The theory of case-control studies for early detection programs

Biostatistics (2003), 4, 3, pp. 411–421 Printed in Great Britain

The theory of case-control studies for early detection programs ORI DAVIDOV∗ Department of Statistics, University of Haifa, Mount Carmel, Haifa 31905, Israel [email protected] MARVIN ZELEN Department of Biostatistics, Harvard University and the Dana Farber Cancer Institute,Boston MA 02115 USA [email protected] S UMMARY Although case-control studies are widely used for evaluating the benefit of early detection programs, the theoretical basis underlying this application has not been well developed. In this paper the properties of chronic disease case-control studies for evaluating early detection programs are investigated. An idealized case-control study is analyzed and the theoretical expression for the odds ratio associated with the benefit of screening is derived. The odds ratio is related to the natural history of disease and the screening program. Our results indicate that case-control studies result in odds ratios that are surprisingly close to unity and consequently have low power. Keywords: Case-control studies; Early detection programs; Odds ratio; Screening.

1. I NTRODUCTION In recent years there has been an increased interest in early detection programs (EDPs) for many chronic diseases. This is especially true in several cancer sites; e.g. mammography to detect breast cancer, prostate specific antigen (PSA) to detect prostate cancer, papanicolaou smears (Pap test) to detect cervical cancer, and fecal occult blood test for colorectal cancer. The expectation is that early diagnosis will result in improved prognosis leading to more cures and longer survival. Consequently, accurate quantification of the benefit associated with an EDP is of major importance. The theoretical properties of EDPs may be studied using mathematical models for the screening process which have been developed by various investigators including Zelen and Feinleib (1969), Prorok (1976a,b), Albert et al. (1978a,b), Schwartz (1978), Eddy (1980, 1984), Yakovlev and Tsodikov (1996), Zelen (1993), and Parmigiani (1993, 1997). The theory and methodology for evaluating the benefit for EDPs is a more recent development. For example, Etzioni et al. (1995) explored some issues related to the design and analysis of screening studies. Hu and Zelen (1997) show that clinical trials for EDPs will require long follow-up times and large sample sizes in order to achieve reasonable power and accuracy. Hence, well designed clinical trials for this application are very expensive. ∗ To whom correspondence should be addressed

c Oxford University Press; all rights reserved. Biostatistics 4(3) 

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An alternative to carrying out clinical trials is to evaluate the benefit of EDPs using case-control studies or other types of observational studies. For example, Baker and Chu (1990) suggest a method, applicable to prospective observational studies, that compares periodically screened individuals with a non-randomized control group. More commonly, case-control designs have been used to assess the benefit of screening (Weiss, 1994). The review article by Cronin et al. (1999) summarized current methodological approaches using approximately fifty recently published studies. In particular, they focus on (a) the selection of cases and controls, (b) the definition of exposure (c) common biases. Typical biases are length bias and lead time bias (Zelen and Feinleib 1969, Habbema et al. 1983, Sasco 1988, Church 1999). Connor et al. (2000) used a large-scale simulation study to examine biases resulting from exposure definition, matching, and other factors. To date, the theoretical basis for the use of case-control studies for evaluating the benefit of screening has not been well developed (Hosek, 1996). In this paper we investigate the theoretical properties of case-control studies for early detection programs. The paper is organized in the following way. In Section 2 we formulate the problem and define the notation for the disease and screening process. In Section 3, formulae for the odds ratio of an idealized case-control study are derived. The results are investigated numerically in Section 4. Section 5 serves as a summary and discussion.

2. D ISEASE AND SCREENING PROCESS Consider an idealized model for the natural history of disease in which an individual can be in one of four states denoted by Sh , Sp , Sc and Sd . An individual in the healthy state, Sh , is either disease-free, or has disease which cannot be detected. An individual in the preclinical state, Sp , has detectable disease but is unaware of it. An individual in the clinical state, Sc , has been diagnosed with disease. Finally, the state Sd corresponds to death due to disease. The natural history of disease is assumed to be progressive and is represented by the sequence Sh → Sp → Sc → Sd . Let qi (x) denote the probability density function of the sojourn time distribution in the ith state. The corresponding survival functions are Q i (x). The sojourn time distribution conditional on being in Si , also known as the forward recurrence time, is denoted qi0 (x). Define the probability of being in Si at age t to be Pi (t). The probability of detecting the disease conditional on being in Sp is denoted by β. Clearly, early detection followed by an effective treatment interferes with the natural history of the disease. Screening is beneficial because the disease is diagnosed in an earlier stage compared to usual care. We model the effect of early detection by modifying the sojourn time in the clinical state Sc . Note that screendetected distribution cases are treated in the preclinical state. It is assumed that in the absence of an early detection program, all, individuals would have ultimately entered the clinical state. Consequently there is a guaranteed survival time for all screen-detected cases. This is the time from early diagnosis until the theoretical time one would have entered the clinical state. The guarantee time is a random variable which is incorporated into our modeling. Let qcs (x) denote the sojourn time distribution in the clinical state, for individuals diagnosed by a screening program. Since early detection may improve survival, we assume that Q sc (x)
Q c (x) for all x. As pointed out by a referee, our model does not include a distinct cure state. We consider a benefit (or cure) to be any (positive) modification of the overall survival time. Note that our model may be embedded within a semi-Markov framework as described by Parmigiani et al. (2002) in the context of screening, and Davidov (1999), Davidov and Zelen (2000) for cancer prevention trials. In such models one may also consider a separate state for individuals who are ‘cured’ from the disease. Alternatively, we may allow qcs (x) to integrate to less than unity, which is equivalent to a cured state. Additionally a semi-Markov framework may allow transitions form Si → Sd where i = h, p. Note that these modeling strategies do not have a substantial effect on the subsequent theoretical development.

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3. O DDS RATIOS FOR CASE - CONTROL STUDIES 3.1

General

In a typical case-control study, cases and controls are matched on variables related to the outcome, and compared with respect to an exposure. Theoretically, the cases and controls are a random sample drawn from a hypothetical population of cases and controls. The key assumption underlying the case-control design is that both cases and controls are at risk for the outcome and the exposure. In the screening application it is desired to make inference about the dependence of mortality (the outcome) on the history of early detection exams (the exposure). Mortality is the primary endpoint of many EDPs. Note that the definition of an exposure does not depend on whether the disease was actually diagnosed at a screening examination. Cases are individuals that have died from the disease whereas the controls have not. Additionally, both cases and the controls must have been eligible for an early detection exam. We consider two populations of controls. They may be defined as (a) living individuals, or more restrictively as (b) living individuals with clinical disease. We consider theses two cases separately. 3.2

Derivation of the odds ratios

We will initially model a case-control study assuming that individuals may have only one screening examination at a fixed age, τ . We express the theoretical odds ratio in terms of the relevant quantities associated with the screening and disease process. Let u, s and t denote the ages at which the transitions Sh → Sp , Sp → Sc and Sc → Sd occur. Note that s and t, the age of onset of clinical disease and the age of death, respectively, are observable whereas u, the age at which an individual enters the preclinical state, is not. Let E (τ ) = 1 if the individual had an early detection exam at age τ and E (τ ) = 0 otherwise. Define the joint probability density functions of the age of onset and the age of death conditional on the exposure variable by f (s, t|E (τ ) = 0) and f (s, t|E (τ ) = 1). A non screened individual who is incident with disease in (s, s + ds) and who dies in (t, t + dt) was (neglecting differential terms) either (a) in Sh at age τ , had a forward recurrence time of u − τ in Sh , a sojourn time of s − u in Sp and a sojourn time of t − s in Sc , or (b) was in Sp at age τ , had a forward recurrence time of s − τ in Sp and sojourn time of t − s time units in Sc . These events are mutually exclusive. Hence, upon adding and integrating over u we have
 s  0 f (s, t|E (τ ) = 0) = Ph (τ |E (τ )) qh (u − τ ) qp (s − u) du qc (t − s) +Pp (τ |E

τ (τ )) qp0 (s

− τ ) qc (t − s) ,

(3.1)

if t > s > τ , and zero otherwise. The quantity Pi (τ |E (τ )) is the conditional probability of being in Si at age τ given that the individual was eligible for screening. An individual is eligible for screening if they are alive and have not entered the clinical stage by age τ . Consequently, Ph (τ |E (τ )) =

Ph (τ ) Ph (τ ) + Pp (τ )

and Ph (τ |E (τ )) + Pp (τ |E (τ )) = 1. The joint conditional probability density function of the age of onset of clinical disease and the age of death for a screened individual is an easy consequence. Recall that screening only affects individuals

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who are in Sp at time τ . Therefore, f (s, t|E (τ ) = 1) is obtained from (3.1) by replacing the density qc , appearing in the second term, with βqcs + (1 − β) qc . Hence,   f (s, t|E (τ ) = 1) = f (s, t|E (τ ) = 0) − β Pp (τ |E (τ )) qp0 (s − τ ) qc (t − s) − qcs (t − s) .

(3.2)

Note that in our formulation the density function qcs may incorporate both treatment and length-biased effects. Furthermore, the conditional density functions (3.1) and (3.2) depend on the values of the state probabilities and forward recurrence times. The state probabilities are  τ Pi (τ ) = wi (z) Q i (τ − z) dz, 0

where wi (z) is the probability of entering Si , in (z, z + dz). Clearly, wi (z) is given by the convolution of densities up to stage i for i = p, c and equals unity if i = h. The forward recurrence distribution is simply τ wi (z) qi (x − z) dz 0 qi (x − τ ) =  0τ , x
τ. 0 wi (z) Q i (τ − z) dz 3.2.1 Unrestricted controls. We start by deriving the odds ratios for the situation where controls are defined as living individuals. Let T denote the follow-up time from the last exam. Initially we assume that the same value of T is used for all subjects. The binary outcome variable is denoted by Dτ (T ). Its value is unity if the individual dies of disease in the interval (τ, T ), and zero otherwise. Recall that E (τ ) denotes the binary exposure variable. It is well known that the odds ratio for both prospective and retrospective studies are the same, therefore the unrestricted odds ratio, θu (τ, T ), is given by P [E (τ ) = 1|Dτ (T ) = 1] P [E (τ ) = 0|Dτ (T ) = 0] P [E (τ ) = 1|Dτ (T ) = 0] P [E (τ ) = 0|Dτ (T ) = 1] P [Dτ (T ) = 1|E (τ ) = 1] P [Dτ (T ) = 0|E (τ ) = 0] = . P [Dτ (T ) = 1|E (τ ) = 0] P [Dτ (T ) = 0|E (τ ) = 1]

θu (τ, T ) =

(3.3)

The retrospective odds are observed in practice whereas the theoretical value is calculated using the prospective odds. It is easy to see that, conditional on exposure, the probability of death in the interval (τ, T ) is simply  P [Dτ (T ) = 1|E (τ )] =

τ

 =

T

τ

T

f (t|E (τ )) dt 

t τ

f (s, t|E (τ )) ds dt,

(3.4)

where f (s, t|E (τ )) is given by (3.1) or (3.2) appropriately. Similarly the conditional probability of being alive at age T , given an early detection history, is  ∞ P [Dτ (T ) = 0|E (τ )] = f (t|E (τ )) dt. (3.5) T

Equations (3.4) and (3.5) are evaluated for both E (τ ) = 0 or 1. Note that (3.4) and (3.5) may be evaluated explicitly. However, the resulting expressions are complicated and are therefore omitted. The odds ratio θu (τ, T ) can now be calculated.

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3.2.2 Restricted controls. In this situation the control population consists of living individuals with clinical disease. Hence the hypothetical population from which the cases and controls are randomly sampled consists of individuals who are either dead (cases) or with disease (controls). More formally, let Cτ (T ) = 1 if an individual is in the clinical state at T , and zero otherwise. In this population Dτ (T ) + Cτ (T ) = 1

(3.6)

for all individuals. The definition of the odds ratio is modified accordingly and is conditioned on the event (3.6). Hence, θr (τ, T ) =

P [Dτ (T ) = 1|E (τ ) = 1, Dτ (T ) + Cτ (T ) = 1] P [Dτ (T ) = 1|E (τ ) = 0, Dτ (T ) + Cτ (T ) = 1] P [Dτ (T ) = 0|E (τ ) = 0, Dτ (T ) + Cτ (T ) = 1] × , P [Dτ (T ) = 0|E (τ ) = 1, Dτ (T ) + Cτ (T ) = 1]

which reduces to θr (τ, T ) =

P [Dτ (T ) = 1|E (τ ) = 1] P [Cτ (T ) = 0|E (τ ) = 0] . P [Dτ (T ) = 1|E (τ ) = 0] P [Cτ (T ) = 0|E (τ ) = 1]

(3.7)

The probabilities P [Dτ (T ) = 1|E (τ )] are calculated using (3.4). An individual is in the clinical state at age T provided they enter Sc at age s < T and remain there for at least T − s units of time. Thus,  T f (s|E (τ )) Q c (T − s) ds, P [Cτ (T ) = 1|E (τ ) = 0] =  P [Cτ (T ) = 1|E (τ ) = 1] = The formula for f (s|E (τ )) = θr (τ, T ).

∞ s

τ

T τ

  f (s|E (τ )) (1 − β) Q c (T − s) + β Q sc (T − s) ds.

f (s, t|E (τ )) dt is especially simple, enabling the calculation of

3.3

Multiple exams

Clearly, studies with multiple screening exposures are more realistic than a single, fixed-age screening study. In this section we briefly outline the analysis for this more general situation. Consider for example a population subjected to τ = (τ1 , . . . , τm ) possible early detection exams. A case diagnosed in the age interval (τi , τi+1 ), i = 1, . . . , m, where τm+1 = ∞, is called an ith-generation case. The ith-generation of cases may be further stratified according to the age (interval) of diagnosis, or in other words according to their eligibility for a screening exam. For example, an ith-generation individual who became incident in the j th interval ( j  i) was eligible for an exam up to age τ j . Therefore, the appropriate control population consists of individuals that are alive at τi+1 and were eligible for a screening exam at τ j . More formally, let Di = 1 if an individual is an ith-generation case. Clearly Di =

i 

Di ( j) ,

j=1

where Di ( j) = 1 if an individual is an ith-generation case diagnosed in the jth interval. Thus τ j indicates the last exam that could have been administered. Consequently, the cases Di ( j) are compared to controls who are alive at τi+1 (denoted Di ( j) = 0) with respect to the exposure E τ j . The analysis of this situation follows directly from our previous derivations and is omitted for brevity. To summarize: a general case-control study involving multiple exams is analyzed by combining m (m + 1) /2 separate simple casecontrol studies.

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Fig. 1. Plot of the unrestricted odds ratio as a function of age for various screening ages and screening exam sensitivities.

4. N UMERICAL EXAMPLE We illustrate our calculations of the odds ratio using a hypothetical breast cancer screening trial. Based on reported results from such trials (Walter and Day, 1984; Shapiro et al., 1982) we assume that the sojourn time distributions in all states are exponential with different parameters. The annual incidence of breast cancer in the general population is approximately 0.002 (Stomper and Gelman, 1989) and the prevalence of preclinical disease is approximately 0.007. Using the well known identity linking prevalence, incidence, and mean duration we estimate the mean sojourn time in Sp to be 3.5 years. The median survival in the clinical state is assumed to be 17 years if the disease was diagnosed at a screen and 10 years otherwise (Shapiro et al. 1985). Finally, we assume that the mean sojourn time in Sh is 56 years. Hence the mean age of clinical diagnosis is approximately 60 years. Note that the odds ratios (3.3) and (3.7) will be smaller than unity whenever the treatment has a positive effect on mortality. We therefore plot the inverse of the odds ratio θ j (τ, T )−1 for j = u and r as a function of age, T , for several screening times, τ , and screening sensitivities, β. Specifically the odds ratios are plotted for screening sensitivities of 0.8, 0.9 and 1.0 and screening ages of 45, 50, 55 and 60. The results for the unrestricted odds are displayed in Figure 1. Note that the odds ratio decays monotonically to unity as the follow-up time T increases. Cases diagnosed at large follow-up times T are likely to have developed the disease after age τ and an early detection exam has no effect for such individuals. This explains why the odds ratio become close to unity as T → ∞. More surprisingly, the odds ratios are unexpectedly close to unity despite the fact that the median survival in the clinical state for a positively screened individual is 17 years compared with 10 years for an non-screened individual. In fact it can be shown that sup θu (τ, T ) = lim θu (τ, T ) = (1 − β) + β

T
τ

T τ

λsc , λc

(4.8)

where λc and λsc are the hazard functions (evaluated at τ ) of the sojourn time distributions in the clinical state for a screen-detected and undetected individual, respectively. If the sensitivity of the exam is unity

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0.0

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Matching Ratio - 1 Matching Ratio - 2 Matching Ratio - 5

0

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800

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Fig. 2. Plot of the power of a case-control study versus the required sample size for three matching ratios.

the limit reduces to the ratio of the hazard rates. For example, if early detection results in a five-fold increase in the mean survival in the clinical state, e.g. from 10 to 50 years for breast cancer, which in practice is tantamount to a complete cure, then there will be only a five-fold increase in the odds ratio. We draw two conclusions from this observation. The first is that relatively modest odd ratios in case-control studies for screening may actually indicate relatively large mortality differences. The second conclusion is that case-control studies result in a significant loss of power because even strong clinical effects are associated with odds ratios having a small departure from unity. The relationship between the power of a case-control study and the required sample size is displayed in Figure 2. The power was computed assuming that the screening age is 50 and the follow-up time is 5 years. In addition we assume that the screening test has unit sensitivity and is administered to p = 20% of the population (the results are insensitive to p). Our results indicate that large case-control studies are required for reasonable power. For example, if a single control is matched to every case then over 400 pairs are necessary for the power to exceed 50% and almost 1000 pairs are necessary for the power to exceed 80%. Similar results are obtained when more than one control is matched for each case. Next we display plots for the restricted odds ratio. The restricted odds ratio behaves somewhat differently from the unrestricted odds ratio. Although the restricted odds ratio also satisfies (4.8) the restricted odds ratio tends to be further from unity, compared with the unrestricted odds ratio. However, the difference between them is relatively small. More interestingly, the value of the restricted odds ratio does not approach unity as T increases. In both situations, the small departure from unity of the odds ratio is explained by the fact that screening is not a regular exposure in the context of case-control studies. Screening has no effect on the survival of an unaffected individual or of an affected but undetected individual. The probability of dying, given a history of screening, is calculated using (3.4) and is the sum of three terms. The first is the probability of dying conditional on being in Sh at time τ . The second and third terms are the death probabilities conditional on being in Sp at time τ and being either undetected or detected, respectively. The first two terms provide no information about the benefit of screening. Only the third term does. However, the first two terms are larger and dominate the third. A similar effect is observed for the probability of living beyond time T with screening. Consequently, the odds ratio does not compare the quantities which

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Fig. 3. Plot of the restricted odds ratio as a function of age for various screening ages and screening exam sensitivities.

directly reflect the effect of early diagnosis. A more meaningful comparison would be for individuals who were undetected in Sp at time τ with matched detected individuals. Unfortunately, these are unobservable quantities. The analysis that we actually perform may be thought of as the collapsing of a contingency table, i.e. we group individuals according to their screening history regardless of whether screening leads to early diagnosis. This is similar to analyzing misclassified data. It is well known (e.g. Davidov et al., 2003) that misclassification typically biases the estimates of treatment effects. Finally, we note that θu (τ, T ) = θr (τ, T ) = 1 for all values of T whenever β = 0 or qc = qcs . 5. S UMMARY AND DISCUSSION In this paper we have investigated the theoretical foundation of case-control studies to evaluate the benefit of early detection programs. We do not address estimation problems using data collected from a case-control study. For those aspects see Brookmeyer et al. (1986) and Brookmeyer and Day (1987). The theoretical expression for the odds ratio, based on single exam, is derived for two different definitions for the control population. We indicate how to carry out case-control studies with multiple exams as the exposure by relating them to single exam studies. The mathematical expression for the odds ratio is investigated numerically for a hypothetical example based on a breast cancer screening program. Our calculations show that clinically meaningful reductions in mortality may result in odds ratios which do not differ substantially from unity. This fact is important when the results are interpreted and explained. The small departure of the odds ratio from unity is explained by the fact that screening is not a ‘regular’ exposure. In fact, screening has no effect on the survival of a non-diseased individual or of a diseased individual not detected by a screening exam. This property was noticed by several investigators including Weiss et al. (1992). Consequently, case-control studies for this application have low power and therefore relatively large studies are necessary to show that EDPs are beneficial. Additionally, the magnitude of the odds is affected primarily by the time from screening rather than the age at which screening was performed. Hence, individuals from various age groups can be combined when the data are analyzed and matched only on the time from last screening exam.

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The analysis of our simplified case-control study helps clarify more general situations. In particular, it is apparent that the odds ratio in any case-control study would be close to unity. In addition, our methodology provides the tools to assess the design of a future case-control study. For example, it is possible within this framework to investigate theoretically the effect of various definitions of the terms: ‘case’, ‘control’ and ‘exposure’. For more details see the review by Cronin et al. and the references therein. It is important to realize that there are many practical difficulties in the application of the casecontrol methodology to evaluate EDPs. For example, it may be difficult to recognize self-selection bias, to distinguish between symptomatic and diagnostic tests or to verify eligibility criteria. It is well known that these complicating factors may bias the odds ratio considerably (Connor et al., 2000). In fact the bias is of the order of the theoretical effect. This result further undermines the reliability of such studies. We note that although the odds ratio may not be the best measure of clinical efficacy of early detection programs, no other is available in the case-control setting. Ideally, comparisons should be made of diseaserelated mortality for individuals diagnosed on a planned early-detection exam (no signs or symptoms) relative to individuals having their usual care. Direct comparisons of survival are biased because the early-detection cases will have a lead time advantage and be influenced by length bias. Both biases will result in larger observed survival for screen-detected individuals. This problem may be addressed using a model-based approach for evaluating the survival of cancer cases detected by screening (e.g. Habbema et al., 1983; Walter and Stitt, 1987). However, the conclusions of model-based approaches depend on the specific, usually unverifiable, modeling assumptions regarding both observable and unobservable quantities such as the lead time. Finally, it should be stressed that our investigation shows that the case-control methodology for evaluating the benefit of early detection programs in a non-symptomatic population is a poor statistical tool. Even when the effect of early treatment is highly beneficial, very large numbers of observations are required to have acceptable statistical power. The interpretation of all case-control studies in this application should be viewed with caution. ACKNOWLEDGEMENTS This investigation was partly supported by Public Health Service Grant CA-78607 from the National Cancer Institute, National Institutes of Health, Department of Health and Human Services, Bethesda, MD. We also thank the two reviewers whose comments improved the paper.

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