Risk assessment of radioisotope contamination for aquatic living resources in and around Japan

Significance

Quantification of contamination risk caused by radioisotopes released from the Fukushima Dai-ichi nuclear power plant is useful for excluding or reducing groundless rumors about food safety. Our new statistical approach made it possible to evaluate the risk for aquatic food and showed that the present contamination levels of radiocesiums are low overall. However, some freshwater species still have relatively high risks. We also suggest the necessity of refining data collection plans to reduce detection limits in the future, because a small number of precise measurements are more valuable than many measurements that are below detection limits.

Abstract

Food contamination caused by radioisotopes released from the Fukushima Dai-ichi nuclear power plant is of great public concern. The contamination risk for food items should be estimated depending on the characteristics and geographic environments of each item. However, evaluating current and future risk for food items is generally difficult because of small sample sizes, high detection limits, and insufficient survey periods. We evaluated the risk for aquatic food items exceeding a threshold of the radioactive cesium in each species and location using a statistical model. Here we show that the overall contamination risk for aquatic food items is very low. Some freshwater biota, however, are still highly contaminated, particularly in Fukushima. Highly contaminated fish generally tend to have large body size and high trophic levels.

The Fukushima Dai-ichi Nuclear Power Plant (FDNPP) accident caused by the catastrophic earthquake and tsunami on 11 March 2011 caused immense damage to human society in and around Japan by releasing large amounts of radioisotopes to the environment (1–5). This accident has raised great public concern about food safety. The government of Japan has therefore been monitoring intensively the γ-emitting radioisotopes in various foods since March 2011, to prevent highly contaminated foods from being distributed in the market. The current monitoring targets are two cesium isotopes (radiocesiums), ${}^{134}\text{C}\text{s}$ and ${}^{137}\text{C}\text{s}$, which were the main radioisotopes released from FDNPP, with long physical half-lives of 2.0652 and 30.167 y, respectively (4).

Because of the continual leakage of contaminated water with traces of radioisotopes into the ocean from FDNPP after the accident, as reported by the Tokyo Electric Power Company (TEPCO) in August 2013 (6), concerns about radioisotope contamination of aquatic foods are being raised. Some reports have indicated that demersal fish off Fukushima are at the highest risk of all aquatic foods. Some researchers have reported that contamination levels in demersal fish had not decreased even a year after the accident (7), and the decrease was much slower than predicted at the end of 2012 (8). In contrast, others have reported that the concentration in most marine organisms, even including demersal fish in the Fukushima coastal waters, had decreased exponentially, although the radiocesium introduced into the ocean was rapidly transferred to marine organisms (9). These contradictory statements must confuse the general public.

The Ministry of Health, Labor, and Welfare (MHLW) in Japan has reported the inspection results of radioisotope contamination in various foods every month since the FDNPP accident (10). We extracted radiocesium measurement data for aquatic foods from the MHLW database. In addition to pairs of ${}^{134}\text{C}\text{s}$ and ${}^{137}\text{C}\text{s}$ measurements, each observation has species name, prefecture, and date of official announcement of measurement (see Materials and Methods for details). We used the data collected from 1April 2011 to 31 March 2015. Overall, we have 1,646 combinations of species and prefectures. However, the analysis of the radiocesium measurement data is not straightforward. Many measurement results have “N.D.” (not detected). This N.D. information does not mean that the item is free from contamination but that any radiocesium concentration is below the detection limit. The limit depends on the measurement conditions but is typically defined as the concentration that gives counts within 3 SDs of the counting error (9). These N.D. measurements are, in fact, missing data and do not occur at random. Therefore, when the contamination is low, they not only reduce precision, but also cause bias (11). In the most extreme case, all measurements for a specific species in a specific prefecture are N.D. Then it is more difficult to evaluate the contamination risk.

We developed a statistical method for quantifying the spatial and temporal contamination risk for foods. This method can handle missing data caused by detection limits. When parameter estimation was difficult because of small sample size, missing data, and shortage of data contrasts, a random-effects model was used, in which the data of similar species with similar sample locations (i.e., prefectures) are used to increase the amount of information available. The risk is defined as the probability that the sum of ${}^{134}\text{C}\text{s}$ and ${}^{137}\text{C}\text{s}$ is greater than the threshold D Bq/kg (D = 20, 50, and 100)—that is, $\text{Pr}({}^{134}\text{Cs}+{}^{137}\text{Cs}>D\hspace{0.17em}\text{Bq}/\text{kg})$, on a specific date—and it is calculated for each combination of species and prefecture. When all of the data were below detection limits, we used a minimum replacement method, which is a kind of worst-case scenario. In this method, the minimum detection limit is treated as if it was the observed value (not the detection limit). The contamination risk map classified into species and prefecture categories is generated using a binomial regression model with random effects. See Materials and Methods for a full exposition of methods.

Results and Discussion

The simple regression analysis for the difference between $\text{log}\left({}^{137}\text{Cs}\right)$ and $\text{log}\left({}^{134}\text{Cs}\right)$ indicates that the observed contamination levels of the last few years are almost completely explained by the explosion of FDNPP and that the influence of other disasters such as the Chernobyl nuclear power plant accident (12) would be relatively very small (Materials and Methods and Fig. S1). Our statistical model generally fitted the data quite well (Dataset S1). The predicted species-specific and prefecture-specific risks [i.e., $\text{Pr}\left({}^{134}\text{Cs}+{}^{137}\text{Cs}>D\hspace{0.17em}\text{Bq}/\text{kg}\right)$] on 1 September 2015 were very small overall (Datasets S1 and S2). When the risks were evaluated with the minimum replacement method, sometimes overly high and unreasonable risks even for probably safe aquatic foods were predicted (Datasets S1 and S2). This unreasonably high risk indicates that the detection limit should be set as low as possible. For statistical risk analysis, one value above the detection limit is much more informative than many values below the detection limit. The current monitoring program should be refined with lower detection limits.

Fig. S1.

Linear regression of the difference between $\text{log}\left({}^{137}\text{Cs}\right)$ and $\text{log}\left({}^{134}\text{Cs}\right)$ against the cumulative days from 1 April 2011 for the observed data above detection limits.

The risk of cesium contamination in Fukushima has steadily decreased from 1 April 2011 to 1 September 2015 (Fig. 1). The contamination risks of marine species are much smaller than those of freshwater species. The median $\text{Pr}\left({}^{134}\text{Cs}+{}^{137}\text{Cs}>100\hspace{0.17em}\text{Bq}/\text{kg}\right)$ for recent days is almost zero for both marine and freshwater species, whereas the median $\text{Pr}\left({}^{134}\text{Cs}+{}^{137}\text{Cs}>20\hspace{0.17em}\text{Bq}/\text{kg}\right)$ of freshwater species is still greater than zero and that of marine species is almost zero. Similar trends are observed in other prefectures; that is, higher contamination risks for freshwater species (Figs. S2 and S3). It is well known that freshwater fish have longer biological half-lives of radiocesium than marine fish because of differences in osmoregulation systems (13, 14). This characteristic would have caused the long-term contamination of radiocesium in freshwater biota as with the Chernobyl accident (15).

Fig. 1.

Temporal risk changes for freshwater and marine species in Fukushima prefecture. Risks are evaluated by $\text{Pr}\left({}^{134}\text{Cs}+{}^{137}\text{Cs}>D\right)$ for freshwater species [freshwater fish, diadromous fish, freshwater crustaceans, and freshwater molluscs: (A) $D=20$, (B) $D=50$, and (C) $D=100$] and marine species [demersal fish, pelagic fish, marine crustaceans, and marine molluscs: (D) $D=20$, (E) $D=50$, and (F) $D=100$] from 1 April 2011 to 1 September 2015. Red lines are (unweighted) medians, and dashed lines are 90% CIs for the risks.

Fig. S2.

Temporal risk changes for freshwater and marine species for prefectures to the north of Fukushima (Aomori, Iwate, and Miyagi). Risks are evaluated by Pr(¹³⁴Cs+¹³⁷Cs > D) for freshwater species [freshwater fish, diadromous fish, freshwater crustaceans, and freshwater molluscs; (A) D = 20, (B) D = 50, (C) D = 100] and marine species [demersal fish, pelagic fish, marine crustaceans, and marine molluscs; (D) D = 20, (E) D = 50, (F) D = 100] from April 1, 2011 to September 1, 2015. Red lines are (unweighted) medians and dashed lines are 90% CIs for the risks.

Fig. S3.

Temporal risk changes for freshwater and marine species for prefectures to the south of Fukushima (Gunma, Saitama, Tochigi, Ibaraki, and Chiba). Risks are evaluated by Pr(¹³⁴Cs+¹³⁷Cs > D) for freshwater species [freshwater fish, diadromous fish, freshwater crustaceans, and freshwater molluscs; (A) D = 20, (B) D = 50, (C) D = 100] and marine species [demersal fish, pelagic fish, marine crustaceans, and marine molluscs; (D) D = 20, (E) D = 50, (F) D = 100] from April 1, 2011 to September 1, 2015. Red lines are (unweighted) medians and dashed lines are 90% CIs for the risks.

For the contamination risk map, we predicted the probability of aquatic foods with $\text{Pr}\left({}^{134}\text{Cs}+{}^{137}\text{Cs}>100\hspace{0.17em}\text{Bq}/\text{kg}\right)>{10}^{-6}$ on 1 September 2015 for species and prefectures using a binomial regression model with random effects (Materials and Methods). The number of groups with relatively high risk is largest in Fukushima, and the prefectures to the south of Fukushima generally tend to have higher risks than the northern prefectures (Fig. 2 and Figs. S2 and S3). This relatively high risk in the southern prefectures would result from the higher radioactive deposition accompanied by radioactive plume transport during mid-March 2011 (1, 16) and the polluted water mass transport southward from the FDNPP (17). Contamination risk tends to decrease as the distance from Fukushima increases. The risks for freshwater fish and freshwater crustaceans are high compared with those for other species groups such as marine fish. As expected from previous studies (7, 8), the contamination risk of demersal fish is highest in marine fish and much higher than that for pelagic fish. However, the conspicuous risk for demersal fish is limited to Fukushima (Fig. 2). The spatial and temporal risk transition indicates that the contamination of marine fish has rapidly dispersed even at the bottom of the sea since 11 March 2011 (Fig. 1, Dataset S1, and Figs. S2 and S3).

Fig. 2.

Spatial radioisotope contamination risks for species groups and prefecture groups around Fukushima. Risk is evaluated by $\text{Pr}\left({}^{134}\text{Cs}+{}^{137}\text{Cs}>100\hspace{0.17em}\text{Bq}/\text{kg}\right)>{10}^{-6}$.

The predicted contamination risk higher than ${10}^{-6}$ [$=\text{Pr}\left({}^{134}\text{Cs}+{}^{137}\text{Cs}>100\hspace{0.17em}\text{Bq}/\text{kg}\right)>{10}^{-6}$] on 1 September 2015 was 3% in all species and prefectures, excluding species with extremely small sample size (Dataset S2). In particular, the risk for Salvelinus leucomaenis leucomaenis (whitespotted char) was higher than ${10}^{-6}$ in four prefectures (Fukushima, Miyagi, Iwate, and Gunma), and the risk for Anguilla japonica (Japanese eel) was higher than ${10}^{-6}$ in three prefectures (Fukushima, Ibaraki, and Chiba). Although seaweed in Fukushima has relatively high risk (Fig. 2), this is because one of six samples (Eisenia bicyclis) showed slightly higher risk ($1.48\times {10}^{-6}$) than the criterion.

Table 1 shows the risks for representative fish in Fukushima. We selected those species somewhat subjectively. Sebastes cheni (Japanese white seaperch) and Hexagrammos otakii (fat greenling) are well known as highly contaminated demersal fish (7, 9, 18). Salvelinus leucomaenis leucomaenis and Anguilla japonica are highly contaminated freshwater and diadromous fish from our analysis. Salvelinus leucomaenis leucomaenis and Anguilla japonica in Fukushima have high risks compared with the highly contaminated demersal fish in Fukushima. Finally, Trachurus japonicus (Japanese jack mackerel) is a typical pelagic fish and selected as a reference. The information for all species in all prefectures is provided in Dataset S2.

Table 1.

Radiocesium contamination risk for representative species in the Fukushima prefecture

Scientific name	English name	n	m	Te	k	Pr(134Cs + 137Cs > 20)	Pr(134Cs + 137Cs > 50)	Pr(134Cs + 137Cs > 100)
Sebastes cheni	Japanese white seaperch	367	719.04	314.1	0.981	0.45502	0.09668	0.00749
Hexagrammos otakii	Fat greenling	1,077	475.06	203.7	0.820	0.00089	2.70 × 10−7	2.13 × 10−12
Salvelinus leucomaenis leucomaenis	Whitespotted char	921	56.76	465.6	0.640	0.09046	0.01063	0.00076
Anguilla japonica	Japanese eel	10	189.69	561.0	1.585	0.62447	0.06672	0.00013
Trachurus japonicus	Japanese jack mackerel	298	49.13	205.8	0.936	1.93 × 10−33	1.74 × 10−78	3.11 × 10−150

n is the sample size. m, Te, and k are the parameters of the Weibull distribution (m, initial contamination level; Te, ecological half-life; k, dispersion parameter).

We conducted multiple regression for the parameters [$\log \left(m\right)$, ρ, and $\log \left(k\right)$] of the Weibull distribution (Materials and Methods). We used trophic level (TL), the logarithm of asymptotic length [$\log \left(\text{Lmax}\right)$], which is usually related to trophic level (19), species group, and prefecture group as explanatory variables. Because TL and Lmax are consistently collected only for fish species, the analysis was limited to fish species. All parameters kept species group and prefecture group as explanatory variables after model selection using the Akaike information criterion (AIC) (20). The AIC best model of $\log \left(m\right)$ did not include TL and $\log \left(\text{Lmax}\right)$, whereas the AIC best models of ρ and $\log \left(k\right)$ included negative trends for $\text{log}\left(\text{Lmax}\right)$ and TL, respectively (Fig. 3). Because TL and $\log \left(\text{Lmax}\right)$ have a positive correlation (Fig. 3A), this suggests that the initial contamination was not significantly different among fish with different TLs, although there are differences in the species and prefectures probably caused by the massive explosion. The dispersion rates are different for different TLs or body sizes, probably because of bioaccumulation (13, 21). In fact, S. leucomaenis leucomaenis and A. japonica have relatively large asymptotic body sizes and high trophic levels (Dataset S2). We also performed the model selection using the Bayesian information criterion (BIC) and the AIC with small-sample bias adjustment (AICc) (20). In either case, the two variables, $\log \left(\text{Lmax}\right)$ for ρ and TL for $\log \left(k\right)$, were selected as the significant variables.

Fig. 3.

The relationship between the Weibull distribution parameters, trophic level, and $\text{log}\left(\text{Lmax}\right)$ [(A) TL vs. $\text{log}\left(\text{Lmax}\right)$, (B) $\log \left(m\right)$ vs. TL, (C) ρ vs. $\text{log}\left(\text{Lmax}\right)$, and (D) $\text{log}\left(k\right)$ vs. TL]. Red lines are predicted trends for TL or $\text{log}\left(\text{Lmax}\right)$. Neither TL nor $\log \left(\text{Lmax}\right)$ for $\log \left(m\right)$ was selected by AIC.

The contamination risks of freshwater fish, freshwater crustaceans, and diadromous fish are relatively high, as indicated in earlier studies (22). However, freshwater fish used as food in Japan are usually not wild but cultured. Thus, because the radiocesium concentration of cultured fish tends to be low (10), the high contamination risk of wild freshwater fish around Fukushima should not be a serious food concern but is a vital problem for recreational fisheries and tourism industries because even leisure fishing is restricted or prohibited if a fish exceeding the limit (100 Bq/kg) is caught. Although the Japanese freshwater system is very complex with many short rivers with small flows, continuing careful monitoring and increasing sample sizes for these species is important as an indicator of environmental contamination (22).

In conclusion, our analysis showed that the present contamination levels of radiocesiums were low overall, even for demersal fish, although some freshwater species still have relatively high risks. Although many N.D. data have made radiocesium risk assessment of foods difficult, our method made it possible to evaluate the risk for each food item and to produce a big picture of contamination risk. Our methodology could contribute greatly to decision-making for quick and effective recovery from the FDNPP accident. Whereas our method can quantify risks even when all data are N.D., high detection limits tend to cause biased results. We therefore recommend data collection plans with lower detection limits in the future.

Materials and Methods

Aquatic Food Data.

The radioisotope contamination data published by the MHLW in Japan come from monthly inspections and contain various food items including drinking water, farm products, dairy products, stock farm products, and seafood. Each observation in the data has species name, prefecture, date of official announcement of measurement, and measured values of ${}^{134}\text{C}\text{s}$ and ${}^{137}\text{C}\text{s}$. We focus on aquatic food here.

The sampling was carried out by research ships and commissioned fishing boats of the Japanese and prefectural governments for major fishery items, including 472 species on major fishery sites (23). The numbers of species for main prefectures are shown in Dataset S3. The researchers selected the sampled species based on their habitats, including the surface layer, middle layer, deep layer, and seaweed, in each fishery season. Consequently, the sampling covered the relevant geographical areas of sea. The area of sea for each prefecture was defined by the area enclosed between the extended boundaries of the prefecture and the line marking 200 nautical miles off the Japanese main islands.

The data used in this article were collected from 1 April 2011 to 31 March 2015. We set the starting date in our analysis to 1 April 2011 because the peak ocean discharge occurred 1 mo after the earthquake (1, 24). We then excluded aquacultured fish and processed foods like fried fish before analysis. As a result, the data consist of 68,894 measurements with independent inspections. We have 1,646 combinations of species and prefectures.

The measurements of radiocesiums were sometimes recorded as less than a threshold value (detection limit), for instance, $<10$. We can lower the detection limit by taking measurements for longer time and increasing the quantity of measured sample. However, for species that are difficult to sample and in prefectures having inadequate measuring instruments, the detection limit tends to increase. The Japanese government instructed each prefecture to ensure that their detection limit was 20 Bq/kg or less for the sum of radiocesiums ${}^{134}\text{C}\text{s}+{}^{137}\text{C}\text{s}$ (25). The maximum of detection limits achieved for either radiocesium was 25 Bq/kg.

Before the risk analysis, we regressed the difference between $\text{log}\left({}^{137}\text{Cs}\right)$ and $\text{log}\left({}^{134}\text{Cs}\right)$ against the cumulative days from 1 April 2011 for the observed data above detection limits (Fig. S1). The intercept in the regression was 0.055, which means that the ratio of ${}^{137}\text{C}\text{s}$ to ${}^{134}\text{C}\text{s}$ was almost 1 at the original scale. The regression coefficient (trend) was 0.0008. This value is close to the value expected from the physical half-lives for ${}^{134}\text{C}\text{s}$ and ${}^{137}\text{C}\text{s}$ [$\text{log}\left(2\right)/\left(2.0652\times 365.25\right)-\text{log}\left(2\right)/\left(30.167\times 365.25\right)=0.000856$]. These results indicate that the observed contamination levels of the last few years are almost completely explained by the explosion of FDNPP, whereas the influence of other disasters such as the Chernobyl nuclear power plant accident (12) would be relatively very small.

Basic Model Structure.

The probability distribution for contamination of radiocesium was modeled by a Weibull distribution, a two-parameter extension of the exponential distribution that allows flexible modeling. This distribution is widely used in survival analysis (26)

$$f\left(x\right)=\frac{k}{\mu }{\left(\frac{x}{\mu }\right)}^{k-1}{e}^{-{\left(x/\mu \right)}^k}\hspace{1em}\text{for}\hspace{0.17em}x\ge 0,$$ [1]

where x is the contamination level for either ${}^{134}\text{C}\text{s}$ or ${}^{137}\text{C}\text{s}$, μ is the scale parameter, and k is the shape parameter. To take into account the temporal change of contamination level, the scale parameter μ of the Weibull distribution was modeled using

$$\mu =m{e}^{-\left(\lambda +\rho \right)t}.$$ [2]

Here, m is the parameter related to the initial contamination level on 1 April 2011 {the mean of a Weibull distribution is $\mu \mathrm{\Gamma }\left[k/\left(k+1\right)\right]$}. The variable t is the cumulative days from 1 April 2011. This variable was calculated using sampling dates obtained by subtracting 7 d from the announcement dates to adjust for the delay between announcement and measurement. The coefficient λ is the dispersion rate caused by radioactive decay, which is calculated by $\text{log}\left(2\right)/T_p$, where $T_p$ is the physical half-life ($T_p=2.0652\times 365.25$ d for ${}^{134}\text{C}\text{s}$ and $T_p=30.167\times 365.25$ d for ${}^{137}\text{Cs}$) (4). The coefficient ρ is the dispersion rate caused by ecological processes, which includes all decays other than physical decay, including biological and environmental factors, and is calculated by $\text{log}\left(2\right)/T_e$, where we call $T_e$ the ecological half-life (27). We have a vector of ${}^{134}\text{C}\text{s}$ and ${}^{137}\text{C}\text{s}$ values as an observation for one measurement. The likelihood function is therefore the product of the likelihood values for ${}^{134}\text{C}\text{s}$ and ${}^{137}\text{C}\text{s}$. When an observed datum is below its detection limit, the Weibull density in the likelihood function is replaced by the cumulative distribution according to notions about the likelihood for censored data (26). The likelihood function for one observation is then

$$L=f{\left(x\right)}^{c}F{\left(x\right)}^{\left(1-c\right)},$$ [3]

where c is an indicator variable that represents whether the observation is above the detection limit or not (above: $c=1$, below: $c=0$). $F\left(x\right)$ is the cumulative distribution of $f\left(x\right)$

$$F\left(x\right)=1-e^{-{\left(x/\mu \right)}^k}.$$ [4]

Because the physical half-life is known, the parameters to be estimated are m, ρ, and k. Estimation of parameters is done using a maximum likelihood method (26). Although m values can be different between ${}^{134}\text{C}\text{s}$ and ${}^{137}\text{C}\text{s}$, we assumed that they were the same, because we know that the ratio of ${}^{134}\text{C}\text{s}$ and ${}^{137}\text{C}\text{s}$ was 1:1 during the first month of radioactive release (4).

Risk Assessment.

After the parameters— m, ρ, and k—have been estimated by maximizing the log-likelihood function, the probability that the sum of ${}^{134}\text{C}\text{s}$ and ${}^{137}\text{C}\text{s}$ is greater than D Bq/kg on a specific date is calculated using the convolution operation for summation of random variables (28)

$${\displaystyle {\int }_D^{\infty }\left[{\displaystyle {\int }_0^{\infty }{f}_{134}\left(x-u\right)f_{137}\left(u\right)du}\right]dx},$$ [5]

where $f_{134}$ and $f_{137}$ are the Weibull density functions for ${}^{134}\text{C}\text{s}$ and ${}^{137}\text{C}\text{s}$, respectively, and the criterion D is set to 20, 50, or 100. The limit of radiocesium concentration in Japan was reduced to 100 from 500 Bq/kg on 1 April 2012. Currently, 100 Bq/kg is the standard limit for general foods where food safety is basically secured (29, 30); 20 Bq/kg is the maximum detection limit of the sum of radiocesiums recommended by the Japanese government (25), and 50 Bq/kg is the standard limit for infant foods (30).

Extrapolation Through the Random-Effects Model for the Data Poor Cases.

The parameter estimation using the Weibull model did not always converge and sometimes produced extremely unrealistic time trends when the number of parameters was greater than the number of observed data values and/or the time span of the data were short, making it difficult to estimate ρ. For those cases, we first estimated the ρ parameter depending on species group and prefecture group using the random-effects model with a multivariate normal distribution for the converged outcomes, and then substituted the estimated ρ in the model and estimated k and m. Here we used 13 species groups based on similarities of biological and ecological characteristics. The groups are freshwater fish, diadromous fish, demersal fish, pelagic fish, sharks and rays, freshwater crustaceans, marine crustaceans, freshwater molluscs, marine molluscs, cephalopods, aquatic mammals, aquatic invertebrates, and seaweeds. The prefectures were also divided into 13 groups based on their closeness to Fukushima and the sample sizes. The groups are Hokkaido, Aomori, Iwate, Miyagi, Fukushima, Ibaraki, Tochigi, Gunma, Chiba, Saitama, Tokyo, Kanagawa, and Others (Fig. 2 and Dataset S2). The probability model is given by

$$\rho \sim \text{N}\left(\nu ,{\sigma }^2\right),$$ [6]

where $\nu =z_{ij}$, $i,j=1,\dots ,13$, i corresponds to species group, and j corresponds to prefecture group. $z_{ij}$ is a random effect and has a multivariate normal distribution $\text{N}\left(0,\mathrm{\Sigma }\right)$. $\mathrm{\Sigma }$ is a variance–covariance matrix related to species group. The best linear unbiased predictors (BLUPs) of random effects (25) are used as the predicted value of ρ for observations that cannot estimate ρ by themselves. If the parameter estimation does not converge even when ρ is given, we estimated the k parameter depending on species group and prefecture group using the random-effects model with a multivariate normal distribution for the converged outcomes, and then substituted the estimated k in the model and estimated only m. The probability model for $\log \left(k\right)$ is the same as that for ρ. Thus, we could estimate the parameters for all datasets except those in which all data are N.D., in which case the minimum replacement method that follows was used.

Minimum Replacement Method.

Here we focus on the situation where all of the data are reported as detection limit values. The maximum likelihood estimate for the mean μ is then always 0 because the likelihood function is strictly monotone decreasing. This case is known as an improper problem for maximum likelihood. However, the inference would lead to severe bias if we disregarded such data. We therefore propose a reasonable method for estimating μ. We hypothetically assume that the minimum of detection limit values, say $x_{\text{min}}$, was the actual observed value. This counterfactual is one of the most conservative cases for the estimation of μ because the censored observation, or the interval data $\left[0,x_{\text{min}}\right]$, is replaced by the observed value $x_{\text{min}}$. Thus, the maximum likelihood estimate based on the counterfactual uniquely exists in the interval between 0 and the minimum in the Weibull model. We call this procedure the minimum replacement method.

When applying this method to our data, we must make the data revert to the hypothetical measurement on a common specific date. Using the estimate ρ or $\log \left(2\right)/T_e$ obtained from the random-effects model, the datum is transformed by multiplying $2^{t\times \left[1/\left(T_p\times 365.25\right)+\rho \right]}$ ($T_p=2.0652$ y for ${}^{134}\text{C}\text{s}$ and $T_p=30.167$ y for ${}^{137}\text{C}\text{s}$), where t is the cumulative days from 1 April 2011. We then assume that the minimum in the transformed data was an observed value. Given ρ and k from the random-effects models, we can estimate m using the maximum likelihood method.

The Risk Map of Radioisotope Contamination.

The risk map of radioisotope contamination was generated using a binomial regression model for binary outcomes with 1 if $\text{Pr}\left({}^{134}\text{Cs}+{}^{137}\text{Cs}>100\hspace{0.17em}\text{Bq}/\text{kg}\right)>{10}^{-6}$ and 0 otherwise. Here species group and prefecture group were also treated as random effects with a multivariate normal distribution in the linear predictor through a logit-link function. The probability model for binary outcome (y) is given by

$$y\sim \text{Bin}\left(1,\omega \right),$$ [7]

where ω is the expected value of binary outcomes, $\text{logit}\left(\omega \right)=z_{ij}$, $i,j=1,\dots ,13$, i corresponds to species group, and j corresponds to prefecture group. $z_{ij}$ is a random effect and has a multivariate normal distribution $N\left(0,\mathrm{\Gamma }\right)$. $\mathrm{\Gamma }$ is a variance–covariance matrix related to species group. The BLUPs of random effects are used as the predicted values of $\text{logit}\left(\omega \right)$. The inverse-logit transformation of $\text{logit}\left(\omega \right)$ (i.e., ω) is the expected value of risk that $\text{Pr}\left({}^{134}\text{Cs}+{}^{137}\text{Cs}>100\hspace{0.17em}\text{Bq}/\text{kg}\right)$ is greater than ${10}^{-6}$. The threshold ${10}^{-6}$ was selected as the inverse of a rough estimate of the number of aquatic fish caught annually in each prefecture (31).

Multiple Linear Regression Between the Weibull Parameters and Trophic Level.

To examine the relationship between radiocesium contamination and biological characteristics, we performed a multiple linear regression between the Weibull parameters (m, ρ, and k) and trophic level or asymptotic body length ($\text{Lmax}$). This analysis was limited to fish species because reliable estimates of trophic levels and asymptotic body lengths were available only for those species from the FishBase website (www.fishbase.org/). We used the Weibull parameters as response variables and trophic level (TL), the logarithm of the asymptotic body length [$\log \left(\text{Lmax}\right)$], species group [four levels for fish groups (freshwater fish, diadromous fish, demersal fish, pelagic fish, and sharks and rays)], and prefecture group (13 levels) as explanatory variables. The expected value for $\log \left(m\right)$ is given by

$$\text{E}\left[\log \left(m\right)\right]={\alpha }_0+{\alpha }_1\text{TL}+{\alpha }_2\log \left(\text{Lmax}\right)+{\displaystyle \sum {\alpha }_k\text{Sp}}+{\displaystyle \sum {\alpha }_j\text{Pref}},$$ [8]

where αs are regression parameters, and $\text{Sp}$ and $\text{Pref}$ are indicator variables that represent species group and prefecture group, respectively. Similarly, the expected values for $\rho =\text{log}\left(2\right)/T_e$ and $\text{log}\left(k\right)$ are respectively given by

$$\text{E}\left(\rho \right)={\beta }_0+{\beta }_1\text{TL}+{\beta }_2\log \left(\text{Lmax}\right)+{\displaystyle \sum {\beta }_k\text{Sp}}+{\displaystyle \sum {\beta }_j\text{Pref}},$$ [9]

and

$$\text{E}\left(\text{log}\left(k\right)\right)={\gamma }_0+{\gamma }_1\text{TL}+{\gamma }_2\log \left(\text{Lmax}\right)+{\displaystyle \sum {\gamma }_k\text{Sp}}+{\displaystyle \sum {\gamma }_j\text{Pref}}.$$ [10]

The variable selection for each model was done using AIC (20). As a sensitivity test, we also used BIC or AICc. When the coefficients for TL and $\log \left(\text{Lmax}\right)$ are selected, the predicted values are calculated using partial dependence plots (32).

All statistical analyses and figures were made using the programs R (version 3.20) and AD Model Builder, version 2.11.1 (33, 34). We provided the SEs for the risks when we could estimate all three parameters—m, ρ, and k—from data without depending on the random-effects model. The SEs were evaluated using the Hessian matrix and the delta method (26) whenever possible (Dataset S2). However, when we could not estimate ρ and/or k without the random-effects model, we were not able to evaluate the SEs for the risks. Because we put more importance on the overall assessment of radiocesium contamination, estimation uncertainty was ignored in the overall assessment of contamination risk because most estimations are based on extrapolation and the minimum replacement method. In fact, estimation uncertainty has been ignored in many post hoc analyses; nevertheless, such analyses have continued to provide important knowledge for us (35). However, ignoring estimation errors can result in an inadequate recognition of the uncertainty and may compromise the soundness of results and confidence in them (36). Therefore, developing a new approach to incorporate estimation uncertainty into our method appropriately will be a first priority in the future.