Likelihood Ratio Test process for Quantitative Trait Loci detection.

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Preprint submitted on 1 Oct 2009

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Likelihood Ratio Test process for Quantitative Trait Loci detection.

Charles-Elie Rabier, Jean-Marc Azaïs, Céline Delmas

To cite this version:

Charles-Elie Rabier, Jean-Marc Azaïs, Céline Delmas. Likelihood Ratio Test process for Quantitative

Trait Loci detection.. 2009. �hal-00421215�

Likelihood Ratio Test process

for Quantitative Trait Loci detection

Charles-Elie Rabier

Institut de Mathématiques de Toulouse, Toulouse, France.

INRA UR631, Auzeville, France.

Jean-Marc Azaïs

Institut de Mathématiques de Toulouse, Toulouse, France.

Céline Delmas

INRA UR631, Auzeville, France.

Summary. We consider the likelihood ratio test (LRT) process related to the test of the ab- sence of QTL on the interval[0, T]representing a chromosome (a QTL denotes a quantitative trait locus, i.e. a gene with quantitative effect on a trait). We give the asymptotic distribution of this LRT process under the null hypothesis that there is no QTL on[0, T]and under the general alternative that there existmQTL on[0, T]. We propose to estimate the number of QTL, their positions and their effects by penalized likelihood. Our results are extended to the case where individuals are structured into families.

Keywords: Gaussian process, Likelihood Ratio Test, Mixture models, Nuisance parameters present only under the alternative, QTL detection,χ²process.

1. Introduction

n

Y_j, j= 1, ..., n

! "

# $ A×(A×B) AB ^{% &}

'!(!(

[0, T] [0, T]⁾

$ t=t Y

* $ +

p(t)f_(µ+q,σ)(.) +{1−p(t)}f_(µ−q,σ)(.) ^!

f_(µ,σ)(.) ^, µ σ² (µ, q, σ)

$ - t ∈ [0, T] ^$

. “q = 0” ^! n Y₁, ..., Y_n

Λ_n(t)

/ 0 p_j(t) p(t) ¹

{Λ_n(t), t∈[0, T]} ^{2$ 3 $}

* .

*

4 5 !

6 !(7( . [0, T]

8&$ 9

!((7 6 * % 8

&$ -%:9 - "55"-%:

"55( / .

.: !((;

<!(==<!(7= *

>,

4

4 ) ? .

[0, T] [0, T]q= 0

t [0, T]

* !t=t

.

*m [0, T]t₁,· · · , t_m q¹,· · · , q^m

* M = 2^{m +} M

α=1

p_αf_(mα,σ)(.)

p_αm_α^{$ $} µmt₁t_m q¹q^{m &.}

, $

$ % $

6 . 2

3 .

*

# $ .

.:!((;.:!((@ 2

3 % ?

*

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9 "55( 8 $

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?

.: *

% $

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AA!((7 4

2. Model

[0, T] K ^$

* t₁ = 0 < t₂ < ... < t_K =T ^$

2 3t X(t) ^'!(!(

+ N(t) ^B X(t) ¹₂ (δ₁+δ₋₁)

X(t) = (−1)^N(t)X(t₁) ^'!(!(C r: [0, T]²−→

0, ¹₂

+

r(t, t) =P(X(t)X(t) =−1) =P(|N(t)−N(t)| ) = 1

2 (1−e⁻²|t−t|)

¯

r(t, t) 1−r(t, t)

Y X(t)t ∈[t1, t_K]

D+

Y_j =µ + X(t)q + σε

ε^, q

6 2 3 $

t₁, t₂, ..., t_K X_j(t) X(t) ¹

/ (Y_j, X_j(t1), ..., X_j(t_K))

% t ^$ 3. Only 2 genetic markers

$(K= 2) 0T ⁺ 0 =t₁<

t2=T ^{-* $} t ∈[t1, t2]

t ∈[t1, t2] ⁴ p(t) ^Dp(t) =P

X(t) = 1X(t1), X(t2)

9 *X(t1) =X(t2) = 1⁶⁺

P

X(t) = 1X(t1) = 1, X(t2) = 1

= (1/2)P{N(t)−N(t1)}P{N(t2)−N(t)} (1/2)P{N(t₂)−N(t₁)}

/ ∀t∈]t1, t2[⁺

p(t) =Q¹_t^,11_X(t1)=11_X(t2)=1 + Q¹_t^,−11_X(t1)=11_X(t2)=−1

+Q⁻_t^1,11_X(t1)=−11_X(t2)=1 + Q⁻_t^1,−11_X(t1)=−11_X(t2)=−1 ^"

+

Q¹_t^,1= ¯r(t₁, t) ¯r(t, t₂)

¯

r(t₁, t₂) , Q¹_t^,−1= r(t¯ ₁, t)r(t, t₂) r(t₁, t₂) Q⁻_t^1,1= r(t₁, t) ¯r(t, t₂)

r(t1, t2) , Q⁻_t^1,−1= r(t₁, t)r(t, t₂)

¯ r(t1, t2)

$+

Q⁻_t^1,−1= 1−Q¹_t^,1 Q⁻_t^1,1= 1−Q¹_t^,−1

6 p(t₁) = 1_X(t1)=1 p(t₂) = 1_X(t2)=1 ^/ p(t) t₁

t₂

θ= (q, µ, σ) t^D*θ₀= (0, µ, σ)

H₀ ^$ (Y, X(t1), X(t2))

λ⊗N ⊗N λ N N

∀t∈[t₁, t₂]⁺

L(θ, t) =

p(t)f_(µ+q,σ)(y) +{1−p(t)}f_(µ−q,σ)(y)

g(t) ⁰

g(t) =1 2

r(t¯ ₁, t₂) 1_X(t1)=11_X(t2)=1 + r(t₁, t₂) 1_X(t1)=11_X(t2)=−1

+1 2

r(t₁, t₂) 1_X(t1)=−11_X(t2)=1 + ¯r(t₁, t₂) 1_X(t1)=−11_X(t2)=−1

$ L_n(θ, t) n n

θˆ= (ˆq, µ,ˆ σ)ˆ ^{*$ )E} θ

&H0 [t1, t2] ⁶ H1

tt∈[t1, t2]

D H1 ^{/ +}

H_at :² t q=a/√

na∈R ³

q q=a/√

n ^9!(7FC

3.1. Results

S_n(.)⇒Z(.) Λ_n(.)^F.d.→ {Z(.)}²

H₀ H_at

• S_n(.) n

• ⇒ ^F.d.→

• Z(.) ∀(t, t)∈[t₁, t₂]²

Γ(t, t) = 4E{p(t)p(t)} −1

E

{2p(t)−1}² E

{2p(t)−1}²

∀(t, t)∈[t1, t2]²

• H₀m(t) = 0

• H_at

m_t(t) = aE[X(t){2p(t)−1}] σ

E

{2p(t)−1}²

Z(.)Z(.)

∀t∈[t₁, t₂]

Z(t) ={α(t)Z(t1) + β(t)Z(t2)}/

E

{2p(t)−1}²

∀t∈]t1, t2[ α(t) =Q¹_t^,1+Q¹_t^,−1−1 β(t) =Q¹_t^,1−Q¹_t^,−1α(t1) = 1β(t1) = 0 α(t2) = 0β(t2) = 1{Z(t1), Z(t2)}=e^−2t2

! ∀(t, t)∈[t₁, t₂]²

m_t(t) ={α(t)m_t(t₁) + β(t)m_t(t₂)}/

E

{2p(t)−1}²

"# E

{2p(t)−1}^{2 $%&'(}

) % E{p(t)p(t)} ^{* %} E[X(t){2p(t)−1}] ^$%+'

( ) %

D

% "55=

?! Γ(t, t) m_t(t) T

0.2^{) $}

9 -%:"55F-%:"55( t Γ(t, t)

Y ^{2 3} X(t1) X(t2) ^-

) 9

Fig. 1.Mean function and Covariance function (a= 4,σ= 1,T= 0.2M)

$ .

∀t∈[t1, t₂]⁺

Λ_n(t) ={W_n(t)}²+o_Pθ

0(1) ={S_n(t)}²+o_Pθ

0(1)

W_n(t)S_n(t) n

/ =! o_Pθ

0(1)

% H₀

- ( / =!+

W_n(t) =√ nqˆ

E

{2p(t)−1}² /σ , S_n(t) =ⁿ

j=1

(y_j−µ) (2p_j(t)−1)

√n σ

E

{2p(t)−1}²

;

# σ σˆ ^/$C

µ µˆ ^B σ

ˆσ ^{/$C # $}

D ;

! ' !

!

- $ +

S_n(t) ={α(t)S_n(t₁) + β(t)S_n(t₂)}/

E

{2p(t)−1}^{2 @}

9

S_n⁰(t₁), S_n⁰(t₂)

=e^−2t2 S_n⁰(.) H₀

4 +

Λ_n(t) ={α(t)S_n(t₁) + β(t)S_n(t₂)}²/E

{2p(t)−1}² + o_Pθ

0(1)

={α(t)W_n(t₁) + β(t)W_n(t₂)}²/E

{2p(t)−1}² + o_Pθ

0(1)

6 !/ =!o_Pθ

0(1)

% H_at ^. t

$

$

# $

Λ_n(.)

E) )EC

$

3.2. Remarks

* S_n(.)Λ_n(.) V_n(.)

S_n(.)⁺

•

• ^%

) +

V_n(t) =

t2−t

t2 S_n(t1) + t

t2 S_n(t2)

/

τ(t) ^F

τ(t) =V t₂−t

t2 S_n⁰(t₁) + t

t2 S_n⁰(t₂)

=

t₂−t t2

2

+ 2 t(t₂−t)

(t2)² e^−2t2 + t

t2

2

4 τ(t)= 0 ∀t ∈[t₁, t₂] V_n(.) ^,

H₀ ⁹

S_n⁰(t₁), S_n⁰(t₂)

=e^−2t2

/ V_n(.)⁺

- !! / =!

V_n²(.) ^$

* +

p(t) = 1_X(t1)=11_X(t2)=1 + t₂−t

t₂ 1_X(t1)=11_X(t2)=−1 + t

t₂ 1_X(t1)=−11_X(t2)=1 ⁼

$ * D

" /

* $

V_n²(.) ^% H₀ ^{.: !((@+}

*

n¯r(t₁, t₂)/2 nr(t₁, t₂)/2 nr(t₁, t₂)/2 n¯r(t₁, t₂)/2 _n

j=11_Xj(t1)=11_Xj(t2)=1n

j=11_Xj(t1)=11_Xj(t2)=−1n

j=11_Xj(t1)=−11_{Xj(t2)=1 n}

j=11_Xj(t1)=−11_Xj(t2)=−1

6 ! / =! H_at V_n(.)

H₀ m˜_t(t)

m˜_t(t) ⁺

˜ m_t(t) =

t₂−t

t₂ m_t(t₁) + t

t₂ m_t(t₂)

/ τ(t)

V_n(.) ^{D 9}

S_n⁰(t1), S_n⁰(t2)

=e^{−2t2 4}

S⁰_n(t1)S_n⁰(t2)τ(.) ⁴

V_n²(.)^{% .:}

!((@ S_n⁰(t₁)S_n⁰(t₂) E

{2p(t)−1}² = 0

p(t)^{D =}

4. Several markers : the “Interval Mapping‘’ of Lander and Botstein (1989)

4 K ^$0 = t1 < t2 < ... < t_K =T

t t t ^$

$ ? t ∈[t1, t_K]\Tk

Tk={t1, ..., t_K}^Dt t^r+

t=sup{t_k ∈Tk:t_k< t} , t^r=inf{t_k∈Tk :t < t_k}

4 t ^2)$3(t, t^r)

%

E

{2p(t)−1}² , E{p(t)p(t)} , E[X(t){2p(t)−1}] , α(t) , β(t)

,Z(.) Z(t) =

α(t)Z(t) + β(t)Z(t^r) /

E

{2p(t)−1}²

∀k∀k{Z(t_k), Z(t_k)}=e^{−2|tk−tk|}

! m_t(t) m_t(t) =

α(t)m_t(t) + β(t)m_t(t^r) /

E

{2p(t)−1}²

* -

# ∀k ∀kΓ(t_k, t_k) =e^{−2|tk−tk| 4 8&$}

6 !(7(9 !((7

6 / 0!+

∀k W_n(t_k) =√

nq/σˆ , S_n(t_k) = n j=1

(y_j−µ) (2 1_Xj(tk)=1−1) σ √

n

Λ_n(t) =

α(t)S_n(t) + β(t)S_n(t^r)2

/E

{2p(t)−1}² + o_Pθ

0(1) ⁷

=

α(t)W_n(t) + β(t)W_n(t^r)2

/E

{2p(t)−1}² + o_Pθ

0(1)

# ∀k∀k⁹

S_n⁰(t_k), S_n⁰(t_k)

=e^{−2|tk−tk|}

6 *7"o_Pθ

0(1) ^%

H_at

4.1. Remarks

V_n(.)^{/ 0"%}

$ % H₀

.:!((; * 70

8

* X(t) ^$

/

8&$ $

$ M_n(.) ^-