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<Sps utu 4 iyai bh? d? 4 y PI-' Amxjucan Journal ok Epidemiology Copyright < 1981 by The Johns Hopkins University School of Hygiene and Public Health All rights reserved L. * Vol. 114, No. 5 Printed in USA ` *S` ^ K? I \y ' p POWER CONSIDERATIONS IN EPIDEMIOLOGIC STUDIES OF VINYL CHLORIDE WORKERS , ,s JAMES J. BEAUMONT*1 and NORMAN E. BRESLOW1 V* Launone Beaumont, J. J. (NIOSH, Cincinnati, OH 45226) and N. E. Breslow. Power considerations in epidemiologic studies of vinyl chloride workers. Am J Epidemiol -1981;114:725-34. Nine retrospective mortality studies of workers exposed to vinyl chloride were reviewed to determine whether differences in their hypothesis testing results might be due to differences in statistical power. Where possible, the power of each study was calculated for cancer of the lung, brain and liver. When power was taken into consideration, the results for liver and brain cancer were found to be consistent with an etiologic role for vinyl chloride. For lung cancer, the data were not consistent with an etiologic role, in that two studies with very high power yielded negative results. !-> at epidemiologic methods; respiratory tract neoplasms; brain neoplasms; liver neoplasms; vinyl chloride The nine mortality studies of workers exposed to vinyl chloride that have been completed to date disagree in their hy pothesis testing results for some causes of sites of cancer--liver, brain and lung-- was calculated, and their hypothesis test ing results, positive or negative, were considered in the context of power. death, and for those causes it is difficult to draw conclusions about excess risk (1-12). The studies that are negative for Methods Power calculations a cause of death are especially difficult to interpret, since they may or may not have had the statistical power to detect an ex cess risk if, in fact, one existed. To aid in the interpretation of the studies, their statistical power with respect to three Statistical power is the probability of not overlooking an excess risk, i.e., of not making a Type II statistical error. To aid in the interpretation of this frequently confusing subject, a brief review of Type I and Type II errors follows. Investigators are fnost familiar with Received for publication November 14, 1980, and in final form April 13, 1981. 1 Industrywide Studies Branch F-8, National In stitute for Occupational Safety and Health, 4676 Columbia Parkway, Cincinnati, OH 45226 (address for reprint requests). 1 Dept, of Biostatistics, School of Public Health and Community Medicine, U. of Washington, Seat tle, WA 98195, The authors thank Jay Lubin, Richard Wax- weiler, Thomas Fischbach, Noel Weiss and William Halperin for their valuable advice. Their special thanks go m Philip Landrigan for encouraging this approach. This paper was presented at the Conference to Re evaluate the Toxicity of Vinyl Chloride, Polyvinyl Chloride and Structural Analogues, Rockville, MD, March 20-21, 1980. Type I errors. If the true state of nature is the null hypothesis (no increase in risk), then a Type I error is wrongly rejecting the null hypothesis and declaring that there is an increase in risk. The probabil ity of a Type I error is known as alpha, or more commonly, the "level of signifi cance." For example, when an inves tigator finds an excess and declares it "significant at the 0.05 level," in a sense what he or she is saying is that there is less than a 5 per cent chance of having made a Type I error. Type II errors are less familiar but, GENC 00730 725 For Distribution bv CM A SPECIAL ?AO; V.S DIVISION Ref. No. e PntA l2ll<t>rS1 726 BEAUMONT AND BRESLOW nevertheless, very important (13). Under lying this concept is the fact that when the true state of nature is the alternative hypothesis (increased risk), the research goal is to detect it (i.e., reject the null hy pothesis). Suppose one has specified an alpha level (0.05, for example) for testing a null hypothesis of no increased risk for a particular exposure. If an exposure is harmful with a relative risk off?, then not rejecting the null hypothesis (i.e., wrongly accepting the exposure as harm less) is called a Type II error. The proba bility of a Type II error is usually denoted beta (13). Conversely, the probability of correctly rejecting the null hypothesis and therefore of detecting the excess risk, is called the power and is equal to 1 -- f3. Thus, power quantifies the ability of a particular study to detect an excess risk that truly exists. It is intuitively clear that with a fixed amount of data there is a greater likelihood of detecting a large ex cess risk. Similarly, an increase in the amount of data increases the chance of observing a given risk, i.e., increases the power. The approximate power 1 - (3 of the vinyl chloride studies to detect a relative risk R at the alpha level of significance was calculated from the following for mula, which uses the fact that the square root transformation stabilizes the vari ance of the Poisson distribution (see the statistical appendix): Z_a = Z,, - 2(VS" - 1 )(VE) Here Za denotes the upper 100a per centile of the standard normal distribu tion and E the expected number of cancer deaths based on general population rates. This approximation agrees well with exact power calculations based on Poisson probabilities made by Cutler et al. (14, 15). When the discreteness of the exact test based upon Poisson probabilities is accounted for, the approximate and exact power curves are virtually identical. A family of power curves based upon the approximation is shown in figure 1, where each curve is for a different rela tive risk. It can be seen that as the as sumed relative risk increases for a given number of expected deaths, the power also increases. Similarly, as the expected number of deaths increases for a given relative risk, the power also increases. Application to vinyl chloride literature The assumed relative risks for the power calculations were the median (ap proximate) standardized mortality ratios reported for each cancer site in the vinyl chloride literature. Power calculations are usually performed with a somewhat arbitrarily chosen relative risk, but since relative risk information was available from the vinyl chloride studies, it was thought best to make use of these data. Separate assumptions were made for analyses considering all person-years at risk and for analyses considering only person-years after a minimum latency (time since first exposure), because in the former case the risk was diluted by the 10 to 20 years that are often required for cancer to develop after exposure to a car cinogen. The median reported mortality ratio was not used in one instance: for overall lung cancer, where the median was 1.03, essentially no excess risk. Since some risk needs to be assumed to calcu late power, the arithmetic mean (approx imately 1.5) was used. The statistical powers of mortality studies of workers exposed to vinyl chloride were calculated with respect to three sites of cancer: liver, brain and lung. These three sites were chosen be cause they have been the subject of most of the discussion of causality in the vinyl chloride literature. The powers of the studies were subsequently plotted on power curves for two purposes: to show the variability in powers, and to relate positive and negative findings to power. For the purposes of this review, a positive finding was defined as an excess risk for a >1 1 cc LU 5o c. Figure. I ratio srjcie; particular cance at t Poisson t; It was was carci should be power she the studi general, 1 exposure of R aero assumpti cause, as differed i: The hi vinyl chi in table ! tions var study hv POWER CONSIDERATIONS IN VINYL CHLORIDE STUDIES 100 110< 5.0, 80 3.0, 60 2.0. 40 1.5 20 727 EXPECTED DEATHS Figure. 1. Approximate power curves for various assumed relative risks in standardized mortality ratio studies (for one-sided comparisons with alpha = 0.05). particular cancer with statistical signifi cance at the 0.05 level, using a one-sided Poisson test. It was expected that if vinyl chloride was carcinogenic for an organ site, there should be a pattern: the studies with high power should, in general, be positive, and the studies with low power should, in general, be negative. This assumed that exposure would result in a constant value of R. across the various populations. The assumption was somewhat tenuous be cause, as discussed below, the populations differed in a number of respects. Results The historical prospective studies of vinyl chloride exposed workers are listed in table 1. The sizes of the study popula tions varied considerably, from 255 in the study by Nicholson et al. (3) to 9677 in the Equitable Environmental Health study (6, 7). The reports were not entirely in dependent; for example, many of the workers in the study by Ott et al. (10) were included in the Equitable Environ mental Health study, and there was an overlap of about 800 workers in the studies by Waxweilet et al. (4) and Waxweiler (5). It should be noted that the studies were dissimilar in many other ways. As can be seen in table 1, a minimum exposure of 1-5 years was required in some studies, while others required only one day. Some studies reported findings based upon all person-years at risk, some included person-years at risk only after a mini mum time since first exposure, and some reported both types of analysis. The con trol (standard) populations used were dif ferent: four of the five American studies -r; GENC 007632 7 2 8 BEAUMONT AND BRESL0W Table 1 Historical prospective (standardized mortality rntia) studies of vinyl chloride exposed workers SLurly Fox nnd Collior (1, 2) Nicliotaon et al. (3> VVaxwcilcr et al. (d) Waxweilcr {5)* Ef|ititl>le Environmental Health (G, 7) Beinl et al. (E) Butner et al. (9) 0LL et al. (10) Byren el al. (11) Duck et al. (12) Yeor 1976, 1977 1975 197G 1978 1974, 1978 1978 1979 197G 197G 1975 Cohort she 7717 255 1291 '1806 9677 7021 -(fit 522 750 2120 Espnrnire minim nm (yenrs) >0 5 5 >0 1 >0 >0 >0 >0 >0 Person-years considered All Minimum latency XX X X XX XX X XX X XX X Cancer ratios reported Lung Brain Liver X XX X X XI XX XX X X X X * One of four plants in 1JV76 study* t Liver cancer not retried (Waxweilcr, personal communication, 1980). ^ro 2.a oV 0t^o a" ro rr *oq ET a & ro* to o cT S'a on o p ro roa o ro rS Oo a --r *-t p ST- r* (U'O p a o 2 a TJ a oq XT crq CO tr <T> Isis's.! tr !- EJS no O2 S tSo 31 3 1 9 ^ 3 v" i-1 * 5 3 oo 10 p '2 2 ^ p >-. o gr"fl d <t> P 5 P- H *0 2 -g 3 3 3 Q ^p. r . aj to S-l a ro f5 i-h. 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CL g ?*g ?s ? 10 ro i ro *-* o 33 ^F3 SS-2 td ^ s|s 5l 65 *0 * * +sS 4h ^ si 5 +& POWER CONSIDERATIONS IN VINYL CHLORIDE STUDIES 729 Table 2 Liver cancer: results and powers of epidemiologic studies of vinyl chloride Author Observed deaths Expected deaths Mortality ratio Excess p < 0,05? Power (%) IfRR = 5.0 If RR = LO.O Fox and Collier (1, 2) Waxweiler (5) (10+ latency) Waxweiler et al. (4) (15+ latency, 5 + exposure) Byren et al. (11) (10+ latency) Reinl et al. (8) 4 1.6 2.44 10 2.3 4.27 8 0.5 15.09 7 0.4 16.08 4 1.0 4 0.7 4.13 5.89 12 0.9 15.23 - + + + + + 93 98 80 76 92 86 98 EXPECTED DEATHS Figure 2. Approximate powers of vinyl chloride studies reporting results for liver cancer; 10.0 = assumed relative risk after a minimum latency, 5.0 = assumed relative risk overall. REINL = Reinl et al. (8); BYREN = Byren et al. (11); FOX = Fox and Collier (1, 2); WAX = Waxweiler et al. (4) and Waxweiler (5). two yielded statistically significant re sults (4, 5, 9) (table 4 and figure 4). Again, there was a wide range of powers, from 14 per cent to almost 100 per cent, but here the studies with very high power, were negative, an unlikely occurrence if vinyl chloride is indeed a human lung carcin ogen. Discussion The data regarding liver cancer in vinyl chloride exposed workers suggested l !'4Mm saftg \ttes s&3(*? m #s MM| vl&P ISfi iff j$^ rMm# 3$& %twr )$m wm >v3&f C o o o -.1 o- otf tfl ?B.i 's'l&M Ta hl 3 <1 CO o AuLlior Fox and Collier (1,2) Wuxvvciler (5) (10+ ln(cncy) Wnxweilcr ct al. (4) (15 + latency, 5-t exposure) Equitable Environmental Health ((>, 7) (1 + exposure) Hyren et a). {1 U Ueinl et nl. (8) {tenths 2 9 B ;i 32 2 2 Expected deathn 3-7 43 3.1 09 r>.9 0.3 1.3 Mortality rutin 0 53 2 09 2 50 3 2!) 24K1 r> 12 1.02 Excess p < 0.05? -f -1 - + + - Power (%) intB - 20 [fRU = 3.0 48 53 82 40 64 12 24 tB M > O 'Z H <23. ft ft f^t fst w (S' 5 o a 3 t^2. 3"f3t 2^ *1 to ^ 2 s.ir|LmM g x g. p 3S 5 PX c+ *t3 ft ft ft ft 2. ft ^ s'5---'* 1(a+* r-^e1 aa a- cro 3 s s 3 g" 3 E ft ft 2 3p ft*oa ffiw ft re 3c ft fto3- o xp 2. 53 ft cr1 o 'I "D- S -3 ap O r+ Xi3 ft o Eicrq 7 3p SJ 5sf3t^*. M X 2 Sjv3i re < n o "ft os Cr2T- P4 ET. 3ppi ooor*, fti w a- g ,2 2. " -f re ~ C cl. e CL. 3 -c ^ ft ft p re .!<B ar 3 sr M* ft g. ~ " o H S- ft J1J a- Q tn ^ ft re 3^ J? w r+ *3 w tO n p y v& ft a pP g p ^ ft ffPtt-j a-S: w 0,2- ? " a o< XO ff a, ^ < p S3 3 P P^u ^P CO ft K- p o c. o< 3 ee n j-r W s: g o ft ^ K-. 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O CL a- p ft ft cn 3 ft ft Cl '< P rCtO- ^P t rt- ft P* * PO ^3 < 3' 3 2 (jq p- r t-t So Oo ^p S 2ft P CL r -i' po rp,- pa *-t *0 ft co ft (A 1 T3 rt* H` O p p a' 2- rre ft ^ o *> s- "* c f<t VCl cl 2* sr 2. 3 re cl. ^ a Oi to ^ o a 3 C S. p tr ^tn 3 o CL. re .- . re i S' rt ^ a St ft X V ft M fl> T3 o co cn _* rt t-1 n* ^p a ^ O ft ^ eL S- ar re p & CL ^ ^erq re HW Cl Cn 5^ a- rJ CO rt- - > go -- ar 3" P croq co p fpt-oaq m co ft P^ --I fCtl pr a aq ft o o >-- ft Cl rt w ft >^ ^ ^ 2. co ff. a- pP~- prt - p ocr o p < "O ft ft g cn to ^ 3 2 P 3-*f o P m, n P fxt n ft X P3 i ft P' < ft ft -1 Cn a ro rt- cn aO> tro+ r* ^ r+ ^e ft Cr -j p H EC 3 ft 2ft af>o ft a- o' ro B p 2. &: o 13 ? o- O Cd ?d w C/3 5 cr jd a- a Sre' 2a. g g re trer oMntr o S 3rcLr. jj crcrcr** m a 3-* S m- 5 gn:.`5O* 33,, d a a p^_ n(23h ^t S'*1 ^mf(,tf a- < re 3- ^ rja -1 00 re a <n < re rine re 5' J *f ft CO o R1 uq <i x p rt! O fl rt- ft P a o ^ -- ^ 2. .2* -OSS'? G'/2 n ^r< p II H *T M POWER (%) POWER CONSIDERATIONS IN VINYL CHLORIDE STUDIES 731 EXPECTED DEATHS Figure 3. Approximate powers of vinyl chloride studies reporting results for brain cancer 3.0 = assumed relative risk after a minimum latency, 2.0 = assumed relative risk overall. BYREN = Byren et al. (11); REINL = Reinl et al. (S); FOX - Fox and Collier (1, 2), WAX = Waxweiler et al. (4) and Waxweiler (5), EQUIT = Equitable Environmental Health (6, 7). tive" hypothesis testing results in the light of statistical power is only one of several approaches to reviewing the evi dence of carcinogenicity for a substance. For example, one can compute a combined relative risk for each cancer site (by com bining the observed and expected num bers from the studies), and then, with the help of homogeneity testing, determine which studies are compatible with the combined risk estimate (see the Appendix for the homogeneity test). Liver cancer in the present review has a combined overall relative risk of 5.17 (p < 0.00001), and also a significant (p = 0.002) result in homogeneity testing, largely due to the Reinl (8) relative risk of 15.23. The com bined relative risk for overall brain cancer is 1.74 ip < 0.01); here the results are more homogeneous (p = 0.10), al though Byren (11) is somewhat of an out lier with a relative risk of 6.12. Finally, the combined relative risk for overall lung cancer is 1.06 (nonsignificant), with some evidence (homogeneity p = 0.06) that the results from Waxweiler (4, 5) and Buffler (9) are out of step with the others (relative risks of 1.49 and 2.89, respec tively). Close examination of the individual studies can also be helpful in searching for reasons for differing results. For example, the expected number of deaths from liver cancer in the Byren study (11) is unusually large relative to the expected numbers for brain and lung. Detailed examination of this anomaly might be informative. There are other consid- >i>sff$P Jiiw&s fpSfi o m o o Onsj CO j -VtlibiSftis TAIiT.E -1 i,m,g nrneer.- resr/ffs and powers of epulevuolnpw studies of vinyl chSorult Study Fox anti Collier O, 21 (15+ latency) WuxwcUor (5) OO f- latency) Wuxwciler ct at. (d) (15 i latency, 5 I- exposure) Erpjitnlrle Environmental Health (fi, 7) (1 l exposure) (15) latency, 1 I exposure^ Humer el al. (3) (6 h latency, 2.20 t exposure) Byi'en et al. 111) Duck el al. 112) Hein! cl. al. (R) Oil et al. 110) Olw'rved ctcnltia 4S 28 -12 :w 11 45 41 5 4 0 1R 22 4 l'lxjaa ir4 ilrnlhs 51.2 26 0 28.2 24 3 5.7 44.:) 09.2 L7 1.0 18 18 5 24.6 5.2 Mortal 4y mtio 0 9(1 1.08 1.49 1.56 1 91 1.02 ] 05 2 89 3 81 1 R8 1.09 0.95 0 77 Fixer ns p < 0 057 Power <%) unit - 1.5 If HR *, 2.0 94 100 77 99 63 91 100 H 21 15 55 75 27 tnd > g o H D rO1 m T POWER CONSIDERATIONS IN VINYL CHLORIDE STUDIES 733 Figure 4. Approximate powers of vinyl chloride studies reporting results for lung cancer: 2.0 = assumed relative risk after a minimum latency, 1.5 = assumed relative risk overall. BUFF = Bufiler et al. (9r, BYREN = Byren et al. (11); OTT = On et al. (10); DUCK = Duck et al. (12); REINL = Reinl et al. (S); WAX = Waxweiler et al. (4, 5); EQUIT = Equitable Environmental Health (6, 7); FOX = Fox and Collier (1, 2) erations, such as dose-response, degree of exposure, concomitant exposures, and confidence limits for risk ratios, that also need to be kept in mind. While consid eration of statistical power cannot pro vide a complete answer, it is one more useful way of looking at epidemiologic evidence. References 1. Fox AJ, Collier PF. Low mortality rates in in dustrial cohort studies due to selection for work and survival in the industry. Br J Prev Soc Med 1976;30:225-30. 2. Fox AJ, Collier PF. Mortality experience of workers exposed to vinyl chloride monomer in the manufacture of polyvinyl chloride in Great Britain. Br J Ind Med 1977;34:1-10. 3. Nicholson WJ, Hammond EC, Seidman H, et al. Mortality experience of a cohort of vinyl chloride-polyvinyl chloride workers. Ann NY Acad Sci 1975;246:225-30. 4. Waxweiler RJ, Stringer W, Wagoner JK, et al. Neoplastic risk among workers exposed to vinyl chloride. Ann NY Acad Sci 1976;271:40-8. 5. Waxweiler RJ. An epidemiologic investigation of lung cancer in a multixenobiotic environ ment. Ph.D. Dissertation. Chapel Hill, NC: University of North Carolina, 1978. 6. Tabershaw IR, GafTey WR. Mortality study of workers in the manufacture of vinyl chloride and its polymers. J Occjip Med 1974;16:509-18. 7. Equitable Environmental Health, Inc. Epidemi ologic study of vinyl chloride workers. Final re port submitted to Manufacturing Chemists As sociation, 1978. 8. Reinl W, Weber H, Greiser E. Diseases caused by vinyl chloride. Special report of the State In dustrial Physician, Dusseldorf, West Germany, 1978. 9. Buffler PA, Wood S, Clayton E, et al. Mortality experience of workers in a vinyl chloride monomer production plant. J Occup Med 1979;21:195-203. 10. Ott MG, Langner RR, Holder BB. Vinyl chloride exposure in a controlled industrial environ ment. Arch Environ Health 1975;30:333-9. 11. Byren D, Engholm G, Englund A, et al. Mortal ity and cancer morbidity in a group of Swedish 734 BEA VSlONT and breslow VCM and PVC production workers' Environ Health Perspect 1976;17:167-70. 12. Duck BW, Carter JT, "Coobes EJ. Mortality study of workers in a polyvinylchloride produc tion plant. Lancet 1975;2:1197-9. 13. Freiman JA, Chalmers TC, Smith H, et al. The importance of beta, the Type II error, and sam ple size in the design and interpretation of the randomized control trial. N Engl J Med 1978;299:690-94. 14. Cutler SJ, Schneiderman MA, Greenhouse SW. Some statistical considerations in the study of cancer in industry. Am J Public Health 1954;44:1159-66. 15. Molina EC. Poisson's exponential binomial limit. Huntington, NY; Robert E. Krieger Pub lishing Company, 1973. 16. Armitage P. Statistical methods in medical re search. Oxford: Blackwell Scientific Publica tions, 1971. Appendix The approximate power formula assumes that the observed number of deaths from the cause of interest follows a Poisson distribution with mean RE, where E is the ex pected number based on standard population rates and R is the ratio of rates (relative risk) for the study as opposed to the standard population. It follows that for large values of RE, the square root of the observed deaths is approximately normally distributed with mean Vjj and variance % (16). Now, if a random quantity has a normal distri bution with mean p and variance <r2, its power 1 -- /3 to reject the null hypothesis p = po at the a level of significance is given by 6v\-0 _ '7 P* Mo (j Substitution of p = \',RE, p0 = V', and cr = % into this expression yields the re quired result. The homogeneity test for the combined relative risk uses the chi-square distribu tion with Ti--l degrees of freedom. The equation is i y (Q< - tesi"i 8El where n O, E, and 8 the number of studies, the observed deaths from the ith study, the expected deaths from the ith study, VO./T#,, the combined relative risk estimate. A MI* Copyngm , All rights THE yei scJ r mi su cia po tor an de tin ho ph da re me of 9 major ui a rare d mated :c to affect commur ing and The illn ability, ment ar. Little schizcpr Receive in final fc 1 Dept, i of Medicir School of ! Dept. School of 3 Dept, i of Medici: Addres. 1006, Jo) Baltimor: This w grants fre GENC 00743?