Analytic MSSN for a TDT1-NGS Sequencing Study

Computes the prospective minimum sample size necessary (MSSN), measured in complete father-mother-affected-child trios, for the same single-variant TDT1-NGS design as tdt_ngs_power().

tdt_ngs_mssn(
  power,
  pd,
  R1,
  coverage = NULL,
  seq_error = NULL,
  alpha = 0.05,
  verbose = TRUE,
  father_coverage = coverage,
  mother_coverage = coverage,
  child_coverage = coverage,
  epsilon0 = seq_error,
  epsilon1 = seq_error,
  pheno_misclass = FALSE,
  prev = NULL,
  pi01 = 0,
  locus_het = FALSE,
  heter_rate = 0
)

Arguments

power: A single finite target power strictly between 0 and 1.
pd: A single finite disease/risk-allele frequency strictly between 0 and 1. The supplied allele labeling is retained.
R1: A single finite positive heterozygote genotype relative risk under the multiplicative model. The homozygote relative risk is $R_2 = R_1^2$.
coverage: A single finite integer greater than or equal to 2. Common fixed-depth shorthand for the father, mother, and affected child. It may be omitted when all three member-specific depths are supplied.
seq_error: A single finite symmetric per-read sequencing-error probability in $[0,0.5)$. Internally, the directional error parameters default to $\epsilon_0 = \epsilon_1 =$ seq_error. It may be omitted when both directional parameters are supplied.
alpha: A single finite significance level in $(0,1)$. Defaults to 0.05.
verbose: Logical scalar. If TRUE, print a concise result summary.
father_coverage, mother_coverage, child_coverage: Optional fixed depths for each trio member. An explicitly supplied member depth overrides coverage; otherwise that member inherits coverage.
epsilon0: Directional probability that a reference-allele read is observed as the alternative allele. Defaults to seq_error.
epsilon1: Directional probability that an alternative-allele read is observed as the reference allele. Defaults to seq_error.
pheno_misclass: Logical scalar. If TRUE, add a phenotype- misclassification sensitivity scenario using the ordinary-TDT NCP bridge.
prev: Disease prevalence in $(0,1)$. Required when either phenotype misclassification or locus heterogeneity is requested. It is used only by the ordinary-TDT bridges and does not alter the raw-read likelihood.
pi01: Probability that an unaffected individual is misclassified or ascertained as affected, in $[0,1)$. It must be zero unless pheno_misclass = TRUE.
locus_het: Logical scalar. If TRUE, add an independent locus- heterogeneity sensitivity scenario using the ordinary-TDT NCP bridge.
heter_rate: Heterogeneous trio fraction in $[0,1]$. The ordinary- TDT linked/homogeneous fraction is $\pi=1-$heter_rate. It must be zero unless locus_het = TRUE.

Value

Invisibly, an object of class "tdt_ngs_mssn" containing the target, continuous and integer trio requirements, achieved power and NCP, per-trio NCP coefficient, multiplicative-model parameters, efficient information, the 11 by 11 information matrix, numerical diagnostics, effective sequencing-design metadata, model metadata, and scenario-level base and independently adjusted coefficients and sample sizes. Legacy coverage and seq_error fields retain supplied shorthand.

Details

The calculation uses Kim's Appendix B TDT1-NGS noncentrality parameter $$\lambda = N \delta^2 I_{eff},$$ where $N$ is the number of complete trios, $I_{eff}$ is the nuisance-adjusted per-trio information under the null, and under the multiplicative model $\delta = \log(R_1)$. The target one-degree-of- freedom NCP, $\lambda_*$, is obtained by numerical inversion of the noncentral chi-square power function. Trio MSSN is then solved analytically: $$N_{continuous} = \frac{\lambda_*}{\log(R_1)^2 I_{eff}}.$$ The planned MSSN is the ceiling of this quantity, with achieved power and the immediately smaller design checked at the integer boundary. For the optional phenotype scenario, the ordinary-TDT attenuation factor $r_{pheno}$ described in tdt_ngs_power multiplies the per-trio coefficient before inversion: $$N_{continuous,pheno}= \frac{\lambda_*}{\log(R_1)^2 I_{eff}r_{pheno}}.$$ The bridge fixes $R_2=R_1^2$, $\delta'=1$, and $\theta_1=p_d$. It does not modify Kim's likelihood or information matrix and is not a TDTae or simulation model.

For locus heterogeneity, $r_{het}$ is the ratio of the ordinary-TDT heterogeneous and homogeneous per-trio NCP coefficients, with $\pi=1-$heter_rate. The adjusted coefficient is $\log(R_1)^2 I_{eff}r_{het}$ before inversion. Phenotype and heterogeneity requests remain separate scenarios; no product of their attenuation factors is used.

If target power is no greater than alpha, the target NCP is zero and the minimum supported design is one trio. If R1 = 1 and target power exceeds alpha, no finite MSSN exists because the transmission effect is zero.

Each member uses its resolved fixed coverage, and the directional read model is $q_G=\epsilon_0+(1-\epsilon_0-\epsilon_1)G/2$. See tdt_ngs_power for parameter precedence and interpretation. Coverage 1 is unsupported for every member because efficient information is not identifiable under the implemented 11-parameter nuisance model. This prospective calculation uses raw sequencing read-count information and latent trio genotype states; it performs no simulation, genotype calling, or EM fitting.

References

Kim, W. (2015). Transmission disequilibrium tests based on read counts for low-coverage next-generation sequence data. Human Heredity, 80(1), 36–49. doi:10.1159/000434645 .

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 .

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