Longitudinal and Multilevel Methods for Multinomial Logit David K. Guilkey

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1 Longitudinal and Multilevel Methods for Multinomial Logit David K. Guilkey

2 Focus of this talk: Unordered categorical dependent variables Models will be logit based Empirical example uses data from the Indonesian Family Life Survey: Two outcomes: Binary indicator for whether the respondent uses contraception Unordered categorical variable for method choice

3 Data Set Overview Four waves of data: 1993, 1997, 2000, and 2007 Individual level information on fertility, education, migration Community and facility level data on health and family planning providers Data from 321 enumeration areas we will consider these communities IFLS Longitudinal Sample Size Initial Participation Cohort Survey Year total Wave 1 Cohort Wave 2 Cohort Wave 3 Cohort Wave 4 Cohort total observations 20362

4 IFLS Summary Statistics mean s.d. Dependent Variables Contraceptive Use Method Choice no method.412 temporary modern.397 long Lasting modern.168 traditional.023 Independent Variables highest ed grade school highest ed high school highest ed college age muslim number of posyandus Observations 20,000

5 Basic Model for Longitudinal Logit: Where: PY ( = 1 µ ) ti i ln = X β + α P + Zδ + ρµ ti ti i i PY ( = 0 µ ) ti i Y ti : observed binary variable (respondent i from time period t) X ti : time varying explanatory variables (age and education level) P ti : time varying program variable (posyandus) Z i : time invariant regressors (Muslim) i=1,2, N (individuals) t=1,2, T i (observations per individual -- unbalanced panel)

6 Assumptions: µ i ~ N(0,1) for the parametric logit in STATA (xtlogit, melogit, and one variant of GLLAMM) and: E( Xtiµ i) = E( Ptiµ i) = E( Ziµ i) = 0 Note that observations for the same individual will be correlated because of the time invariant error sometimes referred to as unobserved heterogeneity Given the assumptions, estimation options are: 1. Simple logit yields consistent point estimates but incorrect SE s 2. Simple logit with cluster option corrects SE s 3. Parametric or semi-parametric maximum likelihood allows estimation of random effects parameters and typically more efficient

7 The likelihood function for this model is derived as follows: e PY ( = 1 µ ) = 1 + e X β + α P + Zδ + ρµ ti ti i i ti i X β + α P + Zδ + ρµ ti ti i i This is the probability that individual i at time t is using contraception conditional on time invariant heterogeneity. For individual i, we observe T i binary responses that we can write as: Y i = (1,0,0,1) for a woman that is observed for 4 time periods and used contraception at time periods 1 and 4.

8 Let Y i be the set of observed outcomes for individual i, then: Joint probability must be approximated -- approximating the area under a curve. With the assumption of normality the approximation method is Gaussian Quadrature or Hermite integration Points: Ti Yti Yti 1 ( ) = ( 1 ) (1 ( 1 ) ( ) i ti i ti i i i t = = = 1 PY PY µ PY µ f µ dµ 1. More accurate with more Hermite points but execution time is longer. Adaptive quadrature is an option in GLLAMM 2. You need more points as T i gets larger.

9 Hermite integration replaces the integral with a sum: M Ti Yti Yti PY ( ) = w PY ( = 1 µ ) (1 PY ( = 1 µ ) i m= 1 m t= 1 ti m ti m 1 where the weights (w m s) and the masspoints (μ m s) are known because of the assumption of normality Alternative: The discrete factor approximation searches over weights and mass points along with the other parameters of the model. Must impose a normalization; 1. Weights sum to one 2. Either set one mass point to zero (fortran program) or set mean of distribution to zero (GLLAMM)

10 Multilevel Panel Models Basic Form of the model: ln PY ( = 1 µ, λ ) tij ij j PY ( = 0 µ, λ ) tij ij j = X β+ αp + Z δ+ ρµ + ρλ tij tij j 1 ij 2 j where j=1,2,,j (communities) i=1,2,,n j (individuals from community j) t=1,2,,t ij (observations for person i for community j)

11 X tij : individual level variables (some could be fixed through time) P tij : time varying program variable Z j : time invariant community level variables μ ij : time invariant individual level unobserved heterogeneity λ j : time invariant community level unobserved heterogeneity This model allows observations on the same individual to be correlated and observations from the same community to be correlated.

12 Assumptions: EX ( λ ) = EP ( λ ) = EZ ( λ ) = 0 tij j tij j ij j EX ( µ ) = EP ( µ ) = EZ ( µ ) = 0 tij ij tij ij ij ij 1. Simple logit yields consistent point estimates but incorrect SE s 2. Simple logit with cluster option corrects SE s (at community level) 3. Parametric or semi-parametric maximum likelihood -- estimates random effects and more efficient Multilevel Maximum likelihood estimator is a straight forward extension of the basic longitudinal data model: e PY ( = 1 µ, λ ) = 1 + e X β+ αp + Z δ+ ρµ + ρλ tij tij j 1 ij 2 j tij ij j X β+ αp + Z δ+ ρµ + ρλ tij tij j 1 ij 2 j

13 You need the unconditional joint probability of the observed set of outcomes for the set of individuals in each community: Conditional on the unobservables at the community level, the probability of the set of observed outcomes for person i from community j are: Tij Ytij Ytij 1 ( ) = ( 1, ) (1 ( 1, ) ( ) ij j tij ij j tij ij j ij ij t = = = 1 PY λ PY µ λ PY µ λ f µ dµ The unconditional joint probability of the set of observed outcomes for all individuals in community j is then: N i PY ( ) PY ( λ ) f( λ ) dλ = = ij tij j j j i 1 We then either use Hermite integration or the discrete factor method to approximate the integral.

14 Parametric Maximum Likelihood gllamm cont_use posyandus age grade_school high_school college muslim, i(ind_id com_id) family(binomial) link(logit) nip(20) ip(g) trace dot number of level 1 units = number of level 2 units = 9394 number of level 3 units = 313 gllamm model log likelihood = cont_use Coef. Std. Err. z P> z [95% Conf. Interval] posyandus age grade_school high_school college muslim _cons Variances and covariances of random effects ***level 2 (ind_id) var(1): ( ) ***level 3 (com_id) var(1): ( )

15 Non-parametric Maximum Likelihood gllamm cont_use posyandus age grade_school high_school college muslim, i(ind_id com_id) family(binomial) link(logit) nip(3) ip(f) trace dot number of level 1 units = number of level 2 units = 9394 number of level 3 units = 313 gllamm model log likelihood = cont_use Coef. Std. Err. z P> z [95% Conf. Interval] posyandus age grade_school high_school college muslim _cons Probabilities and locations of random effects ***level 2 (ind_id) loc1: , , var(1): prob: 0.295, , ***level 3 (com_id) loc1: ,.65457, var(1): prob: , ,

16 Basic Multinomial Logit with 4 Choices: U = X β + α P + Zδ + ε 1ti ti 1 1 ti i 1 1ti U = X β + α P + Zδ + ε 2ti ti 2 2 ti i 2 2ti U = X β + α P + Zδ + ε 3ti ti 3 3 ti i 3 3ti U = X β + α P + Zδ + ε 4ti ti 4 4 ti i 4 4ti Individual i at time t time makes choice 3 (for example) if : P( U > U and U > U and U > U ) 3ti 1ti 3ti 2ti 3ti 4ti If we assume that the ε s follow independent extreme value distributions and impose the restriction that:

17 β = α = δ = So that the probabilities sum to one then: PY ( = k) ti i ln PY ( = 1) ti for k=2,3,4. = X β + α P + Zδ ti k k ti i k Let: A = + e β α δ ti 1 4 X ti m m P ti Z + + i m m= 2 Then: PY ( = 1) = ti 1 A ti

18 And: PY ( = k) = ti e X β + α P + Zδ ti k k ti i k A ti Which allows us to write the likelihood function N Ti 4 L= PY = m ti i= 1 t= 1 m= 1 ( ) Btim where B = 1if Y = m and B = 0otherwise tim ti tim Maximize with respect to parameters to find MLE

19 Independence of Irrelevant Alternatives Problem (IIA): PY ( ri = 2) PY ( = 3) Red Bus/Blue Bus Problem ti = e e X β + α P + Zδ ti 2 2 ti i 2 X β + α P + Zδ ti 3 3 ti i 3 Start with red bus and walking and calculate odds Now add blue bus that is exactly like red bus and irrelevant to walking Because of IIA, the probability of walking falls even though the blue bus alternative is irrelevant but perfectly correlated with read bus which multinomial logit model does not allow

20 Extension of longitudinal logit to longitudinal multinomial logit: PY ( = k µ ) ti i ln PY ( = 1 µ ) ti i for k=2,3,4. = X β + αp + Zδ + ρµ ti k k ti i k k i The discrete factor model allows a more general pattern of correlation: ln PY ( = k µ ) ti km PY ( = 1 µ ) ti km = X β + α P + Zδ + µ ti k k ti i k km for m=1,2,m and a common set of weights: allows for correlation in the μ s w m

21 Unfortunately, base version of GLLAMM estimates a needlessly restrictive version of the model: Parametric: ρ = ρ 2 3 If there are more than 3 choices, all ρ s are restricted Non-parametric: µ = µ 2m 3m for all m.

22 This restriction means that IIA is not solved: PY ( = 2) e e ri = = PY ( = 3) e e ti X β + α P + Zδ + ρµ X β + α P + Zδ ti 2 2 ti i 2 i ti 2 2 ti i 2 X β + α P + Zδ + ρµ X β + α P + Zδ ti 3 3 ti i 3 i ti 3 3 ti i 3 Since the unobserved heterogeneity drops out.

23 Extension to Multilevel Panel Model: Parametric: ln PY ( = k µ, λ ) tij ij j PY ( = 1 µ, λ ) tij ij j = X β + α P + Z δ + ρ µ + ρ λ tij k k tij j k 1k ij 2k j Semi-parametric: ln PY ( = k µ, λ ) tij km kn = X β tij k + α P Z k tij + δ j k + µ λ km + kn PY ( = 1 µ, λ ) ti km kn

24 The empirical example estimates a model with four choices: 1= Non use 2=Temporary Methods (pill, condom, injection) 3=Long Lasting Methods (IUD, sterilization) 4=Traditional Methods We show the GLLAMM code to estimate various versions of the model and some selected results: Multinomial logit with SE s corrected at community (outermost) level: mlogit new_method posyandus age grade_school high_school college muslim, cluster(com_id) base(1)

25 Restricted model for longitudinal data and random effect at individual level: Normal distribution: gllamm new_method posyandus muslim, i(ind_id) family(binomial) link(mlogit) b(1) nip(16) ip(g) trace dot Discrete Factor: gllamm new_method posyandus muslim, i(ind_id) family(binomial) link(mlogit) b(1) nip(4) ip(f) trace dot Restricted model for multilevel data and random effects at individual level and community level: Normal distribution: gllamm new_method posyandus muslim, i(ind_id com_id) family(binomial) link(mlogit) b(1) nip(16) ip(g) trace dot

26 Discrete Factor: gllamm new_method posyandus muslim, i(ind_id com_id) family(binomial) link(mlogit) b(1) nip(4) ip(f) trace dot To estimate the less restrictive version of the model, you must expand the data set (as is done in conditional logit)-- with 4 choices each original observation expands to 4 observations: sort ind_id year gen tid = _n expand 4 by tid, sort: gen alt = _n gen choice = (new_method == alt) tab alt, gen(a) eq a2: a2 eq a3: a3 eq a4: a4 eq a5: a2 eq a6: a3 eq a7: a4

27 Less restrictive models for longitudinal and multilevel models (just for normal distribution): Longitudinal: gllamm alt posyandus college muslim, i(ind_id) family(binomial) nrf(3) eq(a2 a3 a4) link(mlogit) expand(tid choice m) b(1) nip(8) ip(g) trace dot Multilevel (use 4 points as starting values for 8 points): gllamm alt posyandus.. muslim, i(ind_id com_id) family(binomial) nrf(3,3) eq(a2 a3 a4 a5 a6 a7) link(mlogit) expand(tid choice m) b(1) nip(4) ip(g) trace dot matrix a=e(b) gllamm alt posyandus muslim, i(ind_id com_id) family(binomial) nrf(3,3) eq(a2 a3 a4 a5 a6 a7) link(mlogit) expand(tid choice m) b(1) nip(8) ip(g) trace dot from(a)

28 Complete results for Longitudinal Multinomial Logit with Normal Unobserved Heterogeneity:

29 alt Coef. Std. Err. z P> z [95% Conf. Interval] c2 posyandus age grade_school high_school college muslim _cons c3 posyandus age grade_school high_school college muslim _cons c4 posyandus age grade_school high_school college muslim _cons Variances and covariances of random effects ***level 2 (ind_id) var(1): ( ) cov(2,1): ( ) cor(2,1): var(2): ( ) cov(3,1): ( ) cor(3,1): cov(3,2): ( ) cor(3,2): var(3): ( )

30 Comparison of Normal Distribution Results with Discrete Factor Results Less Restrictive Multilevel Discrete Factor with 4 points of support Normal Distribution Discrete Factor Variable Coefficient SE Coefficient SE Temporary Modern vs none Temporary Modern vs none Posyandus Age Heterogeneity Com_ Com_ Com_ Ind_ Ind_ Ind_ Long Lasting vs none Long Lasting vs none Posyandus Age Heterogeneity Com_ Com_ Com_ Ind_ Ind_ Ind_3 Traditional vs none Traditional vs none Posyandus Age Heterogeneity Com_ Com_ Com_ Ind_ Ind_ Ind_

31 Probability Weights for Discrete Factor Model Com_ Ind_ Com_ Ind_ Com_ Ind_ Com_ Ind_

32 Summary IIA is not much of an issue at least in this example the basic multinomial logit got almost the same point estimates as the least restrictive random effects model More complex models that do not suffer from IIA make a lot of difference in conditional logit model (see mixlogit application) Estimation of models with multilevels using Gaussian Quadrature is extremely computer intensive when you use the preferred less restrictive models Adaptive Quadrature sounds good in theory but frequently fails to converge The Discrete Factor Model in GLLAMM is easy to use, does not require the normal error heterogeneity assumption, and is much faster computationally Under some circumstances, GLLAMM ran into problems taking derivatives either with Gaussian Quadrature or the Discrete Factor model This was never an issue in fortran which was also light years faster

33 Competing Risk Model Computer software for the discrete time competing risk model with multiple levels is the same as the multilevel multinomial logit Can be more complicated if a multiple spell model Example: Trajectories of Unintended Fertility (Rajan, Morgan, Harris, Guilkey, Hayford, Guzzo) Competing risks were the time until unintended versus intended birth (no birth is the base outcome for the 3 unordered categories) Multiple spell (through three births) but we focus on single spell here

34 Review of discrete time hazard: The variable of interest is: P(t T < t+n T > t) This is the conditional probability that an individual experiences the event between t and t+n given that she has not experienced the event until that time. Example: The dependent variable is the timing of a first birth. Suppose the discrete time interval is a year and we observe each woman from the beginning of her child bearing years: Consider three cases: Person 1: Has a birth in year 1 (time 0 may be age 12) Person 2: Has a birth in year 2 Person 3: Still has not had a birth at the end of the observation period

35 Some important notes: 1. Since we are following the woman from the beginning of her child bearing years, we have eliminated the possibility of left censoring (the event occurs before the observation period). 2. Left censoring combined with unobserved heterogeneity introduces bias into the estimation results. The correction requires the estimation of an initial conditions equation similar to Heckman selection equation which are well known to yield unstable parameter estimates. 3. The third person is right censored. However, right censoring is easily handled as part of the estimation process. 4. The dependent variable in a discrete time hazard model is dichotomous. Logit typically used which can be extended to multinomial logit for competing risk.

36 Difficult to distinguish duration dependence from time variant unobserved heterogeneity so must allow for both and then test Duration dependence This is a concept similar to state dependence in a standard panel data model. Duration dependence occurs when the value of the hazard at any point in time depends on the amount of time that has already elapsed. Examples: Mortality hazard increases with time regardless of the values of the other covariates Unemployment duration hazard of finding employment may decrease as the length of the unemployment spell increases Important Point:

37 Illustrate methods with simulated data so it is easy to compare estimators: Competing risks are 1 and 3 person stays in sample as long as keep drawing 2 Number of communities: 200 Individuals per community: 40 Maximum time until right censoring: 20 Generate the unobserved heterogeneity from both a normal distribution and a bimodal distribution mean zero and variance one in both cases note that there is no duration dependence in the true model

38 Bimodal Distribution for community heterogeneity: Density com

39 Comparison of Normal Distribution Results with Discrete Factor Results for Normal and Bimodal Unobserved Heterogeneity Just 2 versus 1 Variable Coefficient SE Coefficient SE Data Generated using Normal Unobserved Heterogeneity Multinomial Logit Corrected SE Multinomial Logit Assume Normality and use Restricted Model X1 (2) X2 (-0.75) Time (0) Multilevel Multinomial Logit Discrete Factor 4 points Normality with Less Restrictive Model X1 (2) X2 (-0.75) Time (0) Data Generated using Bimodal Unobserved Heterogeneity Multinomial Logit Corrected SE Multinomial Logit Assume Normality and use Restricted Model X1 (2) X2 (-0.75) Time (0) Multilevel Multinomial Logit Discrete Factor 4 points Normality with Less Restrictive Model X1 (2) X2 (-0.75) Time (0)

40 Restrictive versions of the unobserved heterogeneity exhibit duration dependence when none exists GLLAMM with normal errors is extremely slow but faster than GSEM (results not shown) it is not clear how practical it is for large models with many observations A very surprising result is that GLLAMM assuming normality was robust to a very nonnormal error generating process GLLAMM with the discrete factor approximation was fast and gave results almost identical to a fortran program although it was still much slower than fortran GLLAMM with the discrete factor approximation was also robust to the true error distribution process this is not surprising given the Monte Carlo evidence that is available

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