Expected Transmissions (ET*) and Non-Transmissions (ENT*) in the Transmission Disequilibrium Test (TDT)

Source: R/tdt_functions.R

Computes the expected number of transmissions (\(ET^*\)) and non-transmissions (\(ENT^*\)) for a given number of affected trios under heterogeneity and linkage disequilibrium models. Implements Equations 5.31-5.32 from Gordon et al. (2020), Heterogeneity in Statistical Genetics.

tdt_expected_transmission_counts(
  N_star,
  pd,
  prev,
  R1,
  R2,
  delta_prime = 1,
  pi = 1,
  theta1 = NULL,
  digits = 6,
  verbose = TRUE
)

Arguments

N_star

Numeric. Number of affected trios.

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).

pi

Numeric in \([0,1]\). Linked/homogeneous trio fraction (default 1); \(1-\pi\) is the heterogeneous fraction.

theta1

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

digits

Integer. Number of digits for printing (default = 6).

verbose

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

Value

A list containing:

ET_star

Expected transmissions (\(ET^*\)).

ENT_star

Expected non-transmissions (\(ENT^*\)).

C, D

Contrast and LD terms.

f0, f1, f2

Derived penetrance frequencies.

pd, prev, R1, R2, pi, theta1

Input parameters.

Details

The expected transmissions (\(ET^*\)) and non-transmissions (\(ENT^*\)) are computed as:

$$ ET^* = 2N^* \left[ \pi \left( \frac{D(p_+ - \theta_1)C}{\phi_1} + p_d p_+ \right) + (1 - \pi)\left( \frac{D(p_+ - 0.5)C}{\phi_1} + p_d p_+ \right) \right] $$

$$ ENT^* = 2N^* \left[ \pi \left( \frac{D(\theta_1 - p_d)C}{\phi_1} + p_d p_+ \right) + (1 - \pi)\left( \frac{D(0.5 - p_d)C}{\phi_1} + p_d p_+ \right) \right] $$

These are Eqs. 5.31–5.32 in Chapter 5, Section 5.3.3 (pp. 293–294). They are expected counts over N_star affected trios. Dividing by \(2N^*\) gives the corresponding \(g_T^*\) and \(g_{NT}^*\) probabilities. The canonical heter_rate used by tdt_power() and tdt_mssn() equals \(1-\pi\).

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 . PMID: 19883370.

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 1: Homogeneous model (pi = 1)
tdt_expected_transmission_counts(
  N_star = 1000, pd = 0.25, prev = 0.005,
  R1 = 2, R2 = 2,
  delta_prime = 1, pi = 1
)
#> 
#> --- Transmission Disequilibrium Test: Expected ET* and ENT* ---
#> Implements Equations 5.31-5.32 (Heterogeneity model)
#> -----------------------------------------------------------
#> Number of Trios (N*):                        1000
#> Allele Frequency (p_d):                  0.250000
#> Prevalence (phi1):                       0.005000
#> LD Scale (delta_prime):                  1.000000
#> Heterogeneity Parameter (pi):            1.000000
#> Population Allele Freq (theta1):         0.250000
#> -----------------------------------------------------------
#> Contrast Term (C):                       0.002609
#> LD Term (D):                             0.187500
#> -----------------------------------------------------------
#> Expected Transmissions (ET*):          472.826087
#> Expected Non-Transmissions (ENT*):     375.000000
#> -----------------------------------------------------------

# Example 2: Heterogeneous model (pi = 0.7)
tdt_expected_transmission_counts(
  N_star = 1000, pd = 0.25, prev = 0.005,
  R1 = 2, R2 = 2,
  delta_prime = 1, pi = 0.7
)
#> 
#> --- Transmission Disequilibrium Test: Expected ET* and ENT* ---
#> Implements Equations 5.31-5.32 (Heterogeneity model)
#> -----------------------------------------------------------
#> Number of Trios (N*):                        1000
#> Allele Frequency (p_d):                  0.250000
#> Prevalence (phi1):                       0.005000
#> LD Scale (delta_prime):                  1.000000
#> Heterogeneity Parameter (pi):            0.700000
#> Population Allele Freq (theta1):         0.250000
#> -----------------------------------------------------------
#> Contrast Term (C):                       0.002609
#> LD Term (D):                             0.187500
#> -----------------------------------------------------------
#> Expected Transmissions (ET*):          458.152174
#> Expected Non-Transmissions (ENT*):     389.673913
#> -----------------------------------------------------------