Functional Equivalence Test (TOST)

Tests whether two functional means are equivalent within an equivalence margin delta, based on the supremum norm. Uses the approach of Dette & Kokot (2021) with a simultaneous confidence band.

Usage

fequiv.test(
  fdataobj1,
  fdataobj2 = NULL,
  delta,
  mu0 = NULL,
  n.boot = 1000,
  alpha = 0.05,
  method = c("multiplier", "percentile"),
  seed = NULL
)

Arguments

fdataobj1

An object of class fdata (first sample).

fdataobj2

An object of class fdata (second sample), or NULL for a one-sample test.

delta

Equivalence margin (positive scalar). The test checks whether the sup-norm of the mean difference is less than delta.

mu0

Hypothesized mean function for one-sample test. A numeric vector, a single-row fdata, or NULL (defaults to zero).

n.boot

Number of bootstrap replicates (default 1000).

alpha

Significance level (default 0.05). The SCB has coverage 1-2*alpha.

method

Bootstrap method: "multiplier" (Gaussian multiplier, default) or "percentile" (resampling).

seed

Optional random seed for reproducibility.

Value

An object of class fequiv.test with components:

statistic

Observed sup-norm of the mean difference

delta

Equivalence margin used

critical.value

Bootstrap critical value for the SCB

scb.lower

Lower bound of simultaneous confidence band

scb.upper

Upper bound of simultaneous confidence band

diff.mean

Estimated mean difference curve (numeric vector)

reject

Logical; TRUE if equivalence is declared

p.value

Bootstrap p-value

alpha

Significance level used

method

Bootstrap method used

argvals

Argument values (time grid)

n1

Sample size of first group

n2

Sample size of second group (NA for one-sample)

boot.stats

Vector of bootstrap sup-norm statistics

data.name

Description of input data

Hypotheses

H0

sup_t |mu1(t) - mu2(t)| >= delta (NOT equivalent)

H1

sup_t |mu1(t) - mu2(t)| < delta (equivalent)

Equivalence is declared when the entire (1-2*alpha) simultaneous confidence band for the mean difference lies within [-delta, delta].

References

Dette, H. and Kokot, K. (2021). Detecting relevant differences in the covariance operators of functional time series. Biometrika, 108(4):895–913.

Examples

set.seed(42)
t_grid <- seq(0, 1, length.out = 50)
X1 <- fdata(matrix(rnorm(30 * 50), 30, 50), argvals = t_grid)
X2 <- fdata(matrix(rnorm(25 * 50, mean = 0.1), 25, 50), argvals = t_grid)
# \donttest{
result <- fequiv.test(X1, X2, delta = 1, n.boot = 500)
print(result)
#> Functional Equivalence Test (TOST)
#> ===================================
#> Data: X1 and X2 
#> Two-sample test, n1 = 30 , n2 = 25 
#> Bootstrap method: multiplier 
#> Equivalence margin (delta): 1 
#> Significance level (alpha): 0.05 
#> ---
#> Test statistic (sup|d_hat|): 0.5827 
#> Critical value: 0.0442 
#> SCB range: [ -0.6269 , 0.5484 ]
#> P-value: <2e-16 
#> ---
#> Decision: Reject H0 -- equivalence declared
plot(result)

# }