Family-Based (TDT) Minimum Sample Size Necessary

Computes the minimum number of affected trios required to achieve a specified power for the transmission disequilibrium test (TDT) with no effect or one selected effect: phenotype misclassification, locus heterogeneity, or genotype misclassification. input_mode lets the transmission probabilities come either from a genetic model ("model_based", the default) or directly from user-supplied expected transmission and non-transmission counts ("model_free").

tdt_mssn(
  target_power,
  input_mode = c("model_based", "model_free"),
  pd = NULL,
  prev = NULL,
  R1 = NULL,
  R2 = NULL,
  alpha = 0.05,
  delta_prime = 1,
  misclass_rate = 0,
  heter_rate = 0,
  ET = NULL,
  ENT = NULL,
  n_trios = NULL,
  verbose = TRUE,
  effect = c("none", "phenotype_misclassification", "locus_heterogeneity",
    "genotype_misclassification"),
  genotype_misclassification_rate = NULL
)

Arguments

target_power: Numeric in (0,1). Desired power for the TDT.
input_mode: Character. One of "model_based" (default) or "model_free". See Details.
pd: Numeric in (0,1). Frequency of the disease (high-risk) allele at the marker locus. Required for input_mode = "model_based". Optional for "model_free": required only if heter_rate or misclass_rate is non-zero, and if omitted in that case it is solved from ET/ENT (see Details).
prev: Numeric in (0,1). Disease prevalence ($\phi_1$). Required for input_mode = "model_based". Required for "model_free" only if misclass_rate is non-zero.
R1: Numeric $> 0$. Genotype relative risk for heterozygotes. Required for input_mode = "model_based"; unused for "model_free".
R2: Numeric $> 0$. Genotype relative risk for homozygotes. Required for input_mode = "model_based"; unused for "model_free".
alpha: Numeric in (0,1). Significance level for the TDT (default 0.05).
delta_prime: Numeric. Linkage disequilibrium scale parameter $D'$ (default 1). Used for input_mode = "model_based" only.
misclass_rate: Numeric in $[0,1)$. Phenotype misclassification rate for controls ($\pi_{01}$) in the misclassification scenario. Zero means no misclassification and is the default.
heter_rate: Numeric in $[0,1)$. Proportion of trios whose affection status is not due to the locus of interest ($1 - \pi$) in the heterogeneity scenario. Zero means complete homogeneity and is the default.
ET, ENT: Numeric $\ge 0$. Expected transmission and non-transmission counts for input_mode = "model_free".
n_trios: Numeric $> 0$. Number of affected trios that the supplied ET and ENT correspond to. Required for input_mode = "model_free" (unlike the power function, there is no N argument here to reuse, since this function solves for the required number of trios).
verbose: Logical. If TRUE (default), prints a formatted summary of the no-error required number of trios and any requested modifier-specific required counts and percentage increases.
effect: Character. Optional explicit effect selector: "none", "phenotype_misclassification", "locus_heterogeneity", or "genotype_misclassification". The default is no effect. Calls that omit this argument retain the historical separate-sensitivity behavior of misclass_rate and heter_rate.
genotype_misclassification_rate: Numeric in $[0,0.5]$. The Chapter 5 adjacent-genotype TDT error parameter $e$, required only when effect = "genotype_misclassification".

Value

An object of class "tdt_mssn", containing:

alpha, target_power, lambda_star, input_mode: Design targets, target one-df NCP, and input mode.
N: Named no_error, misclassification, and heterogeneity required affected-trio counts.
percent_increase: Modifier-specific conventional percentage inflation, $100(N_{adjusted}/N_{no-error}-1)$, relative to the no-error required count. Values are NA when the no-error required count is not finite and positive.
power_at_N_no_error, power_loss_at_N_no_error: Power and power loss obtained if the no-error design size is used under each modifier.
gT_star, gNT_star: Scenario-specific transmission and non-transmission probabilities used in the MSSN formulas.
model_parameters: Supplied and derived genetic-model, LD, misclassification, heterogeneity, and model-free count-scale information.

Details

With input_mode = "model_based", penetrances are derived from prev, R1, R2, and pd, and $$N^* = \frac{\lambda^* (g_T^* + g_{NT}^*)}{2 (g_T^* - g_{NT}^*)^2}$$ is computed for each of the three scenarios. The no-error calculation follows Eq. 1.25 (p. 27); phenotype misclassification follows Eqs. 5.24–5.27 (pp. 284–285); and locus heterogeneity follows Eqs. 5.30–5.34b (pp. 293–294). Eq. 5.28 is a numerical example. Here heter_rate is the heterogeneous fraction, equal to $1-\pi$ when lower-level formulas use homogeneous fraction $\pi$. Phenotype misclassification and locus heterogeneity are reported as separate sensitivity scenarios; supplying both rates does not create a combined-error scenario.

With input_mode = "model_free", the no-error scenario uses $g_T = ET / (2\,n_{trios})$ and $g_{NT} = ENT / (2\,n_{trios})$. The misclassification and heterogeneity scenarios then use the same closed-form identities as tdt_power (see its Details for the formulas and the pd-solving fallback), applied to this no-error $g_T$/$g_{NT}$ pair.

For effect = "genotype_misclassification", the Chapter 5 single-parameter adjacent-genotype TDT transition model, Mendelian-inconsistency filtering, and Eqs. 5.16–5.22 are the same as in tdt_power. The adjusted NCP is linear in the number of affected trios, so the existing target-NCP is divided by the adjusted per-trio NCP. This model-based branch cannot be combined with phenotype misclassification or locus heterogeneity. TDTae and sequencing/NGS genotype error are not implemented, and the result uses the package's existing nominal asymptotic TDT convention.

References

Spielman, R. S., McGinnis, R. E., & Ewens, W. J. (1993). Transmission test for linkage disequilibrium: the insulin gene region and insulin-dependent diabetes mellitus. American Journal of Human Genetics, 52(3), 506–516. PMID: 8447318; PMCID: PMC1682161.

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

Buyske, S., Yang, G., Matise, T. C., & Gordon, D. (2009). When a case is not a case: Effects of phenotype misclassification on power and sample size requirements for the transmission disequilibrium test with affected child trios. Human Heredity, 67(4), 287–292. doi:10.1159/000194981 .

Chen, C., Yang, G., Buyske, S., Matise, T., Finch, S. J., & Gordon, D. (2009). Transmission disequilibrium test power and sample size in the presence of locus heterogeneity. Statistical Applications in Genetics and Molecular Biology, 8, Article 44. doi:10.2202/1544-6115.1501 .

Examples