Power
Power analysis/Sample Size determination
- https://en.m.wikipedia.org/wiki/Power_(statistics)
- Sample size determination from Wikipedia
- Power and Sample Size Determination http://www.stat.wisc.edu/~st571-1/10-power-2.pdf#page=12
- http://biostat.mc.vanderbilt.edu/wiki/pub/Main/AnesShortCourse/HypothesisTestingPart1.pdf#page=40
- Power analysis and sample size calculation for Agriculture (pwr, lmSupport, simr packages are used)
- Why Within-Subject Designs Require Fewer Participants than Between-Subject Designs
Power analysis for default Bayesian t-tests
http://daniellakens.blogspot.com/2016/01/power-analysis-for-default-bayesian-t.html
Using simulation for power analysis
Power analysis and sample size calculation for Agriculture
http://r-video-tutorial.blogspot.com/2017/07/power-analysis-and-sample-size.html
Power calculation for proportions (shiny app)
https://juliasilge.shinyapps.io/power-app/
Derive the formula/manual calculation
- One-sample 1-sided test, One sample 2-sided test
- Two-sample 2-sided T test ([math]\displaystyle{ n }[/math] is the sample size in each group)
- [math]\displaystyle{ \begin{align} Power & = P_{\mu_1-\mu_2 = \Delta}(\frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\sigma^2/n + \sigma^2/n}} \gt Z_{\alpha /2}) + P_{\mu_1-\mu_2 = \Delta}(\frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\sigma^2/n + \sigma^2/n}} \lt -Z_{\alpha /2}) \\ & \approx P_{\mu_1-\mu_2 = \Delta}(\frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\sigma^2/n + \sigma^2/n}} \gt Z_{\alpha /2}) \\ & = P_{\mu_1-\mu_2 = \Delta}(\frac{\bar{X}_1 - \bar{X}_2 - \Delta}{\sqrt{2 * \sigma^2/n}} \gt Z_{\alpha /2} - \frac{\Delta}{\sqrt{2 * \sigma^2/n}}) \\ & = \Phi(-(Z_{\alpha /2} - \frac{\Delta}{\sqrt{2 * \sigma^2/n}})) \\ & = 1 - \beta =\Phi(Z_\beta) \end{align} }[/math]
Therefore
- [math]\displaystyle{ \begin{align} Z_{\beta} &= - Z_{\alpha/2} + \frac{\Delta}{\sqrt{2 * \sigma^2/n}} \\ Z_{\beta} + Z_{\alpha/2} & = \frac{\Delta}{\sqrt{2 * \sigma^2/n}} \\ 2 * (Z_{\beta} + Z_{\alpha/2})^2 * \sigma^2/\Delta^2 & = n \\ n & = 2 * (Z_{\beta} + Z_{\alpha/2})^2 * \sigma^2/\Delta^2 \end{align} }[/math]
# alpha = .05, delta = 200, n = 79.5, sigma=450 1 - pnorm(1.96 - 200*sqrt(79.5)/(sqrt(2)*450)) + pnorm(-1.96 - 200*sqrt(79.5)/(sqrt(2)*450)) # [1] 0.8 pnorm(-1.96 - 200*sqrt(79.5)/(sqrt(2)*450)) # [1] 9.58e-07 1 - pnorm(1.96 - 200*sqrt(79.5)/(sqrt(2)*450)) # [1] 0.8
Calculating required sample size in R and SAS
pwr package is used. For two-sided test, the formula for sample size is
- [math]\displaystyle{ n_{\mbox{each group}} = \frac{2 * (Z_{\alpha/2} + Z_\beta)^2 * \sigma^2}{\Delta^2} = \frac{2 * (Z_{\alpha/2} + Z_\beta)^2}{d^2} }[/math]
where [math]\displaystyle{ Z_\alpha }[/math] is value of the Normal distribution which cuts off an upper tail probability of [math]\displaystyle{ \alpha }[/math], [math]\displaystyle{ \Delta }[/math] is the difference sought, [math]\displaystyle{ \sigma }[/math] is the presumed standard deviation of the outcome, [math]\displaystyle{ \alpha }[/math] is the type 1 error, [math]\displaystyle{ \beta }[/math] is the type II error and (Cohen's) d is the effect size - difference between the means divided by the pooled standard deviation.
# An example from http://www.stat.columbia.edu/~gelman/stuff_for_blog/c13.pdf#page=3
# Method 1.
require(pwr)
pwr.t.test(d=200/450, power=.8, sig.level=.05,
type="two.sample", alternative="two.sided")
#
# Two-sample t test power calculation
#
# n = 80.4
# d = 0.444
# sig.level = 0.05
# power = 0.8
# alternative = two.sided
#
# NOTE: n is number in *each* group
# Method 2.
2*(qnorm(.975) + qnorm(.8))^2*450^2/(200^2)
# [1] 79.5
2*(1.96 + .84)^2*450^2 / (200^2)
# [1] 79.4
And stats::power.t.test() function.
power.t.test(n = 79.5, delta = 200, sd = 450, sig.level = .05,
type ="two.sample", alternative = "two.sided")
#
# Two-sample t test power calculation
#
# n = 79.5
# delta = 200
# sd = 450
# sig.level = 0.05
# power = 0.795
# alternative = two.sided
#
# NOTE: n is number in *each* group
CRAN Task View: Design of Experiments
- powerAnalysis w/o vignette
- powerbydesign w/o vignette
- easypower w/ vignette
- pwr w/ vignette, https://www.statmethods.net/stats/power.html. The reference is Cohen's book.
- powerlmm Power Analysis for Longitudinal Multilevel/Linear Mixed-Effects Models.
- ssize.fdr w/o vignette
- samplesize w/o vignette
- ssizeRNA w/ vignette
- power.t.test(), power.anova.test(), power.prop.test() from stats package
Russ Lenth Java applets
https://homepage.divms.uiowa.edu/~rlenth/Power/index.html
Bootstrap method
The upstrap Crainiceanu & Crainiceanu, Biostatistics 2018
Multiple Testing Case
Optimal Sample Size for Multiple Testing The Case of Gene Expression Microarrays
Unbalanced randomization
Can unbalanced randomization improve power?
Yes, unbalanced randomization can improve power, in some situations