| powerPoisson {powerMediation} | R Documentation |
Power calculation for simple Poisson regression
Description
Power calculation for simple Poisson regression. Assume the predictor is normally distributed.
Usage
powerPoisson(
beta0,
beta1,
mu.x1,
sigma2.x1,
mu.T = 1,
phi = 1,
alpha = 0.05,
N = 50)
Arguments
beta0 |
intercept |
beta1 |
slope |
mu.x1 |
mean of the predictor |
sigma2.x1 |
variance of the predictor |
mu.T |
mean exposure time |
phi |
a measure of over-dispersion |
alpha |
type I error rate |
N |
toal sample size |
Details
The simple Poisson regression has the following form:
Pr(Y_i = y_i | \mu_i, t_i) = \exp(-\mu_i t_i) (\mu_i t_i)^{y_i}/ (y_i!)
where
\mu_i=\exp(\beta_0+\beta_1 x_{1i})
We are interested in testing the null hypothesis \beta_1=0
versus the alternative hypothesis \beta_1 = \theta_1.
Assume x_{1} is normally distributed with mean
\mu_{x_1} and variance \sigma^2_{x_1}.
The sample size calculation formula derived by Signorini (1991) is
N=\phi\frac{\left[z_{1-\alpha/2}\sqrt{V\left(b_1 | \beta_1=0\right)}
+z_{power}\sqrt{V\left(b_1 | \beta_1=\theta_1\right)}\right]^2}
{\mu_T \exp(\beta_0) \theta_1^2}
where \phi is the over-dispersion parameter
(=var(y_i)/mean(y_i)),
\alpha is the type I error rate,
b_1 is the estimate of the slope \beta_1,
\beta_0 is the intercept,
\mu_T is the mean exposure time,
z_{a} is the 100*a-th lower percentile of
the standard normal distribution, and
V\left(b_1|\beta_1=\theta\right)
is the variance of the estimate b_1 given the true slope
\beta_1=\theta.
The variances are
V\left(b_1 | \beta_1 = 0\right)=\frac{1}{\sigma^2_{x_1}}
and
V\left(b_1 | \beta_1 = \theta_1\right)=\frac{1}{\sigma^2_{x_1}}
\exp\left[-\left(\theta_1 \mu_{x_1} + \theta_1^2\sigma^2_{x_1}/2\right)\right]
Value
power
Note
The test is a two-sided test. For one-sided tests, please double the
significance level. For example, you can set alpha=0.10
to obtain one-sided test at 5% significance level.
Author(s)
Weiliang Qiu <stwxq@channing.harvard.edu>
References
Signorini D.F. (1991). Sample size for Poisson regression. Biometrika. Vol.78. no.2, pp. 446-50
See Also
See Also as sizePoisson
Examples
# power = 0.8090542
print(powerPoisson(
beta0 = 0.1,
beta1 = 0.5,
mu.x1 = 0,
sigma2.x1 = 1,
mu.T = 1,
phi = 1,
alpha = 0.05,
N = 28))