Allowing for random errors in radiation dose estimates for the atomic bomb survivor data

The presence of random errors in the individual radiation dose estimates for the A-bomb survivors causes underestimation of radiation effects in dose-response analyses, and also distorts the shape of dose-response curves. Statistical methods are presented which will adjust for these biases, provided that a valid statistical model for the dose estimation errors is used. Emphasis is on clarifying some rather subtle statistical issues. For most of this development the distinction between radiation dose and exposure is not critical. The proposed methods involve downward adjustment of dose estimates, but this does not imply that the dosimetry system is faulty. Rather, this is a part of the dose-response analysis required to remove biases in the risk estimates. The primary focus of this report is on linear dose-response models, but methods for linear-quadratic models are also considered briefly. Some plausible models for the dose estimation errors are considered, which have typical errors in a range of 30-40% of the true values, and sensitivity analysis of the resulting bias corrections is provided. It is found that for these error models the resulting estimates of excess cancer risk based on linear models are about 6-17% greater than estimates that make no allowance for dose estimation errors. This increase in risk estimates is reduced to about 4-11% if, as has often been done recently, survivors with dose estimates above 4 Gy are eliminated from the analysis.     

Allowance for random dose estimation errors in atomic bomb survivor studies: a revision

Allowing for imprecision of radiation dose estimates for A-bomb survivors followed up by the Radiation Effects Research Foundation can be improved through recent statistical methodology. Since the entire RERF dosimetry system has recently been revised, it is timely to reconsider this. We have found that the dosimetry revision itself does not warrant changes in these methods but that the new methodology does. In addition to assumptions regarding the form and magnitude of dose estimation errors, previous and current methods involve the apparent distribution of true doses in the cohort. New formulas give results conveniently and explicitly in terms of these inputs. Further, it is now possible to use assumptions about two components of the dose errors, referred to in the statistical literature as "classical" and "Berkson-type". There are indirect statistical indications, involving non-cancer biological effects, that errors may be somewhat larger than assumed before, in line with recommendations made here. Inevitably, methods must rely on uncertain assumptions about the magnitude of dose errors, and it is comforting to find that, within the range of plausibility, eventual cancer risk estimates are not very sensitive to these.     

Effect of recent changes in atomic bomb survivor dosimetry on cancer mortality risk estimates

The Radiation Effects Research Foundation has recently implemented a new dosimetry system, DS02, to replace the previous system, DS86. This paper assesses the effect of the change on risk estimates for radiation-related solid cancer and leukemia mortality. The changes in dose estimates were smaller than many had anticipated, with the primary systematic change being an increase of about 10% in gamma-ray estimates for both cities. In particular, an anticipated large increase of the neutron component in Hiroshima for low-dose survivors did not materialize. However, DS02 improves on DS86 in many details, including the specifics of the radiation released by the bombs and the effects of shielding by structures and terrain. The data used here extend the last reported follow-up for solid cancers by 3 years, with a total of 10,085 deaths, and extends the follow-up for leukemia by 10 years, with a total of 296 deaths. For both solid cancer and leukemia, estimated age-time patterns and sex difference are virtually unchanged by the dosimetry revision. The estimates of solid-cancer radiation risk per sievert and the curvilinear dose response for leukemia are both decreased by about 8% by the dosimetry revision, due to the increase in the gamma-ray dose estimates. The apparent shape of the dose response is virtually unchanged by the dosimetry revision, but for solid cancers, the additional 3 years of follow-up has some effect. In particular, there is for the first time a statistically significant upward curvature for solid cancer on the restricted dose range 0-2 Sv. However, the low-dose slope of a linear-quadratic fit to that dose range should probably not be relied on for risk estimation, since that is substantially smaller than the linear slopes on ranges 0-1 Sv, 0-0.5 Sv, and 0- 0.25 Sv. Although it was anticipated that the new dosimetry system might reduce some apparent dose overestimates for Nagasaki factory workers, this did not materialize, and factory workers have significantly lower risk estimates. Whether or not one makes allowance for this, there is no statistically significant city difference in the estimated cancer risk.     

Mechanistic models for radiation carcinogenesis and the atomic bomb survivor data

Recently, Heidenreich et al. (Radiat. Res., 158, 607-617, 2002) suggested that the Radiation Effects Research Foundation (RERF) A-bomb survivor cohort study is not large enough to discriminate between various possible carcinogenic mechanisms. At least with the current follow-up, this is true to some extent, but I think the specific issues are rather different than they suggest. In particular, I do not think it is true-as they further indicate-that various models fit the data about equally well while estimating very different patterns of excess risk, which would imply that these patterns cannot be reasonably well characterized. I will point to specific criticisms of their approach to the data and offer some more general comments on mechanistic modeling approaches. Although there are important distinctions, I suggest on a very optimistic note that the two major approaches may be converging, and soon the main differences may not be in the assumptions made but in the aims of the modeling.     

Flexible dose-response models for Japanese atomic bomb survivor data: Bayesian estimation and prediction of cancer risk

Generalised absolute risk models were fitted to the latest Japanese atomic bomb survivor cancer incidence data using Bayesian Markov Chain Monte Carlo methods, taking account of random errors in the DS86 dose estimates. The resulting uncertainty distributions in the relative risk model parameters were used to derive uncertainties in population cancer risks for a current UK population. Because of evidence for irregularities in the low-dose dose response, flexible dose-response models were used, consisting of a linear-quadratic-exponential model, used to model the high-dose part of the dose response, together with piecewise-linear adjustments for the two lowest dose groups. Following an assumed administered dose of 0.001 Sv, lifetime leukaemia radiation-induced incidence risks were estimated to be 1.11 x 10(-2) Sv(-1) (95% Bayesian CI -0.61, 2.38) using this model. Following an assumed administered dose of 0.001 Sv, lifetime solid cancer radiation-induced incidence risks were calculated to be 7.28 x 10(-2) Sv(-1) (95% Bayesian CI -10.63, 22.10) using this model. Overall, cancer incidence risks predicted by Bayesian Markov Chain Monte Carlo methods are similar to those derived by classical likelihood-based methods and which form the basis of established estimates of radiation-induced cancer risk.     

Temporal analysis of a dose-response relationship: leukemia mortality in atomic bomb survivors

A data analysis that incorporates time dependencies is demonstrated for the dose response of leukemia mortality in the atomic bomb survivors. The time dependencies are initially left unspecified and the data on leukemia mortality--up to the end of 1978--are used to infer them. Several findings based on T65 revised doses (T65DR) are obtained. First, it is shown that the fits to the data of time-dependent L (linear in gamma dose)-Q (quadratic in gamma dose)-L (linear in neutron dose), L-L, and Q-L dose-response models are significantly improved (P less than 0.001) by using the corresponding time-dependent dose-response models. Second, it is shown that the increased risk of leukemia mortality due to gamma irradiation decreases in time while the increased risk due to neutron exposure decreases more slowly, if at all, in time. Consequently, relative biological effectiveness (RBE) of neutrons is shown to increase in time (P = 0.002) and the current definition of RBE as a time-independent quantity is therefore challenged. It is demonstrated with time-dependent models that the L-L model has a poor fit (P = 0.01) to the data for the first 7 years of study, but has an adequate fit for the remaining 21 years. In contrast the Q-L model has an adequate fit for the entire follow-up period (P greater than 0.30).     

The effect of changes in dosimetry on cancer mortality risk estimates in the atomic bomb survivors

In the spring of 1986 the Radiation Effects Research Foundation (RERF) received a new atomic bomb dosimetry system. This report presents the comparisons of leukemia and nonleukemia cancer mortality risk estimates under the old and new dosimetries. In terms of total kerma (essentially whole-body gamma plus neutron exposure), risk estimates for both classes of cancer are 75-85% higher with the new dosimetry. This and other summary comparisons allow for possible nonlinearity at high estimated doses. Changes are also considered in relation to organ doses and assumptions about the relative biological effectiveness (RBE) of neutrons. Without regard to RBE, the risk estimates for total organ dose are essentially unchanged by the dosimetry revision. However, with increasing assumed values of RBE, the estimated low-LET risk decreases much less rapidly under the new dosimetry, due to the smaller neutron component. Thus at an assumed constant RBE of 10, for example, the effect of the dosimetry revision is to increase organ dose risk estimates, relative to those based on the old dosimetry, by 30% for nonleukemia and 80% for leukemia. At an RBE of 20 these increases are 72 and 136%, respectively. A number of other issues are discussed. The city difference in dose is no longer statistically significant, even at an RBE of one. Estimation of RBE is even less feasible with new dosimetry. There is substantial question of the linearity in dose response, in the sense of a leveling off at higher doses. Finally, some indication is given of how risks estimated from this dosimetry and the current data may compare to widely used estimates based largely on the RERF data with the previous dosimetry.     

Type 2 diabetes,obesity, and breast cancer risk among Japanese women of the atomic bomb survivor cohort

BackgroundMuch less is known about diabetes than obesity as a predictor of breast cancer incidence and most previous studies have been conducted in white populations. Therefore, this project within the Radiation Effects Research Foundation’s cohort of Japanese atomic bomb survivors aimed to determine the independent contributions of obesity and diabetes to develop breast cancer.MethodsAfter excluding women with unknown A-bomb radiation dose, a radiation dose of ≥100 mGy, a pre-existing history of breast cancer, and missing body mass index (BMI), the analysis included 29,818 women. Breast cancer status and deaths until 2009 were identified from cancer registries and vital records. Cox regression with age as the time metric was applied to estimate hazard ratios (HR) and 95% confidence intervals (CI) for BMI and diabetes status as time-varying exposures alone and in combination while adjusting for known confounders.ResultsDiabetes prevalence increased from 2.6% to 5.3% and 7.5% from the first to the second and third data collection. During 27.6 ± 12.2 years of follow-up, 703 women had developed breast cancer (mean age of 66.0 ± 12.9 years) and 31 (4.4%) had been diagnosed with diabetes. A diagnosis of diabetes was not significantly associated with breast cancer incidence without (HR 1.12, 95% CI 0.77–1.64) and with BMI (HR 1.01, 95% CI 0.69–1.49) as a covariate. The respective HRs for overweight and obesity were 1.61 (95% CI 1.34–1.93) and 2.04 (95% CI 1.40–2.97).ConclusionsAmong a long-time Japanese cohort, excess body weight but not a diabetes diagnosis was significantly associated with breast cancer risk.     

Effect of radiation dose on the height of atomic bomb survivors: a longitudinal study

We conducted a longitudinal analysis of height after age 20 for atomic bomb survivors in the Adult Health Study (AHS) cohort. The measurements we used were made from July 1958 to June 1998 (AHS examination cycles 1-20). We analyzed only the subjects with known atomic bomb radiation doses, excluding those who were not in the city at the time of bombing (ATB) and those exposed in utero. We also excluded from the analysis measurements made after the occurrence of vertebral fracture. The total number of subjects was 11,862, and the total number of measurements was 109,770; the mean number of measurements per subject was 9.25. Assuming that stature after age 20 is approximately constant, a simple mixed-effects model was fitted to stature after age 20, and linear dose effects for young ATB subjects were modeled for both sexes. The estimated mean heights for subjects born in 1945 in Hiroshima were 166.0 cm for men and 155.4 cm for women. The sex difference in height was 10.6 cm, with men significantly taller than women (P < 0.001). The difference between the cities was not significant (P = 0.162). The birth cohort effects per decade were -1.7 cm for men (P < 0.001) and -2.1 cm for women (P < 0.001). A reduction of stature due to radiation exposure was observed for individuals of both sexes who were below 19 years of age ATB (95% confidence interval, 17-21 years), and the dose effect was larger for women than for men (P = 0.028). The estimated effects per gray for those who were age 0 ATB were -1.2 cm for men and -2.0 cm for women and for those who were age 10 ATB were-0.57 cm for men and -0.96 cm for women.     

Studies of mortality of atomic bomb survivors. Report 13: solid cancer and noncancer disease mortality: 1950-1997

This continues the series of general reports on mortality in the cohort of atomic bomb survivors followed up by the Radiation Effects Research Foundation. This cohort includes 86,572 people with individual dose estimates, 60% of whom have doses of at least 5 mSv. We consider mortality for solid cancer and for noncancer diseases with 7 additional years of follow-up. There have been 9,335 deaths from solid cancer and 31,881 deaths from noncancer diseases during the 47-year follow-up. Of these, 19% of the solid cancer and 15% of the noncancer deaths occurred during the latest 7 years. We estimate that about 440 (5%) of the solid cancer deaths and 250 (0.8%) of the noncancer deaths were associated with the radiation exposure. The excess solid cancer risks appear to be linear in dose even for doses in the 0 to 150-mSv range. While excess rates for radiation-related cancers increase throughout the study period, a new finding is that relative risks decline with increasing attained age, as well as being highest for those exposed as children as noted previously. A useful representative value is that for those exposed at age 30 the solid cancer risk is elevated by 47% per sievert at age 70. There is no significant city difference in either the relative or absolute excess solid cancer risk. Site-specific analyses highlight the difficulties, and need for caution, in distinguishing between site-specific relative risks. These analyses also provide insight into the difficulties in interpretation and generalization of LSS estimates of age-at-exposure effects. The evidence for radiation effects on noncancer mortality remains strong, with risks elevated by about 14% per sievert during the last 30 years of follow-up. Statistically significant increases are seen for heart disease, stroke, digestive diseases, and respiratory diseases. The noncancer data are consistent with some non-linearity in the dose response owing to the substantial uncertainties in the data. There is no direct evidence of radiation effects for doses less than about 0.5 Sv. While there are no statistically significant variations in noncancer relative risks with age, age at exposure, or sex, the estimated effects are comparable to those seen for cancer. Lifetime risk summaries are used to examine uncertainties of the LSS noncancer disease findings.     

Studies of mortality of atomic bomb survivors. Report 13: Solid cancer and noncancer disease mortality: 1950-1997

This continues the series of general reports on mortality in the cohort of atomic bomb survivors followed up by the Radiation Effects Research Foundation. This cohort includes 86,572 people with individual dose estimates, 60% of whom have doses of at least 5 mSv. We consider mortality for solid cancer and for noncancer diseases with 7 additional years of follow-up. There have been 9,335 deaths from solid cancer and 31,881 deaths from noncancer diseases during the 47-year follow-up. Of these, 19% of the solid cancer and 15% of the noncancer deaths occurred during the latest 7 years. We estimate that about 440 (5%) of the solid cancer deaths and 250 (0.8%) of the noncancer deaths were associated with the radiation exposure. The excess solid cancer risks appear to be linear in dose even for doses in the 0 to 150-mSv range. While excess rates for radiation-related cancers increase throughout the study period, a new finding is that relative risks decline with increasing attained age, as well as being highest for those exposed as children as noted previously. A useful representative value is that for those exposed at age 30 the solid cancer risk is elevated by 47% per sievert at age 70. There is no significant city difference in either the relative or absolute excess solid cancer risk. Site-specific analyses highlight the difficulties, and need for caution, in distinguishing between site-specific relative risks. These analyses also provide insight into the difficulties in interpretation and generalization of LSS estimates of age-at-exposure effects. The evidence for radiation effects on noncancer mortality remains strong, with risks elevated by about 14% per sievert during the last 30 years of follow-up. Statistically significant increases are seen for heart disease, stroke, digestive diseases, and respiratory diseases. The noncancer data are consistent with some non-linearity in the dose response owing to the substantial uncertainties in the data. There is no direct evidence of radiation effects for doses less than about 0.5 Sv. While there are no statistically significant variations in noncancer relative risks with age, age at exposure, or sex, the estimated effects are comparable to those seen for cancer. Lifetime risk summaries are used to examine uncertainties of the LSS noncancer disease findings.     

Radiation and smoking effects on lung cancer incidence by histological types among atomic bomb survivors

While the risk of lung cancer associated separately with smoking and radiation exposure has been widely reported, it is not clear how smoking and radiation together contribute to the risk of specific lung cancer histological types. With individual smoking histories and radiation dose estimates, we characterized the joint effects of radiation and smoking on type-specific lung cancer rates among the Life Span Study cohort of Japanese atomic bomb survivors. Among 105,404 cohort subjects followed between 1958 and 1999, 1,803 first primary lung cancer incident cases were diagnosed and classified by histological type. Poisson regression methods were used to estimate excess relative risks under several interaction models. Adenocarcinoma (636 cases), squamous-cell carcinoma (330) and small-cell carcinoma (194) made up 90% of the cases with known histology. Both smoking and radiation exposure significantly increased the risk of each major lung cancer histological type. Smoking-associated excess relative risks were significantly larger for small-cell and squamous-cell carcinomas than for adenocarcinoma. The gender-averaged excess relative risks per 1 Gy of radiation (for never-smokers at age 70 after radiation exposure at age 30) were estimated as 1.49 (95% confidence interval 0.1-4.6) for small-cell carcinoma, 0.75 (0.3-1.3) for adenocarcinoma, and 0.27 (0-1.5) for squamous-cell carcinoma. Under a model allowing radiation effects to vary with levels of smoking, the nature of the joint effect of smoking and radiation showed a similar pattern for different histological types in which the radiation-associated excess relative risk tended to be larger for moderate smokers than for heavy smokers. However, in contrast to analyses of all lung cancers as a group, such complicated interactions did not describe the data significantly better than either simple additive or multiplicative interaction models for any of the type-specific analyses.     

Solid cancer incidence in atomic bomb survivors: 1958-1998

This is the second general report on radiation effects on the incidence of solid cancers (cancers other than malignancies of the blood or blood-forming organs) among members of the Life Span Study (LSS) cohort of Hiroshima and Nagasaki atomic bomb survivors. The analyses were based on 17,448 first primary cancers (including non-melanoma skin cancer) diagnosed from 1958 through 1998 among 105,427 cohort members with individual dose estimates who were alive and not known to have had cancer prior to 1958. Radiation-associated relative risks and excess rates were considered for all solid cancers as a group, for 19 specific cancer sites or groups of sites, and for five histology groups. Poisson regression methods were used to investigate the magnitude of the radiation-associated risks, the shape of the dose response, how these risks vary with gender, age at exposure, and attained age, and the evidence for inter-site variation in the levels and patterns of the excess risk. For all solid cancers as a group, it was estimated that about 850 (about 11%) of the cases among cohort members with colon doses in excess of 0.005 Gy were associated with atomic bomb radiation exposure. The data were consistent with a linear dose response over the 0- to 2-Gy range, while there was some flattening of the dose response at higher doses. Furthermore, there is a statistically significant dose response when analyses were limited to cohort members with doses of 0.15 Gy or less. The excess risks for all solid cancers as a group and many individual sites exhibit significant variation with gender, attained age, and age at exposure. It was estimated that, at age 70 after exposure at age 30, solid cancer rates increase by about 35% per Gy (90% CI 28%; 43%) for men and 58% per Gy (43%; 69%) for women. For all solid cancers as a group, the excess relative risk (ERR per Gy) decreases by about 17% per decade increase in age at exposure (90% CI 7%; 25%) after allowing for attained-age effects, while the ERR decreased in proportion to attained age to the power 1.65 (90% CI 2.1; 1.2) after allowing for age at exposure. Despite the decline in the ERR with attained age, excess absolute rates appeared to increase throughout the study period, providing further evidence that radiation-associated increases in cancer rates persist throughout life regardless of age at exposure. For all solid cancers as a group, women had somewhat higher excess absolute rates than men (F:M ratio 1.4; 90% CI 1.1; 1.8), but this difference disappears when the analysis was restricted to non-gender-specific cancers. Significant radiation-associated increases in risk were seen for most sites, including oral cavity, esophagus, stomach, colon, liver, lung, non-melanoma skin, breast, ovary, bladder, nervous system and thyroid. Although there was no indication of a statistically significant dose response for cancers of the pancreas, prostate and kidney, the excess relative risks for these sites were also consistent with that for all solid cancers as a group. Dose-response estimates for cancers of the rectum, gallbladder and uterus were not statistically significant, and there were suggestions that the risks for these sites may be lower than those for all solid cancers combined. However, there was emerging evidence from the present data that exposure as a child may increase risks of cancer of the body of the uterus. Elevated risks were seen for all of the five broadly classified histological groups considered, including squamous cell carcinoma, adenocarcinoma, other epithelial cancers, sarcomas and other non-epithelial cancers. Although the data were limited, there was a significant radiation-associated increase in the risk of cancer occurring in adolescence and young adulthood. In view of the persisting increase in solid cancer risks, the LSS should continue to provide important new information on radiation exposure and solid cancer risks for at least another 15 to 20 years.     

Investigation on circular asymmetry of geographical distribution in cancer mortality of Hiroshima atomic bomb survivors based on risk maps: analysis of spatial survival data

While there is a considerable number of studies on the relationship between the risk of disease or death and direct exposure from the atomic bomb in Hiroshima, the risk for indirect exposure caused by residual radioactivity has not yet been fully evaluated. One of the reasons is that risk assessments have utilized estimated radiation doses, but that it is difficult to estimate indirect exposure. To evaluate risks for other causes, including indirect radiation exposure, as well as direct exposure, a statistical method is described here that evaluates risk with respect to individual location at the time of atomic bomb exposure instead of radiation dose. In addition, it is also considered to split the risks into separate risks due to direct exposure and other causes using radiation dose. The proposed method is applied to a cohort study of Hiroshima atomic bomb survivors. The resultant contour map suggests that the region west to the hypocenter has a higher risk compared to other areas. This in turn suggests that there exists an impact on risk that cannot be explained by direct exposure.     

Postoperative cataract cases among atomic bomb survivors: radiation dose response and threshold

Recent evidence argues against a high threshold dose for vision-impairing radiation-induced cataractogenesis. We conducted logistic regression analysis to estimate the dose response and used a likelihood profile procedure to determine the best-fitting threshold model among 3761 A-bomb survivors who underwent medical examinations during 2000-2002 for whom radiation dose estimates were available, including 479 postoperative cataract cases. The analyses indicated a statistically significant dose-response increase in the prevalence of postoperative cataracts [odds ratio (OR), 1.39; 95% confidence interval (CI), 1.24-1.55] at 1 Gy, with no indication of upward curvature in the dose response. The dose response was suggestive when the restricted dose range of 0 to 1 Gy was examined. A nonsignificant dose threshold of 0.1 Gy (95% CI, <0-0.8) was found. The prevalence of postoperative cataracts in A-bomb survivors increased significantly with A-bomb radiation dose. The estimate (0.1 Gy) and upper bound (0.8 Gy) of the dose threshold for operative cataract prevalence was much lower than the threshold of 2-5 Gy usually assumed by the radiation protection community and was statistically compatible with no threshold at all.     

Congenital malformations, stillbirths, and early mortality among the children of atomic bomb survivors: a reanalysis

Of all the data sets pertinent to the estimation of the genetic risks to humans following exposure to ionizing radiation, potentially the most informative is that composed of the cohort of children born to atomic bomb survivors. We present here an analysis of the relationship between parental exposure history and untoward pregnancy outcomes within this cohort, using to the fullest extent possible the recently revised estimates of the doses received by their parents, the so-called DS86 doses. Available for study are 70,073 terminations, but DS86 doses have not been or presently cannot be computed on the parents of 14,770. The frequency of untoward pregnancy outcomes, defined as a pregnancy terminating in a child with a major congenital malformation, and/or stillborn, and/or dying in the first 14 days of life, increases with combined (summed) parental dose, albeit not significantly so. Under a standard linear model, when the sample of observations is restricted to those children whose parents have been assigned the newly established DS86 doses (n = 55,303), ignoring concomitant sources of variation and assuming a neutron RBE of 20, the estimated increase per sievert in the predicted frequency of untoward outcomes is 0.00354 (+/- 0.00343). After adjustment for concomitant sources of variation, the estimated increase per sievert in the proportion of such births is 0.00422 (+/- 0.00342) if the neutron RBE is assumed to be 20. A "one-hit" model with appropriate adjustments for extraneous sources of variation results in an almost identical value, namely, 0.00412 (+/- 0.00364). When the sample is extended to include parents lacking the full array of dose parameters necessary to calculate the DS86 dose, but sufficient for an empirical conversion of the previously employed T65DR dose system to its DS86 equivalent, we find under the linear model that the estimated increase per sievert in untoward pregnancy outcomes is some 31% higher than that published previously, 0.00264 (+/- 0.00277), assuming an RBE of 20, after adjustment for extraneous sources of variation. (Since a dose could not be calculated in 367 of the 70,073 outcomes, the n = 69,706). The corresponding value with the one-hit model is 0.00262 (+/- 0.00294).     

Reverse smoothing: a model-free data smoothing algorithm

Biophysical chemistry experiments, such as sedimentation-equilibrium analyses, require computational techniques to reduce the effects of random errors of the measurement process. The existing approaches have primarily relied on assumption of polynomial models and least-squares approximation. Such models by constraining the data to remove random fluctuations may distort the data and cause loss of information. The better the removal of random errors the greater is the likely introduction of systematic errors through the constraining fit itself. An alternative technique, reverse smoothing, is suggested that makes use of a more model-free approach of exponential smoothing of the first derivative. Exponential smoothing approaches have been generally unsatisfactory because they introduce significant data lag. The approaches given here compensates for the lag defect and appears promising for the smoothing of many experimental data sequences, including the macromolecular concentration data generated by sedimentation-equilibria experiments. Test results on simulated sedimentation-equilibrium data indicate that a 4-fold reduction in error may be typical over standard analyses techniques.