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 PASS documentation.

NOTE: 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.

Usage

power.z.poisson(base.rate = NULL, rate.ratio = NULL,
                beta0 = log(base.rate), beta1 = log(rate.ratio),
                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 = TRUE, pretty = 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

log(base.rate) or natural logarithm of the base mean event rate.

beta1

log(rate.ratio) or natural logarithm of the relative increase in the mean event rate for one unit increase in the predictor.

mean.exposure

the mean exposure time (should be > 0). Usually 1

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.

alpha

type 1 error rate, defined as the probability of incorrectly rejecting a true null hypothesis, denoted as \(\alpha\).

alternative

character; direction or type of the hypothesis test: "not equal", "greater", "less".

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

logical; whether the output should be printed on the console. TRUE by default.

pretty

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.

References

Demidenko, E. (2007). Sample size determination for logistic regression revisited. Statistics in Medicine, 26(18), 3385-3397. doi: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 Claim) : Rate Ratio = 1 
#>   H1 (Alt. Claim) : Rate Ratio != 1 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size          = 474  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.8
#> 

## 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 Claim) : Rate Ratio = 1 
#>   H1 (Alt. Claim) : Rate Ratio != 1 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size          = 474  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.8
#> 

## 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 Claim) : Rate Ratio = 1 
#>   H1 (Alt. Claim) : Rate Ratio != 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 Claim) : Rate Ratio = 1 
#>   H1 (Alt. Claim) : Rate Ratio != 1 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size          = 2003  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.8
#> 

## 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 Claim) : Rate Ratio = 1 
#>   H1 (Alt. Claim) : Rate Ratio != 1 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size          = 2003  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.8
#> 

## 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 Claim) : Rate Ratio = 1 
#>   H1 (Alt. Claim) : Rate Ratio != 1 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Sample Size          = 2404  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.8
#>