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Assurance Probability for Limits of Agreement

Source: R/power_agreement_assurance.R
agree_assurance.Rd

[Stable]

Usage

agree_assurance(
  conf.level = 0.95,
  assurance = 0.9,
  omega = NULL,
  pstar = 0.95,
  sigma = 1,
  n = NULL,
  m = 1,
  icc = 0
)

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. With clustered data (m > 1), this is the total SD of the differences (between- plus within-subject variance).

n

sample size (optional): the number of pairs, or the number of subjects when m > 1. If provided, the function will solve for omega rather than sample size.

m

number of paired measurements per subject (default 1, independent pairs). See the "Clustered data" section.

icc

intraclass correlation of the paired differences (between-subject share of the variance of the differences), in [0, 1). Ignored when m = 1. This is not the test-retest ICC of the measurements themselves.

Value

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

  • n: The required sample size (number of pairs, or of subjects when m > 1)

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

  • zp: The normal quantile for the central proportion

  • m, icc: The clustering inputs

  • n.obs: The total number of pairs, n * m

  • 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, or (with m > 1) follow the balanced compound symmetry model in the "Clustered data" section

Clustered data

When each of n subjects contributes m paired measurements (repeated measures, or several pairs per subject), the pairs are not independent and treating n * m pairs as independent overstates precision. Setting m and icc accounts for this under a balanced one-way random-effects (compound symmetry) model for the differences:

  • The mean difference has variance sigma^2 \* DEFF / (n \* m), with design effect DEFF = 1 + (m - 1) \* icc, so the effective sample size for the mean is n \* m / DEFF.

  • The total variance is estimated as S^2 = MSB / m + (m - 1) / m \* MSW, whose distribution is a weighted sum of two independent chi-square variables with n - 1 and n(m - 1) degrees of freedom. This distribution is integrated numerically in place of the single chi-square of the independent case. It is more skewed than a Satterthwaite approximation suggests, which matters when icc is large.

Under these assumptions the calculation is exact (it agrees with simulation to Monte Carlo error), and m = 1 gives exactly the published method. The simple design effect 1 + (m - 1) \* icc applied to the whole sample size overstates the penalty, because the SD has a much smaller design effect, 1 + (m - 1) \* icc^2. Unbalanced designs, other correlation structures, or several levels of nesting are not covered; use simulation for those.

icc is the ICC of the differences, which can be estimated from pilot data with an intercept-only mixed model on the differences (e.g. lme4::lmer(diff ~ 1 + (1 | id))), and sigma is then the square root of the sum of the two variance components.

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. (2007). Agreement between methods of measurement with multiple observations per individual. Journal of Biopharmaceutical Statistics, 17(4), 571-582. doi:10.1080/10543400701329422

Mee, R.W. and Owen, D.B. (1983). Improved factors for one-sided tolerance limits for balanced one-way ANOVA random model. Journal of the American Statistical Association, 78(384), 901-905. doi:10.1080/01621459.1983.10477038

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 = 0.9024825
#>            omega = 2.5
#>            pstar = 0.95
#>            sigma = 1
#>                g = 2.306169
#>               zp = 1.959964
#>                m = 1
#>              icc = 0
#>            n.obs = 115
#> 
#> NOTE: n is the number of pairs. Two measurements per unit; one for each method.
#> 

# Clustered data: 4 pairs per subject, ICC of the differences 0.3
agree_assurance(
  assurance = 0.90,
  omega = 2.5,
  pstar = 0.95,
  m = 4,
  icc = 0.3
)
#> 
#>      Assurance probability & sample size for Limits of Agreement 
#> 
#>                n = 40
#>       conf.level = 0.95
#>        assurance = 0.9
#> actual.assurance = 0.9000544
#>            omega = 2.5
#>            pstar = 0.95
#>            sigma = 1
#>                g = 2.314759
#>               zp = 1.959964
#>                m = 4
#>              icc = 0.3
#>            n.obs = 160
#> 
#> NOTE: n is the number of subjects, each with m = 4 paired measurements (n.obs pairs in total).
#>      Assumes balanced compound symmetry with ICC of the differences = 0.3.
#>