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Assurance Probability for Limits of Agreement

Source: R/power_agreement_assurance.R
agree_assurance.Rd

[Maturing]

Usage

agree_assurance(
  conf.level = 0.95,
  assurance = 0.9,
  omega = NULL,
  pstar = 0.95,
  sigma = 1,
  n = NULL
)

Arguments

conf.level

confidence level for the range of agreement (1 - alpha). The confidence level of the confidence interval of the range of agreement (tolerance interval). Default is 0.95.

assurance

target lower bound of the assurance probability (1 - gamma). The assurance probability is the probability that the study half-width will be less than omega. Common values are 0.80, 0.90, or 0.95.

omega

upper bound of assurance half-width. The sample size guarantees (assures) that 100(1 - gamma)% of interval half-widths will be less than this value. Can be specified in standard deviation units.

pstar

central proportion of the data distribution covered (P*). It is the proportion of observations that fall between the limits. For example, a value of 0.95 indicates that 95% of the variable's values fall between the limits. Must be between 0 and 1. Common values are 0.90 or 0.95.

sigma

population standard deviation of the paired differences. If the true value is unknown, omega can be specified in standard deviation units by setting sigma = 1.

n

sample size (optional). If provided, the function will solve for a different parameter rather than sample size.

Value

An object of class "power.htest" containing the following components:

  • n: The required sample size (number of subject pairs)

  • conf.level: The confidence level (1 - alpha)

  • assurance: The target assurance probability (1 - gamma)

  • actual.assurance: The actual assurance probability achieved (may differ slightly from target due to discrete nature of n)

  • omega: The upper bound of assurance half-width

  • pstar: The central proportion covered (P*)

  • sigma: The population standard deviation

  • g.factor: The Odeh-Owen factor (g”) used to construct the tolerance interval, tabulated in Odeh and Owen (1980)

  • method: Description of the method used

  • note: Additional notes about the analysis

Details

Calculate the sample size necessary for a confidence interval of the Bland-Altman range of agreement when the underlying data distribution is normal. This function uses the assurance probability criterion to determine the optimum sample size, based on the exact confidence interval method of Jan and Shieh (2018), which has been shown to be superior to approximate methods.

Overview

This function implements the exact method for determining sample size based on assurance probability for Bland-Altman limits of agreement, as described in Jan and Shieh (2018). The assurance probability criterion determines an N that guarantees with specified probability (1 - gamma) that the confidence interval half-width will be no more than a boundary value omega.

Technical Details

Suppose a study involves paired differences (X - Y) whose distribution is approximately N(mu, sigma^2). The range of agreement is defined as a confidence interval of the central portion of these differences, specifically the area between the 100(1-p)th and 100p-th percentiles, where p* = 2p - 1.

The exact two-sided, 100(1 - alpha)% confidence interval for the range of agreement is defined as:

Pr(theta_(1-p) < theta_hat_(1-p) and theta_hat_p < theta_p) = 1 - alpha

The equal-tailed tolerance interval recommended by Jan and Shieh (2018) is:

(X_bar - d, X_bar + d)

where d = g * S, g is the Odeh-Owen tolerance factor (tabulated as g” in Odeh and Owen (1980)), and S is the sample standard deviation.

Sample Size Determination

The sample size N is selected to satisfy: Pr(H <= omega) >= 1 - gamma

This leads to the expression:

psi(eta) >= 1 - gamma

where psi() is the CDF of a chi-square distribution with N-1 degrees of freedom and eta = (N-1) \* (omega/(g\*sigma))^2.

The method uses equal-tailed tolerance intervals based on the noncentral t-distribution to construct exact confidence intervals for the range of agreement. The tolerance factor g is calculated such that the interval maintains the specified confidence level under normality.

Jan and Shieh (2018) demonstrated through extensive simulations that this exact method should be adopted rather than the classical Bland-Altman approximate method.

Interpreting Results

Each subject produces two measurements (one for each method being compared). The sample size n returned is the number of subject pairs needed. The actual assurance probability may differ slightly from the target due to the discrete nature of sample size.

For dropout considerations, inflate the sample size using: N' = N / (1 - dropout_rate), always rounding up.

Assumptions

  • The paired differences are normally distributed

  • The variance is constant across the range of measurement

  • Pairs are independent

References

Jan, S.L. and Shieh, G. (2018). The Bland-Altman range of agreement: Exact interval procedure and sample size determination. Computers in Biology and Medicine, 100, 247-252. doi:10.1016/j.compbiomed.2018.06.020

Odeh, R.E. and Owen, D.B. (1980). Tables for Normal Tolerance Limits, Sampling Plans, and Screening. Marcel Dekker, Inc., New York.

See also

agree_expected_half() for sample size determination using expected half-width criterion, power_agreement_exact() for power analysis of agreement tests.

Examples

# Example: Planning a method comparison study
# Researchers want 95% confidence, 90% assurance that half-width
# will be within 2.5 SD units, covering central 95% of differences
agree_assurance(
  conf.level = 0.95,
  assurance = 0.90,
  omega = 2.5,
  pstar = 0.95,
  sigma = 1
)
#> 
#>      Assurance probability & sample size for Limits of Agreement 
#> 
#>                n = 115
#>       conf.level = 0.95
#>        assurance = 0.9
#> actual.assurance = 0.9024848
#>            omega = 2.5
#>            pstar = 0.95
#>            sigma = 1
#>                g = 2.306167
#>               zp = 1.959964
#>