LRTae power and sample-size design

What is LRTae?

LRTae is a likelihood-ratio framework for case-control genetic association studies in which phenotype classification, genotype classification, or both may be measured with error. A subset of participants can receive more accurate or gold-standard measurements. These validation data inform the error process and contribute information about the association.

The basic problem is easy to state:

true phenotype and genotype -> fallible measured phenotype and genotype -> potentially distorted apparent association

Validation provides a second path:

gold-standard measurements on a subset -> information about misclassification -> improved efficient information for the association test

paweh implements the prospective design calculations lrtae_ncp(), lrtae_power(), and lrtae_mssn(). They calculate expected operating characteristics before data collection. The package does not implement the observed-data iterative likelihood fitting or association test from the historical LRTae spreadsheet and software.

Four validation tiers

Each planned participant belongs to one mutually exclusive measurement tier.

Tier Phenotype Genotype Fixed-design argument MSSN allocation argument
Fallible only Fallible Fallible n_fallible prop_fallible
Phenotype validated Gold standard Fallible n_pheno_validated prop_pheno_validated
Genotype validated Fallible Gold standard n_geno_validated prop_geno_validated
Both validated Gold standard Gold standard n_both_validated prop_both_validated

The four fixed counts sum to the total sample size. The four MSSN proportions must be nonnegative and sum to one.

Matrix orientation is important: for both misclassification matrices, rows are true categories and columns are observed categories.

A reproducible prospective design

Consider a research team planning a case-control association study. Routine phenotype ascertainment is imperfect, routine genotyping has a small classification error, and gold-standard clinical adjudication and confirmatory genotyping are too expensive for every participant.

The following deterministic inputs come from a historical LRTae development workbook. They are used here as a reproducible worked design, not as observed patient data and not as a realized NCP benchmark.

p_control <- c(0.5645454545, 0.3739393939, 0.0615151515)
p_case <- c(0.36, 0.48, 0.16)
q <- c(0.5, 0.5)

phenotype_error <- matrix(
  c(0.98, 0.02,
    0.04, 0.96),
  nrow = 2,
  byrow = TRUE
)
genotype_error <- matrix(
  c(0.965, 0.025, 0.010,
    0.025, 0.950, 0.025,
    0.010, 0.025, 0.965),
  nrow = 3,
  byrow = TRUE
)

model_inputs <- list(
  true_control_genotype_frequencies = p_control,
  true_case_genotype_frequencies = p_case,
  true_sampling_proportions = q,
  phenotype_misclassification_matrix = phenotype_error,
  genotype_misclassification_matrix = genotype_error
)
tier_counts <- list(
  n_fallible = 248,
  n_pheno_validated = 135,
  n_geno_validated = 72,
  n_both_validated = 45
)

The sampling proportions are in control/case order. The genotype-frequency vectors use the same three-category order as the rows and columns of genotype_error.

Computing the NCP

historical_ncp <- do.call(lrtae_ncp, c(model_inputs, tier_counts))
historical_ncp
#> [1] 22.40827

The noncentrality parameter (NCP) summarizes the expected separation between the alternative and null distributions. Larger values imply greater expected ability to detect association, holding the reference distribution fixed.

Technically, the implementation evaluates

\[ \lambda = \Delta^\mathsf{T} I_{\mathrm{eff}} \Delta, \]

where \(\Delta\) contrasts the free control and case genotype frequencies and \(I_{\mathrm{eff}}\) is their efficient information after nuisance parameters are eliminated by a Schur complement. This is an expected-information design quantity; it is not the likelihood-ratio statistic from one simulated or observed data set.

Computing power

historical_power <- do.call(
  lrtae_power,
  c(model_inputs, tier_counts, list(alpha = 0.05))
)
historical_power
#> $power
#> [1] 0.9925041
#> 
#> $ncp
#> [1] 22.40827
#> 
#> $df
#> [1] 2
#> 
#> $alpha
#> [1] 0.05
#> 
#> $critical_value
#> [1] 5.991465
#> 
#> $n_fallible
#> [1] 248
#> 
#> $n_pheno_validated
#> [1] 135
#> 
#> $n_geno_validated
#> [1] 72
#> 
#> $n_both_validated
#> [1] 45
#> 
#> $N_total
#> [1] 500
#> 
#> attr(,"class")
#> [1] "lrtae_power"

For three genotype categories, the test has 3 - 1 = 2 degrees of freedom. The returned object records power, NCP, degrees of freedom, significance level, critical value, all four tier counts, and total sample size.

Computing minimum sample size

The workbook allocation corresponds to proportions 248/500, 135/500, 72/500, and 45/500. The same planned allocation can be used to ask how many total participants are required for 80% power.

historical_mssn <- do.call(
  lrtae_mssn,
  c(
    model_inputs,
    list(
      target_power = 0.80,
      alpha = 0.05,
      prop_fallible = 248 / 500,
      prop_pheno_validated = 135 / 500,
      prop_geno_validated = 72 / 500,
      prop_both_validated = 45 / 500
    )
  )
)
historical_mssn
#> $MSSN_total
#> [1] 216
#> 
#> $target_power
#> [1] 0.8
#> 
#> $achieved_power
#> [1] 0.8021865
#> 
#> $achieved_ncp
#> [1] 9.685026
#> 
#> $df
#> [1] 2
#> 
#> $alpha
#> [1] 0.05
#> 
#> $critical_value
#> [1] 5.991465
#> 
#> $prop_fallible
#> [1] 0.496
#> 
#> $prop_pheno_validated
#> [1] 0.27
#> 
#> $prop_geno_validated
#> [1] 0.144
#> 
#> $prop_both_validated
#> [1] 0.09
#> 
#> $n_fallible
#> [1] 107
#> 
#> $n_pheno_validated
#> [1] 58
#> 
#> $n_geno_validated
#> [1] 31
#> 
#> $n_both_validated
#> [1] 20
#> 
#> attr(,"class")
#> [1] "lrtae_mssn"

MSSN_total is the minimum integer total under this allocation rule. The Hamilton largest-remainder rule assigns integer tier counts that sum exactly to the selected total, so the achieved proportions can differ slightly from the requested proportions. The returned achieved power is therefore normally just above, rather than exactly equal to, the target.

Why validation allocation matters

Here four designs have the same total sample size (N = 220) and identical biological and error assumptions. Only the allocation across validation tiers changes. The low-validation design retains a small number in every validation tier; complete absence of validation can leave the error model unidentified and its information matrix singular.

allocations <- data.frame(
  design = c(
    "Low validation", "Phenotype focused", "Genotype focused", "Balanced"
  ),
  n_fallible = c(198, 154, 154, 110),
  n_pheno_validated = c(9, 44, 11, 33),
  n_geno_validated = c(9, 11, 44, 33),
  n_both_validated = c(4, 11, 11, 44),
  stringsAsFactors = FALSE
)

design_results <- lapply(seq_len(nrow(allocations)), function(index) {
  counts <- as.list(allocations[index, 2:5])
  do.call(lrtae_power, c(model_inputs, counts, list(alpha = 0.05)))
})
allocations$N_total <- rowSums(allocations[2:5])
allocations$NCP <- vapply(design_results, function(x) x$ncp, numeric(1))
allocations$Power <- vapply(design_results, function(x) x$power, numeric(1))
allocations
#>              design n_fallible n_pheno_validated n_geno_validated
#> 1    Low validation        198                 9                9
#> 2 Phenotype focused        154                44               11
#> 3  Genotype focused        154                11               44
#> 4          Balanced        110                33               33
#>   n_both_validated N_total       NCP     Power
#> 1                4     220  9.230733 0.7817215
#> 2               11     220  9.536493 0.7956772
#> 3               11     220  9.538300 0.7957574
#> 4               44     220 10.021618 0.8163019
old_par <- par(mar = c(4.3, 8.2, 1.0, 1.0))
barplot(
  allocations$Power,
  names.arg = allocations$design,
  horiz = TRUE,
  xlim = c(0, 1),
  xlab = "Prospective power",
  las = 1,
  col = "#355C7D",
  border = NA
)
abline(v = 0.80, lty = 2, col = "#7A4F5B")
Prospective LRTae power for four validation allocations with the same total sample size and error assumptions.

Prospective LRTae power for four validation allocations with the same total sample size and error assumptions. 

par(old_par)

Validation is therefore not merely a data-cleaning exercise: it contributes statistical information. The comparison does not establish that one allocation is universally optimal. Phenotype and genotype validation can trade off differently under other frequencies, error matrices, costs, or effect sizes.

Reading the misclassification matrices

For the phenotype matrix,

\[ \begin{pmatrix} 0.98 & 0.02 \\ 0.04 & 0.96 \end{pmatrix}, \]

98% of true controls are observed as controls and 2% as cases; 4% of true cases are observed as controls and 96% as cases. Again, true categories are rows and observed categories are columns.

The genotype matrix is read the same way. For example, a participant in the first true genotype category is observed in categories one, two, and three with probabilities 0.965, 0.025, and 0.010. Each row must sum to one, and its category order must match both genotype-frequency vectors.

When to use these functions

The current LRTae functions are appropriate when:

  • a case-control genetic association study is being planned;
  • expected true control and case genotype distributions are available;
  • phenotype or genotype measurement may be imperfect;
  • validation subsamples are planned; and
  • the investigator wants an NCP, power, or minimum total sample size.

Important limitations are:

  • these are prospective calculations, not observed-data association fitting;
  • error matrices are treated as known design assumptions;
  • validity depends on the assumed error model;
  • validation allocation affects both information and identifiability;
  • this implementation is not LTTae; and
  • it is not a sequencing/read-level error model.

References

Barral S, Haynes C, Stone M, Gordon D. LRTae: improving statistical power for genetic association with case/control data when phenotype and/or genotype misclassification errors are present. BMC Genetics. 2006;7:24. https://doi.org/10.1186/1471-2156-7-24.

Ji F, Yang Y, Haynes C, Finch SJ, Gordon D. Computing asymptotic power and sample size for case-control genetic association studies in the presence of phenotype and/or genotype misclassification errors. Statistical Applications in Genetics and Molecular Biology. 2005;4(1):Article 37. https://doi.org/10.2202/1544-6115.1184.

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()
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