Expected Transmission Probability (gT*) in the Transmission Disequilibrium Test (TDT)

Computes the expected transmission probability (\(g_T^*\)) for the Transmission Disequilibrium Test (TDT), incorporating effects of linkage disequilibrium (LD), relative risks, and phenotype misclassification. Implements Equation 5.24 from Gordon et al. (2020), Heterogeneity in Statistical Genetics.

tdt_expected_transmission_probability(
  pd,
  prev,
  R1,
  R2,
  delta_prime = 1,
  pi01 = 0,
  theta1 = NULL,
  digits = 6,
  verbose = TRUE
)

Arguments

pd

Numeric. Frequency of the disease-associated allele.

prev

Numeric. Disease prevalence (\(\phi_1\)).

R1

Numeric. Relative risk for heterozygotes.

R2

Numeric. Relative risk for homozygotes.

delta_prime

Numeric. Linkage disequilibrium scale parameter (\(D'\)) (default = 1).

pi01

Numeric. Misclassification rate (default = 0).

theta1

Numeric. Population allele frequency (defaults to pd if NULL).

digits

Integer. Number of digits for printed results (default = 6).

verbose

Logical. If TRUE, prints intermediate quantities (default = TRUE).

Value

A list containing:

gT_star

Expected transmission probability (\(g_T^*\)).

C, D

Contrast and LD terms.

f0, f1, f2

Derived penetrance frequencies.

pd, phi1, phi0, theta1

Input and derived parameters.

Details

The function computes the penetrance-weighted genotype frequencies \(f_0,f_1,f_2\) and the contrast term \(C\), representing genotype-specific effects. Linkage disequilibrium is modeled as \(D = D' p_d p_+\).

The expected transmission probability is given by:

$$ g_T^* = \frac{ p_d p_+ \phi_1 + D(p_+ - \theta_1)C + \pi_{01}(p_d p_+ \phi_0 + D(\theta_1 - p_+)C) }{\phi_1 + \pi_{01}\phi_0} $$

When \(\pi_{01}=0\), this reduces to the standard non-misclassified case. This is Eq. 5.24 in Chapter 5, Section 5.2.6 (p. 284). gT_star is a probability, not an expected count.

References

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 . PMID: 19172087.

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 .

See also

tdt_expected_nontransmission_probability, tdt_expected_transmission_counts, and tdt_power.

Examples

# Example 1 – no misclassification
tdt_expected_transmission_probability(
  pd = 0.25, prev = 0.005,
  R1 = 2, R2 = 2,
  delta_prime = 1, pi01 = 0
)
#> 
#> --- Transmission Disequilibrium Test: Expected Transmission (gT*) ---
#> Implements Equation 5.24 from Gordon et al. (2020)
#> -----------------------------------------------------------
#> Allele Frequency (p_d):                  0.250000
#> Prevalence (phi1):                       0.005000
#> LD Scale (delta_prime):                  1.000000
#> Misclassification Rate (pi01):           0.000000
#> Population Allele Freq (theta1):         0.250000
#> -----------------------------------------------------------
#> Contrast Term (C):                       0.002609
#> LD Term (D):                             0.187500
#> Expected Transmission (gT*):             0.236413
#> -----------------------------------------------------------

# Example 2 – 10% misclassification
tdt_expected_transmission_probability(
  pd = 0.25, prev = 0.005,
  R1 = 2, R2 = 2,
  delta_prime = 1, pi01 = 0.1
)
#> 
#> --- Transmission Disequilibrium Test: Expected Transmission (gT*) ---
#> Implements Equation 5.24 from Gordon et al. (2020)
#> -----------------------------------------------------------
#> Allele Frequency (p_d):                  0.250000
#> Prevalence (phi1):                       0.005000
#> LD Scale (delta_prime):                  1.000000
#> Misclassification Rate (pi01):           0.100000
#> Population Allele Freq (theta1):         0.250000
#> -----------------------------------------------------------
#> Contrast Term (C):                       0.002609
#> LD Term (D):                             0.187500
#> Expected Transmission (gT*):             0.189606
#> -----------------------------------------------------------