Power Calculation for Bland-Altman Limits of Agreement

Computes the sample size, power, maximal allowable difference, or standard deviation for a method comparison study analyzed with the Bland-Altman limits of agreement, using the method of Lu et al. (2016). Agreement is concluded when the confidence intervals of both limits of agreement fall within [-delta, delta].

Usage

power_agreement_limit(
  n = NULL,
  delta = NULL,
  mu = 0,
  sigma = NULL,
  power = NULL,
  conf.level = 0.95,
  agree.level = 0.95,
  m = 1,
  icc = 0
)

Arguments

n: Number of pairs (sample size), or the number of subjects when m > 1.
delta: Maximal allowable difference: the methods agree if the confidence limits of the limits of agreement lie within [-delta, delta].
mu: Mean of the paired differences (default 0).
sigma: Standard deviation of the paired differences. With clustered data (m > 1), the total SD of the differences.
power: Target power.
conf.level: Confidence level of the confidence intervals of the limits of agreement (default 0.95).
agree.level: Agreement level of the limits of agreement: the proportion of differences the limits should contain (default 0.95).
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", a list with components:

n: Sample size (pairs, or subjects when m > 1)
delta: Maximal allowable difference
mu: Mean of differences
sigma: Standard deviation of differences
conf.level: Confidence level of the CIs of the limits of agreement
agree.level: Agreement level of the limits of agreement
power: Power of the procedure
m, icc: The clustering inputs
n.obs: Total number of pairs, n * m
method: Description of the method
note: Additional notes

Details

Specify three of n, delta, sigma, and power; the fourth is calculated.

The limits of agreement are mu +/- z \* sigma, with z the (1 + agree.level) / 2 quantile of the standard normal, and their confidence intervals are those of Bland and Altman (1999), with standard error sigma \* sqrt(1/n + z^2 / (2 (n - 1))). This is the interval from agreement_limit() with loa_calc = "blandaltman". The power is Lu et al. (2016), eq. 5:

power = 1 - T(t, n - 1, tau1) - T(t, n - 1, tau2),

where T is the CDF of the non-central t distribution, t is the 1 - (1 - conf.level) / 2 quantile of the central t distribution with n - 1 df, tau1 = (delta - mu - z \* sigma) / SE, and tau2 = (delta + mu - z \* sigma) / SE. Negative values are set to 0. These are the values reported by PASS (Chapter 427), which notes that the sample sizes in Table 1 of Lu et al. (2016), based on their approximate eq. 6, are sometimes slightly too large.

A sample size can only be found when the true limits of agreement lie within the bounds, i.e. |mu| + z \* sigma < delta; otherwise power goes to 0 as n increases and an error is returned.

With clustered data (m > 1), the standard error of the limits uses the effective sample size n \* m / (1 + (m - 1) \* icc) for the mean difference and a Satterthwaite df for the SD, under a balanced compound symmetry model (see agree_expected_half() for the model). This is an approximation; m = 1 gives the original results.

For a power curve over a range of sample sizes, see blandPowerCurve().

References

Lu, M. J., et al. (2016). Sample Size for Assessing Agreement between Two Methods of Measurement by Bland-Altman Method. The International Journal of Biostatistics, 12(2). doi:10.1515/ijb-2015-0039

Bland, J. M., and Altman, D. G. (1999). Measuring agreement in method comparison studies. Statistical Methods in Medical Research, 8(2), 135-160. doi:10.1177/096228029900800204

NCSS (PASS Sample Size Software). Chapter 427: Bland-Altman Method for Assessing Agreement in Method Comparison Studies.

Examples

# Sample size for 80% power (PASS Example 1: n = 60)
power_agreement_limit(delta = 7, mu = 0.5, sigma = 2.5, power = 0.80)
#> 
#>      Power for Bland-Altman Limits of Agreement (Lu et al. 2016) 
#> 
#>               n = 60
#>           delta = 7
#>              mu = 0.5
#>           sigma = 2.5
#>      conf.level = 0.95
#>     agree.level = 0.95
#>           power = 0.8059154
#>               m = 1
#>             icc = 0
#>           n.obs = 60
#> 
#> NOTE: H0: CIs of the 95% limits of agreement not within [-delta, delta]
#>      H1: CIs of the limits of agreement within [-delta, delta]
#>      n is the number of pairs. Two measurements per unit; one for each method.
#> 

# Power at a given sample size (PASS Example 2)
power_agreement_limit(n = 201, delta = 2.5, mu = 0.2, sigma = 1)
#> 
#>      Power for Bland-Altman Limits of Agreement (Lu et al. 2016) 
#> 
#>               n = 201
#>           delta = 2.5
#>              mu = 0.2
#>           sigma = 1
#>      conf.level = 0.95
#>     agree.level = 0.95
#>           power = 0.800283
#>               m = 1
#>             icc = 0
#>           n.obs = 201
#> 
#> NOTE: H0: CIs of the 95% limits of agreement not within [-delta, delta]
#>      H1: CIs of the limits of agreement within [-delta, delta]
#>      n is the number of pairs. Two measurements per unit; one for each method.
#> 

# Largest SD for which 90% power is reached with 100 pairs
power_agreement_limit(n = 100, delta = 7, mu = 0.5, power = 0.90)
#> 
#>      Power for Bland-Altman Limits of Agreement (Lu et al. 2016) 
#> 
#>               n = 100
#>           delta = 7
#>              mu = 0.5
#>           sigma = 2.577867
#>      conf.level = 0.95
#>     agree.level = 0.95
#>           power = 0.9
#>               m = 1
#>             icc = 0
#>           n.obs = 100
#> 
#> NOTE: H0: CIs of the 95% limits of agreement not within [-delta, delta]
#>      H1: CIs of the limits of agreement within [-delta, delta]
#>      n is the number of pairs. Two measurements per unit; one for each method.
#> 

# Clustered data, 3 pairs per subject, ICC of the differences 0.15
power_agreement_limit(delta = 7, mu = 0.5, sigma = 2.5, power = 0.80,
                      m = 3, icc = 0.15)
#> 
#>      Power for Bland-Altman Limits of Agreement (Lu et al. 2016) 
#> 
#>               n = 23
#>           delta = 7
#>              mu = 0.5
#>           sigma = 2.5
#>      conf.level = 0.95
#>     agree.level = 0.95
#>           power = 0.8119145
#>               m = 3
#>             icc = 0.15
#>           n.obs = 69
#> 
#> NOTE: H0: CIs of the 95% limits of agreement not within [-delta, delta]
#>      H1: CIs of the limits of agreement within [-delta, delta]
#>      n is the number of subjects, each with m = 3 paired measurements (n.obs pairs in total).
#>      Assumes balanced compound symmetry with ICC of the differences = 0.15.
#>