Computing plausibility functions

ngrid_in <- 10
ngrid_out <- 100
nperms <- 100000
n1 <- 30
n2 <- 30
set.seed(1301)
x1 <- rnorm(n1, mean = 0, sd = 1)
x2 <- rnorm(n2, mean = 3, sd = 1)
y1 <- rnorm(n1, mean = 0, sd = 1)
y2 <- rnorm(n2, mean = 0, sd = 2)
z1 <- rnorm(n1, mean = 0, sd = 1)
z2 <- rnorm(n2, mean = 3, sd = 2)

The concept of plausibility functions pertains to assessing the \(p\)-value of a set of null hypotheses and to plot this \(p\)-value surface on the domain defined by the set of null hypotheses. The idea behind is that, if such a plausibility function is available, you can deduce from it point estimates or confidence interval estimates for parameters used to define the nulls or extract a single \(p\)-value for a specific null of interest (Martin 2017; Fraser 2019; Infanger and Schmidt-Trucksäss 2019). In particular, there is another R package dedicated to plausibility functions called pvaluefunctions.

Plausibility function for the mean

null_spec <- function(y, parameters) {
  map(y, ~ .x - parameters)
}
stat_functions <- list(stat_t)
stat_assignments <- list(delta = 1)

pf <- PlausibilityFunction$new(
  null_spec = null_spec, 
  stat_functions = stat_functions, 
  stat_assignments = stat_assignments, 
  x1, x2,  
  seed = 1234
)
pf$set_nperms(nperms)
pf$set_point_estimate(mean(x2) - mean(x1))
pf$set_parameter_bounds(
  point_estimate = pf$point_estimate, 
  conf_level = pf$max_conf_level
)
pf$set_grid(
  parameters = pf$parameters, 
  npoints = ngrid_in
)

pf$set_alternative("two_tail")
pf$evaluate_grid(grid = pf$grid)
df <- rename(pf$grid, two_tail = pvalue)

pf$set_alternative("left_tail")
pf$grid$pvalue <- NULL
pf$evaluate_grid(grid = pf$grid)
df <- bind_rows(
  df, 
  rename(pf$grid, left_tail = pvalue)
)

pf$set_alternative("right_tail")
pf$grid$pvalue <- NULL
pf$evaluate_grid(grid = pf$grid)
df <- bind_rows(
  df, 
  rename(pf$grid, right_tail = pvalue)
)

pf$set_grid(
  parameters = pf$parameters, 
  npoints = ngrid_out
)

df_mean <- tibble(
  delta = pf$grid$delta, 
  two_tail = approx(df$delta, df$two_tail, delta)$y, 
  left_tail = approx(df$delta, df$left_tail, delta)$y, 
  right_tail = approx(df$delta, df$right_tail, delta)$y, 
) %>%
  pivot_longer(-delta)

df_mean %>%
  ggplot(aes(delta, value, color = name)) +
  geom_line() +
  labs(
    title = "P-value function for the mean", 
    subtitle = "t-statistic", 
    x = expression(delta), 
    y = "p-value", 
    color = "Type"
  ) +
  geom_hline(
    yintercept = 0.05,
    color = "black",
    linetype = "dashed"
  ) +
  geom_vline(
    xintercept = mean(x2) - mean(x1),
    color = "black"
  ) +
  geom_vline(
    xintercept = stats::t.test(x2, x1, var.equal = TRUE)$conf.int,
    color = "black",
    linetype = "dashed"
  ) +
  scale_y_continuous(breaks = seq(0, 1, by = 0.05), limits = c(0, 1))

Plausibility function for the variance

null_spec <- function(y, parameters) {
  map(y, ~ .x / parameters)
}
stat_functions <- list(stat_f)
stat_assignments <- list(rho = 1)

pf <- PlausibilityFunction$new(
  null_spec = null_spec, 
  stat_functions = stat_functions, 
  stat_assignments = stat_assignments, 
  y1, y2,  
  seed = 1234
)
pf$set_nperms(nperms)
pf$set_point_estimate(sd(y2) / sd(y1))
pf$set_parameter_bounds(
  point_estimate = pf$point_estimate, 
  conf_level = pf$max_conf_level
)
pf$set_grid(
  parameters = pf$parameters, 
  npoints = ngrid_in
)

pf$set_alternative("two_tail")
pf$evaluate_grid(grid = pf$grid)
df <- rename(pf$grid, two_tail = pvalue)

pf$set_alternative("left_tail")
pf$grid$pvalue <- NULL
pf$evaluate_grid(grid = pf$grid)
df <- bind_rows(
  df, 
  rename(pf$grid, left_tail = pvalue)
)

pf$set_alternative("right_tail")
pf$grid$pvalue <- NULL
pf$evaluate_grid(grid = pf$grid)
df <- bind_rows(
  df, 
  rename(pf$grid, right_tail = pvalue)
)

pf$set_grid(
  parameters = pf$parameters, 
  npoints = ngrid_out
)

df_sd <- tibble(
  rho = pf$grid$rho, 
  two_tail = approx(df$rho, df$two_tail, rho)$y, 
  left_tail = approx(df$rho, df$left_tail, rho)$y, 
  right_tail = approx(df$rho, df$right_tail, rho)$y, 
) %>%
  pivot_longer(-rho)

df_sd %>%
  ggplot(aes(rho, value, color = name)) +
  geom_line() +
  labs(
    title = "P-value function for the standard deviation", 
    subtitle = "F-statistic", 
    x = expression(rho), 
    y = "p-value", 
    color = "Type"
  ) +
  geom_hline(
    yintercept = 0.05,
    color = "black",
    linetype = "dashed"
  ) +
  geom_vline(
    xintercept = sqrt(stats::var.test(y2, y1)$statistic),
    color = "black"
  ) +
  geom_vline(
    xintercept = sqrt(stats::var.test(y2, y1)$conf.int),
    color = "black",
    linetype = "dashed"
  ) +
  scale_y_continuous(breaks = seq(0, 1, by = 0.05), limits = c(0, 1))

Plausibility function for both mean and variance

Assume that we have two r.v. \(X\) and \(Y\) that differ in distribution only in their first two moments. Let \(\mu_X\) and \(\mu_Y\) be the means of \(X\) and \(Y\) respectively and \(\sigma_X\) and \(\sigma_Y\) be the standard deviations. We can therefore write

\[ Y = \delta + \rho X. \]

In this case, we have

\[ \begin{cases} \mu_Y = \delta + \rho \mu_X \\ \sigma_Y^2 = \rho^2 \sigma_X^2 \end{cases} \Longleftrightarrow \begin{cases} \delta = \mu_Y - \frac{\sigma_Y}{\sigma_X} \mu_X \\ \rho = \frac{\sigma_Y}{\sigma_X} \end{cases} \]

In the following example, we have \(\delta = 3\) and \(\rho = 2\).

null_spec <- function(y, parameters) {
  map(y, ~ (.x - parameters[1]) / parameters[2])
}
stat_functions <- list(stat_t, stat_f)
stat_assignments <- list(delta = 1, rho = 2)

pf <- PlausibilityFunction$new(
  null_spec = null_spec, 
  stat_functions = stat_functions, 
  stat_assignments = stat_assignments, 
  z1, z2,  
  seed = 1234
)
pf$set_nperms(nperms)
pf$set_point_estimate(c(
  mean(z2) - sd(z2) / sd(z1) * mean(z1), 
  sd(z2) / sd(z1)
))
pf$set_parameter_bounds(
  point_estimate = pf$point_estimate, 
  conf_level = pf$max_conf_level
)

# Fisher combining function
pf$set_aggregator("fisher")
pf$set_grid(
  parameters = pf$parameters, 
  npoints = ngrid_in
)
pf$evaluate_grid(grid = pf$grid)
grid_in <- pf$grid
pf$set_grid(
  parameters = pf$parameters, 
  npoints = ngrid_out
)
if (requireNamespace("interp", quietly = TRUE)) {
  Zout <- interp::interp(
    x = grid_in$delta,
    y = grid_in$log_rho,
    z = grid_in$pvalue,
    xo = sort(unique(pf$grid$delta)),
    yo = sort(unique(pf$grid$log_rho))
  )
  pf$grid$pvalue <- as.numeric(Zout$z)
} else
  pf$grid$pvalue <- rep(NA, nrow(pf$grid))

df_fisher <- pf$grid

# Tippett combining function
pf$set_aggregator("tippett")
pf$set_grid(
  parameters = pf$parameters, 
  npoints = ngrid_in
)
pf$evaluate_grid(grid = pf$grid)
grid_in <- pf$grid
pf$set_grid(
  parameters = pf$parameters, 
  npoints = ngrid_out
)
if (requireNamespace("interp", quietly = TRUE)) {
  Zout <- interp::interp(
    x = grid_in$delta,
    y = grid_in$log_rho,
    z = grid_in$pvalue,
    xo = sort(unique(pf$grid$delta)),
    yo = sort(unique(pf$grid$log_rho))
  )
  pf$grid$pvalue <- as.numeric(Zout$z)
} else
  pf$grid$pvalue <- rep(NA, nrow(pf$grid))

df_tippett <- pf$grid

df_fisher %>% 
  ggplot(aes(delta, log_rho, z = pvalue)) + 
  geom_contour_filled(binwidth = 0.05) + 
  labs(
    title = "Contour plot of the p-value surface", 
    subtitle = "Using Fisher's non-parametric combination", 
    x = expression(delta), 
    y = expression(log(rho)), 
    fill = "p-value"
  ) + 
  theme_minimal()

df_tippett %>% 
  ggplot(aes(delta, log_rho, z = pvalue)) + 
  geom_contour_filled(binwidth = 0.05) + 
  labs(
    title = "Contour plot of the p-value surface", 
    subtitle = "Using Tippett's non-parametric combination", 
    x = expression(delta), 
    y = expression(log(rho)), 
    fill = "p-value"
  ) + 
  theme_minimal()

References

Fraser, D. A. S. 2019. “The p-Value Function and Statistical Inference.” The American Statistician 73 (sup1): 135–47. https://doi.org/10.1080/00031305.2018.1556735.

Infanger, Denis, and Arno Schmidt-Trucksäss. 2019. “P Value Functions: An Underused Method to Present Research Results and to Promote Quantitative Reasoning.” Statistics in Medicine 38 (21): 4189–97. https://doi.org/10.1002/sim.8293.

Martin, Ryan. 2017. “A Statistical Inference Course Based on p-Values.” The American Statistician 71 (2): 128–36. https://doi.org/10.1080/00031305.2016.1208629.