american journal of epidemiology vol. 129, … · 192 armstrong and sloan the cumulative odds and...

AMERICAN JOURNAL OF EPIDEMIOLOGYCopyright © 1989 by The Johns Hopkins University School of Hygiene and Public HealthAU rights reserved

Vol. 129, No. 1Printed in U.S.A

ORDINAL REGRESSION MODELS FOR EPIDEMIOLOGIC DATA

BEN G. ARMSTRONG AND MARGARET SLOAN

Armstrong, B. G., and M. Sloan (School of Occupational Health, McGill U.,Montreal, Quebec, Canada H3A 1A3). Ordinal models for epidemiologic data. AmJ Epidemiol 1989; 129:191-204.

Health status is often measured in epidemiologic studies on an ordinal scale,but data of this type are generally reduced for analysis to a single dichotomy.Several statistical models have been developed to make full use of informationin ordinal response data, but have not been much used in analyzing epidemiologicstudies. The authors discuss two of these statistical models—the cumulativeodds model and the continuation ratio model. They may be interpreted in termsof odds ratios, can account for confounding variables, have clear and testableassumptions, and have parameters that may be estimated and hypotheses thatmay be tested using available statistical packages. However, calculations ofasymptotic relative efficiency and results of simulations showed that simplelogistic regression applied to dichotomized responses can in some realisticsituations have more than 75% of the efficiency of ordinal regression models,but only if the ordinal scale is collapsed into a dichotomy dose to the optimalpoint The application of the proposed models to data from a study of chest x-rays of workers exposed to mineral fibers confirmed that they are easy to useand interpret, but gave results quite similar to those obtained using simple logisticregression after dichotomizing outcome in the conventional way.

asbestosis; regression analysis; statistics

Health status is often measured in epi-demiologic studies on an ordinal scale. Forexample, extent of symptoms of physicaldisease may be measured on a scale such as"absent" to "mild" to "severe", and psycho-metric evaluations are frequently ordinal.In a particular example pursued further inthis article, chest x-rays of men exposed todust are commonly read for small opacitieson a four- or 12-point ordered scale. Dataof these type are sometimes analyzed asnumerical scores, but such an approach is

Received for publication August 6, 1987, and infinal form June 9, 1988.

From the School of Occupational Health, 1130 PineAvenue West, Montreal, Quebec, Canada H3A 1A3.(Send reprint requests to Dr. Ben G. Armstrong atthis address.)

Research supported by the Institut de recherche ensante et en securite du travail du Quebec, through itsfunding of the Equipe Associee de Recherche en epi-demiologie des lesions professionelles.

only strictly valid if intervals between con-secutive points on the scale can be consid-ered equivalent. To avoid this assumption,the scales are often dichotomized and ana-lyzed using standard techniques for binarydata. Although valid, this approach losesinformation by collapsing some categoriesof the original scale. Statistical methodswhich respect the ordinal nature of thiskind of response data have been developed,and have recently received considerable at-tention in the statistical journals. Althoughsome of these articles have used epidemio-logic data to illustrate methods, the meth-ods do not appear to have been widely usedin substantive reports of epidemiologicstudies. The purpose of the present articleis to provide an introduction to these tech-niques which is accessible to epidemiolo-gists, and to evaluate their usefulness forepidemiologic research.

191

192 ARMSTRONG AND SLOAN

THE CUMULATIVE ODDS ANDCONTINUATION RATIO MODELS

Statistical models of the dependence ofan ordinal variable on one or more explan-atory variables are termed "ordinal regres-sion models" (1). Two of these statisticalmodels, the cumulative odds model and thecontinuation ratio model, will be discussedin detail here. They will be illustrated firstby the data presented in table 1 from ahypothetical survey of symptoms measuredon a three category ordinal scale (none,mild, severe) in two groups.

Following the notation of McCullagh andNelder (2), the integers 1, k act as labelsfor the k ordered response categories; andiTj,j = 1, k are the multinomial probabil-ities of being in each category. In all ordinalregression models, the Vj depend on thevalues of a vector of explanatory variablesx through regression parameters.

The cumulative odds model

Walker and Duncan (3), in a naturalextension of the logistic model for binaryresponse data, proposed the model:

logit(7,) = ln(7,/(l - 7,)) ( 1 )

= 0j- PTx, j = 1....A - 1

where -y, = *\ +....+ *-, are cumulative prob-abilities of being in one of the first j cate-gories, ln( •) is the natural logarithmic func-tion, and dj and /3 are unknown parameters.McCullagh (1) considers this model ingreater detail, calling it the "proportionalodds" model. The parameters 0, representthe baseline logits of cumulative responseprobabilities in a person for whom x = 0,and /S represents the "regression" parame-ter through which the effect of the explan-atory variables is mediated.

The cumulative odds model is sometimesdescribed as "grouped-continuous", since itmay be derived by assuming an unobservedunderlying continuous response variable,the observed ordinal response variablebeing formed by taking contiguous inter-vals of the continuous scale, with cut-pointsunknown. This interpretation motivated

the development of these models, and maybe useful in some applications, but is notnecessary. McCullagh (1) describes somefurther generalizations within the groupedcontinuous class of models.

Model 1 may be motivated from our il-lustrative data. The table may be simplifiedto a 2 x 2 table by dichotomizing the re-sponse scale either by collapsing levels 2and 3, or levels 1 and 2 (these constituenttables are also shown in table 1). The for-mer gives an estimated odds ratio of 1.71(95 per cent logit confidence interval (CI)= 0.89-3.29) and a likelihood ratio againstthe null hypothesis of no association of 2.68(p = 0.05, one-sided), the latter gives anodds ratio of 2.1 (95 per cent CI = 0.69-6.42) and a likelihood ratio of 1.83 (p =0.09, one-sided). Either analysis is a validapproach to investigating the dependenceof the response on group membership, buteach one discards some information whichis pertinent to that dependence. The cu-mulative odds model assumes that in the

TABLE 1

Hypothetical two group example of ordinal responsedata

Original data

Response categoryTotal

None Mild Severe

Group 1 80 15 5 100

Group 2 70 20 10 100

Constituent tables of the cumulative odds model

1 2. , Mild or None _None , Severesevere or mild

Group 1 80 20 95 5

Group 2 70 30 90 10

Constitutent tables of the continuation ratio model

1 2. t Mild or . . . . . „None Mild Severesevere

Group 1 80 20 15 5Group 2 70 30 20 10

ORDINAL REGRESSION MODELS FOR EPIDEM1OLOGIC DATA 193

hypothetical population from which thissample was drawn, the odds ratios fromeach of the two possible dichotomies arethe same. If x = 0 for group 1, and x = 1for group 2, this common odds ratio isobtained from expression l a s e " . Becausethe data of the 2 X 2 tables obtained fromeach dichotomy are correlated, estimationof the common odds ratio and inferenceconcerning it cannot be done using tech-niques (Mantel-Haenszel, simple logisticregression, etc.) for obtaining summary in-formation from independent strata. The pa-rameters 8j and /J may, however, be esti-mated by maximizing the multinomial like-lihood induced by the model 1, andinference effected using likelihood ratios,or the asymptotically equivalent scoreor Wald (j(/3 - po)/SE(p)\2) chi-squaredtests. Other adequate approaches to signif-icance testing are possible, for example inthe two-sample case the score statistic isthe same as Wilcoxon's average rank sta-tistic (1); these tests are not stressed here,since our emphasis is on estimation. Con-fidence intervals may be estimated mostconveniently from standard errors on theassumption that the sampling distributionof parameters is normal. Computational is-sues are discussed below.

The data of table 1 give a maximumlikelihood estimate for 0 of 0.56, and hencefor the common odds ratio, e", of 1.74 (95per cent CI = 0.92-3.33). This estimate liesbetween those obtained from each of the 2X 2 tables after dichotomizing the three-point ordinal outcome, as is intuitively rea-sonable. The likelihood ratio against thenull hypothesis that 0 = 0 is 2.91 (p = 0.04,one-sided), slightly higher than the largestof the two likelihood ratios from the 2 x 2tables.

The continuation ratio model

Replace the cumulative probability ofbeing in one of the first j categories in thecumulative model (y,) by the probability ofbeing in category j conditional on being incategory; or greater, 5; = x ;/(l — T,-I). This

gives:

j=l....k-l, (2)

where

logit («,-) = lnfVU - «,)}= ln|7r,/(l -

This model of conditional odds has beencalled the continuation ratio model (4), andit is essentially the proportional hazardsmodel proposed by Cox (5) for survival datain discrete time. Note that, although theparameters 0, and /? in model 2 play ap-proximately the same role as those giventhe same letters in model 1 (representingbaseline category parameters and a regres-sion parameter, respectively), they are notdirectly comparable, since one model pre-dicts cumulative probabilities, and theother conditional probabilities ("hazards").

For the illustrative data, the continua-tion ratio model corresponds to partition-ing the table into two 2 x 2 tables, but in adifferent manner than for the cumulativeodds model. The tables are shown in table1: the first is obtained by collapsing re-sponse levels 2 and 3 as before; but thesecond is obtained by discarding the re-sponses in the first category, and compar-ing response category 3 to category 2. Thislatter gives an odds ratio of 1.5 (95 per centCI = 0.42-5.32) and a likelihood ratio of0.40 (p > 0.20, one-sided). In the contin-uation ratio model, it is assumed that theunderlying population odds ratios fromthese two tables are the same and equal toef. Estimation of the parameters 0, and @,and inference concerning them, may be car-ried out by maximum likelihood. Followingthe argument of Cox (5, 6), the strata in-duced by the continuation ratio model maybe considered "conditionally independent",so that estimation and inference may pro-ceed as for independent strata of binaryresponse data.

The continuation ratio model fitted tothe data in table 1 gives a maximum like-lihood estimate for /3 of 0.51, and hence forthe common odds ratio, eP, of 1.67 (95 per


cent CI = 0.93-2.97). The likelihood ratioagainst H0:P = 0, is 2.8 (p = 0.05, one-sided).

Generalizations

The cumulative odds model, 1, and thecontinuation ratio model, 2, have similarforms, and can be generalized in parallelways.

First, the right hand side of expressions1 and 2 can be generalized. As they stand,the models can include any number of ex-planatory variables ("covariates") in thevector x, and can thus be used to accountfor confounding variables and examine in-teractions between explanatory variables inthe same way as the simple logistic regres-sion model for dichotomous outcome (withanalogous assumptions); explanatory vari-ables may be numerical (discrete or contin-uous) as well as categorical. The basic as-sumption of the models, that covariate ef-fects (0Tx) are additive (on the logisticscale) to "level" effects (0,) may be relaxedby allowing the regression parameter £ todepend on level (i.e., allow interaction be-tween covariates and levels). Thus, for thecumulative odds model:

(3)

= 6j + 0/x. (4)

Fitting model 3 or 4 to a simple two groupdata set would give a "saturated" model(one in which all cell frequencies are fittedexactly), because there would be as manyparameters as cells; however, with morecomplex covariate structures, this need notbe the case.

In most applications, the usefulness ofmodel 3 or 4 is to test the assumption ofproportionality (also called parallelism)which may be formally stated as the hy-pothesis: /Si = /92 = . . . = /3*-i. Where severalexplanatory variables are included in themodel, so that /3 is a vector, it may beinteresting to consider models in which theeffect of some explanatory variables is par-

and for the continuation ratio model:

allel, but the effect of others is not; forexample, we may wish to test for parallel-ism of variable effects separately.

Second, the logit "link" function of bothmodels may be replaced by others, as withbinary outcome data. The probit functionis very similar to the logit; the complimen-tary log-log link, /(•) = ln[ln{l - (•)}]makes the cumulative odds equivalent tothe continuation ratio model (7); the log-log link, /(•) = ln{— ln(•)} has also beensuggested. In some circumstances, log andidentity links, corresponding to relativerisk and risk difference models with di-chotomous outcome (8), may also be useful.

Testing model assumptions

The goodness of fit of the cumulativeodds or continuation ratio model to a dataset may be examined in various ways. First,the likelihood ratio between the proposedmodel and the "saturated" multinomialmodel in which every cell count is fittedexactly (the "residual deviance") asymptot-ically follows a chi-squared distribution ifall systematic variation has been accountedfor by the proposed model. A residual de-viance very much higher than the residualdegrees of freedom is thus indicative of apoor fit. The approximation of the residualdeviance to a chi-squared distribution ispoor when cell frequencies are low, how-ever, and in particular if one or moreexplanatory variables is continuous, thismethod should not be relied on (2). ThePearson chi-square (2{(0 - E)2/E}) for cellfrequencies has similar properties. Second,a more directed test of model assumptionsis achieved by comparing the fit of a model(usually via the likelihood ratio) with aspecific generalization, such as one of thosediscussed in the previous section. Thus,choosing the appropriate combination ofcovariates and interactions between themmay proceed in exactly the same way as forlogistic regression with dichotomous out-come; testing proportionality (parallelism)may be done by comparing the log likeli-hood achieved by model 1 or 2 with thatachieved by model 3 or 4. (In simple situa-

ORDINAL REGRESSION MODELS FOR EPIDEMIOLOGIC DATA 195

tions, such as the two group case, this isequivalent to examining the residual devi-ance). Comparing the fit of models withdifferent links, or completely different or-dinal regression models (e.g., the cumula-tive odds vs. the continuation ratio model),can be carried out informally by comparinglikelihoods (9), or more formally by nestingthe two models in a more general "mixture"model (10). Analysis of residuals, whichmay also be useful, is discussed by Mc-Cullagh (1) but not described here.

For the data of table 1, the fitted cumu-lative odds and the continuation ratiomodels gave residual deviances of 0.18 and0.03, respectively, on one degree of freedom.Viewing these statistics as general indica-tors of goodness of fit, or of tests of propor-tionality or parallelism (& = /32 in models3 and 4), we see that both models fit well,and that there is no evidence for differentunderlying odds ratios in the two constitu-ent tables defined by the two possible cut-points.

Computational issues

The cumulative odds model may be fittedto data with fairly general covariate struc-ture (numerical or categorical, includinginteractions), using the SAS supplementallibrary program PROC LOGIST (11). Theprogram output gives likelihood ratio andscore statistics as well as maximum likeli-hood estimates (with standard errors) ofthe 8j and /?. Neither residual deviance nora test for parallelism is provided withPROC LOGIST. However, these test sta-tistics can be obtained with modest addi-tional ad hoc calculations (Appendix 1).

The cumulative odds model may beshown to be a generalized linear model witha "composite link" (12). Thus, the GLIMpackage (13) may be adapted to fit thismodel, and Hutchison (14) has published aGLIM MACRO for this purpose. Thismethod allows testing for parallelism, butis more cumbersome to use than PROCLOGIST, and works only for fairly simplecovariate structures (it cannot be used withcontinuous covariates).

As mentioned above, the continuationratio model may be fitted by consideringthe tables into which the data are parti-tioned as independent strata of binary re-sponses. If the data are restructured in thisway, any package capable of logistic regres-sion can be used to estimate parameters ()9and dj), and to calculate likelihood ratios.The GLIM package is well suited to this,allowing links other than the logit to beused, as well as having convenient mecha-nisms for handling categorical as well asnumerical variables. Iyer (15) has givendetails of how continuation-ratio modelsmay be fitted using GLIM. In addition tothe explanatory variables of interest, a cat-egorical variable ("LEVEL") indexing eachof the strata formed by partitioning is re-quired from which the level parameters, dj,are estimated. Parallelism may be tested bytesting the interaction of "LEVEL" withthe explanatory variable(s).

EXAMPLE: FIBROSIS IN MINERS

Data from a cross-sectional study of ra-diographic abnormalities in 244 past andpresent miners exposed to tremolite fibers(16) are used as an example. The responseof interest was the profusion of small opac-ities on the chest x-ray, read according tothe International Labour Office 1980 clas-sification (17) on a 12-point ordered scale,conventionally labelled; 0/-, 0/0, 0/1, 1/0,1/1,1/2, 2/1, 2/2, 2/3,3/2, 3/3,3/4. A majorobjective of the study was to examine therelation between profusion of small opaci-ties and exposure to fibers (here measuredas a cumulative exposure index in units offiber/ml-years). Potentially confoundingvariables included age and smoking habit.

Table 2 shows profusion score brokendown by cumulative exposure, age, andsmoking habit. As is common in this typeof study, men are concentrated in the lowercategories of the scale. Three of the profu-sion categories (0/-, 2/1,3/4) were not usedin the study, and thus could be droppedfrom consideration without losing infor-mation. Estimates of parameters from verysparse data may not be consistent (18).


TABLE 2

Example: profusion of small opacities in miners exposed to fibers

No.Profusion of small opacities

0/- 0/0 0/1 1/0 1/1 1/2 2/1 2/2 2/3 3/2 3/3 3/4

All men 244 182 17 16 16 1

By cumulative exposure (fiber/ml-yeare)0-10

-20-100-200>200

9264531618

794838

89

64502

44350

24325

11210

01101

01101

01000

00001

Although inconsistency is unlikely to be aproblem here, computational difficultieswere experienced in maximizing likelihoodswhen the higher categories with very smallfrequencies were left separate; therefore,scores of 2/3 and above were collapsed intoa single category, leaving seven. We fittedthe cumulative odds, continuation ratio,and simple logistic model, the latter bydichotomizing the profusion score between0/1 and 1/0, as is customary to define "nor-mals" and "abnormals". Age and cumula-tive exposure were included in the modelsas continuous explanatory variables, andsmoking as a factor of three levels (neversmokers, ex-smokers, and current smok-ers).

Table 3 shows the components of theestimated regression parameter /S (with allvariables in the model), the correspondingodds ratios with 95 per cent confidenceintervals, and the likelihood ratio againstthe null hypothesis that the relevant com-ponents of (8 are zero. As noted above, theregression parameter /}, and hence the com-mon odds ratio, has a different interpreta-tion for the cumulative odds model than forthe continuation ratio model; those for thesimple logistic regression model are com-parable to those of the cumulative oddsmodel if the latter model holds.

Profusion score was significantly associ-ated with age, smoking, and cumulativeexposure under all three forms of analysis,although slightly less so (judging by thelikelihood ratio tests) in the simple logisticthan the two ordinal regression models.Likelihood ratios for smoking and cumula-

tive exposure were similar for cumulativeodds and continuation ratio models butthat for age was higher in the continuationratio model. The global likelihood ratio forall three factors simultaneously was 57.0,75.1, and 76.8 for the simple logistic,cumulative odds, and continuation ratiomodels, respectively.

There was no consistent pattern of dif-ference between the odds ratios estimatedfrom the simple logistic and from the cu-mulative odds model. Standard errors ofthe regression parameters were slightlysmaller in the cumulative odds model thanin the simple logistic model, reflecting someincrease in efficiency in the former. Theodds ratios from the continuation ratiomodel were lower than the other two. Thisis an expected result due to the differentmeaning of /3 and hence the common oddsratio in this model (see "Discussion" sec-tion); the differences are small, however,relative to the width of the confidence in-tervals.

The overall multinomial log likelihoodwas very similar for the cumulative odds(—200.4) and the continuation ratio(—199.5) models, which have an equal num-ber of parameters (10). The binomial loglikelihood for the simple logistic model(-88.1) is not comparable. None of theoverall likelihoods should be interpreted asmeasures of goodness of fit, since continu-ous exposure variables are used- Likelihoodratio tests of model assumptions againstspecific alternatives are shown in table 4.There appears to be no evidence for non-linearity of age and cumulative exposure


TABLE 3

Example: fibrosis in miners—results of model fitting

Factor Model (3 (SE*)Oddsratio (95% CIt) Likelihood ,

ratio (df*) P v a l u e

Age (by 10-year age groups)

Smoking (relative to neversmokers)

Ex-smokers

Current smokers

Cumulative exposure (per 100fiber/ml-years)

Simple logisticCumulative odds

Continuation ratio


Continuation ratio


Continuation ratio


Continuation ratio

0.84 (0.18)0.76 (0.15)

0.68 (0.12)

0.82 (0.69)1.05 (0.60)

0.71 (0.44)

1.70 (0.71)1.88 (0.62)

1.39 (0.45)

0.35 (0.12)0.38 (0.09)

0.24 (0.06)

2.322.14

1.97

2.272.86

2.03

5.476.55

4.01

1.421.46

1.27

(1.63-3.30)(1.59-2.87)

(1.58-2.50)

(0.59-8.78)(0.88-9.26)

(0.86-4.82)

(1.36-22.01)(1.94-22.09)

(1.66-9.70)

(1.12-1.80)(1.23-1.74)

(1.12-1.44)

27.231.2

39.1

8.212.5

12.6

10.917.3

17.8

(1)(1)

(1)

(2)§(2)§

(2)§

(1)(1)

(1)

<0.001<0.001

<0.001

0.O20.002

0.002

<0.001<0.001

<0.001

• SE, standard error. t CI, confidence interval. X df, degrees of freedom.§ Likelihood ratio for removing both smoking variables.

TABLE 4

Example: testing model assumptions

Additional term in the model

Linearity of effectsAge2

Cumulative exposure2

Interactions between explanatory variablesAge«8mokingSmoking* cumulative exposureAget* cumulative exposure

Interactions between level and explanatoryvariables (separate slopes models)

LeveWageLevel* smokingLeveWcumulative exposure

Level* (age + smoking + cumulative exposure)(all effects allowed to depend on level)

dft

11

221

585

18

Simplelogistic

0.30.5

0.10.10.1

--

-

-

Likelihood ratio

Cumulativeodds

0.40.6

0.40.10.2

6.6§6.3§5.4§

20.1

Continuationratio

2.23.3

0.20.20.0

8.08.02.5

18.8

t df, degrees of freedom.X Age dichotimized at 60 years to define interaction.§ Approximate values (see Appendix 1).

effects in the simple logistic or cumulative of conventional statistical significance (podds model. The likelihood ratio for a quad- = 0.07). There was no evidence for inter-ratic cumulative exposure effect in the con- action between age, smoking, and cumula-tinuation ratio model is on the borderline tive exposure in any of the three models,


nor for interaction between outcome leveland these variables in the cumulative oddsor continuation ratio models (i.e., againstparallelism of effects of age, smoking, orcumulative exposure).

Subsets of the data were also investigatedusing the three models, with similar generalresults. In no case did major issues of inter-pretation depend on choice of model. Thereduction in standard error of /S from usingthe cumulative odds rather than the simplelogistic model was never greater than thatin the analysis shown.

THE EFFICIENCY OF ANALYSES BASEDON THE CUMULATIVE ODDS MODELRELATIVE TO THOSE BASED ON THE

SIMPLE LOGISTIC MODEL WITHDICHOTOMIZED OUTCOME

The primary motivation for use of anordinal regression model such as 1 or 2,rather than collapsing the outcome variableto just two groups and using conventionalmethods, is a desire to use all availableinformation, and hence gain more power ininvestigating the effect of risk factors onoutcome. However, in the example consid-ered, this power gain was not very pro-nounced, and use of ordinal regressionwould not have changed interpretation ofresults. We have examined theoreticalpower gain by calculating, for various situ-ations, the asymptotic relative efficiency of

the logistic regression analysis using dichot-omized outcome, compared with an analy-sis using the cumulative odds model.

Asymptotic relative efficiency is definedas the limit, as sample sizes increase, of theratio of the sample sizes required for thetwo methods, in order that each achieve thesame power (or equivalently the same pre-cision) when close to the null hypothesis(19). In this case, this is also equal to theratio of variances of the two estimates of 0.For simplicity, we consider the comparisonof response in two equally sized groups.Because we are initially trying to show themaximum power gain by using the cumu-lative odds model, we look first at the sit-uation in which under the null hypothesiseach category of response is equally prob-able, since it is in this situation that thecumulative odds model is most powerful.Under these assumptions, a simple formulafor asymptotic relative efficiency can bederived (see Appendix 2). The value of this,expressed as a percentage, is shown in table5 for ordinal outcome variables with 3-10levels. Cumulative odds regression with koutcome levels is compared with each of (k- 1) simple logistic regressions, where thedichotomous outcome for the latter anal-yses are created by taking the single cut-point after each of levels 1,..., (k — 1) inturn. Thus, for example, if the outcome hasfour levels, simple logistic regression has 60per cent of the efficiency of cumulative odds

TABLE 5

Asymptotic relative efficiency (per cent) of simple logistic regression compared with cumulative odds regression,comparison of two groups, each category of outcome equally probable

>1>2>3>4>5>6>7>8

(2)t

(100)

3

7575

4

608060

No. of levels of outcome {k)

5

50757550

6

4369776943

7

386375756338

8

33577176715733

9

3053687575685330

* Definition of positive outcome for simple logistic regression.t At two levels of outcome, cumulative odds and simple logistic regression are identical.


regression if the dichotomy is made by cut-ting after level 1 or level 3, but 80 per centrelatively efficient if the cut is after level 2.This exercise shows that if the dichotomyfor simple logistic regression is close to itsoptimal point (creating equal numbers of"positive" and "negative" responders), thenpower gain using the cumulative oddsmodel is modest, since the relative effi-ciency of the simple logistic regression isbetween 75 and 80 per cent, depending onthe numbers of categories used.

Also discernible indirectly from table 5is the fact that power gain on increasingthe number of categories of outcome suffersfrom a law of quickly diminishing returns.In fact, if we assume an underlying contin-uous outcome, so that cut-points may bechosen arbitrarily, we may directly inves-tigate the efficiency of cumulative oddsmodels as a function of the number of levelsused (again, assumed chosen to given equalprobability in each category). Maximumprecision of estimated /S (equivalent tomaximum power) in large samples may beconsidered as the limit as number of out-come levels increase. Relative to this limit,simple logistic regression (dichotomizing atthe median outcome level) is asymptoti-cally 75 per cent efficient, and the relativeefficiency of ordinal regression using fromthree to nine groups is 89, 94, 96, 97, 98,98, and 99 per cent, respectively.

Since the variance of /3 estimated fromthe cumulative odds model is minimum ifTT,J = l/k (see Appendix 2), potential gainfrom using ordinal regression methodswould be expected to be smaller if the vtJ

vary. For example, if the probability offalling in each of seven levels was the sameas for the real example presented above(i.e., 75, 7, 7, 7, 2, 1, and 2 per cent), thenthe efficiency of simple logistic regressionrelative to cumulative odds ordinal regres-sion is 97 per cent if the cut-point for thesimple logistic regression is after the firstlevel used, 77 per cent if the cut-point isafter the second level (the conventional >1 /0 used in the example), and 54, 26, 16, and10 per cent if the cut-point is after levels 3,4, 5, and 6, respectively.

To test the applicability of asymptoticefficiencies to small data sets with substan-tial risk differences between groups, a smallsimulation study was carried out. Onehundred data sets, each containing 100 sub-jects divided into two exposure groups, weregenerated according to the cumulative oddsmodel. The parameters for the simulationwere based on those found in a subset ofdata from our example. Specifically, fiveresponse categories, 0/0, 0/1, 1/0, 1/1, 1/2+, were assumed, with probabilities of0.904, 0.040, 0.027, 0.029, 0.007, respec-tively, in the less exposed group. The effectparameter /3 was taken as 1.10, equivalentto an odds ratio of three between exposuregroups. Each data set was then analyzedusing a simple logistic model (defining"positive" as >l/0 as before), a cumulativeodds model, and a continuation ratio model.

Results of the simulations, shown in ta-ble 6, broadly confirm the applicability ofasymptotic efficiencies to this situation.The empirical efficiency of the simple lo-gistic regression estimate of # relative to

TABLE 6

Results of simulations comparing analyses by the simple logistic, cumulative odds, and continuation ratio models

ModelMean of estimated 0*

(CIt)Standard deviation of

estimated /3 (CIt)Mean of test statistic^

(CIt)significantat p < 0.05

Simple logisticCumulative oddsContinuation ratio

1.19 (1.08-1.30)1.16 (1.06-1.26)0.85 (0.77-0.93)

0.55 (0.48-0.63)0.50 (0.44-0.58)0.42 (0.37-0.49)

6.7 (5.7-7.7)8.8 (7.6-10.0)7.5 (6.3-8.7)

758476

* True /3 -» 1.1.t CI, 95 per cent confidence interval.X Likelihood ratio test against null hypothesis: /3 • 0.


that obtained from the cumulative oddsmodel (informally estimated as the ratio ofthe variances of the two estimates of /3 overthe 100 simulations) is 83 per cent, veryclose to the asymptotic efficiency of 82 percent. A comparison of the two mean likeli-hood ratios against the null, /3 = 0, alsoreflect a modest information gain from us-ing the cumulative odds model. If data weresimulated to follow a model other than thecumulative odds, we would generally expectto see results less favorable to this model.

Direct comparison of analyses based onthe continuation ratio model with thosebased on the other two is not possible,because of differences between the modelsreflected in the different interpretation ofthe parameter /9; however, we expect thecontinuation ratio model to have propertiessimilar to the cumulative odds model inthis respect. Results for the continuationratio model from the simulations (whichshow efficiency intermediate between thetwo other methods) would be more favora-ble if the data had been simulated to followthis model.

DISCUSSION

Other models for ordinal response data

Polytomous logistic regression modelsfor nominal (unordered categorical) re-sponses are gaining popularity among bio-statisticians and epidemiologists (20, 21)and may also be applied to ordinal re-sponses. In its general form, this type ofmodel may be fitted with repeated simplelogistic regressions (22) and is available instandard statistical packages such as SASand SPSSX. A polytomous regressionmodel for x-ray category on age, smoking,and cumulative exposure was fitted (usingSAS PROC LOGIST) to the example de-scribed above. The model required 30 pa-rameters. The six representing the effectsof cumulative exposure (for increase in logodds per 100 fiber/ml-years of having pro-fusion score 0/1, 1/0, 1/1, 1/2, 2/2, 2/3+relative to having score 0/0) are shown intable 7. The (Wald) chi-square for cumu-

TABLE 7

Example: application of polytomous logistic regressionmodel for nominal data

Level

0/00/11/01/11/22/12/22/3+

Regressionparameter (SE")(per 100 fiber/ml-

yeara)

-

0.01 (0.26)0.05 (0.25)0.46 (0.15)

-0.04 (0.64)-

0.46 (0.23)0.74 (0.21)

Odds ratio (95% CIt)(relative to level 0/0)

1.0 (baseline)1.01 (0.61-1.68)1.05 (0.64-1.72)1.58 (1.18-2.13)0.96 (0.27-3.37)

-1.58 (1.01-2.49)2.10 (1.38-3.16)

* SE, standard error.t CI, confidence interval.

lative exposure was 15.1 on six degrees offreedom—a clearly significant effect (p =0.02), but less strongly so than for the moresparsely parameterized ordinal models dis-cussed above. The ordinality of the scale issuggested by the trend in odds ratios, butthe confidence intervals are wide, so that itwould be inappropriate to pay great atten-tion to deviations from this trend. Theoverall log likelihood was —191.5—almostexactly the improvement in fit over thecumulative odds and continuation ratiomodels that would be expected by chance,given the 20 extra parameters.

To overcome the problem of over-parameterization (in general, (p + 1) {k —1) parameters are required for p explana-tory variables and k response categories),most authors have suggested reduced formsof the general model. Some of these, whilenot implying ordinality, are especially suit-able for reflecting patterns typical in ordi-nal response data. Cox and Chuang (23)explore a number of possibilities alongthese lines, and they also illustrate the useof cumulative odds and continuation ratiomodels. Anderson (24) proposes a family of"stereotype regression models" for ordinalresponse data, which lie within the polyto-mous logistic regression model framework.Particular advantages of these models arethat order of response categories need notbe specified a priori, and tests for distin-


guishability of response categories are im-mediate.

We concur with the view of McCullagh(1, 25) and McCullagh and Nelder (2) thatif order of categories may be specified withconfidence a priori, models making thisordering a strong assumption (such as thecumulative odds and continuation ratiomodels) are preferable to the more flexiblelogistic models such as that of Anderson(24) because of their simpler interpretation.However, we acknowledge that this view iscontentious among statisticians. Perhapsthe attitude that data analysts take on thisissue will reflect the relative emphasis thatthey wish to give to a priori considerationsand to information contained in the data.A more practical impediment to wider useof general polytomous logistic models is theabsence of easy to use software for fittingthe more interesting reduced forms.

A substantial body of statistical litera-ture on the analysis of ordinal data hasbeen published in the contingency tabletradition (26). Since these models do notgenerally distinguish between explanatoryand dependent variables, they are less suit-able for the regression context that is con-sidered here. One approach in this traditionthat could nevertheless be applied is toassign a score to each category of outcome,and to include this score in the specificationof a log-linear model for cell probabilities.The assignation of scores introduces anundesirable arbitrary element into theanalysis, but, to avoid this, Chuang andAgresti (27) proposed taking the scores asunknown parameters that are estimatedfrom the data.

Choice of ordinal regression model

The cumulative odds and continuationratio models have several properties whichwe think make them attractive to epide-miologic data analysts:

1) Both models may be interpreted interms of odds ratios, which are familiar toepidemiologists.

2) The basic underlying assumption of

each model—equality of fts in models 3 or4—can in principal be tested simply (al-though this requires some computing be-yond that available in the major packagesfor the cumulative odds models), and canbe intuitively grasped as the equality ofodds ratios in 2 X 2 tables.

3) Packages exist (SAS PROC LOGISTand GLIM) which can estimate parametersand carry out significance tests.

4) Statistical models should be plausiblebiologically. However, in few epidemiologiccontexts is enough known about biologicmechanisms for this criterion to be evalu-ated. The grouped continuous interpreta-tion of the cumulative odds model is con-sistent with a mechanism in which regres-sor variables shift the location of anunderlying continuous response that fol-lows a logistic distribution.

Nature may not always be so obliging asto present investigators with data that con-form to convenient models. However, datawill often be insufficient to distinguish be-tween models—see for example Cox andChuang (23) or our example—and interpre-tation will depend little on the model used,in which case, choice of the most conven-ient model seems justified.

We turn now to the choice between thecumulative odds and continuation ratiomodels. If the strict cumulative odds model,1, holds (i.e., cumulative log odds ratios areconstant, say /S*), then the conditional logodds ratios (ft) of the more general contin-uation ratio model 4, will start at /3i = /}*,but then tend toward 0 asy increases. Thus,the cumulative odds model has been pro-posed as an alternative to the proportionalhazards (i.e., continuation ratio) model forsurvival data when hazard rates of groupsare thought likely to converge with time(28). (Essentially the same argument ex-plains why if the cumulative odds and con-tinuation ratio models 1 and 2 are bothfitted to a set of data, the estimated contin-uation odds ratio will be less than the cu-mulative odds ratio, as seen in the fibrosisexample.) In many data sets, the fit of thetwo models (models 1 and 2) is similar,


however, so that choice between them mustbe on a priori grounds.

A feature of the cumulative odds modelbut not of the continuation ratio model isthat it is unchanged if the ordering of thecategories is reversed. In some applications(such as to survival data), the direction ofordering appropriate for the continuationratio (proportional hazards) model is clearfrom the context. In other applications, thisis not so, so that the fact that differentresults may be obtained according to thisarbitrary choice is an undesirable featureof the model (although differences willoften be marginal; on reversing the orderfor the data in table 1, the estimate of #from the continuation ratio model changedfrom 0.51 to 0.52, and the likelihood ratioagainst the null changed from 2.8 to 2.9).

Concluding remarks

We concluded from our study of effi-ciency that simple logistic regression ap-plied to a dichotomized response measurehas only slightly less power than the ordinalregression methods discussed if the cut-point is made close to its optimal point.However, there may be reasons other thanefficiency for not dichotomizing (for ex-ample, investigating whether level-specificeffects are present). Furthermore, choice ofexactly where to make the cut-point forsimple logistic regression constitutes an un-desirable arbitrary element in the analysis.Sometimes (for example, with chest x-rays), the choice is strongly suggested byconvention, but the conventional cut-pointmight not be close to that optimal forachieving power. Thus, where avoiding lossof information is a major consideration, orif choosing a single cut-point to define apositive response involves an arbitrarychoice, the use of ordinal regression meth-ods are indicated.

REFERENCES

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2. McCullagh P, Nelder JA. Generalized linearmodels. London: Chapman Hall, 1983.

3. Walker SH, Duncan DB. Estimation of the prob-ability of an event as a function of several inde-pendent variables. Biometrika 1967;54:167-79.

4. Fienberg SE. The analysis of cross-classified data.2nd ed. Cambridge, MA: MIT Press, 1980.

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6. Cox DR. Partial likelihood. Biometrika 1975;62:269-76.

7. Laara E, Matthews JNS. The equivalence of twomodels for ordinal data. Biometrika 1985;72:206-7.

8. Wacholder S. Binomial regression in GLIM: esti-mating risk ratios and risk differences. Am JEpidemiol 1986;123:174-84.

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10. Atkinson AC. A method for discriminating be-tween models (with discussion). J R Stat Soc [B]1970;32:323-53.

11. SAS Institute Inc. SUGI supplemental libraryuser's guide, version 5 edition. Cary, NC: SASInstitute Inc., 1986.

12. Thompson R, Baker RJ. Composite link functionsin generalized linear models. Appl Stat 1981;54:167-79.

13. Payne CD, ed. The GLIM system release 3.77manual. Oxford: Numerical Algorithms Group,1985.

14. Hutchison D. Ordinal variable regression usingthe McCullagh (proportional odds) model. GLIMNewsletter 1985;9:9-17.

15. Iyer R. Continuation-odds model in ordinal vari-able regression. GLIM Newsletter 1985;10:4-8.

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17. International Labour Office. Guidelines for theuse of ILO international classification of radio-graphs of pneumoconioses. Geneva: ILO, 1980.(Occupational Safety and Health, series 22, rev.80).

18. McCullagh P. On the elimination of nuisance pa-rameters in the proportional odds model. J R StatSoc [B] 1984;46:250-€.

19. Cox DR, Hinkley DV. Theoretical statistics. Lon-don: Chapman and Hall, 1974:304-97.

20. Marshall RT, Chisholm EM. Hypothesis testingin the polychotomous logistic model with an ap-plication to detecting gastrointestinal cancer. StatMed 1985;4:337-44.

21. Thomas DC, Golberg M, Dewar R, et al. Statisticalmethods for relating several exposure factors toseveral diseases in case heterogeneity studies. StatMed 1986;5:49-60.

22. Begg CB, Gray R. Calculation of polychotomouslogistic regression parameters using individualizedregressions. Biometrika 1984;71:11-18.

23. Cox C, Chuang C. A comparison of chi-squarepartitioning and two logit analyses of ordinal paindata from a pharmaceutical study. Stat Med1984;3:273-85.

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25. McCullagh P. In: Discussion of Anderson JA. 27. Chuang C, Agresti A. A new model for ordinalRegression and ordered categorical variables. J R pain data from a pharmaceutical study. Stat MedStat Soc [B] 1984;46:22-3. 1986;5:15-20.

26. Agresti A. Analysis of ordinal categorical data. 28. Bennett S. Analysis of survival data by the pro-New York: John Wiley and Sons, 1984. portional odds model. Stat Med 1983;2:279-85.

APPENDIX 1

Calculating log likelihoods, residual deviances, and tests for parallelism in the cumulative odds model

It is useful to define a general multinomial model in which probability x^ of being in category j is a functionof a vector of explanatory variables x, and unknown parameters. A complete multinomial observation vector isdenoted y, = Oy.i,—.y.*). where yv is the count in the jth category of all persons with the same vector ofexplanatory variables x,. (When x includes a continuous variable, it is usually sensible to index each person i —l...n, so that for all i, yg =• 0 for all but one j , for which it is 1.) The log likelihood for the total data \y, ) is then

' = E I y. In (TJ) (1.1)t 1

where ln(-) denotes the natural logarithmic function. In the cumulative odds model, iu is dependent onparameters Q, and /3 through the cumulative probabilities yv according to the equations:

»« = 7« ~ 7.0-u. forj = 2 . . . . k - 1 (1.2)

xu, - 1 - 7«*-i>.

and 7U is given by

yv = 1/|1 + exp(-fl; + PT xv)\. j = 1 k - 1 (1.3)

The log likelihood can be maximized using any general maximization routine, for example, the BMDP programP3R (see reference 23 and the BMDP manual for some further details). Since use of such general programsrequires greater mathematical and computing expertise than does the use of SAS PROC LOGIST, it is worthconsidering ad hoc methods for calculating the useful statistics which PROC LOGIST does not provide—theresidual deviance and the test for parallelism.

The residual deviance is equal to twice the difference between the log likelihood in the model underconsideration and that in the "saturated" model in which all cells are fitted exactly. Since PROC LOGISTprovides the former, it is necessary only to calculate the latter. This is obtained by replacing X(, in expression1.1 by y^fZ yv i.e., the actual observed proportion of observations with this combination of explanatory variables

which fall into category j . Thus, for the illustrative data in table 1 the saturated log likelihood (/) is

-80 X ln(80/100) - 15 X ln(15/100) + •••• = -141.47.

Since the log likelihood for the cumulative odds model 1 is (from PROC LOGIST) 141.55, the residual devianceis 2 X (141.55 - 141.47) = 0.16.

To obtain the test for parallelism of /3s in the general cumulative odds model, we must first fit the "separateslopes" model 3, and compare the log likelihood obtained using this model with that (given by PROC LOGIST)for the more restricted cumulative odds model 1. Fitting the model in which no variable effects are assumedparallel, and hence all components of /3 are allowed to depend on level j , may be carried out by fitting ft - 1separate simple logistic regression equations with the dependent variable defined at each possible cut-point inturn. Since the intercept and slope terms of the simple logistic regression dichotomizing between levels j and j+ 1 correspond to $j and —ft, respectively, in model 3, equations 1.3, 1.2, and 1.1 can be used to calculate a loglikelihood.

For the data in table 1, simple logistic regression dichotomizing between levels 1 and 2 gives intercept Si =1.386 and slope jS, = 0.539; dichotomizing between levels 2 and 3 gives 92 = 2.944 and ft = 0.747. Applying 1.3and 1.2, we obtain from 1.1:

x,, = -y,, = l/ji + exp (-1.386 + 0.539 x 0)| = 0.80,

xu «« etc.

and / - - 80 x In (0.80) - . . . . = -141.47.

Hence, the likelihood ratio test for parallelism is 2 X (141.55 - 141.47) = 0.16. This is the same as the residualdeviance, because, in this simple case, the separate slopes model is the "saturated" model.


In practice, it is usually easier to deal with fitted cumulative counts from the simple logistic regressions

and to calculate the log likelihood as

where

We may wish also to obtain the fitted parameters and the log likelihood for models in which some componentsof /3 are assumed the same for all levels, whereas others depend on level—for example, to test parallelism forspecific variables only. Exact ad hoc calculations for this are tedious, but approximate estimates may beobtained by keeping those components of /3 which are assumed the same for all levels fixed at the estimatedvalue given for model 1 by PROC LOGIST, and offsetting these effects in fitting separate estimates forremaining components of /? as above.

The calculations above may be performed in simple cases by hand held calculators, or programmed quitesimply using a high level language. We have found the GLIM language suitable for this.

Finally, we should note a theoretical problem with model 3 that can sometimes hinder estimation. Since the0i vary with j , graphs of logit (7,) against x will intersect at some value of z, so that cumulative probabilitycannot be a monotone increasing function of; for all x. Thus, negative probabilities T, will be predicted forsome 1 values. Usually, the points of intersection do not occur within the domain of the observed x, so that thisdoes not present a problem.

APPENDIX 2

Derivation of asymptotic relative efficiencies

For the cumulative odds model 1 applied to the comparison of outcome in two groups (of size nx and n-2, nt

+ n? = n), the asymptotic variance of /S (here, the log odds ratio) close to the null hypothesis, (3 = 0, may beshown (for example, by McCullagh (1)) to be equal to

(nin2/3n) |1 - I x/|. (2.1)

(Since /? a 0, r, is approximately equal in the two groups.)Simple logistic regression after dichotomizing between levels m and m + 1 may be seen as a special case of

the cumulative odds model, giving asymptotic variance near the null

( r W 3 n ) / | l - ( £ r,V ~ ( 1 r,f\. (2.2)

Asymptotic relative efficiency of simple logistic regression relative to the cumulative odds model is thusexpression 2.2 divided by expression 2.1.

As McCullagh (1) observes, it is clear from expression 2.1 that variance is minimum if *-, = x2 = .... = rk =(1/k). Substituting this in 2.2 and 2.1 and dividing, we obtain the asymptotic relative efficiency for this specialcase:

|1 - (m/kf - [(k - m)/kf\/\l - H\/kV\ = 3m(k - m)/(kl - 1).

american journal of epidemiology vol. 129, … · 192 armstrong and sloan the cumulative odds and...

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