1 Generalized Linear Latent and Mixed Models with Composite Links and Exploded Likelihoods 1 2 Anders Skrondal and Sophia Rabe-Hesketh 1 [email protected] Norwegian Institute of Public Health, Oslo ( ) 2 University of California, Berkeley Abstract: Applications of composite links and exploded likelihoods f or general- ized linear latent and mixed models are explored. Keywords: Generalized linear latent and mixed models; Composite link ; Ex- ploded likelihood. 1 Introduction Instead of linking the expectation of each observation with a single linear predictor as in generalized linear models, it is often usefu l to link it with a composite function of several linear predictors. Moreover , each likelihood ms. contribution can sometimes be exploded into a product of ter Linear We explore how these tools can be used to extend ‘Generalized l and Latent And Mixed Models’ or GLLAMMs (Rabe-Hesketh, Skronda Pickles, 2004a; Skrondal and Rabe-Hesketh, 2004). Applica tions consid- dels for ordinal ered include discrete time frailty models, item response mo items, unfolding models for attitudes, small area estimati on with census in- inuous latent formation, measurement models combining discrete and cont variables, ability testing with guessing, sensitivity ana lysis of the assump- tion of normal random effects, and zero-inflated Poisson mode ls. 2 Generalized Linear Models y be the response and Let explanatory variables for unit i , and define x i i the conditional expectation of the response given the covar iates as μ , i.e. i x E[ y μ | ≡ ]. Generalized linear models can be specified as i i i − 1 g = μ ( ν , ) i i − 1 ′ where ( g ) is an inverse link function, ν is a linear predictor and = x · β i i β are fixed effects. The specification is completed by choosing a condi- tional distribution for the responses y given the conditional expectations i μ ), from the exponential family. , f ( y μ | i i i

2 2 Generalized Linear Latent and Mixed Models with Composite Links and Exploded Likelihoods 3 Exploded likelihoods and composite links 3.1 Exploded likelihoods multivariate responses Generalized linear models can be extended to handle =1 , for each unit. The responses may be of mixed types com- , ,...,T t y it bining different links and families, for instance a Poisson d istributed count and a logistically distributed dichotomous response. Depe ndence can be or factors) in modelled by including latent variables (random effects and/ ng vectors of the linear predictors; see Section 4. Given the correspondi (which depend on the latent variables), the joint con- μ conditional means i ditional distribution of the vector of responses y is i T ∏ μ y ) = | Pr( (1) . f ) ( y μ | it t i it i t =1 We now distinguish between two types of artificial multivari ate responses where the response is univariate but individual likelihood contributions are nevertheless ‘exploded’ into product terms: Phantom responses y can in some cases be rep- A univariate response i S phantom responses y entering the likelihood (1) as if they resented by it were truly multivariate responses. Phantom responses can be used for the Luce-Plackett model fo r rankings t of successive where the likelihood contribution of a ranking is the produc multinomial logit choice probabilities among remaining al ternatives (e.g. val analysis Skrondal and Rabe-Hesketh, 2003). Another example is survi oportional based on data exploded into risk sets, for instance the Cox pr plementary hazard model implemented via Poisson regression and the com abe-Hesketh, log-log model for discrete time hazards (e.g. Skrondal and R 2004, Ch.2). y can sometimes Mutually exclusive responses A univariate response i S y be represented by one of having distri- mutually exclusive responses it f ( butions | μ ) from generalized linear models. For the case of T =2 the y it it t likelihood can be written as 1 − δ δ i i ( y Pr( | μ y | μ , ) = f ( y | μ ) f ) 1 1 i 2 i i i 2 i 1 2 i where the indicator δ picks out the appropriate component. i A simple example is a log-normal survival model with right-c ensoring. Let ′ β be the linear predictor, x the log survival time if the event is observed y i 1 i i ( δ for = 0) and y = δ the censoring time if the event is censored ( 2 i i i 1). The likelihood contribution then becomes either a norma l distribution ′ 2 with identity link and linear predictor x β , f )= ( y ,σ μ | μ ; y ), or a φ ( 1 i 1 i i 1 1 i i ′ ar predictor x β , Bernoulli distribution with a (scaled) probit link and line i ′ x β − y 2 i i f ) is the cumulative standard normal ) = Φ( ( y · ). Here, Φ( | μ 2 2 i 2 i σ is treated as an offset. − y distribution and i 2

3 Anders Skrondal and Sophia Rabe-Hesketh 3 3.2 Composite links Thompson and Baker (1981) suggested linking the expectatio with a n μ i a function of a composite function of several linear predictors instead of single linear predictor as in generalized linear models. is a weighted In this case the expectation μ Simple composite links i w , sum of inverse links with known weights ir ∑ − 1 w , g = μ ) ( ν ir ir i r r − 1 r th linear predictor for unit i and g where ν · ( is the ) an inverse link ir r function. r categorical A simple example of composite links are cumulative models fo ordered response categories s = 1 ,...,S , which can be responses with S expressed as − 1 >s | x 1 ) = g y Pr( ( ν − − κ ,...,S ) , s = 1 s i i i are threshold parameters and the inverse link function is a c umu- κ where s ogistic or extreme lative distribution function such as the standard normal, l value distributions. The response probabilities can be wri tten as a compos- ite link, 1 − 1 − ( x g | s ( ν ) − g = y ) = ν ) , ν Pr( = ν − κ , s = 1 ,...,S, (2) 1 i is is s i i i,s − − 1 − 1 ν ) = 0. An = ∞ so that g where −∞ ( = ν ( ) = 1 and g and κ κ 0 i iS S 0 right-censoring, advantage of the composite link formulation is that left and accommodated. or even interval censoring of an ordinal response are easily . This is particularly useful for discrete time survival data Bilinear composite links A first extension is to consider unknown linear with products functions of inverse links, replacing the known constants w ir α of the constants and unknown parameters , giving r ∑ − 1 = α w μ g ) ( ν . i r ir ir r r A second extension is to let the expectation be some (not nece ssarily linear) h {·} of the above sum, function ∑ 1 − ν ( . } α ) w g { h = μ ir ir r i r r 1 − ( f g In this case general functions General composite links )] [ ν ir ir r − 1 ν g ) in the above expressions. w ( replace ir ir r

4 4 Generalized Linear Latent and Mixed Models with Composite Links and Exploded Likelihoods 4 Generalized Linear Latent and Mixed Models 4.1 Generalized Linear Mixed Models (GLMMs) e responses of A crucial assumption of generalized linear models is that th . This assumption is i x different units are independent given the covariates i nature with units often unrealistic since data are frequently of a multilevel i nested in clusters j , for instance repeated measurements (units) nested in (clusters). There subjects (clusters) or subjects (units) nested in families will often be unobserved heterogeneity at the cluster level inducing depen- dence among the units, even after conditioning on covariate s. In generalized linear mixed models (e.g. Breslow and Clayton, 1993) unobse rved hetero- (2) geneity is modeled by including random effects η in the linear predictor, mj M ∑ (2) (2) ′ ( x (3) z β ) = η μ g + ν . = ij ij ij mj mij ︸ ︷︷ ︸ m =1 ︸ ︸ ︷︷ Fixed part Random part (2) (2) (2) (2) (2) ′ | are random , z y , η E[ ] where η ≡ x = ( η ) , ··· ,η μ Here, ij ij ij 1 j j j ij M,j (2) z effects varying at level 2 and corresponding covariates. Specifically, ij (2) (2) z η is a random effect of covariate for cluster j , a random intercept mj mij (2) if =1. It is typically assumed that the random effects are multiv ariate z mij normal. 4.2 Extending GLMMs to GLLAMMs The basic idea of factor or IRT models Multilevel factor structures is that one or more unobserved variables, latent traits or fa ctors ‘explain’ the dependence between different observed measurements for a subject, in ent given the the sense that the measurements are conditionally independ factor(s). A simple example of a unidimensional factor model is the two- parameter logistic item response model often used in ability testing. Examinees j i , i = 1 ,...,I , giving responses y answer test items equal to 1 if the ij answer is correct and 0 otherwise. The probability of a corre ct response is modelled as a function of the examinee’s latent ability η , j exp( ν ) ij , ν . λ + β (4) = η y ) = η | = 1 Pr( i j i ij j ij ) ν 1 + exp( ij The latent ability η are factor is assumed to have a normal distribution, λ i j loadings or discrimination parameters (with λ =1) signifying how well the 1 items discriminate between examinees with different abilit ies, and - β /λ i i are item ‘difficulties’.

5 Anders Skrondal and Sophia Rabe-Hesketh 5 generalized We can specify models of this form by extending the two-level iplied not linear mixed model in (3) to allow each random effect to be mult just by a single variable but by a linear combination of varia bles. To obtain e dichotomous the two-parameter logistic item response model, we stack th responses into a single response vector and define dummy variables y ij { 1 if = i p = d pi 0 otherwise itten as The linear predictor of the item response model can then be wr ∑ ∑ = d ν β η + = λ η . d λ + β ij j pi p pi i j p i p p ctor model can be The linear predictor for a three-level multidimensional fa expressed as M M 2 3 ∑ ∑ (3) (2) (3) (2) (3) (2) ′ ′ ′ + , η z λ λ z + η β = x ν ijk m m ijk m m ijk jk m ijk m k 2 3 2 2 3 3 ︷︷ ︸ ︸ m =1 m =1 3 2 ︸ ︷︷ ︸ ︸ ︷︷ ︸ Fixed part Level-3 random part Level-2 random part (3) (2) z z are vectors of dummy variables with correspond- where and ijk m ijk m 2 3 (3) (2) λ λ ing vectors of factor loadings, . See Rabe-Hesketh, Skrondal and m m r model with and Pickles (2004a) for an application of a multilevel facto dichotomous responses. The response model can be further generalized Discrete latent variables η by allowing the latent variables to have discrete distributions. This is j useful if the level 2 units are believed to fall into a number o f groups or ‘latent classes’ within which the latent variables do not va ry. If the number of latent classes, or masses, is chosen to maxim ize the like- lihood the nonparametric maximum likelihood estimator (NP MLE) can be achieved (e.g. Rabe-Hesketh, Pickles and Skrondal, 2003 ), relaxing the assumption of multivariate normal latent variables. Multilevel structural equations Continuous latent variables (random coefficients and/or factors) can be regressed on covariates ( see Section 6) and other latent variables at the same or higher levels, gene ralizing con- ventional structural models to a multilevel setting. If the latent variables are discrete, the masses, component weights or latent class probabilities can depend on covariates via multinomial logit models. See S krondal and Rabe-Hesketh (2004, Ch.4).

6 6 Generalized Linear Latent and Mixed Models with Composite Links and Exploded Likelihoods 5 Composite links and exploded likelihoods in GLLAMMs An outline is given of some extensions of GLLAMMs arising fro m plugging o composite in linear predictors with latent variables from GLLAMMs int links and exploded likelihoods. Discrete time frailty models If we let the linear predictor in (2) be ′ and use a logit link we can obtain a proportional odds ν β + = x η j ij ij model with frailty (see Skrondal and Rabe-Hesketh, 2004, Ch .12). Item response models for ordinal items Letting the linear predictor = in (2) be β ν + λ as in the two parameter IRT model (4) and the η j i ij i sponse model thresholds be item-specific, we obtain Samejima’s graded re for ordinal items (see Skrondal and Rabe-Hesketh, 2004, Ch. 10). Unfolding or ideal point models In standard item response models the probability of a positive response for an item is a monotonic function of the . This assumption may be violated for attitude items where latent trait η j respondents are asked to rate their agreement as ‘disagree’ or ‘agree’, or s =1 ,...,S more generally in terms of ordered categories. For instance, as sentiments favouring capital punishment i ncrease from neg- nt ‘capital pun- ative infinity, the probability of agreeing with the stateme ncreases from ishment seems wrong but is sometimes necessary’ initially i 0, reaches a maximum when the latent trait is in the ‘ambiguou s’ zone (at the ‘ideal point’) and then declines as the latent trait goes to infinity. It has been argued (e.g. Roberts and Laughlin, 1996) that a re spondent may give a particular rating of an attitude item for two reasons. Considering trait is below ‘disagree’, he can ‘disagree from below’ because his latent the position of the item or ‘disagree from above’ because it e xceeds the position. These two possibilities can be expressed in terms of ‘subjective z = ; such that z s ratings’ if the respondent ‘disagrees from below’ and ij ij S +1 − s if he ‘disagrees from above’. z =2 ij z , are not observed, the probabilities of the observed rating Since the y ij ij η given the latent trait , can be written as the sum of the probabilities of j the two disjunct ‘subjective ratings’ corresponding to the observed rating. We propose using a cumulative model (2) for the subjective ra tings Pr( y (5) = s | η ) = ) = Pr( z η = s | η | ) + Pr( z s =2 S +1 − ij ij j j ij j [ ] [ ] 1 − 1 − − 1 1 − κ ) ( ν − κ ) + κ g − ( ν − g ) − g ν ( ( ν − κ − ) g , s ij 1 s ij ij ij − 2 2 S − S +1 − s s where ν β as in (4). For identification, the thresholds must be + λ = η j i ij i κ = − κ =0. , s =1 ,...,S , and κ constrained as for instance s S − s S 2 Importantly, embedding the models in the GLLAMM framework p roduce a wide range of novel unfolding models. The latent trait can f or instance be regressed on same or higher level latent variables and/or regressed on covariates as demonstrated in Section 6.

7 Anders Skrondal and Sophia Rabe-Hesketh 7 Small area estimation Rindskopf (1992) emphasizes that composite link erved counts functions are useful for modelling count data where some obs different kinds represent sums of counts for different groups of units, due to ese ideas have been of missing or partially observed categorical variables. Th used by Tranmer et al. (2004) in random effects modeling and em pirical Bayes prediction of area specific odds-ratios, for instance for the association y marginal between ethnicity and unemployment. They make use of one-wa rate and ethnic tables from the census ‘tabular output’, e.g. unemployment reas as usual composition, in addition to borrowing strength from other a in empirical Bayes prediction. Models combining discrete and continuous latent variables Latent class models can be specified by modeling the ‘complete’ data (including latent class membership) using log linear models. Since lat ent class mem- bership is unknown, we must sum over the latent classes to obt ain expected counts for the observed response patterns. For a two-class m odel with three = 1 , i dichotomous observed responses ,..., 3, a log-linear model with y i mbership can be conditionally independent responses given latent class me written as ∑ ∑ , cα y log μ β = ν y + + cα + β = i c y y 0 y 0 y y i c i i y 3 3 1 1 2 2 i i = 0 μ 1 is the latent class indicator, c where is the expected count for , y y c y 3 1 2 y and latent class ,y 3 ,y ,..., response pattern c , and β = 0 and α p , p 1 2 p 3 μ of the observed counts are are parameters. The expected values y y y 1 2 3 modeled as the sum of the class-specific expected counts, ) . μ = exp( ν ν ) + exp( y y y y y 0 y y 1 y y 3 1 2 2 3 3 2 1 1 Qu, Tan and Kutner (1996) include continuous random effects η within a j the responses latent class model to relax conditional independence among given latent class membership. To incorporate subject-spe cific random ef- fects in the model, we expand the data to obtain counts (0 or 1) for each response and latent class pattern for each subject j . The model can then be written as ∑ ∑ cα y log + β y = ν μ + cα + β = i cj 0 y y 0 i y i cj i y y y 2 3 1 1 3 2 i i ∑ ∑ ( + y cλ (1 − c ) λ y , + ) η 0 i i i i 1 j i i where η j ’s propensity to have a ‘1’ (e.g. can be interpreted as subject j gnosed by a score positively on a diagnostic test, have a symptom, be dia λ λ for those who are healthy and rater), with item-specific effects for 1 i 0 i son j is fixed those who have the disease. Since the total count for each per del at 1, we can estimate the multinomial logit version of this mo ) exp( ν cj y y y 1 2 3 ∑ y Pr( y y c | j ) = 3 1 2 ν ) exp( y cj y y 1 2 3 y c y y 1 2 3

8 8 Generalized Linear Latent and Mixed Models with Composite Links and Exploded Likelihoods j , so the likelihood contribution for subject c Again, we do not know be- comes ) ν ) + exp( exp( ν y 1 y y j y j y 0 y 3 2 1 2 3 1 ∑ | j y y Pr( ) = . y 3 2 1 exp( ν ) y y y cj 1 2 3 c y y y 3 1 2 This is a composite link model if each multinomial logit term is viewed as onditional an inverse link. Note that this set-up makes it easy to relax c effects of the independence among pairs of items by including interaction y β y form in the linear predictors. 2 1 12 Item response models accommodating guessing If it is possible to guess the right answer of an ‘item’ in ability testing, as whe n multiple response model choice questions are used, the two-parameter logistic item in (4) is sometimes replaced by the three-parameter model ) ν exp( ij . ) = + (1 c c ) − η | =1 Pr( y i j ij i 1+exp( ) ν ij are often called ‘guessing parameters’ and can be interpret c The ed as the i probability of a correct answer on item i for an examinee with ability minus infinity. w , the response If we fix the guessing parameters to some common constant model can be expressed as a generalized linear model with a co mposite link − 1 − 1 | η ) ) = wg Pr( y ν (1) + (1 − w ) g , =1 ( ij j ij 2 1 = is the identity link and where g α be g w is the logit link. If we let 1 1 2 a free parameter, we have a simple example of a bilinear compo site link model. have ‘natural The above kind of model (without latent variables) is said to se bioassay. responsiveness’ or ‘nonzero background’ in quantal respon Log-normal random effects If the random effects distribution is skewed, we may want to specify a linear mixed model with log-normal ra ndom effects ′ = μ x η ) + exp( η + exp( ) z , β ij 2 1 ij j j ij which can be accomplished using the composite link ′ μ x = . β + log( η )) z ) + exp( η + exp( ij j ij 1 j 2 ij This is also a useful way of conducting a sensitivity analysi s of the conven- e GLLAMM tional normality assumption for the random effects. Using th formulation, we can also have log-normal common factors. If we use a bilinear composite link, we can include log-norma l random effects in generalized linear mixed (and item response) mode ls as well, ′ ))] = . [ x μ z β + exp( η + log( η ) + exp( h j 2 j 1 ij ij ij

9 Anders Skrondal and Sophia Rabe-Hesketh 9 Zero-inflated Poisson (ZIP) models The likelihood of ZIP models xploded can be expressed using a combination of composite links and e likelihoods. ulation is The ZIP model is a finite mixture model for counts where the pop assumed to consist of two components, a component c =0 where the count can only be zero and a component c = 1 where the count has a Poisson nt component is distribution. The probability of belonging to the zero-cou modelled as ′ γ exp( z ) i π = (6) 0 i ′ z ) γ 1 + exp( i and the Poisson distribution for the other component is k ′ = k | x Pr( ,c ) =1) = exp( − μ β ) μ y . /k ! , μ x = exp( (7) i i i i i i i The probability of a non-zero count becomes k ! = k > 0 | z /k , x ) exp( ) = Pr( y μ = k > 0 ,c ) =1) = (1 − π Pr( μ y − 0 i i i i i i i i ) ( ] [ 1 k − ! /k = exp( μ μ ) i i ′ γ z 1 + exp( ) i and the probability of a zero count Pr( y ) =0 | z x , x , ) = Pr( y z =0 ,c | =0 | z =1 , x ,c ) + Pr( y =0 i i i i i i i i i i i π + (1 − π ) exp( − μ ) = 0 0 i i i ) ( 1 ′ ′ [exp( z = γ x exp( − ) + exp( β ))] . i i ′ γ ) z 1 + exp( i For a non-zero count, the probability is the product of the pr obability of ′ 0 in a logistic regression model with linear predictor z γ and the Poisson i ′ k with a log link and linear predictor x β . There- probability of a count i fore, for non-zero counts, we obtain the correct likelihood by creating two responses, 0 and and specifying a mixed response (logistic and Poisson) k model. For a zero count, we again create a 0 response, modelled as a lo gistic re- composite gression, for the first term. For the second term, we specify a link, 1 − 1 − ′ ′ ′ ′ [exp( z exp( ) + exp( − x γ ))] = ( g z β γ g x ) + ( β , ) i i i i 1 2 g is the log link and g where the log-log link. If we create a 1 response 1 2 and specify a Bernoulli distribution with this composite li nk, we obtain the required term. This set-up also makes it fairly straightforward to include random effects in ZIP models to capture dependence induced by clustered dat a. For in- stance, in modeling the number of alcoholic drinks consumed by respon- dents nested in regions, we could include region-specific ra ndom effects in both (6) and (7) to model variations in the prevalence of non- drinking and in the amount consumed among drinkers, with possible cor relations between these random effects.

10 10 Generalized Linear Latent and Mixed Models with Composit e Links and Exploded Likelihoods 6 Unfolding attitudes to female work participation e USA were In the 1988 and 2002 General Social Surveys respondents in th g female work presented with the following attitude statements regardin participation: [famhapp] A woman and her family will all be happier if she goe s to work [twoincs] Both the husband and wife should contribute to the family income [warmrel]: A working mother can establish just as warm and se cure a relation- ship with her children as a woman who does not work [jobindep] Having a job is the best way for a woman to be an inde pendent person for pay [housewrk] Being a housewife is just as fulfilling as working [homekid] A job is alright, but what most women really want is a home and children [famsuff] All in all, family life suffers when the woman has a fu ll-time job [kidsuff] A pre-school child is likely to suffer if his or her mo ther works look after the [hubbywrk] A husband’s job is to earn money; a wife’s job is to home The respondents rated each statement as either ‘disagree co mpletely’ (1), ‘disagree’ (2), ‘agree somewhat’ (3), ‘agree’ (4), or ‘agre e completely’ (5). In 2002, the ‘disagree completely’ and ‘disagree’ response options were col- lapsed into a single ‘disagree’ option. We use the unfolding model proposed in Section 5, with g as scaled probit σ links with item-specific scale parameters (estimated on the log-scale), i ) ( κ − η λ + β i s j i 1 − − 1 ( g ) = Φ ν . ijs σ i mposite links In 2002, the composite link for ‘disagree’ is the sum of the co for ‘disagree’ and ‘disagree completely’. To investigate if sentiments in favour of female work partic ipation η (loosely j referred to as ‘feminism’) have changed from 1988 to 2002, we specify the structural model η , = γ ) w ,ψ + ζ N(0 , ζ ∼ j j 1 j j where is a dummy variable for year being [2002]. w j Maximum likelihood estimates based on data from 1462 respon dents are given in Table 1 where the items have been ordered from the mos t positive ̂ to the most negative according to their estimated scale valu β . Since es i the magnitude of ̂ γ is negligible, mean ‘feminism’ does not appear to have 1 changed.

11 Anders Skrondal and Sophia Rabe-Hesketh 11 TABLE 1. Estimates for scaled probit unfolding model Item parameters β ln σ λ i i i Est SE Est SE i Item Est SE 0.30 0.04 -0.24 0.05 [famhapp] -2.32 0.08 0.29 0.05 -0.06 0.05 [twoincs] -1.60 0.07 1 – 0 – -0.99 0.07 [warmrel] -0.27 0.14 1.15 0.15 0.64 0.05 [jobindep] [housewrk] 0.54 0.08 0.22 0.06 1.29 0.08 [homekid] 0.76 0.06 -0.06 0.04 2.11 0.07 2.19 0.08 -0.29 0.05 [famsuff] 1.43 0.09 [kidsuff] 2.24 0.08 1.49 0.09 -0.46 0.06 2.42 0.09 1.14 0.09 -0.11 0.05 [hubbywrk] − κ = κ Thresholds s 2 S − s s (categories) Est SE 1 (‘disagree completely’/‘disagree’) 3.43 0.11 2 (‘disagree’/‘agree somewhat’) 2.36 0.08 3 (‘agree somewhat’/‘agree’) 1.67 0.06 4 (‘agree/‘agree completely’) 0.72 0.03 Latent trait regression Est SE [2002] -0.04 0.04 γ 1 ψ 0.62 0.08 Variance phically. Following Roberts and Laughlin (1996) we assess model fit gra ̃ of respondent j relative ν First, we estimate the position or ‘dominance’ ij i (how much more ‘feminist’ the respondent is than the item) to item ̃ η by plugging in the empirical Bayes prediction of the latent trait and j the parameter estimates into the linear predictor. Substit uting this into ry for each the unfolding model, we obtain the expected response catego person-item pair. Grouping the ̃ into approximately homogeneous groups ν ij of size 30 for each item and plotting the corresponding avera ge observed and expected frequencies versus the average ̃ ν for each item gives Figure ij 1. Our unfolding model appears to fit quite well. Although the expected response takes the form of a single-pe aked function consistent with an unfolding process when all items are cons idered together, none of the individual items exhibit single-peaked behavio ur with the pos- sible exception of [jobindep]. Using conventional item res ponse models that assume monotonicity might therefore be appropriate if eith er (1) reversing the coding of the appropriate items can be based on a priori in formation or (2) the model accommodates negative factor loadings.

12 12 Generalized Linear Latent and Mixed Models with Composit e Links and Exploded Likelihoods [twoincs] [warmrel] [famhapp] 4 3 2 1 0 [jobindep] [homekid] [housewrk] 4 3 2 1 0 [hubbywrk] [kidsuff] [famsuff] 4 3 2 1 0 4 0 4 2 0 −2 2 4 0 −2 −2 2 ̃ ν ij expected observed Graphs by item FIGURE 1. Mean expected and observed responses as a function of ‘dominance’ ν ̃ of person j over item i ij 7 Conclusions likelihoods Although simple to implement, composite links and exploded have been demonstrated to be remarkably powerful tools for s pecifying novel GLLAMMs. Indeed, we do not purport to exhaust potentia l applica- tions in this paper. ional composite A further useful extension would be to generalize the tradit products links suggested by Thompson and Baker (1981) to accommodate of inverse links. A simple variant is of the form ∑ ∏ 1 − α = g μ ( ν ) . i r irt rt t r A composite link with products can be used for additive relat ive risk mod- in the Poisson μ els with random effects. The risk or rate parameter ij distribution is specified as ′ μ β = exp( + η x )[1 + β ] , j 0 ij ij x correspondingly not a constant. Note does not include a 1 and β where ij + that the baseline risk when 0 becomes exp( β = η ) > 0. It follows that x 0 ij j the ‘relative risk’ RR , the risk when the covariate vector is x relative to ij ij the baseline risk, is ′ β , RR = 1 + x ij ij an additive function of the covariates.

13 Anders Skrondal and Sophia Rabe-Hesketh 13 Maximum likelihood estimation and of GLLAMMs and empirical Bayes Skrondal and prediction using adaptive quadrature (e.g. Rabe-Hesketh, software running in Stata . Pickles, 2004b) are implemented in the gllamm for further information. See References Breslow, N.E., and Clayton, D.G. (1993). Approximate infer ence in gener- Journal of the American Statistical As- alized linear mixed models. sociation 88 , 9-25. , Qu, Y., Tan, M., and Kutner, M.H. (1996). Random effects model s in la- ests. tent class analysis for evaluating accuracy of diagnostic t Bio- metrics 52 , 797-810 , recting for co- Rabe-Hesketh, S., Pickles, A., and Skrondal, A. (2003). Cor parametric variate measurement error in logistic regression using non Statistical Modelling , 3 , 215-232. maximum likelihood estimation. eralized mul- Rabe-Hesketh, S., Skrondal, A. and Pickles, A. (2004a). Gen tilevel structural equation modeling. Psychometrika , in press. Rabe-Hesketh, S., Skrondal, A. and Pickles, A. (2004b). Max imum likeli- hood estimation of limited and discrete dependent variable models with nested random effects. Journal of Econometrics , in press. a analysis with Rindskopf, D. (1992). A general approach to categorical dat te links. missing data, using generalized linear models with composi Psychometrika , 57 , 29-42. item response Roberts, J.S., and Laughlin, J.E. (1996). A unidimensional response model for unfolding responses from a graded disagree-agree scale. Applied Psychological Measurement , 20 , 231-255. Skrondal, A., and Rabe-Hesketh, S. (2003). Multilevel logi stic regression for polytomous data and rankings. Psychometrika , 68 , 267-287. Skrondal, A., and Rabe-Hesketh, S. (2004). Generalized Latent Variable Mod- eling. Multilevel, Longitudinal and Structural Equation M odels . Boca Raton, FL: Chapman & Hall/CRC. Thompson, R., and Baker, R.J. (1981). Composite link functi ons in gener- alized linear models. Journal of the Royal Statistical Society, Series C , 30 , 125-131. Tranmer, M., Pickles, A., Fieldhouse, E., et al. (2004). The case for small area microdata. Journal of the Royal Statistical Society, Series A , in press.

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