Using observational data to estimate an upper bound on the reduction in cancer mortality due to periodic screening

We derive a simple formulation of PSE and show that it gives an upper bound on the estimated reduction in population cancer mortality.

PSE requires two types of data from subjects who receive two or more screenings at regular intervals. The first type of data are the numbers of subjects who receive each screen and who are detected with cancer as a result of screening or in the interval between screens. (See Tables 1,2,3,4) The second type of data are numbers of subjects with cancer who die from cancer and are in the risk set each year after diagnosis. (See Table 5)

PSE involves three steps: (1) estimate the age-specific incidence of cancer associated with different types of detection: first screen, interval between screens, subsequent screen, refusers, and controls, if available, (2) estimate cancer fatality rates after cancer detection, and (3) combine these estimates to estimate the reduction in population cancer mortality associated with periodic cancer screening.

Because PSE requires regular intervals, the analysis is restricted to screenings that occur "on-time", namely, within a window of time close to the length of interval. The length of the window for on-time screenings is somewhat arbitrary. A very wide window discards relatively few subjects; however it may introduce bias into calculations that are based on assuming the screening interval equals the midpoint of the window. Alternatively, a narrow window might discard too much data, increasing the chance for bias from nonrandom exclusion of subjects. In designing a study for PSE analysis, the screening intervals should be as regular as possible. Without loss of generality in the following discussion, we assume a regular interval between screens of 1 year.

PSE requires estimates of age-specific cancer incidence for the following types of cancer detection: type F, detection on the first screening; type I, detection in the interval between screenings; type S, detection on screening subsequent to the first; and type A, detection in the absence of screening. The incidence of type F detection at age a is q _F (a) = x _F (a) / n _F (a), where x _F (a) is the number of subjects detected with cancer as a result of a first screening at age a and n _F (a) is the number who received the first screening at age a. The incidence of type I detection at age a is q _I (a) = x _I (a) / n _I (a) where x _I (a) is the number of cases in the interval after screening at age a and n _I (a) is the number at risk at the start of the interval. The incidence of type S detection at age a is q _S (a) = x _S (a) / n _S (a) where x _S (a) is the number of subjects detected with cancer as a result of an "on-time" screening after a previous screening at age a and n _S (a) is the number of subjects who received an "on-time" screening after the previous screening at age a. Although a type S detection occurs on screening at age a + 1, for mathematical convenience, it is associated with screening at age a. We cannot observe type A detection from data on subjects screened. However, as derived in [8], one can estimate the probability of type A detection by

q _A (a) = q _F (a) + q _I (a) + q _S (a) - q _F (a + 1),     (1)

if the following key assumption holds.

Once a subject is detectable on screening the subject will always be detectable on screening.

(The quantity q _F (a + 1) in (1) is the cancer detection rate on the first screening among subjects age a + 1 at first screening.) The graphical proof of (1) in Figure 1 generalizes the graphical proof in [8] to allow some subjects with preclinical cancer to be missed on screening. In Figure 1, Assumption 1 corresponds to δ = 0, where δ is the probability that some individuals with preclinical cancer would be detected if screened at age a but missed if screened at age a + 1. As an example of Assumption 7, consider a woman who would have been detected with breast cancer at age 50, but is not screened at age 50 for reasons unrelated to the screening or any possibility of cancer. Under Assumption 1, if the woman were screened at age 51, she would be detected with cancer. There is sometimes confusion about how this relates to sensitivity of the screening test. As shown in Figure 1, Assumption 1 implies that the sensitivity of the screening test equals 1 if a previous screening test would have detected cancer.

Ideally q _F (a + 1) estimates the probability of detecting cancer on a first screen at age a + 1 one year after the start of the study. Because there are no data on subjects first screened after a one year delay, we compute q _F (a + 1) from subjects age a + 1 at the start of the study. This procedure requires the following additional assumption.

Given age, year of birth provides no additional information for predicting cancer incidence on the first screen.

PSE also requires estimates of cancer fatality-rates among cases. The estimated probability of death from cancer within 5 years of type d cancer detection at age a is

pr (cancer death | type d detection at age a)

where h _di is the estimated case-fatality rate from cancer in year i after type d detection, and Surv(a, a + i) is the probability of surviving competing risks from age a to age a + i. See also Gooley et al [18]. For year j after type d detection (d = F, I, S), the estimated case-fatality rate from the cancer under study is h _dj = x _dj / r _dj , where x _dj is the number of cancer deaths among cases at year j after type d detection, and r _dj the number of cases with type d detection who are at risk at year j since detection.

We approximate Surv(a, a + i) by the probability of surviving from age a to age a + i obtained from demographic data stratified by sex [19]. We approximate the probability of surviving competing risks each year over five years as the probability of surviving competing risks at the midpoint of two years, i.e., Surv(a, a + i) = Surv(a, a + 3) for i = 1,2,3,4,5. This lets us approximate (2) by

pr (cancer death in cases | type d detection at age i) = Surv(a, a + 3) m _d ,     (3)

where is the estimated probability of cancer death within five years of type d detection conditional on no death from competing risk and Surv(a, a + 3) is the approximate probability of surviving competing risks within five years of type d detection.

A major challenge is how to estimate m _A , the probability of cancer fatality within five years of type A detection (i.e. in the absence of screening) conditional on no death from competing risk. Previous approaches [1‐3] used data from refusers, substituting m _R for m _A . However this requires a strong unreasonable assumption as well as data from refusers, which is often not available.

As an alternative, we estimate an upper bound on the reduction in the population cancer mortality rate from screening by estimating the cancer fatality rate in the absence of screening using data from interval cancers, namely, substituting m _I = m _A . The reason this is an upper bound is that cancers arising in the absence of screening are composed of cancers that would have arisen in the interval after a negative screening (had there been screening) and cancers that would have been detected on a previous screening (had there been screening). The latter cases are presumably slower growing (a type of length-biased sampling) with better survival, so using only the interval cancers artificially increases the estimated cancer-fatality rate in the absence of screening. (One caveat is that the survival of interval cancers may be improved due to increased awareness of treatment options that would occur as part of a screening program. If the effect of length bias is relatively small, substituting m _I for m _A might not be an upper bound, although we believe it would be a reasonable approximation.)

PSE estimates the reduction in population cancer mortality rates due to starting periodic cancer screening at age a instead of age b, where ages a and b lie in the range of ages at initial screening. (This estimate accounts for competing risks through the use of Surv(a, a + 3)). To avoid different rates of overdiagnosis between comparison groups, PSE compares population cancer mortality rates in two hypothetical scenarios involving full compliance: Scenario , periodic screening from age a until age b and Scenario , no periodic screening from age a to age b-1 followed by screening at age b. Scenario involves either detection on the first screen at age a or detection in the interval or on subsequent screens to age b. Scenario involves either detection in the absence of screening from ages a to b - 1 or detection on a first screen at age b. Screening at age b in both Scenarios and avoids differential overdiagnosis rates because, if Assumption 1 holds, both scenarios and specify equal probabilities of detecting cancer by age b.

More formally we can write the reduction in population cancer mortality rates associated with starting periodic cancer screening at age a instead of age b as

g = pr (cancer mortality under Scenario )

- pr (cancer mortality under Scenario ), (4)

where

pr (cancer mortality under Scenario )

= pr (cancer death in cases | type A detection at age i)

+ q _F (b) pr(cancer death in cases | type F detection at age b)

pr (cancer mortality under Scenario )

= q _F (a) pr(cancer death in cases | type F detection at age a)

pr (cancer death in cases | type I detection at age i)

pr (cancer death in cases | type S detection at age i)

and cancer death in cases refers to death from cancer in cases within five years of cancer detection. Accounting for deaths from competing risks, the estimated probability of type d detection at age i, conditional on being alive at age a, is

p (type d detection at age i) = Surv(a, i) q _d (i)     (5)

Substituting (5) and (3) into (4) gives

To simplify (6), we define

Substituting (7) into (4) gives the following simple estimate,

_basic = (Q _A m _A + q _F1(basic) m _F ) - (q _F0(basic) m _F + Q _I m _I + Q _S m _S .     (8)

To increase the stability of _basic , we used averages over k = 3 intervals, for the probabilities of detection on the first screenings at age a and at age b,

Substituting (9) into (8) gives the modified estimate

_modified = (Q _A m _A + q _F1 m _F ) - (q _F0 m _F + Q _I m _I + Q _S m _S ).     (10)

Invoking (1), we substitute q _F0 + Q _I + Q _S - Q _F1 for Q _A (10). As discussed previously to obtain an upper bound, we also set m _A = m _I . This gives the following estimated upper bound in the reduction in population cancer mortality from periodic screening and its asymptotic variance

_upp = (q _F0 - q _F1) (m _I - m _F ) + Q _S (m _I - m _S ),     (11)

There are two basic scenarios in which Assumption 1 could be violated. First the cancer, or at least the detectable part of cancer, could regress over time. This would most likely occur if the cancer were at a very early stage. Of course, early lesions are the principal targets of screening tests. Second, chance fluctuations in the results of the screening test might mask cancer detection, particularly if the interval between the screening tests were small. For example if the screening test were based on a sampling of cells, the screening test may, by chance, not include any of the tumor cells. For many screening modalities Assumption 1 may not hold.

As shown in Figure 1, if Assumption 1 did not hold, PSE would estimate the cancer incidence rate in the absence of screening as q _A (a) + δ, for δ > 0, instead of q _A (a). Thus if Assumption 1 were not satisfied, PSE would overestimate the cancer incidence rate in the absence of screening which overestimates the reduction in the population cancer mortality rate. On the other hand, a violation of Assumption 1 would also imply that some cancers detected on screening in would not have been detected on the last screening in which would lower the reduction in the population cancer mortality rate in (4). However we think this latter situation would have a small impact relative to the former which involves the entire span of ages at screening and not just the last age at screening.

Between 1975 and 1978 investigators randomized approximately 45,000 subjects to either 5 annual fecal occult blood screenings, 3 biennial screenings, or no screening [6, 7]. Due to a lower than expected death rate among controls, the investigators resumed screening between 1982 and 1986. After a hiatus of screening of between 3 and 5 years, the annual screened group received 5 additional annual screenings and the biennial screened group received 3 additional biennial screenings. Approximately 14 percent of subjects randomized to screening did not receive screening. Each screening cycle consisted of six Hemocult slides with planned definitive work-up if any slide showed evidence of occult blood. Screenings for the annual groups were labeled as on time if they were done in the 9 to 15 month time window since the previous screening. Screenings for the biennial group were labeled as on-time if they were done 20 to 28 months since the previous screening. (The longer time window was used to keep the loss of data arbitrarily to no more than 15%) Excluding the first screening after the resumption of screening, approximately 93 percent of the annual subsequent screenings and 85 percent of the biennial subsequent screenings were on-time. The age range for the analysis was 50 to 75. For estimating the age-specific incidence of cancer among controls, we used data collected up to the time of the last screen, which was 16 years after the start of the study. We increased the precision of the estimated age-specific cancer incidence on the first screen by pooling data on the first screening in the annual and biennial arms. For annual cancer screening, age was divided into intervals of 1 year. For biennial screening, age was divided into intervals of 2 years.

Starting in 1963, approximately 60,000 women were randomly assigned to either a study group invited for four annual mammograms and physical examinations or to a control group that received no screening within the study [8]. Approximately 1/3 of the subjects in the study group refused the first screening and received no screenings. Screenings were labeled on-time if they were done 9 to 15 months after a previous screening. Approximately 79 percent of second screenings, 76 percent of third screenings, and 73 percent of fourth screenings were on time. The age range for the analysis was 40 to 64. For estimating the age-specific incidence of cancer among controls, we used data collected up to the time of the last screen, which was 4 years after the start of the study.

Between 1971 and 1976 approximately 9,200 male heavy smokers who tested negative on a prevalence (initial) screening were randomized to either a study group urged to undergo radiologic and cytological screening examinations every 4 months for 6 years or a control group that at study entry received a recommendation for annual chest X-rays with no further reminders [9, 10]. Approximately 7 percent of the study group subjects did not receive any screenings.

Because PSE requires a single screening time at each round of screening, we restricted PSE to the screenings in which the time between cytology and x-ray was less than 3 weeks. Screenings were labeled as on-time if they were done within 3.5 to 5.5 months of the previous screening and the time between cytology and x-ray was less than 3 weeks. Approximately 85 percent of the subsequent screenings were on-time.

Only yearly age data were available. Because the screenings in the Mayo Lung Project were scheduled at 4 month intervals, yearly cancer incidence data for types I and S detection are approximated using the sums of counts for three successive screens. Because all subjects in the control group had an initial screening, we pooled data for detection rates on initial screenings in the study and control groups.

Unfortunately the initial screening in the control group greatly complicated the validation, which requires that no screening be performed in the control group. In order to better approximate a control group that received no screening, we only used data in controls starting 6 years after randomization. The underlying assumption is that by 6 years, most cancers detected on the prevalence screening would have progressed to clinical cancer in the absence of intervention. (This may not be true because of the likely possibility of overdiagnosis [10] and lead times that may exceed 6 years, but it may serve as a useful approximation if the amount of overdiagnosis is small). Due to the 6-year wash-out period, we start the age range for PSE at 51 instead of 45. We chose 6 years for the washout period as a compromise. We thought a longer washout period would have greatly restricted the age range under study and a shorter washout period would have had a much more limited effect.

We illustrate the calculations for the analysis of the HIP data on breast cancer screening. The probability of a women surviving competing risk from age 40 to each successive age up to age 64 is 1., 1., 0.99, 0.99, 0.99, 0.99, 0.99, 0.98, 0.98, 0.98, 0.97, 0.97,0.96, 0.96,0.95, 0.95,0.94, 0.93,0.92, 0.92,0.91, 0.9, 0.89, 0.87. Using these probabilities and data from Tables 1 and 5, we computed q _F = .00166, q _Fs = .00373, Q _I = .0151, Q _S = .0338, m _I = .334, and m _S = .123, Substituting into (11) gave _upp = .00676. To estimate the variance we computed v _I = .00497, v _S = .00189, V _F0 = .00000056, V _F1 = .0000023. Substituting into (12) gave var( _upp ) = .0000090.

There is a large overlap in the confidence intervals for the PSE estimated upper bound from screened subjects and PSE estimates from all subjects (Figure 2). Also, the 95% confidence interval for the estimated difference includes zero (Figure 3), indicating that the upper bound estimate is reasonable. The estimated mortality reduction in the screening program is similar to the PSE estimates (Figure 2) because the long-duration of screening (5 annual or 3 biennial screenings followed by 3–5 years hiatus followed by 5 annual or 3 biennial screenings) in the trial approximated the 21 years of periodic screening.

As in the previous example, there is a large overlap in the confidence intervals for the PSE estimated upper bound from screened subjects and PSE estimates from all subjects (Figure 2). Also, similar to the previous example, the 95% confidence interval for the difference included zero (Figure 3) indicating that the upper bound estimate is reasonable. The PSE estimates were higher than the estimated effect of the screening program in the trial because the former was based on 24 annual screenings and the latter was based on only 4 annual screenings.

Unlike the other examples the confidence intervals for PSE estimates from screened subjects and PSE estimates from all subjects differed considerably (Figure 2). We think Assumption 1 may not have held due to the short interval between screens and to the fact that the performance of sputum cytology screening depends on sampling of the tumor cells. Although the 95% confidence interval for the difference included zero (Figure 3), its large width means that a substantial bias cannot be ruled out. The PSE estimates were higher than the estimated effect of the screening program in the trial because (i) the former is based on 24 years of screening while the latter is based on only 6 years and (ii) the effect in the latter was reduced by a prevalence screening in the controls. PSE for screened subjects does not use any data from the control group. With PSE for all subjects, we assumed a wash-out period to try to remove the effect of the prevalence screen.

The results indicate that a PSE estimated upper bound based on subjects screened is not unreasonable when compared to the PSE estimate based on all subjects in the randomized trial. Because of sampling variability it is not surprising that the point estimate of the upper bound can be smaller than the point estimate based on all subjects.

We caution that violations of Assumption 2 could have a substantial impact. Assumption 2 depends on the cumulative effect of birth cohort from ages a to b. According to Moran [21] the relative bias due to violation of Assumption 2 is particularly large if the age-specific incidence on the first screen changes little with age and interval and subsequent cancers are relatively rare. In that case Moran advised that other methods be applied. One way to reduce bias from Assumption 2 is to only estimate the effect of screening for at most 5 years. That way the cumulative birth cohort effect would be limited to only 5 years.