of subjects by age group, sex and genotype

Genotype 13(cid:2)34 35(cid:2)64 65(cid:2)99 100(cid:1) Total / / / / Male Female Male Female Male Female Male Female Allele 6 ((cid:4)(cid:4)) 37 60 70 57 55 95 32 82 488 / / ((cid:4)(cid:1),  31 38 60 51 61 81 31 48 401 / / / / ((cid:1)(cid:1)) 3 9 9 6 20 12 5 12 76 / / Allele 7 ((cid:4)(cid:4)) 44 70 102 69 110 130 44 105 674 / / ((cid:4)(cid:1),  27 33 36 43 25 53 21 33 271 / / / / ((cid:1)(cid:1)) 0 4 1 2 1 5 3 4 20 / / Allele 8 ((cid:4)(cid:4)) 53 86 108 91 102 146 48 115 749 / / ((cid:4)(cid:1),  17 18 30 20 34 41 18 27 205 / / / / ((cid:1)(cid:1)) 1 3 1 3 0 1 2 0 11 / / Allele 9 ((cid:4)(cid:4)) 42 59 83 79 89 124 48 90 614 / / ((cid:4)(cid:1),  27 43 47 30 43 60 20 51 321 / / / / ((cid:1)(cid:1)) 2 5 9 5 4 4 0 1 30 / / Allele 10* ((cid:4)(cid:4)) 49 76 83 69 88 111 45 73 594 / / ((cid:4)(cid:1),  21 27 53 41 43 65 20 56 326 / / / / ((cid:1)(cid:1)) 1 4 3 4 5 12 3 13 45 / / Allele 10 ((cid:4)(cid:4)) 66 102 135 111 134 184 68 134 934 / / ((cid:4)(cid:1),  5 5 4 3 2 4 0 7 30 / / / / ((cid:1)(cid:1)) 0 0 0 0 0 0 0 1 1 / /

of log hazard of death is additive on the estimated relative risk. In Fig. 1, we plot the observed and the estimated frequencies of alleles 8 and 10* carriers by age. While frequency for allele eight carriers is almost constant (Fig. 1A), the frequency

heterozygous genotypes for allele 10* is approxi- mately the square root of the risk of the 10*10* homozygous genotype, thus suggesting a dose- effect for this allele. According to the model specification, allele 10* phenotypic value in term

Table 2

Estimated allele frequency and relative risks by genotype for each allele

Allele Frequency Heterozygous genotype Homozygous genotype Gene(cid:2)sex interaction / Risk SE P-value Risk SE P-value Risk SE P-value 6 0.290 1.055 0.034 0.107 1.058 0.055 0.295 0.906 0.042 0.025 7 0.161 1.026 0.038 0.501 0.993 0.074 0.927 0.956 0.049 0.365 8 0.118 1.039 0.043 0.356 1.147 0.141 0.298 0.899 0.049 0.040 9 0.210 0.983 0.034 0.618 1.409 0.178 0.022 1.064 0.052 0.218 10* 0.201 0.920 0.032 0.013 0.851 0.053 0.005 1.111 0.054 0.039 10 0.036 0.916 0.080 0.291 2.548 1.077 0.151

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

Freq.Freq, 0.30 (A)    Est.A Obs.0.25A& 4A0.200.15 —      20 40 60 80 100Age(B)0.450.40 xx0.35Est.3K Obs.0.30 x20 40 60 80 100Age

method, a simulation study has been performed by generating 10,000 data sets having exactly the same data structure (size, age and sex composi- tions) as our sample. To be simple, we assume there is one allele at a locus with relative risk on hazard of death r(cid:3)0.9 for heterozygous carriers / and r2 (cid:3)0.81 for homozygous carrier. We also / assume the allele has no sex-specific effect in that r (cid:3)1. Frequency for the allele is set to 0.2. For g(cid:5)s / each of the 10,000 data sets, the method described above is applied to retrieve the parameters. Although we assume the risk of heterozygous genotype is the square root of the risk of homo- zygous genotype in the simulation, we assign two irrelevant risk parameters for the two genotypes in the estimation to see if our model can correctly capture their corresponding true values. Fig. 2 presents the results. The median of allele frequency from the 10,000 estimates is 0.200 and 95% range {0.180, 0.220} (Fig. 2A). The medians for esti- mates on relative risks are 0.895 for heterozygous genotype with 95% range {0.822, 0.970}, 0.801 for homozygous genotype with 95% range {0.687, 0.934} and 1.001 for gene(cid:2)sex interaction with / 95% range {0.875, 1.138} (Fig. 2B(cid:2)D). All the / parameter estimates converge to their true values set in the simulation. It is interesting to see that the medians of risk estimates on heterozygous and homozygous carriers of the allele are related as 0.8952 (cid:3)0.801. This shows again that the model / retrieves the parameters used for the simulation.

Fig. 1. Observed and estimated frequencies for carriers of allele 8 (A) and allele 10* (B) by age.

for 10* allele carriers climbs up as age advances (Fig. 1B) revealing obvious survival advantages for 10* allele carriers.

As for the other alleles, allele 9 seems to be recessive and detrimental, because the homozy- gotes for this allele tend to have increased hazard of death over the ages (r(cid:3)1.409 with SE(cid:3)0.178, / / P(cid:3)0.022). However, such a risk value becomes / non-significant after adjustment for multiple com- parisons.

4. Discussion

Lastly, as for gene(cid:2)sex interaction, we see from / Table 2 that alleles 6, 8 seem to decrease and allele 10* to increase the risks of death in males. However, also in this case, the P values become non-significant after adjustment for multiple com- parisons.

Given the complexity of the longevity trait, a demanding issue is to develop new models that are capable to capture the various gene trajectories relating to the trait which occur as the population ages and survival selection operates. The present study could serve as a hint in this direction, because it introduces an analytical method which permits to reveal the interaction between the alleles of a gene (TH) associated with human longevity. The method is applied to a large data set, so that

Note in Table 2, we only estimate one risk for carriers of allele 10 (non-significant) because we only observe one homozygous carrier of the allele in the sample (Table 1).

1500100050000.17 0.19 0.21 0.23Allele Frequency1200800400(C)800800600400200(B)0.8 0.9 1.0 1.1Risk Heter.(D)  05 06 07 08 0.9 1.0Risk Homo.08 09 1.0 1.1 1.2 1.3Risk GxS

Fig. 2. Parameter estimates from 10,000 simulated data for: (A) allele frequency (true value 0.2); (B) risk of heterozygous genotype (true value 0.9); (C) risk of homozygous genotype (true value 0.81); and (D) risk of gene(cid:2)sex interaction (true value 1). /