Correcting for Lead Time and Length Bias in Estimating the Effect of Screen Detection on Cancer Survival

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1 American Journal of Epidemiology ª Te Autor Publised by te Jons Hopkins Bloomberg Scool of Public Healt. All rigts reserved. For permissions, please Vol. 168, No. 1 DOI: /aje/kwn120 Advance Access publication May 25, 2008 Practice of Epidemiology Correcting for Lead Time and Lengt Bias in Estimating te Effect of Screen Detection on Cancer Survival Stepen W. Duffy 1, Iris D. Nagtegaal 2, Mattew Wallis 3, Fay H. Cafferty 1, Nemat Houssami 4, Jane Warwick 1, Prue C. Allgood 1, Olive Kearins 5, Nancy Tappenden 5, Emma O Sullivan 5, and Gill Lawrence 5 1 Cancer Researc UK Centre for Epidemiology, Matematics and Statistics, Wolfson Institute for Preventive Medicine, London, United Kingdom. 2 Department of Patology, Dutc Cancer Society, Nijmegen, te Neterlands. 3 Cambridge Breast Unit, Addenbrooke s Hospital, Cambridge, United Kingdom. 4 Screening and Test Evaluation Program, Scool of Public Healt, University of Sydney, Sydney, Australia. 5 West Midlands Breast Screening Quality Assurance Reference Centre, University of Birmingam, Birmingam, United Kingdom. Received for publication December 21, 2007; accepted for publication April 8, Determination of survival time among persons wit screen-detected cancer is subject to lead time and lengt biases. Te autors propose a simple correction for lead time, assuming an exponential distribution of te preclinical screen-detectable period. Assuming two latent categories of tumors, one of wic is more prone to screen detection and correspondingly less prone to deat from te cancer in question, te autors ave developed a strategy of sensitivity analysis for various magnitudes of lengt bias. Here tey demonstrate tese metods using a series of 25,962 breast cancer cases ( ) from te West Midlands, United Kingdom. bias (epidemiology); breast neoplasms; mass screening; models, statistical; survival Abbreviation: CI, confidence interval. In disease screening, te concepts of lead time and lengt bias ave been familiar for decades (1, 2). Lead time is te amount of time by wic te diagnosis as been advanced by screening. In analysis of survival from diagnosis, it constitutes an artificial addition to te survival time of screendetected cases. Lengt bias is te penomenon wereby more slowly growing tumors, wit less capacity to prove fatal, may ave a longer presymptomatic screen-detectable period and will terefore be more likely to be screendetected. Tis again confers an artificial survival advantage to screen-detected cases. Te extreme form of lengt bias is overdiagnosis, defined as diagnosis by screening of cancers wic would not ave come to clinical attention in te ost s lifetime ad screening not taken place. It is tougt, for example, tat some in situ cancers (cancers confined to te ducts wic ave not yet invaded te surrounding tissue) detected by screening migt never ave become invasive or given rise to symptoms in te absence of screening (3). To avoid tese biases, investigators in randomized trials of cancer screening compare mortality rates from te disease in question in te wole population randomized to screening wit tose in te wole control population, instead of comparing survival rates of disease cases. In addition, te time origin is taken as te point of randomization, not te point of diagnosis (4, 5). However, te effect of screen detection on case survival is often of interest, particularly in te case of mammograpic screening for breast cancer, wic is now in te post-trial epoc. Te empasis Correspondence to Dr. Prue Allgood, Cancer Researc UK Centre for Epidemiology, Matematics and Statistics, Wolfson Institute for Preventive Medicine, Carterouse Square, London WC1M 6BQ, United Kingdom ( prue.allgood@cancer.org.uk). 98

2 Correcting for Lead Time and Lengt Bias 99 now is on evaluation of routine screening services and on assessing screening programs in special risk groups for wic randomized trials may not be feasible or etical (6). We ave developed a simple metod of correction for lead time in analysis of survival including screen-detected cases and an approac to sensitivity analyses for lengt bias. Te lead-time metod owes muc to te earlier work of Walter and Stitt (7) but is rater simpler and less general tan teir metod. Te lengt-bias approac involves assuming two latent tumor populations, one wit bot a iger probability tan te oter of being screen-detected and a correspondingly lower probability of fatality, weter symptomatic or screen-detected. Note tat tese metods are not tools wit wic to evaluate te efficacy of cancer screening. Te appropriate metod for determining weter a cancer screening strategy works is te randomized controlled trial, wit mortality as te endpoint. Here we apply tese metods to results from a large series of breast cancer cases from te West Midlands, United Kingdom. METHODS Correction for lead-time bias Correction for lead-time bias involves estimation of te additional follow-up time observed purely as a result of lead time in a case of screen-detected cancer. We assume an exponential distribution of te sojourn time (8), te period during wic te tumor is asymptomatic but screen-detectable, wit a rate of transition to symptomatic disease k. Tus, 1/k is te mean sojourn time. Consider first a screen-detected cancer wic as resulted in breast cancer deat at time t after diagnosis. Te additional follow-up time cannot be greater tan t. Te expected additional follow-up time, s, due to lead time is te expectation of te lead time conditional on its being less tan t, tat is, R t EðsÞ 0 xe kx dx 1 e kt 1 e kt kt kte kð1 e kt : ð1þ Þ For a patient wit a screen-detected cancer known to be alive at time t after diagnosis, we ave EðsÞPðstÞEðsjstÞþPðs > tþt: So in tis case, EðsÞ ð1 e kt Þ R t 0 xe kx dx 1 e kt þ te kt 1 e kt : ð2þ k Tus, a correction for lead time would involve subtracting E(s) from te observed survival time, or time to last live follow-up, of patients wit screen-detected cases. By expansion of te exponential terms, it can be seen tat E(s) is never greater tan t, so te correction never gives negative survival times. Sensitivity analyses to adjust for lengt bias Lengt-biased sampling occurs wen te cance of an observation s being in a sample is proportional to a particular caracteristic of te observation. In te context of screening, lengt bias is used to refer to te penomenon wereby slower-growing, less aggressive tumors ave a longer preclinical screen-detectable period and are terefore more likely to be screen-detected tan faster-growing, more aggressive cancers. Let us assume tat tumors can fall into two categories, one wit a probability p s of being screendetected (category A) and te oter wit a probability p s / of being screen-detected, were < 1 and p s / < 1 (category B). Suppose tat a proportion q of te tumors is in category A and te complementary proportion 1 q is in category B. Te probabilities of being screen-detected will depend on te screening regimen offered in terms of frequency and sensitivity and te rate of participation in screening. We assume tese to be uniform witin te particular screening program under study. Category B tumors ave a greater cance of being screendetected. If we assume tat tis is because te tumors are slow-growing (tis group migt include a large proportion of in situ cases), it seems reasonable to assume tat tey are correspondingly less likely to cause deat. Suppose, terefore, tat te probability of deat from breast cancer during te period of observation from a category A symptomatic tumor is p and te corresponding probability for category B symptomatic cancers is p. Tus, te tumors more likely to be screen-detected are postulated to be correspondingly less likely to cause deat a priori te classic manifestation of lengt bias in a screening context. Assume furter tat te true relative risk of deat from te cancer in question for screen-detected versus symptomatic tumors in a population offered screening, independent of lengt bias, is u. Tis means tat witin eac category, screen-detected cancers are u times as likely to cause deat as symptomatic tumors. Correspondingly, we assume tat witin eac detection mode (screening and symptomatic), patients wit category B tumors are times as likely to die from breast cancer as patients wit category A tumors. Tus, deat rates from breast cancer in te four categories are as follows: Category A symptomatic tumors: p. Category B symptomatic tumors: p. Category A screen-detected tumors: up. Category B screen-detected tumors: up. Te value of u will depend on caracteristics of te symptomatic tumors as well as on te effectiveness of te screening. Symptomatic tumors will typically include bot interval cancers (cancers arising symptomatically among screening participants in te intervals between screens) and cancers diagnosed symptomatically in women wo cose not to attend screening (for brevity we sall refer to tese women as nonattenders). If, as as been observed in te past, cancers arising in nonattenders ave particularly poor outcomes, u will depend on te proportion of nonattender cancers among te symptomatic tumors, and terefore on te attendance rate in te screening program under study. Te observed probability of cancer deat for symptomatic tumors in a population offered screening will be

3 100 Duffy et al. pðdjsymptþp 1 qð1 p sþp þð1 qþ 1 p s p qð1 p s Þþð1 qþ 1 p s pðq p sq þ q p s þ qp s Þ qp s ð1 qþp s pðqð1 Þþð p sþþ qp s ð1 qþp s : Te probability of cancer deat for screen-detected tumors in te same population will be observed to be pdjasympt ð Þ p 2 ðqp sup þð1 qþp s upþ qp s þð1 qþ p s p s up qp s þð1 qþp s up q þ 1 q : Te observed relative risk of cancer deat will terefore be R p 2 p 1 up q þ 1 q, pðqð1 Þþð p s ÞÞ qp s ð1 qþp s uð qp s ð1 qþp s Þ ðq þ 1 qþðqð1 Þþð p s ÞÞ : Tis will be smaller tan u, provided tat q and are less tan unity, so it will overestimate te reduction conferred by screen detection. Te overall probability of screen detection is pðasymptþ p 3 qp s þð1 qþ p s p s ðq þ 1 qþ: We can estimate p s, p, and u in terms of p 1, p 2, p 3,, and q, as follows: ˆp s p 3 ðq þ 1 qþ ; ˆp p 1ð qp s ð1 qþp s Þ ðqð1 Þþð p s ÞÞ : After substitution of p s and some algebra, we ave ˆp p 1 ðq þ 1 qþð1 p 3 Þ ðq þ 1 qþð þ qð1 ÞÞ p 3 : ð3þ ð4þ Finally, û p 2ðq þ 1 qþ p p 2fðq þ 1 qþð þ qð1 ÞÞ p 3 g : ð5þ p 1 ð1 p 3 Þ We can estimate p 1, p 2, and p 3 from screening data. A range of assumed values can be posited for and q to obtain a likely range of true values for u. An estimate of te relative azard assuming exponential survival is EXAMPLE ŵ lnð1 û ˆpÞ lnð1 ˆpÞ : In collaboration wit breast cancer screening units, te West Midlands Cancer Intelligence Unit collected clinicopatologic, diagnostic, and follow-up data on cancers diagnosed in te county of West Midlands, United Kingdom, among women aged years from 1988 to 2001 and among women aged years from 2002 to We ad data on 10,100 screen-detected breast cancers and 15,862 symptomatic breast cancers (6,009 interval cancers and 9,853 tumors in nonattenders). Tus, p 3, te observed probability of screen detection, was Tere were 4,935 deats among te symptomatic cases (4,620 witin 10 years of diagnosis) and 929 deats among te screen-detected cases (819 witin 10 years). Te observed 10-year case fatality for symptomatic tumors was p , and for screen-detected tumors it was p Tis produces a relative risk of 0.34 (95 percent confidence interval (CI): 0.31, 0.37). Te corresponding relative azard from Cox regression is 0.27 (95 percent CI: 0.25, 0.30). First, we correct for lead time. In te Swedis Two- County Trial, a study of breast screening, Tabar et al. (9) estimated k as 0.27 for te age group years and 0.24 for te age group years, wit a weigted average of Figure 1 sows te uncorrected and lead-timecorrected survival for te screen-detected cancers as compared wit symptomatic cancers. Te survival of symptomatic cases is uncanged, but te correction as led to a lower survival estimate in te screen-detected cases. After te correction, tere were 906 breast cancer deats witin 10 years among te screen-detected cases. Te 10-year case fatality for te screen-detected cases, corrected for lead time, was Te relative risk was 0.49 (95 percent CI: 0.45, 0.53), and te Cox regression relative azard was 0.40 (95 percent CI: 0.37, 0.44). For te correction for lengt bias, we do not know te values of q and, so we calculate te corrected results for a range of plausible values. Results are sown in table 1. Te correction is more dependent on, te relative rate of screen detection and fatality in te lengt-bias group, tan on q, te complement of te group s size. Tis range of values yields estimates of te true relative risk ranging from 0.49 to 0.59, wit a median of 0.51, and estimates of te relative azard

4 Correcting for Lead Time and Lengt Bias 101 FIGURE 1. Survival of women wit symptomatic and screendetected cases of breast cancer, before and after correction for lead time, West Midlands, United Kingdom, ranging from 0.43 to 0.52, wit a median of Furter analyses sowed tat for te lengt bias to account for te entire difference in survival, we would require q 0.7 and 0.2; tat is, te lengt-bias group would ave to comprise at least 30 percent of te tumor population and be five times more likely to be screen-detected and five times less likely, a priori, to cause deat from breast cancer. Te above analysis is based on all tumors in te series. Excluding te 2,102 cases of in situ carcinoma, te uncorrected 10-year survival in te screen-detected cases was 86 percent and tat in te symptomatic cases was 64 percent, a relative risk of 0.39 (95 percent CI: 0.36, 0.42). Te Cox regression relative azard was 0.31 (95 percent CI: 0.28, 0.33). Correcting for lead time, te 10-year survival in te screen-detected cases was 81 percent, giving a relative risk of 0.53 (95 percent CI: 0.49, 0.57); te relative azard was 0.45 (95 percent CI: 0.42, 0.49). Sensitivity analysis for lengt bias using te range of values of q and in table 1 gave a range of values for te relative risk from 0.53 to 0.63, wit a median of Te relative azard ranged from 0.47 to 0.56, wit a median of Te results of bot lead-time and lengt-bias corrections are summarized in table 2. DISCUSSION Te above work demonstrates a simple correction for lead-time bias in analysis of cancer survival data involving screen-detected cases, as well as a relatively simple approac to sensitivity analysis for lengt bias. A STATA routine for te lead-time correction and an Excel spreadseet for te lengt-bias sensitivity analyses are available from te autors. Te modeling requires strong assumptions but as te advantage of being easy to carry out. Part of te simplicity of te lengt-bias analysis lies in te fact tat we ave modeled it as an effect on relative risk, te ratio of probabilities of dying of te disease witin a specified time, rater tan an effect on relative azard, te ratio of te instantaneous rates of deat. Tis means tat te relative azards calculated in te lengt-bias analyses are approximate, being dependent on te absolute probabilities of dying and terefore on te period of observation. In addition, tere is te assumption tat in te lengt-bias population (category B), te proportional increase in screen detection propensity is equal to te proportional decrease in causespecific fatality. Lengt bias is a well-known penomenon, but it is difficult to estimate or oterwise quantify. Te metod demonstrated ere at least provides a means of estimating a likely range for its effect. We reiterate tat te metods sown ere are not intended for use in establising weter or not screening works in principle. Tis is determined by randomized trials wit mortality as te endpoint. It is also wort bearing in mind tat te relative risk estimated ere is for screen-detected cases as compared wit symptomatic cases. In a population randomized controlled trial, te mortality relative risk for te study group offered screening as compared wit a control group not offered screening would dilute te relative risk estimated ere by te cancers in te study arm diagnosed in nonattenders or as interval cancers. Te expected dilution migt be around 35 percent (10). Tus, te corrected 45 percent reduction in risk (u 0.55, above) would translate to 29 percent ( percent) a relative risk of 0.71, wic is compatible wit trial results (4, 10). Te corrected relative azard of 0.55 is similar to te 0.57 observed for first screen-detected cancers compared wit control symptomatic cancers in te Swedis Two-County Trial, adjusting for size, lymp node status, and grade, wic would tend to adjust out te effects of lead time and lengt bias, in addition to some of te genuine survival advantage of screen detection. Note tat tere are oter biases in comparing survival among screen-detected cases wit tat among symptomatic cases, particularly te fact tat outside of te randomized trial setting, tere may be ealty volunteer bias in tat te symptomatic cases may include persons wo ave declined te offer of screening. Tis population may be less ealtconscious and terefore more likely to die of teir disease regardless of screening. Tis can be addressed by comparing screen-detected cancers wit interval cancers, tose wic arise in screening participants but symptomatically in te intervals between screens, or by using publised metods for correction for te ealty volunteer bias (10). It is interesting tat in te example considered, te leadtime correction makes a substantial difference in te estimated relative risk of dying from breast cancer, but te lengt-bias adjustment makes a smaller difference. Note tat te lengt-bias correction is dependent primarily on te values of and q and is not affected by te fact tat te lead-time correction was carried out first. To make a difference wic would correct te relative risk to unity in tis example would require lengt bias of an implausible magnitude. In tis metod, we ave treated lengt bias and lead time as two independent penomena, but in fact tey are mutually associated, since a lengt-bias case will ave a longer sojourn time and potentially, terefore, a longer lead time. Our metod adjusts for te expected lead time based

5 102 Duffy et al. TABLE 1. Estimates of p s, p, and u for varying values of u and q, for lead-time-corrected fatality rates of 0.17 and 0.35, and for an observed relative risk of 0.49 among screendetected and symptomatic cases of breast cancer, West Midlands, United Kingdom, Relative screen detection and fatality rate, Mixing fraction, q Category A screen detection probability, p s Category A symptomatic tumor fatality rate, p True relative risk, u Approximate relative azard on te average sojourn time first, and ten carries out a series of sensitivity analyses for different magnitudes of eterogeneity around te average sojourn time. Te mean sojourn estimate wic we used for te lead time correction was 4 years, from te Swedis Two-County Trial (9). Oter studies yield similar or smaller estimates (11 14), wit only one exception (15), so it is likely tat 4 years is accurate and may even be conservative. We can use algebra similar to tat of te lengt-bias analyses above to estimate te effect of te most extreme form of lengt bias, overdiagnosis. Suppose tat in category B, te tumors would never ave become symptomatic and would never ave caused deat, as is almost certainly te case for some in situ tumors. Ten te observed case fatality rate of te symptomatic tumors would be p 1 pqð1 p sþ qð1 p s Þ p: Te observed proportion of cases tat were screen-detected would be p 3 qp s þ 1 q: Te observed case fatality rate of persons wit te screendetected tumors would be p 2 qp sup qp s þ 1 q : Tis would give u Rðqp s þ 1 qþ : qp s Our estimates of overdiagnosis tend to be small, on te order of 10 percent or less (13), but we assume a more extreme case for demonstration purposes. If we assume tat 25 percent of screen-detected cases are overdiagnosed, p 0.39 gives 0:25 1 q 0:39 ; wic gives q 0.9. Tis in turn gives p s 0.32 and u 0.66.

6 Correcting for Lead Time and Lengt Bias 103 TABLE 2. Results of survival analysis, bot uncorrected and corrected for lead time and lengt bias, among screen-detected and symptomatic cases of breast cancer, West Midlands, United Kingdom, Parameter measured Symptomatic breast cancers Screen-detected breast cancers Relative risk 95% CI* or range Hazard ratio Cox regression relative azard 95% CI or range Total no. of breast cancers 15,862 10,100 NA* NA No. of in situ breast cancers 535 1,567 No. of microinvasive and invasive breast cancers 15,327 8,533 Total no. of deats 4, Total no. of deats at 10 years 4, Total no. of deats at 10 years after correction for lead time 4, Total no. of in situ breast cancer deats at 10 years 3 1 Total no. of microinvasive and invasive breast cancer deats at 10 years 4, year case fatality (all cases) , , year survival (all cases) (%) year case fatality (excluding in situ cases) , , year survival (excluding in situ cases) (%) Results obtained after application of lead-time correction 10-year case fatality (all cases) , , year survival (all cases) (%) year case fatality (excluding in situ cases) , , year survival (excluding in situ cases) (%) Results obtained after additional application of lengt-bias correction 10-year case fatality (all cases) y y 10-year survival (all cases) (%) year case fatality (excluding in situ cases) y y 10-year survival (excluding in situ cases) (%) * CI, confidence interval; NA, not applicable. y For lengt-bias adjustment, absolute ranges of values from sensitivity analyses are given instead of 95% confidence intervals. Te lead time correction is applied as a constant, so te 95 percent confidence intervals will be sligtly anticonservative, since tey do not reflect te additional uncertainty from estimation of k. However, te correction is applied to a minority of observations, te variance of our estimate of k is small (9), and wit large data sets suc as tis, te confidence intervals would be narrow even if tey did incorporate uncertainty in estimation of k. Te corrections demonstrated in tis paper are specific to te screening program providing te data. In te example sown, tey pertain to te United Kingdom program, wic trougout te period of observation was mainly using twoview mammograpy every 3 years. A program wit more intensive or more frequent screening would ave a different probability of screen detection and terefore different correction factors. In addition, te lead-time correction applied is based on te average sojourn time over all cancers. It could be argued tat a better correction would be based on eac individual tumor s stage at diagnosis. Tis, owever, would involve considerable algebraic and analytic complexity. Te effect of te policy of screening is best evaluated using population mortality from te disease in question. However, te effect of screen detection on case survival is of considerable interest to te clinicians treating cancer patients and to te patients temselves. Te work above provides a means of estimating survival rates taking into account te major biases inerent in suc estimates. ACKNOWLEDGMENTS Dr. Prue Allgood was supported by a grant from te Princess Grace Hospital, London. Dr. Iris Nagtegaal was supported by a fellowsip from te Dutc Cancer Society. Te West Midlands Screening Histories Project was supported by a grant from te Breast Cancer Researc Trust. Te autors are grateful to te National Healt Service Trusts, private ospitals, and National Healt Service breast screening services in te West Midlands for providing

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