On Intra-individual Variations in Hair Minerals in Relation to Epidemiological Risk Assessment of Atopic Dermatitis

Abstract

We have conducted a cohort study of 834-mother-infant pairs to determine the association between hair minerals at one month and the onset of atopic dermatitis (AD) at ten months after birth. Thirty-two minerals were measured by PIXE (particle induced X-ray emission) method. (Yamada et al., J. Trace Elem. Med. Bio. 27, 126-131, 2013, [11]) described a logistic model with explanatory variables Selenium (Se), Strontium (Sr) and a family history of AD whose performance in predicting the risk of AD was far better than that of any similar study. However, as discussed in (Saunders et al., Biometrie und Medizinische Informatik Greifswalder Seminarberichte, 18, 127-139, 2011, [9]), intra-individual variations in those minerals were large and could have degraded the regression coefficients of Sr and Se in the logistic model. Therefore, (Yamada et al., Biometrie und Medizinische Informatik Greifswalder Seminarberichte, 2013, [12]) examined the intra-individual variations of Sr levels in the mothers (Mother-Sr) assuming log-normality and obtained a regression coefficient of Mother-Sr corrected for the variations. This paper addresses Sr levels in the babies (Baby-Sr) which are not distributed as log-normal and require more sophisticated modeling of the variations. Here we elaborate on the “true-equivalent sample” (TES) method, developed in (Yamada et al., Biometrie und Medizinische Informatik Greifswalder Seminarberichte, 2013, [12]) and determine the distribution of Baby-Sr. The revised TES method presented here will be useful for determining the distribution type for minerals whose distributions are zero-inflated, thereby obtaining a risk estimate corrected for the intra-individual variations. This will allow hair mineral data to play a more important role in medical and epidemiological research.

Appendix

1.1 Transportation of Measurement Error Variance of Sr

Let $X_{ki}$ denote an observed value of Sr of $k$th experiment for $i$th subject. We assume the following random effect model:

$$ X_{ki}=\tau _{k}+\rho _{k}(Z_{i}+\varepsilon _{ki})=\tau _{k}+\rho _{k} Z_{i}+\rho _{k} \varepsilon _{ki}, \>\quad k=1,2\, ; \quad i=1,\cdots , n $$

• $Z_{i}$ is the true value of $i$th subject,

• $\tau _{k}$, $\rho _{k}$ represent calibration effects for the $k$th experiment,

• $\varepsilon$’s are independent random variables with $E(\varepsilon _{ki})=0$ and $V(\varepsilon _{ki})=\sigma _{e}^2$,

• where $\varepsilon$’s represent intra-individual variations

The ratio of the variance of $X$ explained by the variation of $Z$ to the variance of $X$ is defined as a reliability index of $X$ and usually denoted as $\lambda$ [3, 10].

As for the $k$th experiment, since

$$ \textit{Var}(X_{ki} \mid k)=\rho _{k}^2 \textit{Var}(Z)+\rho _{k}^2 \sigma _{e}^2, $$

the variance of $X$ due to $Z$ is $\rho _{k}^2 Var(Z)$ and that due to $\varepsilon$, or the measurement error variance, $\rho _{k}^2 \sigma _{e}^2$. Therefore,

$$\begin{aligned} \lambda =\rho _{k}^2 \textit{Var}(Z)/\textit{Var}(X_{k})=\textit{Var}(Z)/\{\textit{Var}(Z)+\sigma _{e}^2\} \end{aligned}$$

Thus, $\lambda$ is independent of the calibration effect $\tau _{k}$ and $\rho _{k}$. That is, $\lambda$ is a parameter intrinsic to the sample determined by the inter and intra individual variance $\textit{Var}(Z)$ and $\sigma _{e}^2$, respectively. Hereafter, for the sake of notational simplicity, $\textit{Var}(X_{ki} \mid k)$ and $E(X_{ki} \mid k)$ will be simply denoted by $\textit{Var}(X_{k})$ and $E(X_{k})$, respectively.

It is straightforward to show that

$$ E(X_{k})=\tau _{k}+\rho _{k}\overline{Z}, $$

$$ X_{ki}-E(X_{k})=\rho _{k}(Z_{i}-\overline{Z}+\varepsilon _{ki}) $$

and

$$ (\rho _2/\rho _1)^2=\textit{Var}(X_2)/\textit{Var}(X_1). $$

Define

$$ X_{1i}^\alpha = E(X_2)+(\rho _2/\rho _1)\{X_{1i}-E(X_1)\}. $$

Then,

$$\begin{aligned} X_{1i}^\alpha -X_{2i}= & {} E(X_2)+(\rho _2/\rho _1)\{X_{1i}-E(X_1)\}-X_{2i}\\= & {} (\rho _2/\rho _1)[\rho _1\{Z_{i}-\overline{Z}+\varepsilon _{1i}\}-\rho _2 (Z_{i}-\overline{Z})-\varepsilon _{2i}]\\= & {} \rho (\varepsilon _{1i}-\varepsilon _{2i}). \end{aligned}$$

Let $D^\alpha =\sum _{i}(X_{1i}^\alpha -X_{2i})^2/2m$, then

$$ E(D^\alpha )=\rho _2^2E(\sum _{i}(\varepsilon _{1i}-\varepsilon _{2i})^2/2m)=\rho _2^2\sigma _{e}^2 $$

Thus, $D^\alpha$ is an unbiased estimate of the measurement error variance of $X_{k}$.

Therefore, let

$$ X_{1i}^*=\overline{X}_2+\{\textit{Var}(X_2)/\textit{Var}(X_1)\}^{1/2}(X_{1i}-\overline{X}_1) $$

then

$$ D^*=\sum _{i}(X_{1i}^*-X_{2i})^2/2m $$

is an asymptotically unbiased estimate for $\rho _2^2\sigma _{e}^2$.