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Looking backwards from outcome to exposure to assess cancer latency
0021-9681/87 S3.OO + 0.00 Copyright © 1987 Pergamon Journals Ltd
J Cbron Dls Vol. 40, SuppJ. 2, pp. 113S-117S, 1987 Printed in Great Britain. All rights reserved
LOOKING BACKWARDS FROM OUTCOME TO EXPOSURE TO ASSESS CANCER LATENCY ALLAN
H.
SMITH
University of California, Berkeley, CA 94720, U.S.A.
Abstract-Assessing time relationships in cancer epidemiology involves linking exposure and cancer incidence or mortality on a time scale. The traditional approach is to make some exposure criterion the referent point, such as date of first exposure. However, the time relationships between exposure and cancer may also be usefully explored using the date of diagnosis (or death) as the referent point in time. This has been accomplished to date in industry-based case-control studies by estimating the etiologic fraction among ever exposed cases for years prior to death from cancer, and in the serially additive expected dose (SAED) model by estimating excess case exposure dose for each year prior to cancer death. Each of these measures assesses exposure in time windows (of arbitrary width) prior to cancer death or diagnosis. Both measures are mathematically related (by constants) to a measure of the fraction of excess case exposure dose achieved in each time window, where "excess" means beyond that expected on the basis of non-cases or controls. This particular measure, the retrospective excess exposure fraction (REEF) in time windows, may be valuable in exploring time relationships between exposure and cancer. It is concluded that retrospective measures between cancer occurrence and exposure warrant further attention to maximize the informativeness of cancer epidemiology studies.
difference in exposure between cases with a disease and controls without it is of equal value The traditional approach to epidemiological to evidence that those who were exposed are data is to look forward from exposure to out- more likely to get a disease than those who were come, and to calculate measures such as relative not exposed, since the one implies the other. In risk or risk difference, or the standardized mor- addition, when considering exposure in a partality ratio. Even in case-control studies, a ticular patient, it may be useful to know in what backward looking study design involving com- time period prior to diagnosis one would expect parison of exposure between the case and the the exposure to have occurred if it were causal. control group, the emphasis has been on forThese observations would not be of much ward risk estimation. The odds ratio, which is importance if it were not for the fact that in the ratio of the odds for having been exposed in certain settings, backward looking measures of the case and the control group and as such is a association between exposure and disease may backward looking measure of association, is yield information which is difficult to obtain justified because it approximates relative risk or with forward looking measures, or which reis an estimate of incidence density in certain quire statistical assumptions not needed with settings. the backward looking measures. One of these There is no question that risk estimation is an settings is the study of the relationship in time important part of epidemiological studies, par- between exposure and disease incidence such as ticularly when assessing the public health in the study of cancer latency. significance of a presumed causal association In this paper, reasons will be given why between exposure and disease. However, as far backward looking measures of association beas hypothesis testing is concerned, and ignoring tween exposure and disease incidence should be issues of bias and data validity, evidence for a considered for certain types of cancer latency INTRODUCTION
Il3S
114S
ALLAN
H . SMITH
analysis. A description of backward looking measures which have been used in latency analyses will be given. A new backward looking measure of association for exploring latency will be presented which involves a small modification of measures which have been used , but which has the advantage of intuitive meaning and wider applicability. Finally suggestions will be given for future development in this area.
WHY CONSIDER BACKWARD MEASURES OF ASSOCIATION FOR EXPLORING CANCER LATENCY?
Thomas has examined differences between prospective and retrospective views of latent periods [1]. In his prospective view of latency, the age at exposure was held constant and the age at risk varied. In the retrospective view, the age at risk was held constant and the age at exposure varied. He noted that the two excess relative risk curves differed in shape and location. Further theoretical development is needed concerning such differences which may indicate advantages or disadvantages with the retrospective or backward measures. However, the reasons which led to the consideration of the use of backward measures were more practical than theoretical. Exposure variability
The assessment of cancer latency is straightforward for exposures occurring at a single point in time such as a nuclear explosion. It is also relatively straightforward for exposures of constant intensity for all exposed persons. However, most human carcinogenic exposures are highly variable over time. This is particularly apparent in occupational settings. Workers move from job to job within an industry with different exposures. Even in the same job, processes and chemicals used may change over time, and there may be changes in industrial hygiene with new ventilation systems, or installation of closed systems of production. Workers may be exposed for a period of months or years , then have no exposure at a new job, and become exposed again in later years at the same or another company. It is much simpler in such situations to fix as a referent point in time the outcome, such as disease diagnosis or death from cancer, than it is to define exposure relevant points in time.
Fixing points in time for forward looking measures Cancer latency is most frequently considered, and often defined, from the point in time of'first exposure, although there has also been interest in the last point in time of exposure [2] and the mid-point of exposure [3]. Polednack [4] suggested dividing the incubation time into a period in which the development of cancer in an individual is not inevitable, and a period during which irreversible changes have already taken place extending up to the point in time of diagnosis. It is well known, and indeed obvious, that the last exposures an individual experiences prior to the onset of disease may not be relevant. In addition, it is apparent that some individuals in a study may have had incidental exposure to a chemical and have got cancer for other reasons . For example, not every asbestos exposed worker who gets lung cancer got it due to that exposure. What is not so obvious is that the first exposures experienced in an intermittent sequence of exposure may also not be relevant. It is possible that changes from the first exposure may be completely repaired during an interim nonexposed period. Even if this is not the case, a subsequent cancer could arise in cells not affected by the first exposure. Thus the point in time of first exposure may not be relevant to the biological incubation period for some individuals in a study involving variable intensity exposure, particularly if it is intermittent. The effect of this on latency analyses using forward looking measures is to overestimate the true incubation period. The same problems arise with considering latency from the point in time certain cumulative exposure measures have been experienced. Indeed there are added problems in that cumulative exposure measures assume a linear dose-response relationship [5] and latency may also vary with intensity of exposure. An attempt can be made to deal with some of these problems by forward looking ~easures of association by mathematical modeling or stratification of individuals with different exposure patterns. However, backward looking assessment of exposure should also be considered for certain types of information relevant to latency. Backward looking assessment of exposure
Backward looking measures of exposure to investigate cancer latency seem to have first
115S
Assessing Cancer Latency
been proposed for occupational cancer studies constants. Before doing so, it will help to intro[6]. In the first instance, odds ratios were calcu- duce the terminology presented by Rothman lated for each year prior to death of cancer cases who referred to the retrospective examination of which were matched with controls from the the incubation period as being in time windows same occupational cohort. However, the odds of variable width [11]. In the case-control analyratio became unstable many years prior to sis presented by Goldsmith [7] time windows of death, and in the few years before death when 5 years' width were utilized with a moving very few study subjects were working. Because average approach: thus the etiologic fraction of this, the next step involved using an estimate among ever exposed cases was calculated for of the etiologic fraction among all cases which each year prior to death based on the 5-year is a more stable measure. The disadvantage of period of which it was the mid-point. In the this measure is that it is influenced by the SAED example of angiosarcoma among vinyl number of cases ever exposed. Thus the measure chloride exposed workers, the width of the time would be low for a study in a company in which window used was 1 year [8]. Using this terminology, the new REEF the exposure was very rare, and higher if the exposure were common and proportionally measure is the fraction of the excess exposure of more cases were attributable to that exposure. the cancer cases compared with non-cases (conFor this reason it was decided to make the trols) accumulated in a particular time window denominator ever exposed cases rather than all prior to death. As such it has a more straightcases. This measure, the etiologic fraction forward intuitive meaning lacking from the among ever exposed cases (or attributable risk retrospective measures used so far. among the exposed), was subsequently used in The only difference between this measure and a study of prostate cancer in the rubber industry that previously used in the SAED analysis is to [7] and has been presented and discussed in a divide the annual dose difference by the sum of separate paper in this issue (D. F. Goldsmith, all annual dose differences. Figure I is based on a figure previously published [8], but with the p. 119S). Similar ideas were incorporated into a latency ordinal axis scale changed to the new measure analysis for use with the SAED model [8,9]. In and the data presented in histogram form. It can this approach, for each year a case worked in be seen, for example, that 52% of the excess the industry under study, their expected ex- exposures were accumulated between 5 and 15 posure to a particular chemical was estimated years prior to diagnosis. In other words, over from all comparable members of the cohort. In half the excess exposure of angiosarcoma cases the latency analysis, the average difference be- occurred in the period 5-15 years to diagnosis. To demonstrate the similarity of the REEF tween observed and expected exposure dose is calculated for each year prior to death, since measure with that used in the matched case-control studies, one can consider the exdates of diagnosis were not available. The SAED model has been criticized because posure dose to be I if a worker was exposed in of possible bias, and it has been shown that it a given time window, and 0 otherwise. The can be modified and embedded in a classical calculations in the Appendix show that the Cox proportional hazards model [10]. However, numerator of the two measures are then identithe extent of possible bias is small, and several cal, and the denominators are both constants advantages of the SAED model have been for a particular study. pointed out, including the fact that latency and dose-response relationships can be explored 0.27 without the assumptions of additive or multiplicative risk models [9]. Proposed measure for backwards looking examination of latency
A new measure, the retrospective excess exposure fraction (REEF), is proposed for backwards examination of cancer latency. It will be shown that the measure described for matched case-control studies, and the measure used with the SAED model are related to this measure by
o
5
10 15 20 25 Years prior to death
Fig. I.
30
116S
ALLAN
H.
SMITH
experiencing excess exposure even after their development of cancer has become inevitable. The backward looking analysis is a response The effect of such serial autocorrelation on to a straightforward question: In what time forward risk estimates is that risk appears to period in relation to death (or preferably diag- continue to increase more than it really does nosis) did the cancer cases have exposures with continuing exposure; or, put another way, greater than what one would expect? It is essen- risk due to earlier exposure is underestimated. tially a descriptive investigation of this question. The same applies at the other end of the specIt in no way supersedes other types of latency trum: that there appears to be some excess analysis based on risk estimates, which ask and exposure accumulated 25-30 years prior to answer different questions. However, it has death in Fig. 1 may merely be because someone several useful features. There is no requirement exposed in later years is more likely to have to fix an exposure point in time from which worked in the same job earlier than someone latency is to be measured, such as the time of else. first exposure or the time some cumulative exposure amount is reached. In addition, there CONCLUSION are virtually no statistical assumptions as are There are various advantages in using backrequired in forward looking risk analysis of latency: for example assumptions that risks are ward looking measures to examine the readditive or multiplicative. Also, the time re- lationship in time between exposure and cancer lationship between excess exposure and disease incidence. In particular the methods presented can be assessed without stratification. For ex- provide simple descriptive analyses with miniample, the vinyl chloride example presented mal statistical assumptions. This approach does involved only 10 cases, yet confounding vari- not supersede methods of analysis based on ables such as year of hire, age at hire, were taken forward looking risk estimates, but answers a into account without the need to stratify into different question which is relevant, particularly different times from first exposure to death as in the context of occupational studies using would be required for most forward looking other cohort members for comparison with latency analyses. cases. It is suggested that the most appropriate Although not discussed in this paper, the measure to use in such settings is the retroretrospective view of cancer latency method can spective excess exposure fraction or REEF also be used to identify the calendar years in which could be used for a variety of study types. which excess exposures were accumulated by cases. An example has been presented elsewhere in this issue by Goldsmith (see also [7]). The REFERENCES REEF measure would also be appropriate for this application. I. Thomas DC. Temporal effects and interactions in cancer: implications of carcinogenic models. In: PreSome of the apparent problems of the backntice RL, Whittemore AS. Eds. Environmental Epiward view of cancer latency are actually shared demiology: Risk Assessment. Philadelphia: Siam Instiby forward looking analyses. For example, in tute for Mathematics and Society, 1982, pp. 107-121. 2. Armenian HK, Lilienfeld AM. The distribution of anyone study, cases may subsequently be found incubation periods of neoplastic diseases. Am J Eplwith a longer latency. Thus although Fig. 1 demiol 1974; 99: 92-100. shows that 93% of the excess exposure was 3. Cobb S, Miller M, Wald N. On the estimation of the incubation period in malignant disease. The brief accumulated within 25 years of death, cases may exposure case, leukemia. J Chron Dis 1959;9: 385-393. be discovered later with longer latency. 4. Polednack AP. Latency periods in neoplastic disease. Another problem shared with forward lookAm J Epidemlol 1974; 100: 354-356. 5. Smith AH. Problems in dose-response interpretation ing measures is due to serial autocorrelation. in occupational epidemiology. J Univ Occup Environ This is because a person exposed on anyone Health 1983; 5(suppl): 189-195. given day is more likely to be exposed on the 6. Smith AH, Checkoway H, Goldsmith DF, Wolf PH, Tyroler HA. An analytic procedure for investigation of next day than a person not exposed on the given cancer latency in matched case-control studies illusday. For example, one cannot infer anything trated with occupational data. Am J Epidemiol 1978; from the fact that 15% of the excess exposures 108: 226. were experienced in the 5-year period prior to 7. Goldsmith OF, Smith AH, McMichael AJ. A case-control study of prostate cancer within a cohort death in Fig. 1. Workers continue to be exposed of rubber and tire workers. J Occup Med 1980; 22: merely because they tend to stay in the same job 533-541. ADVANTAGES OF THE BACKWARD LOOKING MEASURES FOR LATENCY ANALYSIS
Assessing Cancer Latency Smith AH , Waxweiler RJ, Tyro ler HA . Epidemiologic investigation of occupational carcinogenesis using a serially additive expected dose mode l. Am J Epidemiol 1980; 112: 787-797 . 9. Waxweiler RJ, Smith AH . The serially additi ve expected dose (SAED) model in occupational cancer epiemiology. In: Harrington JM, Ed . Recent Advances In Occupational Health, Vol. 2. Edinburgh: Churchill Livingstone, 1984. pp . 131 -142. [0. Lubin JH . A reformulation of the serially additive expected dose method for occupational cohort data. Am J Epldemlol 1983; 118: 592-598. II. Rothman KJ. Induction and latent periods. Am J Epidemlol [981; 114: 253-259. 8.
APPENDIX
Relationship Between the Measure Previously Used for Case-Control Studies and the REEF The backward examination of latency for case-control studies has been accomplished by calculating the etiologic fraction among ever exposed persons in time windows [7]. (See also paper by Goldsmith D . F ., p. 119S in this issue.)
117S
Consider time window i (e.g. 10-15 years prior to d iagnosis), for which the following two by two table applies: exposed cases controls
OJ
c,
not exposed bi d,
The etiologic fraction (EF) for this time window with ever exposed cases as the denominator is: EF; = (aj - bjc;!d,)/'k a,.
(AI)
N ow, if we make the exposure dose = I if a person is ever exposed in a time window, and zero otherwise, then the excess exposure dose for this time window is a, - b.c.id., which is the numerator of the above expression. The total excess exposure for all time windows is: L(a; - bfcdd f).
Hence, the REEF (retrospective excess exposure fraction) , which is the proportion of total excess exposure experienced in the time window is: (a f - b,c;/d,)/L(a; - bfc;/df).
(A2)
Now , since 'kGj and also L(a,- b,c;/df ) are fixed for any one study, the two expressions (AI) and (A2) are related by constants.
J Cbron Dls Vol. 40, SuppJ. 2, pp. 113S-117S, 1987 Printed in Great Britain. All rights reserved
LOOKING BACKWARDS FROM OUTCOME TO EXPOSURE TO ASSESS CANCER LATENCY ALLAN
H.
SMITH
University of California, Berkeley, CA 94720, U.S.A.
Abstract-Assessing time relationships in cancer epidemiology involves linking exposure and cancer incidence or mortality on a time scale. The traditional approach is to make some exposure criterion the referent point, such as date of first exposure. However, the time relationships between exposure and cancer may also be usefully explored using the date of diagnosis (or death) as the referent point in time. This has been accomplished to date in industry-based case-control studies by estimating the etiologic fraction among ever exposed cases for years prior to death from cancer, and in the serially additive expected dose (SAED) model by estimating excess case exposure dose for each year prior to cancer death. Each of these measures assesses exposure in time windows (of arbitrary width) prior to cancer death or diagnosis. Both measures are mathematically related (by constants) to a measure of the fraction of excess case exposure dose achieved in each time window, where "excess" means beyond that expected on the basis of non-cases or controls. This particular measure, the retrospective excess exposure fraction (REEF) in time windows, may be valuable in exploring time relationships between exposure and cancer. It is concluded that retrospective measures between cancer occurrence and exposure warrant further attention to maximize the informativeness of cancer epidemiology studies.
difference in exposure between cases with a disease and controls without it is of equal value The traditional approach to epidemiological to evidence that those who were exposed are data is to look forward from exposure to out- more likely to get a disease than those who were come, and to calculate measures such as relative not exposed, since the one implies the other. In risk or risk difference, or the standardized mor- addition, when considering exposure in a partality ratio. Even in case-control studies, a ticular patient, it may be useful to know in what backward looking study design involving com- time period prior to diagnosis one would expect parison of exposure between the case and the the exposure to have occurred if it were causal. control group, the emphasis has been on forThese observations would not be of much ward risk estimation. The odds ratio, which is importance if it were not for the fact that in the ratio of the odds for having been exposed in certain settings, backward looking measures of the case and the control group and as such is a association between exposure and disease may backward looking measure of association, is yield information which is difficult to obtain justified because it approximates relative risk or with forward looking measures, or which reis an estimate of incidence density in certain quire statistical assumptions not needed with settings. the backward looking measures. One of these There is no question that risk estimation is an settings is the study of the relationship in time important part of epidemiological studies, par- between exposure and disease incidence such as ticularly when assessing the public health in the study of cancer latency. significance of a presumed causal association In this paper, reasons will be given why between exposure and disease. However, as far backward looking measures of association beas hypothesis testing is concerned, and ignoring tween exposure and disease incidence should be issues of bias and data validity, evidence for a considered for certain types of cancer latency INTRODUCTION
Il3S
114S
ALLAN
H . SMITH
analysis. A description of backward looking measures which have been used in latency analyses will be given. A new backward looking measure of association for exploring latency will be presented which involves a small modification of measures which have been used , but which has the advantage of intuitive meaning and wider applicability. Finally suggestions will be given for future development in this area.
WHY CONSIDER BACKWARD MEASURES OF ASSOCIATION FOR EXPLORING CANCER LATENCY?
Thomas has examined differences between prospective and retrospective views of latent periods [1]. In his prospective view of latency, the age at exposure was held constant and the age at risk varied. In the retrospective view, the age at risk was held constant and the age at exposure varied. He noted that the two excess relative risk curves differed in shape and location. Further theoretical development is needed concerning such differences which may indicate advantages or disadvantages with the retrospective or backward measures. However, the reasons which led to the consideration of the use of backward measures were more practical than theoretical. Exposure variability
The assessment of cancer latency is straightforward for exposures occurring at a single point in time such as a nuclear explosion. It is also relatively straightforward for exposures of constant intensity for all exposed persons. However, most human carcinogenic exposures are highly variable over time. This is particularly apparent in occupational settings. Workers move from job to job within an industry with different exposures. Even in the same job, processes and chemicals used may change over time, and there may be changes in industrial hygiene with new ventilation systems, or installation of closed systems of production. Workers may be exposed for a period of months or years , then have no exposure at a new job, and become exposed again in later years at the same or another company. It is much simpler in such situations to fix as a referent point in time the outcome, such as disease diagnosis or death from cancer, than it is to define exposure relevant points in time.
Fixing points in time for forward looking measures Cancer latency is most frequently considered, and often defined, from the point in time of'first exposure, although there has also been interest in the last point in time of exposure [2] and the mid-point of exposure [3]. Polednack [4] suggested dividing the incubation time into a period in which the development of cancer in an individual is not inevitable, and a period during which irreversible changes have already taken place extending up to the point in time of diagnosis. It is well known, and indeed obvious, that the last exposures an individual experiences prior to the onset of disease may not be relevant. In addition, it is apparent that some individuals in a study may have had incidental exposure to a chemical and have got cancer for other reasons . For example, not every asbestos exposed worker who gets lung cancer got it due to that exposure. What is not so obvious is that the first exposures experienced in an intermittent sequence of exposure may also not be relevant. It is possible that changes from the first exposure may be completely repaired during an interim nonexposed period. Even if this is not the case, a subsequent cancer could arise in cells not affected by the first exposure. Thus the point in time of first exposure may not be relevant to the biological incubation period for some individuals in a study involving variable intensity exposure, particularly if it is intermittent. The effect of this on latency analyses using forward looking measures is to overestimate the true incubation period. The same problems arise with considering latency from the point in time certain cumulative exposure measures have been experienced. Indeed there are added problems in that cumulative exposure measures assume a linear dose-response relationship [5] and latency may also vary with intensity of exposure. An attempt can be made to deal with some of these problems by forward looking ~easures of association by mathematical modeling or stratification of individuals with different exposure patterns. However, backward looking assessment of exposure should also be considered for certain types of information relevant to latency. Backward looking assessment of exposure
Backward looking measures of exposure to investigate cancer latency seem to have first
115S
Assessing Cancer Latency
been proposed for occupational cancer studies constants. Before doing so, it will help to intro[6]. In the first instance, odds ratios were calcu- duce the terminology presented by Rothman lated for each year prior to death of cancer cases who referred to the retrospective examination of which were matched with controls from the the incubation period as being in time windows same occupational cohort. However, the odds of variable width [11]. In the case-control analyratio became unstable many years prior to sis presented by Goldsmith [7] time windows of death, and in the few years before death when 5 years' width were utilized with a moving very few study subjects were working. Because average approach: thus the etiologic fraction of this, the next step involved using an estimate among ever exposed cases was calculated for of the etiologic fraction among all cases which each year prior to death based on the 5-year is a more stable measure. The disadvantage of period of which it was the mid-point. In the this measure is that it is influenced by the SAED example of angiosarcoma among vinyl number of cases ever exposed. Thus the measure chloride exposed workers, the width of the time would be low for a study in a company in which window used was 1 year [8]. Using this terminology, the new REEF the exposure was very rare, and higher if the exposure were common and proportionally measure is the fraction of the excess exposure of more cases were attributable to that exposure. the cancer cases compared with non-cases (conFor this reason it was decided to make the trols) accumulated in a particular time window denominator ever exposed cases rather than all prior to death. As such it has a more straightcases. This measure, the etiologic fraction forward intuitive meaning lacking from the among ever exposed cases (or attributable risk retrospective measures used so far. among the exposed), was subsequently used in The only difference between this measure and a study of prostate cancer in the rubber industry that previously used in the SAED analysis is to [7] and has been presented and discussed in a divide the annual dose difference by the sum of separate paper in this issue (D. F. Goldsmith, all annual dose differences. Figure I is based on a figure previously published [8], but with the p. 119S). Similar ideas were incorporated into a latency ordinal axis scale changed to the new measure analysis for use with the SAED model [8,9]. In and the data presented in histogram form. It can this approach, for each year a case worked in be seen, for example, that 52% of the excess the industry under study, their expected ex- exposures were accumulated between 5 and 15 posure to a particular chemical was estimated years prior to diagnosis. In other words, over from all comparable members of the cohort. In half the excess exposure of angiosarcoma cases the latency analysis, the average difference be- occurred in the period 5-15 years to diagnosis. To demonstrate the similarity of the REEF tween observed and expected exposure dose is calculated for each year prior to death, since measure with that used in the matched case-control studies, one can consider the exdates of diagnosis were not available. The SAED model has been criticized because posure dose to be I if a worker was exposed in of possible bias, and it has been shown that it a given time window, and 0 otherwise. The can be modified and embedded in a classical calculations in the Appendix show that the Cox proportional hazards model [10]. However, numerator of the two measures are then identithe extent of possible bias is small, and several cal, and the denominators are both constants advantages of the SAED model have been for a particular study. pointed out, including the fact that latency and dose-response relationships can be explored 0.27 without the assumptions of additive or multiplicative risk models [9]. Proposed measure for backwards looking examination of latency
A new measure, the retrospective excess exposure fraction (REEF), is proposed for backwards examination of cancer latency. It will be shown that the measure described for matched case-control studies, and the measure used with the SAED model are related to this measure by
o
5
10 15 20 25 Years prior to death
Fig. I.
30
116S
ALLAN
H.
SMITH
experiencing excess exposure even after their development of cancer has become inevitable. The backward looking analysis is a response The effect of such serial autocorrelation on to a straightforward question: In what time forward risk estimates is that risk appears to period in relation to death (or preferably diag- continue to increase more than it really does nosis) did the cancer cases have exposures with continuing exposure; or, put another way, greater than what one would expect? It is essen- risk due to earlier exposure is underestimated. tially a descriptive investigation of this question. The same applies at the other end of the specIt in no way supersedes other types of latency trum: that there appears to be some excess analysis based on risk estimates, which ask and exposure accumulated 25-30 years prior to answer different questions. However, it has death in Fig. 1 may merely be because someone several useful features. There is no requirement exposed in later years is more likely to have to fix an exposure point in time from which worked in the same job earlier than someone latency is to be measured, such as the time of else. first exposure or the time some cumulative exposure amount is reached. In addition, there CONCLUSION are virtually no statistical assumptions as are There are various advantages in using backrequired in forward looking risk analysis of latency: for example assumptions that risks are ward looking measures to examine the readditive or multiplicative. Also, the time re- lationship in time between exposure and cancer lationship between excess exposure and disease incidence. In particular the methods presented can be assessed without stratification. For ex- provide simple descriptive analyses with miniample, the vinyl chloride example presented mal statistical assumptions. This approach does involved only 10 cases, yet confounding vari- not supersede methods of analysis based on ables such as year of hire, age at hire, were taken forward looking risk estimates, but answers a into account without the need to stratify into different question which is relevant, particularly different times from first exposure to death as in the context of occupational studies using would be required for most forward looking other cohort members for comparison with latency analyses. cases. It is suggested that the most appropriate Although not discussed in this paper, the measure to use in such settings is the retroretrospective view of cancer latency method can spective excess exposure fraction or REEF also be used to identify the calendar years in which could be used for a variety of study types. which excess exposures were accumulated by cases. An example has been presented elsewhere in this issue by Goldsmith (see also [7]). The REFERENCES REEF measure would also be appropriate for this application. I. Thomas DC. Temporal effects and interactions in cancer: implications of carcinogenic models. In: PreSome of the apparent problems of the backntice RL, Whittemore AS. Eds. Environmental Epiward view of cancer latency are actually shared demiology: Risk Assessment. Philadelphia: Siam Instiby forward looking analyses. For example, in tute for Mathematics and Society, 1982, pp. 107-121. 2. Armenian HK, Lilienfeld AM. The distribution of anyone study, cases may subsequently be found incubation periods of neoplastic diseases. Am J Eplwith a longer latency. Thus although Fig. 1 demiol 1974; 99: 92-100. shows that 93% of the excess exposure was 3. Cobb S, Miller M, Wald N. On the estimation of the incubation period in malignant disease. The brief accumulated within 25 years of death, cases may exposure case, leukemia. J Chron Dis 1959;9: 385-393. be discovered later with longer latency. 4. Polednack AP. Latency periods in neoplastic disease. Another problem shared with forward lookAm J Epidemlol 1974; 100: 354-356. 5. Smith AH. Problems in dose-response interpretation ing measures is due to serial autocorrelation. in occupational epidemiology. J Univ Occup Environ This is because a person exposed on anyone Health 1983; 5(suppl): 189-195. given day is more likely to be exposed on the 6. Smith AH, Checkoway H, Goldsmith DF, Wolf PH, Tyroler HA. An analytic procedure for investigation of next day than a person not exposed on the given cancer latency in matched case-control studies illusday. For example, one cannot infer anything trated with occupational data. Am J Epidemiol 1978; from the fact that 15% of the excess exposures 108: 226. were experienced in the 5-year period prior to 7. Goldsmith OF, Smith AH, McMichael AJ. A case-control study of prostate cancer within a cohort death in Fig. 1. Workers continue to be exposed of rubber and tire workers. J Occup Med 1980; 22: merely because they tend to stay in the same job 533-541. ADVANTAGES OF THE BACKWARD LOOKING MEASURES FOR LATENCY ANALYSIS
Assessing Cancer Latency Smith AH , Waxweiler RJ, Tyro ler HA . Epidemiologic investigation of occupational carcinogenesis using a serially additive expected dose mode l. Am J Epidemiol 1980; 112: 787-797 . 9. Waxweiler RJ, Smith AH . The serially additi ve expected dose (SAED) model in occupational cancer epiemiology. In: Harrington JM, Ed . Recent Advances In Occupational Health, Vol. 2. Edinburgh: Churchill Livingstone, 1984. pp . 131 -142. [0. Lubin JH . A reformulation of the serially additive expected dose method for occupational cohort data. Am J Epldemlol 1983; 118: 592-598. II. Rothman KJ. Induction and latent periods. Am J Epidemlol [981; 114: 253-259. 8.
APPENDIX
Relationship Between the Measure Previously Used for Case-Control Studies and the REEF The backward examination of latency for case-control studies has been accomplished by calculating the etiologic fraction among ever exposed persons in time windows [7]. (See also paper by Goldsmith D . F ., p. 119S in this issue.)
117S
Consider time window i (e.g. 10-15 years prior to d iagnosis), for which the following two by two table applies: exposed cases controls
OJ
c,
not exposed bi d,
The etiologic fraction (EF) for this time window with ever exposed cases as the denominator is: EF; = (aj - bjc;!d,)/'k a,.
(AI)
N ow, if we make the exposure dose = I if a person is ever exposed in a time window, and zero otherwise, then the excess exposure dose for this time window is a, - b.c.id., which is the numerator of the above expression. The total excess exposure for all time windows is: L(a; - bfcdd f).
Hence, the REEF (retrospective excess exposure fraction) , which is the proportion of total excess exposure experienced in the time window is: (a f - b,c;/d,)/L(a; - bfc;/df).
(A2)
Now , since 'kGj and also L(a,- b,c;/df ) are fixed for any one study, the two expressions (AI) and (A2) are related by constants.