2.4. Statistical method

The statistical method we are going to apply is based on a relative risk approach that has been recently proposed and described by Yashin et al. (1998, 1999b, 2000), Tan et al. (2001b). We define the relative risk of one observed genotype r as the ratio of hazard of death for carriers, m(x, r), to that for the non-carriers or the baseline hazard, m (x). Then, according to the proportional hazard 0 assumption, m(x, r)(cid:3)rm (x). The corresponding / 0 survival function for the carriers is,

2.1. Samples

The data used in this study is an updated version of the data by which TH gene/longevity associa- tion was first discovered (De Benedictis et al., 1998). All individuals in the sample are unrelated, clinically healthy and are from various socio- economic backgrounds. For each individual, ori- gin and residence in Italy were checked by inter-

Q. Tan et al. / Mechanisms of Ageing and Development 123 (2002) 1403(cid:2)1410 /

where mn i(x), fn i(x) are numbers of males and females at age x in each of the three genotypes. Parameters that maximize the likelihood function are estimated by the maximum likelihood proce- dure in Gauss software (Aptech System, 1996).

Here, s (x) is the baseline survival distribution 0 corresponding to the baseline hazard function, H (x) is the cumulative hazard at age x. Although 0 r can take any value greater than zero, a genotype with r larger than 1 increases the hazard of death, while a genotype with r smaller than 1 reduces it.

There are two important features in this ap- proach. First, no parametric form of the baseline survival function is imposed. The estimation procedure is based on a two-step algorithm described by Tan et al. (2001b). In this procedure, the sex-specific baseline survivals are calculated non-parametrically from Eq. (2) for given risk and frequency parameters. The so obtained distribu- tion-free baseline survivals are then introduced in Eq. (3) for parameter estimation by maximizing the likelihood function. The procedure is iterated until the maximum likelihood is achieved. Second, the risk parameters for heterozygous and homo- zygous genotypes are specified separately, which allows for detecting recessive and additive effects in gene action.

For a given single allele, there are three geno- types that can be observed among the subjects, that is, individuals with no such allele designated as ((cid:4)(cid:4)), individuals with one such allele   / / / / / (cid:4)) and individuals with two such alleles ((cid:1)(cid:1)). If / / / frequency of the allele is p, then the corresponding genotype frequencies can be calculated as (1(cid:4)p)2, / 2p(1(cid:4)p) and p2 according to the binomial dis- / tribution. Risks on hazard of death are defined for each genotype but on a relative base, that is, we take the genotype ((cid:4)(cid:4)) as the reference with / / relative risk 1, and relative risks for heterozygous (—+/+-) and homozygous ((cid:1)(cid:1)) genotypes r / / / / / / 1 and r , respectively. Further, we define the risk of 2 gene(cid:2)sex interaction, r , as the risk of male / g(cid:5)s carriers to that of female carriers. Then the average survival at age x for a mixed population is (Vaupel

3. Results

Table 1 shows the distribution of subjects according to age group, sex and HUMTHO1.STR genotype. Table 2 shows, for each allele, the estimated initial allele frequency and the relative risks by genotype. Significance levels for risk parameters (relative risks for the allele and for gene(cid:2)sex interaction) were determined by testing / the statistical differences between the estimated risks and 1, with the null hypothesis H :r(cid:3)1. The 0 / probability of a Type I error is a(cid:3)0.05. /

Allele 10* is interesting in that both heterozy- gous and homozygous genotypes display beneficial influences that reduce the carriers’ hazards of death. Corresponding relative risks are 0.920 (p (cid:4)value(cid:3) /0.013) and 0.851 (p (cid:4)value(cid:3) /0.005) for the heterozygous and homozygous genotypes, respectively. However, after correction for multi- ple comparisons, we obtain a probability of making type I error a?(cid:3)1(cid:4)(1(cid:4)0.013)6 :0.076 / / / / for the risk of allele 10* heterozygous genotype, that is borderline significant, while the risk for the 10*10* homozygous genotype remains significant (a?(cid:3)0.030). Interestingly, the relative risk of the /

m / f / males and females obtainable from population statistics. The corresponding genotype frequencies at age x for the three genotypes in males are p (x)(cid:3)(p2s (x)r 2r g(cid:5)s=s¯ (x)); p (x)(cid:3) m 2 0;m m m 1 (2p(1(cid:4)p)s (x)r 1r g(cid:5)s=s¯ (x)); p (x)(cid:3)((1(cid:4) 0;m m m 0 p)2s (x)=s¯ (x)): In the same way, we can calcu- 0;m m late p (x), p (x) and p (x) for females. Based on f 2 f 1 f 0 multinomial distribution (Evans et al., 1993), the likelihood function can be written as,

of subjects by age group, sex and genotype