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Poisson binomial test for the rate of GRIM consistency

Source: R/binomial.R
grim_binomial.Rd

[Experimental]

Given the output of grim_map(), grim_binomial() tests whether the rate of GRIM-consistent value sets is above what random decimal digits would produce. This approach has not been tested extensively; use it with care.

Usage

grim_binomial(
  data,
  alternative = c("greater", "less", "two.sided"),
  alpha = 0.05
)

Arguments

data

Data frame returned by grim_map(). The output of grim_map_seq() or grim_map_total_n() is rejected: its rows are hypothetical value sets, not reported ones.

alternative

String (length 1). One of "greater" (the default), "less", and "two.sided".

alpha

Numeric (length 1). Significance level of the test. The interval around estimate has a confidence level of 1 - alpha. Default is 0.05.

Value

Tibble with the columns estimate (observed consistency rate), statistic (number of consistent cases), p.value, parameter (number of cases), conf.low, conf.high, conf.level (1 - alpha), method, and alternative.

Details

Under the null hypothesis, the decimal digits of each mean are random, so its probability of consistency is 1 - exp_incon, with exp_incon from grim_exp_incon(). Since this differs between value sets, the test uses the Poisson binomial distribution rather than averaging the probabilities. Value sets whose consistency or exp_incon is NA are left out; exp_incon can be NA even where the value set has a definite result (see grim_exp_incon()). Value sets with data_type = "continuous" have an exp_incon of 0, so they count as certain successes: they leave the p-value unchanged but raise estimate and the interval. Filter them out first to compare only the others.

The two-sided p-value is twice the smaller one-sided one, capped at 1.

Correct value sets are always consistent, so a collection of them should be more consistent than random digits even if some are wrong. There is no corresponding reason to expect fewer consistent sets than random, hence the one-sided default.

The confidence interval is the exact Clopper-Pearson interval for the consistency rate, as in stats::binom.test(), with the same alternative as the test. It treats the value sets as sharing one probability, so it is an interval for their average. Compare it to the mean of 1 - exp_incon.

Examples

pigs1 |>
  grim_map(digits_x = 2, items_averaged = 1, data_type = "integer") |>
  grim_binomial()
#> # A tibble: 1 × 9
#>   estimate statistic p.value parameter conf.low conf.high conf.level method     
#>      <dbl>     <int>   <dbl>     <int>    <dbl>     <dbl>      <dbl> <chr>      
#> 1    0.333         4   0.453        12    0.123         1       0.95 Poisson bi…
#> # ℹ 1 more variable: alternative <chr>