Multivariate Falconer Minimum Sample Size

Finds the minimum integer total sample size for Pillai MANOVA or the minimum selected case-control sample for a joint-threshold genotype chi-square test.

qtl_multivariate_mssn_full(
  power,
  alpha,
  qtl_var,
  tau,
  pd,
  cor_matrix,
  test = c("pillai", "threshold_chisq"),
  x_upper = NULL,
  x_lower = NULL,
  k = 1,
  verbose = TRUE
)

Arguments

power

Target power in (0, 1).

alpha

Significance level in (0, 1).

qtl_var

Numeric vector of phenotype-specific QTL variances in (0, 1).

tau

Numeric vector of phenotype-specific dominance/additivity ratios.

pd

Shared increaser-allele frequency in (0, 1).

cor_matrix

Positive-definite phenotype correlation matrix. Its order must match qtl_var and tau.

test

Either "pillai" or "threshold_chisq".

x_upper, x_lower

Vectors of upper- and lower-tail percentages. A case satisfies every upper threshold (joint AND); a control satisfies every lower threshold (joint AND).

k

Control/case ratio for the threshold chi-square design.

verbose

Logical; whether to print a polished summary.

Value

An object of class "qtl_multivariate_mssn_full". Both modes include test, target and achieved power, alpha, and the full falconer model. The Pillai result includes integer total N, historical_fractional_mssn, genotype counts, NCP, degrees of freedom, critical value, and all pillai intermediates. The threshold result includes selected case/control/total MSSNs, target and achieved NCPs, internal S, thresholds and integration diagnostics, expected genotype counts, sparse-cell diagnostics, and separately labelled expected source-population screening counts.

Details

Pillai MSSN is found by an integer search because its denominator degrees of freedom depend on sample size. For threshold selection, the statistical MSSN is kept separate from the expected population screening burden. The Pillai branch recomputes its noncentral F distribution at each candidate total N; the returned integer is the first design attaining the target power. The historical fractional root is retained only for comparison. The threshold branch uses direct multivariate-normal rectangle integration from Eqs. 6.11–6.14 and genotype chi-square Eq. 1.22; it solves for selected cases and controls and reports source-population screening expectations separately. The two branches therefore return different sample-size units.

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 6.2, Eqs. 6.9–6.14, pp. 332–333; validation in Section 6.2.4, pp. 336–339. doi:10.1007/978-3-030-61121-7 .

Gordon, D., Londono, D., Patel, P., Kim, W., Finch, S. J., & Heiman, G. A. (2017). An analytic solution to computation of power and sample size for genetic association studies under a pleiotropic mode of inheritance. Human Heredity, 81(4), 194–209. doi:10.1159/000457135 .

Pillai, K. C. S. (1955). Some new test criteria in multivariate analysis. Annals of Mathematical Statistics, 26(1), 117–121. doi:10.1214/aoms/1177728599 .

Genz, A., & Bretz, F. (2009). Computation of Multivariate Normal and t Probabilities. Springer. doi:10.1007/978-3-642-01689-9 .

See also

qtl_multivariate_power_full, qtl_falconer_parameters, and qtl_threshold_chisq_mssn.

Examples

cor_matrix <- matrix(c(1, 0.15, 0.15, 1), 2, byrow = TRUE)
qtl_multivariate_mssn_full(
  power = 0.95, alpha = 5e-8, qtl_var = c(0.01, 0.005),
  tau = c(0, 0.5), pd = 0.25, cor_matrix = cor_matrix,
  test = "pillai", verbose = FALSE
)