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Mixed Logit Model on Multivariate Binary Response using Maximum Likelihood Estimator and Generalized Estimating Equations



Jaka Nugraha
 
ABSTRACT

This study presents discussion on the effects of correlation among response respect to estimator properties in mixed logit model on multivariate binary response. It is assumed that each respondent was observed for T response. Yit is the tth response for the ith individual/subject and each response is binary. Each subject has covariate Xi (individual characteristic) and covariate Zijt (characteristic of alternative j). Individual response i that is represented by Yi = (Yi1,....,YiT), Yit is tnd response on ith individual/subject and the response is binary. In order to simplify, one of individual characteristic was and alternative characteristics. We studied effects of correlations using data simulation. Methods of estimations used in this study are Generalized Estimating Equations (GEE) and Maximum Likelihood Estimator (MLE). We generate data and estimate parameters using software R.2.10. From simulation data, we conclude that MLE on mixed logit model is better than GEE. The higher correlation among utility, the higher deviation estimator to parameter.

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Jaka Nugraha , 2011. Mixed Logit Model on Multivariate Binary Response using Maximum Likelihood Estimator and Generalized Estimating Equations. Asian Journal of Mathematics & Statistics, 4: 113-127.

DOI: 10.3923/ajms.2011.113.127

URL: https://scialert.net/abstract/?doi=ajms.2011.113.127
 
Received: February 27, 2011; Accepted: April 14, 2011; Published: May 30, 2011

INTRODUCTION

Discussion on binary response modeling was reported by Sarkar and Midi (2010), Zadkarami (2008), Sarkar et al. (2011) and the models usually adopted probit model and logit model (Alpu and Fidan, 2004; Damisa et al., 2007). Generally for panel data observations towards same subjects and variables are conducted repeatedly. The model was constructed based on approach of fixed effect, random and dynamic models (Lechner et al., 2005). Simulation study of an efficient estimation of a mean true effect using a fixed effects model and a random effects model in order to establish appropriate confidence interval estimation have been reported (Antoine et al., 2007). Method for estimating parameter on panel data modeling used was Maximum Likelihood Estimating (MLE), moment method and Generalized Estimating Equation (GEE) (Diggle et al., 1996). The simplest model is independent model which the assumption taken based on the response on the same and inter subjects is independent. Based on this assumption, combined probability is result of multiplication of its marginal probability. Liang and Zeger (1986) suggest that logistic and probit analysis that are ignoring the correlation and produce consistence estimation parameter but still have close relation becomes inefficient. Chaganty and Joe (2004) have proposed the use of equations of the correlation estimator for probit models in GEE.

In many cases data are multivariate or correlated (e.g., due to repeated observations on a study subject or for subjects within centers) and it is appealing to have a model that maintains a marginal logistic regression interpretation for the individual outcomes. An alternative is to use a marginal analysis that avoids complete specification of the likelihood (Lipsitz et al., 1990). Prentice (1988) demonstrate modeling strategic to obtain a consistent and normal asymptotic regression coefficient estimator by using GEE. Double regression was not needed in this calculation.

Development of model by controlling unobserved and homogeneous individual characteristic towards replication was reported. However, for heterogeneous individual characteristic, problem would be found is that estimation parameter becomes bias (Greene, 2005). Double integral was needed in probit model. From several simulations, Hajivassiliou et al. (1996) reported that Geweke-Hajivasilou-Keane (GHK) was the best simulation. Geweke et al. (1997) also find double integral calculation by using Monte Carlo simulation known as unbias and consistence GHK.

Eventually, several different independent variables are observed simultaneously. This observation produces multivariate data. Nugraha (2000) examined properties of parameter estimators on bivariate logistic regression by using MLE and GEE method. These method produce a consistence estimator. Nugraha et al. (2006) reported that for logistic model on multivariate binary, smaller variance estimator gained by GEE and compared to independent assumption approach. Nugraha et al. (2009a) evaluated the comparison of GEE and MLE method for multivariate binary logistic. It was concluded that GEE model has smaller bias. However, the estimator for correlation parameter is always underestimate. Akanda et al. (2005) proposed an alternative procedure for goodness of fit test based on GEE where, the correlation between two responses was considered.

Nugraha et al. (2008) presented that analytically, both first and second derivation of log-likelihood binary multivariate model can be calculated. Based on its first derivation and second derivation, estimation on the parameter can be executed by Newton-Raphson iteration procedure. Nugraha et al. (2009b) obtained the estimator of binary multivariate probit model by Simulated Maximum Likelihood Estimator (SMLE) based on GHK simulation.

During the last decade, practitioners and researchers are willing to move from the multinomial and nested logit models, that were the standard until only a few years ago, towards these more general models, specifically towards Mixed Logit (Munizaga and Alvarez-Daziano, 2004). In this present study, correlation effect on mixed logit model towards binary multivariate data is discussed. Estimation of parameter is conducted by MLE and GEE method and comparative study for simulation data is subjected. Both simulation data and estimation of parameter were performed utilizing R.2.10.0. program. (www.R-project.org).

MULTIVARIATE BINARY RESPONSE

It is assumed that Yit is binary response, Yit = 1 as the subject i at the response of t choosing the alternative 1 and Yit = 0 if the subject of i at the response of t choosing the alternative of 2. Each individual has covariate Xi as individual characteristics i and covariate Zijt as characteristic of choice/alternative j at the individu of i.

Utility of subject i selecting the alternative of j on response t is:

(1)

with:

Vijt = αjtjt XitZijt

By assuming that decision maker select the alternative based on the maximum utility value, the model can be expressed in the form of difference utility:

(2)
with:
Vit = (Vi1t–Vi0t) = (α1t0t)+(β1t0t) Xit (Zi1t-Zi0t) = αtt XitZit

and εit = (εiti0t). The probability of subject of i selecting (yi1 = 1, ...., yiT = 1) is:

(3)

With εi = (εi1, ..., εiT)'. Value of probability is calculated by multiple integral T depend on the parameter θ = (θ1', θ2', ..., θT') as well as distribution of ε (Train, 2003).

GEE ON MODEL LOGIT

GEE on multivariate binary response can be expressed in the following form :

(4)

Wi is called as observation matrix:

and Δi = diag (πil (1-πil) ... πiT (1-πiT)) Si is matrix TxT and can be expressed as:

It is defined that and Ri is “working” matrix with the correlation Yi in the size of TxT.

Paired correlation coefficient between response of s and response of r, ρisr= corr (Yis, Yir) for i = 1, 2, ..n and s, r = 1, 2, …T. In order to estimate Ri, empirical correlation, ri is defined as vector with the size of T (T-1)/2 with the elements:

Empirical correlation rist that is contained parameter πis and πit is unbias estimator for ρist (Liang and Zeger, 1986). As it is assumed that ρist = ρst for all of i, the estimator is:

(5)

Equation 4 and 5 can be solved at the same time to get the parameter estimator θ and ρ.

Theorem 1: If is GEE estimator that is solution of Eq. 4, so is normal multivariate distributed:

 

where,

MIXED LOGIT MODEL

Mixed logit model is combination of extreme value distribution and normal distribution assumption. Based on equation of utility difference (2), the individual effect δit is added:

(6)

εit has extreme value type I distribution. δit is ith individual effect of tth response and is independent respect to εit. It is assumed that δi = (δi1, ...., δiT) has multivariate normal, δit ~ N (0, 1) and δi~N (0, Σ). Parameter that will be estimated on this mixed Logit model are the parameter of regression coefficient (αt, βt, γt) and Σ. Parameter Σ are estimated by the Cholesky factor, C.

Individual effect δi can be can be modified in the following form:

(7)

where, C is matrix of Cholesky factor of Σ:

CC’ = Σ and ξi ~ N (0, 1)

I is identity matrix with the size of TxT. Generally, for t = 1, …, T, the utility function can be constructed:

(8)

where,

extreme value type I, ctl is an element of cholesky matrix. Based on Eq. 8 conditional logit model is derived with expression:

(9)

Marginal probability (for t and i) in the mixed logit model is:

(10)

φ (ξit) is standard normal density.

The joint conditional probability is:

(11)

where, c = (c11, c12, c13, c22, c23, c33) and ξiil, ..., ξiT). The joint probability is:

(12)

φ (ξi) is normal standard density.

Theorem 2: If it is known the utility model as represented in Eq. 8 that is fit to regularity condition, so MLE for parameter of θt = (αtt, γt)' and is the solution of estimator equation:

(a)
(b)

where, l = 1, ..., T. with:

and can be obtained by using iteration process of methods BHHH ((Berndt, Hall, Hall, Hausman) or BFGS (Broyden, Fletcher, Goldfarb dan Shanno) (Chong and Zak, 1996).

SIMULATION DATA

Simulation data was simulated by taking case T = 3 and n = 1000. Utility equation is:

Ui0t = α0t0t Xit Zi0ti01 dan Ui1t = α1t1t Xit Zi1ti1t
(13)

For i = 1, ..., n; t = 1, 2, 3 and j = 0, 1. Equation of utility (13) can be transformed in the difference of utility: Uit = Ui1t–Ui0t and the corresponding model is:

Ui1 = Vi1i1; Ui2 = Vi2i2; Ui3 = Vi3i3

With Vit = (Vi1t–Vi0t) = αtt Xit Zit; Zit = (Zi1t –Zi0t); βt = β0t1t; αt = (α0t1t). Data was generated on parameter value of αt = -1, βt = 0.5 and γt = 0.3. Structure of correlation/covariances to be examined are r12 = ρ and r13 = r32 = 0:

Utility for t = 1 is in correlation with the utility for t = 2 and the correlation order ρ = 0, 0.1, 0.2,..., 0.9. The value of observation variable Xi and Zijt taken from normal distribution is:

Generated data with the assumption of εijt is extreme value distributed and δi~N (0, σ2). Estimation towards parameter was conducted by utilizing GEE and MLE for mixed logit model.

In order to estimate parameter using MLE, double integration was required and for this purpose, Halton sequence can be implemented. Simulation program for Halton can be adopted from library randtoolbox. For optimizing log-likelihood function, optim function by available BFGS method in R.2.10 program was used (R Development Core Team, 2009).

RESULT OF SIMULATION

Figure 1-21 depicted the effect of correlation value to the each estimator bias. From results, it can be concluded that if assumption by using GEE does not appropriate, the utilities that are correlated each other will be bias. Utility 1 (Ui1) is in correlation with utility 2 (Ui2) and both are not correlated to utility 3 (Ui3). Therefore, value of correlation only influences on parameters within Ui1 and Ui2. On both utilities, the estimator bias appreciable and comparable respect to the correlation values (Fig. 1-9).

Estimation by using MLE is more accurate compared to as in using GEE. At bigger correlation between utilities, estimator on GEE will has higher deviation (Fig. 1-9) and this is caused misassumption. Model Logit model estimated by using GEE is based on the assumption that error has variance of π2/3 while for data, the variance is (σ22/3). As the assumption fulfilled, Mixed Logit model can estimate all parameter appropriately (Fig. 11-20)
Correlation parameter estimator for MLE is better (smallet bias) compared to as or GEE (Fig. 20, 21) more over, Correlation parameter estimator for GEE tends to be underestimate (Fig. 10)
In low correlation (less than 0.5), MLE is robust respect to assumption deviation and can estimate precisely. Deviation of assumption in GEE method produce in bias estimator

Fig. 1: Box plot α1 (t = 1) in GEE

Fig. 2: Box plot α2 (t = 2) in GEE

Fig. 3: Box plot α3 (t = 3) in GEE

Fig. 4: Box plot β1 (t = 1) in GEE

Fig. 5: Box plot β2 (t = 2) in GEE

Fig. 6: Box plot β3 (t = 3) in GEE

Fig. 7: Box plot γ1 (t = 1) in GEE

Fig. 8: Box plot γ2 (t = 2) in GEE

Fig. 9: Box plot γ3 (t = 3) in GEE

Fig. 10: Box plot ρ in GEE

Fig. 11: Box plot α1 (t = 1) in MLE

Fig. 12: Box plot α2 (t = 2) in MLE

Fig. 13: Box plot α3 (t = 3) in MLE

Fig. 14: Box plot β1 (t = 1) in MLE

Fig. 15: Box plot β2 (t = 2) in MLE

Fig. 16: Box plot β3 (t = 3) in MLE

Fig. 17: Box plot γ1 (t = 1) in MLE

Fig. 18: Box plot γ2 (t = 2) in MLE

Fig. 19: Box plot γ3 (t = 3) in MLE

Fig. 20: Box plot ρ in MLE

Fig. 21: Graph of a bias ρ in GEE and MLE

CASE STUDY

Data are obtained from library Ecdat in R.2.10.0 which are data Heating. Data are resulted from survey of room heater household in California. From 850 families, there are several variables observed. Yi is heater system utilized, that are gc (gas central), gr (gas room), ec (electric central) and er (electric room). X1i is income of each family (income). X2i is age of familyhed (Agehed). X3i is amount of room in house (rooms) where, I = 1, ..., 850. Furthermore, multivariate approach with two responses was utilized. Yi1 is source of energy (g = 1) and electricity (e = 0). Yi2 is application type, central controled (c = 1) and room controled (r = 0). Characteristic variable for each response is defined in Table 1. From those data, tabulated data between energy source (Y1) and application type (Y2) (Table 2).

Based in Chi-squared Pearson’s test, obtained value χ2 = 95.8792 (p-value<2.2e-16), means that energy source choosen by respondents and application type are not independent.

If there is correlation among responses, parameter estimator will be bias. On this condition, Mixed Logit model is more accurate in estimating parameter compared with as in discrete choice model. For three approximations used; univariate, GEE, mixed logit model, instalation cost and operational cost are the significant variables influence to the Y variable. In contrast, effect of individual charactersitics of X1i, X2i and X3i to the response is not significant.

Equation of marginal probability on independent model and GEE for energy source is as follow:

and the probability equation for application type is:

Estimator value for each parameter is presented in Table 3.

Table 1: Characteristic variabel of election

Table 2: Crossed tabulation of energy source and application type

Table 3: Estimator for three models
Values in blanket is statistic Wald

Equation of marginal probability for heater system on Logit model is:

with:

ξi has normal distribution with density φ (ξi) and:

Equation of marginal probability for the variable of application type is:

With:

Equation of combined probability on mixed logit model is:

If Yi1 = g, so yi1 = 1 and if Yi1 = e so yi1 = 0. As Yi2 = c, yi2 = 1 and as Yi2 = r yi2 = 0.

By using mixed logit model, correlation value between energy source and application source is 0.628 otherwise by using GEE correlation value is 0.313. From simulation result, it is noted that corelation estimator for GEE tend to be underestimate, therefore, estimator correlation on Mixed Logit model is more trusted.

CONCLUSION

Mixed Logit model is based on assumption that error component has extreme value distribution and parameter has normal distribution. Based on data simulation, if there is a violation to the assumption (specifically related to variance) GEE estimator will be bias. Estimator bias will be higher at higher variance value. Estimator bias in GEE can be reduced by MLE method. The correlation of parameter estimator in GEE tends to be underestimate but MLE can be used for estimating correlation between parameters accurately.

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