Statistical Power for the Generic Z-Test

Calculates power for the generic Z-Test with (optional) Type 1 and Type 2 error plots.

power.z.test(mean = NULL, sd = 1, null.mean = 0, null.sd = 1,
             alpha = 0.05, alternative = c("two.sided",
                                           "one.sided", "two.one.sided"),
             plot = TRUE, verbose = TRUE, pretty = FALSE, ...)

Arguments

mean: mean of the alternative.
sd: standard deviation of the alternative. Do not change this value except when some sort of variance correction is applied (e.g. as in logistic and Poisson regressions).
null.mean: mean of the null. When alternative = "two.one.sided", the function expects two values in the form c(lower, upper). If a single value is provided, it is interpreted as the absolute bound and automatically expanded to c(-value, +value).
null.sd: standard deviation of the null. Do not change this value except when some sort of correction is applied.
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: "one.sided", "two.sided", or "two.one.sided". "two.one.sided" is used for equivalence and minimal effect testing.
plot: logical; FALSE switches off Type 1 and Type 2 error plot. TRUE by default.
verbose: logical; whether the output should be printed on the console. TRUE by default.
...: legacy inputs will be mapped to their corresponding arguments (silent). e.g. ncp
pretty: logical; whether the output should show Unicode characters (if encoding allows for it). FALSE by default.

Value

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 \((1-\beta)\).

Examples

# two-sided
# power defined as the probability of observing z-statistics
# greater than the positive critical t value OR
# less than the negative critical t value
power.z.test(mean = 1.96, alpha = 0.05,
             alternative = "two.sided")

#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Generic Z-Test
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : mean = null.mean 
#>   H1 (Alt. Claim) : mean != null.mean 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.500
#>   Statistical Power    = 0.5  <<
#> 

# one-sided
# power is defined as the probability of observing z-statistics
# greater than the critical t value
power.z.test(mean = 1.96, alpha = 0.05,
             alternative = "one.sided")

#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Generic Z-Test
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : mean <= null.mean 
#>   H1 (Alt. Claim) : mean > null.mean 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.376
#>   Statistical Power    = 0.624  <<
#> 

# equivalence
# power is defined as the probability of observing a test statistic
# greater than the upper critical value (for the lower bound) AND
# less than the lower critical value (for the upper bound)
power.z.test(mean = 0, null.mean = c(-2, 2), alpha = 0.05,
             alternative = "two.one.sided")

#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Generic Z-Test
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : mean <= min(null.mean) or 
#>                     mean >= max(null.mean) 
#>   H1 (Alt. Claim) : mean > min(null.mean) and 
#>                     mean < max(null.mean) 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.722
#>   Statistical Power    = 0.278  <<
#> 

# minimal effect testing
# power is defined as the probability of observing a test statistic
# greater than the upper critical value (for the upper bound) OR
# less than the lower critical value (for the lower bound).
power.z.test(mean = 2, null.mean = c(-1, 1), alpha = 0.05,
             alternative = "two.one.sided")

#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Generic Z-Test
#> 
#> ---------------------------------------------------
#> Hypotheses
#> ---------------------------------------------------
#>   H0 (Null Claim) : mean >= min(null.mean) and 
#>                     mean <= max(null.mean) 
#>   H1 (Alt. Claim) : mean < min(null.mean) or 
#>                     mean > max(null.mean) 
#> 
#> ---------------------------------------------------
#> Results
#> ---------------------------------------------------
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.831
#>   Statistical Power    = 0.169  <<
#>