The original cohort sample consisted of 834 mother-infant pairs, who were recruited by their obstetricians at the infants’ first one-month national health checkups starting in November 2005, and who participated in their ten-month national follow up health checkups [2]. The subjects lived in Fukuoka city and voluntarily presented their hair samples at both checkups, which were performed by 13 obstetricians and 77 pediatricians at 90 hospitals throughout Fukuoka. Fukuoka is located on Kyushu Island, faces the Sea of Japan, and has a population of about 2 million. It was elected as the best city to live in Asia several times, because of the moderate climate, good infrastructure and non-polluting industry.

In 2011, six years after the initial research, we sampled 209 then 6-year-old children from the original cohort using PIXE. Hair samples were collected from the base of the scalp close to the occipital region at a length not exceeding 5 cm using a pair of stainless steel scissors. For target preparation, the root-sides of the hairs were attached to the bottom of a holder with adhesive tape and then fixed on the topside in order to avoid overlapping. The target samples were analyzed by the standard-free PIXE method described in S2 Text, in the Nishina Memorial Cyclotron Center, Iwate Medical University, Japan.

Each of the 209 children’s hair strand samples was divided into two specimens for PIXE analysis to obtain two independent measurements for each subject. We statistically analyzed the measurements based on a simple statistical model described S1 Text.

One issue particular to statistical analysis of hair minerals is how to treat “0”, since 0 does not necessarily mean exactly 0, rather it indicates a small amount less than the detection limit. When the two measurements of a pair are both 0, we call the pair Zero. Since the difference between two 0’s is exactly 0, it seems operationally relevant to consider their true value to be also 0 and disregard 0’s in an analysis of differences between pair of measurements to avoid analyzing variations in measurements below the detection limit. Treatment of Zero in applications is addressed in the Discussion.

First we examined the differences between the two measurements of each pair; see S1 Text for the statistical model and calculations. When the differences were approximately normally distributed for a mineral, the mineral was termed valid, since the differences were considered to be caused solely by chance. The Shapiro-Wilk W test was used for testing the normality [33]. Pairs where the differences between the two measurements were too large to fit the normal distribution were termed outliers and excluded in the testing for normality. Pairs other than outliers were termed valid. However, since outliers sometimes carry the most significant information [34], they should not be simply disregarded. Treatment of the outliers in applications is addressed in the Discussion.

Only valid pairs were treated in the following statistical analysis. For a valid pair, the mean of the two measurements is more reliable than the individual measurements. Thus, we obtained a histogram of the mean of the two measurements of each pair and determined a parametric distribution that well fit it. Since extreme values are inevitable for some minerals due to unusual food consumption, medication, hair treatments, environment, and other exposures [1], a few extremely large values were disregarded in the fitting analysis. Kolmogorov D test and Cramer-von Mises W test were used for testing the fitness of the log-normal and Weibull distribution, respectively [33]. When a parametric distribution is well fitted for a mineral, we call the mineral statistically tractable. S1 Fig illustrates the flow of the statistical analysis.

This study was approved by the Institutional Review Board of Kyushu University and the informed consent was provided on a written document and signed by each mother.

Table 1 summarizes the results. Among the 32 minerals, Ag, Cd, I and Ba were not detected in any pairs and excluded. Table 1 column 2 shows the number of positive pairs, and column 3 the proportion of them. 17 minerals (60%) and 27 minerals (96%) have a positive rate greater than or equal to 97% and 68%, respectively.

S3 Fig shows the histogram of the differences, overwritten with a normal distribution, for each mineral that is associated with mean, SD, sample size, skewness, kurtosis, minimum, maximum and p-value by the Shapiro-Wilk W test of normality. A two-tailed p<0.05 is considered significant, and all minerals were confirmed to be valid. No pair was excluded for the 6 minerals, less than 2% were excluded for 17 minerals and less than or equal to 5% were excluded for 25 minerals. As described in S1 Text, SD²/2 is an unbiased estimate of the intra-individual variance σ², or the variance due to locations within a subject. The 4th column of Table 1 indicates that three minerals, Ca, Ti and Cu, were log transformed to confirm the validity. The 5th and 6th columns indicate the number of valid pairs and the proportion of the excluded pairs, or outliers, in the positive pairs.

S4 Fig presents the histogram of the mean of the two measurements of a pair, overwritten with a parametric distribution for each mineral. Associated with them are mean, median, SD, minimum, maximum, type of distribution, parameter values, p-value obtained from the statistical test for the fitness. The fitness for the Normal, Weibull, and Lognormal distributions were tested by Kolmogorov D, Cramer-von Mises W and Kolmogorov D tests, respectively. The SD will be denoted by SD_B to distinguish it from the SD shown in S3 Fig. The 7^th column shows that Cr, Co and Sr were square-root transformed to analyze the tractability. The 8^th column shows the type of distribution fitted for the mineral. The 9^th column shows the number of pairs used to confirm the tractability and the 10^th the proportion of excluded pairs among the valid pairs. No pair was excluded for 20 minerals, less than or equal to 2.5% were excluded for 24 minerals and less than 10% were excluded for 26 minerals. All excluded pairs were too large to fit the distribution, and all minerals were confirmed to be tractable. Distribution types used for the fitting are Normal, Log-normal, Weibull, √-Normal or √-Weibull. As described in S1 Text, the inter-individual variance σ_B², the variance due to individual differences, is obtained by SD_B²—SD²/2.

The results described in this section apply to positive measurements. In other words, comparing positive measurements among populations is performed using the two parameters of those distributions. On the other hand, for 0, the proportion of 0’s is compared among populations.

Table 1 reveals that some elements have low detection rates of positive pairs such as Mg, Al, P, V. Further studies should investigate the reason for these low rates from a biomedical or physical point of view to determine whether these elements should or should not be used for main analysis.

Measurements below LOD are frequently observed with hair element analysis. The appropriate treatment of them depends on the objective of the study. According to Molina-Villalba et al. [12], values below the LOD were “assigned the LOD divided by the square root of 2“. However, Varrica et al. [16] and Dongarrà et al. [18] set “values below the detection limit… at one-third the detection level and treated (them) as real values”.

In our study, we obtained hair strands from each subject and separated them to make two analytes from different locations. The difference in the true value between the two locations is the intra-individual variation that is estimated using the difference between measurements obtained from the analytes. We assigned measurements <LOD “0”. The results of this study suggested that the differences between observed values in the two locations were regarded as random errors, implying that the magnitude of the difference between the true value and “0” was relatively negligible as compared to that of the difference in observed values between the two locations.

To answer the first and the second questions described in the Introduction, we sampled two hair-strand specimens from each subject to obtain two statistically independent hair mineral measurements. If the distribution of the differences between the two measurements is approximately normally distributed then the differences are regarded as normal random errors; the measurements are considered valid and follow a simple random model described in S1 Text. These measurements are qualified for statistical analysis assuming this model.

A valid mineral is termed tractable when the distribution of the mean of the two measurements, after excluding the outliers, is well approximated by a parametric distribution. In this study, all minerals were confirmed tractable. That is, all of the minerals are approximated by either Normal, Lognormal or Weibull distribution, and it will be quite useful in applications that each distribution is determined by only two parameters. For instance, if exposures such as medical treatments, environmental changes, or dietary intakes affect some hair minerals, then the effects of the exposure can be assessed by comparing the two parameters of the distribution with those of the distribution without exposure. This answers the third question. It is also straightforward to calculate any coverage interval and percentile points from a parametric distribution. Coverage intervals adjusting for those exposures may also be obtained by modifying parameter values. This answers the fourth question.

The 0’s are not used in fitting a parametric distribution, since interpretation regarding the biological abnormality of 0 depends on the objective of the study. For an essential mineral in clinical medicine, 0 should be regarded as abnormally low, whilst for a toxic element in an environmental science, 0 should be considered normal. In certain applications, it may be useful to consider jointly the number of 0’s and the parameter values of a distribution fitted for the positive pairs.

When the differences between the two measurements are normally distributed, the pairs are termed valid. Conversely, when a difference is too large to fit the normal distribution, a detailed examination might find particular factors causing the unusually large differences between the measurements in the same subject. They could be experimental errors or unexpected responses.

In fitting a parametric distribution, too large measurements were excluded as outliers. However, those measurements may carry unobserved but significant information [34], unless they are caused simply by mistakes. Appropriate treatment of those measurements should be considered according to the objective of the study. Conversely, determining the distribution that most subjects follow makes it possible to identify outliers.

When examining the reliability of hair mineral results from laboratories, a standard procedure is to split the hair sample of a single healthy volunteer, send them to different laboratories, compare the reported results, and conclude on the reliability of the hair mineral analysis of the laboratories [35–37]. This design of reliability analysis treats hair like blood. Most minerals are uniformly distributed in the blood in the same density regardless of volume. However, most minerals are not uniformly distributed in hair strands and the density may vary considerably between hair strands within a subject. Thus, to comprehensively understand the variations in hair mineral measurements, appropriate statistical analysis taking into account intra-individual variations is necessary. However, many studies rarely considered this aspect. They ascribe the differences in measurements between laboratories to systematic factors such as variations in sample preparations or calibration standards between laboratories.

We demonstrated a statistical analysis assuming a simple statistical model to assess the reliability of hair mineral measurements. The model requires two measurements from each subject. The results of this analysis verified that the differences are regarded as normal random errors. In other words, the hair mineral measurements obtained from PIXE analysis in this study are reliable. Therefore, results obtained from applying an appropriate statistical model to the measurements are also reliable.

Large intra-individual variations in hair minerals cause troubles in various applications. Dulgaszek et al. [5] concluded “Taking into account the high variability of results, further research on the content of elements in human hair should be continued”. Kempson et al. [1] overview possible causes of the variability of hair minerals, and suggest much more normalization and accurate quality controls for hair mineral analysis. Wołowiec et al. [4] review associations between the mineral composition of hair, and physical or mental disorders, and insist on the standardization of sample preparation. Mikulewicz et al. [38] review reference values of elements in human hair. They screened 52 studies to leave only five as eligible for their review. However, since the reported reference values still varied, they concluded it is necessary to further elaborate the standard procedures to validate hair mineral analysis.

To surmount these large intra-individual variations, it is crucial to understand the statistical nature of the variations in hair mineral measurements. We must first define the true value of hair mineral measurements for a subject, and then define the intra-individual and inter-individual variations. Statistical analysis using a simple random model (S1 Text) revealed that the intra-individual variations are mostly random, rather than caused by any temporal accident or systematic biases. More interestingly, the inter-individual variations are approximated by ordinary two-parameter distributions. In other words, those variations are statistically tractable, or controllable. The ratio of the inter-individual variance to the intra-individual variance will be particularly useful for designing studies taking into account the variability of hair mineral measurements.

Intractable diseases refer mainly to rare diseases with unidentified preventions or treatments. Early and accurate diagnosis is critically important for those patients; however, the current situation for specific rare disease identification is not optimistic [39]. Tang and Makuuti [39] suggest that biomedical research on rare diseases will provide insights into underlying mechanisms, which may ultimately reveal possible avenues to therapeutics. Since a cohort study with the endpoint being the onset of a rare disease is difficult to perform, a case/control study is the only possible study design for finding risk factors for those diseases. The most difficult part in implementing the case/control study is the need for accurate information on the subject's environment or health condition a few months to a few years before the onset. In that respect, hair minerals are ideal biomarkers for case/control studies, since hair strands are usually easy to obtain, and hair minerals are measured using PIXE or ICP methods. Thus, it is expected that hair minerals will contribute to providing useful information for studies on underlying mechanisms of rare diseases, if a standard method for the analysis of hair mineral measurements is established.

If a study is designed to confirm hair minerals to be significant biomarkers for intractable diseases, then information on the nature of measurement errors and inter-individual distributions of hair minerals would be of critical importance [40, 41].

Dataset used in this study is presented in S1, S2 and S3 Dataset.