Skip to contents

Sample Size for Limits of Agreement Using Expected Half-Width

Source: R/power_agree_expected_half.R
agree_expected_half.Rd

[Maturing]

Usage

agree_expected_half(
  conf.level = 0.95,
  delta = 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.

delta

target upper bound of expected half-width (delta). The sample size guarantees that the expected half-width of the confidence interval will be no more 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, delta 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)

  • delta.target: The target upper bound of expected half-width

  • delta.actual: The actual upper bound of expected half-width achieved (may differ slightly from target due to discrete nature of n)

  • 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)

  • c.factor: The bias correction factor used in the expected half-width calculation

  • 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 expected half-width 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 expected half-width for Bland-Altman limits of agreement, as described in Jan and Shieh (2018). The expected half-width criterion determines an N that guarantees that the expected half-width of the confidence interval is less than a boundary value delta.

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 Hahn and Meeker (1991)), and S is the sample standard deviation.

Sample Size Determination

The sample size N is selected to satisfy: E(H) <= delta

where H is the half-width of the confidence interval. This leads to the expression:

g(P*, 1-alpha, N-1) / c <= delta / sigma

where c is a bias correction factor:

c = (Gamma((N-1)/2) * sqrt((N-1)/2)) / Gamma(N/2)

The expected half-width is E(H) = (g/c) * sigma. This accounts for the fact that the sample standard deviation S is a biased estimator of sigma, requiring correction when computing expected values.

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.

Comparison with Assurance Probability Method

This function uses the expected half-width criterion, which ensures that E(H) <= delta. An alternative approach is the assurance probability criterion (see agree_assurance()), which ensures that Pr(H <= omega) >= 1 - gamma.

The expected half-width criterion:

  • Controls the average half-width across repeated sampling

  • Generally requires smaller sample sizes than assurance probability

  • Is appropriate when the average performance is of primary interest

The assurance probability criterion:

  • Provides a probability guarantee about the half-width

  • Generally requires larger sample sizes

  • Is appropriate when a stronger guarantee is needed for planning purposes

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 expected half-width 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

Bland, J.M. and Altman, D.G. (1986). Statistical methods for assessing agreement between two methods of clinical measurement. The Lancet, 327(8476), 307-310.

Hahn, G.J. and Meeker, W.Q. (1991). Statistical Intervals: A Guide for Practitioners. John Wiley & Sons, New York.

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_assurance() for sample size determination using assurance probability criterion, power_agreement_exact() for power analysis of agreement tests.

Examples

# Example 1: Reproduce Jan and Shieh (2018), page 251
# Expected half-width criterion with P* = 0.95
agree_expected_half(
  conf.level = 0.95,
  delta = 2.25 * 19.61,
  pstar = 0.95,
  sigma = 19.61
)
#> 
#>      Expected half-width and sample size for limits of agreement 
#> 
#>               n = 155
#>      conf.level = 0.95
#>    target.delta = 44.1225
#>    actual.delta = 44.1113
#>           pstar = 0.95
#>           sigma = 19.61
#>               g = 2.253084
#>               c = 1.001625
#>              zp = 1.959964
#> 
# Expected result: n = 155

# Example 2: Planning a method comparison study
# Researchers want 95% confidence with expected half-width
# no more than 2.4 SD units, covering central 90% of differences
agree_expected_half(
  conf.level = 0.95,
  delta = 2.4,
  pstar = 0.90,
  sigma = 1
)
#> 
#>      Expected half-width and sample size for limits of agreement 
#> 
#>               n = 26
#>      conf.level = 0.95
#>    target.delta = 2.4
#>    actual.delta = 2.382023
#>           pstar = 0.9
#>           sigma = 1
#>               g = 2.405957
#>               c = 1.010047
#>              zp = 1.644854
#>