Power Analysis for Poisson Regression Coefficient (Wald's z Test)

Calculates power or sample size (only one can be NULL at a time) to test a single coefficient in poisson regression. power.z.poisson() and power.z.poisreg() are the same functions, as well as pwrss.z.poisson() and pwrss.z.poisreg(). The distribution of the predictor variable can be one of the following: c("normal", "poisson", "uniform", "exponential", "binomial", "bernouilli", "lognormal"). The default parameters for these distributions are

distribution = list(dist = "normal", mean = 0, sd = 1)
distribution = list(dist = "poisson", lambda = 1)
distribution = list(dist = "uniform", min = 0, max = 1)
distribution = list(dist = "exponential", rate = 1)
distribution = list(dist = "binomial", size = 1, prob = 0.50)
distribution = list(dist = "bernoulli", prob = 0.50)
distribution = list(dist = "lognormal", meanlog = 0, sdlog = 1)

Parameters defined in list() form can be modified, but the names should be kept the same. It is sufficient to use distribution's name for default parameters (e.g. dist = "normal").

Formulas are validated using Monte Carlo simulation, G*Power, and tables in the PASS documentation.

Usage

power.z.poisson(
  base.rate = NULL,
  rate.ratio = NULL,
  beta0 = NULL,
  beta1 = NULL,
  n = NULL,
  power = NULL,
  r.squared.predictor = 0,
  mean.exposure = 1,
  alpha = 0.05,
  alternative = c("two.sided", "one.sided"),
  method = c("demidenko(vc)", "demidenko", "signorini"),
  distribution = "normal",
  ceiling = TRUE,
  verbose = 1,
  utf = FALSE
)

Arguments

base.rate: the base mean event rate.
rate.ratio: event rate ratio. The relative increase in the mean event rate for one unit increase in the predictor (similar to odds ratio in logistic regression).
beta0: the natural logarithm of the base mean event rate.
beta1: the natural logarithm of the relative increase in the mean event rate for one unit increase in the predictor.
n: integer; sample size.
power: statistical power, defined as the probability of correctly rejecting a false null hypothesis, denoted as \(1 - \beta\).
r.squared.predictor: proportion of variance in the predictor accounted for by other covariates. This is not a pseudo R-squared. To compute it, regress the predictor on the covariates and extract the adjusted R-squared from that model.
mean.exposure: the mean exposure time (should be > 0), usually it is 1.
alpha: type 1 error rate, defined as the probability of incorrectly rejecting a true null hypothesis, denoted as \(\alpha\).
alternative: character; the direction or type of the hypothesis test: "two.sided" or "one.sided".
method: character; calculation method: "demidenko(vc)" stands for Demidenko (2007) procedure with variance correction; "demidenko" stands for Demidenko (2007) procedure without variance correction; "signorini" stands for Signorini (1991) procedure. "demidenko" and "signorini" methods produce similar results but "demidenko(vc)" is more precise.
distribution: character; distribution family. Can be one of the c("normal", "poisson", "uniform", "exponential", "binomial", "bernouilli", "lognormal").
ceiling: logical; whether sample size should be rounded up. TRUE by default.
verbose: 1 by default (returns test, hypotheses, and results), if 2 a more detailed output is given (plus key parameters and definitions), if 0 no output is printed on the console.
utf: logical; whether the output should show Unicode characters (if encoding allows for it). FALSE by default.

Value

parms: list of parameters used in calculation.
test: type of the statistical test (Z-Test).
mean: mean of the alternative distribution.
sd: standard deviation of the alternative distribution.
null.mean: mean of the null distribution.
null.sd: standard deviation of the null distribution.
z.alpha: critical value(s).
power: statistical power \((1-\beta)\)
n: sample size.

Details

NB: The pwrss.z.poisson() and its alias pwrss.z.poisreg() are deprecated. However, they will remain available as wrappers for the power.z.logistic() function during a transition period.

References

Demidenko, E. (2007). Sample size determination for logistic regression revisited. Statistics in Medicine, 26(18), 3385-3397. https://doi.org/10.1002/sim.2771

Signorini, D. F. (1991). Sample size for Poisson regression. Biometrika, 78(2), 446-450.

Examples

# predictor X follows normal distribution

## regression coefficient specification
power.z.poisson(beta0 = 0.50, beta1 = -0.10,
                alpha = 0.05, power = 0.80,
                dist = "normal")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Poisson Regression Coefficient (Wald's Z-Test)
#> 
#>   Method          : Demidenko (Variance Corrected)
#>   Predictor Dist. : Normal
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : Rate Ratio (RR)  = 1
#>   H1 (Alternative) : Rate Ratio (RR) != 1
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 474  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.800
#> 

## rate ratio specification
power.z.poisson(base.rate = exp(0.50),
                rate.ratio = exp(-0.10),
                alpha = 0.05, power = 0.80,
                dist = "normal")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Poisson Regression Coefficient (Wald's Z-Test)
#> 
#>   Method          : Demidenko (Variance Corrected)
#>   Predictor Dist. : Normal
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : Rate Ratio (RR)  = 1
#>   H1 (Alternative) : Rate Ratio (RR) != 1
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 474  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.800
#> 

## change parameters associated with predictor X
dist.x <- list(dist = "normal", mean = 10, sd = 2)
power.z.poisson(base.rate = exp(0.50),
                rate.ratio = exp(-0.10),
                alpha = 0.05, power = 0.80,
                dist = dist.x)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Poisson Regression Coefficient (Wald's Z-Test)
#> 
#>   Method          : Demidenko (Variance Corrected)
#>   Predictor Dist. : Normal
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : Rate Ratio (RR)  = 1
#>   H1 (Alternative) : Rate Ratio (RR) != 1
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 318  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.199
#>   Statistical Power    = 0.801
#> 


# predictor X follows Bernoulli distribution (such as treatment/control groups)

## regression coefficient specification
power.z.poisson(beta0 = 0.50, beta1 = -0.10,
                alpha = 0.05, power = 0.80,
                dist = "bernoulli")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Poisson Regression Coefficient (Wald's Z-Test)
#> 
#>   Method          : Demidenko (Variance Corrected)
#>   Predictor Dist. : Bernoulli
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : Rate Ratio (RR)  = 1
#>   H1 (Alternative) : Rate Ratio (RR) != 1
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 2003  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.800
#> 

## rate ratio specification
power.z.poisson(base.rate = exp(0.50),
                rate.ratio = exp(-0.10),
                alpha = 0.05, power = 0.80,
                dist = "bernoulli")
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Poisson Regression Coefficient (Wald's Z-Test)
#> 
#>   Method          : Demidenko (Variance Corrected)
#>   Predictor Dist. : Bernoulli
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : Rate Ratio (RR)  = 1
#>   H1 (Alternative) : Rate Ratio (RR) != 1
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 2003  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.800
#> 

## change parameters associatied with predictor X
dist.x <- list(dist = "bernoulli", prob = 0.30)
power.z.poisson(base.rate = exp(0.50),
                rate.ratio = exp(-0.10),
                alpha = 0.05, power = 0.80,
                dist = dist.x)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Poisson Regression Coefficient (Wald's Z-Test)
#> 
#>   Method          : Demidenko (Variance Corrected)
#>   Predictor Dist. : Bernoulli
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : Rate Ratio (RR)  = 1
#>   H1 (Alternative) : Rate Ratio (RR) != 1
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 2404  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.800
#>