Basic Functions for Power Analysis pwr

Description

Power calculations along the lines of Cohen (1988) using in particular the same notations for effect sizes. Examples from the book are given.

Details

This package contains functions for basic power calculations using effect sizes and notations from Cohen (1988) : pwr.p.test: test for one proportion (ES=h) pwr.2p.test: test for two proportions (ES=h) pwr.2p2n.test: test for two proportions (ES=h, unequal sample sizes) pwr.t.test: one sample and two samples (equal sizes) t tests for means (ES=d) pwr.t2n.test: two samples (different sizes) t test for means (ES=d) pwr.anova.test: test for one-way balanced anova (ES=f) pwr.r.test: correlation test (ES=r) pwr.chisq.test: chi-squared test (ES=w) pwr.f2.test: test for the general linear model (ES=f2) ES.h: computing effect size h for proportions tests ES.w1: computing effect size w for the goodness of fit chi-squared test ES.w2: computing effect size w for the association chi-squared test cohen.ES: computing effect sizes for all the previous tests corresponding to conventional effect sizes (small, medium, large)

Author(s)

Stephane Champely, based on previous works by Claus Ekstrom and Peter Dalgaard, with contributions of Jeffrey Gill, Stephan Weibelzahl, Clay Ford, Aditya Anandkumar and Robert Volcic.

Maintainer: Helios De Rosario-Martinez <helios.derosario@gmail.com>

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

See Also

power.t.test,power.prop.test,power.anova.test

Examples

## Exercise 8.1 P. 357 from Cohen (1988)
pwr.anova.test(f=0.28,k=4,n=20,sig.level=0.05)

## Exercise 6.1 p. 198 from Cohen (1988)
pwr.2p.test(h=0.3,n=80,sig.level=0.05,alternative="greater")

## Exercise 7.3 p. 251
pwr.chisq.test(w=0.346,df=(2-1)*(3-1),N=140,sig.level=0.01)

## Exercise 6.5 p. 203 from Cohen (1988)
pwr.p.test(h=0.2,n=60,sig.level=0.05,alternative="two.sided")

Effect size calculation for proportions

Description

Compute effect size h for two proportions

Usage

ES.h(p1, p2)

Arguments

Details

The effect size is 2*asin(sqrt(p1))-2*asin(sqrt(p2))

Value

The corresponding effect size

Author(s)

Stephane CHAMPELY

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

See Also

pwr.p.test, pwr.2p.test, pwr.2p2n.test, power.prop.test

Examples

## Exercise 6.5 p. 203 from Cohen 
h<-ES.h(0.5,0.4)
h
pwr.p.test(h=h,n=60,sig.level=0.05,alternative="two.sided")

Effect size calculation in the chi-squared test for goodness of fit

Description

Compute effect size w for two sets of k probabilities P0 (null hypothesis) and P1 (alternative hypothesis)

Usage

ES.w1(P0, P1)

Arguments

Value

The corresponding effect size w

Author(s)

Stephane CHAMPELY

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

See Also

pwr.chisq.test

Examples

## Exercise 7.1 p. 249 from Cohen 
P0<-rep(1/4,4)
P1<-c(0.375,rep((1-0.375)/3,3))
ES.w1(P0,P1)
pwr.chisq.test(w=ES.w1(P0,P1),N=100,df=(4-1))

Effect size calculation in the chi-squared test for association

Description

Compute effect size w for a two-way probability table corresponding to the alternative hypothesis in the chi-squared test of association in two-way contingency tables

Usage

ES.w2(P)

Arguments

Value

The corresponding effect size w

Author(s)

Stephane CHAMPELY

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

See Also

pwr.chisq.test

Examples

prob<-matrix(c(0.225,0.125,0.125,0.125,0.16,0.16,0.04,0.04),nrow=2,byrow=TRUE)
prob
ES.w2(prob)
pwr.chisq.test(w=ES.w2(prob),df=(2-1)*(4-1),N=200)

Conventional effects size

Description

Give the conventional effect size (small, medium, large) for the tests available in this package

Usage

cohen.ES(test = c("p", "t", "r", "anov", "chisq", "f2"),
    size = c("small", "medium", "large"))

Arguments

Value

The corresponding effect size

Author(s)

Stephane CHAMPELY

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

Examples

## medium effect size for the correlation test
cohen.ES(test="r", size="medium")

## sample size for a medium size effect in the two-sided correlation test
## using the conventional power of 0.80
pwr.r.test(r=cohen.ES(test="r",size="medium")$effect.size,
  power=0.80, sig.level=0.05, alternative="two.sided")

Plot diagram of sample size vs. test power

Description

Plot a diagram to illustrate the relationship of sample size and test power for a given set of parameters.

Usage

## S3 method for class 'power.htest'
plot(x, ...)

Arguments

Details

Power calculations for the following tests are supported: t-test (pwr.t.test(), pwr.t2n.test()), chi squared test (pwr.chisq.test()), one-way ANOVA (pwr.anova.test(), standard normal distribution (pwr.norm.test()), Pearson correlation (pwr.r.test()), proportions (pwr.p.test(), pwr.2p.test(), pwr.2p2n.test()))

Value

These functions are invoked for their side effect of drawing on the active graphics device.

Note

By default it attempts to use the plotting tools of ggplot2 and scales. If they are not installed, it will use the basic R plotting tools.

Author(s)

Stephan Weibelzahl <weibelzahl@pfh.de>

See Also

pwr.t.test, pwr.p.test, pwr.2p.test, pwr.2p2n.test, pwr.r.test, pwr.chisq.test, pwr.anova.test, pwr.t2n.test

Examples

## Two-sample t-test
p.t.two <- pwr.t.test(d=0.3, power=0.8, type="two.sample", alternative="two.sided")
plot(p.t.two)
plot(p.t.two, xlab="sample size per group")

Power calculation for two proportions (same sample sizes)

Description

Compute power of test, or determine parameters to obtain target power (similar to power.prop.test).

Usage

pwr.2p.test(h = NULL, n = NULL, sig.level = 0.05, power = NULL, 
    alternative = c("two.sided","less","greater"))

Arguments

Details

Exactly one of the parameters 'h','n', 'power' and 'sig.level' must be passed as NULL, and that parameter is determined from the others. Notice that the last one has non-NULL default so NULL must be explicitly passed if you want to compute it.

Value

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

Note

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

Author(s)

Stephane Champely <champely@univ-lyon1.fr> but this is a mere copy of Peter Dalgaard work (power.t.test)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

See Also

ES.h, pwr.2p2n.test, power.prop.test

Examples

## Exercise 6.1 p. 198 from Cohen (1988)
pwr.2p.test(h=0.3,n=80,sig.level=0.05,alternative="greater")

Power calculation for two proportions (different sample sizes)

Description

Compute power of test, or determine parameters to obtain target power.

Usage

pwr.2p2n.test(h = NULL, n1 = NULL, n2 = NULL, sig.level = 0.05, power = NULL,
    alternative = c("two.sided", "less","greater"))

Arguments

Details

Exactly one of the parameters 'h','n1', 'n2', 'power' and 'sig.level' must be passed as NULL, and that parameter is determined from the others. Notice that the last one has non-NULL default so NULL must be explicitly passed if you want to compute it.

Value

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

Note

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

Author(s)

Stephane Champely <champely@univ-lyon1.fr> but this is a mere copy of Peter Dalgaard work (power.t.test)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

See Also

ES.h, pwr.2p.test, power.prop.test

Examples

## Exercise 6.3 P. 200 from Cohen (1988)
pwr.2p2n.test(h=0.30,n1=80,n2=245,sig.level=0.05,alternative="greater")

## Exercise 6.7 p. 207 from Cohen (1988)
pwr.2p2n.test(h=0.20,n1=1600,power=0.9,sig.level=0.01,alternative="two.sided")

Power calculations for balanced one-way analysis of variance tests

Description

Compute power of test or determine parameters to obtain target power (same as power.anova.test).

Usage

pwr.anova.test(k = NULL, n = NULL, f = NULL, sig.level = 0.05, power = NULL)

Arguments

Details

Exactly one of the parameters 'k','n','f','power' and 'sig.level' must be passed as NULL, and that parameter is determined from the others. Notice that the last one has non-NULL default so NULL must be explicitly passed if you want to compute it.

Value

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

Note

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

Author(s)

Stephane Champely <champely@univ-lyon1.fr> but this is a mere copy of Peter Dalgaard work (power.t.test)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

See Also

power.anova.test

Examples

## Exercise 8.1 P. 357 from Cohen (1988) 
pwr.anova.test(f=0.28,k=4,n=20,sig.level=0.05)

## Exercise 8.10 p. 391
pwr.anova.test(f=0.28,k=4,power=0.80,sig.level=0.05)

power calculations for chi-squared tests

Description

Compute power of test or determine parameters to obtain target power (same as power.anova.test).

Usage

pwr.chisq.test(w = NULL, N = NULL, df = NULL, sig.level = 0.05, power = NULL)

Arguments

Details

Exactly one of the parameters 'w','N','power' and 'sig.level' must be passed as NULL, and that parameter is determined from the others. Notice that the last one has non-NULL default so NULL must be explicitly passed if you want to compute it.

Value

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

Note

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

Author(s)

Stephane Champely <champely@univ-lyon1.fr> but this is a mere copy of Peter Dalgaard work (power.t.test)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

See Also

ES.w1,ES.w2

Examples

## Exercise 7.1 P. 249 from Cohen (1988) 
pwr.chisq.test(w=0.289,df=(4-1),N=100,sig.level=0.05)

## Exercise 7.3 p. 251
pwr.chisq.test(w=0.346,df=(2-1)*(3-1),N=140,sig.level=0.01)

## Exercise 7.8 p. 270
pwr.chisq.test(w=0.1,df=(5-1)*(6-1),power=0.80,sig.level=0.05)

Power calculations for the general linear model

Description

Compute power of test or determine parameters to obtain target power (same as power.anova.test).

Usage

pwr.f2.test(u = NULL, v = NULL, f2 = NULL, sig.level = 0.05, power = NULL)

Arguments

Details

Exactly one of the parameters 'u','v','f2','power' and 'sig.level' must be passed as NULL, and that parameter is determined from the others. Notice that the last one has non-NULL default so NULL must be explicitly passed if you want to compute it.

Value

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

Note

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

Author(s)

Stephane Champely <champely@univ-lyon1.fr> but this is a mere copy of Peter Dalgaard work (power.t.test)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

Examples

## Exercise 9.1 P. 424 from Cohen (1988)
pwr.f2.test(u=5,v=89,f2=0.1/(1-0.1),sig.level=0.05)

Power calculations for the mean of a normal distribution (known variance)

Description

Compute power of test or determine parameters to obtain target power (same as power.anova.test).

Usage

pwr.norm.test(d = NULL, n = NULL, sig.level = 0.05, power = NULL, 
    alternative = c("two.sided","less","greater"))

Arguments

Details

Exactly one of the parameters 'd','n','power' and 'sig.level' must be passed as NULL, and that parameter is determined from the others. Notice that the last one has non-NULL default so NULL must be explicitly passed if you want to compute it.

Value

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

Note

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

Author(s)

Stephane Champely <champely@univ-lyon1.fr> but this is a mere copy of Peter Dalgaard work (power.t.test)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

Examples

## Power at mu=105 for H0:mu=100 vs. H1:mu>100 (sigma=15) 20 obs. (alpha=0.05) 
sigma<-15
c<-100
mu<-105
d<-(mu-c)/sigma
pwr.norm.test(d=d,n=20,sig.level=0.05,alternative="greater")

## Sample size of the test for power=0.80
pwr.norm.test(d=d,power=0.8,sig.level=0.05,alternative="greater")

## Power function of the same test
mu<-seq(95,125,l=100)
d<-(mu-c)/sigma
plot(d,pwr.norm.test(d=d,n=20,sig.level=0.05,alternative="greater")$power,
    type="l",ylim=c(0,1))
abline(h=0.05)
abline(h=0.80)

## Power function for the two-sided alternative
plot(d,pwr.norm.test(d=d,n=20,sig.level=0.05,alternative="two.sided")$power,
    type="l",ylim=c(0,1))
abline(h=0.05)
abline(h=0.80)

Power calculations for proportion tests (one sample)

Description

Compute power of test or determine parameters to obtain target power (same as power.anova.test).

Usage

pwr.p.test(h = NULL, n = NULL, sig.level = 0.05, power = NULL,
    alternative = c("two.sided","less","greater"))

Arguments

Details

These calculations use arcsine transformation of the proportion (see Cohen (1988))

Exactly one of the parameters 'h','n','power' and 'sig.level' must be passed as NULL, and that parameter is determined from the others. Notice that the last one has non-NULL default so NULL must be explicitly passed if you want to compute it.

Value

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

Note

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

Author(s)

Stephane Champely <champely@univ-lyon1.fr> but this is a mere copy of Peter Dalgaard work (power.t.test)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

See Also

ES.h

Examples

## Exercise 6.5 p. 203 from Cohen 
h<-ES.h(0.5,0.4)
h
pwr.p.test(h=h,n=60,sig.level=0.05,alternative="two.sided")

## Exercise 6.8 p. 208
pwr.p.test(h=0.2,power=0.95,sig.level=0.05,alternative="two.sided")

Power calculations for correlation test

Description

Compute power of test or determine parameters to obtain target power (same as power.anova.test).

Usage

pwr.r.test(n = NULL, r = NULL, sig.level = 0.05, power = NULL,
    alternative = c("two.sided", "less","greater"))

Arguments

Details

These calculations use the Z' transformation of correlation coefficient : Z'=arctanh(r) and a bias correction is applied. Note that contrary to Cohen (1988) p.546, where zp' = arctanh(rp) + rp/(2*(n-1)) and zc' = arctanh(rc) + rc/(2*(n-1)), we only use here zp' = arctanh(rp) + rp/(2*(n-1)) and zc' = arctanh(rc).

Exactly one of the parameters 'r','n','power' and 'sig.level' must be passed as NULL, and that parameter is determined from the others. Notice that the last one has non-NULL default so NULL must be explicitly passed if you want to compute it.

Value

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

Note

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

Author(s)

Stephane Champely <champely@univ-lyon1.fr> but this is a mere copy of Peter Dalgaard work (power.t.test). The modified bias correction is contributed by Jeffrey Gill.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

Examples

## Exercise 3.1 p. 96 from Cohen (1988)
pwr.r.test(r=0.3,n=50,sig.level=0.05,alternative="two.sided")
pwr.r.test(r=0.3,n=50,sig.level=0.05,alternative="greater")

## Exercise 3.4 p. 208
pwr.r.test(r=0.3,power=0.80,sig.level=0.05,alternative="two.sided")
pwr.r.test(r=0.5,power=0.80,sig.level=0.05,alternative="two.sided")
pwr.r.test(r=0.1,power=0.80,sig.level=0.05,alternative="two.sided")

Power calculations for t-tests of means (one sample, two samples and paired samples)

Description

Compute power of tests or determine parameters to obtain target power (similar to power.t.test).

Usage

pwr.t.test(n = NULL, d = NULL, sig.level = 0.05, power = NULL, 
    type = c("two.sample", "one.sample", "paired"),
    alternative = c("two.sided", "less", "greater"))

Arguments

Details

Exactly one of the parameters 'd','n','power' and 'sig.level' must be passed as NULL, and that parameter is determined from the others. Notice that the last one has non-NULL default so NULL must be explicitly passed if you want to compute it.

Value

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

Note

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

Author(s)

Stephane Champely <champely@univ-lyon1.fr> but this is a mere copy of Peter Dalgaard work (power.t.test)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

See Also

power.prop.test

Examples

## One sample (power)
## Exercise 2.5 p. 47 from Cohen (1988)
pwr.t.test(d=0.2,n=60,sig.level=0.10,type="one.sample",alternative="two.sided")

## Paired samples (power)
## Exercise p. 50 from Cohen (1988)
d<-8/(16*sqrt(2*(1-0.6)))
pwr.t.test(d=d,n=40,sig.level=0.05,type="paired",alternative="two.sided")

## Two independent samples (power)
## Exercise 2.1 p. 40 from Cohen (1988)
d<-2/2.8
pwr.t.test(d=d,n=30,sig.level=0.05,type="two.sample",alternative="two.sided")

## Two independent samples (sample size)
## Exercise 2.10 p. 59
pwr.t.test(d=0.3,power=0.75,sig.level=0.05,type="two.sample",alternative="greater")

Power calculations for two samples (different sizes) t-tests of means

Description

Compute power of tests or determine parameters to obtain target power (similar to as power.t.test).

Usage

pwr.t2n.test(n1 = NULL, n2= NULL, d = NULL, sig.level = 0.05, power = NULL,
 alternative = c("two.sided", 
        "less","greater"))

Arguments

Details

Exactly one of the parameters 'd','n1','n2','power' and 'sig.level' must be passed as NULL, and that parameter is determined from the others. Notice that the last one has non-NULL default so NULL must be explicitly passed if you want to compute it.

Value

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

Note

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

Author(s)

Stephane Champely <champely@univ-lyon1.fr> but this is a mere copy of Peter Dalgaard work (power.t.test)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

Examples

## Exercise 2.3 p. 437 from Cohen (1988)
pwr.t2n.test(d=0.6,n1=90,n2=60,alternative="greater")