Let i = 0, 1 represent not-carrying and carrying hereafter; and j as not-exposed and exposed hereafter. Let p11 be the proportion of x-year-old individuals who are a carrier and not-exposed, p10 and p01 are non-carrier and not-exposed, and neither carrier nor exposed. Note that the sum of p11, p10, p01, and p00 of FOXO3A and whether carrying the minor allele of 1A-209, and the estimates of the odds ratios of survival probability (RS ij/00) from middle age to ages 100+ measuring the independent and joint effects of the genotypes.

If the OR of survival RS ij/00 smaller than 1.0, the ij combination of the gene-fixed exposure is positively or negatively associated with longevity. The frequency distribution of the genotypes defined by combinations of whether carrying the minor allele of one of the three SNPs of FOXO3A and whether carrying the minor allele of 1A-213, and the estimates of the odds ratios of survival probability (RS) from middle age to ages 100+ measuring the independent and joint effects of the genotypes are shown in Table 2.

The mathematical and demographic derivation is presented in Appendix 1: p-values and 95% confidence intervals (CIs) (23) can be performed to assess whether the estimates of an OR of survival (RS in Formula (1) and RS in Formula (2)) is significantly different from 1.0, using a simple procedure in standard statistical software (e.g., STATA, SAS, or SPSS). Note that RS10/00, RS01/00, and RS11/00 represent the independent impact of the genetic variant (or another fixed exposure) on survival from age x to x+n in the absence of another fixed exposure (including genetic variant), and that RS11/00 represents the joint impact of the genetic variant and the fixed exposure on survival from age x to x+n when both of them are present.

The basic and extended FAD methods are based on the fundamental demographic insight that the prevalence of a fixed attribute (e.g., genetic variant, birth weight, childhood conditions, and serious disease(s) suffered earlier in life) and combinations of two fixed attributes in a population can change with age, even though no individual can change his or her variant. Therefore, much can be learned about the impact of the fixed attribute (including genetic variant) and the combinations of the fixed attributes on survival.

In Formula (2), individuals are identified as carriers or non-carriers, and a carrier could have 1 or 2 copies of the minor allele of the SNPs in the application of our present study.study. Hence, we are implicitly assuming that the minor allele is dominant rather than recessive or additive. A modified version of the FAD approach and application is required to analyze recessive or additive models. Gene-environment interaction (GE) is defined as a different effect of an environmental exposure on disease and health risk in persons with different genotypes, or alternatively, a different effect of a genotype on disease and health risk in persons with different environmental exposures (24, 25). Similarly, gene-gene interaction (GG) is defined as a different effect of a genotype on disease and health risk in persons with and without another genotype. Applying the standard definition and method for estimating the GE or GG interactions (25), we present in Appendix 2 the procedures for estimating the effects of gene-fixed exposure interactions on survival from age t to x + n, using the ORs of survival (RS,i=0,1;j=0,1) derived by ij/00 Formula (2). For cases and younger controls, the estimates of the logistic regression not adjusted for mathematical proof and the empirical evidence, regression without controlling for covariates are not presented here due to space limitation but available upon request. This fact verifies the statistical property that much the probability of survival from age t to x + words, the FAD method provides clearer interpretations and presents meaningful syntheses of the case-control genetic analysis on longevity. As the two approaches produce exactly the same formulas (1) and (2) can be used to analyze cross-sectional data from different cohorts of long-lived cases and younger controls or follow-up data from the same cohort. In the case of analyzing the cross-sectional data, we actually connect the younger cohort and the long-lived cohort into what the demographic textbooks call a hypothetical cohort and estimate the OR of survival based on proportions of the genetic variant and fixed exposure among the long-lived and younger cohorts (i.e., at advanced and younger ages in the hypothetical cohorts).

It is important to note that if one uses cross-sectional data, both the FAD method expressed in Formulas (1) and (2) and the conventional case-control statistical genetic approach assume that the initial distributions of the genetic variant, the fixed exposure, and their combinations, and their association with longevity do not differ substantially between cohorts of long-lived individuals and younger controls, that is, there is no substantial problem of population stratification (see more detailed discussions in Appendix 1 on potential bias if this basic assumption is violated). If cohort follow-up data are used, one needs to assume that the loss to follow-up from age t to x + n is not correlated with respondents' combination of the genetic variant and the fixed exposure under study. Serious attention must be paid to this issue in applying either the conventional statistical genetic approach or the FAD method for identifying genetic contributions to longevity.