Efficient Association Mapping of Quantitative Trait Loci with Selective Genotyping
2007; Elsevier BV; Volume: 80; Issue: 3 Linguagem: Inglês
10.1086/512727
ISSN1537-6605
Autores Tópico(s)Genetic Syndromes and Imprinting
ResumoSelective genotyping (i.e., genotyping only those individuals with extreme phenotypes) can greatly improve the power to detect and map quantitative trait loci in genetic association studies. Because selection depends on the phenotype, the resulting data cannot be properly analyzed by standard statistical methods. We provide appropriate likelihoods for assessing the effects of genotypes and haplotypes on quantitative traits under selective-genotyping designs. We demonstrate that the likelihood-based methods are highly effective in identifying causal variants and are substantially more powerful than existing methods. Selective genotyping (i.e., genotyping only those individuals with extreme phenotypes) can greatly improve the power to detect and map quantitative trait loci in genetic association studies. Because selection depends on the phenotype, the resulting data cannot be properly analyzed by standard statistical methods. We provide appropriate likelihoods for assessing the effects of genotypes and haplotypes on quantitative traits under selective-genotyping designs. We demonstrate that the likelihood-based methods are highly effective in identifying causal variants and are substantially more powerful than existing methods. Mapping genes associated with quantitative traits is an important step toward genetic dissection of complex human diseases. Because the disease genes are unlikely to have very large effects on quantitative traits, power is a major concern in association studies, especially with the need to adjust for multiple testing. Despite the continuing improvements in genotyping efficiency, it is still highly expensive to genotype a large number of individuals, particularly in genomewide association studies. A cost-effective strategy is to preferentially genotype individuals whose trait values deviate from the population mean. Known as "selective genotyping," this approach can result in a substantial increase in power (relative to random sampling with the same number of individuals), because much of the genetic information resides in individuals with extreme phenotypes.1Laitinen T Kauppi P Ignatius J Ruotsalainen T Daly MJ Kääriäinen H Kruglyak L Laitinen H de la Chapelle A Lander ES et al.Genetic control of serum IgE levels and asthma: linkage and linkage disequilibrium studies in an isolated population.Hum Mol Genet. 1997; 6: 2069-2076Crossref PubMed Scopus (87) Google Scholar, 2Slatkin M Disequilibrium mapping of a quantitative-trait locus in an expanding population.Am J Hum Genet. 1999; 64: 1765-1773Abstract Full Text Full Text PDF Scopus (39) Google Scholar, 3van Gestel S Houwing-Duistermaat JJ Adolfsson R van Duijn CM van Broeckhoven C Power of selective genotyping in genetic association analyses of quantitative traits.Behav Genet. 2000; 30: 141-146Crossref PubMed Scopus (77) Google Scholar, 4Xiong M Fan R Jin L Linkage disequilibrium mapping of quantitative trait loci under truncation selection.Hum Hered. 2002; 53: 158-172Crossref PubMed Scopus (15) Google Scholar, 5Chen Z Zheng G Ghosh K Li Z Linkage disequilibrium mapping of quantitative-trait loci by selective genotyping.Am J Hum Genet. 2005; 77: 661-669Abstract Full Text Full Text PDF PubMed Scopus (39) Google Scholar, 6Cornish KM Manly T Savage R Swanson J Morisano D Butler N Grant C Cross G Bentley L Hollis CP Association of the dopamine transporter (DAT1) 10/10-repeat genotype with ADHD symptoms and response inhibition in a general populations sample.Mol Psychiatry. 2005; 10: 686-698Crossref PubMed Scopus (189) Google Scholar, 7Wallace C Chapman JM Clayton DG Improved power offered by a score test for linkage disequilibrium mapping of quantitative-trait loci by selective genotyping.Am J Hum Genet. 2006; 78: 498-504Abstract Full Text Full Text PDF PubMed Scopus (37) Google Scholar Slatkin2Slatkin M Disequilibrium mapping of a quantitative-trait locus in an expanding population.Am J Hum Genet. 1999; 64: 1765-1773Abstract Full Text Full Text PDF Scopus (39) Google Scholar suggested genotyping a selected sample of individuals with unusually high values of the quantitative trait, together with a random sample from the study population. Because selection depends on the phenotype, standard statistical methods that assume random sampling are not applicable. Slatkin2Slatkin M Disequilibrium mapping of a quantitative-trait locus in an expanding population.Am J Hum Genet. 1999; 64: 1765-1773Abstract Full Text Full Text PDF Scopus (39) Google Scholar developed two tests: one comparing the allele frequencies between the selected sample and the random sample and one comparing the mean trait values among individuals with different genotypes in the selected sample. The two tests are approximately independent, so their P values can be combined to form an overall test. Slatkin2Slatkin M Disequilibrium mapping of a quantitative-trait locus in an expanding population.Am J Hum Genet. 1999; 64: 1765-1773Abstract Full Text Full Text PDF Scopus (39) Google Scholar used simulation to show that his tests are more powerful than the simple t test (when the latter is applied to a random sample with the same number of individuals). Chen et al.5Chen Z Zheng G Ghosh K Li Z Linkage disequilibrium mapping of quantitative-trait loci by selective genotyping.Am J Hum Genet. 2005; 77: 661-669Abstract Full Text Full Text PDF PubMed Scopus (39) Google Scholar recommended replacement of the random sample with a selected sample of individuals with unusually low trait values and described two sampling schemes to obtain the selected samples. They demonstrated through a simulation study that, with Slatkin's three tests, their designs are more efficient than Slatkin's original design. In a recent Science report on obesity,8Herbert A Gerry NP McQueen MB Heid IM Pfeufer A Illig T Wichmann HE Meitinger T Hunter D Hu FB et al.A common genetic variant is associated with adult and childhood obesity.Science. 2006; 312: 279-283Crossref PubMed Scopus (581) Google Scholar one of the replication studies genotyped individuals from the 90th–97th percentile of the BMI distribution and those from the 5th–12th percentile, and another replication study genotyped individuals from the top and bottom quartiles. In both studies, the individuals with high and low BMI values were treated as cases and controls, respectively, and case-control methods (i.e., testing for allele-frequency differences between the two selected groups) were used for analysis. Case-control methods disregard the actual trait values and are thus inefficient. Slatkin's tests2Slatkin M Disequilibrium mapping of a quantitative-trait locus in an expanding population.Am J Hum Genet. 1999; 64: 1765-1773Abstract Full Text Full Text PDF Scopus (39) Google Scholar do not make full use of the available data either—individuals who are homozygous for the minor allele are discarded, and the trait values in the random sample or the low-trait-value sample are not used at all. Recently, in this journal, Wallace et al.7Wallace C Chapman JM Clayton DG Improved power offered by a score test for linkage disequilibrium mapping of quantitative-trait loci by selective genotyping.Am J Hum Genet. 2006; 78: 498-504Abstract Full Text Full Text PDF PubMed Scopus (37) Google Scholar proposed a Hotelling's T2 test for normal traits, which they showed through simulation has increased power over Slatkin's tests.2Slatkin M Disequilibrium mapping of a quantitative-trait locus in an expanding population.Am J Hum Genet. 1999; 64: 1765-1773Abstract Full Text Full Text PDF Scopus (39) Google Scholar Wallace et al.'s test,7Wallace C Chapman JM Clayton DG Improved power offered by a score test for linkage disequilibrium mapping of quantitative-trait loci by selective genotyping.Am J Hum Genet. 2006; 78: 498-504Abstract Full Text Full Text PDF PubMed Scopus (37) Google Scholar which is essentially the standard t test in the case of a single marker, ignores the biased sampling nature of the selective-genotyping design and thus may not be optimal. Furthermore, none of the existing methods deals with haplotype-based testing or estimation of genetic effects. In this report, we show how to properly and efficiently map QTLs with selective genotyping. We derive appropriate likelihoods that make full use of the available data and that properly reflect trait-dependent sampling. The corresponding inference procedures are valid and efficient. Our methods can be used to perform both genotype-based and haplotype-based association analyses. Their advantages over the existing methods are demonstrated through extensive simulation studies. We consider two very general selective-genotyping designs. Under design 1, the quantitative trait is measured on a random sample of N individuals from the study population, and a subset of n individuals is selected for genotyping; the selection probabilities depend on the trait values. Under design 2, a random sample of n individuals whose trait values fall into certain regions is selected for genotyping, and the trait values are retained for only those individuals. Thus, the main difference between the two designs is that the trait values on those individuals who are not selected for genotyping are retained under design 1 but not under design 2. Under design 2, it is not necessary to specify N or to ascertain the individuals outside the selection regions. Let Yi be the trait value of the ith individual and Gi be the corresponding multilocus genotype denoting the number of minor alleles at each SNP site. The association between Gi and Yi is characterized by the conditional density function P(Yi|Gi;θ) indexed by a set of parameters θ. In the special case of a single locus with the additive mode of inheritance, P(Yi|Gi;θ) may take the familiar form of the linear regression modelYi=α+βGi+ɛi,(1) where ɛi is zero-mean normal with variance σ2. In this case, θ=(α,β,σ2). Under the dominant (or recessive) mode of inheritance, Gi in equation (1) is replaced by the indicator of whether the ith individual has at least one minor allele (or, for the recessive model, two minor alleles). If there are multiple loci, then βGi in equation (1) is replaced by an appropriate linear combination of individual genotype scores and (possibly) their cross-products. We denote the probability function of the genotype by P(G;γ), where γ represents the (multilocus) genotype frequencies. Under design 1, the data consist of (Yi,Gi)(i=1,…,n) and Yi(i=n+1,…,N). (Without loss of generality, the data are arranged so that the first n records pertain to the n individuals who are selected for genotyping and the remaining (N−n) records to the unselected individuals.) The corresponding likelihood for θ and γ can be written as∏i=1nP(Yi|Gi;θ)P(Gi;γ)∏i=n+1N∑GP(Yi|G;θ)P(G;γ),(2) where the summation over G is taken over all possible genotypes; a derivation is given in appendix A. Under design 2, the data consist only of(Yi,Gi)(i=1,…,n),which are a random sample from all the individuals whose trait values belong to a particular setC. We can use the likelihood for θ and γ,∏i=1nP(Yi,Gi|Yi∈C)=∏i=1nP(Yi|Gi;θ)P(Gi;γ)∑GP(Yi∈C|G;θ)P(G;γ),(3) or the likelihood for θ,∏i=1nP(Yi|Gi,Yi∈C)=∏i=1nP(Yi|Gi;θ)P(Yi∈C|Gi;θ).(4) If only the individuals whose trait values are less than the lower threshold cL or larger than the upper threshold cU are selected for genotyping, then, under equation (1),P(Yi∈C|Gi;θ)=1−Φ(cU−α−βGiσ)+Φ(cL−α−βGiσ),where Φ is the cumulative distribution function of the standard normal distribution. We refer to expression (2) as the full likelihood and to equations (3) and (4) as the conditional likelihoods. These likelihoods properly reflect the selective-genotyping designs and use all the available data. Note that expression (2) is the same as the likelihood for a prospective study of size N in which genotype data are missing on N−n individuals. Under design 1, one may disregard the trait values of those individuals who are not selected for genotyping and use the conditional likelihoods, provided that the genotyped individuals are a random sample from setC. The maximum-likelihood estimators can be obtained by the standard Newton-Raphson algorithm. As shown in appendix A, the maximizations of equations (3) and (4) yield the same estimator of θ. By the likelihood theory, the maximum-likelihood estimators are approximately unbiased, normally distributed, and statistically efficient. Association testing can be performed by using the familiar likelihood-ratio, score, or Wald statistics. The above description pertains to the analysis of genotype-phenotype association. It is also desirable to assess haplotype-phenotype association.9Lin DY Zeng D Millikan R Maximum likelihood estimation of haplotype effects and haplotype-environment interactions in association studies.Genet Epidemiol. 2005; 29: 299-312Crossref PubMed Scopus (105) Google Scholar, 10Schaid DJ Rowland CM Tines DE Jacobson RM Poland GA Score tests for association between traits and haplotypes when linkage phase is ambiguous.Am J Hum Genet. 2002; 70: 425-434Abstract Full Text Full Text PDF PubMed Scopus (1566) Google Scholar Let Hi denote the diplotype of the ith individual. The effects of haplotypes on the trait are characterized by the conditional density function P(Yi|Hi;θ) indexed by a set of parameters θ. If we are interested in assessing the effect of a particular haplotype h*, then P(Yi|Hi;θ) may take the formYi=α+βZ(Hi)+ɛi,(5) where Z(Hi) is the number of occurrences of h* in Hi under the additive mode of inheritance, the indicator of whether Hi contains at least one h* under the dominant mode of inheritance, and the indicator of whether Hi contains two copies of h* under the recessive mode of inheritance. One may also define P(Yi|Hi;θ) in such a way that multiple haplotypes are compared with a reference in a single model.9Lin DY Zeng D Millikan R Maximum likelihood estimation of haplotype effects and haplotype-environment interactions in association studies.Genet Epidemiol. 2005; 29: 299-312Crossref PubMed Scopus (105) Google Scholar Because haplotypes are not directly observed, it is necessary to impose some restrictions, such as Hardy-Weinberg equilibrium (HWE), on the diplotype distribution. For k=1,…,K, let hk denote the kth possible haplotype in the population and let πk denote the population frequency of hk. Under HWE,P[Hi=(hk,hl)]=πkπl(k,l=1,….K).We denote the diplotype probability function by P(Hi;γ), where γ=(π1,…,πK). Inference on haplotype effects must properly account for phase ambiguity. Note thatP(Yi,Gi)=ΣH∈S(Gi)P(Yi|H;θ)P(H;γ),whereS(Gi) is the set of diplotypes compatible with genotype Gi.9Lin DY Zeng D Millikan R Maximum likelihood estimation of haplotype effects and haplotype-environment interactions in association studies.Genet Epidemiol. 2005; 29: 299-312Crossref PubMed Scopus (105) Google Scholar Thus, the full likelihood and conditional likelihood analogous to expressions (2) and (3) are∏i=1n∑H∈S(Gi)P(Yi|H;θ)P(H;γ)∏i=n+1N∑HP(Yi|H;θ)P(H;γ)(6) and∏i=1n∑H∈S(Gi)P(Yi|H;θ)P(H;γ)∑HP(Yi∈C|H;θ)P(H;γ),(7) where the second summation in expression (6) and the summation in the denominator of expression (7) are taken over all possible diplotypes. The maximizations of expressions (6) and (7) can be performed by the expectation-maximization (EM) algorithm or the Newton-Raphson algorithm; see appendix A. The maximum-likelihood estimators are approximately unbiased, normally distributed, and statistically efficient. Note that β pertains to genetic effect in equation (1) and to haplotype effect in equation (5). If we are concerned with one SNP at a time, however, the models in equations (1) and (5) are the same. In that case, likelihoods of expressions (6) and (7) differ from expressions (2) and (3) in that the former impose HWE and allow missing genotype values, whereas the latter do not impose HWE and exclude subjects with missing genotype values. Thus, the former yield more efficient analyses, provided that HWE is a reasonable assumption. We conducted extensive simulation studies to assess the performance of the proposed methods. We considered both designs 1 and 2. Specifically, we generated a random sample of N=5,000 individuals from the joint distribution of the trait value and genotype, and we identified the subset of all the individuals whose trait values are cU. We then selected a random sample of n=500 individuals from that subset. By setting the genotypes of the unselected individuals to "missing," we obtained the data under design 1; by deleting the unselected individuals altogether, we obtained the data under design 2. We evaluated both the full-likelihood and conditional-likelihood methods. These evaluations provided information about the relative efficiency of using full likelihood versus conditional likelihood under design 1 or, equivalently, the relative efficiency of design 1 versus design 2. For comparison, we also evaluated the standard methods, which are based on the prospective likelihoods. For genotype-based analysis, the prospective likelihood7Wallace C Chapman JM Clayton DG Improved power offered by a score test for linkage disequilibrium mapping of quantitative-trait loci by selective genotyping.Am J Hum Genet. 2006; 78: 498-504Abstract Full Text Full Text PDF PubMed Scopus (37) Google Scholar is simply Πi=1nP(Yi|Gi;θ); for haplotype-based analysis, the prospective likelihood is the first term in expression (6).10Schaid DJ Rowland CM Tines DE Jacobson RM Poland GA Score tests for association between traits and haplotypes when linkage phase is ambiguous.Am J Hum Genet. 2002; 70: 425-434Abstract Full Text Full Text PDF PubMed Scopus (1566) Google Scholar In addition, we evaluated the case-control tests, which regard the upper and lower trait values as cases and controls, respectively. In our first study, we generated the trait values from equation (1) with α=0, σ2=1, and β=0, 0.1, 0.2, 0.3, 0.4, and 0.5. We set (cL,cU) to (−0.5,0.5), (−1.0,1.0), (−1.5,0.5) or (−2.0,1.0). Under the condition that β=0, the thresholds of −2.0, −1.5, −1.0, −0.5, 0.5, and 1.0 correspond approximately to the 2nd, 7th, 16th, 31st, 69th, and 84th percentiles of the trait distribution, respectively. We considered three modes of inheritance—additive, dominant, and recessive—and various values of the minor-allele frequency (MAF). The genotypes were generated under HWE, and the analyses were performed both with and without this assumption. The results without the HWE assumption are summarized in table 1. The results with HWE are similar and thus omitted.Table 1Bias, SE, Average SEE, Coverage Probability of 95% CI (CP), and Power at the .05 Nominal Significance Level at a Candidate Locus Under Additive (A) and Dominant (D) Models with MAFs of .05 and Recessive (R) Model with MAF of .2Full LikelihoodConditional LikelihoodProspective LikelihoodModel, β, and cLcUBiasSESEECPPowerBiasSESEECPPowerBiasSESEECPPowerCC PowerA: 0: −.5.5.001.12.1295.35.0.001.12.1295.25.0.002.18.1895.04.95.1 −1.01.0.001.09.0995.35.0.001.09.0995.35.0.003.23.2395.05.05.0 −1.5.5.009.12.1295.45.1.010.12.1295.25.1.001.19.1995.04.94.5 −2.01.0.014.11.1195.85.3.015.12.1195.65.3.002.20.2095.24.85.0 .2: −.5.5.001.12.1295.040.9.003.12.1295.040.9.111.18.1891.140.634.7 −1.01.0.003.10.1095.059.0.004.10.1095.359.0.291.22.2375.658.655.0 −1.5.5.011.13.1395.040.0.014.13.1394.840.0.079.15.1795.835.827.1 −2.01.0.016.13.1395.042.2.020.13.1394.942.2.084.14.1897.134.525.7 .3: −.5.5.002.12.1295.272.9.004.12.1295.573.0.159.17.1886.172.864.0 −1.01.0.003.10.1095.490.3.004.10.1095.390.2.403.20.2255.790.087.7 −1.5.5.010.13.1394.670.7.014.14.1394.770.6.084.14.1796.266.551.7 −2.01.0.016.14.1494.775.2.022.14.1495.075.1.076.12.1798.568.655.2D: 0: −.5.5.001.12.1295.35.0.002.12.1295.24.9.002.19.1995.14.95.1 −1.01.0.001.10.1095.35.0.001.10.1095.35.0.003.24.2495.14.94.9 −1.5.5.010.12.1295.35.2.010.12.1295.15.2.001.20.1995.05.04.6 −2.01.0.014.12.1195.85.2.015.12.1295.65.2.002.21.2195.24.85.0 .2: −.5.5.001.12.1294.938.4.003.12.1294.938.5.112.19.1990.938.332.5 −1.01.0.002.10.1095.355.6.003.10.1095.355.6.292.23.2476.955.152.2 −1.5.5.009.13.1395.236.8.012.13.1395.036.8.080.16.1895.633.226.2 −2.01.0.016.14.1394.940.1.021.14.1395.040.0.090.15.1896.932.924.7 .3: −.5.5.002.12.1294.769.9.004.12.1294.969.8.162.18.1986.369.760.9 −1.01.0.004.10.1095.288.2.006.10.1095.388.2.417.22.2356.388.085.0 −1.5.5.009.14.1494.767.6.013.14.1494.767.5.091.15.1895.763.449.7 −2.01.0.018.14.1494.772.0.024.15.1495.172.0.090.13.1798.165.553.0R: 0: −.5.5−.001.19.1995.35.4−.001.19.1995.35.4−.002.29.2994.75.24.8 −1.01.0.005.15.1595.94.9.005.15.1595.94.9.011.37.3795.24.75.0 −1.5.5.024.20.1995.45.5.026.20.1995.45.5−.000.30.3094.95.14.8 −2.01.0.041.20.1996.05.4.043.20.1995.85.5.004.32.3295.54.54.7 .4: −.5.5.005.20.1994.558.1.009.20.1995.058.0.201.27.2890.257.348.9 −1.01.0.018.17.1695.579.0.022.17.1795.679.0.524.31.3567.978.273.9 −1.5.5.024.22.2293.956.6.031.22.2294.456.5.087.20.2698.848.229.0 −2.01.0.037.23.2394.060.2.048.23.2394.260.1.066.17.2599.746.027.4 .5: −.5.5.010.20.1994.677.7.014.20.2095.277.7.242.26.2888.377.166.9 −1.01.0.021.17.1795.693.5.026.18.1795.693.4.600.28.3457.193.189.4 −1.5.5.018.22.2294.274.9.027.22.2294.574.7.068.18.2599.268.045.1 −2.01.0.027.22.2394.179.0.039.23.2394.478.9.027.15.2499.868.046.7Note.—Each entry is based on 10,000 simulated data sets. CC = case-control analysis. Open table in a new tab Note.— Each entry is based on 10,000 simulated data sets. CC = case-control analysis. Both the full and conditional likelihoods provide (virtually) unbiased estimators of genetic effects and correct type I error. The SE estimators (SEEs) accurately reflect the true variations, and the CIs have proper coverages. The conditional likelihood has nearly the same power as the full likelihood. As expected, the power is substantially higher under the additive and dominant models than under the recessive model (given the same MAF and the same effect size). The power increases as selection becomes more extreme. Also, the power tends to be higher when cL and cU are of the same distance from the population mean (as opposed to unequal distances), which implies that the optimal sample-size ratio between the upper and lower ends should be ∼1:1 (as in the case of the case-control design). In practice, the population mean may be unknown, or it may be easier to recruit subjects with high trait values than those with low trait values, or vice versa. Thus, it may not be feasible to set cL and cU the same distance from the population mean. In the presence of a causal variant, both the estimator of the genetic effect and the SEE based on the prospective likelihood are biased upward, and the coverages of the CIs may be substantially below or above the desired levels. The prospective likelihood appears to preserve the type I error. The power of the prospective likelihood tends to be lower than that of the full and conditional likelihoods, especially when (cL,cU)=(−2,1) and under the recessive mode of inheritance. When (cL,cU)=(−2,1), the full and conditional likelihoods have power of ∼75% to detect effect size of 0.3 under the additive and dominant models with MAF=0.05, and they have power of ∼80% to detect effect size of 0.5 under the recessive model with MAF=0.2. By contrast, the prospective likelihood has 0(k=1,…,K) into the calculations, we define π*k=πk/πK and ηk=logπ*k. For notational convenience, denote σ2 as v. Let η=(η1,…,ηK−1) and ϑ=(β,v,η). Then the log-likelihood isℓ(ϑ)=−n2logv+∑i=1nlog∑(hk,hl)∈S(Gi)exp{−(2v)−1[Yi−βTZ(hk,hl)]2+ηTW(hk,hl)}−nlog∑k,leηTW(hk,hl){1−Φ[cU−βTZ(hk,hl)v]},whereW(hk,hl)=[I(hk=h1)+I(hl=h1)⋮I(hk=hK−1)+I(hl=hK−1)].LetQikl(ϑ)=exp{−[Yi−βTZ(hk,hl)]22v+ηTW(hk,hl)},RklL(ϑ)=cL−βTZ(hk,hl)v,RklU(ϑ)=cU−βTZ(hk,hl)v,andS(ϑ)=∑k,l{1−Φ[RklU(ϑ)]+Φ[RklL(ϑ)]}eηTW(hk,hl).Also, let a⊗2=aaT, and let ϕ be the standard normal density function. Then∂ℓ(ϑ)∂v=−n2v+∑i=1nΣ(hk,hl)∈S(Gi)Qikl(ϑ)[Yi−βTZ(hk,hl)]22vΣ(hk,hl)∈S(Gi)Qikl(ϑ)−nΣk,l{ϕ[RklU(ϑ)]RklU(ϑ)−ϕ[RklL(ϑ)]RklL(ϑ)}eηTW(hk,hl)2vS(ϑ),∂ℓ(ϑ)∂β=∑i=1nΣ(hk,hl)∈S(Gi)Qikl(ϑ)v[Yi−βTZ(hk,hl)]Z(hk,hl)Σ(hk,hl)∈S(Gi)Qikl(ϑ)−n∑k,l{ϕ[RklU(ϑ)]−ϕ[RklL(ϑ)]}Z(hk,hl)veηTW(hk,hl)S(ϑ),∂ℓ(ϑ)∂η=∑i=1nΣ(hk,hl)∈S(Gi)Qikl(ϑ)W(hk,hl)Σ(hk,hl)∈S(Gi)Qikl(ϑ)−n∑k,l{1−Φ[RklU(ϑ)]+Φ[RklL(ϑ)]}eηTW(hk,hl)W(hk,hl)S(ϑ),∂2ℓ(ϑ)∂v2=n2v2+∑i=1n(Σ(hk,hl)∈S(Gi)Qikl(ϑ){[Yi−βTZ(hk,hl)]44v4−[Yi−βTZ(hk,hl)]2v3}Σ(hk,hl)∈S(Gi)Qikl(ϑ)−{Σ(hk,hl)∈S(Gi)Qikl(ϑ)[Yi−βTZ(hk,hl)]22v2Σ(hk,hl)∈S(Gi)Qikl(ϑ)}2)−n[Σk,l(ϕ[RklU(ϑ)][[RklU(ϑ)3]−3RklU(ϑ)]−ϕ[RklL(ϑ)]{[RklL(ϑ)3]−3RklL(ϑ)})eηTW(hk,hl)4v2S(ϑ)−(Σk,l{ϕ[RklU(ϑ)]RklU(ϑ)−ϕ[RklL(ϑ)]RklL(ϑ)}eηTW(hk,hl)2vS(ϑ))2],∂2ℓ(ϑ)∂v∂β=∑i=1n[Σ(hk,hl)∈S(Gi)Qikl(ϑ)Z(hk,hl){[Yi−βTZ(hk,hl)]32v3−[Yi−βTZ(hk,hl)]v2}Σ(hk,hl)∈S(Gi)Qikl(ϑ)]−({Σ(hk,hl)∈S(Gi)Qikl(ϑ)[Yi−βTZ(hk,hl)]22v2Σ(hk,hl)∈S(Gi)Qikl(ϑ)}×{Σ(hk,hl)∈S(Gi)Qikl(ϑ)[Yi−βTZ(hk,hl)]Z(hk,hl)vΣ(hk,hl)∈S(Gi)Qikl(ϑ)})−n([Σk,l{ϕ(RklU(ϑ))(RklU(ϑ)2−1)−ϕ[RklL(ϑ)][RklL(ϑ)2−1]}eηTW(hk,hl)2v3/2Z(hk,hl)S(ϑ)]−{Σk,l{ϕ[RklU(ϑ)]RklU(ϑ)−ϕ[RklL(ϑ)]RklL(ϑ)}eηTW(hk,hl)2vS(ϑ)}{Σk,l(ϕ[RklU(ϑ)]−ϕ[RklL(ϑ)])eηTW(hk,hl)Z(hk,hl)vS(ϑ)}),∂2ℓ(ϑ)∂ββT=∑i=1n(Σ(hk,hl)∈S(Gi)Qikl(ϑ){[Yi−βTZ(hk,hl)]2v2−v−1}Z⊗2Σ(hk,hl)∈S(Gi)Qikl(ϑ)−{Σ(hk,hl)∈S(Gi)Qikl(ϑ)[Yi−βTZ(hk,hl)]Z(hk,hl)vΣ(hk,hl)∈S(Gi)Qikl(ϑ)}⊗2)−n{Σk,l{ϕ[RklU(ϑ)]RklU(ϑ)−ϕ[RklL(ϑ)]RklL(ϑ)}eηTW(hk,hl)[Z(hk,hl)v]⊗2S(ϑ)−[Σk,l{ϕ[RklU(ϑ)]−ϕ[RklL(ϑ)]}eηTW(hk,hl)Z(hk,hl)vS(ϑ)]⊗2},∂2ℓ(ϑ)∂v∂η=∑i=1n(Σ(hk,hl)∈S(Gi)Qikl(ϑ)W(hk,hl)[Yi−βTZ(hk,hl)]22v2Σ(hk,hl)∈S(Gi)Qikl(ϑ)−{Σ(hk,hl)∈S(Gi)[Yi−βTZ(hk,hl)]22v2Qikl(ϑ)Σ(hk,hl)∈S(Gi)Qikl(ϑ)}{Σ(hk,hl)∈S(Gi)Qikl(ϑ)W(hk,hl)Σ(hk,hl)∈S(Gi)Qikl(ϑ)})−n(Σk,l{ϕ[RklU(ϑ)]RklU(ϑ)−ϕ[RklL(ϑ)]RklL(ϑ)}eηTW(hk,hl)2vW(hk,hl)S(ϑ)−{Σk,l{ϕ[RklU(ϑ)]RklU(ϑ)−ϕ[RklL(ϑ)]RklL(ϑ)}eηTW(hk,hl)2vS(ϑ)}{Σk,l{1−ϕ[RklU(ϑ)]+Φ[RklL(ϑ)]}eηTW(hk,hl)W(hk,hl)S(ϑ)}),∂2ℓ(ϑ)∂β∂ηT=∑i=1n(Σ(hk,hl)∈S(Gi)Qikl(ϑ)[Yi−βTZ(hk,hl)]Z(hk,hl)vW(hk,hl)TΣ(hk,hl)∈S(Gi)Qikl(ϑ)−{Σ(hk,hl)∈S(Gi)Qikl(ϑ)[Yi−βTZ(hk,hl)]Z(hk,hl)vΣ(hk,hl)∈S(Gi)Qikl(ϑ)}{Σ(hk,hl)∈S(Gi)Qikl(ϑ)W(hk,hl)Σ(hk,hl)∈S(Gi)Qikl(ϑ)}T)−n[Σk,l{ϕ[RklU(ϑ)]−ϕ[RklL(ϑ)]}Z(hk,hl)veηTW(hk,hl)W(hk,hl)TS(ϑ)−(Σk,l{ϕ[RklU(ϑ)]−ϕ[RklL(ϑ)]}Z(hk,hl)veηTW(hk,hl)S(ϑ))(Σk,l{1−Φ[RklU(ϑ)]+Φ[RklL(ϑ)]}eηTW(hk,hl)W(hk,hl)S(ϑ))T],and∂2ℓ(ϑ)∂ηηT=∑i=1n{Σ(hk,hl)∈S(Gi)Qikl(ϑ)W(hk,hl)⊗2Σ(hk,hl)∈S(Gi)Qikl(ϑ)−[Σ(hk,hl)∈S(Gi)Qikl(ϑ)W(hk,hl)Σ(hk,hl)∈S(Gi)Qikl(ϑ)]⊗2}−n(Σk,l{1−Φ[RklU(ϑ)]+Φ[RklL(ϑ)]}eηTW(hk,hl)W(hk,hl)⊗2S(ϑ)−{Σk,l{1−Φ[RklU(ϑ)]+Φ[RklL(ϑ)]}eηTW(hk,hl)W(hk,hl)S(ϑ)}⊗2).
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