Falconer Parameters for a Single Quantitative Trait

Source: R/falconer_quantitative.R

Calculates genotype-specific normal-mixture parameters for a standardized continuous trait under the single-locus Falconer model.

qtl_falconer_parameters(qtl_var, tau, pd, verbose = TRUE)

Arguments

qtl_var

Variance in the standardized trait attributable to the QTL. Must be in \((0,1)\).

tau

Finite dominance-to-additivity ratio \(\delta/a\).

pd

Frequency of the increaser allele. Must be in \((0,1)\).

verbose

Logical. If TRUE, prints a formatted parameter summary. Set to FALSE to suppress all console output.

Value

An object of class "qtl_falconer_parameters". The list contains qtl_var, tau, pd, p_plus, additive effect a, dominance effect delta, centering constant m, genotype means mu, mixing proportions pi, residual_variance, residual_sd, weighted_mean, and total_variance.

Details

Genotypes are ordered as zero, one, and two copies of the increaser allele. Their Hardy-Weinberg mixing proportions are \(((1-p_d)^2, 2p_d(1-p_d), p_d^2)\). The population-centering constant makes the genotype-weighted trait mean zero, and the residual variance is \(1 - V_{QTL}\).

This is a classical, Falconer-style quantitative-genetic parameterization as formulated for this study-design framework by Gordon, Finch, and Kim (2020), Chapter 6, Section 6.1. The population mixture is Eq. 6.1 (p. 324); the additive effect, dominance effect, and centering formulas on p. 325 are unnumbered. The function name does not assert that every exact implemented expression is a numbered equation in Falconer and Mackay.

References

Gordon, D., Finch, S. J., & Kim, W. (2020). Heterogeneity in Statistical Genetics: How to Assess, Address, and Account for Mixtures in Association Studies. Springer, Chapter 6, Section 6.1, pp. 324–325. doi:10.1007/978-3-030-61121-7 .

Fisher, R. A. (1918). The correlation between relatives on the supposition of Mendelian inheritance. Transactions of the Royal Society of Edinburgh, 52, 399–433. doi:10.1017/S0080456800012163 .

Lynch, M., & Walsh, B. (1998). Genetics and Analysis of Quantitative Traits. Sinauer Associates.

Falconer, D. S., & Mackay, T. F. C. (1996). Introduction to Quantitative Genetics, 4th ed. Longman.

Examples

qtl_falconer_parameters(qtl_var = 0.025, tau = 0.5, pd = 0.15)
#> 
#> ==========================================================================
#> Falconer Quantitative Trait Parameters
#> ==========================================================================
#> 
#> Input Parameters
#> QTL variance:                                        0.0250
#> Dominance/Additivity ratio (tau):                    0.5000
#> Increaser allele frequency:                          0.1500
#> --------------------------------------------------------------------------
#> 
#> Derived Parameters
#> Additive effect (a):                                 0.2280
#> Dominance effect (delta):                            0.1140
#> Centering mean (m):                                  0.1305
#> Residual variance:                                   0.9750
#> Residual SD:                                         0.9874
#> --------------------------------------------------------------------------
#> 
#> Genotype-Specific Quantities
#>                                  Genotype 0     Genotype 1     Genotype 2
#> Mixing proportion:                   0.7225         0.2550         0.0225
#> Trait mean:                         -0.0975         0.2445         0.3585
#> --------------------------------------------------------------------------
#> 
#> Validation
#> Weighted population mean:                          0.000000
#> Total variance:                                    1.000000
#> --------------------------------------------------------------------------