TDT Minimum Sample Size Necessary from Genetic Model Parameters

Computes the required number of affected trios ($N^*$) needed to achieve a specified statistical power in the Transmission Disequilibrium Test (TDT), given model parameters for allele frequency, relative risks, disease prevalence, and heterogeneity. Implements the probability and MSSN chain in Eqs. 5.33–5.34b (Chapter 5, Section 5.3.3, pp. 293–294) of Gordon, Finch, and Kim (2020).

tdt_mssn_from_model(
  power,
  alpha,
  df,
  pd,
  prev,
  R1,
  R2,
  delta_prime = 1,
  pi = 1
)

Arguments

power: Numeric. Desired power (e.g., 0.8).
alpha: Numeric. Significance level (e.g., 0.05).
df: Integer. Degrees of freedom (typically 1 for TDT).
pd: Numeric. Frequency of the disease-associated allele.
prev: Numeric. Disease prevalence.
R1: Numeric. Relative risk for heterozygotes.
R2: Numeric. Relative risk for homozygotes.
delta_prime: Numeric. Linkage disequilibrium (LD) scale factor (default = 1).
pi: Numeric in $[0,1]$. Linked/homogeneous trio fraction; 1 is complete homogeneity and $1-\pi$ is the heterogeneous fraction.

Value

A list containing:

lambda_star: Non-centrality parameter ($\lambda^*$).
gT_star: Expected transmission probability.
gNT_star: Expected non-transmission probability.
N_star: Required number of trios.

Details

This function determines the non-centrality parameter ($\lambda^*$) via root-finding (uniroot) such that the test power equals the desired level. It then computes the expected transmission ($gT^*$) and non-transmission ($gNT^*$) probabilities, followed by the required number of trios using: $$N^* = \frac{\lambda^*}{2} \frac{(gT^* + gNT^*)}{(gT^* - gNT^*)^2}$$

The expected transmission and non-transmission components are calculated under allele frequency and penetrance model assumptions in Eq. 5.33. Their NCP and MSSN are Eqs. 5.34a–b.

References

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 .

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 .

Examples

# Example: compute required trios for 80% power at alpha = 0.05
tdt_mssn_from_model(
  power = 0.8, alpha = 0.05, df = 1,
  pd = 0.25, prev = 0.005, R1 = 2, R2 = 2,
  delta_prime = 1, pi = 1
)
#> 
#> --- Transmission Disequilibrium Test (Trios) ---
#> Equation: 5.34b  |  Computes Required Number of Trios (N_star)
#> -----------------------------------------------------------
#> Desired Power:                              0.800  |  Significance Level (alpha):  0.050
#> Allele Frequency (p_d):                     0.250  |  Prevalence (phi1):  0.005
#> Relative Risks (R1,R2):                      2, 2
#> Heterogeneity Parameter (pi):               1.000
#> -----------------------------------------------------------
#> Non-Centrality Parameter (lambda_star):     7.8488
#> Expected Transmission (gT_star):          0.26087
#> Expected Non-Transmission (gNT_star):     0.16304
#> Required Number of Trios (N_star):            174
#> -----------------------------------------------------------