入門統計学 検出力

危険率 0.05% として，標準正規分布の上側95%点のz値は，

qnorm(0.95)

[1] 1.644854

[[]]

上図の\(N(\mu_0, \sigma^2)\) での a 点は，\( a = \mu_0 + 1.645 \sigma / \sqrt{n} \).

\(N(\mu, \sigma^2)\)でのa点は，\( a = \mu + z_{\a} \sigma / \sqrt{n} \).

\(z_{a} = \frac{a-\mu}{\sigma/\sqrt{n}} = 1.645 - (\mu - \mu_0)/\sigma/\sqrt{n} = 1.645 - \frac{\Delta\mu}{\sigma/\sqrt{n}}\)

検出力 \(1-\beta\) は，

\( 1-\beta = P(z>=z_{a}) \) と計算できる。

危険率 0.05% として，標準正規分布の上側97.5%点のz値は，

qnorm(0.975)

[1] 1.959964

[[]]

上図の\(N(\mu_0, \sigma^2)\) での a 点は，\( a = \mu_0 + 1.96 \sigma / \sqrt{n} \), b 点は，\( b = \mu_0 - 1.96 \sigma / \sqrt{n} \).

\(N(\mu, \sigma^2)\)でのa点は，\( a = \mu + z_{a} \sigma / \sqrt{n} \), b点は，\( b = \mu - z_b{} \sigma / \sqrt{n} \).

\(z_{a} = \frac{a-\mu}{\sigma/\sqrt{n}} = 1.96 - (\mu - \mu_0)/\sigma/\sqrt{n} = 1.96 - \frac{\Delta\mu}{\sigma/\sqrt{n}}\)

\(z_{b} = \frac{b-\mu}{\sigma/\sqrt{n}} = -1.96 - (\mu - \mu_0)/\sigma/\sqrt{n} = -1.96 - \frac{\Delta\mu}{\sigma/\sqrt{n}}\)

検出力 \(1-\beta\) は，

\( 1-\beta = P(z>=z_{a}) + P(z <= z_b \) と計算できる。

Compute the power of the one- or two- sample t test, or determine parameters to obtain a target power.

power.t.test(n = NULL, delta = NULL, sd = 1, sig.level = 0.05, power = NULL, type = c("two.sample", "one.sample", "paired"), alternative = c("two.sided", "one.sided"), strict = FALSE, tol = .Machine$double.eps^0.25)

n

number of observations (per group)

delta

true difference in means

sd

standard deviation

sig.level

significance level (Type I error probability)

power

power of test (1 minus Type II error probability)

type

string specifying the type of t test. Can be abbreviated.

alternative

one- or two-sided test. Can be abbreviated.

strict

use strict interpretation in two-sided case

tol

numerical tolerance used in root finding, the default providing (at least) four significant digits.

Exactly one of the parameters n, delta, power, sd, and sig.level must be passed as NULL, and that parameter is determined from the others. Notice that the last two have non-NULL defaults, so NULL must be explicitly passed if you want to compute them.

If strict = TRUE is used, the power will include the probability of rejection in the opposite direction of the true effect, in the two-sided case. Without this the power will be half the significance level if the true difference is zero.

Object of class "power.htest", a list of the arguments (including the computed one) augmented with method and note elements.

uniroot is used to solve the power equation for unknowns, so you may see errors from it, notably about inability to bracket the root when invalid arguments are given.

Peter Dalgaard. Based on previous work by Claus Ekstrøm

t.test, uniroot

 power.t.test(n = 20, delta = 1)

     Two-sample t test power calculation 

              n = 20
          delta = 1
             sd = 1
      sig.level = 0.05
          power = 0.8689528
    alternative = two.sided

NOTE: n is number in *each* group

power.t.test(power = .90, delta = 1)

     Two-sample t test power calculation 

              n = 22.0211
          delta = 1
             sd = 1
      sig.level = 0.05
          power = 0.9
    alternative = two.sided

NOTE: n is number in *each* group

power.t.test(power = .90, delta = 1, alternative = "one.sided")

     Two-sample t test power calculation 

              n = 17.84713
          delta = 1
             sd = 1
      sig.level = 0.05
          power = 0.9
    alternative = one.sided

NOTE: n is number in *each* group

[Package stats version 4.0.3 Index]