conditional risk models for ordinal response data: simultaneous logistic regression analysis and...

Journal of Statistical Planning andInference 108 (2002) 201–217

www.elsevier.com/locate/jspi

Conditional risk models for ordinal responsedata: simultaneous logistic regression analysis

and generalized score testsLimin Fua, Douglas G. Simpsonb; ∗

aFreddie Mac, MacLean, VA 22102-3110, USAbDepartment of Statistics, University of Illinois, 101 Illini Hall 725 So. Wright St., Champaign,

IL 61820, USA

Abstract

A general class of conditional risk models is introduced for ordinal regression. Special casesinclude the cumulative logit models, continuation ratio models and adjacent category odds mod-els. A simultaneous logistic regression (SLR) approach is introduced for 1tting the models in auni1ed fashion. Inferences are obtained by adapting the theory of generalized estimating equa-tions. SLR is fully e3cient for the continuation ratio model and has high e3ciency in othercases. The general approach applies to other link functions such as ordinal probit analysis aswell. Rao-type generalized score tests are developed for model assessment within this frame-work. These tests are useful in testing for parallelism within the general class of models. Realdata examples illustrate the uni1ed modeling made possible by this approach.c© 2002 Elsevier Science B.V. All rights reserved.

MSC: primary 62J12; secondary 62H17

Keywords: Adjacent category odds; Continuation ratios; Empirical logit plot; Logistic regression;Proportional odds model; Rao score test

1. Introduction

Ordinal data are common in social science research and increasingly common inother areas such as the biological sciences. There are di=erent strategies for modelingordinal response data. Commonly used models include the proportional odds model

∗ Corresponding author. Tel.: +1-217-333-2167.E-mail address: [email protected] (D.G. Simpson).

0378-3758/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0378 -3758(02)00279 -3

202 L. Fu, D.G. Simpson / Journal of Statistical Planning and Inference 108 (2002) 201–217

(McCullagh, 1980), the adjacent category odds model (see, for example, Clogg and Shi-hadeh, 1994), and the continuation ratio model (see Feinberg, 1980; Agresti, 1990).Simpson et al. (1996) discussed maximum likelihood estimation and marginal ana-lysis subject to interval censoring for these three types of models. Heagerty andZeger (1996) proposed a GEE approach to the analysis of clustered ordinal datausing the proportional odds model. Maximum likelihood estimation (MLE) iscommonly used to get parameter estimates, but direct maximization of the likeli-hood is somewhat complicated when di=erent model forms are used. One of ourgoals is to develop a uni1ed approach for a broad class of ordinal regressionmodels.

This article introduces a general class of models that includes well-known ordinalregression models as special cases, and it introduces a convenient class of estimatorsbased on simultaneous logistic regression (SLR). These estimators are e3cient forthe continuation ratio model and nearly e3cient for the cumulative odds and adjacentodds models. We consider regression analysis for independent ordinal responses, butwe anticipate that the approach taken here will also facilitate the analysis of correlatedordinal data.

Begg and Gray (1984) studied the technique of individualized logistic regressionfor calculating polychotomous logistic regression parameters. They 1t separate logisticregressions to binary indicators for the di=erent response categories. In a simulationstudy they found that the resulting estimators have reasonably high e3ciency in com-parison with maximum likelihood. Ordinal models usually impose constraints on theparameters. For example, the proportional odds model assumes the same slopes acrossall response levels. For ordinal data, all the models mentioned above can be viewedas a set of logistic regressions at di=erent levels. Each ordinal response contributes tothese logistic regressions according to its observed value and the model form. The dataare represented by two vectors of binary indicator variables and the model form bytheir expectations.

We consider SLR, an extension of individualized logistic regression that allows con-sistent estimation of the parameters and provides valid large sample inferences. Theestimates may be obtained using standard logistic regression software after some pre-processing of the data. Another advantage is that one can enforce linear constraints onthe parameters conveniently by forming the corresponding design matrix. For example,one can use this method to 1t a proportional odds model, a partial proportional oddsmodel (Peterson and Harrell 1990), and an unrestricted model in a uni1ed fashion.It is clear that the collection of indicator variables generated from a given ordinalresponse are correlated with each other. In order to perform valid inferences these cor-related binary response variables can be treated as if they are clustered in a generalizedestimating equation analysis. Clayton (1995) considered this approach for the propor-tional odds model. We extend the SLR method to a general class of ordinal regressionmodels.

We also consider a modi1ed SLR approach, which incorporates the correlation struc-ture for the ordinal model and refer to this method as e3cient logistic regression (ELR).This is a version of e3cient GEE estimation as described by Liang and Zeger (1986).In our case the ordinal responses are independent, and the correlation structure of the

L. Fu, D.G. Simpson / Journal of Statistical Planning and Inference 108 (2002) 201–217 203

multiple binary response variables (the “clusters”) is a known function of the ordinalregression parameters.

In addition to computational considerations, another advantage of the SLR and ELRmethods is that they lend themselves to the development of generalized score testsfor composite hypotheses about the model. Like the e3cient score tests pioneered byRao (1948), generalized score tests require only the null parameter estimates, and theyare invariant to full-rank di=erentiable transformations of the parameters. Boos (1992)developed generalized score tests in the context of GEE.

Section 2 introduces the conditional risk regression models and a convenient binarycoding. Section 3 develops the SLR method for the conditional risk models. Section 4discusses the use of fully e3cient weights. Section 5 evaluates the asymptotic relativee3ciencies of the parameter estimates. In Section 6 we develop generalized scoretests for SLR and ELR. These score tests can be applied to test proportionality inproportional odds models and other hypotheses about the model. In Section 7 wepresent two examples to illustrate the methodology.

2. Conditional risk models for ordinal regression

Ordinal regression models are designed to model the probability distribution of theordinal score, Yi, as a function of covariate information represented by a vector, xi. Weassume without loss of generality that Yi takes values in {0; 1; 2; : : : ; S}. It is su3cientto model the probabilities for {1; 2; : : : ; S}, because the probability of 0 is obtained bysubtraction. We consider the following general class of models:

Pr(Yi ∈Ts |Yi ∈Rs) = H (�s + xTi �s); s = 1; 2; : : : ; S; (1)

where Rs is the risk set, Ts is the target set, H is a cumulative distribution function,and it is assumed that Ts ⊂ Rs. This class of models includes well-known forms suchas the proportional odds model, ordinal threshhold probit regression, adjacent categorylogit regression, and ordinal regression based on continuation ratios.

In this general framework we introduce binary codings of the events (Yi ∈Rs) and(Yi ∈Ts). This binary coding facilitates a uni1ed approach to parameter estimation andinference. Assume that Y=(Y1; : : : ; Yn)T represents a vector of ordinal measurements forn cases, Yi represents the ith observation of Y, taking on the values s= 0; 1; : : : ; S¿ 1.We represent Yi by two vectors of binary indicator variables Y?

i = (Y?i1 ; : : : ; Y?

iS )T

and W?i = (W?

i1 ; : : : ; W?iS )T. We use Y?

is to specify the contribution of Yi in the logisticregression at level s, and W?

is to specify whether we would include Y?is in the regression

at level s. The values of the indicator variables of an ordinal response are determinedby its observed value and the model form. Let xi denote the covariate associated withYi. Denote by � the vector of all unknown parameters. Three examples are used toillustrate this coding method.

The proportional odds model, developed by McCullagh (1980), assumes parallele=ects for di=erent levels. In the proportional odds models for the marginal means, itis assumed that

logit{Pr(Yi¿ s)}= �s + xTi �; s = 1; : : : ; S:


We can naturally represent the ordinal measure Yi through a vector of cumulativeindicator variables

Y?is =

{1 if Yi¿ s;

0 if Yi ¡ sand W?

is = 1; s = 1; : : : ; S:

An alternative to the proportional odds model, which can model nonparallel e=ects,is the continuation ratio model given by

logit{Pr(Yi = s |Yi6 s)}= �s + xTi �s:

Feinberg (1980, p. 86) referred to the conditional odds associated with this modelas “continuation ratios”. If Yi is a discrete survival time, then the equivalent ordinalregression model for Pr(Yi =s |Yi¿ s) provides a discrete version of hazard regression;see Heagerty and Zeger (1999). If we were to perform an individual logistic regressionat level s, we would pick all the responses whose scores are less than or equal to s.We call this set of responses the risk set at level s. So for Yi at level s, there are threepossible outcomes: being 1 or 0 in the sth risk set, or excluded from the sth risk set.So we can de1ne the indicator variables of Yi by

Y?is =

{1 if Yi = s;

0 otherwise;and W?

is =

{1 if Yi6 s;

0 otherwise; s = 1; : : : ; S:

Another possibility is to model the adjacent category odds in the log-linear model(see Clogg and Shihadeh, 1994).

log{

Pr(Yi = s)Pr(Yi = s− 1)

}= �s + xT

i �s:

This model is parametrically equivalent, that is �s = �s and �s = �s, to the conditionalmodel

logit{Pr(Yi = s | s− 16Yi6 s)}= �s + xTi �s:

Since logit{Pr(Yi = s | s − 16Yi6 s)} = log{Pr(Yi = s)=Pr(Yi = s − 1)}. Similar tothe continuation ratio model, we can form the risk set at each level, and de1ne theindicator variables as

Y?is =

{1 if Yi = s;

0 otherwiseand W?

is =

{1 if Yi = s or Yi = s− 1;

0 otherwise;s = 1; : : : ; S:

We illustrate the binary coding for all three forms in the case where the responsetakes on the four possible values s = 0; 1; 2; 3. Table 1 lists the values of the indicatorvariables for all the possible responses for the three types of logit models.

The three types of models described above can be treated in a uni1ed fashion byspecifying the conditional link function model: E(Y?

is |W?is = 1) = H (�s + xT

i �s). Wemake the convention that W?

is = 0 implies Y?is = 0, i.e., the binary response is de-

1ned to be zero outside of the risk set. We then have the full speci1cation of the


Table 1Indicator variables for ordinal responses

Cumulative odds Continuation ratio Adjacent category

Yi Y?i1 Y?

i2 Y?i3 Y?

i1 Y?i2 Y?

i3 Y?i1 Y?

i2 Y?i3

0 0 0 0 0 0 0 0 0 01 1 0 0 1 0 0 1 0 02 1 1 0 0 1 0 0 1 03 1 1 1 0 0 1 0 0 1

Yi W?i1 W?

i2 W?i3 W?

i1 W?i2 W?

i3 W?i1 W?

i2 W?i3

0 1 1 1 1 1 1 1 0 01 1 1 1 1 1 1 1 1 02 1 1 1 0 1 1 0 1 13 1 1 1 0 0 1 0 0 1

conditional model,

P?is :=E(Y?

is |W?is ) = W?

is H (�s + xTi �s): (2)

It is convenient to express (2) as

P?i =WiH (Xi�); (3)

where P?i = (P?

i1 ; : : : ; P?iS )T, Wi = diag(W?

i1 ; : : : ; W?iS ),

XTi =

(e1 e2 · · · eS

e1 ⊗ xi e2 ⊗ xi · · · eS ⊗ xi

)with ej =

0...1...0

S×1

← jth position;

and � = (�1; : : : ; �S ; �11; : : : ; �1p; : : : ; �S1; : : : ; �Sp; )T. A common modeling assumption isthat the slope parameters for the explanatory variables are constant across severitylevels. Under this parallel slopes assumption, e.g., in the proportional odds model, thepseudo-design matrix has the simpli1ed form

XTi =

(e1 e2 · · · eS

xi xi · · · xi

):

Observe that the S binary responses generated by observation i appear as if they wereseparate responses in a binary regression with augmented design matrix. In performinginferences we adjust for their correlation using GEE theory as described by Diggleet al. (1994, Chapter 8). The marginal model corresponding to (2) is given by

E(Y?is ) = E(W?

is )H (�s + xTi �s): (4)

In a fully parametric approach, the term E(W?is ) may have a complicated dependence

on the parameters. In the development that follows we avoid this complication throughthe use of conditionally unbiased estimating equations.


3. Simultaneous logistic regression

To derive conditionally unbiased estimating equations we start with (2) and observethat

[Y?is |W?

is ] ∼ Bernoulli(P?is ); (5)

where P?is = W?

is H (�s + xTi �). If W?

is = 0, then, by de1nition Y?is = 0 as well, so that

(5) holds trivially. De1ning 0 log(0) = 0 through the usual limiting argument, we havethe marginal pseudo-log-likelihood:

n∑i=1

S∑s=1

Y?is log(P∗

is) + (1− Y?is ) log(1− P?

is ): (6)

Now W?is Y?

is =Y?is . Moreover, if W?

is = 0, then P?is = 0 and log(1−P?

is ) = 0. Therefore,the criterion in (6) is equal to the weighted logistic regression criterion,

n∑i=1

S∑s=1

W?is {Y?

is log(His) + (1− Y?is ) log(1− His)}; (7)

where His = H (�s + xTi �). The criterion in (7) corresponds to the estimating equation

n∑i=1

XTi Wi{Yi − H (Xi�)}= 0; (8)

where Y?i =(Y?

i1 ; : : : ; Y?iS )T, and where Xi, Wi and � have the same form as in Eq. (3).

Eq. (2) implies that (8) is a conditionally unbiased, and hence marginally unbiased,estimating equation. For further discussion of conditional versus marginal unbiasedestimating equations see KQunsch et al. (1989).

Because the weights are 0–1 valued, the maximizer of (7) can be computed byordinary logistic regression of the binary responses in the risk sets. We refer to theresulting estimates as the SLR estimates. At each level s, model (8) can be viewed as alogistic regression on the “risk” set at level s. Fitting this logistic regression using SLRallows for pooling information. Simpson et al. (1996) introduced the proportional oddsversion of SLR, referring to the method as “pseudo-strata”. They used the psuedo-strataestimates as starting values for maximum likelihood, but did not discuss the possibilityof basing inferences on them. The conditional odds version of SLR is in fact equivalentto maximum likelihood estimation. A number of authors have used this fact aboutcontinuation ratios to simplify the computations for that special case; see, e.g., Agresti(1990).

The SLR estimation produces consistent, asymptotically normal estimates, as can beshown by adapting results of Liang and Zeger (1986). Except for the special caseof the conditional odds model, the inverse of the pseudo-information matrix will giveinconsistent estimates of the asymptotic variance of the parameter estimates. The theoryof estimating equations provides a consistent estimator for the asymptotic covarianceof � and �, namely, the sandwich estimator,

ˆvarslr(�; �) = H1(�; �)−1H2(�; �)H1(�; �)−1; (9)


where

H1(�; �) =n∑

i=1

XTi W

?i �iW?

i Xi ;

H2(�; �) =n∑

i=1

XTi W

?i (Y?

i − H i)(Y?i − H i)TW?

i Xi ;

Hi =(H (�1 +�T1 xi); : : : ; H (�S +�T

S xi))T, and �i =diag{H ′(�1 +xTi �1); : : : ; H ′(�S +xT

i �S)}.For background and further references on estimating equations see Diggle et al. (1994).

Computationally, one can exclude the inactive indicator variables and use a standardlogistic regression procedure on the active indicator variables. The SLR method iseasy to implement and, as will be seen, it can be highly e3cient. Inferences based onSLR provide a fast method for model selection. Furthermore, SLR can provide goodstarting values for the more e3cient ELR approach, described below, and maximumlikelihood.

4. E�cient logistic regression

Liang and Zeger (1986), Zeger and Liang (1986) and Prentice (1988) have devel-oped moment-based GEE methods for regression models for longitudinal categoricalresponses, where the repeated measurements on the same individual are correlated.Similarly, we can treat the indicator variables obtained from one observation as fromone cluster, and consider the correlations among them. In the present case it is possibleto work out the exact covariance functions for the coded response variables. We referto the resulting method as ELR. It is known that ELR is e3cient for the cumula-tive odds model; see Clayton (1995). ELR coincides with SLR for continuation ratiomodels. The ELR for � and � solve the estimating equation

n∑i=1

XTi W

?i �iV−1

i W?i (Y?

i − Hi) = 0; (10)

where Vi = E[W?i (Y?

i − Hi)(Y?i − Hi)TW?

i ]. The ELR approach leads to unbiasedestimating equations under the following conditions.

Remark 1. If Vi is diagonal or W∗i is nonstochastic for i = 1; : : : ; n; then (10) is an

unbiased estimating equation.

These conditions are satis1ed by the cumulative odds model and the continuationratio model, but not by the adjacent category odds model.

Under the conditions of Remark 1, and because the assumed covariance is correctunder the model, the covariance matrix of � and � is consistently estimated by

ˆvarelr(�; �) =

(n∑

i=1

XTi W

?i �V

−1i �iW?

i Xi

)−1

: (11)


For the three model types mentioned previously Vi has elements vist (s; t ∈{1; : : : ; S}),where

proportional odds : vist = His(1− Hit) if s¿ t;

conditional probability : vist =

His

S∏j=s

(1− Hij) if s = t;

0 if s �= t;

adjacent odds : vist =

�is{His + (1− His)2} if s = t;

−�is(1− His)Hi;s+1 if t = s + 1;0 if t ¿ s + 1;

where �is =∏s

j=1 Hij{1 +∑S

j=1

∏jk=1 Hik}−1 and vits = vist .

5. Asymptotic relative e�ciencies

Here we address the issue of asymptotic e3ciencies of the SLR estimates. The e3-ciencies are generally high for the SLR method, and they yield fully e3cient estimatesfor special cases

5.1. Cumulative odds models

In general, it is di3cult to compare the e3ciency of SLR estimates to MLE analyt-ically, so we consider 2×2 cross-sectional design con1gurations with di=erent numberof response levels. Here xi1 and xi2 are the dichotomous covariates indicating groupmembership for the ith individual. The parameters are �1 =1; �2 =2; and three designswith 3, 6, and 10 response levels are selected:

(i) �1 = 0; �2 =−2;(ii) �1 = 0; �2 =−0:5; �3 =−1; �4 =−1:5; �5 =−2;

(iii) �1 = 0; �2 = −0:25; �3 = −0:5; �4 = −0:75; �5 = −1; �6 = −1:25; �7 = −1:5;�8 =−1:75; �9 =−2.

The e3ciencies were calculated in Mathematica. We assume balance in all designs,that is each of the four combinations occurs with probability 0.25. Table 2 lists the

Table 2Asymptotic relative e3ciencies of SLR for the cumulative odds model

Design �1 �2 �1 �2 �3 �4 �5 �6 �7 �8 �9

(i) 0.97 0.99 0.98 0.98(ii) 0.93 0.96 0.96 0.95 0.95 0.96 0.96(iii) 0.92 0.95 0.95 0.94 0.94 0.94 0.94 0.94 0.95 0.95 0.95


asymptotic relative e3ciencies for the three designs. All of the asymptotic e3cienciesare high, and the minimum e3ciency is 92%.

The log-likelihood of the indicator variables, Y?i = {Y?

i1 ; : : : ; Y?iS }, can be written as

l(Y?i ) = (1− Y?

i1 ) log(1− P?i1) + (Y?

i1 − Y?i2 ) log(P?

i1 − P?i2)

+ · · ·+ (Y?iS−1 − Y?

iS ) log(P?iS−1 − P?

iS ) + Y?iS log(P?

iS );

which follows an exponential family distribution. By the theory of generalized linearmodels (McCullagh and Nelder, 1989), the ELR estimation is equivalent to MLE forthe cumulative odds models.

5.2. Continuation ratio models

For the continuation ratio model. We 1rst consider the multinomial representation.Let ns; s=0; 1; : : : ; S denote the response count in each cell and let n=

∑Ss=0 ns. De1ne

by qs = Pr(Y = s |Y 6 s). The multinomial mass function has factorization

b(n; nS ; qS) b(n− nS ; nS−1; qS−1) · · · b(n− nS − · · · − n2; n1; q1);

where b(n; y; q) denote the binomial probability of y “successes” in n trials, whenthe success probability is q on each trial (see Agresti, 1990). So the log-likelihoodfunction is

l =n∑

i=1

S∑s=1

W?is {Y?

is log(P?is ) + (1− Y?

is ) log(1− P?is )}:

which is the same as (7). Therefore the SLR is equivalent to MLE for the continuationratio models, and it is asymptotically e3cient.

5.3. Adjacent category odds models

Using the same design con1gurations above, we compare the asymptotic relativee3ciency of SLR to MLE. The results are listed in Table 3. In our examples, we haveattempted to assess the inTuence of the number of categories. In general, the asymptoticrelative e3ciencies are high throughout, although occasionally SLR is ine3cient forindividual parameters. E3ciencies seem to decrease as the number of response levelsincreases.

Table 3Asymptotic relative e3ciencies of SLR for the adjacent category model

Design �1 �2 �1 �2 �3 �4 �5 �6 �7 �8 �9

(i) 0.99 0.98 0.99 0.99(ii) 0.96 0.88 0.98 0.95 0.95 0.98 0.96(iii) 0.92 0.79 0.99 0.98 0.96 0.96 0.96 0.94 0.94 0.97 0.92


6. Goodness of 't and score tests

Rao (1948) introduced the score test for general parametric testing problems. Thetest statistic has the form

S(�)TI−1(�)S(�); (12)

where S(�) is the vector of partial derivatives of the log-likelihood function, I(�) isthe Fisher information matrix and � is the maximum likelihood estimate subject tothe constraints implied by the null hypothesis. The score test is locally asymptoti-cally equivalent to Wald and likelihood ratio statistics (SerTing, 1980, p. 156), but itonly requires computation of � under the null hypothesis. It also has the advantageof invariance to full-rank di=erentiable transformations of the parameter vector. Gen-eralizations of Rao’s score test have been developed by Boos (1992), for inferencebased on general estimating equations, and by White (1982), for inference based on amisspeci1ed model. These generalizations are able to account for lack of knowledgeabout the correlation structure by using semiparametric variance estimates. We developgeneralized score tests for the SLR approach and the ELR approach to assess goodnessof 1t aspects of the various ordinal regression models.

In the test for parallelism we consider model (1) as the full model, and we testthe hypothesis H0:g(�) = 0, where g : Rp → Rr is a continuous vector function of �such that its Jacobian at �, G(�) = @g(�)=@�, is 1nite with full row rank r, against thealternative H1: g(�) �= 0.

Let � solve the constrained maximization problem

max�∈)

Q(�)

s:t: g(�) = 0;

and let

S(�) =@Q(�)

@�:

With the SLR approach, Q(�) is given by (7), and S(�) is given by (8) with unre-stricted parameters. We need to solve (8) subject to g(�) = 0 to get �, the restrictedparameter estimates. The estimates � which maximizes Q(�) subject to H0 satis1es

S(�)− G(�)T+ = 0; g(�) = 0;

where + is an r × 1 vector of Lagrange multipliers. This form is general but maynot be easy to implement using existing software. For some special cases this canbe accomplished by changing the design matrix to simplify the computation of re-stricted parameter estimation. For instance, for testing parallelism discussed before, therestricted parameterization is that �1=�2=· · ·=�S . The parameter estimation for the re-stricted model is found by using the design matrix assuming common slope parameter.The score test statistic is as follows:

Sslr = S(�)TH1(�)−1G(�)T[G(�)H1(�)−1H2(�)H1(�)−1G(�)T]−1

×G(�)H1(�)−1S(�): (13)


Under H0 and suitable regularity conditions, Sclr → ,2r . As for the ELR approach for

the proportional odds model, the score test statistic has the form

Selr = S(�)TI−1(�)−1S(�): (14)

Under H0 and suitable regularity conditions, Selr → ,2r . The SLR approach and its

generalized score test provide a Texible and simple way of modeling ordinal responsedata and check for lack of 1t.

As an application, we can use generalized score tests to assess proportionality inproportional odds models and parallelism of slope parameters in other models. Considerthe general model without making the parallel line assumption P?

i =WiH (Xi�) where� = (�1; : : : ; �s; �11; : : : ; �1S ; : : : : : : ; �p1; : : : ; �pS)T. Under the null hypothesis H0: �k1 =�k2 = · · · = �kS ; k = 1; : : : ; p, there is a single common slope parameter for each ofthe p explanatory variables. Let �1; : : : ; �p be the common slope parameters underH0. Let �1; : : : ; �S ; and �1; : : : ; �p be the estimated parameters under H0. So the scorestatistics Sslr and Selr have an asymptotic chi-square distribution with p(S − 1) degreeof freedom. The score tests can also be used as model selection statistics for testingindividual variables not in the model.

7. Examples

This section presents two examples to illustrate the methodology. S-Plus (MathSoft,Inc.) code for these examples is available by request to the authors.

7.1. Analysis of mental health data

To illustrate the methodology, we 1rst consider data from Srole et al. (1962) on therelationship between an individual’s mental health status and the socioeconomic statusof his or her parents. The data are given in Table 5. These data were also analyzed byAgresti (1990, p. 289). The probability bar plot in Fig. 1 shows a decreasing trend ofmental health as the socioeconomic status goes down. It is not immediately clear whichmodel is likely to provide a better description of these data. We therefore examinethe empirical logit transformations using di=erent logits. We plot the empirical logitsversus the response levels. This technique is simple and often useful for choosing theparsimonious model form. Fig. 2 shows the empirical logit transformation plots for thethree types of logits. Each curve connects the transformed empirical logits of contiguouslevel for each socioeconomic group. All the logit transformation plots reveal a parallelrow e=ects. So we 1t all the three models with common slopes parameters cross levelsand compute the score test statistics for testing the parallelism assumption.

The 10 degree of freedom score tests of parallel slope parameters for the proportionalodds model using both the ELR and SLR are Selr = 7:78; pelr = 0:65, and Sslr =7:89; pslr = 0:64. The score test of parallel slopes for the continuation ratio model isS = 6:89; p= 0:74. And the score test of parallel slopes for the adjacent category oddsmodel is S = 5:41; p = 0:86. Table 4 gives the 1tted counts using the three models.Parameter estimates and their standard errors are displayed in Table 4.


ImpairedModerateMildWell

A (high) B C D E F (low)0.0

0.5

1.0

1.5

Fig. 1. Bar plot of mental health data.

Response Level

Cum

ulat

ive

Logi

ts

1 3

-1

0

1

2

123456

Response Level

Con

tinua

tion-

ratio

Log

its

1

-1

0

1123

456

Response LevelAdj

acen

t Cat

egor

ies

Logi

ts

1 3

-1

0

1

1

123456

1 ---- A (high)

2 ---- B

3 ---- C

4 ---- D

5 ---- E

6 ---- F (low)

2 2 3

2

.....

.

...... ......

........... ......

......

............

Fig. 2. Empirical logit transformation plot of mental health data.


Table 4Parameter estimates and standard errors for mental health data

Model form �1 �2 �3 �1 �2 �3 �4 �5

Cumulative ELR 2.03 0.33 − 0.68 − 0.83 − 0.85 − 0.62 − 0.53 − 0.26(0.13) (0.12) (0.13) (0.17) (0.17) (0.16) (0.15) (0.17)

SLR 2.04 0.34 − 0.67 − 0.84 − 0.86 − 0.63 − 0.53 − 0.26(0.13) (0.12) (0.13) (0.17) (0.17) (0.16) (0.15) (0.17)

Continuation SLR 1.16 − 0.47 − 0.75 − 0.70 − 0.70 − 0.53 − 0.47 − 0.28(0.12) (0.11) (0.11) (0.14) (0.14) (0.13) (0.14) (0.14)

Adjacent SLR 0.99 − 0.22 0.35 − 0.48 − 0.47 − 0.35 − 0.29 − 0.13(0.13) (0.12) (0.13) (0.15) (0.15) (0.15) (0.14) (0.15)

For further simpli1cation, we assign scores to the six socioeconomic status levelswith A = 5, B = 4, C = 3, D = 2, E = 1, and F = 0 regress on the resulting quantitativevariable. We 1t parallel slope models using the quantitative status score as the predictor.The 14 degree of freedom score tests for the proportional odds model, the continuationratio model, and the adjacent category odds model are 7.91, 6.88, and 5.41, respectively.The 1tted values are given in Table 5. All three of the simpli1ed models adequately1t the mental health data.

7.2. Analysis of chicken embryo data

As a second example, Table 6 lists a subset of data from Jarrett et al. (1981).The data were also given in Morgan (1992, p. 10, Table 1.7). The objective was toinvestigate the e=ects of arboviruses injected into chicken embryos and to quantifythe potency of arboviruses. In the experiments, eggs were inoculated with a range ofviruses and several inoculation levels and candled daily for 14 days to check viablity.The surviving embryos were then examined for gross abnormalities and the results werereported 4 days later; see Jarrett et al. (1981) and McPhee et al. (1984) for more details.The resulting data are for the control group and two arboviruses, the Facey’s Paddockvirus and the Tinaroo virus. There are three levels of possible responses—death, alivebut deformed, and alive but not deformed. The need to examine the dependence of theresponses on the amount of injected viruses leads to an ordinal regression analysis.

A conventional approach is to model the exposure data together with the controldata using a cumulative odds model. Morgan (1992) suggested using log10-transformeddoses to improve the 1tting and reduce the inTuence of the highest doses. However,for the Tinaroo and control data, with the log10-scaled dose entering the conventionalproportional odds model, the control response rates at both deformed and death sever-ities should be zero. This contradicts the fact that there is an observed death in thecontrols. In order to use the cumulative odds model, McPhee et al. (1984) and Morgan(1992) argued that the observed control death rate ( 1

18 ) was small and could thereforebe ignored, i.e., they omitted the control observations. However, as indicated by Mor-gan (1992, p. 120), the proportional odds model does not 1t the data. Xie and Simpson(1999) pointed out that ignoring low incidence rates at high severity levels may lead


Table 5Observed and 1tted values for mental health data with linear scores

Parent’s Mental health statussocioeconomicstatus Well Mild Moderate Impaired

symptom symptomformation formation

64 94 58 465 (66.1)a (65.9)b (104.0) (103.7) (48.8) (48.9) (43.1) (43.5)

(68.9)c (66.7)d (99.0) (104.8) (47.8) (49.5) (46.3) (41.0)

57 94 54 404 (54.3) (54.3) (95.0) (94.8) (49.3) (49.3) (46.4) (46.6)

(55.9) (55.0) (92.2) (95.2) (48.4) (49.5) (48.5) (45.3)

57 105 65 603 (55.6) (55.7) (107.5) (107.3) (61.6) (61.6) (62.2) (62.4)

(56.3) (56.2) (106.5) (107.3) (61.0) (61.5) (63.3) (61.9)

72 141 77 942 (64.8) (65.0) (137.4) (137.2) (87.0) (86.9) (94.8) (94.9)

(63.9) (65.0) (138.8) (136.7) (87.2) (86.4) (94.1) (95.8)

36 97 54 781 (38.7) (39.0) (89.5) (89.5) (62.6) (62.4) (74.1) (74.1)

(37.1) (38.4) (92.3) (89.0) (63.8) (61.9) (71.9) (75.7)

21 71 54 710 (27.4) (27.6) (68.5) (68.6) (52.7) (52.6) (68.4) (68.2)

(25.2) (26.6) (71.9) (68.0) (55.0) (52.2) (64.9) (70.3)aFitted values with the proportional odds model using the SLR method.bFitted values with the proportional odds model using the ELR method.cFitted values with the continuation ratio model using the SLR method.dFitted values with the adjacent category odds model using the SLR method.

to the failure of the model. They used a ordinal regression model with nonzero controlresponse probability on the chicken embryo data.

We 1rst consider to model the exposure data and the control data for each virusseparately. The models we used are the cumulative odds model, the continuation ratiomodel, and adjacent category odds model. We use the score statistics with the SLRmethod, described in Section 5, for testing goodness of 1t. The results are listed inTable 7. The proportional odds model and adjacent category odds model do not 1tthe Tinaroo and control data, whereas, the continuation ratio model 1ts well. For theFacey’s Paddock and control data, all three models 1t.

Since both experiments share the same control data, it is appropriate to analyze theentire data set (two viruses and control). We 1t a model with three sets of parameters:a set of spontaneous baseline parameters, and two sets of virus-speci1c intercept andslope parameters. For each of the three forms summarized in Table 7, the full model


Table 6Chicken embryo data and 1tted values using continuation ratio regression

Virus Inoculum titre Number Number Number of(PFU=egg) not deformed deformed deaths

Control 0 17 (16.15) 0 (0.46) 1 (1.38)

Tinaroo 3 18 (17.05) 0 (1.71) 1 (0.24)20 17 (15.48) 0 (3.05) 2 (0.47)

2400 2 (5.43) 9 (7.36) 4 (2.06)88,000 0 (1.58) 10 (9.29) 9 (8.12)

Facey’s 3 13 (14.34) 1 (0.60) 3 (2.06)

Paddock 18 14 (10.45) 1 (1.75) 4 (6.80)30 9 (8.09) 2 (2.11) 8 (6.80)90 2 (3.64) 1 (2.48) 17 (13.88)

Table 7Tests of parallelism for chicken embryo regression models

Model form Data set Score Degrees of p-Valuestatistics freedom

Cumulative Tinaroo and control 13.3 7 0.065Paddock and control 1.3 7 0.98All data 79.9 10 5:3 × 10−13

Continuation ratio Tinaroo and control 1.6 7 0.97Paddock and control 6.9 7 0.44All data 3.8 10 0.95

Adjacent category Tinaroo and control 30.6 7 7:4 × 10−5

Paddock and control 9.1 7 0.25All data 20.5 10 0.024

can be expressed as follows:

E(Y?is |W?

is = 1) = H (�s0 + �s1xi1 + �s2xi2 + �1xi1di + �2xi2di); s = 1; 2;

where

xi1 =

{1 Tinaroo;

0 otherwise;xi2 =

{1 Facey′s Paddock;

0 otherwise

and di is log(dose+1) for the ith observation. The score test results show that, assumingparallel slopes across severity categories, only the continuation ratio model 1ts the entiredata set. The 1tted cell counts are listed in Table 6.


8. Discussion

In this paper we have described a uni1ed form for di=erent logit models of ordinalresponses. The methods we propose avoid the need to develop di=erent model-1ttingprocedures for di=erent logit models. Both the SLR and the ELR approaches are consis-tent provided that the model for the mean has been correctly speci1ed. The asymptoticrelative e3ciency calculations demonstrate that the SLR approach is often nearly e3-cient. In an e=ort to obtain more e3cient estimates, the ELR approach to consider thecorrelations among the indicator variables has been developed. In addition, we adaptedthe generalized score test to provide fast tests for parallelism and other model selectionhypotheses.

Finally, we note that a potential use of our work is to provide a framework for mod-eling correlated ordinal data. Various correlation patterns can be incorporated by mod-eling the covariance between the indicator variables from the correlated observations.There are two main methods in the literature on measures of association between corre-lated ordinal data. Using the correlation coe3cient as a measure of association, Milleret al. (1993) have used the GEE method for proportional odds model. Williamson etal. (1995), and Heagerty and Zeger (1996) used the global odds ratios to measure theassociation. Several authors have developed extensions of the continuation ratio model,i.e., the discrete hazard model, to multivariate responses; see Guo and Lin (1994),TenHave and Uttal (1994), Shih (1998) and Heagerty and Zeger (1999). A promis-ing direction for further research is to build a general longitudinal GEE approach forordinal modeling on the binary coding approaches of SLR and ELR.

Acknowledgements

This research was supported in part by NSF Grant DMS-0073044 and by the Uni-versity of Illinois Research Board. We thank the referees for insightful suggestions andadditional references.

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