Start with a raw genetic score. For individual \(i\), multiply the coded genotype at each causal marker by that marker’s effect and add the contributions:
\[ c_i = \sum_{j=1}^{m} x_{ij} b_j, \qquad g_i = \sqrt{\pi}\,\frac{c_i-\bar c}{s_c}. \tag{A.1}\]
Read the second expression in three steps. Subtracting the mean score \(\bar c\) centers the scores on zero. Dividing by their sample standard deviation \(s_c\) gives variance one. Multiplying by \(\sqrt{\pi}\) gives variance \(\pi\), because variance changes by the square of a multiplier. In the first chapter, \(m=20\) is the number of causal markers and \(\pi=0.5\) is the genetic allocation. \(x_{ij}\) is individual \(i\)’s genotype at marker \(j\), and \(b_j\) is its raw effect.
In default fixed mode, normal random values \(z_i\) undergo the same process:
\[ e_i = \sqrt{1-\pi}\,\frac{z_i-\bar z}{s_z}, \qquad y_i=g_i+e_i. \tag{A.2}\]
The bar denotes the sample mean and \(s_z\) the sample standard deviation. The multiplier gives residual sample variance \(1-\pi\). This fixes the sample mean and variance, so the adjusted residuals are no longer independent normal observations. In random mode the package instead draws independent normal residuals with mean parameter zero and variance parameter \(1-\pi\), without these adjustments. The resulting sample mean and variance fluctuate.
These are package definitions: .component_raw(), .genetic_matrix(), and .draw_residual() implement these steps. They describe the first chapter’s single additive layer, with no vQTL or other components.
This appendix derives the identities used in the first chapter directly from sample definitions. It does not assume random mating, Hardy-Weinberg equilibrium, linkage equilibrium, or a particular distribution for the realized vectors. Use paired finite vectors with \(n>1\); variance ratios additionally require a nonzero phenotypic sample variance.
Here \(s_y^2\), \(s_g^2\), and \(s_e^2\) denote sample variances, and \(s_{ge}\) denotes sample covariance. A bar denotes a sample mean; \(n\) is the number of individuals. Each sum adds one term for each individual.
For \(y_i=g_i+e_i\), subtract the corresponding sample means and expand the square:
\[ \begin{aligned} s_y^2 &=\frac{1}{n-1}\sum_i\left[(g_i-\bar g)+(e_i-\bar e)\right]^2\\ &=\frac{\sum_i(g_i-\bar g)^2}{n-1} +\frac{\sum_i(e_i-\bar e)^2}{n-1} +2\frac{\sum_i(g_i-\bar g)(e_i-\bar e)}{n-1}\\ &=s_g^2+s_e^2+2s_{ge}. \end{aligned} \tag{A.3}\]
Centering and dividing a nonconstant vector by its sample standard deviation gives sample variance one. Multiplication by \(\sqrt{\pi}\) then gives sample variance \(\pi\). Applying this argument to the first chapter’s two components in fixed mode gives \(s_g^2=\pi\) and \(s_e^2=1-\pi\). Substituting these into the variance identity gives
\[ R_G = \frac{s_g^2}{s_y^2} = \frac{\pi}{1+2s_{ge}}. \tag{A.4}\]
The two variance allocations add to one; the covariance supplies the remaining term in the denominator. \(R_G\) is the genetic-to-phenotypic sample variance ratio. This reduced formula applies to fixed mode in this example. In random mode use the full identity, because \(s_e^2\) need not equal \(1-\pi\).
These are algebraic derivations. The normalization steps themselves are package definitions, with code provenance in the evidence register.
The following deliberately negatively associated vectors disprove the claim that every realized genetic-to-phenotypic sample variance ratio must lie in \([0,1]\). They do not represent a typical independent residual draw from the package.
g <- c(-1, 0, 1)
e <- -0.75 * g
y <- g + e
tibble::tibble(
quantity = c(
"Genetic variance", "Residual variance", "Covariance",
"Phenotypic variance", "Genetic / phenotypic variance"
),
value = c(var(g), var(e), cov(g, e), var(y), var(g) / var(y))
)| quantity | value |
|---|---|
| Genetic variance | 1.0000 |
| Residual variance | 0.5625 |
| Covariance | -0.7500 |
| Phenotypic variance | 0.0625 |
| Genetic / phenotypic variance | 16.0000 |
The counterexample concerns the diagnostic ratio. It does not redefine the usual population heritability under a model with the required variance partition.