Background
Although there has been a upsurge of interest in jointly modelling longitudinal and event data during the last decade 123456789, no statistical models have been developed to characterize the shared genetic basis for these two types of traits. In biomedicine, the identification of specific genetic variants responsible for an HIV patient's time-dependent CD4 count and for the time to onset of AIDS symptoms can help to design individualized drugs to control this patient's progression to AIDS. Similarly, in studies of prostate cancer, a shared genetic basis between prostate specific antigen, repeatedly measured for patients following treatment for prostate cancer, and the time to disease recurrence can be used to make optimal treatment schedules for patients. In plants, knowledge about whether the genetic loci for reproductive behaviors, such as the time to first flower and the time to form seeds, also govern growth rates and sizes of plants helps to understand the etiology of plant's adaptation to the environment in which they are grown.
The genetic mapping of quantitative trait loci (QTL) that are responsible for longitudinal traits has long been a difficult issue because of the dynamic features of these traits. More recently, part of this difficulty has been solved by integrating the statistical analysis of longitudinal data into a QTL mapping framework, leading to a so-called functional mapping strategy 10111213141516. Statistical models for functional mapping were established on the belief that biological processes can be described by mathematical functions. One of the most significant examples for this is the use of S-shaped logistic curves to model growth trajectories. West et al. 17 indicated from fundamental principles of biophysical processes that logistic forms of growth are biologically crucial for the maintenance of optimal metabolic level and, thereby, the best use of available resources for an organism from birth to adulthood. Because of the embedment of fundamental biological principles within the modelling model, functional mapping provides a quantitative framework for testing biologically relevant hypotheses at the interplay between gene actions and development.
The concept of functional mapping can be further extended to jointly mapping a longitudinal variable and a time-to-event by incorporating statistical theories developed to characterize the relationships between longitudinal response and event processes 123456789. However, original functional mapping models for a dynamic trait reply upon explicit mathematical functions that describe the development of the trait. In practice, there are also many situations in which no appropriate curves can be used to describe a biological process. To model an arbitrary shape of curves, a different statistical model based on nonparametric theory should be formulated. Polynomial analyses that can be specified by varying orders have power to fit curves with arbitrary shapes. As shown by Kirkpatrick and Heckman 18, Legendre polynomials have several favorable properties for curve fitting which include: (1) the functions are orthogonal, (2) it is flexible to fit sparse data, (3) higher orders are estimable for high levels of curve complexity and (4) computation is fast because of good convergence.
The purpose of this article is to develop a joint statistical model for nonparametric functional mapping of longitudinal trajectories based on the Legendre polynomials, integrated with time-to-events. This joint model is constructed within the maximum likelihood context, including simultaneously modelling of the mean vector (based on nonparametric approaches) and covariance matrix (based on parametric approaches). By analyzing stem volume growth data in an example of a forest tree, we will demonstrates the implications of our joint model. Lastly, the advantages of our model in general biomedical and biological research and the areas in which the model can be further refined are discussed.
The Model
The likelihood function
Consider a mapping population of size n for which a number of molecular markers are genotyped, aimed to identify QTL for a longitudinal trait and time-to-event. Every individual of the mapping population is measured for the longitudinal trait at multiple (say T) time points (y) and a time-to-event (z). Variable z can be the time to first flower, the timing of cancer malignance, the time of mortality, or the events that happen at a time. The inference of unknown QTL genotypes for the phenotypic traits based on observed marker information (M) can be made due to co-segregation between the QTL and markers.
Suppose there are two segregating QTL for longitudinal and event traits in the mapping population, each with genotypes 2, 1 and 0. These two QTL are assumed to be linked or associated with and, therefore, can be inferred from, markers. The joint likelihood function of the two types of phenotypic data and marker information at the two underlying QTL is written as
L
(
Ω
|
y
,
z
,
M
)
=
∏
i
=
1
n
∑
j
1
=
0
2
∑
j
2
=
0
2
[
ϖ
j
1
j
2
|
i
f
j
1
j
2
(
y
i
,
z
i
;
u
j
1
j
2
,
v
j
1
j
2
,
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)
]
,
(
1
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGmbatcqGGOaakiiqacqWFPoWvcqGG8baFieqacqGF5bqEcqGGSaalcqWG6bGEcqGGSaalcqGFnbqtcqGGPaqkcqGH9aqpdaqeWbqaamaaqahabaWaaabCaeaacqGGBbWwiiGacqqFwpGDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaliabcYha8jabdMgaPbqabaGccqWGMbGzdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaGccqGGOaakcqGF5bqEdaWgaaWcbaGaemyAaKgabeaakiabcYcaSiabdQha6naaBaaaleaacqWGPbqAaeqaaOGaei4oaSJae4xDau3aaSbaaSqaaiabdQgaQnaaBaaameaacqaIXaqmaeqaaSGaemOAaO2aaSbaaWqaaiabikdaYaqabaaaleqaaOGaeiilaWIaemODay3aaSbaaSqaaiabdQgaQnaaBaaameaacqaIXaqmaeqaaSGaemOAaO2aaSbaaWqaaiabikdaYaqabaaaleqaaOGaeiilaWIaeyyeIuUaeiykaKIaeiyxa0faleaacqWGQbGAdaWgaaadbaGaeGOmaidabeaaliabg2da9iabicdaWaqaaiabikdaYaqdcqGHris5aaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqGH9aqpcqaIWaamaeaacqaIYaGma0GaeyyeIuoaaSqaaiabdMgaPjabg2da9iabigdaXaqaaiabd6gaUbqdcqGHpis1aOGaeiilaWIaaCzcaiaaxMaadaqadaqaaiabigdaXaGaayjkaiaawMcaaaaa@81DF@
where Ω is the unknown vector that defines the QTL positions, time-dependent QTL effects (uj1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWF1bqDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@335D@ and vj1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG2bGDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@3359@) and covariance matrix (Σ), ϖj1j2|i
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFwpGDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaliabcYha8jabdMgaPbqabaaaaa@369F@ is the mixture proportion expressed as the conditional probability of a joint genotype j1j2 for the longitudinal and event QTL given marker genotypes for individual i and fj1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGMbGzdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@3339@ is the (T + 1)-dimensional multivariate normal distribution function with mean vector (uj1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWF1bqDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@335D@, vj1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG2bGDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@3359@) and covariance matrix Σ.
Conditional probabilities
There are different descriptions of the conditional probability, depending on the type of the mapping population. If the mapping population is an experimental cross initiated with two contrasting parents, such as the F2 or backcross, the conditional probability is described in terms of the recombination fractions between the markers and QTL 1219. If the two QTL are bracketed by different pairs of markers, the conditional probability of joint QTL genotypes given the marker intervals can be expressed as the product of the corresponding conditional probabilities for QTL genotypes given a single marker interval. If the two QTL are located at the same marker interval, the conditional probabilities should be derived using the principle of 4-point analysis. For a natural population, the association between the QTL and markers can be described by the coefficients of linkage disequilibria 17
Modelling the mean vector
The choice of a mean function for a longitudinal trait is based on theory or past experience that suggests a certain mathematical form for the time-dependent mean. However, it would be essential to derive a general approach that can fit any kind of curves. By choosing different orders of orthogonal polynomials, the Legendre function has potential to approximate the functional relationships between trait values and times to any specified degree of precision. The Legendre polynomials are solutions to a very important differential equation, the Legendre equation,
(
1
−
x
2
)
d
2
z
d
x
2
−
2
x
d
z
d
x
+
r
(
r
+
1
)
z
=
0.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGGOaakcqaIXaqmcqGHsislcqWG4baEdaahaaWcbeqaaiabikdaYaaakiabcMcaPmaalaaabaGaemizaq2aaWbaaSqabeaacqaIYaGmaaGccqWG6bGEaeaacqWGKbazcqWG4baEdaahaaWcbeqaaiabikdaYaaaaaGccqGHsislcqaIYaGmcqWG4baEdaWcaaqaaiabdsgaKjabdQha6bqaaiabdsgaKjabdIha4baacqGHRaWkcqWGYbGCcqGGOaakcqWGYbGCcqGHRaWkcqaIXaqmcqGGPaqkcqWG6bGEcqGH9aqpcqaIWaamcqGGUaGlaaa@4F6C@
The polynomials may be denoted by Pr(x), called the Legendre polynomial of order r. The polynomials are either even or odd functions of x for even or odd orders r.
The general form of a Legendre polynomial of order k is given by the sum,
P
r
(
x
)
=
∑
k
=
0
K
(
−
1
)
k
(
2
r
−
2
k
)
!
2
r
k
!
(
r
−
k
)
!
(
r
−
2
k
)
!
x
r
−
2
k
,
(
2
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@64B5@
where K = r/2 or (r - 1)/2 whichever is an integer. This polynomial is defined over the interval [-1, 1]. From Eq. 5, we show the first few polynomials as
P
0
(
x
)
=
1
P
1
(
x
)
=
x
P
2
(
x
)
=
1
2
(
3
x
2
−
1
)
P
3
(
x
)
=
1
2
(
5
x
3
−
3
x
)
P
4
(
x
)
=
1
8
(
35
x
4
−
30
x
2
+
3
)
P
5
(
x
)
=
1
8
(
63
x
5
−
70
x
3
+
15
x
)
P
6
(
x
)
=
1
16
(
231
x
6
−
315
x
4
+
105
x
2
−
5
)
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@ABA9@
In this modelling, independent variable x is expressed as time t, which is adjusted, to rescale the measurement times to the range of the orthogonal function [-1, 1], by
t
*
=
−
1
+
2
(
t
−
t
min
)
t
max
−
t
min
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG0baDcqGGQaGkcqGH9aqpcqGHsislcqaIXaqmcqGHRaWkdaWcaaqaaiabikdaYiabcIcaOiabdsha0jabgkHiTiabdsha0naaBaaaleaacyGGTbqBcqGGPbqAcqGGUbGBaeqaaOGaeiykaKcabaGaemiDaq3aaSbaaSqaaiGbc2gaTjabcggaHjabcIha4bqabaGccqGHsislcqWG0baDdaWgaaWcbaGagiyBa0MaeiyAaKMaeiOBa4gabeaaaaGccqGGSaalaaa@4AFC@
where tmin and tmax are respectively the first and last time points.
Our aim is to model the time-dependent genotypic values for different QTL genotypes j1j2, using the orthogonal Lengedre polynomial with a particular order r. A family of such polynomials is denoted by
P
→
r
(
t
*
)
=
[
P
0
(
t
*
)
,
P
1
(
t
*
)
,
⋯
,
P
r
(
t
*
)
]
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGqbaugaWcamaaBaaaleaacqWGYbGCaeqaaOGaeiikaGIaemiDaqNaeiOkaOIaeiykaKIaeyypa0Jaei4waSLaemiuaa1aaSbaaSqaaiabicdaWaqabaGccqGGOaakcqWG0baDcqGGQaGkcqGGPaqkcqGGSaalcqWGqbaudaWgaaWcbaGaeGymaedabeaakiabcIcaOiabdsha0jabcQcaQiabcMcaPiabcYcaSiabl+UimjabcYcaSiabdcfaqnaaBaaaleaacqWGYbGCaeqaaOGaeiikaGIaemiDaqNaeiOkaOIaeiykaKIaeiyxa0faaa@4F02@
and a vector of genotypic values, which is time-independent, denoted by
w
→
j
1
j
2
=
(
w
j
1
j
2
0
,
w
j
1
j
2
1
,
⋯
,
w
j
1
j
2
r
)
′
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5323@
The time-dependent genotypic values uj1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG1bqDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@3357@(t) can be described as a linear combination of w→j1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG3bWDgaWcamaaBaaaleaacqWGQbGAdaWgaaadbaGaeGymaedabeaaliabdQgaQnaaBaaameaacqaIYaGmaeqaaaWcbeaaaaa@336D@ weighted by the family of the polynomials, i.e.,
u
j
1
j
2
(
t
)
=
P
→
r
(
t
*
)
w
→
j
1
j
2
.
(
3
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG1bqDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaGccqGGOaakcqWG0baDcqGGPaqkcqGH9aqpcuWGqbaugaWcamaaBaaaleaacqWGYbGCaeqaaOGaeiikaGIaemiDaqNaeiOkaOIaeiykaKIafm4DaCNbaSaadaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaGccqGGUaGlcaWLjaGaaCzcamaabmaabaGaeG4mamdacaGLOaGaayzkaaaaaa@49D7@
Substituting the mean vector of the likelihood (1) by the above expression (3), we will need to estimate time-invariant genotypic values for the longitudinal trait, w→j1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG3bWDgaWcamaaBaaaleaacqWGQbGAdaWgaaadbaGaeGymaedabeaaliabdQgaQnaaBaaameaacqaIYaGmaeqaaaWcbeaaaaa@336D@, and the genotypic mean for the event trait, vj1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG2bGDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@3359@.
Modelling the covariance matrix
A general form for the covariance matrix among longitudinal trajectories and development event in the likelihood (1) is expressed as
∑
=
(
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∑
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z
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4
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHris5cqGH9aqpdaqadaqaauaabeqaciaaaeaacqGHris5daWgaaWcbaGaemyEaKhabeaaaOqaaiabggHiLpaaBaaaleaacqWG5bqEcqWG6bGEaeqaaaGcbaGaeyyeIu+aaSbaaSqaaiabdQha6jabdMha5bqabaaakeaaiiGacqWFdpWCdaqhaaWcbaGaemOEaOhabaGaeGOmaidaaaaaaOGaayjkaiaawMcaaiabcYcaSiaaxMaacaWLjaWaaeWaaeaacqaI0aanaiaawIcacaGLPaaaaaa@46FB@
where Σy and σz2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGaemOEaOhabaGaeGOmaidaaaaa@3112@ are the covariance matrix and variance for the longitudinal and event traits, respectively, and Σyz = ∑′zy
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuGHris5gaqbamaaBaaaleaacqWG6bGEcqWG5bqEaeqaaaaa@3180@ is the covariance matrix between these two types of traits. The structures of Σy and Σyz can be empirically modelled on the basis of prior knowledge or results. Several approaches for parametric modelling of the covariance matrix, reviewed by Zimmerman and Nunez-Anton 20, can be utilized.
The most common approach for modelling the covariance structure is based on a variance-correlation specification, in which functions for the responses' variances and correlations are specified. In previous QTL mapping 1011121314, the covariance structure for longitudinal traits is modelled by the simplest, most parsimonious and most flexible first-order autoregressive (AR(1)) model in which there are two parameters, stationary variance (σy2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGaemyEaKhabaGaeGOmaidaaaaa@3110@) and correlation (ρy). Relaxing the stationary variance assumption for growth data, Wu et al. 15 adopted a transform-both-sides (TBS) model to obtain an empirically homogeneous variance. The results from simulation studies suggest that the TBS-based mapping model provides more precise estimates for curve parameters and residual variance-correlation than the untrans-formed model.
The TBS-based model displays the potential to relax the assumption of variance stationarity, but the covariance stationarity issue remains unsolved. Zimmerman and Núñez-Antón 20 proposed a so-called structured antedependence (SAD) model to model the age-specific change of correlation in the analysis of longitudinal traits. The SAD model has been employed in several studies and displays many favorable properties 2122.
The emergence of a developmental event (z) at time t* can be correlated with the longitudinal trait. For example, larger tumor sizes may be likely to lead to earlier malignance of cancer than smaller tumor sizes. An AIDS patient would die when his/her HIV load accumulates to a particularly high level. In plants, first flowering only appears after some investment of vegetative growth. All such common knowledge suggests that the correlation between the event trait at time t* and longitudinal trait measured at time t (before t*) decays with time difference (t* - t). In fact, a similar pattern of correlation should also hold for t > t* because of the autocorrelation nature. With all this consideration, the correlation between the event and longitudinal traits can be modelled by the power equation, expressed as
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGJbWycqqGVbWBcqqGYbGCcqqGYbGCcqGGOaakcqWG5bqEcqGGOaakcqWG0baDcqGGPaqkcqGGSaalcqWG6bGEcqGGOaakcqWG0baDcqGGQaGkcqGGPaqkcqGGPaqkcqGH9aqpdaGabaqaauaabaqaceaaaeaadaGabaqaauaabaqaciaaaeaaiiGacqWF3oaAdaahaaWcbeqaaiabcIcaOiabdsha0jabcQcaQmaaCaaameqabaGae83UdWgaaSGaeyOeI0IaemiDaq3aaWbaaWqabeaacqWF7oaBaaWccqGGPaqkcqGGVaWlcqWF7oaBcqGHRaWkcqaIXaqmaaGccqGGSaalaeaacqWF7oaBcqGHGjsUcqaIWaamaeaacqWF3oaAdaahaaWcbeqaaiGbcYgaSjabc6gaUjabcIcaOiabdsha0jabcQcaQiabc+caViabdsha0jabcMcaPiabgUcaRiabigdaXaaakiabcYcaSaqaaiab=T7aSjabg2da9iabicdaWiabcYcaSaaaaiaawUhaaiaaxMaacqqGMbGzcqqGVbWBcqqGYbGCcqqGGaaicqWG0baDcqGGQaGkcqGH+aGpcqWG0baDcqGGSaalaeaadaGabaqaauaabaqaciaaaeaacqWF3oaAdaahaaWcbeqaaiabcIcaOiabdsha0naaCaaameqabaGae83UdWgaaSGaeyOeI0IaemiDaqNaeiOkaOYaaWbaaWqabeaacqWF7oaBaaWccqGGPaqkcqGGVaWlcqaH7oaBcqGHRaWkcqaIXaqmaaGccqGGSaalaeaacqWF7oaBcqGHGjsUcqaIWaamaeaacqWF3oaAdaahaaWcbeqaaiGbcYgaSjabc6gaUjabcIcaOiabdsha0jabc+caViabdsha0jabcQcaQiabcMcaPiabgUcaRiabigdaXaaakiabcYcaSaqaaiab=T7aSjabg2da9iabicdaWiabcYcaSaaaaiaawUhaaiaaxMaacqqGMbGzcqqGVbWBcqqGYbGCcqqGGaaicqWG0baDcqGH+aGpcqWG0baDcqGGQaGkaaaacaGL7baacaWLjaGaaCzcamaabmaabaGaeGynaudacaGLOaGaayzkaaaaaa@AC82@
where 0 ≤ η ≤ 1. Equation (5) suggests that the event is correlated with the longitudinal trait, to the same extent, before and after its emergence. The event trait should be individual-specific when it is the timing of development, such as the time to first flower. In this case, Equation (5) and, therefore, the covariance matrix (4), expressed as Σi, should be individual-specific. If Σy is modelled by the AR(1) model, one can derive the explicit expressions of the determinant and inverse of Σi specified by (σy2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGaemyEaKhabaGaeGOmaidaaaaa@3110@, ρy, σz2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGaemOEaOhabaGaeGOmaidaaaaa@3112@, η, λ).
Computational algorithms
The unknown parameters (Ω) contained with the mixture model (1) include three types, QTL-marker recombination fractions for a pedigree or QTL-marker linkage disequilibria for a natural population reflected in the conditional probabilities ϖj1j2|i
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFwpGDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaliabcYha8jabdMgaPbqabaaaaa@369F@, the curve parameters (w→j1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG3bWDgaWcamaaBaaaleaacqWGQbGAdaWgaaadbaGaeGymaedabeaaliabdQgaQnaaBaaameaacqaIYaGmaeqaaaWcbeaaaaa@336D@, vj1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG2bGDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@3359@) that model the mean vector, and the parameters (σy2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGaemyEaKhabaGaeGOmaidaaaaa@3110@, ρy, σz2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGaemOEaOhabaGaeGOmaidaaaaa@3112@, η, λ) that model the structure of the covariance matrix. We derived the EM algorithm to estimate these parameters. Using the (prior) conditional probability and the likelihood, we define the posterior probability for individual i to bear on a QTL genotype j1j2 as
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHpis1daWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaliabcYha8jabdMgaPbqabaGccqGH9aqpdaWcaaqaaGGaciab=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@A2CD@
The posterior probabilities are then used to derive a closed-form maximum likelihood estimates of the QTL locations, expressed as the ratio of recombination fractions, for linkage analysis or QTL-marker haplotype frequencies for linkage disequilibrium analysis 17. For functional mapping, in which the mean vectors and covariance matrix are modelled by mathematical parameters based on non-linear equations, it is impossible to derive the closed forms for these parameters, the simplex algorithm, widely used in operations research, is found to provide a fast and precise estimation of the curve parameters and the parameters that model the residual covariances 23. Thus, we implement the simplex algorithm in the maximization process of the EM algorithm.
For linkage analysis based on an experimental cross, ϖj1j2|i
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFwpGDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaliabcYha8jabdMgaPbqabaaaaa@369F@'s are expressed in the recombination fraction between the QTL and two flanking markers. In practical computations, the QTL position parameter can be viewed as a fixed parameter because a putative QTL can be searched at every 1 or 2 cM on a map interval bracketed by two markers throughout the entire genome. The amount of support for a QTL at a particular map position is often displayed graphically through the use of likelihood maps or profiles, which plot the likelihood ratio test statistic as a function of map position of the putative QTL. The peak of the profile corresponds to the position of the QTL over the genome.
For linkage disequilibrium analysis of a natural population, we have derived a closed form for the EM algorithm to estimate QTL-marker haplotype frequencies. From the estimated haplotype frequencies, the allele frequencies of QTL and QTL-marker linkage disequilibria can be estimated. How the markers are associated with the underlying QTL in the population can be tested for the significance of QTL-marker linkage disequilibria.
After the point estimates of parameters are obtained by the EM algorithm, the approximate variance-covariance matrix and the sampling errors of the estimates (Ω^
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiqacuWFPoWvgaqcaaaa@2E4F@) can be estimated. The techniques for so doing involve calculation of the incomplete-data information matrix which is the negative second-order derivative of the incomplete-data log-likelihood. The incomplete-data information can be calculated by extracting the information for the missing data from the information for the complete data 24.
Order selection
For a QTL to be detected, we need to determine the optimal order for the Legendre polynomial that fits the data. We propose using the AIC information criterion to select the best model. The AIC value at a particular order, r, is calculated by
AIC = -2 ln L(Ω^
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiqacuWFPoWvgaqcaaaa@2E4F@|r) + 2 dimension(Ω|r), (7)
where (Ω^
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiqacuWFPoWvgaqcaaaa@2E4F@|r) is the the MLE of parameters for the Legendre polynomial of order r and dimension (Ω|r) represents the number of independent parameters under order r.
Also, Bayesian Information Criterion (BIC) 25 is used to determine the optimal order of the Legendre function, which is calculated by
BIC = -2 ln L(Ω^
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiqacuWFPoWvgaqcaaaa@2E4F@|r) + 2 dimension(Ω|r) ln (nT). (8)
As compared to AIC, BIC adjusts the effects of sample size and the number of time points measured.
Hypothesis tests
Our model allows for a number of hypothesis tests to examine the genetic control of growth processes 14. All these tests are helpful to address biological questions related to the genetic control mechanisms of growth. Testing whether specific QTL exist to affect the longitudinal and event processes is a first step toward the understanding of the detailed genetic architecture of complex phenotypes. This can be tested by formulating the following hypotheses,
H0 : uj1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWF1bqDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@335D@ = u and vj1j2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG2bGDdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabigdaXaqabaWccqWGQbGAdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@3359@ = v vs. H1: Not all equalities in H0 hold. (9)
The H0 states that there is no QTL affecting longitudinal and event processes (the reduced model), whereas the H1 proposes that such a QTL does exist (the full model). The test statistic for testing the hypotheses is calculated as the log-likelihood ratio of the reduced to the full model:
LR = -2[ln L0(Ω˜
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiqacuWFPoWvgaacaaaa@2E4E@|y, z) - ln L1 (Ω^
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiqacuWFPoWvgaqcaaaa@2E4F@|y, z, M)], (10)
where Ω˜
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiqacuWFPoWvgaacaaaa@2E4E@ and Ω^
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiqacuWFPoWvgaqcaaaa@2E4F@ denote the MLEs of the unknown parameters under H0 and H1, respectively. Because the LR calculated by equation (9) may not be asymptotically χ2-distributed with eleven degrees of freedom due to violation of regularity conditions, an empirical approach for determining the critical threshold based on permutation tests is used. By repeatedly shuffling the relationships between marker genotypes and phenotypes, a series of the maximum log-likelihood ratios are calculated, from the distribution of which the critical threshold is determined.
After the QTL are detected to be significant for both longitudinal and event traits, we need to test whether the detected QTL are significant separately for each trait. We assume two different genetic settings:
(1) Longitudinal and event traits are under control of the same QTL;
(2) Each process is controlled by different QTL that are linked on the same chromosomal region.
For the first setting, only after it is significant in two separate tests for longitudinal and event traits can the tested QTL be thought to be pleiotropic in affecting both types of traits. For the second setting, we assume that two tested QTL are located in the same interval bracketed by two markers. The comparison of the first and second setting can examine how the detected QTL jointly affect the differentiation in longitudinal and event traits. First, we can test how two genetic mechanisms, pleiotropy or close linkage, contribute to the correlation between these two types of traits. If the two QTL are detected to be significant for both, we then test whether such a correlation is due to pleiotropy or close linkage. Second, when two QTL exist, we can test how they epistatically interact to affect longitudinal trajectories and developmental events. Wu et al. 14 formulated a procedure for testing the epistatic effects on developmental trajectories.