power_linear: Power and sample size estimation for linear models
In postcard: Estimating Marginal Effects with Prognostic Covariate Adjustment
power_linear R Documentation
Power and sample size estimation for linear models
Description
variance_ancova provides a convenient function for estimating a
variance to use for power and sample size approximation.
The power_gs and samplesize_gs functions calculate the Guenther-Schouten
power approximation for ANOVA or ANCOVA.
The approximation is based in (Guenther WC. Sample Size Formulas for Normal Theory T Tests.
The American Statistician. 1981;35(4):243–244) and (Schouten HJA. Sample size formula with
a continuous outcome for unequal group sizes and unequal variances. Statistics in Medicine.
1999;18(1):87–91).
The function power_nc calculates the power for ANOVA or ANCOVA based on the
non-centrality parameter and the exact t-distributions.
See more details about each function in Details and in sections after Value.
Usage
variance_ancova(formula, data, inflation = 1, deflation = 1)
power_gs(variance, ate, n, r = 1, margin = 0, alpha = 0.025)
samplesize_gs(variance, ate, r = 1, margin = 0, power = 0.9, alpha = 0.025)
power_nc(variance, df, ate, n, r = 1, margin = 0, alpha = 0.025)
Arguments
formula
an object of class "formula" (or one that can be coerced to that class):
a symbolic description used in stats::model.frame() to create a data.frame with
response and covariates. This data.frame is used to estimate the R^2, which is
then used to find the variance. See more in details.
data
a data frame, list or environment (or object
coercible by as.data.frame to a data frame),
containing the variables in formula. Neither a matrix nor an
array will be accepted.
inflation
a numeric to multiply the marginal variance of the response by.
Default is 1 which estimates the variance directly from data. Use values above 1
to obtain a more conservative estimate of the marginal response variance.
deflation
a numeric to multiply the R^2 by.
Default is 1 which means the estimate of R^2 is unchanged. Use values
below 1 to obtain a more conservative estimate of the coefficient of determination.
See details about how R^2 related to the estimation.
variance
a numeric variance to use for the approximation. See more details
in documentation sections of each power approximating function.
ate
a numeric minimum effect size that we should be able to detect.
n
a numeric with number of participants in total.
From this number of participants in the treatment group is n1=(r/(1+r))n
and the control group is n1=(1/(1+r))n.
r
a numeric allocation ratio r=n1/n0. For one-to-one randomisation r=1.
margin
a numeric superiority margin
(for non-inferiority margin, a negative value can be provided).
alpha
a numeric significance level. Due to regulatory guidelines when
using a one-sided test, half the specified significance level is used.
Thus, for standard significance level of 5%, the default is alpha = 0.025.
power
a numeric giving the desired power when calculating the sample size
df
a numeric degrees of freedom to use in the t-distribution.
Details
This details section provides information about relation between arguments to
functions and the formulas described in sections below for each power
approximation formula.
Note that all entities that carry the same name as an argument and in the formula
will not be mentioned below, as they are obviously linked (n, r, alpha)
-
ate: \beta_1-\beta_0
-
margin: \Delta_s
-
variance: \widehat{\sigma}^2(1-\widehat{R}^2)
Finding the variance to use for approximation
The variance_ancova function estimates \sigma^2(1-R^2) in data and
returns it as a numeric that can be passed directly as the variance
in power_gs. Corresponds to estimating the power from using an lm with
the same formula as specified in variance_ancova.
The user can estimate the variance any way they see fit.
Value
All functions return a numeric. variance_ancova returns a numeric with
a variance estimated from data to used for power estimation and sample size
estimation. power_xx and samplesize_xx functions return a numeric with
the power or sample size approximation.
Guenther-Schouten power approximation
The estimation formula in the case of an ANCOVA model with multiple covariate adjustment is (see description for reference):
n=\frac{(1+r)^2}{r}\frac{(z_{1-\alpha}+z_{1-\beta})^2\widehat{\sigma}^2(1-\widehat{R}^2)}{(\beta_1-\beta_0-\Delta_s)^2}+\frac{(z_{1-\alpha})^2}{2}
where \widehat{R}^2= \frac{\widehat{\sigma}_{XY}^\top \widehat{\Sigma}_X^{-1}\widehat{\sigma}_{XY}}{\widehat{\sigma}^2},
we denote by \widehat{\sigma^2} an estimate of the variance of the outcome,
\widehat{\Sigma_X} and estimate of the covariance matrix of the
covariates, and \widehat{\sigma_{XY}} a p-dimensional column vector consisting of
an estimate of the covariance
between the outcome variable and each covariate.
In the univariate case R^2 is replaced by \rho^2
Power approximation using non-centrality parameter
The prospective power estimations are based on (Kieser M. Methods and Applications of Sample Size Calculation and Recalculation in Clinical Trials. Springer; 2020).
The ANOVA power is calculated based on the non-centrality parameter given as
nc =\sqrt{\frac{r}{(1+r)^2}\cdot n}\cdot\frac{\beta_1-\beta_0-\Delta_s}{\sigma},
where we denote by \sigma^2 the variance of the outcome, such that the power can be estimated as
1-\beta = 1 - F_{t,n-2,nc}\left(F_{t, n-2, 0}^{-1}(1-\alpha)\right).
The power of ANCOVA with univariate covariate adjustment and no interaction is calculated based on the non-centrality parameter given as
nc =\sqrt{\frac{rn}{(1+r)^2}}\frac{\beta_1-\beta_0-\Delta_s}{\sigma\sqrt{1-\rho^2}},
such that the power can be estimated as
1-\beta = 1 - F_{t,n-3,nc}\left(F_{t, n-3, 0}^{-1}(1-\alpha)\right).
The power of ANCOVA with either univariate covariate adjustment and interaction or multiple covariate adjustement with or without interaction is calculated based on the non-centrality parameter given as
nc =\frac{\beta_1-\beta_0-\Delta_s}{\sqrt{\left(\frac{1}{n_1}+\frac{1}{n_0} + X_d^\top\left((n-2)\Sigma_X\right)^{-1}X_d \right)\sigma^2\left(1-\widehat{R}^2\right)}}.
where X_d = \left(\overline{X}_1^1-\overline{X}_0^1, \ldots, \overline{X}_1^p-\overline{X}_0^p\right)^\top, \widehat{R}^2= \frac{\widehat{\sigma}_{XY}^\top \widehat{\Sigma}_X^{-1}\widehat{\sigma}_{XY}}{\widehat{\sigma}^2},
we denote by \widehat{\sigma^2} an estimate of the variance of the outcome,
\widehat{\Sigma_X} and estimate of the covariance matrix of the
covariates, and \widehat{\sigma_{XY}} a p-dimensional column vector consisting of
an estimate of the covariance
between the outcome variable and each covariate.
Since we are in the case of randomized trials the expected difference between the covariate
values between the to groups is 0. Furthermore, the elements of \Sigma_X^{-1} will be small, unless the variances are close to 0, or the covariates exhibit strong linear dependencies, so that the correlations are close to 1.
These scenarios are excluded since they could lead to potentially serious problems regarding inference either way. These arguments are used by Zimmermann et. al
(Zimmermann G, Kieser M, Bathke AC. Sample Size Calculation and Blinded Recalculation for Analysis of Covariance Models with Multiple Random Covariates. Journal of Biopharmaceutical Statistics. 2020;30(1):143–159.) to approximate
the non-centrality parameter as in the univariate case where \rho^2 is replaced by R^2.
Then the power for ANCOVA with d degrees of freedom can be estimated as
1-\beta = 1 - F_{t,d,nc}\left(F_{t, d,0), 0}^{-1}(1-\alpha)\right).
Examples
# Generate a data set to use as an example
dat_gaus <- glm_data(Y ~ 1+2*X1-X2+3*A,
X1 = rnorm(100),
X2 = rgamma(100, shape = 2),
A = rbinom(100, size = 1, prob = 0.5),
family = gaussian())
# Approximate the power using no adjustment covariates
va_nocov <- var(dat_gaus$Y)
power_gs(n = 200, variance = va_nocov, ate = 1)
# Approximate the power with a model adjusting for both variables in the
# data generating process
## First estimate the variance sigma^2 * (1-R^2)
va_cov <- variance_ancova(Y ~ X1 + X2 + A, dat_gaus)
## Then estimate the power using this variance
power_gs(n = 100, variance = va_cov, ate = 1.8, margin = 1, r = 2)
# Approximate the sample size needed to obtain 90% power with same model as
# above
samplesize_gs(
variance = va_cov, ate = 1.8, power = 0.9, margin = 1, r = 2
)
# No adjustment covariates
power_nc(n = 200, variance = va_nocov, df = 199, ate = 1)
# Adjusting for all covariates in data generating process
power_nc(n = 200, variance = va_cov, df = 196, ate = 1.8, margin = 1, r = 2)
postcard documentation built on April 12, 2025, 1:57 a.m.
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| power_linear | R Documentation |
Power and sample size estimation for linear models
Description
variance_ancova provides a convenient function for estimating a
variance to use for power and sample size approximation.
The power_gs and samplesize_gs functions calculate the Guenther-Schouten
power approximation for ANOVA or ANCOVA.
The approximation is based in (Guenther WC. Sample Size Formulas for Normal Theory T Tests.
The American Statistician. 1981;35(4):243–244) and (Schouten HJA. Sample size formula with
a continuous outcome for unequal group sizes and unequal variances. Statistics in Medicine.
1999;18(1):87–91).
The function power_nc calculates the power for ANOVA or ANCOVA based on the
non-centrality parameter and the exact t-distributions.
See more details about each function in Details and in sections after Value.
Usage
variance_ancova(formula, data, inflation = 1, deflation = 1)
power_gs(variance, ate, n, r = 1, margin = 0, alpha = 0.025)
samplesize_gs(variance, ate, r = 1, margin = 0, power = 0.9, alpha = 0.025)
power_nc(variance, df, ate, n, r = 1, margin = 0, alpha = 0.025)
Arguments
formula |
an object of class "formula" (or one that can be coerced to that class):
a symbolic description used in |
data |
a data frame, list or environment (or object
coercible by |
inflation |
a |
deflation |
a |
variance |
a |
ate |
a |
n |
a |
r |
a |
margin |
a |
alpha |
a |
power |
a |
df |
a |
Details
This details section provides information about relation between arguments to functions and the formulas described in sections below for each power approximation formula.
Note that all entities that carry the same name as an argument and in the formula will not be mentioned below, as they are obviously linked (n, r, alpha)
-
ate:\beta_1-\beta_0 -
margin:\Delta_s -
variance:\widehat{\sigma}^2(1-\widehat{R}^2)
Finding the variance to use for approximation
The variance_ancova function estimates \sigma^2(1-R^2) in data and
returns it as a numeric that can be passed directly as the variance
in power_gs. Corresponds to estimating the power from using an lm with
the same formula as specified in variance_ancova.
The user can estimate the variance any way they see fit.
Value
All functions return a numeric. variance_ancova returns a numeric with
a variance estimated from data to used for power estimation and sample size
estimation. power_xx and samplesize_xx functions return a numeric with
the power or sample size approximation.
Guenther-Schouten power approximation
The estimation formula in the case of an ANCOVA model with multiple covariate adjustment is (see description for reference):
n=\frac{(1+r)^2}{r}\frac{(z_{1-\alpha}+z_{1-\beta})^2\widehat{\sigma}^2(1-\widehat{R}^2)}{(\beta_1-\beta_0-\Delta_s)^2}+\frac{(z_{1-\alpha})^2}{2}
where \widehat{R}^2= \frac{\widehat{\sigma}_{XY}^\top \widehat{\Sigma}_X^{-1}\widehat{\sigma}_{XY}}{\widehat{\sigma}^2},
we denote by \widehat{\sigma^2} an estimate of the variance of the outcome,
\widehat{\Sigma_X} and estimate of the covariance matrix of the
covariates, and \widehat{\sigma_{XY}} a p-dimensional column vector consisting of
an estimate of the covariance
between the outcome variable and each covariate.
In the univariate case R^2 is replaced by \rho^2
Power approximation using non-centrality parameter
The prospective power estimations are based on (Kieser M. Methods and Applications of Sample Size Calculation and Recalculation in Clinical Trials. Springer; 2020). The ANOVA power is calculated based on the non-centrality parameter given as
nc =\sqrt{\frac{r}{(1+r)^2}\cdot n}\cdot\frac{\beta_1-\beta_0-\Delta_s}{\sigma},
where we denote by \sigma^2 the variance of the outcome, such that the power can be estimated as
1-\beta = 1 - F_{t,n-2,nc}\left(F_{t, n-2, 0}^{-1}(1-\alpha)\right).
The power of ANCOVA with univariate covariate adjustment and no interaction is calculated based on the non-centrality parameter given as
nc =\sqrt{\frac{rn}{(1+r)^2}}\frac{\beta_1-\beta_0-\Delta_s}{\sigma\sqrt{1-\rho^2}},
such that the power can be estimated as
1-\beta = 1 - F_{t,n-3,nc}\left(F_{t, n-3, 0}^{-1}(1-\alpha)\right).
The power of ANCOVA with either univariate covariate adjustment and interaction or multiple covariate adjustement with or without interaction is calculated based on the non-centrality parameter given as
nc =\frac{\beta_1-\beta_0-\Delta_s}{\sqrt{\left(\frac{1}{n_1}+\frac{1}{n_0} + X_d^\top\left((n-2)\Sigma_X\right)^{-1}X_d \right)\sigma^2\left(1-\widehat{R}^2\right)}}.
where X_d = \left(\overline{X}_1^1-\overline{X}_0^1, \ldots, \overline{X}_1^p-\overline{X}_0^p\right)^\top, \widehat{R}^2= \frac{\widehat{\sigma}_{XY}^\top \widehat{\Sigma}_X^{-1}\widehat{\sigma}_{XY}}{\widehat{\sigma}^2},
we denote by \widehat{\sigma^2} an estimate of the variance of the outcome,
\widehat{\Sigma_X} and estimate of the covariance matrix of the
covariates, and \widehat{\sigma_{XY}} a p-dimensional column vector consisting of
an estimate of the covariance
between the outcome variable and each covariate.
Since we are in the case of randomized trials the expected difference between the covariate
values between the to groups is 0. Furthermore, the elements of \Sigma_X^{-1} will be small, unless the variances are close to 0, or the covariates exhibit strong linear dependencies, so that the correlations are close to 1.
These scenarios are excluded since they could lead to potentially serious problems regarding inference either way. These arguments are used by Zimmermann et. al
(Zimmermann G, Kieser M, Bathke AC. Sample Size Calculation and Blinded Recalculation for Analysis of Covariance Models with Multiple Random Covariates. Journal of Biopharmaceutical Statistics. 2020;30(1):143–159.) to approximate
the non-centrality parameter as in the univariate case where \rho^2 is replaced by R^2.
Then the power for ANCOVA with d degrees of freedom can be estimated as
1-\beta = 1 - F_{t,d,nc}\left(F_{t, d,0), 0}^{-1}(1-\alpha)\right).
Examples
# Generate a data set to use as an example
dat_gaus <- glm_data(Y ~ 1+2*X1-X2+3*A,
X1 = rnorm(100),
X2 = rgamma(100, shape = 2),
A = rbinom(100, size = 1, prob = 0.5),
family = gaussian())
# Approximate the power using no adjustment covariates
va_nocov <- var(dat_gaus$Y)
power_gs(n = 200, variance = va_nocov, ate = 1)
# Approximate the power with a model adjusting for both variables in the
# data generating process
## First estimate the variance sigma^2 * (1-R^2)
va_cov <- variance_ancova(Y ~ X1 + X2 + A, dat_gaus)
## Then estimate the power using this variance
power_gs(n = 100, variance = va_cov, ate = 1.8, margin = 1, r = 2)
# Approximate the sample size needed to obtain 90% power with same model as
# above
samplesize_gs(
variance = va_cov, ate = 1.8, power = 0.9, margin = 1, r = 2
)
# No adjustment covariates
power_nc(n = 200, variance = va_nocov, df = 199, ate = 1)
# Adjusting for all covariates in data generating process
power_nc(n = 200, variance = va_cov, df = 196, ate = 1.8, margin = 1, r = 2)
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