longitudinal analysis of deciduous tooth emergence: ii. parametric survival analysis in bangladeshi,...

Longitudinal Analysis of Deciduous Tooth Emergence:II. Parametric Survival Analysis in Bangladeshi, Guatemalan,Japanese, and Javanese Children

DARRYL J. HOLMAN1* AND ROBERT E. JONES2

1Department of Anthropology and Population Research Institute,Pennsylvania State University, University Park, Pennsylvania 168022Center for Demography and Ecology, University of Wisconsin-Madison,Madison, Wisconsin 53706

KEY WORDS dental anthropology; hazards analysis; studydesign; eruption; agenesis

ABSTRACT We present a form of parametric survival analysis thatincorporates exact, interval-censored, and right-censored times to deciduoustooth emergence. The method is an extension of common cross-sectionalprocedures such as logit and probit analysis, so that data arising from mixedlongitudinal and cross-sectional studies can be properly combined. Weextended the method to incorporate and estimate a proportion of agenic teeth.While we concentrate on deciduous tooth emergence, the method is relevant tostudies of permanent tooth emergence and other developmental events.

Deciduous tooth emergence data were analyzed from four longitudinalstudies. The samples are 1,271 rural Guatemalan children examined everythree months up to age two and every six months thereafter as part of theINCAP study; 397 rural Bangladeshi children examined monthly to age oneand quarterly thereafter as part of the Meheran Growth and DevelopmentStudy; 468 rural Indonesian children examined monthly as part of theNgaglik study; and 114 urban Japanese children examined monthly in studiesfrom 1910 and 1920. Although all four studies were longitudinal, manyobservations from the Guatemala and Bangladesh studies were effectivelycross-sectionally observed. Three different parametric forms were used tomodel the eruption process: a normal distribution, a lognormal distribution,and a lognormal distribution with age shifted to shortly after conception. Allthree distributions produced reliable estimates of central tendencies, but theshifted lognormal distribution produced the best overall estimates of shape(variance) parameters. Estimates of emergence were compared to otherstudies that used similar methods. Japanese children showed relatively fastemergence times for all teeth. Bangladeshi and Javanese children showedemergence times that were slower than are found in most previous studies.

Estimates of agenesis were not significantly different from zero for mostteeth. One or two central incisors showed significant agenesis that rangedfrom 0.1 to 0.8% in three of the samples; even so, failure to model the agenicproportion did not seriously bias the estimates. Am J Phys Anthropol105:209–230, 1998. r 1998 Wiley-Liss, Inc.

The study of dentition has a long traditionin physical and biological anthropology (Hat-ton, 1955; Jelliffe and Jelliffe, 1973; Smith etal., 1994; Tanner, 1986; Townsend and Ham-mel, 1990). While both deciduous and perma-nent tooth emergence have been examined

Contract grant sponsor: NSF; contract grant number BNS-8115586; Contract grant sponsor: NIH; contract grant numbersR01-HD26899-01 (REJ), F32-HD07994-02 (DJH); Contract grantsponsor: NICHD Population Center; contract grant numbers1-HD28263-01 (PSU), HD-05876 (UW).

*Correspondence to: Darryl J. Holman, 601 Oswald Tower,Population Research Institute, Pennsylvania State University,University Park, PA 16802. E-mail: [email protected]

Received 7 October 1996; accepted 31 October 1997.

AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 105:209–230 (1998)

r 1998 WILEY-LISS, INC.

in many human populations, important is-sues surrounding these biological mile-stones remain unresolved. Questions con-cerning the degree of sexual dimorphism,variability among populations, effects ofhealth and nutritional status of the motheror child on emergence, and the appropriateage standard for emergence remain unan-swered. Resolution of these issues has beenstymied in the past by the use of differentstudy designs and statistical procedures.

Differences in study design, such as cross-sectional vs. longitudinal, usually result fromexogenous factors like access to the studypopulation and time or cost constraints. Yetthese differences in study design shouldhave no inherent effect on statistical esti-mates of emergence times, provided properstatistical methods are used.

Here, we present a statistically properand powerful method that is applicable forthe analysis of tooth emergence data arisingfrom longitudinal studies. We also demon-strate that this method is an extension ofcommon cross-sectional procedures such aslogit and probit analysis, so that data aris-ing from mixed longitudinal and cross-sectional studies can be properly combined.A second focus extends the method to incor-porate and estimate the proportion of agenicteeth. Accordingly, we follow the lead ofHayes and Mantel (1958) who described anonparametric method for estimating me-dian times to emergence along with an agenicproportion. An agenic fraction introduces noanalytic difficulty if all individuals are fol-lowed longitudinally until all teeth that willemerge do so. When some observations areincomplete, the true proportion of agenicindividuals is unknown but can still beestimated. Estimating means or medianswhen observations are incomplete withoutseparate estimation of the agenic proportioncan potentially lead to serious biases.

We analyze tooth emergence data in fourculturally and biologically distinct popula-tions. Three of the four samples are fromrural developing country settings (Indone-sia, Bangladesh, and Guatemala), and thefourth is from Tokyo in the early 1900s.Study designs were similar in the four stud-ies so that the resulting emergence data areeither completely longitudinal observations

or a mixture of longitudinal with some cross-sectional observations. The methods of eachstudy were well documented and all birthand examination dates are exact.

Finally, we revisit the issue of the properparametric distribution to use for the analy-sis of tooth emergence (Klein and Palmer,1937; Kihlberg and Koski, 1954; Hayes andMantel, 1958). Specifically, whether a nor-mal, lognormal, or a shifted lognormal distri-bution1 best describes tooth emergence. Wepreviously used life tables to produce nonpar-ametric distributions for tooth emergencefrom two of the samples analyzed here (Hol-man and Jones, 1991; Jones and Holman,1991). The resulting nonparametric distribu-tions can be used to assess how well eachparametric form estimates parameters likemeans, medians, and variances. We are ad-dressing the same issues that Hayes andMantel (1958) did when they compared anonparametric cross-sectional analysismethod to probit analysis, except that wecompare two longitudinal methods: life tableestimates and parametric survival analysis.

STUDY DESIGNS

Study design can be divided into cross-sectional and longitudinal types, with a thirdcategory defined as longitudinal that alsoincludes some cross-sectional observations.Cross-sectional studies of tooth emergenceexamine children of different known ages,each at a single point in time. These studiesare easier and less expensive than longitudi-nal studies, which involve tracking of eachindividual, repeat examinations, and accu-rate record-keeping over an extended periodof time. In cross-sectional studies, observa-tions for which the event of interest (i.e.,emergence of a particular tooth) has oc-curred at or before the time of measurementare called responders. Non-responders areobservations for which the subject has notyet experienced the event. We usually think

1The term ‘‘shifted lognormal distribution’’ refers to a lognor-mal distribution for which the start of emergence is assumed tobegin prenatally. We use one month post-conception as the startof emergence, although conception is another reasonable startingpoint. Shifted lognormal distributions are estimated after addingsome constant (eight or nine months) to the emergence timesmeasured from birth. Other distributions, such as the logisticdistribution, have been used to describe tooth emergence as well.These distributions are reasonable alternatives that work un-changed with the methods given here.

210 D.J. HOLMAN AND R.E. JONES

of cross-sectional studies as involving a singleexamination of each subject, but an explicitor implicit observation is also required atthe start of the process leading to emer-gence. For most tooth emergence studies thebirth date and, less commonly, the estimateddate of conception is assumed to be thebeginning of the process of tooth eruption.

A longitudinal study follows a particularcohort over time, with examinations sepa-rated by some interval of time. Typically,longitudinal studies of emergence specifyone-month intervals between examinationsover a 2–4 year period. Under some studydesigns, children may be enrolled into thestudy at ages other than birth, and some ofthese subjects may have emerged one ormore of their teeth before the first examina-tion. These individuals would only contrib-ute one informative observation—their firstexamination—for the teeth that have al-ready emerged. Longitudinal studies thatinclude observations of this type are calledmixed longitudinal studies with ‘‘mixed’’ de-noting the presence of cross-sectional obser-vations.

The virtues of different designs for toothemergence studies have been discussed in anumber of previous publications with a vari-ety of conclusions rendered (see, e.g., Dahl-berg and Menengaz-Bock, 1958; Lysell et al.,1962; Smith, 1991). It seems to us that manyof the problems discussed arise from anincomplete distinction between differentstudy designs and the proper methods ofanalysis for the data collected under particu-lar designs.

For example, Dahlberg and Menengaz-Bock (1958) prefer a cross-sectional ap-proach to represent population characteris-tics because of the larger sample sizepossible. This conclusion is only true if cross-sectional observations are statisticallyequivalent to the same number of longitudi-nal observations. Smith (1991:150) assertsthat probit analysis takes no special advan-tage of longitudinal records and treats allsuch data as if they are cross-sectional, sothat there is no reason to prefer longitudinalrecords for analysis of growth events. Smithis correct in pointing out that probit analysistakes no special advantage of longitudinalrecords. The reason, however, is that probit

or logit models are not the proper methods touse with longitudinal data. When propermethods are used, a longitudinal study re-quires fewer observations than does a cross-sectional study to produce a given amount ofstatistical inference. That is, each longitudi-nal observation provides at least the sameand usually much more statistical informa-tion than does each cross-sectional observa-tion. We demonstrate and expand on thisrelationship in the Appendix.

Lysell and colleagues (1962) question theappropriateness of combining longitudinaland cross-sectional studies because of diffi-culties evaluating the combined results. Withcurrent statistical methods, however, longi-tudinal and cross-sectional observations canindeed be analyzed together. The preferencefor a longitudinal, cross-sectional, or mixedlongitudinal study need not hinge on themethod of analysis.

Our intention is not to criticize the work ofthese investigators. Many of the statisticalmethods we discuss in this article have beendeveloped only recently or have been littleknown outside of the specialized statisticalliterature of survival or event history analy-sis. Smith (1991), in fact, provides a usefuland comprehensive review of these differentstudy designs and analysis methods for dif-ferent types of dental event chronologies.She suggests that the best methods of analy-sis produce a ‘‘cumulative tooth emergencecurve.’’ She also discusses the difficultiesintroduced by missing data, truncation ofthe curve, and unequal observation inter-vals (see also Smith et al., 1994). These areimportant issues for proper analysis of toothemergence and other developmental eventdata. We address each of these issues andshow how survival analysis (a cumulativetooth emergence method) produces statisti-cally proper estimates from all availableinformation: cross-sectional data, longitudi-nal data over large or small intervals (offixed or varying lengths), as well as right-censored data.

MESSY DATA

Apractical problem with longitudinal stud-ies is that they usually produce ‘‘messy’’data. This occurs when some subjects exitthe study before they experience the event of

211DECIDUOUS TOOTH EMERGENCE IN FOUR POPULATIONS

interest (i.e., emergence of a tooth). An exitmay mean a child dies, the family movesaway, or for whatever reason is unavailablefor follow-up before all teeth emerge. Thestatistical literature refers to this type ofincomplete observation as right-censored.Proper accommodation of right censoring isimperative; analysis by conventional meth-ods that do not accommodate right censor-ing leads to exclusion bias (Elandt-Johnsonand Johnson, 1980). Consider, for example,an emergence study that follows a cohort ofinfants from birth to three years of age anddrops from the analysis children who do notemerge a second molar by the end of thattime. Dropping those children biases down-ward the estimate of the mean emergencetime for the second molar because the chil-dren who are dropped from the analysis aremore likely to have slower eruption. A simi-lar problem exists for children who, forwhatever reason, drop out before the studyfinishes. Dropping children whose tooth doesnot emerge from the analysis biases down-ward the estimate of mean emergence timefor that tooth.

Another messy aspect of longitudinal datais that observations are nearly always col-lected over intervals so that exact emer-gence times (i.e., times to the day, for ex-ample) are rarely ascertained.2 The intervalin which emergence is recorded is defined bytwo time points: the last visit for which thechild is seen without the tooth and the firsttime the child is seen with the tooth protrud-ing through the gingiva. A protocol of equallyspaced visits is rarely adhered to, either byplan or by execution. In many longitudinalemergence studies children are visitedmonthly, so emergence times are only knownto have fallen within an interval of 30 days.In other studies the length of the intervalchanges with time. For example, in theINCAP study from Guatemala (Delgado etal., 1975) the length of the observation inter-val changes with time. Children are exam-ined every three months for the first twoyears and every six months thereafter. Inter-vals of observation can differ by individual

as well. For example, in the Ngaglik study(Ngaglik Study Team, 1978) examinationswere scheduled every 35 days. Even so, oneor more of the visits were sometimes missedbefore emergence of a particular tooth, sothat a few intervals were 70 days, 105 days,or longer. A statistical method useful foremergence studies must accommodate alltypes of interval-censored observations andstill give unbiased statistical results.

A common method used to correct forinterval censoring is to take the midpoint ofthe observation interval as the emergencedate (Dahlberg and Menegaz-Bock, 1958).This correction has great intuitive appeal,but from a statistical point of view it has lessjustification and must be classified as an adhoc method. Still, when small intervals areencountered the correction probably resultsin small or undetectable biases in estimatesof mean or median emergence times. Wheninterpolation takes place over larger inter-vals, however, the biases in resulting esti-mates can become substantial. If the analyti-cal method assumes exact (instead ofinterval-censored) emergence times the mid-point correction method also biases stan-dard errors downward. Thus, midpoint cor-rection makes statistical comparisons amongteeth somewhat dubious within or amongstudies; t-tests evaluating statistical differ-ences will too often tend to reject the hypoth-esis of equal means. Rather than usingmidpoint interpolation, statistical methodsthat explicitly accommodate observationsover intervals should be used. The methodsgiven in this paper provide for proper treat-ment of right censoring and equal or un-equal interval censoring.

The methods we describe come from thefield of survival analysis (Elandt-Johnsonand Johnson, 1980; Namboodiri and Suchin-dran, 1987; Nelson, 1983; Wood et al., 1992).Survival methods have been developed spe-cifically for analysis of times to events, andto accommodate the types of incompletedata that arise from studies of times toevents. The methods given here are paramet-ric and can be used when the process under-lying the time to the event (i.e., tooth emer-gence) follows some known distribution.Covariates, or the effects of independentvariables, can be estimated with interval-

2We use the term ‘‘exact’’ loosely. It is unlikely that any studymeasures tooth emergence exactly. Rather, some studies rely onparents reports of date of emergence. These ‘‘exact’’ reports maycorrespond to some smaller interval of a day to a few weeks.


censored observations in a manner analo-gous to regression models, a topic that willbe discussed in a later paper.3

MATERIALS AND METHODSSubjects

Tooth emergence data used in this articlecome from four longitudinal studies wherechildren were followed over a period of sev-eral years. In each study, a tooth was consid-ered emerged if any part of the tooth had, ondirect inspection of the mouth, pierced thegum line (clinical emergence). The designsand protocols of all four studies were broadlysimilar, although each study had uniquecharacteristics such as age at recruitment,number and timing of visits, study length,and dropout patterns.

Japanese children. These data comefrom published reports (in Japanese) byKitamura (1917, 1942) and include longitu-dinal emergence times for two birth cohorts:49 children born in January 1910 and 65children born in January 1920. The childrenresided in the Ushigome-Ku, Yotsuya-Ku, orKoishikawa-Ku areas of Tokyo. Householdsin these areas were selected randomly withrespect to household income and father’soccupation.

Children were visited at monthly inter-vals for up to three years in both studies.Visits terminated at the end of a three-yearperiod, after all deciduous teeth hademerged, if the child died, or if the parentswithdrew the child from the study. Timesand the reasons for leaving were docu-mented. Each child’s nutritional status wascategorized as either good (39 cases), me-dium (54 cases), or bad (21 cases), but noobjective criteria were given for this particu-lar ranking. Kitamura recalculated the datain various ways but did not publish sum-mary statistics of emergence times. We pre-

viously estimated emergence times fromthese data using a nonparametric life tablemethod (Jones and Holman, 1991).

Javanese children. These data come fromthe Ngaglik project, a longitudinal investiga-tion of maternal and child health and nutri-tion, breast-feeding, and birth spacing dy-namics carried out in the late 1970s inCentral Java, Indonesia (Ngaglik StudyTeam, 1978; Hull, 1983; Jones, 1988). Adescription of the dental data and a life tableanalysis of deciduous tooth emergence arepresented elsewhere (Holman and Jones,1991). Briefly, 468 children in two ruralvillages were examined prospectively overthe two and one-half year course of thestudy. Children recruited into the studywere from birth to six months of age, butnone had emerged a tooth at the first visit.Exact birth dates and visit dates are known.Interviews were scheduled every 35 days(one Javanese month); however, 309 out of6,017 (5%) teeth emerged over a largerinterval because one or more visits weremissed.

Bangladeshi children. These data werecollected as part of the Meheran Growth andDevelopment Study (Khan et al., 1979) con-ducted between 1974 and 1977 in the ruralvillage of Meheran, located about 50 kmsoutheast of Dhaka. Dental records of 397children are available from the study. Clini-cal examinations were conducted monthlyfor the first year of the study and quarterlythereafter. Most of the children were re-cruited at birth; however, all children wereless than one year of age at the beginning ofthe study. Exact dates of birth are known forall subjects because of a continuous demo-graphic surveillance of the region conductedsince the mid-1960s by the InternationalCentre for Diarrhoeal Disease Research,Bangladesh (ICDDR,B, 1992). A descriptionof the dental data is found in Khan andCurlin (1981) and Khan et al. (1981), whoanalyzed the effect of health and nutritionon the mean number of teeth emerged byage. Emergence statistics for individual teethhave not been published.

Emergence information was collected foreach tooth based on periodic examinations of

3This article discusses parametric methods, but several typesof nonparametric methods are appropriate for analysis of longitu-dinal tooth emergence data as well. Life tables (including piece-wise exponential or piece-wise logistic models) provide usefulnonparametric approaches that incorporate right- and fixed-interval censoring (Holman and Jones, 1991; Finkelstein, 1986).Piece-wise exponential and logistic models can also incorporateestimates of the effects of covariates on the underlying emer-gence process. The product-limit or Kaplan-Meier survival modelis appropriate for exact emergence times (Kaplan and Meier,1958).


the children by a trained paramedic. Teethwere checked by fingertip palpitation of thegum by the paramedic during scheduledvisits or when a sick child visited a healthclinic (ICDDR,B, 1990:4). When a new toothemerged, the parent was asked to providethe date of the emergence as precisely aspossible. We did not use the parent’s re-ported dates of emergence for the presentanalysis, but instead used the date from thefirst visit prior to emergence (or birth, ifthere was no previous visit) and the date ofthe visit at which emergence was first diag-nosed by the paramedic to capture the inter-val in which tooth emergence took place.There are two reasons for preferring inter-vals based on paramedic reports over par-ent’s estimates of emergence in this study.First, the dates reported by parents showedconsiderable digit heaping. Of 4,743 datesgiven by parents, 21% fell on the 15th of themonth; 41% fell on either the 10th, 15th or20th of the month. Second, literacy andnumeracy in this rural village were quitelow at the time of the study (see Shaikh andBecker, 1985). Parents were unlikely to havechecked daily for an event like tooth emer-gence and had they observed that a toothemerged would have been unlikely, due tothe high rate of illiteracy, to record the datewith precision. We use the known clinicalexamination dates with survival analysis toproduce unbiased estimates of mean or me-dian emergence times.

The Bangladeshi dental records werecoded by tooth type (deciduous or perma-nent), tooth class (first incisor through thirdmolar), and quadrant (upper left or lowerright). Coding in this way resulted in aconsiderable number of miscoding errors.Fortunately, few errors occurred for any onechild. Most errors resulted in recording onetoo many of a particular tooth and a missingrecord for another tooth. In nearly all casesit was possible to correct the error withconfidence. The antimeres of the missingrecord and duplicate teeth clearly indicatedwhich tooth was incorrectly coded. A smallnumber of dental records with unresolvableerrors were dropped from the analysis.

Guatemalan children. These data comefrom the Institute of Nutrition of Central

America and Panama (INCAP) longitudinalstudy of nutrition and mental developmentcarried out in a chronically malnourishedpopulation in four farming villages in ruralGuatemala between 1968 and 1977 (Readand Habicht, 1993; Habicht and Martorell,1993). The sample includes dental recordsfor 1,277 children, most of whom were en-rolled in the study at birth. Children wereexamined every three months from birth totwo years of age and every six monthsthereafter. A description of the dental dataas well as an analysis of nutritional statuson counts of teeth and by individual teethcan be found in Delgado et al. (1975). Statis-tics for the emergence of individual teethhave not been published.

Older children were sometimes recruitedinto the Guatemala and Bangladesh stud-ies, including some who had emerged and,perhaps, exfoliated one or more deciduousteeth before recruitment. Consequently,these studies might be classified as mixedlongitudinal. Thus, for some children whowere recruited at later ages emergence andexfoliation of deciduous teeth at the firstregular visit had to be assumed based onemergence of the homologous permanenttooth. Left dentitions were used for all esti-mates.

Statistical methods

We produced estimates for the median,mean, and the parametric standard devia-tion for three distributions: a normal distri-bution, a lognormal distribution, and ashifted lognormal distribution. For theshifted lognormal distribution, we assumedthe process of tooth eruption begins eightmonths before birth, corresponding to thetime just prior to when the dental lamina isformed (Ten Cate, 1985).

Two parameters were estimated for eachof the three distributions: a location param-eter denoted µ, and a scale parameter de-noted s. Estimates of these parameters aredenoted µ and s. For the normal distribu-tion, the µ parameter is also the median andthe mean of the distribution, and s2 is theparametric variance of the distribution. Theparameter estimates for a lognormal andshifted lognormal distribution are not inter-pretable in the same easy way as they are


for the normal distribution. In order to com-pare estimates for the three distributions,we converted µ and s into medians, means,and parametric standard deviations for thelognormal and shifted lognormal distribu-tions.

Estimates of standard errors are neededto facilitate statistical comparisons amongmeans, medians, and standard deviations.Here, V(µ) is defined as the variance of µ,and the square root of V(µ) is the standarderror of µ. Likewise, V(s) is defined as thevariance of s and the square root of V(s) isthe standard error of s. Software that esti-mates parameters like µ and s usually pro-vides variances of the parameter estimatesalong with the covariance between the pa-rameters. The parameters and their stan-dard errors can be used to compute standarderrors of the median, mean, and standarddeviation. Equations for converting µ and sfrom the lognormal and shifted lognormaldistribution into means, medians, and stan-dard deviations and methods for estimatingtheir standard errors are found in the Appen-dix.

The following notation is used: N is thenumber of observations, f (t; µ, s) is theprobability density function (PDF) for emer-gence at time t for a tooth, and is either anormal, lognormal, or a shifted lognormaldistribution. For brevity, a subscript to de-note which tooth the PDF describes is notshown. Also, we usually drop the intrinsicparameters and write the PDF as f (t). Thecumulative density functions for f (t) is (1)

F (t) 5 e0

tf (x) dx, (1)

and the survival density function (SDF) attime t is (2)

S(t) 5 1 2 F(t) 5 et

`f (x) dx. (2)

Estimates for µ and s are found by maxi-mum likelihood. Introductions to maximumlikelihood estimation can be found else-where (Edwards, 1972; Konigsberg andFrankenberg, 1994; Pickles, 1985; Wood etal., 1992). The basic idea of the method is tocompute a probability or a likelihood foreach observation given some underlyingprobability model. The overall likelihood isthe product of these individual likelihoods.

The parameter values that globally maxi-mize the overall likelihood are taken as themaximum likelihood estimates (MLEs).Maximum likelihood estimators have a num-ber of desirable properties—they are asymp-totically unbiased, consistent, and are asstatistically efficient as possible. The MLEsare usually found with the help of computerprograms that iteratively try combinationsof parameters until those that maximize thelikelihood are found. One reason likelihoodis a useful method is that incomplete datalike those discussed above can be readilyaccommodated. To do so, one must specifythe likelihood for an incomplete observationunder the specified probability model. Inwhat follows, details of specifying likeli-hoods for different types of data are given.We do this to show how interval censoring isaccommodated, to demonstrate how probitanalysis is a special case of interval-cen-sored survival analysis, and to extend themethod for simultaneous estimation of anagenic proportion.

Likelihood for exact emergence ages.When a tooth is known to emerge at someexact time, the likelihood of that observationis simply the probability density at theemergence age,4 f (t; µ, s). Figure 1, f (t; µ, s).Figure 1, panel a shows this likelihoodgraphically. The overall likelihood for emer-gence at exact ages of N teeth is the productof the individual likelihoods (3)

L 5 pi51

N

f (ti ; µ, s). (3)

Likelihood for cross-sectional observa-tions. Cross-sectional studies assign sub-jects to one of two categories based on emer-gence status: either a responder or anonresponder. A likelihood for a responder isconstructed by specifying the probabilitythat an individual aged t emerged a tooth atsome unknown time in the past. As can beseen in Figure 1, panel b, this probability isthe entire area under the PDF to the left oftime t, which is the cumulative density atage t, F(t).

4We simplify presentation by saying the ‘‘likelihood is equalto. . .’’ when, in fact, it is proportional down to a multiplicativeconstant. The constant is unimportant for purposes of estima-tion.


A nonresponder is a tooth that has notemerged at or before the observation at aget. If we can assume that all teeth willeventually emerge (an assumption we relaxlater), then the likelihood is the area underthe PDF from age t to infinity (Fig. 1, panelc), that is, the survivorship at age t, S(t).

An overall likelihood can be computedfrom all cross-sectionally sampled individu-als who are classified as responders (r) ornonresponders (nr) by taking the product ofthe likelihoods from all responders and allnonresponders as

L 5 pi[nr

S (ti ) pi[r

F (ti). (4)

Likelihood for interval-censored obser-vations. These observations arise whenthe only information known is that a toothemerged at some time after age tu, andbefore age te, where the subscripts denoteunemerged (u) and emerged (e). The prob-ability of this event is the area under thePDF from tu to te (Fig. 1, panel d), which canbe found as S(tu)2S(te). Thus, the likelihoodfor N interval-censored teeth is (5)

L 5 pi51

N

[S(tui) 2 S(tei

)]. (5)

Likelihood for right-censored observa-tions. These observations arise when achild exits the study at age t before emer-

gence of a tooth. Assuming the tooth willeventually emerge, the likelihood is the areaunder the PDF to the right of t. The indi-vidual likelihood is exactly the same as anonresponder in a cross-sectional study(Fig. 1, panel c).

An examination of equations (4) and (5)and Figure 1 suggests an interesting rela-tionship between cross-sectional methodslike probit analysis and survival analysisusing the same underlying distribution. Con-sider the result in (5) and panel d when tuoccurs earlier and earlier until it approachesbirth (time zero). Since S(0) 5 1, the likeli-hood for an observation that occurs in theinterval between birth (tu 5 0) and time tebecomes [12S(te)] 5 F(te), which is the samecontribution to the likelihood as a responderin the cross-sectional likelihood. Thus, cross-sectional responders are a special case ofinterval censoring in which the last un-emerged observation (tu) was birth (or con-ception if that is considered the start ofemergence). Likewise, when te gets later andlater, S(te) approaches 0, and likelihood (5)becomes S(tu), which is the same likelihoodas a nonresponder or a right-censored case.This permits us to consider cross-sectionalnonresponders and right-censored observa-tions as special cases of interval-censoredobservations, in which emergence is onlyknown to occur between the last unemergedobservation and time `.

Following Wood et al. (1992:Appendix), ageneral likelihood can be written that accom-modates all of these types of observations,provided: 1) birth dates are known, 2) a negli-gible proportion of teeth are agenic, 3) nosecular trend exists in the period(s) of obser-vation, that is, the process is stationary, and4) the rate of emergence is independent ofthe child’s probability of death or right cen-soring. We can combine equations (3) through(5) into a single equation taking advantageof the relationship between cross-sectionaland longitudinal observations. The overalllikelihood for N teeth is:

L 5pi51

N

5[S(tui) 2 S(tei

)]12d(tui,tei

) f (tei)d(tui

,tei)6 (6)

where d(x, y) is the Kronecker’s delta func-tion, which returns one when x 5 y and

Fig. 1. Areas under a probability density functioncorresponding to different types of observations. Theareas define the probability for each type of observationat time t (or times tu and te) given some probabilitydensity function. The area in panel a is f (t)dt. Areasunder the other three panels are denoted by horizontallines. From Holman (1996).


returns zero when x Þ y, so for exact emer-gence times, the individual likelihood on theright side of the equation is used, otherwisethat on the left is used. The age tu is set toage zero initially and thereafter is set to thelatest age the child is examined and does nothave the tooth. The age teis initially set toinfinity—an age by which we are certain thetooth will emerge. Then if the child is everobserved as having the tooth, teis set to theage the observation was made. Definitionsfor tu and te under the different types ofobservations are summarized in Table 1.

Modeling an agenic proportion. Stan-dard survival methods assume all teeth willeventually emerge, even if after the end ofthe study. Nevertheless, some fraction ofteeth in a population may be agenic orotherwise never emerge (e.g., dental anky-losis). If complete records are available forall teeth, one can simply remove ageniccases from analyses and estimate emer-gence for only those teeth that emerge.Studies can rarely collect this type of long-term detail so that special methods must beused to estimate the agenic proportion. Inthis section we extend the likelihood givenin (6) to provide for simultaneous estimationof an agenic proportion p.

The effect of an agenic fraction on thesurvival distribution is seen in Figure 2. Theleft panel shows the survival distributionafter all agenic individuals are removed;survivorship is one at the start of emergenceand approaches zero on the right. When theagenic subgroup, making up fraction p, isincluded in the sample survivorship beginsat one, but now approaches p rather thanzero. With time, the dentitions still underobservation are composed of a larger andlarger fraction of agenic dentitions untilonly the agenic subgroup remains.

Call Sf (t), Ff (t), and ff (t) the SDF, CDF,and PDF, respectively, for the non-agenic

fraction of dentitions. From Figure 2, it is clearthat the fraction of dentitions that have notemerged (survived) at time t is equal to Sf (t)weighted by the fraction that is not agenic,(12p), and a second fraction of eternal survi-vors weighted by p. The overall survivaldistribution that includes both subgroups is:

S(t) 5 (1 2 p)Sf (t) 1 p. (7)

Likewise, the PDF is composed of fraction12p individuals who fail with probabilityff(t) weighted by fraction p individuals whofail with probability 0, so that overall theprobability density is

f (t) 5 (1 2 p) ff (t) 1 p 3 05 (1 2 p) ff (t).

(8)

Individual likelihoods can be constructed asweighted means for the four types of observa-tions given in Figure 1: for teeth that weknow emergence age exactly, the individuallikelihood, from (8), is (12p)f (t). Cross-sectional responders cannot include anyagenic teeth (because a tooth is observed);the likelihood, from (8), is (12p)[12F(t)].Nonresponder or right-censored observa-tions include contributions from both sub-groups; following (7), the individual likeli-hood for these observations is (12p)S(tu) 1p. An interval-censored observation for whichthe tooth was observed to emerge must

TABLE 1. Definitions of tu and te for four types of observations

Class Known age before emergence, tu Known age after emergence, te

Exact emergence Exact emergence age Exact emergence ageResponder 0 Age at only observationNonresponder and right-

censoredAge at only observation

Interval censored Age at latest observation prior toemergence

Age at first observation after emergence

Fig. 2. The effect of an agenic subgroup on thesurvival distribution, S(t). The subgroup makes upfraction p of the initial population. The left panel showssurvivorship (i.e., the proportion that has not yetemerged) for the non-agenic subgroup alone. The rightpanel shows the same distribution contaminated by theagenic subgroup. From Holman (1996).


include no contribution from the agenic frac-tion; the individual likelihood is (12p )[S(tu)2S(te)]. Since te is infinity for nonre-sponders or right-censored observations(Table 1), the likelihood for all types ofobservations is

L 5 pi51

N

5(1 2 p) f (tei)d(tui

,tei)

· [S(tui) 2 S(tei

)]12d(tui,tei

) 1 pd(tei, `)6 .

(9)

Time definitions given in Table I hold forthis likelihood.

Numerical methods. Likelihoods in theform of (6) can be estimated in severalstatistical packages—in particular, the SASLIFEREG procedure can estimate param-eters from interval and right-censored obser-vations, exact observations, and cross-sec-tional responders and nonresponders (SASInstitute, 1985). Other packages like GLIM,S1, or GAUSS can be used to construct andestimate likelihoods like (9) that include anagenic proportion. We used software writtenby one of the authors (DJH). The estimationprogram and numerical methods have beenvalidated using simulated data and againstthe SAS LIFEREG procedure with real data(Holman, 1996; Wood et al., 1992). Standarderrors of the parameter and covariate esti-mates were computed from the inverse ofthe observed Fisher’s information matrix(Nelson, 1983:423). The computer programis available from the first author.

Estimating effective Ns. The number ofdentitions that go into an estimate of a meanand standard deviation differs among thefour samples. Additionally, for any giventooth there are potentially several types ofobservations that contribute differentamounts of statistical information to theestimate—right-censored, interval-censored,exact emergence times and cross-sectionalresponders. A property of the normal distri-bution can be used to estimate the effectivenumber of observations that contribute tothe estimate of the emergence distribution.In essence, an estimate is found of thenumber of individuals with exact emergencetimes necessary to produce the equivalentstandard error of the mean (SEM). If we had

recorded exact times and had no censored,cross-sectional or interval-censored observa-tions, the parametric standard deviationand the standard error of the mean arerelated as s/ÎN 5 SEM. Rearranging andreplacing s by its estimate S we compute theeffective N as Neff5(S/SEM)2 . Naturally,Neff will be less than N whenever interval orright-censored observations are included inthe estimates; N and Neff are equal only ifexact emergence times are recorded for allindividuals. If all observations are interval-censored, we expect Neff to approach N asintervals become smaller. Furthermore, Neffis expected to be smallest when all observa-tions are estimated from cross-sectional data.

RESULTS AND DISCUSSION

Results for each of the three parametricdistributions are presented in Tables 2through 5 as estimates of medians, means,standard deviations, effective Ns, and theproportion of agenic individuals.

We evaluate each parametric distributionagainst the nonparametric (life table) esti-mates by a number of comparisons. Table 2includes the life table estimates of mediansfor the Javanese children. Medians pro-duced by the normal distribution, with theexception of m2, were closest to the life tablemedians. In general, parametric medians werenot significantly different from the nonpara-metric medians. The exceptions were for thenormal m1; the lognormal i1, m1, i1, and m1;and the shifted lognormal i1 and m1.

For the Japanese children (Table 3), thenormal distribution produced medians clos-est to the life table medians for three teeth,and the lognormal distribution producedmedians closest to the life table medians forthe remaining seven teeth. Yet, the differ-ences were small and none of the parametricmedians was significantly different from thenonparametric medians.

The nonparametric distributions of emer-gence, taken from life table estimates, arecompared to the corresponding parametricdistributions for upper teeth (Fig. 3) andlower teeth (Fig. 4) of the Javanese (lowerpanels) and Japanese children (upper pan-els). The Japanese parametric distributionsare nearly indistinguishable and almostnever fall outside of the 95% confidence


limits of the nonparametric estimates. TheJavanese parametric distributions are simi-lar and almost indistinguishable from eachother except that the normal distributionappears aberrant for i2. Nevertheless, theparametric distributions frequently fall out-side of the nonparametric confidence limits.In particular, the parametric distributionspredict a higher proportion of emergence

over the earlier part of the distributionswhen about 20% of the teeth have emerged.Some of the differences for the first incisorsmay be because the life table method doesnot account for agenicity.

Another way to evaluate goodness-of-fitfor the parametric models is to comparemeans and standard deviations among thedifferent estimates for a given tooth and

TABLE 2. Survival analysis estimates of means, medians, and standard deviations for three distributions and thenonparametric life table means for deciduous tooth emergence (months) in Javanese children

Normal Lognormal Shifted lognormal Life table

Mean(SE mean)

Std dev(SE SD)

pa

(Chi sq) Neff

Median(SE med)

Mean(SE mean)

Std dev(SE SD)

pa

(Chi sq)Median

(SE med)Mean

(SE mean)Std dev(SE SD)

pa

(Chi sq)Median

(SE med)

i1 11.05 2.430 0.0069 328.1 10.82 11.07 2.420 0.0043 10.91 11.00 2.384 0.0055 11.28(0.134) (0.080) (18.66) (0.116) (0.117) (0.093) (4.13) (0.121) (0.119) (0.083) (8.41) (0.111)

i2 13.01 3.104 — 324.4 12.66 13.02 3.093 — 12.79 12.93 3.041 — 12.97(0.172) (0.108) (1.86) (0.146) (0.149) (0.121) (0) (0.152) (0.150) (0.109) (0.0124) (0.195)

c 20.46 3.253 — 346.6 20.24 20.53 3.462 — 20.30 20.44 3.367 — 20.60(0.175) (0.133) (0.00) (0.190) (0.199) (0.137) (0) (0.183) (0.188) (0.134) (0) (0.199)

m1 17.29 2.481 — 359.2 17.12 17.30 2.512 — 17.17 17.26 2.485 — 17.70(0.131) (0.086) (0.00) (0.127) (0.129) (0.098) (0) (0.127) (0.127) (0.092) (0) (0.162)

m2 28.54 3.226 — 93.9 28.62 28.86 3.697 — 28.60 28.73 3.566 — 28.58(0.333) (0.292) (0.00) (0.385) (0.413) (0.355) (0) (0.371) (0.391) (0.336) (0) (0.422)

i1 10.15 2.619 0.0083 330.8 9.85 10.17 2.610 0.0081 9.98 10.08 2.559 0.0082 10.30(0.144) (0.089) (18.47) (0.121) (0.126) (0.106) (8.27) (0.127) (0.126) (0.093) (9.65) (0.14)

i2 16.23 3.676 — 372.3 15.85 16.29 3.884 — 15.97 16.16 3.746 — 16.42(0.191) (0.134) (0.00) (0.188) (0.200) (0.168) (0) (0.184) (0.187) (0.149) (0) (0.237)

c 22.00 3.557 — 316.9 21.79 22.12 3.837 — 21.84 22.01 3.727 — 22.18(0.200) (0.177) (0.00) (0.218) (0.231) (0.200) (0) (0.211) (0.218) (0.190) (0) (0.269)

m1 18.58 2.714 — 359.1 18.39 18.60 2.786 — 18.45 18.55 2.744 — 18.85(0.143) (0.099) (0.00) (0.143) (0.147) (0.111) (0) (0.142) (0.143) (0.106) (0) (0.154)

m2 28.17 3.421 — 112.8 28.22 28.48 3.902 — 28.20 28.35 3.767 — 28.25(0.322) (0.317) (0.00) (0.371) (0.403) (0.384) (0) (0.358) (0.380) (0.364) (0) (0.447)

a Proportion of agenic teeth when significantly different from zero. Numbers in parenthesis report a likelihood ratio test (Nelson, 1983)for the full model and a reduced model in which p is constrained to zero. The test statistic is distributed as x2 with one degree offreedom.

TABLE 3. Survival analysis estimates of means, medians, and standard deviations for three distributions and thenonparametric life table means for deciduous tooth emergence (months) in Japanese children

Normal Lognormal Shifted lognormal Life table

Mean(SE mean)

Std dev(SE SD)

pa

(Chi sq) Neff

Median(SE med)

Mean(SE mean)

Std dev(SE SD)

pa

(Chi sq)Median

(SE med)Mean


pa

(Chi sq)Median

(SE med)

i1 8.43 1.635 — 78.9 8.32 8.49 1.693 — 8.40 8.45 1.693 — 8.13(0.184) (0.153) (0.063) (0.170) (0.166) (0.141) (0.813) (0.182) (0.177) (0.135) (2.08) (0.211)

i2 9.69 2.019 — 67.5 9.50 9.69 1.959 — 9.58 9.64 1.963 — 9.1(0.246) (0.176) (0) (0.206) (0.198) (0.179) (0) (0.222) (0.214) (0.173) (0) (0.274)

c 17.79 2.829 — 96.8 17.56 17.80 2.966 — 17.63 17.74 2.892 — 17.9(0.288) (0.197) (0) (0.317) (0.335) (0.236) (0) (0.302) (0.310) (0.215) (0) (0.351)

m1 17.04 3.242 — 89.2 16.73 17.04 3.292 — 16.83 16.97 3.238 — 17.09(0.343) (0.243) (0) (0.326) (0.339) (0.282) (0) (0.325) (0.327) (0.258) (0) (0.273)

m2 25.86 3.241 — 79.3 25.66 25.86 3.197 — 25.71 25.82 3.196 — 25.44(0.364) (0.251) (0) (0.333) (0.330) (0.257) (0) (0.339) (0.334) (0.252) (0) (0.272)

i1 8.06 1.713 — 79.2 7.89 8.06 1.670 — 7.97 8.02 1.667 — 7.81(0.192) (0.123) (0.139) (0.160) (0.161) (0.130) (0.558) (0.173) (0.169) (0.120) (1.22) (0.18)

i2 9.58 2.053 — 60.8 9.38 9.57 1.943 — 9.46 9.52 1.965 — 9.12(0.263) (0.155) (0) (0.209) (0.201) (0.152) (0) (0.230) (0.222) (0.147) (0) (0.207)

c 17.44 2.301 — 94.7 17.28 17.45 2.453 — 17.33 17.41 2.375 — 17.52(0.236) (0.125) (0) (0.277) (0.287) (0.140) (0) (0.258) (0.263) (0.130) (0) (0.205)

m1 16.55 3.032 — 80.1 16.28 16.54 2.998 — 16.36 16.49 2.978 — 16.05(0.339) (0.221) (0) (0.298) (0.302) (0.238) (0) (0.306) (0.303) (0.224) (0) (0.378)

m2 24.05 3.101 — 86.2 23.86 24.06 3.090 — 23.91 24.02 3.080 — 23.9(0.334) (0.263) (0) (0.314) (0.315) (0.262) (0) (0.316) (0.315) (0.258) (0) (0.49)

a See footnote in Table 2.


identify estimates that appear to be aber-rant. Estimates of means for the normal,lognormal, and shifted lognormal distribu-tions were usually indistinguishable exceptthat the lognormal mean for the Banglade-shi i1 was significantly later and the lognor-mal mean for the Guatemalan i1 was signifi-cantly earlier. With a few exceptions, theestimates for parametric standard devia-tions were indistinguishable among the three

distributions. In Bangladeshi children, thestandard deviations were estimated as sig-nificantly greater for the m2, c1, m1, and m2by the normal and i1 for the lognormal. ForGuatemalan children the normal distribu-tion estimated significantly larger standarddeviations for i1, i2, and m1.

Small differences were found among esti-mates of central tendency by the three distri-butions, suggesting that any of the three

TABLE 4. Survival analysis estimates of means, medians, and standard deviations for three distributionsfor deciduous tooth emergence (months) in Guatemalan children

Normal Lognormal Shifted lognormal

Mean(SE mean)

Std dev(SE SD)

pa

(Chi sq) Neff

Median(SE med)

Mean(SE mean)

Std dev(SE SD)

pa

(Chi sq)Median

(SE med)Mean


pa

(Chi sq)

i1 10.52 3.117 0.0009 281.7 10.19 10.42 2.260 0.00093 10.29 10.36 2.255 0.00097(0.186) (0.082) (7.69) (0.102) (0.103) (0.071) (4.56) (0.106) (0.105) (0.056) (9.59)

i2 11.25 2.995 — 348.4 10.90 11.18 2.558 — 11.02 11.12 2.549 —(0.160) (0.075) (0.00) (0.113) (0.114) (0.088) (0) (0.119) (0.117) (0.072) (0)

c 19.21 3.048 — 477.1 18.97 19.20 3.063 — 19.04 19.16 3.038 —(0.140) (0.081) (0.00) (0.134) (0.137) (0.100) (0) (0.135) (0.135) (0.093) (0)

m1 16.09 3.187 — 269.1 15.90 16.07 2.330 — 15.96 16.03 2.372 —(0.194) (0.084) (0.00) (0.110) (0.109) (0.063) (0) (0.116) (0.115) (0.056) (0)

m2 27.87 3.798 — 461.4 27.62 27.87 3.776 — 27.68 27.83 3.766 —(0.177) (0.111) (0.00) (0.166) (0.166) (0.121) (0) (0.168) (0.167) (0.117) (0)

i1 8.29 2.480 0.0010 385.9 6.99 7.48 2.863 — 8.11 8.19 2.236 0.00097(0.126) (0.057) (35.92) (0.000) (0.039) (0.131) (0.56) (0.100) (0.101) (0.056) (13.65)

i2 13.81 3.222 — 449.3 13.42 13.78 3.215 — 13.57 13.71 3.173 —(0.152) (0.090) (0.00) (0.137) (0.141) (0.117) (0) (0.140) (0.139) (0.102) (0)

c 20.04 3.231 — 484.1 19.78 20.04 3.246 — 19.85 19.98 3.221 —(0.147) (0.093) (0.00) (0.141) (0.144) (0.109) (0) (0.141) (0.142) (0.103) (0)

m1 17.01 3.034 — 331.3 16.73 17.24 4.275 — 17.37 17.69 5.018 —(0.167) (0.076) (0.00) (0.267) (0.267) (0.114) (0) (0.346) (0.343) (0.128) (0)

m2 27.14 5.553 — 254.8 26.75 27.06 4.155 — 26.82 27.01 4.225 —(0.348) (0.093) (0.00) (0.205) (0.202) (0.068) (0) (0.215) (0.212) (0.063) (0)


TABLE 5. Survival analysis estimates of means, medians, and standard deviations for three distributionsfor deciduous tooth emergence (months) in Bangladeshi children

Normal Lognormal Shifted lognormal

Mean(SE mean)

Std dev(SE SD)

pa

(Chi sq) Neff

Median(SE med)

Mean(SE mean)

Std dev(SE SD)

pa

(Chi sq)Median

(SE med)Mean


pa

(Chi sq)

i1 11.76 3.217 0.0034 197.0 11.44 12.28 4.798 — 11.57 11.73 3.359 —(0.229) (0.110) (12.67) (0.000) 0.062 (0.208) (0) (0.216) (0.220) (0.108) (3.65)

i2 13.89 3.857 — 178.0 13.35 13.97 4.282 — 13.53 13.74 3.823 —(0.289) (0.156) (3.27) (0.000) 0.076 (0.293) (0) (0.253) (0.256) (0.152) (0.10)

c 21.00 3.693 — 138.0 20.66 20.97 3.666 — 20.75 20.91 3.645 —(0.314) (0.181) (0.00) (0.286) 0.285 (0.215) (0) (0.291) (0.288) (0.201) (0)

m1 15.90 2.717 0.0032 151.6 15.70 15.94 2.802 — 15.79 15.89 2.780 —(0.221) (0.092) (6.94) (0.206) 0.207 (0.116) (1.27) (0.212) (0.211) (0.107) (2.74)

m2 27.63 7.484 — 75.2 26.71 27.23 5.420 — 26.84 27.15 5.406 —(0.863) (0.240) (0.00) (0.421) 0.424 (0.186) (0) (0.440) (0.437) (0.163) (0)

i1 10.42 2.693 0.0049 200.7 10.08 10.44 2.777 0.0046 10.23 10.33 2.662 0.0047(0.190) (0.111) (27.32) (0.176) 0.188 (0.137) (5.52) (0.173) (0.174) (0.117) (11.68)

i2 16.31 4.107 — 182.0 15.74 16.30 4.382 — 15.93 16.16 4.151 —(0.304) (0.169) (0.00) (0.307) 0.332 (0.219) (0) (0.289) (0.296) (0.193) (0)

c 22.69 5.253 — 106.4 22.06 22.50 4.553 — 22.18 22.43 4.524 —(0.509) (0.159) (0.00) (0.349) 0.352 (0.185) (0) (0.361) (0.359) (0.163) (0)

m1 18.26 4.499 — 89.4 17.65 17.95 3.313 — 17.76 17.91 3.353 —(0.476) (0.141) (0.00) (0.262) 0.260 (0.115) (0) (0.277) (0.274) (0.106) (0)

m2 27.70 7.147 — 59.7 26.91 27.28 4.539 — 27.19 27.44 4.776 —(0.925) (0.246) (0.00) (0.261) 0.221 (0.235) (0) (0.448) (0.442) (0.129) (0)



Fig. 3. Nonparametric and parametric distributions of tooth emergence for the upper dentition inJapanese and Javanese children. Nonparametric distributions and standard errors are from Holman andJones (1991) and Jones and Holman (1991).

Fig. 4. Nonparametric and parametric distributions of tooth emergence for the lower dentition inJapanese and Javanese children. Nonparametric distributions and standard errors are from Holman andJones (1991) and Jones and Holman (1991).


methods may be used for estimating centraltendencies. Greater differences were foundamong the standard deviations—the normaland lognormal distributions occasionallyseemed to produce too large an estimate. Ofthe three distributions we considered withthe parametric method, the normal providesinterpretable summary statistics (mean andits standard error and standard deviationand its standard error). Additional steps areneeded for the lognormal distributions tocompute the median or mean and standarddeviation from the parameter estimates andcomputing standard errors for the mean andstandard deviation is somewhat tedious.5Additional considerations arise from math-ematical constraints imposed by each model.The lognormal model is constrained to posi-tive ages only, so that it cannot properlymodel observations of natal teeth whichrange from 1 in 2,000 to 1 in 3,500(Meredith, 1973; Leung, 1989; King andLee, 1989). The normal distribution can takeon negative emergence times but will alsohave some small mass at times prior toconception. The shifted lognormal distribu-tion is most realistic because it can only beused with positive ages from the start of theemergence process. This issue becomes moreimportant when working with nonhumanprimates because many species are bornwith teeth (Smith et al., 1994). In fact,parametric estimates of emergence can befound using these methods even if most (butnot all) individuals have emerged a tooth bybirth. In effect, there must be enough postna-tal observations of emergence to fit the righttail of the distribution. Those with teeth atbirth are treated as cross-sectional respond-ers measured from conception.

Ideally, selection of the proper distribu-tion is dictated by theory of the etiologicprocess leading to tooth emergence. We know

of no strong etiologic theory that dictatesthat a normal, lognormal, gamma, etc., dis-tribution properly describes tooth emer-gence. A weakly etiologic model is built fromthe central limit theorem of statistics. Ifmany environmental insults and alleles atmany loci each have a small additive effecton a character then the resulting distribu-tion will be normal; if effects are multiplica-tive on a character, the distribution will belognormal (Wright, 1968). Galton (1879) sug-gests the use of the lognormal distributionfor systems in which a constant percentageincrement in growth occurs per unit timerather than an absolute increment. Toothemergence from conception has been de-scribed in this way (Smith et al., 1994). If webelieve that deviations due to variations inmany alleles or environmental conditionsresult in percentage differences, instead ofadditive differences, tooth emergence timesshould be lognormally distributed when mea-sured from the start of emergence. Timingfrom conception was suggested by Kihlbergand Koski (1954) and Hayes and Mantel(1958) and has been used or discussed in afew other studies (Magnusson, 1982; Smithet al., 1994).

The shifted lognormal distribution ap-pears to be the most etiologic model avail-able. That and the good behavior of theshifted lognormal distribution in the foursamples leads us to suggest the use of thatdistribution when testing for differences invariances or when the entire distribution isof interest. Most of what follows uses resultsfrom the analyses using the shifted lognor-mal distribution.

Agenic proportion

The agenic proportions estimated undereach of the three distributions are given inTables 2 through 5. Significant proportionsof agenic teeth were estimated under one ormore parametric models for the Javanese,Guatemalan, and Bangladeshi i1 and i1 andthe Bangladeshi m1. When agenesis was notsignificantly different from zero for a tooth,we give the parameter estimated with noagenesis. Estimates of agenesis were consis-tent among the three distributions for agiven tooth within one population. Four ofthe seven teeth that had a non-zero propor-tion estimated under the normal distribu-

5Another position is to discard all parametric methods and usethe nonparametric life table instead. The life table methodcertainly has the advantage that it is simple. A spreadsheetprogram can be written in a few minutes and the procedure isavailable in many statistical packages. The life table method hasa few limitations. Since the method involves, in effect, estimatinga parameter for each life table interval, the method is not asstatistically efficient as a well-specified parametric model. Lifetables are weak with interval-censored data in which the inter-val widths differ among individuals, and ordinary life tables donot work with cross-sectional responders. Parametric modelsoffer advantages of greater statistical efficiency and betteraccommodation of incomplete data. Changes in interval width,exact emergence times, and agenic proportions are easily handledby the parametric method.


tion also had non-zero proportions esti-mated under the lognormal distributions.

Estimated agenic proportions ranged from0.1 to 0.8%. These small proportions suggestthat agenesis of the deciduous dentition israre, even in undernourished children. Agen-esis of permanent teeth (excluding M3) givessimilar proportions from 0.12% to as high as3.3%, but with most teeth under 1%(Heidmann, 1986).

One reason we estimated the agenic pro-portion was to see if significant biases re-sulted when agenesis is not estimated alongwith other parameters. Table 6 shows theerrors incurred under the shifted lognormaldistribution when agenesis is ignored in theanalyses. Medians and means are consis-tently biased toward later ages by a trivialamount, and always less then 1%. Standarddeviations were consistently estimated 3–5%too large, yet the absolute differences wereon the order of one standard error. Thus itappears that for populations and samplesizes considered here, agenesis does not leadto substantially biased estimates. Even so,the procedure we have given here may beuseful for use with permanent teeth (espe-cially third molars) and disease or toxicologi-cal studies of tooth emergence or other devel-opmental events.

Emergence times

A comparison of emergence times amongthe four populations reveals that the Japa-nese children have significantly faster emer-gence for most teeth than do children of theother three populations. We have shownthat probit analysis is a special case of thesurvival method used here, so that cross-sectional studies employing probit analysisare ideal for comparison to these results.

Figure 5 shows means with 95% confidenceintervals for the four populations analyzedhere and seven comparable studies. Clearly,when comparable methods are used to ana-lyze emergence data, the same deciduousteeth have significantly different mean emer-gence times among populations.

The longitudinal nature of the studyshould not produce biases in emergencetime. This is empirically verified becausefew teeth in the seven cross-sectional stud-ies emerge significantly earlier than theearliest of the four longitudinal studies (ex-ceptions are the Tunisian i1, m2, and m2).Likewise, none of the teeth in the sevenstudies emerges significantly later than thelatest of the four longitudinal studies. Infact, for most teeth the Javanese or Bang-ladeshi children emerged their teeth slowerthan children of the other populations. Fiveof the ten Japanese means were earlier thanthe corresponding means in all of the otherpopulations. Emergence times for the Guate-malan children usually rank near the middleof the 11 populations. The reasons for theextreme differences among populations arecurrently unknown but may be related tonutrition, disease, or genetic factors. Someof these will be examined in a later paper.

The Javanese and Guatemalan childrenshow the mean emergence sequence of i1-i1-i2-i2-m1-m1-c1-c1-m2-m2. This sequence is themost common mean or median sequencefound in most tooth emergence studies.6 In

6Exceptions to the common sequence are found in New Guinea(Malcolm, 1973), where the second molars are reversed; Bougain-ville Island (Friedlaender and Bailit, 1969) and Tunisia (Boutour-line and Tesi, 1972), where the first molars are reversed; oneIndian population (Kaur and Singh, 1992), where the canines arereversed; one US study (Nanda, 1960), where canines and firstmolars are reversed; and Korea (Yun, 1957), where first andsecond molars are reversed.

TABLE 6. Biases resulting from ignoring agenesis in deciduous tooth emergence for teeth in which a significantagenic proportion exists

Shift lognormal est., p 5 0a Median bias Mean bias SD bias

Median Mean SD Absoluteb Percentc Absolute Percent Absolute Percent

Java i1 10.96 11.05 2.47 20.05 0.42 20.05 0.46 20.08 3.39Java i1 10.01 10.12 2.65 20.03 0.31 20.04 0.38 20.09 3.64Incap i1 10.31 10.39 2.32 20.02 0.20 20.03 0.24 20.07 2.92Incap i1 8.13 8.21 2.31 20.02 0.24 20.02 0.30 20.07 3.26Bang i1 10.29 10.41 2.81 20.06 0.61 20.07 0.72 20.15 5.55a Estimates (in months) from the lognormal model, but with the agenic proportion ( p) constrained to zero. These estimates arecompared to the corresponding estimates from Tables 2, 4, and 5.b Absolute bias between the estimate (x) and the estimate with p 5 0 (x8) computed as x 2 x8.c Percent relative bias computed as 100 0 x 2 x8 0 /x.


the Japanese sample, we find three unusualdifferences: the lateral incisors are reversed,the canines are reversed, and the first mo-lars are reversed. Nevertheless, in all threeof these reversals the differences betweenthe pairs of means are not significant. Infact, the Japanese life table did not reversethe incisors, but did reverse the molars. The

small sample of children followed and therelatively fast emergence in the Japanesechildren makes differences in mean isomeremergence times difficult to detect. TheBangladeshi children show a reversal of thefirst molars and a reversal of the secondmolars by the lognormal distributions.Again,the difference between the mean emergencetimes are not significant.

To examine differences among popula-tions in variability of tooth emergence, coef-ficients of variation (CVs) in tooth emer-gence from this and other comparable studiesare graphed in Figure 6. A reasonably consis-tent pattern is found for both jaws. Variabil-ity in emergence for incisors is larger thanfor other teeth. The variability in emergencefor the molars and canines are usually simi-lar for a given population. The Guatemalanm1 appears to be an exception to this patternfor which we have no obvious explanation.That exception aside, we were surprised bythe similarities among the four populationsin the CVs for a given tooth. A slight increaseis noted by comparing the Japanese CVs

Fig. 5. Mean or median emergence times with 95%confidence interval from studies in which comparablestatistical methods were used. Means from Japan, Gua-temala, Java, and Bangladesh are from this study.India: probit medians from a cross-sectional study of1,643 girls (Kaur and Singh, 1992); Australia: probitmedians from a cross-sectional study of 513 children,pooled sides (Roche et al., 1964); Iraq: probit means froma cross-sectional study of 1,017 children, pooled sides(Baghdady and Ghose, 1981); New Guinea: probit meansfrom a cross-sectional study of 58 children (Malcolm,1970, 1973); Tunisia: probit medians from a cross-sectional study of 1,450 children, left side (Boutourlineand Tesi, 1972); Iceland: probit means from a cross-sectional study of 927 children, pooled sides (Magnus-son, 1982); Bougainville Island: probit medians from across-sectional study of 947 children (Friedlaender andBailit, 1969). Error bars are underestimated from stud-ies in which teeth were pooled from both sides of themouth.

Fig. 6. Coefficient of variation from studies in whichcomparable statistical methods were used. Sources arelisted in Figure 5.


with the Bangladeshi CVs, but we had ex-pected CVs to show radical differences bythe nutritional status of populations. Thissuggests that if nutritional stress affectstooth emergence, it acts largely through aslowing of the rate of emergence, instead ofincreasing the relative variance.

Effective sizes

It is interesting to examine how the start-ing number of observations and the observa-tional regime of each study contributes toeffective number of observations (Neff) foreach tooth. In the Japanese sample, therewere relatively few censored cases, and ob-servations were always across one-monthintervals. The Neff ranged from 61 to 97 fromanalyses of 114 dentitions. The Javanesechildren were more frequently right-cen-sored; so, beginning with 468 observations,Neff ranged from 94 for a tooth with muchcensoring to 372 for a fast-emerging toothwith little censoring. The Bangladesh studyincluded a number of older children wholook like cross-sectional responders and re-duce the effective sample size. Thus, from397 dentitions effective numbers of observa-tions ranged from 60 to 201. The Guatema-lan study included many older children (re-sponders) and had a lower proportion ofright-censored cases compared to the Java-nese sample. For 1,277 individuals used forthe analysis, Neff ranged from 255 to 484.

To compare effective numbers to a purelycross-sectional study, an Neff was computedfor each tooth from summary statistics (SDand SEM) given in a cross-sectional study(Magnusson, 1982). Unfortunately, left andright sides were pooled in the original study,which somewhat obscures the comparisonby inflating Neff.7 Analyses of 498 boys gavean average Neff of 63 (from 41 for m2 to 91 fori1). An analysis of 429 girls gave an averageNeff of 58 (from 29 in c1 to 87 in i2). In otherwords, the longitudinal study of 114 Japa-nese children followed at monthly intervalswas about as informative as this cross-sectional study of 927 children.

In the Appendix, we show how statisticalinformation can be computed for observa-tions under different study regimens. For anumerical example, consider the lower ca-nine of the Javanese children, which had amean emergence time of 22.0 months and astandard deviation of 3.6. Say we lastsampled one child at 19 months. One sce-nario samples the child in a longitudinalstudy with 30-day intervals, and another ina cross-sectional study (one visit only). If thechild had not emerged this tooth by 19months then the information under bothstudy scenarios would be Inr(19) 5 Ic(19) 50.22. If the child had emerged the tooth,then the information from the cross-sec-tional responder is Ir(19) 5 1.61, and from thelongitudinal study is Ie(18–19) 5 2.66. So thelongitudinal observation is 1.7 times moreinformative than the cross-sectional observa-tion. If we observed emergence of the toothat 22 months (near the mean) instead, Ir(22) 50.68 and Ie(21-22) 5 2.19; now the longitudinalobservation is 3.2 times more informative.Finally, if we observed emergence of thetooth at 25 months instead, Ir(25) 5 0.21 andIe(24–25) 5 2.44, or 11.4 times more informa-tive.

Methods to compute the expected informa-tion for longitudinal and cross-sectionalstudy designs are given in the Appendix.Figure 7 shows the results of evaluatingE(I2), E(I3), and E(I28) over a range of vari-ances, with an interval width of one month,assuming the proportion of censored observa-

7Pooling left and right sides does not bias the mean emergencetime, but standards errors for both the mean and standarddeviation are biased downward. For example, we found for boththe Guatemalan and Japanese children that the SEM droppedabout 30% when we pooled sides.

Fig. 7. Expected information (negative log-likeli-hood) for each observation in a normal model over arange of variances in squared months. The longitudinalexpectations assume one-month observation intervals.


tions is 0.20 for I28. For a low variance of 2.89(s 5 1.7), each observation in a longitudinalstudy is expected to contribute the equiva-lent of about three cross-sectional observa-tions (assuming 20% censoring). More typi-cal variances are about 9 (s 5 3.0); then,each longitudinal observation is expected tocontribute the equivalent of about four cross-sectional observations.

Improper analyses

We have shown that survival analysisprovides a number of advantages for estimat-ing parameters of tooth emergence, incorpo-rates a variety of messy data, and providesefficient and complete statistical estimates.The method renders obsolete ad hoc meth-ods or improper analyses. In particular, whensubjects are examined more than once, pro-bit analysis is not a proper analyticalmethod, as has been sometimes used in thepast before survival methods were widelyknown or used (e.g., Mayhall et al., 1978;Nystrom, 1977). Use of survival methodshas not always lead to proper analyses,however. A recent paper used survival andprobit analysis to analyze the deciduous andpermanent dentitions of a sample of chim-panzees (Kuykendall et al., 1992). The sam-ple included a longitudinal group with obser-vations taken over large intervals, as well asa cross-sectional sample. The product limitmethod (which assumes exact emergencetimes) was used to analyze the longitudinalsample, but, because of the interval censor-ing with large intervals, the parametricmethod given here or a life table would havebeen a more appropriate analysis. A secondproblem was the way in which cross-sec-tional and longitudinal data were combinedinto a probit analysis.Although probit analy-sis is proper for the cross-sectional data, thisparticular analysis was biased by improperinclusion of the longitudinal sample. First,they dropped longitudinal cases that wereright-censored (which we showed are statis-tically identical to probit nonresponders andcould have been included as such in theanalysis). Second, they included the longitu-dinal observations of emergence in the analy-sis as cross-sectional observations. Point 1likely resulted in a bias toward faster emer-gence times for the same reasons given for

dropping longitudinal-censored cases men-tioned earlier. The second point introduces amore subtle problem. The probit methodassumes that responders emerged their teethat some unknown time from age zero to thetime of examination. The longitudinal ‘‘re-sponders’’ included in probit analysis do notmatch that criterion; rather, those teethwere known to have emerged within somesmaller interval just prior to the time usedin the analysis. In other words, these caseswere selected in a manner different from theassumptions of the analytic method. Theresulting bias is toward faster emergencetimes.

We demonstrate the effect of an improperprobit analysis on the Japanese and Java-nese lower dentitions. Right-censored obser-vations were correctly treated as nonre-sponders. All opening times (tu) for theinterval-censored observations were set tobirth, thereby making each a probit re-sponder. Means from the correct analysisusing a normal distribution and from theimproper analysis are given in Table 7.Means were biased downward from two toeight months; the relative biases rangedfrom 8% to as high as 33%. For contrast, weempirically determined that other commonanalytical errors, like midpoint interpola-tion or dropping right-censored cases fromthe analyses, usually resulted in biasedmeans of under 3%.

The small bias from midpoint interpola-tion is probably because of the small observa-

TABLE 7. Biases resulting from improperly analyzinglongitudinal tooth emergence data as cross-sectional

data using probit analysis

GuatemalaMeana

(normal)Probitmean

Absolutebiasb

Percentbiasc

i1 8.34 5.44 2.90 35i2 13.81 9.85 3.97 29c1 20.04 15.43 4.61 23m1 17.01 13.36 3.65 21m2 27.14 25.00 2.14 8

Japani1 8.06 5.10 2.96 37i2 9.58 6.06 3.52 37c1 17.44 11.98 5.45 31m1 16.55 11.53 5.02 30m2 24.05 16.03 8.03 33

a Mean (months) estimated using a normal distribution.b Absolute bias between the proper estimate (x) and the probitestimate (x8) computed as x 2 x8.c Percent relative bias computed as 100 0 x 2 x8 0 /x.


tion intervals (35 and 30 days). Biases overlarger intervals can be demonstrated bycomputing a point estimate from a singleobservation. For example, suppose we havea single m1 observation from a Javanesechild that is interval-censored around tu 512 and te 5 18 months. We can produce astatistical ‘‘best guess’’ of when the toothemerged by computing the expectation, trun-cated at 12 and 18 months as

E 5e

12

18xf (x) dx

e12

18f (x) dx

(10)

(Namboodiri and Suchindran, 1987:40). Un-der the assumption that m1 emergence isnormally distributed, f (x) in (10) is a normalPDF with µ 5 17.29 and s 5 2.481 (fromTable 2), which gives E 5 15.9. Under mid-point interpolation, f (x) in (10) is a uniformdistribution, which gives E 5 15 months.The bias introduced by midpoint interpola-tion becomes worse as intervals becomelarger or move away from the mean of theunderlying distribution.

Another point of this example is thatmidpoint interpolation makes the paramet-ric assumption that tooth emergence is uni-formly distributed within the interval, yetwe have already modeled the overall processof tooth emergence as normally (or lognor-mally) distributed. The same distributionshould be used within the interval as well.

Although (10) will find an expected age ofemergence for individual observations, itwould be improper to compute expected agesfrom interval-censored observations (underany of the distributions) and then use theexpected ages to estimate means and vari-ances of emergence. The standard errors ofthe resulting estimates would be biaseddownward. Thus, expected ages should notbe used with methods designed for exactages.

CONCLUSIONS

Our aim has been the implementation of astandard method for estimating time toemergence that produces parametric esti-mates of means and variances as well asstandard errors of these estimates.An advan-tage over other methods is that it can be

used with both cross-sectional or longitudi-nal data to produce similar results. Themethod produces unbiased estimates frominterval-censored observations without theneed for midpoint interpolation. Results arenot biased by the size of the intervals overwhich observations are made. We have dem-onstrated the method on deciduous toothemergence in four populations.

Although data analyzed in this paper werecollected longitudinally (but with some cross-sectional observations), we showed math-ematically that a single analytical methodcan be used and will produce directly compa-rable results regardless of study design orspacing of dental examinations. Importantdifferences remain between the various re-search designs, especially logistic consider-ation in carrying out these two types ofstudies, but these differences do not trans-late into analytical differences when time totooth emergence or other developmentalevents are being analyzed.

ACKNOWLEDGMENTS

We thank Akiko Nosaka for Japanesetranslations, and M. Singarimbum (Popula-tion Studies Center, Gadjah Mada Univer-sity, Yogyakarta) and Dr. V. Hull for facilitat-ing work in Indonesia. We thank LyleKonigsberg for helpful comments. For assis-tance with the Bangladesh data, we thankthe International Centre for Diarrhoeal Dis-ease Research, Bangladesh, and a fellow-ship from the American Institute of Bangla-desh Studies (DJH). We thank CDEInformation Services, University of Wiscon-sin, for obtaining the original Japanese ar-ticles.

APPENDIXMeans, medians, and standard deviations

Equations are given for computing themean, median, and standard deviation aswell as estimates of the standard errors forthe lognormal distribution shifted by someconstant h. The equations hold for the un-shifted lognormal distribution by setting hto zero. We assume, µ, V(µ), s, V(s), and thecovariance between µ and s [denoted Cov(µ, s)]can be estimated.

The median of a shifted lognormal distri-bution is M 5 h 1 exp(µ); the mean is given


by X 5 h 1 exp(µ 1 s2/2) the variance andsquare of the standard deviation is: V 5 S2 5exp(2µ 1 s2)[exp(s2)21] (Cohen, 1991).

Estimates for standard errors for M, X,and V are derived by the method of statisti-cal differentials (Elandt-Johnson andJohnson, 1980:71) which uses terms up tothe second order of a Taylor’s series expan-sion. The approximate standard error ofthe shifted lognormal median is V(M) 5V(µ)exp(2µ). The approximate standard er-ror of the shifted lognormal mean is thesquare root of the variance:

V(X ) 5 e2µ1s2

· [V(µ) 1 2sCov(µ, s) 1 s2V(s)].(11)

The standard error of the shifted lognormalstandard deviation is the square root of thevariance:

V(S) 5e2µ1s2

es22 1

· [e2s25V(µ) 1 4sCov(µ, s) 1 4s2V(s)6

2 2es25V(µ) 1 3sCov(µ, s) 1 2s2V(s)6

1 5V(µ) 1 2sCov(µ, s) 1 s2V(s)6].

(12)

Information from completeinterval-censored and cross-sectional

observations

Here we demonstrate that the informa-tion content from exact observations isgreater than or equal to that from interval-censored observations, which is greater thanor equal to that from cross-sectional observa-tions. The term ‘‘information’’ is used in thecommunications theory sense (Kullback,1968) or in the same sense that Edwards(1972) uses the term ‘‘support’’: it is thenegative log-likelihood for an observation.The term corresponds, in a rough sense, tothe common qualitative concept of informa-tion. Specifically, information used here isadditive and begins at zero for a completelyuninformative observation. Informationmeasures and their properties are found inReza (1961).

We consider three regimens under whichobservations are collected: 1) as continuousobservations from a longitudinal study (i.e.,exact times to emergence are recorded); 2) as

observations at the end of intervals of lengthw, so that emergence at time t is only knownto have occurred over the interval (t–w, t];and 3) as one-time observations at time t in across-sectional study.

For each regimen, some individuals willemerge teeth by the end of the study andsome will not. First, consider observationsin which emergence does not occur at tcorresponding to right-censored (subscriptedby c) or nonresponder (subscripted by nr)individuals. The amount of information isthe same under all three regimes:

I1c 5 I2c 5 I3nr 5 2ln 3et

`f (x) dx45 2ln [S(t)].

(13)

If emergence is observed (subscripted by e)or is a responder (subscripted by r), thecontribution under 1 is, I1e 5 2ln[ f (t) dt].Under 2 above the contribution is

I2e 5 2ln 3et2w

tf (x) dx4

5 2ln [S(t 2 w) 2 S(t)].(14)

Notice that as the interval width, w, ap-proaches 0, I2e becomes smaller. At the limitas w = 0, I2e = 2ln [ f (t) dt] (the proof isfound in any calculus text) which is I1e. Sincef (x) is a probability density function, thearea of the integral falls in the range 0 to 1.This implies that I2e starts out small for alarge w and increases as w becomes small.

For a cross-sectional responder, case 3above, information from the observation at tis

I3r 5 2ln 3e0

tf (x) dx4

5 2ln [1 2 S(t)] 5 2ln[F(t)].(15)

Notice that the lower limit of integration isthe only change from equation (14). The areaof integration is never less than that in (14),which implies that I2e cannot be less thanI3r; they are equal only when t2w 5 0.

Expected information frominterval-censored and cross-sectional

observations

This section derives equations for theexpected information for a study in whichindividuals are followed at fixed intervalsand a study in which individuals are ob-


served cross-sectionally. For the interval-censored case, E(I2) is the sum of the informa-tion over all intervals, weighted by theexpected proportion of emerged teeth ineach interval. For simplicity, assume a strictregime of observations followed in intervalsof length w beginning at birth (i 5 0).Initially, we do not consider right-censoredobservations. This leads to the followingexpectation:

E(I2) 5 2 oi51

`

3ei·w2w

i·wf (x)

ln · [S(i · w 2 w) 2 S(i · w)] dx4 .

(16)

For the cross-sectional case, a child is aresponder with probability F (t) and nonre-sponder with probability S(t), with likeli-hoods ln[F(t)] and ln[S(t)], respectively. Theexpectation weights possible times for obser-vations by integrating f (t) over all t:

E(I3) 5 2 e0

`

5S(x) · f (x) ln [S(x)] 1 F(x) ln [F(x)]6 dx.(17)

For a normal distribution with negligiblearea at times less than zero, E(I2) dependson the variance of the distribution, but notthe mean; however, E(I3) reduces to 0.5independent of the variance. Equation (16)can be extended to account for right-censor-ing. For example, if a fixed proportion, c, ofindividuals are right-censored in every inter-val, the expectation becomes

E(I28) 5 2 oi51

`

3ei·w2w

i·wf (x)

· 5c ln [S(x)] 1 (1 2 c)

· ln [S(i · w 2 w) 2 S(i · w)]6 dx4 .

(18)

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