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Screening intervals and the risk of carcinoma in situ of the cervix Phillips, Norman 1994

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SCREENING INTERVALS AND THE RISK OFCARCINOMA IN SITU OF THE CERVIXbyNORMAN PHILLIPSB.A., McGill University, 1978M.A., The University of British Columbia, 1984A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Department of Statistics)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1994© Norman Phillips, 1994In presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.(Signature)_____________-- -Department of____________________The University of British ColumbiaVancouver, CanadaDateAbstractThis study examines the effect of length of interval between routine tests on the risk ofcarcinoma in situ (CIS) of the uterine cervix using cohort data from the B.C. population.CIS is a symptomless disease only detected by screening. Because of this, specialmethods are required for estimating incidence rates. Some case-control studies have usedprevalence odds ratios to estimate the relative risk of disease, usually invasive cancer,from length of screening interval. But duration of disease is related to interval lengthand hence prevalence rates cannot be used to estimate relative risk. A multivariate modelis fit to the incidence data using Poisson regression, and prevalence rates are fit with alogistic regression model. The results for prevalence odds ratios indicate a positiveassociation between screening interval length and risk of disease whereas the results forrelative risk indicate a negative relationship. Theoretical screening models areconsidered to examine the consequences of a case-control paradigm in which controls arematched with cases on the basis of having had a screen near the date of diagnosis of thecase, the matching period. As the matching period shortens, the distribution of intervallengths for controls converges to the underlying distribution, whereas the distribution ofinterval lengths for cases equals the distribution of lengths of intervals which span a pointin time. The latter distribution favours longer intervals. The difference is not due to thesampling of controls but, rather, to the relation between interval length and duration ofdisease. A matched case-control study is simulated with the cohort data, and a11conditional likelihood logistic regression model is fit. The results agree with those of alogistic regression analysis of prevalence rates indicating a positive relation betweeninterval length and risk of disease. When the sampling of controls is weighted byinterval length the odds ratios approximate the relative risk. A possible explanation ofthe surprising result that screening interval length is inversely related to risk of diagnosisof CIS is that more cases are cured with time by the natural regression of disease thanby treatment of earlier stages of disease. On the other hand, incidence rate is negativelyrelated to recency and frequency of prior negative screens, possibly because of theoccurrence of false negative tests. However, the effect of regression predominates andthe unavoidable conclusion is that less frequent screening decreases the risk of diagnosisof CIS.111Table of ContentsAbstract iiList of Tables viList of Figures viiiAcknowledgement ixIntroduction 1Review of Literature 6Epidemiological Methods 21Screening Models 30Fixed Interval Model 31Uniformity model 33Poisson Model 41ivData Analysis 57Conclusion 75Bibliography 78Appendix 1 81Appendix 2 85VList of TablesTable 1. Corrected incidence rate of CIS or worse per 1000 womenyears 9Table 2. Odds ratios for risk factors comparing CIN cases withcontrols 13Table 3. Hypothetical example of sampling matched controls under fixed intervalscreening 32Table 4. Odds ratio for annual vs bi-annual screeners 32Table 5, The distribution of screen intervals of lengths 1-5 years which spanDecember, 1979 by month of start of interval 54Table 6. Theoretical odds ratio of interval length relative to > 120 months, casesvs. controls, by amount of inter-subject screening intensity variability inPoisson model 57Table 7. Poisson regression of incidence rates on length of “last” interval,combined length of two preceding intervals, and abnormality in gappreceding the “last” interval 65Table 8. Logistic regression of prevalence cases on length of “last” interval,combined length of two preceding intervals, and abnormality in gappreceding the “last” interval 69Table 9. Model estimated incidence rates, prevalence odds and the ratio ofviprevalence odds to incidence rates by “last” interval length 70Table 10. Conditional likelihood logistic regression of a simulated matched casecontrol study 72Table 11. Conditional likelihood logistic regression of a simulated matched casecontrol study with sampling of controls weighted by length of “last”interval 74viiList of FiguresFigure 1. Distribution of intervals spanning 1980 for All intervals and intervalsending before 1981, Unweighted and Weighted by interval length 55viiiAcknowledgementI would like to thank Andy Coidman for all the valuable time and excellent supervisionhe has provided, Nhu Le for helping me out in a “crunch”, Brenda Morrison for herwillingness to offer advice and assistance at all times, the B.C. Cancer Agency forsupplying the data, and finally, my wife, Gloria, for taking on my share of the childcare,laundry, and dishes while I finished this project.ixChapter 1IntroductionCervical cancer, i.e. cancer of the uterine cervix, appears to develop in stages. “Thereis a continuous spectrum of histological [cell] abnormalities of the squamous epithelium[type of cell] of the cervix [the opening of the uterus] from dysplasia through carcinomain situ (CIS) to micro-invasive and invasive lesions [cancer]”. It is generally believedthat every case of invasive cancer of the cervix originated in dysplasia and thenprogressed to carcinoma in situ before developing into cervical cancer. Not all cases ofdysplasia develop into invasive cancer, and the time to cancer development in thosewhich do progress is variable.It is generally believed that the earlier cancer is detected and treated, the better theprognosis. Since the stages of dysplasia and CIS are non-symptomatic they are onlydiscovered by accident or by screening. Screening is the practice of testing for thepresence of disease in the absence of outward signs of disease. The rationale behindscreening is to enable the treatment of the disease at a stage when prognosis is better andtreatment may be less invasive. The Papanicolaou (PAP) test is a procedure for detectingcell abnormalities which are believed to be precursors of cervical cancer and is thescreening method of choice for cervical cancer. A representative cell sample is obtained1using an Ayre’s spatula from the transformation zone of the uterine cervix. The sampleis placed on a slide to be examined by microscope. Test results rate the degree of cellabnormality on a scale from one to four, with “class” ones being normal. Abnormalresults may be followed-up with more PAP tests or the individual may be referred forfurther examination.Prior to the introduction of colposcopy in the mid 1970s, treatment generally consistedof cone biopsy or hysterectomy. Both procedures are relatively serious and so treatmentwas not recommended unless there was evidence of persistent abnormality or theabnormality was sufficiently serious. Colposcopy2is a method of visually inspecting thecervix. This allows the localization of the treatment with such methods as cryotherapy(freezing) and laser therapy. Since these methods are relatively non-aversive and havefewer complications, they are applied quite soon after signs of abnormality.The effectiveness of the PAP test as a screen for cervical cancer is unknown sincerandomized trials have never been carried out. Screening was implemented before itseffectiveness was known3. The indirect evidence is sufficiently compelling as to renderfuture randomized controlled trials unethical. “The most persuasive evidence thatscreening for cervical cancer is effective comes from comparisons of cervical cancer inpopulations which introduced mass screening with different intensities and at differenttimes”3. For example, in B.C., the incidence of clinically invasive cancer and associatedmorbidity have decreased by about 75% since the introduction of cervical screening4.2However, inferences must be drawn from observational studies which have variouspotential sources of bias5. Parkins et. al.6 report risk factors such as SES (socialeconomic status) to be related to the probability of screening. However, adjusting forSES does not seem to materially affect the relation between screening patterns and riskof cervical neoplasia7. Knox8 argues that case-control studies are invalid if the factorswhich predispose to disease also affect the likelihood of screening. The same criticismalso applies to cohort studies. In spite of the inherent flaws of observational studies, thesignificant reduction in deaths from cervical cancer has been attributed to screening usingthe PAP test9’3.One of the risks associated with screening is that individuals might undergo unnecessarytreatment which in itself carries some risk. There is some evidence that stages even asfar along as carcinoma in situ (CIS) have significant rates of regression, i.e., return tonormal spontaneously, especially among younger individuals1. Since it is consideredunethical to withhold treatment from patients with cell abnormalities, it is impossible toestimate regression and progression rates directly’°, except in groups which declinetreatment. So indirect methods have been employed to try to estimate regression rates.One method which has been employed for this purpose compares the estimates ofincidence (the rate of new cases in a time interval) and prevalence (the proportion ofpopulation who are cases at a given point in time) at different stages of disease on theassumption that if all cases of disease progress, then current prevalence cases at one stagewould eventually become incidence cases at the next’. We should find that,3Prevalence of carcinoma in situ or worse at time t, +Totalincidence of carcinoma in situ or worse during the interval t,-t2 =Prevalence of carcinoma in situ or worse at time +Totalincidence of clinical cancer during interval t,-t21.Typically such calculations indicate a lower than expected number of cases of moresevere disease. Such shortfalls are attributed to regression. But the results could alsobe due to false negative test results, i.e. results which indicate absence of disease when,in fact, disease is present. Incidence rates of earlier stages of disease may be inflatedand the prevalence rates of the same may be deflated as a result of false negative testresults’. False negatives, along with inadequate screening, have been blamed for someof the fatalities observed”. The may arise from “inadequate cell collection, smearpreparations or smear interpretation”1.Estimates of false negative rates vary from 6%to 55%”.Current research attempts to model the risk of disease, generally invasive cancer, as afunction of screening patterns using data from observational studies and keeping in mindthe potential impact of disease regression and false negative test results. Screen-detecteddiseases require special methods for estimating incidence rates because the time of onsetis unknown. Although multivariate models are not typically employed with incidencerate data, they are well suited for Poisson regression models. Case-control methods havebeen used to approximate risk of disease by prevalence odds ratios. But this is notappropriate when the exposure variable is related to duration of disease. Some casecontrols studies select controls on the basis of screening times, which would seem tointroduce a dependency between exposure variable status and sampling probabilities.4This issue is explored with theoretical screening models. The distributions of intervallengths for cases and controls are derived under the null hypothesis that there is norelation between interval length and risk of disease. Finally these issues will beillustrated with analyses using data from the B.C. screening program.5Chapter 2Review of LiteratureD.A. Boyes, B. Morrison, and colleagues1,undertook a cohort study with data from theBritish Columbia Cervical Cytology Screening Program covering the years 1949-1969.The data consisted of two cohorts of individuals who were born between 1914 and 1918or between 1929 and 1933 and who had been screened at least once prior to Jan. 1,1970. The objective of the study was to provide estimates of prevalence and incidencerates of dysplasia or worse and carcinoma in situ or worse. Incidence rates of carcinomain situ or worse were estimated from the number of cases developing while undersurveillance relative to the total accumulated time at risk in the sample. Prevalence rateswere estimated from “abnormalities discovered at first contact”. An alternative estimateof prevalence is also given by the proportion of the population who are cases at anygiven point in time. These estimates took into account losses due to death and populationvariation due to immigration and emigration. The initial estimates of cumulativeincidence of carcinoma in situ or worse were 17.9 per 1000 for ages 18 to 38 (Cohort2), 19.3 per 1000 for ages 33 to 53 (Cohort 1) and 29.8 per 1000 for ages 18 to 53 (bothcohorts).6Being an observational study, there were concerns about the possibility of sample biases.In the beginning years of the program, testing was done primarily as a support servicefor the diagnosis of symptomatic women. Thus the estimates of incidence and prevalencetend to be inflated. There is a tendency for higher SES females to be screened morereadily although an effort was made to recruit women with low SES backgrounds. It isunclear what effect this bias might have on the estimates. Third, those who enter theprogram at older ages are different from those who enter at younger ages. A variety offactors may be involved in this selection bias.Boyes et. al. examined two potential sources of false negatives, namely errors in the laband sampling (literally) errors. The rate of lab errors was determined to be about 8.5%for Cohort 1 and 15.7% for Cohort 2. These estimates were based on review of testswhich were originally coded negative but the patient subsequently developed the diseaseand interpolated to women who only presented once for examination. This translates intoa dating error of about 26-38.6 months based on the average interval between the dateof the actual previous negative and the original date of diagnosis.Boyes et. al. refer to the remaining false negatives as “residual” false negatives andattempt to infer their rate from “the difference between apparent incidence rates derivedfrom short intervals between examinations and the rates derived from long intervalsbetween examinations”. They also considered the patients history of tests on theassumption that “after several smears have been taken any positive case is likely to be7a genuinely new case since, although a case may have been missed at one examination,it is unlikely to have been missed at two or three successive examinations”. Incidencerates were calculated for various combinations of interval length and numbers of previousnegative test. The authors observed that short intervals with few preceding negative teststend to have the highest rates. So they based their estimates of rates on those obtainedfrom long intervals with many preceding negative tests.The rationale of the method is straightforward. First, casesdiscovered after a short interval from a previous smear are likelyto be due to a classification error on the earlier smear. Secondly,after several smears have been taken any positive case is likely tobe a genuinely new case since, although a case may have beenmissed at examination, it is unlikely to have been missed at 2 or3 successive examinations.Table 1 presents the corrected incidence rate estimates based on Table 18 in Boyes et al.Using these corrected incidence rates, prevalence estimates were revised to adjust forcases “who were not new ‘incidence’ cases but were missed ‘prevalence’ cases”The discrepancy between observed and corrected incidence rates provided an estimate ofthe false negative rate for various age groups. These ranged from 4.1 % in the 45-49 agegroup to 12.9% in the 25-29 age group. Finally, the data were retabulated, correctingfor estimated false negatives and adjusting the date of onset for such cases by the amountobserved for lab false negatives.Next, Boyes et. al. compared prevalence and incidence estimates in an effort to estimate8Table 1. Corrected incidence rate of CIS or worse per 1000 women yearsAge range Cohort Incidence rate25-29 2 0.9830-34 2 1.0435-39 2 0.3640-44 1 0.4645-49 1 0.6050-54 1 0.32the rate of disease regression. They reasoned thatif pre-clinical cancer progresses to clinical cancer, then at the endof any time period the accumulated incidence of pre-clinical cancershould be equal to the prevalence of pre-clinical cancer plus theaccumulated incidence of clinical cancer. If all the possiblesources of error ... have been taken into account, a ratio ofprevalence of pre-clinical cancer plus accumulated incidence ofclinical cancer to accumulated incidence of pre-clinical cancersubstantially lower than unity must imply an excess incidence ofpre-clinical cancer and indicate that regression is part of thenatural history.’Combining the two cohorts, Boyes et. al. estimate this ratio to be 0.59-0.96 forcarcinoma in situ or worse. Thus there is some evidence that regression occurs.The method of estimating incidence rates for varying numbers of previous negative testsand for varying intervals since the last negative tests has become a research paradigm.Several studies utilizing this paradigm in the investigation of incidence rates for invasivesquamous-cell cancer in women who had at least one negative test, have been assembled9in Hakama et. aL’2 The followers of this paradigm frequently speak of “the protectiveeffect” of an interval of length x following y negative tests. The studies include cohortand case-control designs, and include data from Scotland, Iceland, Denmark, Norway,Sweden, Switzerland, Italy, and Canada. The present study will concentrate on screen-detected cases of CIS rather than incidence cases of invasive cancer since there are fewcases of invasive cancer mt the data available. However, analysis of CIS is of interestin its own right in that one of the postulated benefits of PAP test screening is theprevention of CIS with the attendant morbidity associated with its treatment.As previously mentioned, the Pap test is scored on a scale from Class 1 to Class 4, withClass 1 representing no evidence of abnormality, i.e. a negative test result. However,there are discrepancies in the literature concerning the definition of “negative” tests.Boyes et al.’ used a complicated algorithm to identify negative tests. The test result hadto be either a Class 1 or a Class 2 which did not meet any of the following conditions:(a) one of three successive Class 2’s; (b) one of a pair of Class 2’s separated by aninterval of ten months or more; or (c) followed directly by “a test of Class 3 or higher,or by a histological demonstration of dysplasia, carcinoma in situ, or invasivecarcinoma”. The majority of studies collected in Hakama” adapted the criteria used byBoyes et. al.’ “A negative smear is either class I; class II followed by a class-I smear;or class II followed by a class-Il smear within ten months, followed by a class-Ismear”. Some of the studies reported in Hakama adopted a liberal definition of“negative” in that “a negative smear is recorded when neither the cytological nor the10clinical examination leads to further cytological or gynaecological examination apart fromsucceeding screenings”’3.Most cases of CIS aie discovered as a result of follow-up tests to abnormal screenresults. PAP tests taken in this context are called “diagnostic” as opposed to “screens”.The frequency of such diagnostic tests is a different issue from screening frequency.According to Berrino et. al.14A usual way of coping with this problem has been to excludediagnostic smears from the analysis (Clarke & Anderson15; LaVecchia et a17) or to exclude all the positive smears from both theseries of cases and of controls (MacGregor et al’7).Berrino et al. include all tests taken “before the onset of symptoms”. Boyes et alcollapsed any such series with intervals less than 10 months into a single diagnostic test.Thus the results of different studies must be considered in light of the operationaldefinitions of basic constructs such as “negative” tests and screening intervals, as theoutcomes may hinge on these.One of the case-control studies included in the Hakama collection, conducted by Vecchiaet. al.7, included a group of 145 women with cervical intra-epithelial neoplasia (CIN),(a new classification category that combines dysplasia and CIS stages), who wererecruited from women referred to a university gynaecology clinic or the National CancerInstitute of Milan “for routine cervical screening”. That is, they were detected byroutine screening. The authors refer to a “diagnostic” test for the CIN cases. Thispresumably means an abnormal test result which leads to a positive diagnosis. Twenty11three percent of the cases were classified “histologically” as “mild dysplasia” (CIN I),26% as moderate dysplasia (CIN II), and 51 % as severe dysplasia or carcinoma in situ(CIN III). The control group consisted of “women found to have normal cervical smearsat the same screening clinics where CIN subjects had been identified. They were alsomatched for age by 5-year intervals”. It is not clear what definition of “normal” wasbeing used, although one of the tables suggests that class 2 tests were considered“normal”. Since the authors were interested in the effectiveness of different patterns ofroutine screening, any tests obtained for the purposes of diagnosis “because of bleedingor other symptoms suggestive of cervical neoplasia” were excluded. “Subjects werespecifically asked whether they had been screened by Pap tests, the number of times theyhad been screened, and their age at the first, last, and any abnormal test. The referencepoint for timing appears to be date of diagnosis for the cases and date of interview forthe controls, (i.e., the exact definitions are not clear to me). “The odds ratios (and 95%confidence intervals) obtained for various risk factors comparing CIN cases and controlsare presented in Table 2. (Evidently the index category is no previous screens).The results of logistic regression analyses controlling for “age, social class, number ofvisits to doctor or clinic, number of sexual partners, age at first intercourse, education,cigarette smoking, number of previous hospital admission, oral contraceptive use, andhistory of cervical ectopia” are comparable to the univariate results presented in thetable. Controlling for the total number of tests resulted in an odds ratio of 0.37 (95 %CI 0.13-1.03) for last tests 5 years versus >5 years ensuring that the effect was not12Table 2. Odds ratios for risk factors comparing CIN cases with controls.Risk Factor (relative to no previous Odds Ratio 95 % CIsmear)1 previous smear 0.27 0.10-0.712 previous smears 0.12 0.06-0.25<3 years since last smear* 0.09 0.04-0.203, <5 years since last smear 0.31 0.11-0.855 years since last smear 0.45 0.20-0.70Risk Factor (vs class I/TI smears only) Odds Ratio 95 % CINo previous smear 11.76 5.59-24.75One or more abnormal smears within oo -one year of diagnosis/interview***One or more abnormal smears outside of 7.18 2.83-18.19a year prior to diagnosis/interview* Excluding cases with a positive smear less than one year prior to diagnosis (notincluding “diagnostic smear”) results in an odds ratio of 0.07*** Relative to “normal smears only (class I/IT)”entirely due to the total number of tests. In summary,Screening on one occasion, irrespective of woman’s age and timesince the smear was taken, reduced the risk of ... CIN to about aquarter (RR=0.27), The degree of protection increased withincreasing number of previous smears, and with decreasing intervalsince last smear, both trends in risk being highly significant. [And]women with previous positive smears remain at increased risk ofcervical neoplasia.The authors offer the following explanation for the findings with respect to CIN:The finding that screening reduces the risk of CIN may seem13surprising, because strictly speaking Pap smears do not protectagainst the development of CIN. They are used to detect CIN,which if destroyed, may not develop into invasive cancer. Theexplanation for reduction in risk with increasing number of smearscould be that women with a healthy cervix have more opportunityduring their lifetime than do those with disease to accumulatemultiple tests.Berrino16 offers an alternative explanation.Since CIN is a long-lasting disorder it is unlikely that women withnewly diagnosed CIN would have been screened recently; if so,their CIN would have been detected then. Thus, previouslyscreened women are bound to be under-represented among CINcases detected in any given period. This bias may easily explainthe observed association and its quite strong temporal trend withoutpostulating any protective effect of screening.Berrino seems to be referring to the effect of length of screening interval on prevalenceunder constant incidence. The groups defined by interval length are being compared onthe odds of disease which reflects prevalence rather than incidence of disease. The oddsratios which La Vecchia et. al. report are “prevalence11 odds ratios as opposed toincidence rate ratios which are the usual indices of relative risk. In some situationsprevalence odds ratios approximate relative risk, but not when the factor underconsideration is interval length. This subject will be elaborated below.It should also be observed that PAP tests could serve to prevent the development of CIN,at least some forms of CIN. The diagnostic category “CIN” covers a variety of stagesof disease, and although all of them are without symptoms and thus can only be screendetected, protection against advanced stages such as carcinoma in situ can be achievedby detection of earlier stages through screening and subsequent treatment.14Another case-control study reported in the Hakama collection was carried out byMacGregor et. al.’7 following the same paradigm of assessing the relative risk ofdisease as a function of time since last negative test and the number of previous negativetests. They recruited all cases of invasive squamous carcinoma of the cervix whichappeared on the cancer registry in the Grampian region of Scotland between 1968 and1982 who had attended for screening at least once. Eighty of the cases so obtained wereidentified as having been “detected by routine screening”. It should be emphasized thatthe cases were invasive cases and thus probably quite distinct from CIS cases withrespect to properties like duration of disease. The following method was used to samplecontrols:Five random controls, matched for year of birth and with theadditional constraint that each must have entered the study (at thetime of her first negative smear) before the date of diagnosis ofcancer in the patient [and have] been screened within six monthseither side of the date of the screening test at which the cancer wasdiagnosed. This was to ensure that both patients and controls weredrawn from the same population of women - namely, thoseattending for routine (asymptomatic) screening.17The results are reported in terms of “relative protection”, “the inverses of the relativerisks”. “The relative protection decreased progressively with increasing time since lastnegative smear”7. This study would seem to suffer from the same problem ofinterpretation as the preceding one, at least for the screen-detected cases, namely, sincethe “exposure” variable, “time since last negative test”, is related to the duration ofdisease, the prevalence odds ratio is not equal to the incidence oddsj0l•One of the requirements of a valid case-control study is that selection of controls be15independent of the exposure variable’8status. The MacGregor et. al. study uses differentcriteria to select cases and controls. The cases are required to have had one “diagnostic”screen within a 15 year period and at least one screen prior to that, whereas the controlsfor a given case are required to have had one screen prior to and another screen withina year of the case’s “diagnostic” screen. The issue is whether these procedures forselecting cases and controls have the same, if any, sampling biases with respect toscreening frequency, which is one of the “exposure” variables under consideration. Wewill subsequently examine the implications of these methods of selecting cases andcontrols under three theoretical models.Not all studies have reported the same inverse relationship between risk of disease(invasive or otherwise) and screening frequency. For example, van Oortmarssen &Habbema’9 explain the high incidence rate in the first two years following a negativetest for the B.C. data as possibly being due to the testing of symptomatic individualsbecause the “screening program” started as a diagnostic support service. The Manitobastudy, also reported in Hakama20,proposed different explanations for different age groupsfor “the lack of a trend towards increasing risk of developing invasive disease withincreasing time since a negative smear”.The lack of a trend towards increasing risk of developing invasivedisease with increasing time since a negative smear may beattributable to different reasons depending on the age during whichthe woman-years occurred. In the younger women (<35 years ofage), much of the long-term follow-up, especially after only onenegative smear, is in error because many women may havechanged their names, and subsequent smears were recorded undernew names. This would account for the low incidence after only16one smear among women under 35 years of age.Among women under 40 years of age (after a five-year interval,45 years of age) in particular, there is a relatively high rate ofmigration, and the calculation of woman-years at risk does not takethis into account. Thus, the woman-years will be overestimated,and the incidence rates underestimated, by progressively greateramounts with increasing time interval; and there will be asignificant effect after longer time intervals, when the woman-years at risk are already small.A high rate of hysterectomies was experienced in Manitoba amongwomen aged between 40 and 50 years of age, in particular duringthe years 1969-1975. Hysterectomy would have the same effectas migration of underestimating incidence in a progressive mannerwith time. Among screened women over 55 years of age, a largeproportion of cases may be due to false negatives, since theincidence of in-situ cervical cancer in women at these ages is low.A smaller proportion of women of these ages has been screened,and the low sensitivity of the test for detecting invasive casesamong women who were screened because of symptoms wouldhave a considerable effect in masking a trend.2°Some screening programs are more organized than others in the sense that there is aneffort to screen the entire population at regular intervals. Others tend to rely more onindividual choice and thus are more prone to selection biases. Some of the Scandinaviancountries appear to have implemented relatively more organized screening programswhere an attempt is made to screen all individuals at risk at regular intervals. Theresults from the “organized” screening programs tend to indicate increasing incidencerates of invasive cancer with increasing intervals and the increase is more gradual withmore previous negative tests. The “opportunistic” screening programs like B.C. ‘s orManitoba’s do not show the same pattern. In fact the B.C. data suggests higherincidence rates of CIS are associated with shorter screening intervals21, which is17contrary to the hypothesis that frequent screening reduces the risk of developingcarcinoma in situ.The cohort studies reported in Hakama et al. estimated incidence rates of invasive cancerby age, number of previous negative tests, and time since last negative tests by dividingthe number of cases observed in each category by the total number of woman-years atrisk observed for each category22. Apparently no attempts have been made to fitmultivariate models to incidence data, The present study will fit a Poisson regressionmodel, also called a log-linear model, which is a particular case of generalized linearmodels3° The Poisson model is appropriate for modelling counts of independent eventsunder a Poisson-like process with a constant rate. The canonical model, to be use here,assumes that the covariate effects are multiplicative with respect to the expected numberof counts. The logarithm of counts observed in the covariate classes are fit to a linearfunction of the covariates with an optional “offset” which is given a coefficient of one,If, for this example, the offset is taken to be the log of the time-at-risk accumulated ina covariate class, then the log-linear model effectively models the log of the estimatedincidence rate as a linear function of the covariates. If Y, represents the number ofincidence cases in a covariate class defined by a covariate vector x then the log-linearmodel can be written in the form3°log{E(Y)} = offset - /3’x.18The method of analysis used in the case-control studies consisted primarily of logisticregression with conditional likelihood functions”. Logistic regression is also a specialcase of generalized linear models. If ir1 represents the probability of disease for the jthcovariate class, then 7r1/(l-ir) is the odds of disease. Logistic regression models the logof the odds of disease as a linear funcion of the covariates. Using the same notation forcovariates as in the case of Poisson regression, this is expressed symbolically aslog( ) = 13’x.1 ‘ir1The adaptation of the logistic model to case-control studies involves conditioning on anindividual being sampled for the study. Breslow and Day26 demonstrate using Bayes’theorem, that if sampling is independent of exposure status then the model coefficientsfor the exposure variables are the same as for the population as a whole. When controlsare matched with a particular case, the method of conditional likelihoods can be used toeliminate nuisance parameters. This general principle of statistical inference is discussed,for example, in Cox and Hinkley, 197423. The unconditioned likelihood function mayinvolve parameters, such as age, which are known to be important covariates, but whichare not of interest in the given study, i.e., they are nuisance parameters. The methodof conditional likelihood replaces the unconditional likelihood with a conditionallikelihood which conditions on the nuisance parameters. Estimates of the parameters ofinterest are obtained from the maximum likelihood solution of the conditional likelihoodequation. With one-to-many case-control matched designs, a likelihood function for eachmatched set is constructed which conditions on the fact that exactly one of the members19of the set is a case. Following the presentation in Armitage24, if the probability ofdisease of an individual in a matched set, s, is given by exp(a+Bx1),1=0,1,... ,c wherec is the number of controls in each matched set, then the probability that the observedcase is diseased given that exactly one member of the set is diseased is given by,exp(a+ 13x0)exp(a5-t-fx)’1=0,1,... ,c. The term exp(Y5) factors out of the numerator and denominator giving,exp(fir0)exp(13x1)The conditional likelihood is then the product of such terms from each matched set.20Chapter 3Epidemiol.ogical MethodsThe following is a review of the basic principles of sampling in case-control studies’8.One first selects a case series. Often these comprise all cases reported in a given period.It is recommended, however, that the cases form an uetiologicallyfl homogeneous group,i.e. they probably have a common causal history of disease development. Controls areselected from a pool of eligible controls. The crucial criterion for eligibility is that theindividual would have been included as a case had they developed the disease. Controlscan be matched with cases or not. Rothman’8demonstrates how the odds-ratio obtainedfrom a case-control study provides an estimate of relative risk as follows:The relevant data on disease incidence for a time period of lengtht might be summarized asI=--,1 P1tand21i=J?,Potwhere 1 and I are the incidence rates among exposed andunexposed, respectively, a and b are the respective numbers ofindividuals who developed disease during time interval t, and P1and P0 are the respective population sizes. ... The cases in a case-control study are the individuals who became ill during the timeperiod, that is a total of (a + b) individuals. ... If a proportion, k,of the combined exposed and unexposed cohorts is taken ascontrols, and the number of such controls is c for exposed and dfor unexposed, then the incidence rates among exposed andunexposed could be estimated as= k--,Ct[Actually this is an approximation which assumes that the risk ofdisease is small. The correct estimate is a/{t(c/k+a)}.]and [continuing with Rothman’s presentation]4 k.-,the relative incidence, or rate ratio (RR, often referred to asrelative risk), is obtained asRR=.i= .4 bcSince the sampling fraction, k, is identical for both exposed andunexposed, it divides out, as does t. The resulting quantity, ad/bc,is the exposure odds ratio (ratio of exposure odds among cases toexposure odds among controls), often referred to simply as theodds ratio. This cancelation of the sampling fraction for controlsin the odds ratio thus provides an unbiased estimate of theincidence rate ratio from case-control data [Sheehe25;Miettinen26]. The central condition for conducting valid casecontrol studies is that controls be selected independently of22exposure status to guarantee that the sampling fraction can beremoved from the odds ratio calculation.’8Breslow and Day26 give the same caution.One fundamental sampling requirement to which attention is drawnis that the sampling fractions for cases and controls must be thesame regardless of exposure category. If exposed subjects aremore or less likely to be included in the sample than are theunexposed, serious bias can result.The ultimate aim of many epidemiological investigations is to estimate the relative riskof contracting a disease in a given time period for those exposed to some condition incomparison to those not exposed. Relative risk expresses the ratio of risk rates for twogroups, i.e., relative risk is the ratio of the probabilities of developing disease for theexposed and unexposed groups. The odds ratio is the ratio of odds of contracting diseasefor those exposed relative to those not exposed, where the odds of contracting disease isdefined as the probability of contracting disease divided by the probability of notcontracting disease. For rare diseases, the probability of not contracting disease is closeto unity, so the odds ratio reduces to the relative risk.The probability of contracting disease in a given time period, P(t), can be approximatedby the cumulative incidence rate. This is demonstrated by Breslow and Day27 whocredit Elandt-Johnson28 with the following expression for the instantaneous incidencerate, X(t),23X(t) 1 < dP(t),1-P(t) dtand hence,1—P(t) = exp{-A(t)}, (3.1)where,A(t) = X(u)du,the cumulative incidence rate. Taking logarithms gives,A(t) —log{1-P(t)} P(t),when the disease is rare or t is small. Relative risk is defined as the ratio of incidencerates in exposed versus non-exposed individuals. Following the presentation in Breslowand Day, if r = X1/X2, the ratio of two incidence rates, which is the definition of relativerisk adopted by Breslow and Day, is constant over a period of time t, then from (3.1) wehave,24P1(t) = 1 — exp{—A1Q)}= 1 - exp{—X(u)du}= 1 - ex{—r.{X2(u)du}= 1 - [exp{—A2(t)}]’= 1 -rP2(t)providing the disease is rare or the time period is short. Breslow and Day observe that“In general, the ratio of disease risks is slightly less extreme, i.e., closer to unity, thanis the ratio of the corresponding rates”26.The usual way of estimating incidence rates in a cohort study is to divide the number ofcases occurring in a given time period by the number of “person-years” at-riskcontributed by the cohort population. Person-years is the total of all time at-risk fromall individuals. To estimate incidence rates for strata which are time dependent, i.e., anindividual might belong to one strata level at one point in time and to another strata levelat another time, the usual procedure is to assign the case to the strata level at which thedisease “occurs”, and to partition the individual’s total time at risk among the strataaccording to time spent in the strata. The principle at work is stated by Breslow and Dayas follows,The correct assignment of each increment in person-years offollow-up is to that same exposure category to which a death25would be assigned should it occur at that time.This procedure is difficult to apply for screen-detected diseases since the exact time ofdisease incidence is unknown. All that is known is that the disease occurred at sometime between the start and the end of the screen interval. If the disease is rare, this factwill not have much impact on the denominator estimates since they are comprised mostlyof intervals which do not result in diagnosis. The problem is in how to classify thecases, i.e., which numerators to increment, with respect to time dependent covariates,e.g. age, date, etc.One method which has been used is to date the onset at the midpoint between the dateof diagnosis and last negative screen prior to that. The time at risk is partitioned overtime dependent categories as if the case were incident at the midpoint. However thisprocedure violates the principle that time at-risk should be applied to that category towhich the case would have been assigned had it occurred at that time. If the case isassigned to the category containing the midpoint and if portions of time-at-risk areassigned to categories according to overlap, then portions of the interval which overlapcategories other than the one containing the midpoint will be assigned to categories otherthan the one to which the case would have been assigned had disease occurred at thattime. Whatever method one chooses to locate the time of disease (e.g. intervalmidpoint), the principle would seem to imply that the at-risk portion of the intervalshould be applied to the person-years of the same category as the case. How is the26principle to be applied to control intervals? The time at risk should be applied to thesame category that the case would have been assigned had disease occurred. This wouldseem to imply that the entire interval should be assigned to one category, namely the oneto which the case would have been assigned had disease occurred during the interval.This is perhaps the alternative method employed in some analyses by Boyes et. al.,namely,The denominators are obtained by calculating the number of yearsat risk between pairs of successive smears and allocating thisnumber to the age groups within which the mid-point occurs.These years of risk are then aggregated over all pairs of smearsand all women to produce appropriate denominators, expressed asperson-years of risk which can be used in the calculation of theevent rates. 1This description is ambiguous in that reference is made to “the age groups”, i.e. morethan one age group, within which the mid-point occurs. But assuming that the entireinterval is assigned to the unique age group containing the midpoint, this will meancontributions being made to one category from times when individuals actually belongto another category. The categories will tend to be blended. But at least the methoddoes not violate the principle of assigning time-at-risk to the category to which the casewould have been assigned had disease occurred.If instead of counting the number of incident cases in a given time period one were tocount the number of individuals with disease at a given point in time and divide this bythe number without disease, we would be estimating the prevalence odds of disease’8.27The prevalence odds is a function of the average duration of disease, since the numberof individuals found with disease at a given point in time is equal to the number whocontracted the disease in the past and still have it. Under certain circumstances theprevalence odds approximately equals the incident rate multiplied by the average durationof disease18. Rothman gives the following presentation. First the incidence rate equalsthe inverse of the average time until incidence. To see this, imagine following a finitepopulation until everyone gets disease. At that point the incidence can be calculated bythe number of people divided by the sum of their waiting times. Invert this and you getthe average waiting time. Assume the population is in a steady state, the number ofpeople entering the disease pool equals the number of people leaving it, and let N be thetotal number of people, P the number with disease, I the incidence rate, I’ the incidencerate of exiting from the disease pool, then for any time interval & we have, by thesteady state assumption that,It(N—P) = I’I.tP.And hence, since the mean duration in a state equals the reciprocal of the incidence ratefor exiting from that state, i.e., either going from a state of health to a state of diseaseor vice versa, if i5 represents the average duration of disease, then O=i”, and hence,I&(N-P) = (l/D)&Pp=(3.2)N-PThat is, the prevalence odds equals the product of the incidence rate and the averageduration of disease.28If the prevalence rate is small then N-P N and the prevalence odds is approximatelyequal to the prevalence rate which in turn will be equal to the incidence rate (for a steadystate population) when the average duration of disease is one unit of time. Prevalenceodds ratios equal incidence rate ratios i.e. relative risk, when the average duration ofdisease is equal in the two exposure groups under comparison. Or, alternatively, if theaverage duration of disease is estimable for the groups then the prevalence odds ratiocould be adjusted accordingly to provide an estimate for the incidence ratios, i.e. relativerisk.That case-control studies of screen-detected diseases involve prevalence odds ratios ratherthan incidence odds ratios has been reported by Sasco et. al.29, but the implications interms of the findings of such studies as MacGregor et. al. and La Vecchia et. al. havenot been articulated except, perhaps, in Berrino’s interpretation of La Vecchia’s findings.29Chapter 4Screening ModelsThis study addresses two issues raised by case-control studies involving screen-detecteddisease, namely the effects of proposed methods of sampling controls and the use ofprevalence odds ratios to estimate relative risk. We examine these issues in the contextof three screening models: a fixed interval model which is oversimplified but whichserves to illustrate some of the issues involved; a uniformity model which is lessrestrictive than the fixed interval model; and a Poisson model. The primary objectiveis to see whether the proposed method of matching controls, based on having a screenwithin six months of the “date of diagnosis” of a case, affects the resulting odds ratio.Since the independent variable, interval length, may be related to the probability of beingsampled as a control, there is a possibility of sampling bias. Interval length is alsorelated to duration of disease, so by the theory developed around equation (3.2), the oddsratio, which is a prevalence odds ratio, may not approximate relative risk.We will assume that the cases consist of all cases of disease diagnosed within a studyperiod of length L. The controls will be assumed to be matched with the cases on the30basis of having a screen within a matching period of length 1 centred around the date ofthe screen which led to diagnosis for the matched case. In the study by MacGregor et.al.‘7L was 15 years and 1 was one year, The requirements for controls must be modifiedat the two ends of the study period. Since a prerequisite for eligibility as a case or acontrol is that the screening interval ends within the study period. Controls must alsobe at-risk for disease, i.e. currently free of disease and without prior treatment thatwould preclude disease. To simplify matters we will assume that the tests are error-freeand that the disease under consideration does not regress. The null hypothesis is thatscreening does not have any effect other than identifying the presence of disease.Fixed Interval ModelConsider a population which consists of two groups in equal numbers, say 100,000 each,one group of individuals who are screened annually and another who are screened biannually (i.e., every two years), with 50,000 screened in each year. Assume also thatthe incidence rates are constant, small, and equal in the two groups, say 1 per 1000 peryear, and that any diagnosed individuals are immediately replaced by others from anexternal source. Suppose each case occurring within a 2 year period is selected andmatched with a control drawn randomly from those tested in the same year the case isdiagnosed. Assume diagnosis only occurs as a result of a screen and is immediate. Thesituation would be as depicted in Table 3.The reason 100 diagnosed cases occur in the bi-annual screening groups on each testingoccasion is that 50 were incident during the past year while another fifty were left over31Table 3. Hypothetical example of sampling matched controls under fixed intervalscreening.odd year even yearscreeningpattern screened incidence diagnosed sampled screened incidence diagnosed sampledcases cases controls cases cases controlsannually 100,000 100 100 133.5 100,000 100 100 133.5even years 0 50 0 0 50,000 50 100 66.5odd years 50,000 50 100 66.5 0 50 0 0from the previous year since they were not screened and therefore not diagnosed at thetime. Two hundred controls are sampled randomly from the 150,000 who were screenedin a given year, resulting in a 2-1 ratio of annual to bi-annual screeners among thecontrols. If we now compute the odds ratio for screening frequency we have the setupshown in table 4.Table 4. Odds ratio for annual vs bi-annual screeners.annual bi-annualOdds ratio=134/267 1/2200267casescontrols200133The incidence risk ratio is unity, but the case-control methodology employed results in32an odds ratio of 0.5. The inference we would draw from the results of this study, undernormal circumstances, would be that annual screening reduces the risk of disease.Uniformity modelThe preceding example assumed fixed screening intervals. This is probably not realisticsince interval lengths vary within individuals. In the uniformity model we do not assumeany connections between intervals within an individual, however, we do not rule themout either. We only assume that the distribution of screening intervals is stationary, i.e.,the same at any point in time, and “uniform”, i.e., the starting times for intervals of agiven length are uniformly distributed over a period of time which encompasses allpossible starting times of intervals that end in the study period. Thus in the uniformitymodel, the units of analysis are intervals with lengths and starting times which satisfy thestationarity and uniformity conditions. This model, like the fixed interval model, is notintended to be realistic but, rather, hypothetical for the purposes of illustration. We donot attempt to develop a realistic model of screening patterns.Assuming a constant incidence rate of disease I, the probability that an individualdevelops disease during an interval of length U=u is approximately lu, for if D is arandom variable which takes the value 1 when disease occurs during an interval and 0otherwise, then,P(D=1 I U=u) = 1-exp{-A(u)} Iii. (4.1)Assume that U is a continuous random variable representing interval length with density33f, then by Bayes’ theorem and (4.1), the conditional density of U given D= 1, f D=1’is given by,- P(D= 1 I U= u) •f(u) (4.2)fuD=l(u)- fP(D= 1 I U= u) •f(u)dulu f(u)flu .fu)duuf(u)This gives the distribution of interval lengths for cases. Hence, the odds for intervalsof length u conditional on intervals having length u or v and disease occurring is givenby,fuID=l(u) u•f(u) (4.3)fuDl(v) vNext we consider screening histories containing intervals that end within a matchingperiod of length 1 centred at the time of diagnosis of a case. The matching period iscentred at the screening date which “led” to the diagnosis of a case. The case-controlmethodology which is under examination requires cases to be matched with controls whohave a screen within specified limits of the “diagnostic” screen. Since an individual mayhave more than one interval ending in the matching period, for mathematical convenienceand to ensure uniqueness we will require control intervals to span the start of thematching period, i.e. begin before and end after the start of the matching period. Theevent of an interval spanning a point in time t will be represented by a random variable34S which takes the value 1 if an interval spans t, and 0 otherwise.Let the matching period for the selection of controls for a particular case begin at timet and have length 1. The probability that the endpoint of an interval which spans t landsin the matching period follows from the uniformity assumption. Since the starting pointof an spanning interval of length u is uniformly distributed over (t-u, t) and the sub-interval of starting points which land the endpoint in (t, t+l) has length 1 while the entireinterval of possible starting points has length u, the probability of the endpoint landingin (t, t+l) is min(1, i/u). The event of ending in the matching period is represented bythe random variable M which takes the value 1 when an interval ends in the matchingperiod and 0 otherwise. Then we have,P(M= 1 I U=u, 5= 1) = min(1, i/u). (4.4)The probability of an interval having length u given that it spans t (5=1), ends in thematching period (M= 1), and doesn’t lead to diagnosis (D=0) is given by,- P(M=1, S=1 I U=u, D=0) • fuD.o(u) (4.5)fuIM=1,s1,D=o(u)- P(M= 1, S=1 I D=0)— P(M= 1, S=1 U= u) • fuIDo(u)- P(M=1, S=1 I D=0)by Bayes’ theorem and the independence of interval span and endpoint location fromdisease status. The conditional probability of spanning t and ending in the matchingperiod, given that the interval has length u isP(M= 1, S= 1 U= u) = P(M= 1 I S= 1, U= u) • P(S= 1 U= ii). (4.6)35We already have an expression for the first factor, namely min(l, i/u). For theprobability that an interval of length u spans a time t we can assume that t € (0, L-l) ifcases diagnosed within 112 of the end of the study period are matched with controls withintervals ending within 1 of the end of the study period. Also note that the intervalendpoint can range from 0 to L, but not beyond for otherwise if disease were detectedat an endpoint beyond L the individual would not qualify as a case in the study. By theassumption of uniform starting times for intervals of length u over any specified period,it follows that the endpoints of intervals of length u are uniformly distributed over (0, L).Therefore, the probability of spanning a point t in (0, L-l) is given by the ratio of thelength of the region containing favourable endpoints to the length of all endpoints (L).The length of favourable endpoints depends on the relative sizes of u and 1, for if u> 1then for t c (L-u, L-l) the range of endpoints for which the interval spans t will be limitedto (L-t, L). Otherwise the range of favourable endpoints will have length u. Theseresults are summarized as follows,(4.7)- , 0t<L-u, u>iLP(S==l I U=u) = , L-utL-l, u>l.f, luLSubstituting min(l, i/u) for P(M=1 I S= 1, U=u) and (4.7) in (4.6) gives,36(4.8)= I, Ot<L-u, l<uuL LP(M=1, S=1 U=u) = , L-ut<L-l , i<u, luLNext we compute- P(D=O U=u) • f,(u) (4.9)fu1Do(u) —_________________________P(D=O I U=u) • f(u)duf(u)P(D=O I U=u) f(u)dusince P(D=O U=u) 1-lu 1. This assumes that if [a,b] is the support off thenP(D=O I U=u) 1-lu is true for all u e [a,b]. Substituting (4.8) and (4.9) in (4.5)gives,I , Ot<L-u, l<uLfUM=1,S=1,D=O(u) oc f(u)xl(L-t)L-ut<L-l, l<uf, luL37which shows that the distribution of intervals which span t and land in the matchingperiod approaches the distribution of intervals starting from a point t as 1 4’ 0. Nowdividing by the same expression for v in place of u, and assuming without loss ofgenerality that u < v gives,(4.10)1 , 0t<L-v, l<u0t<L-v, ul<vL-vt<L-u, l<uL-tfuM=1,s=1,D=o(u) f(u)<fu1M=1 ,S= 1 ,D=OO’) f(v) l(L-t), L v t <L- 1, u 1< vX , L-ut<L-l, l<uUf, lvVThis expression gives the odds for intervals of length u which meet the selection criteriafor controls when attention is restricted to intervals of length u or v assuming theregularity conditions hold. Hence under these assumptions the ratio of the odds for cases(4.3) to odds for matched controls (4.10) gives the (prevalence) odds ratio, ‘I’,381 , Ot<L-v, l<uOt<L-v, ul<vL-vt<L-u, l<uL-tV , L-vt<L-l, ul<vl(L-t)L-ut<L-l, l<uUf, ivVThus under the given regularity assumptions, the prevalence odds ratio will equal theratio of interval lengths ignoring complications that arise within an interval length of theend of the study period, which may be a sizable period depending on the intervals undercomparison. It might be advisable to stratify the analyses in comparing different intervallengths by the locations of the case diagnosis dates.Ignoring complications arising near the end of the study period, the distribution ofinterval lengths given that they span a point in time t is given by,— P(S== 1 I U= u) • f(u)f = -P(S 1 I U u) • f(u)duu f(u)E(U)’39since P(S= 1 U=u) ulL. This is the same as the distribution derived earlier forcase intervals (equation 4.2). But the controls are required to satisfy the additionalconstraint of landing in an interval of length 1 which we have seen effectively weights thedistribution of controls inversely by interval length.40Poisson ModelPAP tests may be considered rare events, and if screening intervals are independent ofpast intervals, then the screening process may be modelled by a Poisson process. Wewill assume the Poisson process is time homogenous with parameter . This implies thatinterval lengths have an exponential distribution with mean 14t. Once the process hasbeen going for a while the distribution of starting times of intervals relative to thebeginning of the process will be distributed as the sum of independent exponentials.Hence the distribution of starting times within a given small interval will beapproximately uniform. Thus the Poisson model approximately satisfies the assumptionsof the uniformity model in the previous section. However, the Poisson model specifiesthe distribution of the interval lengths. Once again, the objective of this excercise is toderive the odds ratio which results from selecting controls to match cases on the basisof having a screen within a specified time of the date of diagnosis under the nullhypothesis.I. Distribution of Spanning IntervalsWe will first calculate some useful distributions which will be needed for the calculationof case and control screen interval distributions. As before, let t be the start of amatching period. We now define two random variables. Let T0 be the time of the lasttest before the start of the matching period, and let T1 be the time of the first test afterthe start of the matching period. Let N(t) be the number of tests during time t. The41probability that there are n tests before t is given by,e’ t” (4.11)P(N(t)=n) =_____n!For n 1, P(Toe[t0,t+dt0J, T1c[t ,t1+dt1] I N(t) = n)e( n-I(n— 1)!dt0 • e”1 • dt1—__________________ __ _________n!np_____ e”’ dt01.The four factors in the numerator represent the probability of n-i tests before t0, one testin dt0, no tests between t0 and t1, and finally, one test in dt1. We will not consider n=Osince there is then no screen prior to t. Thus multiplying by P(N(t) =n), for n 1, andsumming over n gives,P(T0E[t,t0÷dtJ,T1E[t,t1÷dt], N(t)1) (4.12)(n—i) —= (itY’e dt01,( (n—i)(n- 1)!2e”dt0dt,=2e1’’dt0dt.Let U=TrT0, the length of the smear interval spanning t, then the Jacobian associatedwith the parameter transformation (t1,t0) — (u,t0) is42J(t,t I t,u) = - .fi.fi = i.i - = 101 0 8t0 3u au at0The marginal distribution for U is obtained by integrating (4.12) over t0 where t0 rangesfrom max{(t-u),O} to t, giving,(4.13)dt0, utfu,N(1)1(u)dt0 , u<tu2te , utu<tNow as t — oo (i.e., the interval spans a point distant from the start of the process) andP(N(t) 1)-1, then,fu,N(t)l(u) —f(u)= ii2ue.(4.14)The interval length thus has a Gamma distribution (equal to the sum of two independentexponentials with parameter t).43II. Distribution of Spanning Intervals Ending in the Matching Period.We will now calculate the same distribution, i.e., the distribution of the interval from thelast smear (T0) prior to t to the time of the next smear, T1, subject to the condition thatT1 t+l, i.e., the endpoint of the spanning interval lands in the matching period. ForN(t)l,P(Toe[to,to+dto],T1e[t,tj+dt I N(t)=n,T1<t+l)n-1kI• udt • e_h1_t0) • dt= (n-i)!e’(Lt)• (i—e’)= nIA—edtdt.t 1—e1Multiplying the terms by P(N(t) =n) and summing gives,P(T0c[t,t+d],T1c[t,t+d],N(t)i T1<t+l)2 -(t-t) (4.15)= pe dtdt0 1Again, let U=T1-T0, and integrate over T0 to get the marginal distribution of U. Therange of integration is determined by the possible values of t0 under the constraint on t0and t1 with respect to t. The integral may appear in two different forms depending onwhether 1< t or 1> t. We are typically only interested in the case t> 1, i.e., where thelength of the interval for having a smear is small compared to the time since the process44started. Thus, we have(4.16)I di , Ou<l01+1—udi , lu<tJ l—e’JU,N(t)l(Tl+1k14)— 1-uj+1—uI dt , tu<t+lI 1—e0 , ut÷lHence, (4.17)2ue 0u<l1—el<+u<tf (u = 1—e”JU,NQ)1171+1\ /p2(t+l—u)e, tu<t+l1—e’0 , ut+lWe may now use this expression to derive the distribution of the screen interval lengthfor spanning intervals required to fall within a matching interval. Let t — oo, thenP(N(t) 1) — 1 and,(4.18)2,u<l1—e1fu,N(I)lIt+l(u) —fuIT<I+l(u) =IA2le’ ul1—Note that if 1 is small in (4.18) above (i.e., lxtC’) then for practical purposes we have45f (ii = (4.19)JU17i-fl\ iand the distribution of intervals which terminate at t is the same as the overalldistribution of intervals. This is the same result that was found for the uniformity model,and suggests that the method of sampling controls based on the criterion of having ascreen close to the date of diagnosis of a case, does not affect the distribution of intervallengths.III. Risk of Disease Over the Study Period.The preceding distributions are not yet directly applicable to case-control studies sincethey have not been calculated separately for cases and controls. We will now considerthe derivation of appropriate distributions for cases and controls. We will shift ourattention from matching intervals of length 1 to the entire study period of length L. Lett be the start of the study period. Then the study period consists of the time period (t,t+L). Any individual screened positive in (t, t+L) will be a case in the study. Let T0be the last screen prior to t, M=m be the number of screens in the study period, and 1be the 1th screen after t but before t +L, 1=1,. . . ,m. (see diagram below).;mH46Define U1 =T1-T0, u=i+-1, 1=1,. . . ,rn- 1. Thus the U1’s are the intervals betweenscreens ending in the study period. Note that U1 is a spanning interval and thus wealready have its distribution. All we need to do is weight this distribution by u1 (whereI is the incidence) to obtain the probability of being diagnosed as a case in a spanninginterval of length U1=u1. Thus we will now consider the situations of screen intervalswhich begin and end within the study period.IV. Risk of Disease from Intervals Contained in the Study Period.Consider the joint distribution of two screening times 7, 7÷ in the interval (t, t+L)during which there are M=m screens. For rn 2,P(7[t,t1+dt], 7’ [t1÷, +dt1], M=m){(t—r)}’ 1eIL(ii) {JL(t÷L— t )}m_i 1e -p(z+L—:,,,)= (i- 1)!• dt e(rn-i- 1)!(t.—t)1t+L—t. )m_i_1= me_iL.(i—1)!(rn—i—1)!dt1 , i=1,2,...,rn—1.We wish to obtain the distribution of (] = 1÷-I. Now consider the transformation (1,—(ti, u). The marginal distribution of U,f1,M(uI+l ,rn),is obtained by integrating over t. Now t 7 t+L-U÷1,thus, for i= 1,2,... ,rn-l,But,47t+L-u.fU,,M(u+l,m)=‘dt, •1+L—u(tj_t)(t+L_tj_Uj+i)m_i_ld (L_u1+)m_i(i— i)!(rn—z— 1)! (m— 1)!so that,m(L_ rn-ifU,,M(uI+l,m)=(rn—i)!e’-” , i=l,2,...,m—l.The above is true for 0 u•÷1L. For u1 >L, fU+l,M(uI+l,rn) =0.First notice that the formula for u11 does not depend explicitly upon i so that the samedistribution holds for all u, thus for V rn 2, we may write,rnafi_ rn-iurn= !.Lk1U,JU,M”’ / —1’This (not unexpected) result says that conditional on the number of tests in a diagnosticinterval the distribution of each inter-screen interval is the same. Now we must calculatewhat the probability is that an individual with a screen interval U=u, will develop cancerin the study period. Under the null hypothesis this probability is approximately iii,where I is the incidence rate. Conditional on there being M=rn tests in the diagnosticinterval there will be rn-i test intervals so that the probability that an individual will havecancer diagnosed at the end of an interval of length u in a study period containing rn48screens is approximately given byP(D= 1, U= u, M=rn) (rn-i) ul .fUM(u,rn)where the factor (rn-i) is included since there are (rn-i) intervals. This calculation isapproximate because we are ignoring the possibility that disease develops in earlierintervals when calculating the contribution of later intervals. Thus, for m 2,mit ‘m—1TP(D=i, U==u, M==m) uI iiU,(in- 2)!In order to obtain P(D,u) we must now sum over m.P(D= 1, U= u) uI• -(4.20)j.L2u(L-u)Ie , 0u<L, m20 , otherwise(Notice the sum from rn 2 since there are no screening intervals within the study periodotherwise).V. Risk of Disease from the Spanning Interval.We must now calculate the contribution of spanning intervals in which the person isfound to have disease at their first test after t. We have calculated the distribution forsuch tests conditional on one existing (equation 4.18). In this case we are interested inthe unconditional probabilities490u<L, m1 (4.21)f(u) = 2Le” , uL, m10 , otherwiseVI. Overall Risk of DiseaseIn order to calculate the overall distribution of cases detected at screening in the studyperiod, (t, t+L), we need only multiply (4.21) by ul and add to (4.20), giving,P(D= 1, U= u) =2uILe , u 0 (4.22)But,P(D= 1)=P(D= 1, U= u)du = ILso that the distribution of screening intervals amongst the cases, P(U=u D= 1) is,P(U=u I D=1) = j2ue , u0,i.e., the distribution of the spanning interval (equation 4.14). Surprisingly, the resultdoesn’t depend on L. Under the null hypothesis the distribution of interval lengths is thesame for cases and controls. But we have just found that the distribution of intervallengths for cases is the same as the distribution of spanning intervals, while thedistribution of interval lengths for controls is the underlying distribution. The Poissonand uniformity models both agree on these two results.50VII. Inter-subject Variabifity.The fixed interval model considered previously has no within subjects variation, whilethe Poisson model just examined has no between subjects variation. In the case-controlframework both models lead to biased estimates of relative risk in that cases tend to havelonger intervals than controls under H0. We can expand the Poisson model to includebetween subjects effects by mixing the “p” parameter. That is, assume that there is adistribution g,() of screening intensities within the population. For the cases wehave,P(D= 1, U=u M=) =i2uILefrom equation (4.22). Thus,P(D= 1, U=u, M=1.) =2uILeg(j),P(D= 1, U=u) f uILeg(ji)d1,P(D= 1) = f f2uILeg(J.L)d,du,and thus the conditional distribution isf ueg(j)d (4.23)uID-1j’ [If () is chosen to be conjugate to2ue then P(u I D) will have a simple form.The density can be written in exponential form as,exp[(-ji)u + log(u) + log(2)]Hence the conjugate prior is a gamma distribution (Cox & Hinkley22), say,I’(a)51Evaluating the numerator of (4.23),a a-i (4.24)f2ue• e =____. fLa+1edU1’(a) 1’(a) j= uf3a r(a÷2)F(a) (u+f3)2’= a(a÷1).13a U(u+ I3)2The denominator evaluates to unity, hence, (4.24) is the conditional density, f1 D=i(u).Next we must consider the controls. We found that for short matching intervals thedistribution of screens is approximately ue” (equation 4.19), that is,P(D=O, U=u I M=1L) =Then,P(D=O, U=u, M=t) =p(D=O, U=u) = fP(D=O) = f eg(p)didu,f (u) = f IAeg(J2)dIUID=O.1 1 eg(I)dI1duThis gives,52a (4.25)UIDO— (u+I3)’Summary and ResultsThe uniformity and Poisson models give very similar results. First, the distribution ofintervals which span a point in time converges to the distribution of intervals starting atany point if the spanning intervals are required to end in a decreasing matching period.Second, the distribution of intervals for cases equals the distribution of spanning intervalsunder the null hypothesis.Under the fixed interval model each individual’s screening history can be assumed tohave an interval which spans a given point in time t. Hence the distribution of spanningintervals is equivalent to the distribution of screening frequencies for sequences of fixedlength intervals. In the example, it was assumed that half the population were screenedannually and the other half were screened bi-annually. Thus we would fmd thedistribution of spanning intervals to be a half for intervals of length one and two years,the same as the distribution of intervals for the cases. This differs from the distributionof intervals that end at a given time which favoured one year intervals by a ratio of twoto one.53We used the B.C. screening program data to investigate the reasonableness of theassumption of uniform starting times. Table 5 presents the distribution of screenintervals which span Dec. 1979 by the starting month for intervals of length 1-5 years.Table 5. The distribution of screen intervals of lengths 1-5 years which span December,1979 by month of start of interval.Years Dec Nov Oct Sept Aug Jul Jun May Apr Mar Feb Jan1 1979 52 153 200 122 121 120 175 246 180 169 153 1692 1979 45 68 49 39 23 27 38 46 32 56 53 581978 33 48 48 51 29 28 40 45 46 46 54 493 1979 16 24 22 18 12 11 17 17 9 29 9 181978 11 24 17 16 15 9 17 14 19 33 18 161977 7 16 21 12 9 11 12 17 16 16 17 2041979 0 6 4 9 5 8 8 17 13 9 5 81978 5 11 3 3 11 4 7 9 10 17 11 61977 1 8 9 6 3 1 8 5 10 12 9 51976 7 8 14 8 3 3 12 6 4 16 8 651979 2 9 7 3 5 2 2 2 7 5 2 41978 0 5 5 5 0 1 9 7 6 5 7 21977 2 2 5 3 3 4 7 5 2 8 5 41976 8 4 6 2 2 8 2 2 5 5 6 51975 0 8 7 3 3 3 5 2 9 6 6 12Except for a slight decrease in screening during vacation periods, the distribution ofstarting times appears to be quite uniform.Figure 1 displays the distributions of interval lengths of all intervals which span 1980(Dec. 1979) as well as the distribution of intervals which span 1980 and which end priorto 1981. The third distribution depicted is that of intervals which span 1980 and endprior to 1981, but the frequencies of the intervals have been weighted by the intervallengths.54-2-3>C)cza)c-4a)>3-50-J-6-710 30 40 60Interval length (months)Figure 1. Distribution of intervals spanning 1980 for All intervals and intervals endingbefore 1981, Unweighted and Weighted by interval length.The weighted distribution approximates closely the distribution of all spanning intervals.The distribution of unrestricted spanning intervals corresponds to the distribution of “inprogress” intervals at a given point in time. If each individual had tests repeated atregular intervals but these intervals varied between individuals then this distribution20 5055would correspond to the distribution of screening frequencies in the population. Thedistribution of intervals which end in a short time range approximates the distribution ofintervals that end at a given point in time. The empirical results displayed in Figure 1support the relation between the two distributions as predicted under the uniformityassumptions. That is, the probability of an interval ending in a narrow range equals theprobability of spanning a point in time weighted inversely by the length of the interval.Implications of the Poisson model with inter-subject variablility were explored fordifferent parameter values of the conjugate prior representing different degress ofvariability. The mean and variance of a Gamma distribution, in terms of the parametersand fl, are a/fl and a/fl2, respectivley. We chose to fix the mean screening intensityto be the inverse of the unweighted sample mean of the screen interval lengths within theB.C. cervical screening program data set (to be described below). The mean intervallength for 668,751 intervals was 27.10 months with a standard deviation of 27.39months. Values for a and ft selected to give a mean of 1/27.1 and coefficients ofvariation of 30%, 60%, and 100% to represent low, medium, and high variabilityrespectively. The principle outcome of interest is the resulting odds ratio for variouscategories of interval length. To parallel the analyses to follow, we chose intervalcategories: 10-18 months, 19-30 months, 31-42 months, 43-54 months, 55-120 months,and > 120 months. The odds ratios of interval category “exposure”, (e.g. exposure toa 10-18 month screen interval), relative to the index category, > 120 months, are givenby,56P(UE[l0, 18] j D=l) / P(UE(120, 0°) I D=l)P(UE[1O, 18] I D=0) I P(UE (120, co) I D=0)The probabilities were obtained by integrating the appropriate conditional density overthe specified range. The results are presented in Table 6.Table 6. Theoretical odds ratio of interval length relative to > 120 months, cases vs.controls, by amount of inter-subject screening intensity variability in Poisson model.Variability (coefficient of variation)Interval lengthlow (30%) ] medium (60%) high (100%)10-18 months 0.125 0.280 0.36819-30 months 0.212 0.418 0.51631-42 months 0.307 0.535 0.62943-54 months 0.396 0.622 0.70555-120 months 0.583 0.763 0.818Evidently lower inter-subject variability in screening intensity results in more pronouncedeffects of interval length on prevalence odds ratios.57Chapter 5Data AnalysisB.C. has a centralised cytology screening program which has been in operation since1949. Until the early sixties, it was more of a diagnostic support service than ascreening program. The proportion of women over 20 ever screened was about 3% in1955’. With the widespread use of oral contraceptives in the early sixties, the Pap test,which was applied in conjunction with the dispensing of oral contraceptives, becameincreasingly less of a diagnostic tool and more of a screening test. By 1962 it wasestimated that 53% of women over 20 had ever been screened, and by 1969 this figurerose to 78%’. The samples are obtained by general practitioners and gynaecologists forthe most part, and then sent to a central laboratory where they are interpreted. Patientsare assigned identification numbers which their physicians are supposed to provide alongwith the sample for patient identification, but name and date of birth are also recorded.The test results and other information are entered into a centralized cytology computerfile. The result, along with recommendations for further care, is returned to thereferring physician who is responsible for the care of the patient. Physicians areresponsible for advising appropriate screening intervals for their patients. If, forexample, a test is abnormal and not merely benign atypia, physicians will be sent a58reminder for a repeat test if one is not obtained within four months. Also, for cases withhistories of severe dysplasia or carcinoma in situ, reminders are automatically sentannually.D.A. Boyes, B. Morrison, and colleagues’, undertook a cohort study with data from theBritish Columbia Screening Program covering the years 1949-1969. Their objective wasto provide estimates of prevalence and incidence rates of dysplasia or worse andcarcinoma in situ or worse. Two cohorts were selected to cover as wide a range of ageas possible as well as providing some overlap. “The records of all women who had beenborn in the years 1914-1918, and 1929-1933, and who had had 1 or more cervical tests[prior to 1969], were pulled from the identity files of the British Columbia CentralCytology Laboratory”1 (52,452 and 66,701 women respectively). Extensive recordlinkage procedures were carried out in order to minimize duplications. The followinginformation was extracted from the data base for every qualifying women:(1) Identifying information, i.e. the patient’s surname (first 12 characters), first name(first 8 characters), second initial, month and year of birth, and husband’s first name(first 4 characters);(2) The month, year and cytological class of every smear taken up to the end of 1969 andfor women with subsequent positive histological findings, the original cytological classas assigned when the smear was first read, as well as a reviewed class based upon a re59examination of the specimens;(3) All consequent radiation and surgical procedures, including cervical biopsy andhysterectomy. Hysterectomies performed for reasons that were not the consequence offindings obtained from screening could not be completely documented unless a womancame for a repeat smear following hysterectomy.(4) Histological diagnoses based on biopsies or other surgical procedures.Follow-up on the original two cohorts has been updated to 1992 by Morrison21. Thisprovides a longer period of observation for each woman as well as a greater overlap inage for the two cohorts (30 years instead of 5). The original study involved 121,722woman. Successful linkage was achieved for 43% providing updated data for 71,236women. There were two reasons why linkage failed in about half the women. (1) Fileswhich had a history of either all negative, no histology, no test in past 7 years, or deathbetween 1976 and 1985 were removed from the archives. (2) The linkage method wasconservative.The “raw” data were prepared for analysis as follows: First, the data consisting of PAPtest, histology, and death records were sorted chronologically within subjects. Therecords were then processed sequentially. A pair of records was defined to constitute ascreen interval if the starting record had class 1, the ending record was a Pap test record,60and the interval was at least 10 months. This operational definition of screening intervalswas designed to achieve two objectives. First, we wanted to examine the risk of diseaseassociated with routine screening, as opposed to diagnostic testing which is often timesdone when the presence of disease is suspected. And since we didn’t have anyinformation other than the times and results of tests, we had to resort to the method ofexcluding intervals under 10 months. We decided to exclude intervals that started withan abnormal test result for the same reason. The excluded intervals create “gaps” in anindividual’s screening history. Although the gaps are not studied as screen intervals,they do play a role in the estimation of incidence rates and in the formation of covariateclasses based on conditions preceding screen intervals.Any records following either diagnosis of CIS or worse or a hysterectomy wereexcluded.Each screen interval was assigned to a covariate class to contribute towards thecomparison of cases and non-cases. For the cases this interval was the last intervalbefore diagnosis. So each valid screen interval is included as a “last” interval andcontributes to a covariate class defined by the following factors:(1) Cohort: born 1914-1918 or 1929-1933,(2) Period: the time period when the interval ended was categorized as: pre- 1963, 1963-611975, 1976-1992. These cut points were chosen because the three periods were believedto differ with respect to sample characteristics and/or screening practices. Prior to 1963screening tended to serve primarily as a diagnostic tool. At about 1963 the use of oralcontraceptives became more prevalent and since Pap tests were frequently performed inconjunction with the dispensing of oral contraceptives the test became more of ascreening instrument. Prior to the mid 70s the diagnosis and treatment of abnormalitieswas quite a serious procedure, so it was common practice to wait until the disease hadreached an advanced stage before taking action. However with the introduction of thecolposcope (an instrument which allows the visual inspection of the tissue) in the mid70s, treatment was applied more readily.(3) Interval: screen interval lengths were broken into categories: 10-18, 19-30, 31-42,43-54, 55-119, 120+ months, the idea being to approximate 1,2,3,4, and 5-10 yearintervals. This category is referred to as the “last interval” category since cases wereassigned to the “last” interval.(4) Previous: the combined lengths of the two screen intervals preceding the “last”interval were grouped in the following categories: 10-36, 37-60, 61-84, 85-119, 120+months, no preceding intervals, and one preceding interval. The reason for looking atpreceding intervals is to address the fact of false negative tests, i.e. tests which reportno abnormalities when in fact disease is present. Thus an individual who tests negativetwice is less likely to have disease than one who has tested negative only once. It may62be useful to be able to estimate the risk of disease associated with various patterns ofscreening intervals. The choice of categories for the combined length of the twopreceding intervals is intended to group individuals who were tested at fixed intervals ofone, two, three, and four years. It also identifies individuals who had one or nopreceding screens.(5) Preceding gap abnormality: three categories of abnormality occurring in the “gap”,if any, preceding the “last” interval: one for intervals with no preceding gap (either the“last” interval began with the termination of a screen interval with a class 1 test resultor it was the individual’s first record); a second category for “minor” signs ofabnormality including class 2 or class 9 (inadequate sample) test results or class 1, 2, or9 less than 10 months apart; and a third category for “major” signs of abnormalityincluding class 3 or 4 test results or any histology record. The inclusion of this factorwas motivated by an attempt to control what is probably a very important predictor ofdisease, namely previous abnormalities.The date of diagnosis for cases with CIS or worse was defined to be the endpoint of thelast screen interval preceding diagnosis. The rationale for this definition was that testsdone subsequent to the last screen interval were probably done for diagnostic purposesrather than routine screening. The exact date of onset of disease cannot be identifiedwith much accuracy, so the decision is somewhat arbitrary. The total time-at-riskcontributed by an individual extends from the start of the first screen interval to the end63of the last record or until diagnosis or hysterectomy. It is partitioned into segmentsaccording to starting points of screen intervals. Gaps are assigned to the precedingscreen interval. The segments are then assigned to covariate classes according to thecovariate pattern of the screen interval defining the segment. Period is defined by theendpoints of the screen intervals. This method of assigning time at risk is method B ofBoyes et. al. and follows the principle that time at risk is assigned to the category towhich the case would have been assigned had disease developed at that time.There were a total of 1198 cases of CIS or worse among 1.1 million records fromroughly 120,000 subjects. The numbers of cases and incidence rate estimates bycovariate class are given in Appendix 1. There appears to be a trend for incidence ratesto decrease with increasing interval length.A Poisson regression analysis3°was performed on the incidence data, predicting the logof the number of cases from cohort, period, length of last interval, combined length oftwo intervals preceding the lastu interval, and the degree of abnormality in the gappreceding the “last” interval. Log time-at-risk was included as the offset. Poissonregression fits the log of the estimated incidence rates to a linear combination of thecovariates which has the effect of treating the covariates as having multiplicative effectson incidence rates. The covariates, being categorical, are represented by terms for allbut one of the category levels - the index category. Treatment contrasts were used tocompare the effect of each level with the effect of the index category. The combination64of all index categories is represented by the intercept. Let n represent the number ofcases, t the aggregated time-at-risk and x the vector of factor categories and an intercept,then the model can be expressed as,log {E(n)} = log(t) f3”xThe results of the model fit are presented in Table 7.Table 7. Poisson regression of incidence rates on length of “last” interval, combinedlength of two preceding intervals, and abnormality in gap preceding the “last” interval.Parameter Coef s.e. t value riskIntercept —11.1 0.330 —33.7CohortPeriodborn 1929—331963—19751976—19920.543 0.256 2.120.192 0.200 0.9580.074 0.221 0.336Period*Cohort period*cohortl 0.094period*cohort2 —0.4260.268 0.3520.281 —1.52Index categories: >120 months for last and preceding twointervals and no preceding gap for preceding gap abnormality.The inclusion of an interaction term for cohort and period significantly improved the fitLength of 55—120 months 0.273 0.175 1.56 1.31last interval 43—54 months 0.581 0.168 3.46 1.7931—42 months 0.890 0.166 5.35 2.4419—30 months 0.966 0.161 6.02 2.6310—18 months 1.31 0.159 8.22 3.71Combined 85—120 months 0.217 0.235 0.921 1.24length of 61—84 months —0.051 0.223 —0.230 0.950two intervals 37—60 months —0.478 0.211 —2.26 0.620preceding 10—36 months —0.605 0.210 —2.88 0.546the last one previous 0.259 0.214 1.21 1.30no previous 0.620 0.212 2.92 1.86Preceding gap minor 0.507 0.067 7.59 1.66abnormality major 1.23 0.217 5.70 3.4265and thus was included in the model. No other interactions were significant. The modelwas also fit without an offset, and including log of the time at risk whose coefficient wasfound to be 0.943 with a standard error of 0.052. This implies that the data areconsistent with assumed relation between incidence and time-at-risk. Visual inspectionof residuals plotted against fitted values and by factor did not indicate any blatant signsof misfit. The dispersion parameter was estimated to be 1.1 using formula (6.4) fromMcCullagh & Nelder3°which is given by,(— IJo2 =X21(n-p) = I(n-p).j U1where n is the number of cells with non-zero time-at-risk, p is the number of parametersin the model, y, is the observed count in cell i, fi is the model predicted count for celli. A dispersion value greater than one suggests that there is ‘over-dispersion’ in thedata30. The residual deviance was 406.3 on 552 degrees of freedom.The rates presented in Table 7 are derived from taking exponents of the coefficients.Since “treatment” contrasts were used (i.e., the coefficient of the index factor categorywas set equal to zero), taking the exponents of other categories provides estimates ofincidence rates relative to the index category. The index category was taken to be > 120years for both interval length factors and no preceding abnormality for the preceding gapabnormality factor. The baseline incidence rate for the covariate class defined by allindex categories is given by the exponent of the intercept which equals 0.181 cases per661000 person-years (note that the exponent of the intercept must be multiplied by 12,000to transform the units from person-months to 1000 person-years). The degrees offreedom can be taken to be 552°, Thus the t values can be compared with 1.96 forsignificance at the .05 level. All but one of the interval length levels were significant aswell as the two shortest combined preceding interval lengths, the level representing noprevious screens, and both levels of the preceding condition factor. It would seem thatlength of screen interval is associated with reduced risk of disease, but having had acouple of screens in the recent past is beneficial if the outcome is negative.The Poisson model can be used to estimate incident rates for joint combinations offactors. If we match up categories from the combined lengths of the preceding twointervals with the length of the current interval, we can get an approximation of the riskof disease with annual, bi-annual, tn-annual, and quadra-annual screening patterns.Since if an interval length is to be recommended, it is assumed that it will be followedon a regular basis. Thus, for example, the interval length category of 19-30 months willbe combined with the category of 37-60 months for the combined length of the twoprevious intervals. Together they represent the risk of a bi-annual screening pattern.The incidence rate relative to the combination of baseline categories is estimated by theproduct of the exponents of the respective coefficients. The estimated rates relative tobaseline are: 2.02 for annual screeners, 1.63 for bi-annuals, 2.31 for tn-annuals, and2.22 for individuals who are screened every four years. When the effect of previousscreens is added to the effect of the “last” screen the two effects tend to dilute each67other. But the effect of the “last” evidently wins out since the incidence rate is still twicethe rate of the reference group which consists of individuals whose “last” interval wasgreater than 10 years and whose combined lengths of the previous two intervals is alsogreater than 10 years.Incidence rates are based on numbers of cases occurring in units of time. We alsoexamined the proportion of intervals in each covariate class which resulted in diagnosisof disease. This gives an index of the prevalence of disease in each covariate class.The prevalence rates are given in Appendix 2. Here the reverse trend of rate withrespect to interval length is observed. We modelled the prevalence rates with logisticregression, using the same set of covariates as in the Poisson model. The results aresummarized in Table 8.Treatment contrasts were again used so the exponents of the coefficients give estimatesfor the odds ratios of each of the factor levels with respect to the index levels. Theresults are the same as the Poisson regression results except for the effect of intervallength which in this case indicates decreasing risk with decreasing length rather than theother way around. But this does not necessarily imply that individuals are better off withshorter intervals because you have to make it through relatively more intervals when theyare shorter. The proper index for comparison is the risk per unit time. However thisdoes not quite equal incidence rates as they have been computed here, since simplydividing the prevalence rate for an interval by the length of the interval does not account68Table 8. Logistic regression of prevalence cases on length of “last” interval, combinedlength of two preceding intervals, and abnormality in gap preceding the “last” interval.InterceptParameter Coef s.e. t value risk—5.98 0.329 —18.2CohortPeriodborn 1929—331963—19751976—19920.526 0.257 2.050.210 0.201 1.040.120 0.222 0.539Cohort*Period cohort *perjodl 0.131cohort*period2 -0.4070.269 0.4880.282 —1.45Index categories: >120 months for last and preceding two intervals and nopreceding gap for preceding gap abnormality.for the time between intervals, the “gaps” during which diagnostic testing is presumablyoccurring.By equation (3.2) the ratio of prevalence odds to incidence rate equals the averageduration of disease. We computed model estimates of prevalence odds and incidencerates for each level of the “last” interval category. We then divided the prevalence oddsby the corresponding incidence rate to obtain an estimate of the average duration ofdisease for each “last” interval category. The results are presented in Table 9.Length of 55—120 months —0.430 0.175 —2.46 0.651last interval 43—54 months —0.601 0.167 —3.59 0.54831—42 months —0.652 0.165 —3.94 0.52119—30 months —0.955 0.159 —5.99 0.38510—18 months —1.08 0.158 —6.84 0.340Combined 85—120 months 0.230 0.236 0.973 1.26length of 61—84 months —0.040 0.224 —0.177 0.961two intervals 37—60 months —0.470 0.212 —2.21 0.625preceding 10—36 months —0.613 0.211 —2.91 0.542the last one previous 0.291 0.215 1.35 1.34no previous 0.658 0.213 3.09 1.93Preceding gap minor 0.554 0.067 8.29 1.74abnormality major 1.28 0.219 5.88 3.6069Table 9. Model estimated incidence rates, prevalence odds and the ratio of prevalenceodds to incidence rates by “last” interval length.Incidence rate Prevalence rate Prevalence!“Last” interval length (/1000 person- (/1000 persons) Incidenceyears) (years)10-18 months 0.673 0.860 1.2819-30 months 0.477 0.974 1.4531-42 months 0.442 1.32 2.9943-54 months 0.325 1.39 4.2855-120 months 0.238 1.65 6.93>120 months 0.181 2.53 14.0The categories correspond roughly to intervals of length 1, 2, 3, 4, 5-10, and > 10years. Clearly the estimated duration of disease is related to “last” interval length. Onewould expect the duration of disease to be longer on average if diagnosed after a longscreen interval rather than a short screen interval. However, the average durationestimates are unrealistic in that they seem to correspond quite closely with the averageinterval length within each category, which would imply that on average disease occurredshortly after the start of the screen interval. Perhaps the duration is inflated by theoccurrence of false negative tests.The assumption of constant incidence overlooks one important factor, namely the stageof disease between normality and CIS, namely dysplasia. Assuming a negative test to70be accurate, an individual cannot develop CIS without first going through a period ofdysplasia. There is a period of grace, namely the sojourn time for dysplasia, duringwhich CIS cannot occur since dysplasia must first run its course. Hence the interval overwhich disease can develop is shorter than the nominal interval. This would have theeffect of deflating the ratios of incidence rates to prevalence odds ratios, which mayaccount for the observed results.We also simulated the methodology of the case-controls studies discussed earlier. Eachof the cases was matched with seven controls on the basis of year of birth and to withinfive months of “date of diagnosis”. That is, controls were required to have had a screeninterval end within five months of the endpoint of the last screen interval before diagnosisof the matched case. The data were then subjected to a conditional likelihood logisticregression analysis using the PECAN package31, The same covariates were usedexcluding cohort and period since these were controlled by design. The results arepresented in Table 10.The same pattern of results are observed as for the unconditional logistic regressionmodel, although the two models are not identical. The case-control study matched byyear of birth and date of diagnosis to within 5 months, while the cohort study merelycontrolled for cohort and period of diagnosis to within 1-2 decades. The risk estimatesare somewhat less extreme for interval length and more extreme for the combined length71Table 10. Conditional likelihood logistic regression of a simulated matched case-control study.Parameter Coef s.e. z score riskLength of 55—120 months —0.744 0.228 —3.27 0.475last interval 43—54 months —0.909 0.219 —4.14 0.40331—42 months —0.954 0.220 —4.35 0.38519—30 months —1.31 0.213 —6.16 0.26910—18 months —1.42 0.212 —6.68 0.242Combined 85—120 months 0.014 0.267 0.054 1.02length of 61—84 months —0.157 0.250 —0.628 0.855two intervals 37—60 months —0.703 0.237 —2.96 0.495preceding 10—36 months —0.858 0.235 —3.65 0.424the last one previous 0.044 0.243 —0.180 0.957no previous 0.292 0.244 1.19 1.34Preceding gap minor 0.544 0.076 7.20 1.72abnormality major 1.20 0.272 4.40 3.31Index categories: >120 months for last and preceding twointervals and no preceding gap for preceding gap abnormality.of preceding screen intervals. The standard errors also tend to be slightly larger.Although the two methods of modelling prevalence rates differ to some extent, they bothsuggest the same conclusion, namely that interval length is directly related to risk ofdisease, contrary to results of the analysis of incidence rates.We have seen that prevalence rates, as given by unconditional logistic regression withcohort data or conditional likelihood regression with case-control data, are inflatedrelative to incidence rates by a factor roughly equal to the average length of duration ofdisease. Thus, crude estimates of incidence rates could be obtained from prevalencerates by adjusting for the likely relation between the factor and disease duration.Another possibility would be to weight the sampling of controls, in case-control studies,to adjust for the effect of interval length as was done in Figure 1. This has the effect of72changing the distribution of interval lengths from that of intervals starting or terminatingat a point in time to that of intervals spanning a point, which, as we have seen, under theuniform and Poisson models of screening, corresponds to the distribution of intervallengths for cases under the null hypothesis. This method may have the advantage, oversimply correcting the obtained prevalence odds, of simultaneously correcting odds ratiosobtained for factors which may be associated with interval length.The suggested method was examined in another simulated, matched, case-control studyusing a weighted sample of controls. The criteria for control selection were the same asbefore, only this time the probability of sampling any given control from those eligiblewas weighted inversely by the length of the “last” interval (to the nearest year). Theresults are presented in Table 11.73Table 11. Conditional likelihood logistic regression of a simulated matched case-controlstudy with sampling of controls weighted by length of “last” interval,Parameter Coef. s.e. z score riskLength of 55—120 months —0.062 0.187 —0.335 0.940last interval 43—54 months 1.15 0.186 6.21 3.1731—42 months 0.834 0.183 4.56 2.3019—30 months 0.977 0.177 5.51 2.6610—18 months 1.70 0.179 9.50 5.48Combined 85—120 months 0.146 0.281 0.520 1.16length of 61—84 months —0.216 0.265 —0.818 0.806two intervals 37—60 months —0.671 0.251 —2.67 0.511preceding 10—36 months —0.836 0.250 —3.35 0.433the last one previous —0.031 0.256 —0.122 0.969no previous 0.437 0.258 1.69 1.55Preceding gap minor 0.515 0.076 6.74 1.67abnormality major 1.16 0.293 3.95 3.18Index categories: >120 months for last and preceding two intervalsand no preceding gap for preceding gap abnormality.Weighting the sampling of controls by length of “last” interval succeeded in reversingthe direction of trend in the relation between length of “last” interval and relative risk.The resulting risk estimates are not too far off those produced by the Poisson regressionmodel, although the estimate for the 43-54 month category appears to be a little high.Strangely, the standard errors are also closer those of the Poisson model than the onesproduced by the unweighted sample matched case-control analysis.The ratios of unweightecl sample (prevalence) to weighted sample (“incidence”) oddsratios is 0.044, 0.102, 0.156, 0.127, and 0.505 for “last” interval categoriescorresponding to 1,2,3,4, and 5-9 years respectively. The relative sizes roughlycorrespond to the relative lengths of the intervals except for the “4” category,74Chapter 6ConclusionThe analysis of screen detected disease involves a number of issues with respect to thedesign and analysis of cohort and case-control studies. Non-standard methods arerequired to estimate incidence rates for time-dependent factors such as interval length.Case-control studies of screen detected disease produce estimates of prevalence oddsratios rather than incidence risk ratios. This can be misleading when the exposurevariable is related to duration of disease. In the case of screening interval length, sinceCIS is a long lasting disease, it is not surprising that higher prevalence rates are observedamong individuals with longer screening intervals. But higher prevalence rates do notimply higher incidence rates, since after the completion of one screen interval anotherone begins with, perhaps, the same risk of disease.To examine the effect of a method of sampling controls in a popular case-controlparadigm, whereby controls are required to have had a screen near the time of diagnosisof a case, theoretical screening models were considered. All models imply that thedistributions of interval lengths are different for cases and controls under the nullhypothesis of no effect of screening. The distribution of interval lengths for sampled75controls converges to the underlying distribution, whereas the distribution of intervallengths for cases is the same as the distribution of spanning intervals, favouring longerintervals.Incidence rates evidently cannot be accurately inferred from prevalence rates unless theduration of disease is known. Brookmeyer, Day, and Moss32 have proposed a methodof estimating the duration of disease along with false negative rates. Unfortunately, themethod presupposes that regression does not occur, and this would certainly not be thecase for CIS.With respect to the specific findings of this study, it would appear that frequent screeningdoes not provide protection against the development of CIS. It is beneficial to have hadnegative screens in the past, but the incidence of diagnosed CIS is higher for shorterscreening intervals than for longer ones. One possible explanation for this paradoxicalresult is that the natural regression of the disease may result in some cases, which ariseand then regress during long screen intervals, going undetected, The time that theseindividuals actually had the disease is being erroneously applied to the time-at-risk,deflating the incidence rate. And even more importantly, an incidence case is goinguncounted. Such errors would be less likely to occur under a frequent screening pattern.However, frequent screening should result in some avoidance of disease in that precursorstages of disease, in this case dysplasia, are detected and treated. Thus frequentscreeners should be disease free altogether, except for rapidly developing subtypes.76Another possible explanation for the observation of higher incidence rates in shorterintervals is that individuals who are at risk are more likely to go for frequent tests. Wetried to control for this somewhat by controlling for degree of abnormality occurring inthe gap preceding the “last” interval. However, it is generally believed that high riskindividuals are less likely to be screened frequently.Although recency of previous tests provides some protection against disease, it does notappear to be enough to counter the effect of interval length. It would appear that theonly logical conclusion would be that the best protection against the diagnosis of CIS isinfrequent screening.A shortcoming of the present study is the lack of control or information about thecircumstances surrounding the decision to be tested. The possibility of confoundingvariables is always a concern with observational studies. Even though the present studycasts some doubt on the benefit of frequent screening for the prevention of diagnosis ofCIS, as opposed to invasive cancer, a proper randomized control trial may still beconsidered unethical because of the possible implications for more advanced stages ofdisease.77Bibliography[1] Boyes D.A., Morrison E.G., Knox E.G., Draper G.J., Miller A.B., A cohortstudy of cervical cancer screening in British Columbia. Clinical & InvestigativeMedicine, 1982; 5:1-29.[2] Eliman R., Indications for colposcopy from a UK viewpoint, in Miller A.B.,Chamberlain J., Day N.E., Hakama M., Prorak P.C. (Eds) Cancer Screening,1991, Cambridge University Press.[3] Day N.E., Screening for cancer of the cervix, Journal of Epidemiology andCommunity Health, 1989, 43:103-106.[4] Anderson G.H., Boyes D.A., Benedet J.L., et. al. Organisation and results ofthe cervical cytology screening programme in British Columbia, 1955-85, BritishMedical Journal, 1988, 296:975-978.[5] Task Force Appointed by the Conference of Deputy Ministers of Health: Cervicalcancer screening programs, Canadian Medical Association Journal, 1976,114:1003-1033.[6] Parkins in Hakama M., Miller A.B., Day N.E., (Eds) Screening for Cancer ofthe Uterine Cervix. 1986. IARC Scientific Publications, No. 76: Lyon.[7] La Vecchia C., Franceschi S., Decarli A., et. al. “PAP” smear and the risk ofcervical neoplasia: quantitative estimate from a case-control study, The Lancet,1984, Oct:779-782.[8] Knox G. Case-control studies of screening procedures, Public Health, 1991,105:55-61.[9] Chronic Disease Reports: Deathsfrom cervical cancer - United States 1984-1986.MMWR, 1989, 38:650-659.[10] Miller A.B., Knight J., Narod S. The natural history of cancer of the cervix, andthe implicationfor screening policy, in[11] Hutchinson M.L., Agarwal P., Denault T. et. al. A new look at cervical cytology,Acta Cytologica, 1992, 36:499-504.78[12] Hakama M., Miller A.B., Day N.E. (Eds) Screening for Cancer of the UterineCervix. 1986. IARC Scientific Publications, No. 76: Lyon.[13] Magnus and Langmark in Hakama M., Miller A.B., Day N.E. (Eds) Screeningfor Cancer of the Uterine Cervix. 1986. IARC Scientific Publications, No. 76:Lyon.[14] Berrino in Hakama M., Miller A.B., Day N.E. (Eds) Screeningfor Cancer oftheUterine Cervix, 1986. IARC Scientific Publications, No. 76: Lyon.[15] Clark E.A., Anderson T.W., Does screening by ‘PAP’ smears help preventcervical cancer? The Lancet, 1979, July: 1-4[16] Berrino F., Papanicolaou smears and risk of cervical neoplasia, The Lancet,1984, Nov: 1099-1100.[17] MacGregor J.E., Moss S.M., Parkin D.M., Day N.E., A case-control study ofcervical cancer screening in north east Scotland, British Medical Journal, 1985,290:1543-1546[18] Rothman K.J., Modern Epidemiology, 1986, Little Brown and Company:Toronto.[19] van Oortmarssen G.J., Habbema J.D.F., in Hakama M., Miller A.B., Day N.E.,(Eds.) Screening for Cancer of the Uterine Cervix. 1986. IARC ScientificPublications, No. 76: Lyon.[20] Choi N.W., Nelson N.A. in Hakama M., Miller A.B., Day N.E. (Eds) Screeningfor Cancer of the Uterine Cervix. 1986, IARC Scientific Publications, No.76:Lyon.[21] Morrison B.J., Coldman A.J., Boyes D.A., Anderson G.H., Forty year follow-upof cervical screening: the cohort study of British Columbia, 1994. Unpublishedmanuscript.[22] Habbema J.D.F. in Hakama M., Miller A.B., Day N.E. (Eds) Screening forCancer ofthe Uterine Cervix. 1986. IARC Scientific Publications, No.76: Lyon.[23] Cox D.R., Hinkley D.V. Theoretical Statistics, Chapman and Hall, New York,1974[24] Armitage P., Berry G. Statistical Methods in Medical Research, 1987, BlackwellScientific Publications:London.[25] Sheehe P.R. Dynamic risk analysis in retrospective matched pair studies ofdisease. Biometrics 1962; 18:323-341.79[26] Miettinen O.S. Estimability and estimation in case-referent studies. Am. J.Epidemiol. 1976; 103:226-235.[27] Breslow N.E., Day N.E., Statistical Methods in Cancer Research, vol.1 1980,vol.2 1987 International Agency for Research on Cancer: Lyon.[28] Elandt-Johnson R. Definition of rates: some remarks on their use and misuse.American Journal Of Epidemiology, 1975; 102:267-271.[29] Sasco A.J., Day N.E., Walter S.D., Case-control studies for the evaluation ofscreening, Journal of Chronic Disease, 1986, 39:399-405.[30] McCullagh P., Nelder J.A., Generalized linear models, (2’ Ed.) Chapman andHall: New York, 1989.[311 Storer B. E., Wacholder S., Breslow N.E., Maximum likelihood fitting of generalrelative risk models to stratified data. Applied Statistics. 1983, 32:172-81.[32] Brookmeyer R., Day N.E., Moss S. Case-control studies for estimation of thenatural history of preclinical disease from screening data, Statistics in Medicine,1986, 5: 127-138.80Appendix 1Number of cases and incidence rates per thousand person—yearsby:— Cohort (l=born 1914-1918, 2= born 1929—1933)— Period (l=pre—1963, 2= 1963—1975, 3=1976—1992)— Degree of abnormality in Gap preceding last screen(l=no gap, 2=minor—<lO months or class 2 or 9,3=major-class 3, 4 or histology)— Length of last screen interval,— Combined length of two screens prior to last.- Cells with no time-at-risk indicated with “-“Cohort 1Period 2 10-18Length of last screen19—30 31—42interval43—540.750.520.280.430. 00(months)55—120 >1207 0.326 0.870 0.000 0.001 0.996 0.702 1.19o o.ooo o.oo0 0.00Cohort 1 Length of last screen interval (months)Period 1 10—18 19—30 31—42 43—54 55—120 >120n rate n rate n rate n rate n rate n rate4 0.741 0.560 0.000 0.000 0.000 0.000 —0 0.000 0.000 0.000 0.000 0.00GapiNo prey1 prey10—3637—6061—8485—120>120Gap 2No prey1 prey10—3637—6061—8485—120>120Gap 3No prey1 prey10—3637—6061—8485—120>1203 0.80 1 0.25 0 0.00 0 0.002 2.50 0 0.00 0 0.00 0 —0 0.00 0 0.00 0 0.00 0 —0 0.00 0 0.00 0 0.00 0 —153.33 0 0.00 0 — 0 —0 0.00 0 — 0 — 0 —0 — 0 — 0 — 0 —0 0.00 0 0.00 0 0.00 0 0.000 0.00 0 0.00 0 0.00 0 —0 0.00 0 0.00 0 — 0 —0 0.00 0 0.00 0 — 0 —0 0.00 0 — 0 — 0 —9 1.762 0.740 0.000 0.000 0.000 0.000 0.006 6.090 0.000 0.000 0.000 0.000 0.000 —0 0.000 0.000 —0 —0 —0 —0 —00— 0 0.00 0 — 0 — 0 —— 0 — 0 — 0 — 0 —0 0.000 —0 —0 0.000 —0 —0 —0 0.000 —0 —0 —0 —0 —0 —0000000— 0 0.00— 0 —— 0 —— 0 —— 0 —— 0 —— 0 —0000000n rate n rate n rate n rate n rate n rateGap 1No prey 11 1.42 12 0.98 11 0.96 141 prey 10 0.85 12 0.95 3 0.35 510—36 13 0.35 5 0.34 1 0.20 137—60 13 0.82 2 0.15 1 0.17 261—84 3 0.80 2 0.46 1 0.42 08185—120 2 1.14 1 0.56 2 1.89 2 2.44 0 0.00 0 0.00>120 1 1.65 1 1.79 1 3.16 0 0.00 0 0.00 0 —Gap 2No prey1 prey10—3637—6061—8485—120>120Gap 3No prey1 prey10—3637—6061—8485—120>120Cohort 15 2.212 0.596 0.585 1.133 2.800 0.000 0.002 27.001 12.241 5.771 12.300 0.000 0.000 0.002 1.001 0.404 0.942 0.711 1.071 2.680 0.000 0.001 26.490 0.001 42.860 0.000 0.000 0.0010000003 2.220 0.001 0.840 0.000 0.000 0.00o o.oo0 0.000 0.000 0.000 0.00o o.oo0 0.000 —1.200.000.000.000.003 1.423 1.920 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 —0 —0 —0 0.002 1.881 2.010 0.000 0.000 0.000 0.000 0.000 —0 0.000 0.000 —0 —0 —0 0.000 —0 —0 —0 —0 —0 —Period 3 10-18n rateLength of last screen interval (months)19—30 31—42 43—54 55—120 >120n rate n rate n rate n rate n rateGap 1No prey1 prey10—3637—6061—8485—120>120Gap 2No prey1 prey10—3637—6061—8485—120>120Gap 3No prey1 prey10—3637—6061—8485—120>1200 0.000 0.005 0.205 0.444 1.251 0.520 0.000 0.000 0.003 0.521 0.330 0.000 0.001 2.440 0.000 0.001 1.801 2.431 8.061 12.171 10.770 0.001 1.555 0.333 0.217 1.271 0.324 1.610 0.000 0.004 1.141 0.380 0.000 0.000 0.000 0.00o o.oo0 0.000 0.000 0.000 0.001 32.096 0.314 0.342 0.260 0.001 0.221 0.330 0.001 0.361 0.520 0.001 0.780 0.00o o.oo0 0.000 0.004 4.910 0.002 0.202 0.402 0.650 0.000 0.000 0.002 1.351 0.660 0.001 2.380 0.00o o.oo0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.133 0.242 0.281 0.210 0.000 0.001 2.830 0.001 0.600 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.005 2.390 0.001 0.122 0.153 0.382 0.332 0.410 0.000 0.000 0.002 1.040 0.001 1.620 0.000 0.000 0.00o o.oo0 0.000 0.000 0.000 0.0000000000. 000.000 • 000. 000.0082Cohort 2 Length of last screen interval (months)Period 1 10—18 19—30 31—42 43—54 55—120 >120n rate n rate n rate n rate n rate n rateGap 1No prey1 prey10—3637—6061—8485—120>120Gap 2No prey1 prey10—3637—6061—8485—120>120Gap 3No prey1 prey10—3637—6061—8485—120>1208 2.06 7 1.80 6 2.25 1 0.39 2 1.43 0 0.002 1.52 0 0.00 0 0.00 0 0.00 0 0.00 0 —0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 —0 0.00 0 0.00 0 0.00 0 0.00 0 — 0 —0 0.00 0 0.00 0 0.00 0 — 0 — 0 —0 0.00 0 — 0 — 0 — 0 — 0 —0 — 0 — 0 — 0 — 0 — 0 —2 2.98 1 1.93 2 7.47 0 0.00 0 0.00 0 —0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 —1 23.03 0 0.00 0 0.00 0 — 0 — 0 —0 0.00 0 0.00 0 0.00 0 — 0 — 0 —0 0.00 0 — 0 — 0 — 0 — 0 —0 0.00 0 — 0 — 0 — 0 — 0 —0 — 0 — 0 — 0 — 0 — 0 —0 0.00 0 0.00 0 0.00 0 — 0 — 00 0.00 0 — 0 — 0 — 0 — 00 — 0 — 0 — 0 — 0 — 00 — 0 — 0 — 0 — 0 — 00 — 0 — 0 — 0 — 0 — 00 — 0 — 0 — 0 — 0 — 00 — 0 — 0 — 0 — 0 — 0>120Cohort 2 Length of last screen interval (months)Period 2 10—18 19—30 31—42 43—54 55—120n rate n rate n rate n rate n rate n rateGap 1No prey 321 prey 3110—36 4937—60 3061—84 1285—120 6>120 2Gap 2No prey 321 prey 1810—36 1437—60 1061—84 385—120 4>120 0Gap 3No prey 0lprev 02.14 391.60 281.02 141.30 102.08 62.73 33.02 16.53 192.79 130.99 61.50 41.73 15.72 10.00 00.00 00.00 01.94 28 1.64 38 1.50 32 1.24 6 0.771.43 19 1.50 12 0.94 8 1.12 0 0.000.71 4 0.61 0 0.00 0 0.00 0 0.000.56 1 0.13 2 0.37 0 0.00 0 0.001.05 2 0.69 2 0.88 0 0.00 0 0.001.27 1 0.80 1 1.14 0 0.00 0 0.001.71 0 0.00 0 0.00 0 0.00 0 —4.35 11 3.77 8 2.24 2 0.76 0 0,002.73 7 2.61 1 0.40 0 0.00 1 5.280.99 3 1.52 0 0.00 0 0.00 0 0.000.89 3 1.47 0 0.00 0 0.00 0 0.000.81 0 0.00 2 3.10 0 0.00 0 —2.18 0 0.00 0 0.00 0 0.00 0 0.000.00 0 0.00 0 0.00 0 0.00 0 —0.00 0 0.00 0 0.00 00.00 0 0.00 0 0.00 00.00 0— 0831 5.76 0 0.00 0 0.00 0 0.00 0o o.oo 0 0.00 0 0.00 0 0.00 00 0.00 0 0.00 0 0.00 0 —0 0.00 0 0.00 0 0.00 0 —0 0.00 0 0.00 0 — 0 —10—3637—6061—8485—120>120— 0 —— 0 —0 0.00 00 — 00 — 019—30 31—42 43—54 55—120 >120Cohort 2 Length of last screen interval (months)Period 3 10-18n rate n rate n rate n rate n rate n rateGap 1No prey 0 0.00 0 0.00 0 0.00 0 0.00 1 0.44 8 0.361 prey 1 1.83 3 2.81 1 0.76 3 1.06 2 0.33 7 0.5710—36 29 0.47 10 0.28 5 0.33 6 0.47 2 0.18 0 0.0037—60 10 0.35 14 0.40 9 0.41 5 0.22 1 0.06 0 0.0061—84 3 0.40 5 0.41 3 0.29 5 0.41 3 0.27 1 0.2085—120 3 0.83 3 0.47 3 0.54 9 1.12 1 0.13 0 0.00>120 2 0.78 1 0.25 1 0,26 2 0.35 0 0.00 1 0.61Gap 2No prey 0 0.00 1 23.76 0 0.00 0 0.00 1 1.13 0 0.001 prey 0 0.00 1 2.74 1 3.20 1 1.91 4 2.36 0 0.0010—36 13 0.88 2 0.23 1 0.32 0 0.00 0 0.00 2 0.7037—60 10 1.22 5 0.74 5 1.54 2 0.63 1 0.28 0 0.0061—84 4 2.10 2 1.00 2 1.62 0 0.00 0 0.00 0 0.0085—120 1 1.16 0 0.00 1 1.56 0 0.00 1 1.14 1 2.08>120 1 1.65 1 2.20 0 0.00 2 4.23 0 0.00 0 0.00Gap 3No prey 0 0.00 0 0.00 1 35.29 0 0.00 0 0.00 0 0.00lprev 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 0.0010—36 3 1.96 0 0.00 1 2.85 0 0.00 0 0.00 0 0.0037—60 1 1.14 0 0.00 0 0.00 0 0.00 0 0.00 0 0.0061—84 1 4.74 0 0.00 0 0.00 0 0.00 0 0.00 0 0.0085—120 1 7.72 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00>120 0 0.00 0 0.00 0 0.00 0 0,00 0 0.00 0 0.0084Appendix 2Number of cases and prevalence rates per thousand individualsby:— Cohort (l=born 1914-1918, 2= born 1929-1933)— Period (l=pre—1963, 2= 1963—1975, 3=1976—1992)- Degree of abnormality in Gap preceding last interval(1=no gap, 2=minor—<1O months or class 2 or 9,3=major—class 3 or 4 or histology)— Length of last screen interval,— Combined length of two screens prior to last.— Cells with no time-at-risk indicated with “-“Cohort 1Period 2 10-18Length of last screen interval19—30 31—42 43—54(months)55—120 >120n rate n rate n rate n rate n rate n rateGap 1No prey 11 1.88 12 2.12 11 3.01 14 3.46 7 2.38 6 8.721 prey 10 1.13 12 2.05 3 1.11 5 2.33 6 6.29 2 14.2910—36 13 0.45 5 0.72 1 0.62 1 1.29 0 0.00 0 0.0037—60 13 1.09 2 0.32 1 0.53 2 1.92 0 0.00 0 0.00Cohort 1 Length of last screen interval (months)Period 1 10—18 19—30 31—42 43—54 55—120 >120n rate n rate n rate n rate n rate n rateGap 1No prey 9 2.30 4 1.57 3 2.51 1 1.14 0 0.00 0 0.00lprev 2 0.96 1 1.17 2 7.97 0 0.00 0 0.00 0 —10—36 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 —37—60 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 —61—84 0 0.00 0 0.00 1166.67 0 0.00 0 — 0 —85—120 0 0.00 0 0.00 0 0.00 0 — 0 — 0 —>120 0 0.00 0 — 0 — 0 — 0 — 0 —Gap 2Noprev 6 8.65 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00lprev 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 —10—36 0 0.00 0 0.00 0 0.00 0 0.00 0 — 0 —37—60 0 0.00 0 0.00 0 0.00 0 0.00 0 — 0 —61—84 0 0.00 0 0.00 0 0.00 0 — 0 — 0 —85—120 0 0.00 0 — 0 0.00 0 — 0 — 0 —>120 0 — 0 — 0 — 0 — 0 — 0 —Gap 3Noprev 0 0.00 0 0.00 0 0.00 0 — 0 0.00 0 —lprev 0 0.00 0 — 0 — 0 — 0 — 0 —10—36 0 — 0 — 0 — 0 — 0 — 0 —37—60 0 — 00.000 — 0 — 0 — 0 —61—84 0 — 0 — 0 — 0 — 0 — 0 —85—120 0 — 0 — 0 — 0 — 0 — 0>120 0 — 0 — 0 — 0 — 0 — 08561—84 3 1.06 2 1.00 1 1.32 0 0.00 1 6.90 0 0.0085—120 2 1.54 1 1.21 2 5.95 2 10.81 0 0.00 0 0.00>120 1 2.15 1 3.79 1 9.90 0 0.00 0 0.00 0 —Gap 2No prey 5 3.21 2 2.20 3 7.06 3 6.55 0 0.00 1 14.71lprev 2 0.84 1 0.86 0 0.00 3 8.72 2 13.24 0 0.0010—36 6 0.80 4 2.01 1 2.62 0 0.00 1 14.09 0 0.0037—60 5 1.61 2 1.51 0 0.00 0 0.00 0 0.00 0 0.0061—84 3 3.89 1 2.38 0 0.00 0 0.00 0 0.00 0 0.0085—120 0 0.00 1 5.62 0 0.00 0 0.00 0 0.00 0 —>120 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 —Gap 3No prey 2 40.82 0 0.00 0 0.00 0 0.00 0 0.00 0 0.001 prey 1 21.28 1 62.50 0 0.00 0 0.00 0 — 0 —10—36 1 9.26 0 0.00 0 0.00 0 0.00 0 0.00 0 —37—60 1 17.54 1100.00 0 0.00 0 0.00 0 0.00 0 —61—84 0 0.00 0 0.00 0 0.00 0 — 0 — 0 —85—120 0 0.00 0 0.00 0 0.00 0 — 0 — 0 —>120 0 0.00 0 0.00 0 — 0 — 0 — 0 —Cohort 1 Length of last screen interval (months)Period 3 10—18 19—30 31—42 43—54 55—120 >120n rate n rate n rate n rate n rate n rateGap 1No prey 0 0.00 0 0.00 0 0.00 0 0.00 5 20.66 6 5.531 prey 0 0.00 1 3.42 4 15.94 0 0.00 0 0.00 4 5.4710—36 5 0.25 5 0.70 0 0.00 1 0.60 1 0.90 2 3.8237—60 5 0.57 3 0.45 2 0.63 3 1.10 2 1.13 0 0.0061—84 4 1.60 7 2.77 2 1.26 2 1.26 3 2.75 1 2.9785—120 1 0.69 1 0.69 2 2.02 1 0.94 2 2.46 1 4.33>120 0 0.00 4 3.42 0 0.00 0 0.00 2 3.02 0 0.00Gap 2No prey 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 1 6.41lprev 0 0.00 0 0.00 0 0.00 1 13.51 0 0.00 1 8.6210—36 3 0.72 4 2.43 2 4.26 0 0.00 0 0.00 0 0.0037—60 1 0.45 1 0.81 1 2.09 1 2.72 2 7.69 1 11.2461—84 0 0.00 0 0.00 0 0.00 0 0,00 0 0.00 0 0.0085—120 0 0.00 0 0.00 1 7.75 0 0.00 1 11.77 0 0.00>120 1 3.53 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00Gap 3Noprev 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 —1 prey 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 0.0010—36 1 2.53 0 0.00 0 0.00 0 0.00 0 0.00 0 0.0037—60 1 3.80 0 0.00 0 0.00 0 0.00 0 0.00 0 0.0061—84 1 12.19 0 0.00 0 0.00 0 0.00 0 0.00 0 —85—120 1 15.87 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00>120 1 17.24 1 62.50 0 0.00 0 0.00 0 0.00 0 0.0086Cohort 2 Length of last screen interval (months)Period 1 10—18 19—30 31—42 43—54 55—120 >120n rate n rate n rate n rate n rate n rateGap 1No prey1 prey10—3637—6061—8485—120>120Gap 2No prey1 prey10—3637—6061—8485—120>120Gap 3No prey1 prey10—3637—6061—8485—120>120Cohort 28 2.66 7 3.75 6 7.07 1 1.71 2 9.90 0 0.002 1.93 0 0.00 0 0.00 0 0.00 0 0.00 0 —0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 —0 0.00 0 0.00 0 0.00 0 0.00 0 — 0 —0 0.00 0 0.00 0 0.00 0 — 0 — 0 —0 0.00 0 — 0 — 0 — 0 — 0 —0 — 0 — 0 — 0 — 0 — 0 —2 3.98 1 4.10 2 24.10 0 0.00 0 0.00 00 0.00 0 0.00 0 0.00 0 0.00 0 0.00 01 30.30 0 0.00 0 0.00 0 — 0 — 00 0.00 0 0.00 0 0.00 0 — 0 — 00 0.00 0 — 0 — 0 — 0 — 00 0.00 0 — 0 — 0 — 0 — 00 — 0 — 0 — 0 — 0 — 00 0.00 0 0.00 0 0.00 0 — 0 — 00 0.00 0 — 0 — 0 — 0 — 00 — 0 — 0 — 0 — 0 — 00 — 0 — 0 — 0 — 0 — 00 — 0 — 0 — 0 — 0 — 00 — 0 — 0 — 0 — 0 — 00 — 0 — 0 — 0 — 0 — 0Period 2 10-18n rate n(months)55—120 >120rate n rateLength of last screen interval19—30 31—42 43—54rate n rate n rate n4.23 28 5.21 38 6.96 323.12 19 4.84 12 4.32 81.52 4 1.96 0 0.00 01.22 1 0.42 2 1.66 02.30 2 2.21 2 3.99 02.83 1 2.53 1 5.24 03.73 0 0.00 0 0.00 0Gap 1No prey 321 prey 3110—36 4937—60 3061—84 1285—120 6>120 2Gap 2No prey 321 prey 1810—36 1437—60 1061—84 385—120 4>120 0Gap 3No prey 0lprev 09.198 . 060. 000. 000. 000. 000. 002.89 392.16 281.34 141.77 102.88 63.72 34.08 19.62 194.12 131.42 62.20 42.60 18.62 10.00 00.00 00.00 060000009.590.000. 000. 000. 000. 009.87 11 12.116.12 7 8.552.20 3 4.931.97 3 4.801.81 0 0.004.74 0 0.000.00 0 0.008 10.281 1.830 0.000 0.002 13.890 0.000 0.002 5.680 0.000 0.000 0.000 0.000 0.000 0.00o o.oo1 62.50o o.oo0 0.000 —0 0.000 —0.00 0 0.00 0 0.00 00.00 0 0.00 0 0.00 00.00 0— 08710—3637—6061—8485—120>120100007.41 0 0.00 0 0.00 0 0.00 0 — 0 —0.00 0 0.00 0 0.00 0 0.00 0 — 0 —0.00 0 0.00 0 0.00 0 — 0 0.00 00.00 0 0.00 0 0.00 0 — 0 — 0 —0.00 0 0.00 0 — 0 — 0 — 0 —Cohort 2 Length of last screen interval (months)Period 3 10—18 19—30 31—42 43—54 55—120 >120n rate n rate n rate n rate n rate n rateGap 1No prey 0 0.00 0 0.00 0 0.00 0 0.00 1 3.77 8 6.321 prey 1 2.76 3 6.55 1 2.46 3 5.09 2 2.62 7 9.3510—36 29 0.60 10 0.60 5 1.02 6 2.11 2 1.34 0 0.0037—60 10 0.46 14 0.85 9 1.30 5 0.99 1 0.42 0 0.0061—84 3 0.54 5 0.89 3 0.92 5 1.83 3 1.96 1 2.7085—120 3 1.11 3 1.03 3 1.70 9 5.09 1 0.93 0 0.00>120 2 1.03 1 0.54 1 0.83 2 1.59 0 0.00 1 7.46Gap 2Noprev 0 0.00 147.62 0 0.00 0 0.00 1 8.48 0 0.001 prey 0 0.00 1 6.49 1 10.64 1 9.35 4 18.69 0 0.0010—36 13 1.26 2 0.52 1 1.01 0 0.00 0 0.00 2 10.2037—60 10 1.73 5 1.63 5 4.91 2 2.83 1 2.11 0 0.0061—84 4 2.98 2 2.24 2 5.11 0 0.00 0 0.00 0 0.0085—120 1 1.62 0 0.00 1 5.18 0 0.00 1 8.40 1 27.78>120 1 2.32 1 4.72 0 0.00 2 19.61 0 0.00 0 0.00Gap 3No prey 0 0.00 0 0.00 1200.00 0 0.00 0 0.00 0 0.00lprev 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 0.0010—36 3 2.86 0 0.00 1 9.01 0 0.00 0 0.00 0 0.0037—60 1 1.62 0 0.00 0 0.00 0 0.00 0 0.00 0 0.0061—84 1 6.54 0 0.00 0 0.00 0 0.00 0 0.00 0 0.0085—120 1 12.35 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00>120 0 0.00 0 0.00 0 0.00 0 0.00 0 0.00 0 0.0088

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