https://webpower.psychstat.org/wiki/ 2026-05-25T07:09:26-0400 https://webpower.psychstat.org/wiki/ https://webpower.psychstat.org/wiki/lib/tpl/dokuwiki/images/favicon.ico text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/balanced_one-way_anova?rev=1726763099&do=diff Statistical Power for Balanced One-way ANOVA Description Power for one-way ANOVA analysis. Arguments Among number of groups, sample size, effect size, significance level, and power, one and only one can be left blank. Number of Groups How many groups in the ANOVA analysis? text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/correlation_based_on_z-test?rev=1726763099&do=diff Statistical Power for Correlation Coefficient Conduct the power analysis now Arguments (Parameters) Among sample size, correlation, significance level, and power, one and only one can be left blank. Sample Size Provide the number of participants. Multiple sample sizes can be provided in two ways. First, multiple sample sizes can be supplied separated by white spaces, e.g., text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/diagram_based_power_analysis?rev=1726763099&do=diff Conduct Monte Carlo based power analysis using the path diagram interface Once can conduct power analysis using Monte Carlo based method by drawing a path diagram with population parameters. An example is given in the figure below: Rules to draw the path diagram text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/index?rev=1726763099&do=diff WebPower Manual WebPower is a collection of tools for conducting statistical power analysis online. WebPower can be used by anyone for free. To use the software now, click here. For a short tutorial on how to use WebPower, click here. For the free online manual book (more than 30MB), please [click here]. To order a hard copy of the book, click the book cover below: text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/linear_regression?rev=1726763099&do=diff Power for Linear Regression Description Power calculation / sample size planning for linear regression based on F test. Arguments Among sample size, number of predictors, effect size, significance level, and power, one and only one can be left blank.\[ f^{2}=\frac{R^{2}}{1-R^{2}} \]$R^{2}$$R^{2}$\[ f^{2}=\frac{R_{AB}^{2}-R_{A}^{2}}{1-R_{AB}^{2}} \]$R_{A}^{2}$$R_{AB}^{2}$$f^{2}$$p$$u=p$$v=n-p-1$$p1$$p2$$p2-p1$$v=n-p2-1$$f^{2}$$R^{2}$$f^{2}$$R^2$ text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/logistic_regression?rev=1726763099&do=diff [Click here to see slides on the power analysis]. text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/mediation01?rev=1726763099&do=diff Power for Simple Mediation via Sobel test Description Consider a simple mediation model $$m_i = a_0 + a*x_i + em_i$$ $$y_i = b_0 + b*m_i + c*x_i + ey_i$$ where $em_i \sim N(0, \sigma_{em}^2)$ and $ey_i \sim N(0, \sigma_{ey}^2)$. The mediation effect is $ab = a*b$. The Sobel test statistic is $$ Z = \frac{\hat{a}\hat{b}}{\hat{\sigma}_{ab}} $$ where $ \hat{\sigma}_{ab}^2 = \hat{a}^2 * \hat{\sigma}_b^2 + \hat{b}^2 * \hat{\sigma}_a^2$. From regression analysis, we have $$\hat{\sigma}_a^2 = … text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/mediation02?rev=1726763099&do=diff Sample size determination for testing mediation with the mediation Bayes factor Note that the method is based on Monte Carlo simulation. It can take a while to run. Wait for the program to complete and DO NOT refresh. The model The simple mediation model (standardized) is written as:$$ M = a*X + e_m $$$$ Y = c'*X + b*M + e_y $$$a$$b$$a*b$ text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/mlm01?rev=1726763099&do=diff Cluster randomized trials with 2 arms [Click here to see slides on the power analysis]. text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/mlm02?rev=1726763099&do=diff [Click here to see slides on the power analysis]. text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/mlm03?rev=1726763099&do=diff [Click here to see slides on the power analysis]. text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/mlm04?rev=1726763099&do=diff [Click here to see slides on the power analysis]. text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/modmed7?rev=1726763099&do=diff Algorithm Implementation of Power for Model 7 Conditional Processes Power analysis is conducted for the model below: Mediation X is the predictor and Y is the dependent variable . Variable X’s effect on a second variable Y is said to be mediated by a third variable M if X causally influences M and M in turn causally influences Y, and then M is the mediator in the model. Moderation$$ M=i_{M}+a_{1}X+a_{2}W+a_{3}XW $$$$ M=i_{M}+(a_{1}+a_{3}W)X+a_{2}W $$$$ Y=i_{Y}+c'X+b_{1}M $$$(a_{1}+a{3}W)b_{… text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/modmed8?rev=1726763099&do=diff Algorithm Implementation of Power for Model 8 Conditional Processes Power analysis is conducted for the model below: Mediation X is the predictor and Y is the dependent variable . Variable X’s effect on a second variable Y is said to be mediated by a third variable M if X causally influences M and M in turn causally influences Y, and then M is the mediator in the model.$$ M=i_{M}+a_{1}X+a_{2}W+a_{3}XW $$$$ M=i_{M}+(a_{1}+a_{3}W)X+a_{2}W $$$$ Y=i_{Y}+c'X+b_{1}M+b_{2}W+b_{3}XW $$$$ Y=i_{Y}+b_… text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/modmed14?rev=1726763099&do=diff Algorithm Implementation of Power for Model 14 Conditional Processes Power analysis is conducted for the model below: Mediation X is the predictor and Y is the dependent variable . Variable X’s effect on a second variable Y is said to be mediated by a third variable M if X causally influences M and M in turn causally influences Y, and then M is the mediator in the model.$$ M=i_{M}+a_{1}X $$$$ Y=i_{Y}+c'X+b_{1}M+c_{1}W+c_{2}XW $$$$ Y=i_{Y}+(c'+c_{2}W)X+b_{1}M+c_{1}W $$$(c'+c_{1}W)$$a_{1}b_{1… text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/modmed15?rev=1726763099&do=diff Algorithm Implementation of Power for Model 15 Conditional Processes Power analysis is conducted for the model below: Mediation X is the predictor and Y is the dependent variable . Variable X’s effect on a second variable Y is said to be mediated by a third variable M if X causally influences M and M in turn causally influences Y, and then M is the mediator in the model.$$ M=i_{M}+a_{1}X $$$$ Y=i_{Y}+c'X+b_{1}M+c_{1}W+c_{2}XW+b_{2}MW $$$$ Y=i_{Y}+(c'+c_{2}W)X+(b_{1}+b_{2}W)M+c_{1}W $$$(c'+c… text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/modmed58?rev=1726763099&do=diff Algorithm Implementation of Power for Model 58 Conditional Processes Power analysis is conducted for the model below: The Model $m=\beta_1x +a_1w+c_1xw+e_1$ $y=a_2w+\beta_3x+\beta_2m+c_2mw+e_2$ In model 58, both the first-stage path from the predictor $x$ to the mediator $m$ and the second-stage path from the mediator $m$$y$$w$$x$$y$$\theta_{x \to m \to y}=(\beta_1+c_1w)(\beta_2+c_2w)$ text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/one_sample_proportion?rev=1726763099&do=diff Statistical Power for One Sample Proportion Description The power is to detect that a proportion is different from 0.5. The power calculation is based on the arcsine transformation of the proportion (see Cohen, 1988; p548). Specifically, for a given proportion $p$$\phi=2*arcsin(\sqrt{p})$$p_0=0.5$$$h=2*arcsin(\sqrt{p})-2*arcsin(\sqrt{p_0})=2*arcsin(\sqrt{p})-2*arcsin(\sqrt{0.5}).$$$$\pi = 1-\Phi(C_\alpha - h\sqrt{n})$$$C_\alpha$$2*arcsin(\sqrt{p})-2*arcsin(\sqrt{.5})$$2*arcsin(\sqrt{p})-2*arcs… text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/power_analysis_based_on_rmsea_maccallum_et_al._1996?rev=1726763099&do=diff Power analysis based on RMSEA text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/power_analysis_using_satorra_saris_1985?rev=1726763099&do=diff Statistical Power for SEM based on Satorra & Saris (1985) Arguments (Parameters) Among sample size, degrees of freedom, effect size, significance level, and power, one and only one can be left blank. Sample Size Provide the number of participants. Multiple sample sizes can be provided in two ways. First, multiple sample sizes can be supplied separated by white spaces, e.g., text/html 2025-01-08T09:02:48-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/power_of_nanova?rev=1736344968&do=diff Power for ANOVA with more than one factor and interaction Current have problem with multiple effect sizes input. For now, please only input one effect size at a time. Description Power calculation for two-way ANOVA with interaction, three-way ANOVA with interaction for factorial designs.$J\times K\times L$$3\times 2\times 4 = 24$$3\times 2 = 6$$5\times 24=120$$J-1=3-1=2$$(J-1)\times (K-1) \times (L-1)$$(J-1)\times (K-1)$$(3-1)\times (2-1) \times (4-1) = 6$$f$$f$$\sigma_m$$\sigma$$$ f = \frac{… text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/power_of_rmanova?rev=1726763099&do=diff Power for Repeated-Measures ANOVA Description Power calculation for repeated-measures ANOVA for between effect, within effect, and between-within interaction. Arguments Among Number of groups, Number of measurements, Sample size, Effect size, Correlation across measurements, Nonsphericity correction, significance level, and power, one and only one field can be left blank. We now discuss how to input information for those fields.$3\times 10=30$$f$$\sigma_m$$\sigma$$C$$$ f = \frac{\sigma_m}{\s… text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/power_of_t-test?rev=1726763099&do=diff Power of t test Description Power calculation / sample size planning based on t test. Works for one-sample t test, paired two-sample t test, and two-sample t test with equal group sizes. It uses the R function power.t.test for power calculation. Arguments text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/power_of_t2n-test?rev=1726763099&do=diff Power of t test (Unbalanced sample sizes) Description Power calculation / sample size planning based on t test with different sample sizes. It uses the R function pwr.t.test from the pwr package for power calculation. Arguments Among sample size for group 1, sample size for group 2, effect size, significance level, and power, one and only one can be left blank. text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/sample_size_calculation_for_a_mixed_model_of_repeated_measures_with_general_correlation_structure?rev=1726763099&do=diff Statistical power for a mixed model of repeated measures with a general structure The calculation is based on the method described in Lu, Luo, & Chen (2008) Formula (3) on page 4 and the R function power.mmrm of the package longpower (Donohue & Edland, 2013). text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/sphericity?rev=1726763099&do=diff Sphericity ANOVAs with repeated measures (within-subject factors) are susceptible to the violation of the assumption of sphericity. Sphericity is the condition where the variances of the differences between all combinations of related groups (levels) are equal. Violation of sphericity is when the variances of the differences between all combinations of related groups are not equal. Violation of sphericity can cause the F-test too liberal (i.e., an increase in the Type I error rate).$\epsilon$$\… text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/statistical_power_for_a_mixed_model_of_repeated_measures_with_ar_1_correlation_structure?rev=1726763099&do=diff Statistical power for a mixed model of repeated measures with AR(1) correlation structure The calculation is based on the method described in Lu, Luo, & Chen (2008) and the R function power.mmrm.ar1 of the package longpower (Donohue & Edland, 2013). text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/two_sample_proportion_equal_sample_size?rev=1726763099&do=diff Statistical Power for Two Sample Proportion Description The power calculation is based on the arcsine transformation of the proportion (see Cohen (1988)) Arguments Among sample size, effect size, significance level, and power, one and only one can be left blank. text/html 2024-09-19T12:24:59-0400 Anonymous (anonymous@undisclosed.example.com) https://webpower.psychstat.org/wiki/manual/two_sample_proportion_unequal_sample_size?rev=1726763099&do=diff Statistical Power for Two Sample Proportion with Difference Sample Sizes Description The power calculation is based on the arcsine transformation of the proportion (see Cohen (1988)) Arguments Among sample size 1, sample size 2 effect size, significance level, and power, one and only one can be left blank.