Hypothesis Testing

- A computational model for functional mapping of genes that regulate intra-cellular circadian rhythms

One of the most significant advantages of functional mapping is that it can ask and address biologically meaningful questions about the interplay between gene actions and trait dynamics by formulating a series of hypothesis tests. Wu et al. [20] described several general hypothesis tests for different purposes. Although all these general tests can be used directly in this study, we propose here the most important and specific tests for the existence of QTL that affect mRNA and protein changes pleiotropically or separately, and for the effects of the QTL on the shape of differential functions.

Testing whether a specific QTL is associated with the differential functions (1) is a first step toward understanding the genetic architecture of circadian rhythms. The genetic control of the entire rhythmic process can be tested by formulating the following hypotheses:

H₀ : D = 0 vs. H₁ : D ≠ 0     (6)

H₀ states that there are no QTL affecting circadian rhythms (the reduced model), whereas H₁ proposes that such QTL do exist (the full model). The statistic for testing these hypotheses (6) is calculated as the log-likelihood ratio (LR) of the reduced to the full model:

LR₁ = -2[ln L(, |x, M) - ln L(, |x, M)],     (7)

where the tildes and hats denote the MLEs of the unknown parameters under H₀ and H₁, respectively. The LR is asymptotically χ²-distributed with one degree of freedom.

A similar test for the existence of a QTL can be performed on the basis of these hypotheses, as follows:

H₀ : Θ_uj ≡ Θ_u, j = (2,1,0)     (8)

H₁ : At least one of the equalities above does not hold;

from which the LR is calculated by

LR₂ : -2[ln L(|x) - ln(, |x, M)],     (9)

with the doubled tildes denoting the estimates under H₀ of hypothesis (8). It is difficult to determine the distribution of the LR₂ because the linkage disequilibrium is not identifiable under H₁. An empirical approach to determining the critical threshold is based on permutation tests, as advocated by Churchill and Doerge [21]. By repeatedly shuffling the relationships between marker genotypes and phenotypes, a series of maximum LR₂ values are calculated, from the distribution of which the critical threshold is determined.

After the existence of a QTL that affects circadian rhythms is confirmed, we need to test whether it affects the rhythmic responses of mRNA and protein jointly or separately. The hypothesis for testing the effect of the QTL on the mRNA response is formulated as

H₀ : (r_Mj, q_Mj, k_j, n_j,) ≡ (r_M, q_M, k, n) for j = 0, 1, 2    (10)

H₁ : At least one of the equalities above does not hold.

The log-likelihood values under H₀ and H₁ are calculated, and thus the corresponding LR.

A similar test is formulated for detecting the effect of the QTL on the protein rhythm:

H₀ : (r_Pj, q_Pj, τ_j, m_j,) ≡ (r_P, q_P, τ, m) for j = 0, 1, 2    (11)

H₁ : At least one of the equalities above does not hold.

For both hypotheses (10) and (11), an empirical approach to determining the critical threshold is based on simulation studies. If the null hypotheses of (10) and (11) are both rejected, this means that the QTL exerts a pleiotropic effect on the circadian rhythms of mRNA and protein.

Two different subspaces of parameters are used to define the features of circadian rhythms: {n, m, τ}, determining the nonlinearity and delay in the system, and {r_M, r_P, q_M, q_P}, determining the phase-response curves. The null hypotheses regarding the genetic control of the system's oscillatory behavior and the shape of the rhythmic responses are:

The oscillatory behavior of a circadian rhythm can also be determined by the amplitude of the rhythm, defined as the difference between the peak and trough values; its phase, defined as the timing of a reference point in the cycle (e.g. the peak) relative to a fixed event (e.g. beginning of the night phase); and its period, defined as the time interval between phase reference points (e.g. two peaks). The genetic determination of all thesevariables can be tested.