Statistical Power for the Generic t-Test

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

Usage

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

Arguments

ncp: non-centrality parameter for the alternative.
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.
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

df: degrees of freedom.
ncp: non-centrality parameter under alternative.
ncp.null: non-centrality parameter under null.
t.alpha: critical value(s).
power: statistical power \((1-\beta)\).

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)        : ncp  = null.ncp
#>   H1 (Alternative) : ncp != null.ncp
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.507
#>   Statistical Power    = 0.493  <<
#> 

# 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)        : ncp <= null.ncp
#>   H1 (Alternative) : ncp  > null.ncp
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.381
#>   Statistical Power    = 0.619  <<
#> 

# 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)        : ncp <= min(null.ncp) or
#>                      ncp >= max(null.ncp)
#>   H1 (Alternative) : ncp  > min(null.ncp) and
#>                      ncp  < max(null.ncp)
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   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)        : ncp >= min(null.ncp) and
#>                      ncp <= max(null.ncp)
#>   H1 (Alternative) : ncp  < min(null.ncp) or
#>                      ncp  > max(null.ncp)
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.837
#>   Statistical Power    = 0.163  <<
#>