Why use an affected-child trio design?

The transmission disequilibrium test (TDT) compares an allele transmitted to an affected child with the alternative parental allele that was not transmitted. This within-family comparison can avoid confounding from population stratification that affects unrelated case-control comparisons.

Let ET and ENT denote expected transmitted and non-transmitted counts. The one-degree-of-freedom TDT non-centrality parameter is based on their contrast,

\[ \lambda = \frac{(ET-ENT)^2}{ET+ENT}. \]

In this vignette, N and every reported MSSN are numbers of affected-child trios, not numbers of individual people.

Power and MSSN for trios

The canonical model-based interfaces answer inverse design questions under a common genetic model.

basic_tdt_power <- tdt_power(
  N = 600, input_mode = "model_based",
  pd = 0.30, prev = 0.05, R1 = 1.5, R2 = 2.25,
  alpha = 0.05, delta_prime = 1,
  misclass_rate = 0.01, heter_rate = 0.10,
  verbose = FALSE
)

basic_tdt_mssn <- tdt_mssn(
  target_power = 0.80, input_mode = "model_based",
  pd = 0.30, prev = 0.05, R1 = 1.5, R2 = 2.25,
  alpha = 0.05, delta_prime = 1,
  misclass_rate = 0.01, heter_rate = 0.10,
  verbose = FALSE
)

data.frame(
  scenario = c("No error", "Phenotype misclassification", "Locus heterogeneity"),
  power_at_600_trios = unlist(basic_tdt_power$power),
  required_trios = unlist(basic_tdt_mssn$N),
  row.names = NULL
)
#>                      scenario power_at_600_trios required_trios
#> 1                    No error          0.9967475       214.9086
#> 2 Phenotype misclassification          0.9750532       306.3362
#> 3         Locus heterogeneity          0.9878723       265.3192

The canonical backend reports separate modifier scenarios. It does not claim that the misclassification and heterogeneity components are a combined-error analysis.

Model-based and model-free TDT inputs

Model-based input uses pd, prev, R1, R2, and delta_prime. The lower-level probability and count functions make the connection to ET and ENT visible without requiring a long derivation.

gT <- tdt_expected_transmission_probability(
  pd = 0.30, prev = 0.05, R1 = 1.5, R2 = 2.25,
  delta_prime = 1, verbose = FALSE
)
gNT <- tdt_expected_nontransmission_probability(
  pd = 0.30, prev = 0.05, R1 = 1.5, R2 = 2.25,
  delta_prime = 1, verbose = FALSE
)
expected_counts <- tdt_expected_transmission_counts(
  N_star = 500,
  pd = 0.30, prev = 0.05, R1 = 1.5, R2 = 2.25,
  delta_prime = 1, pi = 1, verbose = FALSE
)

data.frame(
  quantity = c("Transmission probability", "Non-transmission probability", "ET", "ENT"),
  value = c(gT$gT_star, gNT$gNT_star, expected_counts$ET_star, expected_counts$ENT_star)
)
#>                       quantity       value
#> 1     Transmission probability   0.2465217
#> 2 Non-transmission probability   0.2100000
#> 3                           ET 246.5217391
#> 4                          ENT 210.0000000

If a pilot study or external calculation already supplies expected counts, they can be used directly:

count_power <- tdt_power_from_expected_counts(
  ET = expected_counts$ET_star,
  ENT = expected_counts$ENT_star,
  alpha = 0.05
)
#> 
#> --- Transmission Disequilibrium Test (TDT) ---
#> Equation: 1.25  |  Input: Expected Transmissions and Non-Transmissions
#> -----------------------------------------------------------
#> Expected Transmissions (ET):             246.5217
#> Expected Non-Transmissions (ENT):        210.0000
#> Non-Centrality Parameter (lambda):         2.9217
#> Power at alpha = 0.05:                     0.4012
#> -----------------------------------------------------------

canonical_model_free <- tdt_power(
  N = 500, input_mode = "model_free",
  ET = expected_counts$ET_star,
  ENT = expected_counts$ENT_star,
  misclass_rate = 0, heter_rate = 0,
  verbose = FALSE
)

c(lower_level_power = count_power$power,
  canonical_power = canonical_model_free$power$no_error)
#> canonical_power 
#>       0.4011623

For model-free MSSN, n_trios records the trio count represented by the supplied ET and ENT. Applying heterogeneity additionally requires pd; applying phenotype misclassification requires pd and prev. Model-free inputs therefore relocate assumptions rather than eliminate them.

Chapter 5 genotype misclassification

The public effect argument can select "none", "phenotype_misclassification", "locus_heterogeneity", or "genotype_misclassification". The last option implements the Chapter 5 model-based TDT genotype-error model through the scalar argument genotype_misclassification_rate.

For adjacent-genotype error rate (e), with (0 e ), the transition matrix from true to observed genotype is

\[ M(e) = \begin{bmatrix} 1-e & e & 0 \\ e & 1-2e & e \\ 0 & e & 1-e \end{bmatrix}. \]

Thus a true genotype 0 can be observed as 0 or 1, a true genotype 1 can be observed as 0, 1, or 2, and a true genotype 2 can be observed as 1 or 2. Direct errors between genotypes 0 and 2 are not allowed.

The calculation begins with the true Mendelian trio distribution, applies the transition model to the observed trio genotypes, and removes observed Mendelian-inconsistent states. It renormalizes the retained consistent states, calculates their expected transmitted and non-transmitted counts, and uses the resulting TDT non-centrality parameter for prospective power or MSSN.

genotype_error_power <- tdt_power(
  N = 600, input_mode = "model_based",
  pd = 0.30, prev = 0.05, R1 = 1.5, R2 = 2.25,
  alpha = 0.05, delta_prime = 1,
  effect = "genotype_misclassification",
  genotype_misclassification_rate = 0.01,
  verbose = FALSE
)

genotype_error_mssn <- tdt_mssn(
  target_power = 0.80, input_mode = "model_based",
  pd = 0.30, prev = 0.05, R1 = 1.5, R2 = 2.25,
  alpha = 0.05, delta_prime = 1,
  effect = "genotype_misclassification",
  genotype_misclassification_rate = 0.01,
  verbose = FALSE
)

genotype_error_mssn_values <- c(
  genotype_error_mssn$N$no_error,
  genotype_error_mssn$N$genotype_misclassification
)

data.frame(
  scenario = c("No genotype error", "1% adjacent-genotype error"),
  power_at_600 = c(
    genotype_error_power$power$no_error,
    genotype_error_power$power$genotype_misclassification
  ),
  fractional_mssn = genotype_error_mssn_values,
  recruitment_trios = ceiling(genotype_error_mssn_values)
)
#>                     scenario power_at_600 fractional_mssn recruitment_trios
#> 1          No genotype error    0.9967475        214.9086               215
#> 2 1% adjacent-genotype error    0.9916522        248.4951               249

In this example, genotype error reduces power at 600 trios and increases the recruitment target for 80% power. This branch is available only with input_mode = "model_based" and uses one scalar adjacent-genotype error rate; it does not accept a custom transition matrix. It cannot be combined with phenotype misclassification or locus heterogeneity. It is neither TDTae nor the sequencing/NGS genotype-error implementation.

Published-example reproduction: Buyske et al. (2009)

Buyske et al. examined phenotype misclassification in affected-child trios. The validated multiplicative example uses target power 0.80, alpha = 1e-5, pd = 0.50, prevalence 0.01, R2 = 2.5, R1 = sqrt(2.5), maximum positive LD, and phenotype misclassification rate 0.05.

buyske <- tdt_mssn(
  target_power = 0.80, alpha = 1e-5,
  pd = 0.50, prev = 0.01,
  R1 = sqrt(2.5), R2 = 2.5,
  delta_prime = 1,
  misclass_rate = 0.05, heter_rate = 0,
  verbose = FALSE
)

buyske_ratio <- buyske$N$misclassification / buyske$N$no_error
data.frame(
  quantity = c("No-error MSSN", "Misclassification MSSN", "Final sample-size ratio"),
  published = c("approximately 546", "approximately 21,400", "approximately 39-fold"),
  paweh = c(buyske$N$no_error, buyske$N$misclassification, buyske_ratio)
)
#>                  quantity             published       paweh
#> 1           No-error MSSN     approximately 546   545.55101
#> 2  Misclassification MSSN  approximately 21,400 21400.40957
#> 3 Final sample-size ratio approximately 39-fold    39.22715

This is an excellent Published-example reproduction: approximately 546 versus 21,401 affected-child trios, a 39.23-fold final sample-size ratio. A 39.23-fold ratio is a 3,822.7% increase, not a 39.23% increase. The percent_increase component stores this conventional percentage as 100 * ((new / baseline) - 1); the validated final-to-baseline ratio remains 39.23-fold.

Extending Buyske: a misclassification curve

plot_tdt_mssn(
  x_var = "misclass_rate",
  x_values = seq(0, 0.05, by = 0.005),
  scenario = "misclassification",
  input_mode = "model_based",
  target_power = 0.80, alpha = 1e-5,
  pd = 0.50, prev = 0.01,
  R1 = sqrt(2.5), R2 = 2.5,
  delta_prime = 1, heter_rate = 0,
  title = "Required trios vs phenotype misclassification"
)

Required affected-child trios as phenotype misclassification increases under the Buyske model.

The steep curve identifies a fragile design region. It does not represent a confidence interval or a probability distribution over the error rate.

Extending Buyske: prevalence by misclassification

The optional interactive surface can evaluate the same Buyske parameters over a modest prevalence-by-error grid. The example is left unevaluated in the installed vignette to avoid embedding a multi-megabyte Plotly widget; run the copyable code in an interactive R session to explore the surface.

plot_tdt_surface3d(
  metric = "mssn", scenario = "misclassification",
  x = "prev", y = "misclass_rate",
  x_values = c(0.005, 0.01, 0.02, 0.04),
  y_values = seq(0, 0.05, length.out = 5),
  target_power = 0.80, alpha = 1e-5,
  pd = 0.50, R1 = sqrt(2.5), R2 = 2.5,
  delta_prime = 1, heter_rate = 0
)

When run interactively, this surface extends discrete calculations into a sensitivity analysis; it is not an uncertainty distribution.

Published-example reproduction: Chen et al. (2009)

Chen et al. studied TDT design under locus heterogeneity. Here heter_rate is the heterogeneous or unlinked trio fraction, equal to 1 - pi.

chen_no_heterogeneity <- tdt_mssn(
  target_power = 0.80, alpha = 1e-5,
  pd = 0.25, prev = 0.10,
  R1 = sqrt(1.5), R2 = 1.5,
  delta_prime = 1,
  misclass_rate = 0, heter_rate = 0,
  verbose = FALSE
)

chen_half_heterogeneous <- tdt_mssn(
  target_power = 0.80, alpha = 1e-5,
  pd = 0.25, prev = 0.10,
  R1 = sqrt(1.5), R2 = 1.5,
  delta_prime = 1,
  misclass_rate = 0, heter_rate = 0.50,
  verbose = FALSE
)

data.frame(
  heterogeneous_fraction = c(0, 0.50),
  published_required_trios = c("approximately 3430", "approximately 13721"),
  paweh_required_trios = c(
    chen_no_heterogeneity$N$heterogeneity,
    chen_half_heterogeneous$N$heterogeneity
  )
)
#>   heterogeneous_fraction published_required_trios paweh_required_trios
#> 1                    0.0       approximately 3430             3430.696
#> 2                    0.5      approximately 13721            13722.786

This is a near-exact Published-example reproduction. Small differences reflect the paper’s reported integer values and rounding.

Extending Chen: dilution by heterogeneous trios

plot_tdt_mssn(
  x_var = "heter_rate",
  x_values = seq(0, 0.50, by = 0.05),
  scenario = "heterogeneity",
  input_mode = "model_based",
  target_power = 0.80, alpha = 1e-5,
  pd = 0.25, prev = 0.10,
  R1 = sqrt(1.5), R2 = 1.5,
  delta_prime = 1, misclass_rate = 0,
  title = "Required trios vs locus heterogeneity"
)

$Required affected-child trios as the heterogeneous or unlinked fraction increases under the Chen model.$

Required affected-child trios as the heterogeneous or unlinked fraction increases under the Chen model.

Heterogeneous trios dilute locus-specific signal. A flatter curve would indicate a more robust design; a steep curve signals that modest departures from homogeneity require substantially more families.

Assumptions and limitations

The design is an affected-child trio TDT using an asymptotic chi-square approximation.
In the current parameterization, delta_prime lies in [0, 1] and scales maximum positive LD. Negative LD requires another allele-frequency- dependent normalization and is not implemented.
misclass_rate and heter_rate describe distinct reported scenarios.
Chapter 5 genotype misclassification is model-based, uses one scalar adjacent-genotype error rate, and cannot be combined with the other error mechanisms.
TDTae and sequencing/NGS genotype error are outside this implementation.
Model-free ET and ENT are treated as supplied expected counts.
MSSN is measured in trios.
Curves and surfaces vary fixed assumptions; they are not uncertainty bands.

References

Buyske S, Yang G, Matise TC, Gordon D. 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. 2009;67(4):287–292. https://doi.org/10.1159/000194981.

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

Gordon D, Finch SJ, Kim W. Heterogeneity in Statistical Genetics: How to Assess, Address, and Account for Mixtures in Association Studies. Springer; 2020. https://doi.org/10.1007/978-3-030-61121-7.

Session information

sessionInfo()
#> R version 4.6.1 (2026-06-24)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 26.04.1 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.32.so;  LAPACK version 3.12.0
#> 
#> locale:
#>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
#>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
#>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
#>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
#>  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
#> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
#> 
#> time zone: Etc/UTC
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] paweh_0.99.0     BiocStyle_2.41.0
#> 
#> loaded via a namespace (and not attached):
#>  [1] gtable_0.3.6        jsonlite_2.0.0      dplyr_1.2.1        
#>  [4] compiler_4.6.1      BiocManager_1.30.27 tidyselect_1.2.1   
#>  [7] jquerylib_0.1.4     scales_1.4.0        yaml_2.3.12        
#> [10] fastmap_1.2.0       ggplot2_4.0.3       R6_2.6.1           
#> [13] labeling_0.4.3      generics_0.1.4      knitr_1.52         
#> [16] tibble_3.3.1        maketools_1.3.2     bslib_0.12.0       
#> [19] pillar_1.11.1       RColorBrewer_1.1-3  rlang_1.3.0        
#> [22] cachem_1.1.0        xfun_0.61           sass_0.4.10        
#> [25] sys_3.4.3           S7_0.2.2            otel_0.2.0         
#> [28] cli_3.6.6           withr_3.0.3         magrittr_2.0.5     
#> [31] digest_0.6.39       grid_4.6.1          mvtnorm_1.4-2      
#> [34] lifecycle_1.0.5     vctrs_0.7.3         evaluate_1.0.5     
#> [37] glue_1.8.1          farver_2.1.2        buildtools_1.0.0   
#> [40] rmarkdown_2.32      tools_4.6.1         pkgconfig_2.0.3    
#> [43] htmltools_0.5.9

Transmission disequilibrium test study design