5 perods (d*vst unquely dentfes each observaton) Dstrbuton of T_: mn 5% 5% 50% 75% 95% max 1 4 5 5 5 5 5 Freq. Percent Cum. Pattern ---------------------------+--------- 170 85.43 85.43 11111 15 7.54 9.96 1111. 5.51 95.48 1... 3 1.51 96.98 1.111 3 1.51 98.49 111.1 1 0.50 98.99 1.1.. 1 0.50 99.50 1.11. 1 0.50 100.00 111.. ---------------------------+--------- 199 100.00 XXXXX We create a bnary ndcator of breastfeedng for our outcome varable. *** combne groups to gen 0-1 ndcator of breast feedng ***. gen bfbn=1 f bf==1 bf== (564 mssng values generated). replace bfbn=0 f bfbn==. (564 real changes made) For our covarates of nterest n the analyss, we create a centered age varable, an ndcator for male gender and an nteracton between male gender and centered age. ** create a centered age varable. gen agec = age-37.8194 * generate a bnary ndcator for male gender *. gen male=(sex==1). drop sex * defne centered age and male gender nteracton term *. gen agemale=agec*male Subject-specfc models accountng for correlaton (Random Intercept) We run a random ntercept model for breastfeedng status (y j ) where we account for correlaton of the repeated observatons on chldren by ncludng a random ntercept for each chld () and control for chld s age (centered). logtp U ( y = 1 U ) j ~ N( 0, σ ) u = β 0 = ( β 0 + β 1 agec +U ) + β j 1 +U agec j
(At least) two ways to ft a logstc random ntercept model n Stata 1. xtlogt. gllamm (wll be used a lot n the multlevel modelng class) xtlogt We ll ft the xtlogt model frst snce xtlogt s a more smple command Random-effects logstc regresson Number of obs = 947 Group varable (): d Number of groups = 199 Random effects u_ ~ Gaussan Obs per group: mn = 1 avg = 4.8 max = 5 Wald ch(1) = 151.30 Log lkelhood = -19.04379 Prob > ch = 0.0000 bfbn Coef. Std. Err. z P> z [95% Conf. Interval] agec -.3193337.05961-1.30 0.000 -.370168 -.684506 _cons -.11660.309378-6.84 0.000 -.797-1.51033 /lnsgu 1.79395.1716709 1.3997.065864 sgma_u.37488.037981.006643.80991 rho.6314745.0399503.5503486.7057885 Lkelhood-rato test of rho=0: chbar(01) = 16.34 Prob >= chbar = 0.000 Report results on the OR scale: bfbn OR Std. Err. z P> z [95% Conf. Interval] agec.76633.0188643-1.30 0.000.6905846.764563 /lnsgu 1.79395.1716709 1.3997.065864 sgma_u.37488.037981.006643.80991 rho.6314745.0399503.5503486.7057885 Coeffcent estmates have subject-specfc nterpretaton! Interpretaton of OR for agec: For a gven chld, the odds of breastfeedng decreases by 8% for a one month ncrease n that chld s age. If ths result was from a GEE (populaton average model), the nterpretaton would be: On average, for the Nepalese chldren n our study, the odds of breastfeedng decreases by 40% for each one month ncrease n age. 3
Intra-class correlaton n our model Two equvalent nterpretatons: 1. The proporton of the total varance n breastfeedng status that s due to dfferences between chldren (.e., the varance n the random ntercept for chld) after controllng for age.. The correlaton of the repeated measurements of breastfeedng status on the same chld after controllng for age. In our model, rho = 0.63 rh o σ u σ u + σ ε The varance of the random ntercept s (sgma_u)^ = 5.61 Dgresson on usng xtlogt check for adequate fttng of the model xtlogt uses quadrature methods to approxmate the lkelhood functon snce there s no closed form soluton. Ths means xtlogt and other methods that use quadrature can run slowly and we need to check that the specfed quadrature has adequately approxmated the lkelhood.. quadchk, nooutput Refttng model ntponts() = 8 Refttng model ntponts() = 16 Quadrature check Ftted Comparson Comparson quadrature quadrature quadrature 1 ponts 8 ponts 16 ponts Log -180.5843-180.975-180.58976 lkelhood -.44381901 -.0613697 Dfference.0045844.00033971 Relatve dfference bfbn: -.51073187 -.551378 -.50715679 agec -.04050091.00357508 Dfference.0799975 -.00699991 Relatve dfference bfbn: -3.5069938-3.74501-3.465609 _cons -.3550739.04139091 Dfference.06715364 -.0118039 Relatve dfference lnsgu: 3.147561 3.98336 3.108699 _cons.16807743 -.016168 Dfference.05378898 -.00516081 Relatve dfference Relatve dfference > 10^- (1%) (arbtrary STATA rule of thumb) so reft usng > # of ntegraton ponts (default was 1). 4
. xtlogt bfbn agec, (d) or ntponts(18) nolog Random-effects logstc regresson Number of obs = 947 Group varable (): d Number of groups = 199 Random effects u_ ~ Gaussan Obs per group: mn = 1 avg = 4.8 max = 5 Wald ch(1) = 19.57 Log lkelhood = -187.77104 Prob > ch = 0.0000 bfbn OR Std. Err. z P> z [95% Conf. Interval] agec.7013965.018547-11.38 0.000.6598439.7455659 /lnsgu.0503.1814144 1.694754.405886 sgma_u.787541.58501.333518 3.3990 rho.705504.0379108.63377.771189 Lkelhood-rato test of rho=0: chbar(01) = 134.88 Prob >= chbar = 0.000.. quadchk, nooutput Refttng model ntponts() = 14 Refttng model ntponts() = Quadrature check Ftted Comparson Comparson quadrature quadrature quadrature 18 ponts 14 ponts ponts Log -187.77104-188.93518-185.07007 lkelhood -1.1641388.7009735 Dfference.00619978 -.0143844 Relatve dfference bfbn: -.35468188 -.353077 -.38956901 agec.00160917 -.0348871 Dfference -.00453693.09836173 Relatve dfference bfbn: -.37401 -.347957 -.6170863 _cons.0444394 -.4468535 Dfference -.01030346.1031388 Relatve dfference lnsgu:.0503.0176587.330693 _cons -.036619.8030933 Dfference -.0159985.13671491 Relatve dfference We can ncrease ntponts agan for even more assurance that the lkelhood s approprately approxmated.. xtlogt bfbn agec, (d) or ntponts(100) nolog Random-effects logstc regresson Number of obs = 947 Group varable (): d Number of groups = 199 Random effects u_ ~ Gaussan Obs per group: mn = 1 avg = 4.8 5
max = 5 Wald ch(1) = 57.0 Log lkelhood = -180.73476 Prob > ch = 0.0000 bfbn OR Std. Err. z P> z [95% Conf. Interval] agec.608483.0400476-7.55 0.000.53461.69098 /lnsgu 3.04669.3048884.4491 3.6446 sgma_u 4.587545.6993446 3.40668 6.1850 rho.8648116.0356453.778786.908107 Lkelhood-rato test of rho=0: chbar(01) = 148.96 Prob >= chbar = 0.000 Lkelhood-rato test of rho=0: chbar(01) = 149.1 Prob >= chbar = 0.000. quadchk, nooutput Refttng model ntponts() = 67 Refttng model ntponts() = 133 Quadrature check Ftted Comparson Comparson quadrature quadrature quadrature 100 ponts 67 ponts 133 ponts Log -180.60658-180.60658-180.60658 lkelhood.63e-06.45e-07 Dfference -1.53e-08-1.43e-09 Relatve dfference bfbn: -.51104731 -.51104799 -.51104788 agec -6.763e-07-5.76e-07 Dfference 1.33e-06 1.10e-06 Relatve dfference bfbn: -3.489956-3.48930-3.489996 _cons -6.446e-06-4.033e-06 Dfference 1.847e-06 1.156e-06 Relatve dfference lnsgu: 3.15753 3.15756 3.157555 _cons.863e-06.330e-06 Dfference 9.159e-07 7.454e-07 Relatve dfference Note the dfference! Now the OR for agec s 0.6 and the estmated heterogenety s 0.9. GLLAMM generalzed lnear latent and mxed models (gllamm can do a lot more than just logstc models wth a random ntercept!) Install gllamm (you have to be connected to the nternet) ssc descrbe gllamm or update gllamm (replace prevous verson) ssc nstall gllamm, replace 6
Ft the same random ntercept model where we control for age. logtp U ( y = 1 U ) j ~ N( 0, σ ) u = β = ( β. gllamm bfbn agec, (d) l(logt) f(bnom) number of level 1 unts = 947 number of level unts = 199 0 + β 0 1 agec +U ) + β j 1 +U agec bfbn Coef. Std. Err. z P> z [95% Conf. Interval] agec -.481358.067683-7.11 0.000 -.613895 -.3485791 _cons -3.183983.5831388-5.46 0.000-4.36914 -.04105 Varances and covarances of random effects ***level (d) var(1): 15.385845 (4.301343) ** get OR nstead of log(or) **. gllamm, eform bfbn exp(b) Std. Err. z P> z [95% Conf. Interval] agec.618019.041895-7.11 0.000.5414.7056901 Varances and covarances of random effects ***level (d) var(1): 15.385845 (4.301343) gllamm reports the varance of the random ntercept, σ u =15.38 We ll cover gllamm n much more detal 4 th term. Now, we are gong to compare the GEE models (lab 10) to the random ntercept model. Frst, we obtan the predcted log odds from the random ntercept model (gllamm). * get the ftted log odds. predct fttedlo, xb (xb wll be stored n fttedlo) j Next, we assgn values to the random ntercept for each chld usng Emprcal Bayes estmates (we ll cover ths more n 4 th term).. gllapred ebri, u (means and standard devatons wll be stored n ebrim1 ebris1) Produce a varable contanng the ftted probablty of breastfeedng for an average chld (a chld wth a value of the random ntercept = 0).. gen ftted_avgp = exp(fttedlo)/( 1 + exp(fttedlo) ) 7
Produce a varable contanng the ftted probablty of breastfeedng for each chld.. gen ftted_ndp = exp(fttedlo + ebrim1)/( 1 + exp(fttedlo + ebrim1) ) Now we reft GEE models wth three correlaton structures (from lab 10) and store the ftted values.. quetly xtgee bfbn agec, f(bn) lnk(logt) corr(ar1) robust. predct ftted_geear1, mu. label var ftted_geear1 "gee_ar1". ****Ft GEE wth unform corr model ****. quetly xtgee bfbn agec, f(bn) lnk(logt) corr(exc) robust. predct ftted_geeunf, mu. label var ftted_geeunf "gee_unf". ****Ft GEE wth ndependence corr model ****. quetly xtgee bfbn agec, f(bn) lnk(logt) corr(nd) robust. predct ftted_geend, mu. label var ftted_geend "gee_nd" Smooth the observed breastfeedng data versus agec. ksm bfbn agec, lowess bw(.4) ylab(0(.)1) lwdth(10) gen(bfbnsm) Plot the GEE and RI model fts. twoway (scatter bfbn agec, jtter(4)) (lne bfbnsm ftted_geear1 ftted_geeunf ftted_geend ftted_avgp agec, sort pstyle (p1)), ylab(0(.)1) 0..4.6.8 1-40 -0 0 0 40 agec bfbn gee_ar1 gee_nd lowess: bfbn gee_unf ftted_avgp 8
The ft from random ntercept model s qute dfferent from those from margnal model. Ths s to be expected because they are estmatng dfferent thngs and have dfferent nterpretatons. The OR from the random ntercept model s larger n absolute value (the logstc curve s steeper), as s guaranteed by theory. (Check the book!) twoway (lne ftted_ndp agec, c(l) pstyle(p15)) (lne ftted_avgp agec, sort clwdth(thck)) (lne ftted_geear1 agec, sort clwdth(thck)) 0..4.6.8 1-40 -0 0 0 40 agec ftted_ndp gee_ar1 ftted_avgp Transton model Basc dea: nclude the outcome varable at a prevous tme pont as a fxed effect covarate. We condton the response at tme j on the response at tme j-1 or j-, etc Generate a varable that represents breastfeedng status durng the prevous month. We often call ths lag 1 breastfeedng status.. sort d vst. by d: gen bfbn_lag1 = bfbn[_n-1] (199 mssng values generated) A very smple transton model 9
We ll assume condtonal ndependence once we condton on the outcome at the prevous month (frst order Markov chan) ( y = 1 U ) logt P j = β + β agecj + β y( j 1) 0 1. logt bfbn agec bfbn_lag1 Logstc regresson Number of obs = 748 LR ch() = 75.19 Prob > ch = 0.0000 Log lkelhood = -133.01167 Pseudo R = 0.7316 bfbn Coef. Std. Err. z P> z [95% Conf. Interval] agec -.0976751.01661-5.88 0.000 -.1303 -.065101 bfbn_lag1 4.6351.5610318 8.4 0.000 3.53649 5.7853 _cons -3.899484.50714-7.69 0.000-4.89349 -.905539. logstc bfbn agec bfbn_lag1 Logstc regresson Number of obs = 748 LR ch() = 75.19 Prob > ch = 0.0000 Log lkelhood = -133.01167 Pseudo R = 0.7316 bfbn Odds Rato Std. Err. z P> z [95% Conf. Interval] agec.9069436.0150643-5.88 0.000.8778935.936955 bfbn_lag1 101.845 57.1679 8.4 0.000 33.90793 305.776 Interpretaton of OR on agec (populaton average) Comparng chldren of the same breastfeedng status at the prevous month, a one month ncrease n age s assocated wth a 10% decrease n the odds of breastfeedng. Interpretaton of OR on bfbn_lag1 (populaton average) Comparng chldren of the same age, those chldren who were breastfeedng at the prevous month have an odds rato of breastfeedng that s 101 tmes greater than the odds rato of breastfeedng for those chldren who were not breastfeedng at the prevous month. Huge!! Note that the 95% CI for the lag 1 bfbn varable does not nclude 1, so we should keep ths varable n our model. Less smple transton model We ll allow for a correlaton structure on the responses even after we condton on the outcome at the prevous month 10
. xtgee bfbn agec bfbn_lag1, nolog f(bn) l(logt) corr(unst) robust GEE populaton-averaged model Number of obs = 748 Group and tme vars: d vst Number of groups = 194 Lnk: logt Obs per group: mn = 1 Famly: bnomal avg = 3.9 Correlaton: unstructured max = 4 Wald ch() = 16.98 Scale parameter: 1 Prob > ch = 0.0000 (Std. Err. adjusted for clusterng on d) Sem-robust bfbn Coef. Std. Err. z P> z [95% Conf. Interval] agec -.0919938.01988-4.63 0.000 -.1309579 -.053097 bfbn_lag1 5.050811.714671 7.09 0.000 3.654401 6.4471 _cons -4.18535.644153-6.70 0.000-5.40918 -.961519. xtcorr Estmated wthn-d correlaton matrx R: c1 c c3 c4 r1 1.0000 r -0.061 1.0000 r3-0.095-0.0990 1.0000 r4 0.0847 0.0671-0.1096 1.0000 The workng correlaton matrx estmates correlatons that are all relatvely close to zero, We ll keep the unstructured correlaton wth robust as our fnal model.. xtgee bfbn agec male agemale bfbn_lag1, nolog f(bn) l(logt) corr(unst) robust eform GEE populaton-averaged model Number of obs = 748 Group and tme vars: d vst Number of groups = 194 Lnk: logt Obs per group: mn = 1 Famly: bnomal avg = 3.9 Correlaton: unstructured max = 4 Wald ch(4) = 13.63 Scale parameter: 1 Prob > ch = 0.0000 (Std. Err. adjusted for clusterng on d) Sem-robust bfbn Odds Rato Std. Err. z P> z [95% Conf. Interval] agec.90733.06031-3.37 0.001.857307.9600718 male.800136.4475089-0.40 0.690.673589.394601 agemale 1.01493.0366755 0.41 0.68.9455337 1.089419 bfbn_lag1 181.111 131.048 7.19 0.000 43.857 747.911. xtcorr Estmated wthn-d correlaton matrx R: c1 c c3 c4 r1 1.0000 11
r -0.031 1.0000 r3-0.1078-0.1107 1.0000 r4 0.0936 0.0636-0.166 1.0000 Test for a gender effect. test male agemale ( 1) male = 0 ( ) agemale = 0 ch( ) = 1.58 Prob > ch = 0.4538 We stll do not see an effect of gender on the breastfeedng status of the Nepalese chldren. Fnal note on the transtons observed n breastfeedng status:. xttrans bfbn bfbn bfbn 0 1 Total -----------+----------------------+---------- 0 99.06 0.94 100.00 1 13.93 86.07 100.00 -----------+----------------------+---------- Total 6.30 37.70 100.00 Maybe our model stll doesn t adequately represent the data. There are a class of models called mover-stayer models developed n the 1950 s that attempt to represent data of ths sort (we don t cover these models n ths class). You can fnd tons of nformaton on these models wth a Google search. 1