Chi-square processes for gene mapping in a population with family structure

Abstract

Detecting a quantitative trait locus, so-called QTL (a gene influencing a quantitative trait which is able to be measured), on a given chromosome is a major problem in Genetics. We study a population structured in families and we assume that the QTL location is the same for all the families. We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL on the interval [0, T] representing a chromosome. We give the asymptotic distribution of the LRT process under the null hypothesis that there is no QTL in any families and under local alternative with a QTL at $t^{\star }\in [0, T]$ in at least one family. We show that the LRT is asymptotically the supremum of the sum of the square of independent interpolated Gaussian processes. The number of processes corresponds to the number of families. We propose several new methods to compute critical values for QTL detection. Since all these methods rely on asymptotic results, the validity of the asymptotic assumption is checked using simulated data. Finally we show how to optimize the QTL detecting process.

Appendices

Appendix 1: Proof of Theorem 1

1.1 Preliminaries

Let t belong to the interval $[t_1,t_2]$ and let recall Lemma 2.3 of Azaïs et al. (2014).

Lemma 1

The conditional expectation x(t) of X(t) is linear in $X(t_{1}),X(t_{2})$:

$$\begin{aligned} x(t) =\alpha (t) X(t_1) + \beta (t) X(t_2) \end{aligned}$$

with $\alpha (t)= Q^{1,1}_t - Q^{-1,1}_t$ and $\beta (t)= Q^{1,1}_t - Q^{1,-1}_t$.

Then, we have the following relationship

$$\begin{aligned} \mathbb {V}\left\{ x^{2}(t)\right\} =\mathbb {E}\left\{ x^{2}(t)\right\} = \alpha ^{2}(t)+\beta ^{2}(t)+2\alpha (t)\beta (t)\rho (t_1,t_2). \end{aligned}$$

Since the model is regular, we can apply Theorem 5.39 of Van der Vaart (1998). As a result, according to formulae (4) and (5), we have

$$\begin{aligned} \Lambda _{n}(t)&=\sum _{i=1}^{I} \left[ \sum _{j=1}^{n} \frac{(Y_{j}-\mu _{i})x_{j}(t)}{\sigma \sqrt{n\pi _{i}\mathbb {E}\left\{ x^{2}(t)\right\} }}1_{C_{j}=i}\right] ^{2}+o_{P_{\theta _{0}}}(1) \end{aligned}$$

(9)

where $o_{P_{\theta _{0}}}(1)$ denotes a sequence of random vectors that converges to zero in probability under $H_0$.

Let $S_{n}(.,i)$ be the following process, for n observations:

$$\begin{aligned} S_{n}(t,i)= \sum _{j=1}^{n} \frac{(Y_{j}-\mu _{i})x_{j}(t)}{\sigma \sqrt{n\pi _{i}\mathbb {E}\left\{ x^{2}(t)\right\} }}1_{C_{j}=i}. \end{aligned}$$

(10)

According to Lemma 1,

$$\begin{aligned} S_{n}(t,i)&=\left\{ \alpha (t)S_{n}(t_{1},i)+\beta (t)S_{n}(t_{2},i)\right\} \Big /\sqrt{\mathbb {E}\left\{ x^{2}(t)\right\} }. \end{aligned}$$

We will call $Z^{i}(.)$ the limiting process of $S_{n}(.,i)$.

1.2 Study under $H_{0}$

Without loss of generality, let us assume $n=1$ and let us consider the process S(., i) defined in the following way:

$$\begin{aligned} S(t,i)&=\frac{(Y-\mu _{i})x(t)}{\sigma \sqrt{\pi _{i}\mathbb {E}\left\{ x^{2}(t)\right\} }}1_{C=i}= \frac{Y-\mu _{i}}{\sigma \sqrt{\pi _{i}}}1_{C=i}h(t). \end{aligned}$$

where $h(t)=x(t)/\sqrt{\mathbb {E}\left\{ x^{2}(t)\right\} }$.

h(.) is a random process, independent of Y and C. It is easy to see that

$$\begin{aligned} \mathbb {E}\left\{ S(t,i)\right\} =0,\mathbb {V}\left\{ S(t,i)\right\} =\mathbb {E}\left\{ h(t)\right\} ^{2}=1\text {.} \end{aligned}$$

Besides,

$$\begin{aligned} \mathrm {Cov}\left\{ S(t_{1},i),S(t_{2},i)\right\} =\mathbb {E}\left\{ h(t_{1})h(t_{2})\right\} = \rho (t_1,t_2)\text {.} \end{aligned}$$

So, we have

$$\begin{aligned} Z^{i}(t)&=\left\{ \alpha (t)Z^{i}(t_{1})+\beta (t)Z^{i}(t_{2})\right\} \Big /\sqrt{\mathbb {E}\left\{ x^{2}(t)\right\} }, \end{aligned}$$

$\mathbb {E}\left\{ Z^{i}(t)\right\} =0$, $\mathbb {V}\left\{ Z^{i}(t)\right\} =1$ and $\mathrm {Cov}\left\{ Z^{i}(t_{1}),Z^{i}(t_{2})\right\} = \rho (t_1,t_2)$.

A direct application of central limit theorem implies that $Z(t_1)$ and $Z(t_2)$ have a limit distribution which is a Gaussian distribution. According to formula (9), we have $\Lambda _{n}(t)=\sum _{i=1}^{I}S_{n}^{2}(t,i)+o_{P_{\theta _{0}}}(1)$. As a result, $\Lambda _{n}(.)\mathop {\rightarrow }\limits ^{F.d.}\sum _{i=1}^{I}\left\{ Z^{i}(.)\right\} ^{2}$.

Study under $H_{\lambda t^{\star }}$

In this part, we set

$$\begin{aligned} Y=\mu _{i} +\frac{\lambda _{i}}{\sqrt{n}}X(t^{\star })+\sigma \varepsilon ,\text {if}C=i, \end{aligned}$$

(11)

where $\varepsilon$ is a standard normal random variable. Recall that $t^{\star }$ denotes the QTL location.

According to formula (9), we have

$$\begin{aligned} \Lambda _{n}(t) = \sum _{i=1}^{I}S_{n}^{2}(t,i)+o_{P_{\theta _{0}}}(1). \end{aligned}$$

(12)

Recall that under $H_{\lambda t^{\star }}$, if there is a QTL within family i (i.e. $\lambda _{i}\ne 0$), the density of $Y\big |X(t_1),X(t_2),C$ verifies

$$\begin{aligned} p(t^{\star })f_{\left( \mu _{i}+q_{i},\sigma \right) }(Y)+\left\{ 1-p(t^{\star })\right\} f_{\left( \mu _{i}-q_{i},\sigma \right) }(Y) ,\text {if}C=i. \end{aligned}$$

The model with $t^*$ fixed is differentiable in quadratic mean, this implies that the alternative defines a contiguous sequence of alternatives. By Le Cam’s first lemma, relation (12) remains true under the alternative. As a result, $\Lambda _{n}(.)\mathop {\rightarrow }\limits ^{F.d.}\sum _{i=1}^{I}\left\{ Z^{i}(.)\right\} ^{2}$.

Calculations of the mean function of $Z^{i}(.)$, so-called $m_{t^{\star }}^{i}(t)$, can be done using the process $S_{n}(.,i)$. According to formula (16) and (11), we have

$$\begin{aligned} S_{n}(t,i)&=\frac{1}{\sqrt{n\pi _{i}}}\sum _{j=1}^{n}\varepsilon _{j}1_{C_{j}=i}h_{j}(t)+\sum _{j=1}^{n}\frac{\lambda _{i}}{n\sigma \sqrt{\pi _{i}}}1_{C_{j}=i}X_{j}(t^{\star })h_{j}(t) \nonumber \\&= S_{n}^{0}(t,i)+\sum _{j=1}^{n}\frac{\lambda _{i}}{n\sigma \sqrt{\pi _{i}}}1_{C_{j}=i}X_{j}(t^{\star })h_{j}(t) \end{aligned}$$

(13)

where $S_{n}^{0}(.,i)$ is the process obtained under $H_{0}$.

Recall that $h_{j}(.)$ is the equivalent of the process h(.) for the individual j. According to the law of large number:

$$\begin{aligned} \frac{1}{n}\sum _{j=1}^{n}X_{j}(t^{\star })h_{j}(t)1_{C_{j}=i}\rightarrow \pi _{i}\mathbb {E}\left\{ X(t^{\star })h(t)\right\} . \end{aligned}$$

(14)

Besides, we have $\mathbb {E}\left\{ X(t^{\star })h(t_{1})\right\} =\rho (t_1,t^{\star })$ and $\mathbb {E}\left\{ X(t^{\star })h(t_{2})\right\} =\rho (t^{\star },t_2)$.

As a result,

$$\begin{aligned} m_{t^{\star }}^{i}(t_{1})=\lambda _{i}\sqrt{\pi _{i}}\rho (t_1,t^{\star })/\sigma \text {and}m_{t^{\star }}^{i}(t_{2})=\lambda _{i}\sqrt{\pi _{i}}\rho (t^{\star },t_2)/\sigma . \end{aligned}$$

Due to the interpolation, we have

$$\begin{aligned} m_{t^{\star }}^{i}(t)= \left\{ \alpha (t)m_{t^{\star }}^{i}(t_{1}) +\beta (t)m_{t^{\star }}^{i}(t_{2})\right\} \Big / \sqrt{\mathbb {E}\left\{ x^{2}(t)\right\} }. \end{aligned}$$

Study of the supremum of the LRT process

Since the model with t fixed is regular, we have the relationship (cf. section “Study under $H_0$”)

$$\begin{aligned} \Lambda _{n}(t)=\sum _{i=1}^{I}S_{n}^{2}(t,i)+o_{P_{\theta _{0}}}(1) \end{aligned}$$

under the null hypothesis. Our goal is now to prove that the rest above is uniform in t.

Let us consider now t as an extra parameter. Let $t^*,\theta ^*$ be the true parameter that will be assumed to belong to $H_0$. Note that $t^*$ makes no sense for $\theta$ belonging to $H_{0}$. It is easy to check that at $H_0$ the Fisher information relative to t is zero so that the model is not regular.

It can be proved that Assumptions 1, 2 and 3 of Azaïs et al. (2009) holds. So, we can apply Theorem 1 of Azaïs et al. (2009) and we have

$$\begin{aligned} \sup _{(t,\theta ) } l^{n}_t(\theta ) - l^{n}_{t^*}(\theta ^*) = \sup _{d \in \mathcal { D}} \left[ \left\{ \frac{1}{\sqrt{n}} \sum _{j=1}^n d(X_j) \right\} ^{2} 1_{d(X_j) \ge 0 }\right] + o_{P} (1) \end{aligned}$$

(15)

where the observation $X_j$ stands for $Y_j,X_j(t_1),X_j(t_2),C_j$ and where $\mathcal {D}$ is the set of scores defined in Azaïs et al. (2009), see also Gassiat (2002). A similar result is true under $H_0$ with a set $\mathcal {D}_0$. Let us precise the sets of scores $\mathcal {D}$ and $\mathcal {D}_0$. This sets are defined at the sets of scores of one parameter families that converge to the true model $p_{t^*,\theta ^*}$ and that are differentiable in quadratic mean.

It is easy to see that

$$\begin{aligned} \mathcal {D} = \left\{ \frac{\langle U, l'_t( \theta ^*) \rangle }{\sqrt{\mathbb {V}(\langle U, l'_t( \theta ^*) \rangle )}} , U \in \mathbb {R}^{2I+1}, t \in [t_1,t_2]\right\} \end{aligned}$$

where $l'$ is the gradient with respect to $\theta$. In the same manner

$$\begin{aligned} \mathcal {D}_0 = \left\{ \frac{\langle U, l'_t( \theta ^*) \rangle }{\sqrt{\mathbb {V}(\langle U, l'_t( \theta ^*) \rangle )}}, U \in \mathbb {R}^{I+1} \right\} , \end{aligned}$$

where now the gradient is taken with respect to $\mu _1$, ..., $\mu _{I}$ and $\sigma$ only. Obviously, this gradient does not depend on t.

Using the transform $U \rightarrow -U$ in the expressions of the sets of score, we see that the indicator function can be removed in formula (15). Then, since the Fisher information matrix is diagonal (see formula (5)) , it is easy to see that

$$\begin{aligned}&\sup _{d \in \mathcal { D}} \left[ \left\{ \frac{1}{\sqrt{n}} \sum _{j=1}^n d(X_j) \right\} ^2 \right] - \sup _{d \in \mathcal { D}_0} \left[ \left\{ \frac{1}{\sqrt{n}} \sum _{j=1}^n d(X_j) \right\} ^2 \right] \\ \quad&= \sup _{t \in [t_1,t_2] }\left( \left[ \frac{1}{\sqrt{n}} \sum _{j=1}^n \frac{\frac{\partial l_t}{\partial q_{1} } (X_j)\mid _{\theta _{0}} }{ \sqrt{\mathbb {V}\left\{ \frac{\partial l_t}{\partial q_{1} } (X_j)\mid _{\theta _{0}}\right\} }} \right] ^2 + \cdots + \left[ \frac{1}{\sqrt{n}} \sum _{j=1}^n \frac{\frac{\partial l_t}{\partial q_{I} } (X_j)\mid _{\theta _{0}} }{ \sqrt{\mathbb {V}\left\{ \frac{\partial l_t}{\partial q_{I} } (X_j)\mid _{\theta _{0}}\right\} }} \right] ^2 \right) \\ \quad&=\sup _{t \in [t_1,t_2] }\left( \sum _{i=1}^{I} \left[ \frac{1}{\sqrt{n}} \sum _{j=1}^n \frac{\frac{\partial l_t}{\partial q_{i} } (X_j)\mid _{\theta _{0}} }{ \sqrt{\mathbb {V}\left\{ \frac{\partial l_t}{\partial q_{i} } (X_j)\mid _{\theta _{0}}\right\} }} \right] ^2 \right) . \end{aligned}$$

This is exactly the desired result. Since the model with $t^*$ fixed is differentiable in quadratic mean, the alternative defines a contiguous sequence of alternatives. By Le Cam’s first lemma, relation (15) remains true under the alternative.

Appendix 2: Proof of Theorem 2

The proof of the theorem is the same as the proof of Theorem 1 as soon as we can confine our attention to the interval $(t^{\ell },t^{r})$ when considering a unique instant t and to the intervals $(t^{\ell },t^{r})(t'^{\ell },t'^{r})$ when considering two instants t and $t'$. For that we need to prove that

$$\begin{aligned} x(t) = \mathbb {E}\left\{ X(t)| X(t_1),\ldots , X(t_K) \right\} = \mathbb {E}\left\{ X(t)| X(t^\ell ), X(t^r) \right\} \end{aligned}$$

which is a direct consequence of the independance of the increments of Poisson process.

Proof of results introduced in Sect. 8

Recall that $\mathbb {T}_{K^{i}}^{i}=\{t_{1}^{i},\ldots ,t_{K^{i}}^{i}\}$. Let $t\in [t^{i}_{1},t^{i}_{K^{i}}]\backslash \mathbb {T}_{K^{i}}^{i}$. Let define $x^{i}(t)$ the quantity such as $x^{i}(t) = \mathbb {E}\left\{ X(t)\mid X(t^{\ell ,i}),X(t^{r,i}),C=i\right\}$. Besides, $Q^{1,1}_{t,i}$, $Q^{1,-1}_{t,i}$, $Q^{-1,1}_{t,i}$ and $Q^{-1,-1}_{t,i}$ are the following quantities:

$$\begin{aligned}&Q^{1,1}_{t,i}= \frac{\bar{r}(t^{\ell ,i},t)\bar{r}(t,t^{r,i})}{\bar{r}(t^{\ell ,i},t^{r,i})} , Q^{1,-1}_{t,i}= \frac{\bar{r}(t^{\ell ,i},t)r(t,t^{r,i})}{r(t^{\ell ,i},t^{r,i})} \\&Q^{-1,1}_{t,i}=\frac{r(t^{\ell ,i},t)\bar{r}(t,t^{r,i})}{r(t^{\ell ,i},t^{r,i})} , Q^{-1,-1}_{t,i}=\frac{r(t^{\ell ,i},t)r(t,t^{r,i})}{ \bar{r}(t^{\ell ,i},t^{r,i})}. \end{aligned}$$

Lemma 2

We have the following relationship:

$$\begin{aligned} x^{i}(t) =\alpha _{i}(t) X(t^{\ell ,i}) + \beta _{i}(t) X(t^{r,i}) \end{aligned}$$

with

$\alpha _{i}(t)= Q^{1,1}_{t,i} - Q^{-1,1}_{t,i}$

,

$\beta _{i}(t)= Q^{1,1}_{t,i} - Q^{1,-1}_{t,i}$.

Let $S_{n}(.,i)$ be the following process, for n observations:

$$\begin{aligned} S_{n}(t,i)= \sum _{j=1}^{n} \frac{(Y_{j}-\mu _{i})x_{j}^{i}(t)}{\sigma \sqrt{n\pi _{i}\mathbb {E}\left\{ \left( x^{i}(t)\right) ^{2}\right\} }}1_{C_{j}=i}. \end{aligned}$$

(16)

According to Lemma 2

$$\begin{aligned} S_{n}(t,i)&=\left\{ \alpha _{i}(t)S_{n}(t^{\ell ,i},i)+\beta _{i}(t)S_{n}(t^{r,i},i)\right\} \Big /\sqrt{\mathbb {E}\left\{ \left( x^{i}(t)\right) ^{2}\right\} }. \end{aligned}$$

We will call $Z^{i}(.)$ the limiting process of $S_{n}(.,i)$.

Let us consider now the case where the first informative marker does not lie at the beginning of the chromosome ($0<t_{1}^{i}$). Let $t\in [0,t_{1}^{i}[$, we have

$$\begin{aligned} S_{n}(t,i)&= \sum _{j=1}^{n} \frac{(Y_{j}-\mu _{i})\tilde{x}_{j}^{i}(t)}{\sigma \sqrt{n\pi _{i}\mathbb {E}\left\{ \left( \tilde{x}^{i}(t)\right) ^{2}\right\} }}1_{C_{j}=i} \end{aligned}$$

where $\tilde{x}^{i}(t) = 2 \mathbb {P}\left\{ X(t)=1\mid X(t^{i}_{1}),C=i\right\} - 1$. Recall that in the classical situation, when t have two flanking markers: $x^{i}(t) = 2 \mathbb {P}\left\{ X(t)=1\mid X(t^{\ell ,i}),X(t^{r,i}),C=i\right\} - 1$. In our case,

$$\begin{aligned} \tilde{x}^{i}(t)&= 2 \left\{ \bar{r}(t,t^{i}_{1}) 1_{X(t^{i}_{1})=1} + r(t,t^{i}_{1}) 1_{X(t^{i}_{1})=-1}\right\} - 1\\&= 2 \left\{ \rho (t,t^{i}_{1}) + r(t,t^{i}_{1})\right\} -1 =\rho (t,t^{i}_{1}) \end{aligned}$$

Besides, we have

$$\begin{aligned} \sqrt{\mathbb {E}\left\{ \left( \tilde{x}^{i}(t)\right) ^{2}\right\} }= \rho (t,t^{i}_{1}). \end{aligned}$$

As a result,

$$\begin{aligned} \forall t\in [0,t_{1}^{i}[S_{n}(t,i)&= S_{n}(t^{i}_{1},i). \end{aligned}$$

By symmetry, when $t^{i}_{K^{i}}<T$, we have

$$\begin{aligned} \forall t\in ]t^{i}_{K^{i}},T] S_{n}(t^{i}_{K^{i}},i)=S_{n}(t,i). \end{aligned}$$

To conclude, we just have to use same kind of arguments as in formula (9) in order to prove that the LRT process converges asymptotically to the process $\sum _{i=1}^{I}\left\{ Z^{i}(.)\right\} ^{2}$.