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Genes and Longevity

Lessons From Studies of Centenarians

^a Max Planck Institute for Demographic Research, Rostock, Germany
^b Center for Demographic Studies, Duke University, Durham, North Carolina
^c Sanford Institute, Duke University, Durham, North Carolina
^d Departments of Cell Biology, University of Calabria, Italy
^e System Science, University of Calabria, Italy
^f Institute of Statistics and Demography, University of Southern Denmark, Odense
^g Department of Experimental Pathology, University of Bologna, Italy
^h Istituto Nazionale Riposo e Cura Anziani, Ancona, Italy

A.I. Yashin, Max Planck Institute for Demographic Research, Doberaner Strasse 114, 18057 Rostock, Germany E-mail: yashin{at}demogr.mpg.de.

Decision Editor: Jay Roberts, PhD

	Abstract

Top Abstract Materials and Methods Results Discussion Appendix References

 
In population studies of aging, the data on genetic markers are often collected for individuals from different age groups. The idea of such studies is to identify "longevity" or "frailty" genes by comparing the frequencies of genotypes in the oldest and in the younger groups of individuals. In this paper we discuss a new approach to the analysis of such data. This approach, based on the maximum likelihood method, combines data on genetic markers with survival information obtained from standard demographic life tables. This method allows us to evaluate survival characteristics for individuals carrying respective candidate genes. It can also be used in the estimation of the effects of allele–area and allele–allele interaction, either in the presence or absence of hidden heterogeneity. We apply this method to the analysis of Italian data on genetic markers for five autosomal loci and mitochondrial genomes. Then we discuss basic assumptions used in this analysis and directions of further research.

IN genetic studies of human aging and survival, the gene frequency (GF) method is often used ⁽¹⁾⁽²⁾. In this method, the contribution of candidate genes in the survival process is analyzed by comparing gene frequencies in two different age groups of individuals. According to strategies used currently to identify genes in multifactorial traits ⁽³⁾, allele pools are compared between sample groups of extremely old individuals (cases) and younger people (controls) from the same population. The observed case–control differences in allele frequencies are associated with the influence of a respective candidate gene on survival ^{(4)(5)(6)(7)(8)(9)(10)(11)(12)}. To make proper classification on alleles as "frail," "robust," and "neutral," standard statistical methods, which identify differences in observed frequencies among case and control groups for different candidate alleles, are used. These methods, however, do not allow us to evaluate survival characteristics of individuals carrying candidate alleles. Because several alleles for a candidate locus are usually involved in an analysis, multiple comparisons based on the Bonferroni inequality, Scheffé's method, and others have often been used ⁽¹³⁾. The expediency of this procedure was, however, questioned by Rothman ⁽¹⁴⁾. Indeed, when carefully chosen hypotheses about specific alleles are tested, it does not seem reasonable to insist that each be adjusted for the mere presence of the other. However, when the presence of the effect in a selected locus is checked by a large number of tests, then correction for multiple comparison might be relevant. We ⁽¹⁵⁾ suggested the relative risk (RR) method of combining data on genetic markers with demographic information to obtain a more detailed characterization of genetic influence on survival and longevity. In this paper we extended the RR method to calculate the effects of hidden heterogeneity and interaction. We show that taking these effects into account can make the results of hypotheses testing significant after correction for multiple comparison. We discuss basic assumptions of this method and directions of further research.

	Materials and Methods

Top Abstract Materials and Methods Results Discussion Appendix References

 
Data
Five autosomal loci (APOB, REN, SOD1, SOD2, THO) and the mitochondrial locus (mtDNA) were considered. All these loci carry out biological functions that are expected to be crucial in successful aging and longevity. The APOB gene codes for apolipoprotein B, an exclusive protein of low-density lipoprotein (LDL), the main carrier of cholesterol in the blood. The REN locus codes for renin, an aspartylprotease that catalyzes the first step of the biosynthetic cascade leading to angiotensin 2. Both SOD1 and SOD2 code for superoxidodismutases that are involved in the elimination of superoxide radicals. The THO gene codes for tyrosine hydroxylase, the rate-limiting enzyme for the synthesis of catecholamines. Lastly, the mitochondrial genome contains genes for oxidative phosphorylation. The polymorphic systems were as follows: 3'APOB-VNTR [15 alleles ⁽¹⁶⁾], HUMREN.4 [five alleles, ⁽¹⁷⁾], SOD1-D21S223 [nine alleles ⁽¹⁸⁾], SOD2_C/T [C/T alleles ⁽¹⁹⁾], HUMTHO.1 [six alleles ^(17)(20)], mtDNA haplogroups [nine alleles, ⁽²¹⁾].

The data on genetic markers for the group of centenarians (aged 100 years and above) and the group of younger individuals (aged between 5 and 80 years) were obtained from samples collected in both Northern and Southern Italy. Altogether, 662 individuals were involved in the study; among them were 54 male and 143 female centenarians, and 220 male and 245 female younger individuals. Twenty-six male centenarians were from Northern Italy, and 28 were from Southern Italy. The number of female centenarians from the North was 83 and from the South 60. The younger group contained 75 men and 87 women from the North and 145 men and 158 women from the South. The distribution of this data by sex and area is shown in Table 1 .

In earlier research ⁽¹⁵⁾ we studied the effects of area, sex, and candidate allele on mortality and longevity by using the same sample of Italian data. The analysis of the allele–area and allele–allele interaction effects, performed in this paper, involves additional unknown parameters. To be able to produce reliable parameter estimates associated with interaction effects, we decided to aggregate data for men and women in this study and control only for regional differences. To show how data about area of residence and genetic markers may be included in the model, let us consider the hazard rate for an x-year-old individual for whom these data are available. Let us assume that the hazard rate for this individual may be represented in a Cox form as

(22). Here U₁ refers to the region (0 for the North and 1 for the South). Variable U₂ refers to the presence or absence of a candidate allele on a chromosome. Variable U₃ refers to the presence or absence of the same allele on the homologous chromosome, and µ₀(x) is an underlying hazard that characterizes the mortality rate for individuals with all . Thus, the survival function of an x-year-old individual can be represented as

Here

and

In our study we assume an equal contribution of each allele in homologous chromosomes in the survival process. This assumption is natural when one cannot distinguish between the effects of homologous chromosomes. This yields , so the relative risk in individuals homozygous for the candidate allele is RR₂², where RR₂ = e^ß2 is the risk in individuals heterozygous for the candidate allele. This assumption allows us to use one unknown parameter (instead of two) to characterize the effect of a candidate allele on survival. The same effect may be achieved by the introduction of a covariate U which takes values 0, 1, and 2 (0 for the absence of a candidate allele in the locus, 1 for the presence of one candidate allele, and 2 for the presence of two candidate alleles). In case of such a description, the relative risk for individuals with the double copy of a given allele will be squared automatically. We prefer to use our description because it is more convenient for the representation of allele–allele interaction effects.

In the case of diploid (autosomal) loci, we have groups. Each group is characterized by the area (0 for North and 1 for South), and one of three genotypes (0 for the absence of the candidate allele, 1 for the presence of this allele in one of two chromosomes, and 2 for the presence of this allele at both chromosomes). For the mtDNA locus we have groups. It is convenient to represent the initial proportions of individuals in each of six groups in terms of two parameters: p_0n (the initial proportion of individuals from Northern Italy) and p_0g (the initial frequency of the candidate allele in a population). For example, let us consider one of the six groups for the APOB locus (say, group k). Assume that this number refers to the group of individuals from Southern Italy who have one candidate allele, say, APOB31. This group is characterized by the survival function S_k(x) = S₀(x)^RR₁^RR₂. The respective initial frequency will be p_k = 2(1 - p_0n)p_0g(1 - p_0g); here the term (1 - p_0n) is the initial proportion of individuals from Southern Italy, and 2p_0g(1 - p_0g)is the initial proportion of heterozygous genotype in the case of Hardy–Weinberg equilibrium with one APOB31 allele. The other, say, group j of individuals from the North of Italy who have one APOB31 allele has the survival function S_j (x) = S₀(x)^RR₃ and the initial frequency p_j = 2p_0n p_0g(1 - p_0g). Similar representations can be written for each of the other four groups represented in likelihood (A1) in the Aand for the case of mtDNA.

Such representations for the initial proportions of individuals in the groups are based on two assumptions. The first is that the events of having a certain genotype with respect to a candidate allele and being a resident of Northern or Southern Italy for an individual are independent. The second is that the population of individuals in the study is in Hardy–Weinberg equilibrium. The first assumption is natural for populations in which genes under study are equally represented in all regions. This is usually the case in countries with high internal mobility and a relatively small area of residence. The second assumption is traditional in genetic studies. It provides a simple relationship between gene frequencies and genotype frequencies ⁽³⁾. Otherwise, one has to estimate five unknown initial frequencies for these groups. In principle, these assumptions are statistically testable. Unfortunately, the sample size of our data is not large enough to perform such testing.

Interaction Effects
To test for the interaction effects, we have to introduce the allele–area and allele–allele interaction terms in the survival model. In this new model the survival function S_i(x), i = 1,2,...6 is given in terms of the Cox proportional hazard model with conditional hazards

where U₁,U₂,U₃ are defined above, ß₂ = ß₃, and interaction variables U₄ and U₅ are respective combinations of U₁ (area), U₂ (allele), and U₃ (allele at homologous chromosome); that is, U₄ .

Hidden Heterogeneity
Unobserved heterogeneity, also called frailty, is a major concern in a survival analysis, where individual differences cannot be safely ignored. To take hidden heterogeneity in mortality into account, we use the gamma-frailty model with mean 1 and variance ^{2 (24)}. In accordance with this model,

(1)

The functional form of conditional survival function, S (x,U₁,...,U₅), is derived in the A. So, in addition to regression coefficients, ß_i, and initial frequencies, p_0n,p_0g, one has to estimate ². The respective estimation procedure is called the HRR method.

	Results

Top Abstract Materials and Methods Results Discussion Appendix References

 
Separate and Joint Analyses
First we performed a separate analysis of data on different alleles without taking interaction effects into account. Table 3 shows the results of these calculations.

One can see from Table 3 that the relative risks of the area of residence are all greater than 1. This observation allows us to perform the joint analysis of data for all alleles by assuming that the effects of the area of residence are the same for all candidate alleles. The results of this analysis are shown in Table 4 .

Estimation of Interaction Effects
The results of a joint analysis of a model with interaction effects are shown in Table 5 . One can see from Table 5 that the allele–area interaction is significant for the THO10 allele p . This interaction increases the hazard rate of the THO10 carriers if they are residents of the South of Italy. Without such an interaction, THO10 is a robust allele. The THO9 allele becomes neutral in Table 5 with the p value changed from .057 in Table 4 to .728 in Table 5 , but a significant allele–allele interaction effect is detected. This effect substantially increases the hazard of death for individuals with two THO9 alleles (homozygotes). In contrast, the allele–allele interaction for the REN11 allele has a robust effect. It reduces a carrier's death rate by a factor of 0.8, although the allele itself is classified as a frail allele.

View larger version (11K): [in this window] [in a new window]   Figure 1. Graph of the profile of the log-likelihood (LLK) as a function of 2.

Fig. 2 shows the age patterns of the estimated hazards for Italian individuals carrying zero, one, and two D21S1 alleles, respectively. These estimates are obtained without taking unobserved heterogeneity into account. Hazards are shown in a logarithmic scale. One can see from this figure that the hazard rate for carriers of the D21S1 allele is lower than that for noncarriers, so the D21S1 is a robust allele. All hazards have the same shape, as expected in the case of proportional hazard assumption. The estimates of hazard rates obtained in the case of separate and joint analyses practically coincide. Fig. 3 shows the respective graph for the REN11 allele.

View larger version (13K): [in this window] [in a new window]   Figure 2. Age patterns of the estimated hazards for individuals carrying zero (thick solid line), one (dashed line), and two (thin solid line) copies of the D21S1 allele in Italy. These estimates are obtained in the joint analysis of eight alleles without taking unobserved heterogeneity into account. The graph is shown in a logarithmic scale.

View larger version (10K): [in this window] [in a new window]   Figure 3. Age patterns of the estimated hazards for individuals carrying zero (thick solid line), one (dashed line), and two (thin solid line) copies of the REN11 allele in Italy. These estimates are obtained in the joint analysis of eight alleles without taking unobserved heterogeneity into account. The graph is shown in a logarithmic scale.

View larger version (12K): [in this window] [in a new window]   Figure 4. Age patterns of the estimated hazards for individuals carrying zero (thick solid line), one (dashed line), and two (thin solid line) copies of the D21S1 allele in Italy. These estimates are calculated by using a survival model with heterogeneity. The graph is shown in a logarithmic scale.

View larger version (11K): [in this window] [in a new window]   Figure 5. Age patterns of the estimated hazards for individuals carrying zero (thick solid line), one (dashed line), and two (thin solid line) copies of the REN11 allele in Italy. These estimates are calculated by using a survival model with heterogeneity. The graph is shown in a logarithmic scale.

View larger version (9K): [in this window] [in a new window]   Figure 6. Graph of empirical (filled circles) and estimated (solid line) proportions of the carriers of at least one D21S1 allele in Italy. The estimates correspond to the survival model with heterogeneity.

View larger version (9K): [in this window] [in a new window]   Figure 7. Graph of empirical (filled circles) and estimated (solid line) proportions of the carriers of at least one REN11 allele in Italy. The estimates correspond to the survival model with heterogeneity.

	Discussion

Top Abstract Materials and Methods Results Discussion Appendix References

 
Despite evident progress in establishing a connection between specific genes and longevity ⁽²⁶⁾, some aspects of genetic studies for humans deserve additional efforts. In this paper we show that the use of demographic information together with data on genetic markers can substantially improve our knowledge about the role of genes in human mortality and longevity. The possibility of estimating hazard rates and survival functions for candidate genes opens a new avenue for the study of genetic effects on survival. Now, in addition to demographic information, the data and results of epidemiological studies can also be involved in the genetic analysis of longevity. This is because these studies often estimate values of relative risks or odds ratios for individuals of different genotypes at some age intervals. Such estimates must be consistent with the values of the hazard rates of genes or genotypes obtained in genetic studies of aging and longevity. The use of not only genetic but also epidemiological data, together with life-table demographic information, increases the power of the estimation procedures, expands the class of identifiable models, and permits us to address more sophisticated questions about roles of genes and environment in human mortality and longevity. Thus future studies will involve more additional information about direct and indirect influence of genes on health and survival.

When the sample size of genetic data is large enough, one can use nonparametric methods to estimate hazard rates or survival functions of genotypes from genetic and demographic data ⁽²³⁾. In the case of a smaller sample size, the nonparametric estimates become unreliable, and semiparametric or parametric methods of data analysis can be used to improve the power of the estimation procedures. One has to realize, however, that the quality of approximation of real hazards may be compromised when parametric methods are used. The RR method discussed in this paper allows for the semiparametric estimation of the underlying hazards for respective candidate genes or genotypes. This method allows us to control for the effects of observed covariates, interaction effects of genes, and gene–environment interactions. Because the assumption of proportionality of hazard at the entire age interval may be unrealistic, one can fit the RR model at smaller intervals and get a closer approximation of the real hazard rates for candidate alleles. In this case one has to be sure that the sample size of the data is large enough to get reliable parameter estimates at subintervals.

The RR method for combining genetic and demographic information without taking the effects of heterogeneity and interaction into account was suggested earlier ⁽¹⁵⁾. The idea to apply the frailty model to the analysis of data on genetic markers has also been discussed ⁽²³⁾. This paper extends this approach to the analysis of allele–allele and allele–area interaction effects, as well as heterogeneity. The extended model contains more unknown parameters, and hence the power of the estimation procedure may be substantially reduced. To keep the number of unknown parameters at reasonable level, we did not distinguish between male and female mortality rates in this study. So in eight analyses shown in Table 3 and in the next Tables, the gender effect was not taken into account. However, we did control for regional differences. Table 3 and Table 4 show the results of preliminary calculations without interaction effects. The comparison of results from Table 3 and Table 4 with Table 1 in an eralier article ⁽¹⁵⁾ shows the difference in two detected alleles. The D21S6 allele is missed, and the allele THO9 is added in our study. This difference in the two studies may be attributed to the difference in the structure of respective models. All other alleles, qualified as frail and robust, are the same in both studies.

An analysis of the demographic situation in Italy shows that Italian survival presents some significant regional differences. In particular, the regions in Central Italy have a lower mortality than regions in Northern and Southern Italy. Northern Italy is less favorable for men. Southern Italy is less favorable for women ⁽²⁷⁾. Calculations based on our data show a higher relative risk for the residents of Southern Italy. This may be the result of a higher proportion of women from Southern Italy in the control group and a lower proportion of such women in the centenarian group in our sample. Note also that for the effects of area and sex to be represented correctly, the data should be consistent with the demographic structure of respective populations. Otherwise, the estimates of these effects cannot characterize the population. Moreover, this inconsistency in the sample may bias the results of the genetic analysis. The use of Cox's regression model with explicitly represented covariates allows for the separation of the effects of genes from the effects of other influential factors on survival. However, the question of sensitivity of the estimates of genetic parameters to the changes in a composition of a sample of the data deserves special study.

The ability to take unobserved heterogeneity into account without changing the basic estimation procedure is an important advantage of the RR method. Such heterogeneity may exist as a result of the effects of other genes, or environmental factors not included in the analysis. The probability distribution of hidden heterogeneity is usually unknown, so its proper approximation is an important problem in a frailty modeling. The use of gamma-distributed frailty became popular because of its technical convenience, and because of its ability to explain deviant dynamics of mortality rate at old ages ⁽²⁴⁾. Other distributions used in frailty modeling have also been discussed in the literature ⁽²⁸⁾. A recent comparative analysis of several frailty distributions ⁽²⁹⁾ used in genetic studies of susceptibility to death and longevity shows that gamma-frailty is a reasonable model for analyzing the effects of hidden heterogeneity in survival. Fig. 1 shows the profile of the likelihood as a function of the variance of a frailty distribution calculated for a joint analysis of eight candidate alleles. A better strategy would be to estimate individual heterogeneity parameters for each candidate allele. Our attempts to estimate such parameters resulted in large standard errors. It is clear that more data are needed to realize this idea.

The analysis performed in this paper assumes that (i) the initial proportions of genotypes in all cohorts represented in a cross-sectional study are the same and that (ii) the survival functions of individuals carrying candidate alleles do not depend on the birth year of the cohorts. These assumptions were used in all earlier analyses of centenarian data cited in this paper. It is clear, however, that these assumptions are not realistic. In ⁽²³⁾ we performed an analysis of sensitivity of the parameter estimates to the violations of these assumptions. The analysis shows that condition (i) is most sensitive to migration. This condition can be controlled by a historical demographic analysis of the data. The effects of differential total mortality between cohorts on observed gene frequencies [violation of assumption (ii)] depend on the patterns of allele–environment interaction. The details of such an interaction are unknown and cannot be estimated from the data used in our study. These effects may be small, if mortality rates for candidate alleles change proportionally, or large, if such interaction is more complex ⁽²³⁾. Survival follow-up of individuals who provided genetic information will considerably strengthen the data.

The RR method developed in this paper is applied to several alleles in each of selected loci to test whether they are associated with longevity. For a given locus this procedure deals with a multiple testing, and care must be taken in order for the results not to be misinterpreted. Specifically, the need for adjustment for multiple comparisons depends on the question to be addressed by statistical analysis ⁽³⁰⁾. Indeed, let us assume that one is interesting in testing the null hypothesis, H₀, that a selected allele, say APOB31, is a neutral one (i.e., whether the respective relative risk of death is equal to 1). Because only one test is applied to this allele, no adjustment for multiple comparison is needed.

If, however, one would like to test the null hypothesis, H_0APOB, that all alleles in the APOB locus are neutral, then an adjustment for multiple comparison is needed. In this case one analyzes data on each of 15 different alleles available in this locus, and test 15 null hypotheses, H₀j_, j = 1,2,...15, that the selected allele is a neutral one (i.e., whether the respective relative risk of death is equal to 1). Let be a common significance level for each of these tests. It is clear that the significance level for testing the null hypothesis for the APOB locus, _APOB, is related to as follows:

One can see that in the case of high polymorphic loci, the value of must be extremely small to provide an adequate significance level for testing the H₀ hypothesis that all alleles in the APOB locus are neutral. One can see from Table 5 and Table 6 that taking into account interaction and heterogeneity effects decreases the p values for the estimates of risks associated with some alleles. For example, the p value for the THO10 allele was .04 in Table 4 ; then it became .002 in Table 5 when interaction effects were taken into account; and then it became less than .001 in Table 6 when, in addition, hidden heterogeneity was taken into account. This reduction in p values makes it possible to reject the null hypothesis for the THO locus (with six alleles) after correction for multiple testing.

The analysis of genetic data performed in this paper assumes that the mortality rate for some genes is lower (or higher) than that for the others at the entire demographic age interval. In this case the estimates of mortality rates for robust and frail alleles do not intersect, the estimates for proportions are monotone functions of age, and both GF and RR methods may be used for classification of alleles as robust, frail, and neutral. In addition, the RR method allows us to estimate the hazard rates and survival functions for respective candidate genes or genotypes. The empirical patterns of gene frequencies shown in Fig. 4 and Fig. 5 suggest that the proportional hazard assumption may be too simplified and that the age trajectories of gene frequencies are not necessarily monotone functions of age. The reason for this may be an intersection of respective mortality curves for candidate genes. The presence of such intersections has been reported in studies ^(23)(31) where other approaches to the analysis of genetic data on centenarians have been used. If hazard rates for genotypes intersect at a very young or at a very old age, the estimates of hazards calculated by the RR method still can approximate the average genetic effects on survival. However, in more complicated cases, important details related to the genetic regularities of aging process may be missed. For this reason the use of several approaches to the analysis of genetic data is recommended ⁽²³⁾.

The intersection of hazard rates for carriers of different genes or genotypes suggests that survival to age 100 and more is not necessarily related to the presence of "robust genes," as it was generally believed before ⁽⁵⁾. Extended survival might be the result of a more sophisticated process of an organism's adaptation to the stresses of life. As part of this adaptation, genes responsible for a higher mortality at the beginning or in the middle of life may become beneficial at an advanced age ⁽²³⁾. This effect may illustrate the important relationship among the ability to adapt, aging, and life span. Coping with the stresses of life, the organisms of individuals with disadvantageous genotypes are able to develop a higher "adaptation capacity" to the inevitable stresses of aging than those individuals with robust genotypes. If such adaptation mechanisms are in fact in effect, then the candidate genes also have to be searched for among those genes that produce a survival disadvantage earlier in life.

	Acknowledgments

 
This research was partly ( 40%) financed by the Italian Ministero Università Ricerca Scientifica Technologica (MURST) 1998–2000 project, "Longevity Determinants in Humans: the Model of Centenarians," by Instituto Nationale Riposo Cura Anziani (INRCA), Ancona (Italy), and by Grant PO1 AG08761-01 from the National Institutes of Health/National Institute on Aging. The authors are grateful to the reviewers, whose helpful comments improved the paper substantially. They also thank Cecilia Tomassini and Elisabetta Barbi for their qualified expertise concerning regional differences in Italian mortality, and Baerbel Splettstoesser and Karl Brehmer for help in preparing this paper for publication.

Received June 17, 1999

Accepted November 10, 1999

	Appendix

Top Abstract Materials and Methods Results Discussion Appendix References

 
The Likelihood Function
Let _i(x), i = 1,2,...N be the proportion of x-year-old individuals from the ith group in some cross-sectional study performed in year T, and let N_ix be the respective numbers of individuals observed in this study. Then the likelihood function of the data is

(A1)

where

and

(A2)

Here p_i represents the initial proportion of individuals from group i. Note that if data start from age x₀, then p_i denotes respective proportion at this age. For each the survival function S_i(x) is represented in terms of the Cox ⁽²²⁾ proportional hazard model with conditional hazards

(we assume here that ß₂ = ß₃) and with the respective combination of values for U₁, U₂, and U₃.

Estimation Procedure
The likelihood, 1, must be maximized with respect to parameters p_0n, p_0g and risks RR_i = e^ßi, i = 1,2 under the constraint

(A3)

Here the values of survival functions S(x) are taken from the official demographic life tables for the Italian population for 1996. The values of S_j(x) depend on

and RR_i = e^ßi, i = 1,2. The estimation procedure, which takes into account constraint (A3), starts with the maximization of likelihood (A1) with respect to initial proportions p_0n, p_0g and risks RR_i, , taking the initial guess of S₀(x) to be equal to, say, S(x) (which is a known function of x). Then the estimates of p_0n, p_0g and risks RR_i, are substituted into A3, from which the second guess of S₀(x) is calculated. This guess is substituted in A1 with unknown parameters, p_0n, p_0g, and RR_i, i Then likelihood (A1) is maximized again to produce a second guess of these parameters, and the procedure is repeated until convergence occurs.

The Likelihood Function in the Case of Joint Analysis
The joint analysis of data for several candidate alleles makes sense when some parameters in eight likelihood functions are the same. In our analyses we assume the same risk for the area of residence R₁ (i.e., regression coefficient ß₁ is the same for all candidate genes). In this case the likelihood function of joint data is

(A4)

Here

n = 8, and

(A5)

Here p_i^k represents the initial proportion of individuals in group i for the k_th candidate allele.

Survival in Heterogeneous Population
Let U denote vector U₁,U₂,...,U₅, and let S(x|Z,U) = S(x,U₁,U₂,...,U₅,Z) be conditional the survival function corresponding to hazard ⁽¹⁾:

(A6)

where

Let frailty Z be gamma distributed with mean 1 and variance ². Then

(A7)

here

is the probability density function for gamma distribution with the shape parameter k, and the scale parameter , and (k) denotes a gamma function. The mean of this distribution is

and the variance

To calculate A7, note that this equation can be rewritten in the form

(A8)

and because the integral in A8 is equal to 1 we get

Taking into account that

and hence

we get

(A8)

This survival function is used to characterize proportions of individuals in respective groups in the likelihood function.

	References

Top Abstract Materials and Methods Results Discussion Appendix References

 

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