multinomial logistic multilevel models in R

Question

Problem: I need to estimate a set of multinomial logistic multilevel models and can’t find an appropriate R package. What is the best R package to estimate such

Answer 1

There are generally two ways of fitting a multinomial models of a categorical variable with J groups: (1) Simultaneously estimating J-1 contrasts; (2) Estimating a separate logit model for each contrast.

Produce these two methods the same results? No, but the results are often similar

Which method is better? Simultaneously fitting is more precise (see below for an explanation why)

Why would someone use separate logit models then? (1) the lme4 package has no routine for simultaneously fitting multinomial models and there is no other multilevel R package that could do this. So separate logit models are presently the only practical solution if someone wants to estimate multilevel multinomial models in R. (2) As some powerful statisticians have argued (Begg and Gray, 1984; Allison, 1984, p. 46-47), separate logit models are much more flexible as they permit for the independent specification of the model equation for each contrast.

Is it legitimate to use separate logit models? Yes, with some disclaimers. This method is called the “Begg and Gray Approximation”. Begg and Gray (1984, p. 16) showed that this “individualized method is highly efficient”. However, there is some efficiency loss and the Begg and Gray Approximation produces larger standard errors (Agresti 2002, p. 274). As such, it is more difficult to obtain significant results with this method and the results can be considered conservative. This efficiency loss is smallest when the reference category is large (Begg and Gray, 1984; Agresti 2002). R packages that employ the Begg and Gray Approximation (not multilevel) include mlogitBMA (Sevcikova and Raftery, 2012).

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Why is a series of individual logit models imprecise? In my initial example we have a variable (migration) that can have three values A (no migration), B (internal migration), C (international migration). With only one predictor variable x (age), multinomial models are parameterized as a series of binomial contrasts as follows (Long and Cheng, 2004 p. 277):

Eq. 1:  Ln(Pr(B|x)/Pr(A|x)) = b0,B|A + b1,B|A (x) 
Eq. 2:  Ln(Pr(C|x)/Pr(A|x)) = b0,C|A + b1,C|A (x)
Eq. 3:  Ln(Pr(B|x)/Pr(C|x)) = b0,B|C + b1,B|C (x)

For these contrasts the following equations must hold:

Eq. 4: Ln(Pr(B|x)/Pr(A|x)) + Ln(Pr(C|x)/Pr(A|x)) = Ln(Pr(B|x)/Pr(C|x))
Eq. 5: b0,B|A + b0,C|A = b0,B|C
Eq. 6: b1,B|A + b1,C|A = b1,B|C

The problem is that these equations (Eq. 4-6) will in praxis not hold exactly because the coefficients are estimated based on slightly different samples since only cases from the two contrasting groups are used und cases from the third group are omitted. Programs that simultaneously estimate the multinomial contrasts make sure that Eq. 4-6 hold (Long and Cheng, 2004 p. 277). I don’t know exactly how this “simultaneous” model solving works – maybe someone can provide an explanation? Software that do simultaneous fitting of multilevel multinomial models include MLwiN (Steele 2013, p. 4) and STATA (xlmlogit command, Pope, 2014).

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References:

Agresti, A. (2002). Categorical data analysis (2nd ed.). Hoboken, NJ: John Wiley & Sons.

Allison, P. D. (1984). Event history analysis. Thousand Oaks, CA: Sage Publications.

Begg, C. B., & Gray, R. (1984). Calculation of polychotomous logistic regression parameters using individualized regressions. Biometrika, 71(1), 11-18.

Long, S. J., & Cheng, S. (2004). Regression models for categorical outcomes. In M. Hardy & A. Bryman (Eds.), Handbook of data analysis (pp. 258-285). London: SAGE Publications, Ltd.

Pope, R. (2014). In the spotlight: Meet Stata's new xlmlogit command. Stata News, 29(2), 2-3.

Sevcikova, H., & Raftery, A. (2012). Estimation of multinomial logit model using the Begg & Gray approximation.

Steele, F. (2013). Module 10: Single-level and multilevel models for nominal responses concepts. Bristol, U.K,: Centre for Multilevel Modelling.