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: an example based on a stepped wedge study design

https://www.rdatagen.net/post/using-simulation-for-power-analysis-an-example/

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

R package related to power analysis

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?