Skip to contents

Statistical Power for the Generic t-Test

Source: R/generic.t.test.R
power.t.test.Rd

Determines the power, the non-centrality parameter, or the degrees of freedom for the generic t-Test with (optional) Type 1 and Type 2 error plots.

Usage

power.t.test(
  power = NULL,
  ncp = NULL,
  req.sign = "+",
  null.ncp = 0,
  df = NULL,
  alpha = 0.05,
  alternative = c("two.sided", "one.sided", "two.one.sided"),
  plot = TRUE,
  verbose = 1,
  utf = FALSE
)

Arguments

power

statistical power \((1 - \beta)\); either power, ncp or df needs to be NULL (and is then estimated).

ncp

non-centrality parameter for the alternative; ; either power, ncp or df needs to be NULL (and is then estimated).

req.sign

whether ncp is expected to be greater '+1', less than '-1', or within '0' the null.ncp bounds; only relevant if ncp is to be estimated.

null.ncp

non-centrality parameter for 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).

df

degrees of freedom; either power, ncp or df needs to be NULL (and is then estimated).

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", "one.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

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

power

statistical power \((1-\beta)\).

ncp

non-centrality parameter under alternative.

null.ncp

non-centrality parameter under null.

df

degrees of freedom.

alpha

type 1 error rate (user-specified).

t.alpha

critical value(s).

beta

type 2 error rate.

type.s

type S error rate (only for two-tailed test).

type.m

type M error rate (only for two-tailed test).

Examples

# two-sided
# power defined as the probability of observing test statistics greater
# than the positive critical value OR less than the negative critical value
power.t.test(ncp = 1.96, df = 100, alpha = 0.05, alternative = "two.sided")

#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Generic t-Test
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : lambda  = null.lambda
#>   H1 (Alternative) : lambda != null.lambda
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Target NCP (lambda)  = 1.960 (vs. null.lambda = 0)
#>   Degrees of Freedom   = 100
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.507
#>   Statistical Power    = 0.493  <<
#> 
power.t.test(power = 0.80, df = 100, alpha = 0.05, alternative = "two.sided")

#> +--------------------------------------------------+
#> |       MINIMUM DETECTABLE NCP CALCULATION         |
#> +--------------------------------------------------+
#> 
#> Generic t-Test
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : lambda  = null.lambda
#>   H1 (Alternative) : lambda != null.lambda
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Target NCP (lambda)  = 2.829 (vs. null.lambda = 0)  <<
#>   Degrees of Freedom   = 100
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.800
#> 

# one-sided
# power is defined as the probability of observing a test statistic greater
# than the critical value
power.t.test(ncp = 1.96, df = 100, alpha = 0.05, alternative = "one.sided")

#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Generic t-Test
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : lambda <= null.lambda
#>   H1 (Alternative) : lambda  > null.lambda
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Target NCP (lambda)  = 1.960 (vs. null.lambda = 0)
#>   Degrees of Freedom   = 100
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.381
#>   Statistical Power    = 0.619  <<
#> 
power.t.test(power = 0.80, df = 100, alpha = 0.05, alternative = "one.sided")

#> +--------------------------------------------------+
#> |       MINIMUM DETECTABLE NCP CALCULATION         |
#> +--------------------------------------------------+
#> 
#> Generic t-Test
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : lambda <= null.lambda
#>   H1 (Alternative) : lambda  > null.lambda
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Target NCP (lambda)  = 2.503 (vs. null.lambda = 0)  <<
#>   Degrees of Freedom   = 100
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.800
#> 

# 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.t.test(ncp = 0, null.ncp = c(-2, 2), df = 100, alpha = 0.05,
             alternative = "two.one.sided")

#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Generic t-Test
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : lambda <= min(null.lambda) or
#>                      lambda >= max(null.lambda)
#>   H1 (Alternative) : lambda  > min(null.lambda) and
#>                      lambda  < max(null.lambda)
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Target NCP (lambda)  = 0 (vs. null.lambda = -2 and 2)
#>   Degrees of Freedom   = 100
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.723
#>   Statistical Power    = 0.277  <<
#> 
power.t.test(power = 0.80, req.sign = "0", null.ncp = c(-2, 2),
             df = 100, alpha = 0.05, alternative = "two.one.sided")
#> Warning: The target power rate cannot be achieved within the null bounds.

#> +--------------------------------------------------+
#> |       MINIMUM DETECTABLE NCP CALCULATION         |
#> +--------------------------------------------------+
#> 
#> Generic t-Test
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : lambda <= min(null.lambda) or
#>                      lambda >= max(null.lambda)
#>   H1 (Alternative) : lambda  > min(null.lambda) and
#>                      lambda  < max(null.lambda)
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Target NCP (lambda)  = -0.000 (vs. null.lambda = -2 and 2)  <<
#>   Degrees of Freedom   = 100
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.723
#>   Statistical Power    = 0.277
#> 

# 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.t.test(ncp = 2, null.ncp = c(-1, 1), df = 100, alpha = 0.05,
             alternative = "two.one.sided")

#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Generic t-Test
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : lambda >= min(null.lambda) and
#>                      lambda <= max(null.lambda)
#>   H1 (Alternative) : lambda  < min(null.lambda) or
#>                      lambda  > max(null.lambda)
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Target NCP (lambda)  = 2 (vs. null.lambda = -1 and 1)
#>   Degrees of Freedom   = 100
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.837
#>   Statistical Power    = 0.163  <<
#> 
power.t.test(power = 0.80, req.sign = "+", null.ncp = c(-1, 1),
             df = 100, alpha = 0.05, alternative = "two.one.sided")

#> +--------------------------------------------------+
#> |       MINIMUM DETECTABLE NCP CALCULATION         |
#> +--------------------------------------------------+
#> 
#> Generic t-Test
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : lambda >= min(null.lambda) and
#>                      lambda <= max(null.lambda)
#>   H1 (Alternative) : lambda  < min(null.lambda) or
#>                      lambda  > max(null.lambda)
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Target NCP (lambda)  = 3.864 (vs. null.lambda = -1 and 1)  <<
#>   Degrees of Freedom   = 100
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.800
#>