diff --git a/.Rbuildignore b/.Rbuildignore index 0fb6768..cc05900 100644 --- a/.Rbuildignore +++ b/.Rbuildignore @@ -9,3 +9,7 @@ archive ^man-roxygen$ CNAME temp.R +docs/ +package-lock.json +package.json +node_modules/ diff --git a/.github/workflows/docs.yml b/.github/workflows/docs.yml new file mode 100644 index 0000000..c4fe98d --- /dev/null +++ b/.github/workflows/docs.yml @@ -0,0 +1,64 @@ +# Workflow name. +name: docs + +# Events. +on: + # Trigger on push to branch. + push: + branches: [main] + # Trigger manually via the GitHub UI. + workflow_dispatch: + +# Jobs. +jobs: + docs: + # Virtual machine type. + runs-on: ubuntu-latest + + # Job steps. + steps: + # Fetch all commits to get all `.git` info logs. + - name: Checkout repository + uses: actions/checkout@v2 + with: + fetch-depth: 0 + + # Install `Node.js`. + - name: Setup Node.js + uses: actions/setup-node@v1 + with: + node-version: '14' + + # Cache `node_modules`. + - name: Cache dependencies + uses: actions/cache@v2 + id: yarn-cache + with: + path: | + **/node_modules + key: ${{ runner.os }}-yarn-${{ hashFiles('**/yarn.lock') }} + restore-keys: | + ${{ runner.os }}-yarn- + + # Install `Node.js` dependencies if the cache did not hit. + - name: Install dependencies + if: steps.yarn-cache.outputs.cache-hit != 'true' + run: yarn --frozen-lockfile + + # Run build script. + - name: Build VuePress site + run: yarn docs:build + + # Deploy website. + # https://github.com/crazy-max/ghaction-github-pages. + - name: Deploy to GitHub Pages + uses: crazy-max/ghaction-github-pages@v2 + with: + # Deploy to `gh-pages` branch. + target_branch: gh-pages + # Deploy the default output dir of VuePress. + build_dir: docs/.vuepress/dist + # Write the given domain name to the CNAME file. + fqdn: powerly.dev + env: + GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} diff --git a/.gitignore b/.gitignore index 0e78846..8eaf11c 100644 --- a/.gitignore +++ b/.gitignore @@ -6,3 +6,7 @@ archive powerly.Rcheck/ *.tar.gz **/temp* +node_modules/ +.temp +.cache +dist diff --git a/NEWS.md b/NEWS.md index f6f4dfb..27f31d4 100644 --- a/NEWS.md +++ b/NEWS.md @@ -3,6 +3,11 @@ All notable changes to this project will be documented in this file. The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/), and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html). +## development +### Added +- Add starter package documentation via `VuePress`. The deployed documentation + files can be accessed at [powerly.dev](https://powerly.dev). + ## 1.8.2 - 2022-07-22 ### Fixed - Remove square brackets from version numbers in `NEWS.md` to allow CRAN to diff --git a/docs/.vuepress/config.ts b/docs/.vuepress/config.ts new file mode 100644 index 0000000..aaa259f --- /dev/null +++ b/docs/.vuepress/config.ts @@ -0,0 +1,38 @@ +import { defineUserConfig } from "vuepress" +import { defaultTheme } from "vuepress"; +import { sidebar, navbar, head } from "./configs"; +import { katexPlugin } from "@renovamen/vuepress-plugin-katex"; +import { shikiPlugin } from '@vuepress/plugin-shiki'; + +/** + * VuePress config. + */ +export default defineUserConfig({ + base: "/", + lang: "en-US", + title: "powerly", + description: "Sample Size Analysis for Psychological Networks and more...", + head: head, + theme: defaultTheme({ + docsDir: "docs", + docsBranch: "main", + repoLabel: "GitHub", + lastUpdated: true, + contributors: true, + navbar: navbar, + sidebar: sidebar, + sidebarDepth: 2, + logo: "/images/logos/powerly-logo.png", + repo: "https://github.com/mihaiconstantin/powerly", + editLinkText: "Edit this page on GitHub" + }), + plugins: [ + // LaTeX plugin. + katexPlugin(), + + // Syntax highlighting plugin. + shikiPlugin({ + theme: "dark-plus" + }) + ] +}) diff --git a/docs/.vuepress/configs/head.ts b/docs/.vuepress/configs/head.ts new file mode 100644 index 0000000..40a5525 --- /dev/null +++ b/docs/.vuepress/configs/head.ts @@ -0,0 +1,33 @@ +import type { HeadConfig } from '@vuepress/core' + + +/** + * Header links. + */ +export const head: HeadConfig[] = [ + [ + 'link', + { + rel: 'icon', + type: 'image/png', + sizes: '16x16', + href: `/images/icons/favicon-16x16.png`, + }, + ], + [ + 'link', + { + rel: 'icon', + type: 'image/png', + sizes: '32x32', + href: `/images/icons/favicon-32x32.png`, + }, + ], + ['meta', { name: 'application-name', content: 'powerly' }], + ['meta', { name: 'apple-mobile-web-app-title', content: 'powerly' }], + ['meta', { name: 'apple-mobile-web-app-status-bar-style', content: 'black' }], + [ + 'link', + { rel: 'apple-touch-icon', href: `/images/icons/apple-touch-icon.png` }, + ], +] diff --git a/docs/.vuepress/configs/index.ts b/docs/.vuepress/configs/index.ts new file mode 100644 index 0000000..4e6a1f3 --- /dev/null +++ b/docs/.vuepress/configs/index.ts @@ -0,0 +1,3 @@ +export * from './head'; +export * from './sidebar'; +export * from './navbar'; diff --git a/docs/.vuepress/configs/navbar.ts b/docs/.vuepress/configs/navbar.ts new file mode 100644 index 0000000..105f688 --- /dev/null +++ b/docs/.vuepress/configs/navbar.ts @@ -0,0 +1,32 @@ +import type { NavbarConfig } from '@vuepress/theme-default'; + + +/** + * Navbar links. + */ +export const navbar: NavbarConfig = [ + { + text: "Tutorials", + link: "/tutorial/" + }, + { + text: "Reference", + link: "/reference/" + }, + { + text: "Publications", + link: "/publication/" + }, + { + text: "Developer", + link: "/developer/" + }, + { + text: "News", + link: "https://github.com/mihaiconstantin/powerly/blob/main/NEWS.md" + }, + { + text: "CRAN", + link: "https://CRAN.R-project.org/package=powerly" + } +] diff --git a/docs/.vuepress/configs/sidebar.ts b/docs/.vuepress/configs/sidebar.ts new file mode 100644 index 0000000..56e73fe --- /dev/null +++ b/docs/.vuepress/configs/sidebar.ts @@ -0,0 +1,39 @@ +import type { SidebarConfig } from '@vuepress/theme-default'; + + +/** + * Sidebar links. + */ +export const sidebar: SidebarConfig = { + "/tutorial/": [ + "/tutorial/index.md", + "/tutorial/method.md", + { + text: 'Applications', + collapsible: true, + children: [ + "/tutorial/application/power-psychological-networks.md", + "/tutorial/application/power-structural-equation-modeling.md", + "/tutorial/application/power-multilevel-models.md", + ] + }, + { + text: 'FAQ', + collapsible: true, + children: [ + "/tutorial/faq/choosing-the-initial-range.md", + "/tutorial/faq/validating-the-results.md", + "/tutorial/faq/choosing-the-true-model.md", + ] + }, + ], + "/reference/": [ + "/reference/index.md", + ], + "/publication/": [ + "/publication/index.md", + ], + "/developer/": [ + "/developer/index.md", + ] +} diff --git a/docs/.vuepress/public/favicon.ico b/docs/.vuepress/public/favicon.ico new file mode 100644 index 0000000..1f48c71 Binary files /dev/null and b/docs/.vuepress/public/favicon.ico differ diff --git a/docs/.vuepress/public/images/content/powerly-feature-step-1.png b/docs/.vuepress/public/images/content/powerly-feature-step-1.png new file mode 100644 index 0000000..f05af38 Binary files /dev/null and b/docs/.vuepress/public/images/content/powerly-feature-step-1.png differ diff --git a/docs/.vuepress/public/images/content/powerly-feature-step-2.png b/docs/.vuepress/public/images/content/powerly-feature-step-2.png new file mode 100644 index 0000000..658e7e2 Binary files /dev/null and b/docs/.vuepress/public/images/content/powerly-feature-step-2.png differ diff --git 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@include material_shadow_md; +} + +// Home page. +.theme-container.page-home { + .feature img { + @include material_shadow_lg; + } + + .main-text { + margin-top: 2.5rem; + border-top: 1px solid var(--c-border); + transition: border-color var(--t-color); + padding-top: 1.5rem; + + .repo-badges { + text-align: center; + + img { + margin-right: 0.5rem; + border-radius: 0rem; + margin-top: 0.3rem; + } + } + + .main-text-content { + margin-left: 2.5rem; + margin-right: 2.5rem; + text-align: center; + } + } + + @media only screen and (max-width: 719px) { + .main-text { + .main-text-content { + margin-left: 1rem; + margin-right: 1rem; + text-align: center; + } + } + } + + .license { + margin-top: 0.5rem; + } + + .heart { + display: inline-block; + color: rgb(196, 27, 27); + font-size: 1.3rem; + cursor: default; + filter: grayscale(20%); + transition: all .5s; + + &:before { + content: "E"; + font-family: "Comic Sans MS", "Comic Sans", cursive; + font-size: 0.4rem; + text-align: center; + position: absolute; + left: 0; + right: 0; + bottom: 30%; + text-align: center; + transition: all .5s; + } + + &:hover { + filter: grayscale(0%); + transform: rotate(360deg); + + &:before { + color: rgb(220, 186, 186); + } + } + } +} + + +// Tutorial. +.theme-container.page-tutorial { + table { + td { + line-height: 1.7; + + a > code { + color: var(--c-text-accent); + font-weight: 600; + } + } + } + + ol { + li { + margin-left: 1rem; + padding-left: 0.5rem; + } + } + + .showcase-image, .showcase-video { + text-align: center; + + img, video { + width: 80%; + @include material_shadow_md; + } + + video { + padding-top: 0.75rem; + } + } + + .citation { + padding-left: 25px; + text-indent: -25px; + + p:first-of-type { + margin-bottom: 0.5rem; + } + + p:not(:first-of-type){ + margin-top: 0.5rem; + margin-bottom: 0rem; + } + + p:last-of-type { + margin-bottom: 1rem; + } + } + + .question { + margin-top: 1.5rem; + margin-bottom: 1.75rem; + border: 0px solid red; + + .katex { + transition: all .5s; + } + + p { + border: 0px solid blue; + margin-top: 5rem; + width: 80%; + margin: 0 auto; + padding: 1.5rem 2rem; + font-size: 1.1rem; + text-align: center; + border-radius: 0.5rem; + @include material_shadow_lg; + + &:hover { + .katex { + color: var(--c-text-accent); + } + } + } + } + + .step-3-intersection { + img { + width: 60%; + border: 0.5px solid #373737; + } + } + + .table-arguments { + table { + font-size: 0.875rem; + } + } + + @media only screen and (max-width: 719px) { + .question { + p { + padding: 1.5rem; + } + } + } +} diff --git a/docs/.vuepress/styles/palette.scss b/docs/.vuepress/styles/palette.scss new file mode 100644 index 0000000..e02371b --- /dev/null +++ b/docs/.vuepress/styles/palette.scss @@ -0,0 +1,17 @@ +:root { + // brand colors + --c-brand: #5a90c0; + --c-brand-light: #5a90c0d1; + --code-hl-bg-color: #363b46; +} + +html.dark { + // brand colors + --c-brand: #5a90c0; + --c-brand-light: #5a90c0d1; +} + +// Code languages. +$codeLang: 'c' 'cpp' 'cs' 'css' 'dart' 'docker' 'fs' 'go' 'html' 'java' 'js' + 'json' 'kt' 'less' 'makefile' 'md' 'php' 'py' 'r' 'rb' 'rs' 'sass' 'scss' 'sh' + 'styl' 'ts' 'toml' 'txt' 'vue' 'yml' !default; diff --git a/docs/CNAME b/docs/CNAME deleted file mode 100644 index 2b7ae67..0000000 --- a/docs/CNAME +++ /dev/null @@ -1 +0,0 @@ -powerly.dev \ No newline at end of file diff --git a/docs/developer/index.md b/docs/developer/index.md new file mode 100644 index 0000000..f50f8ee --- /dev/null +++ b/docs/developer/index.md @@ -0,0 +1,7 @@ +--- +pageClass: page-developer +--- + +# Developer + +*Content to be added.* diff --git a/docs/index.md b/docs/index.md new file mode 100644 index 0000000..dbd84d0 --- /dev/null +++ b/docs/index.md @@ -0,0 +1,74 @@ +--- +title: Welcome +home: true +heroImage: /images/logos/powerly-logo.png +heroAlt: Sample Size Analysis Tools +heroText: powerly +tagline: Sample Size Analysis for Psychological Networks ...and more +actions: + - text: Get Started + link: /tutorial/ + type: primary + - text: Reference + link: /reference/ + type: secondary +footer: Made with ♥ by Mihai Constantin
MIT licensed
+footerHtml: true +pageClass: page-home +--- + +
+ +
+ Repository status + + CRAN version + CRAN RStudio mirror downloads + Code coverage + R-CMD-check + CRAN checks +
+ +
+ +`powerly` is an `R` package for conducting sample size analysis that implements +the method by [Constantin et al. (2021)](https://psyarxiv.com/j5v7u). The +implementation takes the form of a three-step recursive algorithm designed to +find an optimal sample size given a model specification and an outcome measure +of interest. It starts with a Monte Carlo simulation step for computing the +outcome at various sample sizes. It continues with a monotone curve-fitting step +for interpolating the outcome. The final step employs stratified bootstrapping +to quantify the uncertainty around the fitted curve. For more details, consult +the [manuscript](https://psyarxiv.com/j5v7u) or check out the +[tutorial](/tutorial/) section. + +
+
+ + +
+ +
+

Step 1

+

+ powerly Step 1 + +

+
+ +
+

Step 2

+

+ + powerly Step 2 +

+
+ +
+

Step 3

+

+ + powerly Step 3 +

+
+
diff --git a/docs/publication/index.md b/docs/publication/index.md new file mode 100644 index 0000000..00b5c1f --- /dev/null +++ b/docs/publication/index.md @@ -0,0 +1,7 @@ +--- +pageClass: page-publication +--- + +# Publication + +*Content to be added.* diff --git a/docs/reference/index.md b/docs/reference/index.md new file mode 100644 index 0000000..661bb46 --- /dev/null +++ b/docs/reference/index.md @@ -0,0 +1,7 @@ +--- +pageClass: page-reference +--- + +# Reference + +*Content to be added.* diff --git a/docs/tutorial/application/power-multilevel-models.md b/docs/tutorial/application/power-multilevel-models.md new file mode 100644 index 0000000..3e81612 --- /dev/null +++ b/docs/tutorial/application/power-multilevel-models.md @@ -0,0 +1,8 @@ +--- +pageClass: page-tutorial +title: Multilevel Models +--- + +# Multilevel Models + +*Content to be added.* diff --git a/docs/tutorial/application/power-psychological-networks.md b/docs/tutorial/application/power-psychological-networks.md new file mode 100644 index 0000000..c0a2327 --- /dev/null +++ b/docs/tutorial/application/power-psychological-networks.md @@ -0,0 +1,8 @@ +--- +pageClass: page-tutorial +title: Psychological Networks +--- + +# Psychological Networks + +*Content to be added.* diff --git a/docs/tutorial/application/power-structural-equation-modeling.md b/docs/tutorial/application/power-structural-equation-modeling.md new file mode 100644 index 0000000..515675b --- /dev/null +++ b/docs/tutorial/application/power-structural-equation-modeling.md @@ -0,0 +1,8 @@ +--- +pageClass: page-tutorial +title: Structural Equation Modeling +--- + +# Structural Equation Modeling + +*Content to be added.* diff --git a/docs/tutorial/faq/choosing-the-initial-range.md b/docs/tutorial/faq/choosing-the-initial-range.md new file mode 100644 index 0000000..7b5766c --- /dev/null +++ b/docs/tutorial/faq/choosing-the-initial-range.md @@ -0,0 +1,8 @@ +--- +pageClass: page-tutorial +title: Choosing the Initial Range +--- + +# Choosing the Initial Range + +*Content to be added.* diff --git a/docs/tutorial/faq/choosing-the-true-model.md b/docs/tutorial/faq/choosing-the-true-model.md new file mode 100644 index 0000000..fcc4f6d --- /dev/null +++ b/docs/tutorial/faq/choosing-the-true-model.md @@ -0,0 +1,8 @@ +--- +pageClass: page-tutorial +title: Choosing the True Model +--- + +# Choosing the True Model + +*Content to be added.* diff --git a/docs/tutorial/faq/validating-the-results.md b/docs/tutorial/faq/validating-the-results.md new file mode 100644 index 0000000..8416f16 --- /dev/null +++ b/docs/tutorial/faq/validating-the-results.md @@ -0,0 +1,8 @@ +--- +pageClass: page-tutorial +title: Validating the Results +--- + +# Validating the Results + +*Content to be added.* diff --git a/docs/tutorial/index.md b/docs/tutorial/index.md new file mode 100644 index 0000000..22e8580 --- /dev/null +++ b/docs/tutorial/index.md @@ -0,0 +1,203 @@ +--- +pageClass: page-tutorial +header-includes: + - \usepackage{bm} +--- + +# Introduction + +`powerly` is an `R` package developed to provide support for researchers who +*wish* to perform power analysis. At its core, `powerly` is based on the method +proposed by [Constantin et al. (2021)](https://psyarxiv.com/j5v7u), which takes +the form of a three-step algorithm designed to iteratively search for an optimal +sample size, given some user input. It takes as input a set of hypothesized true +model parameters (i.e., an effect size), a performance measure of interest and +its corresponding target value (e.g., a sensitivity of $0.6$), and a statistic +that describes *how* the performance measure value should be reached (e.g., +reaching a sensitivity of $0.6$ with a probability of $0.8$). The search for the +optimal sample size is conducted in three steps, starting with a Monte Carlo +simulation step for computing the performance measure and statistic at various +sample sizes within a candidate sample size range. It continues with a monotone +curve-fitting step for interpolating the statistic. The final step employs +stratified bootstrapping to quantify the uncertainty around the fitted curve. +For more details, consult the [manuscript](https://psyarxiv.com/j5v7u) or check +out the remainder of the [Tutorial Section](/tutorial/). + +Next, I will provide a quick overview on how to get started with `powerly`, list +the main functions available in the package, and highlight the main steps +involved in using `powerly` for running a power analysis. + +::: tip +If you are interested in building upon or extending `powerly`, check out the +[Developer Section](/developer/) where the package `API` and the design choices +are discussed in detail. +::: + +## Installation + +You can install the latest version of `powerly` from +[`CRAN`](https://cran.r-project.org/web/packages/powerly/index.html) (i.e., the +official `R` package repository) running the following command: + +```r:no-line-numbers +install.packages("powerly") +``` + +Alternatively, you can install the latest development version from `GitHub` via: + +::: details Click to expand optional step. +```r +# Install the `remotes` package if necessary. +install.packages("remotes") +``` +::: + +```r:no-line-numbers +remotes::install_github("mihaiconstantin/powerly") +``` + +::: tip +You may consult the +[News](https://github.com/mihaiconstantin/powerly/blob/main/NEWS.md) or the +[`GitHub` releases](https://github.com/mihaiconstantin/powerly/releases) for an +overview of the latest changes in the development version. +::: + +## Functions + +`powerly` exports three functions available to researchers for conducting sample +size analysis. + +| Function Name | Description | +| :-------------------------------------------- | :------------------------------------------------------------------------------------------------------------------------------------ | +| [`generate_model`](/reference/generate-model) | Used for generating true models parameters when manually specifying them is non-trivial. | +| [`powerly`](/reference/powerly) | The *main* function of the package implementing the sample size calculation method described above. | +| [`validate`](/reference/validate) | Used for assessing the quality of the sample size recommendation obtained after running the [`powerly`](/reference/powerly) function. | + +::: tip +Check out the [Reference Section](/reference/functions/) corresponding to each +function for detailed information about the available arguments and the output +provided. +::: + +## Usage + +We start by attaching the `powerly` package. + +```r +# Load the package. +library(powerly) +``` + +Suppose we want to find the sample size required to observe a sensitivity of +$0.6$ with a probability of $0.8$ for an arbitrary Gaussian Graphical Model +(GGM) consisting of $10$ nodes with an edge density of $0.4$. + +We start by specifying the true model (i.e., in this case an edge weights matrix +encoding of partial correlation coefficients), or we can generate one using the +`generate_model` function. For example, this can be achieved by running: + +::: details Click to see the seed used for reproducibility. +We may set a seed to ensure that we generate the same true model. While this +will not lead to the exact same sample size recommendation for each repeated run +of the method (i.e., due to the randomness involved in the Monte Carlo +procedure), the method will still, on average, provide the optimal sample size +value (i.e., around $525$ for this particular example). + +```r +# Set a seed. +set.seed(20031993) +``` +::: + +```r +# Generate a `GGM` model. +true_model <- generate_model( + type = "ggm", + nodes = 10, + density = .4 +) +``` + +Then, we can supply the `true_model` object to the `powerly` function to start +the search for the sample size. Alongside the `true_model` we may also specify a +number of other arguments that constitute the method parameters. + +::: tip +You may also run `?powerly` in `R` to bring up the documentation and learn more +about the role of each argument in the `powerly` function. +::: + +```r{12} +# Run the method. +results <- powerly( + range_lower = 300, + range_upper = 700, + samples = 30, + replications = 40, + measure = "sen", + statistic = "power", + measure_value = .6, + statistic_value = .8, + model = "ggm", + model_matrix = true_model, + cores = 7 + verbose = TRUE +) +``` + +At this point we can print the results. + +```txt:no-line-numbers +Method run completed (2.943 sec): + - converged: yes + - iterations: 1 + - recommendation: 531 +``` + +Or better visualize them. + +```r +# Plot method run. +plot(results) +``` + +Which will produce a plot similar to the one below. + +
+ Example of Step 3 output for powerly package +
+ +Optionally, we may also validate the sample size recommendation using the +`validate` function by passing in the output obtained from the `powerly` +function, followed by plotting the validation results. + +```r{3} +# Run validation. +validation <- validate( + method = results, + replications = 3000, + cores = 9 +) + +# Plot validation. +plot(validation) +``` + +Which will result in the following: + +
+ Example of validation output for powerly package +
+ +## Next Steps + +This introduction is intended to provide the bare minimum information needed to +get started with `powerly`. If you want to learn more about where the numbers +come from, I encourage you to check out the +[manuscript](https://psyarxiv.com/j5v7u), or read the [next +post](/tutorial/method) where I succinctly describe the terminology and the +steps of the method. From there, you may continue with applications of the +methods to different models (e.g., psychological networks, structural equation +models etc.), or by checking out the posts under the FAQ, where I discuss tips +and tricks for how to efficiently use `powerly`. diff --git a/docs/tutorial/method.md b/docs/tutorial/method.md new file mode 100644 index 0000000..0ff6358 --- /dev/null +++ b/docs/tutorial/method.md @@ -0,0 +1,381 @@ +--- +pageClass: page-tutorial +header-includes: + - \usepackage{bm} +--- + +# The Method + +The goal of this post is to succinctly explain the main concepts used in the +sample size computation method that `powerly` is based on. In doing so, I aim to +help get you acquainted with terminology used throughout the rest of the posts +in the tutorial section. You may regard this post as a summary of the key +things discussed in [Constantin et al. (2021)](https://psyarxiv.com/j5v7u). + +## Input +To start the search for the optimal sample size, `powerly` requires three +inputs, namely, a true model, a performance measure, and a statistic. + +### True Model +The **_true model_** represents the set of hypothesized parameter values (i.e., +the effect size) used to generate data during the first step of the method. +These true parameter values are collected in a model matrix denoted +$\bm{\Theta}$. One may either manually specify $\bm{\Theta}$ or generate it +according to some hyperparameters via the +[`generate_model`](/reference/generate-model) function. The meaning and function +of $\bm{\Theta}$ changes depending on context in which it is used. For example, +in the context of psychological networks, $\bm{\Theta}$ encodes the edge weights +matrix, where the $\theta_{ij}$ entries may represent partial correlation +coefficients. However, in the Structural Equation Modeling (SEM), for example, +$\bm{\Theta}$ may encode the model implied covariance matrix. + +### Performance Measure +The **_performance measure_** can be regraded as a quality of the estimation +procedure relevant for the research question at hand. It takes the form of a +function $f\left(\bm{\Theta},\widehat{\bm{\Theta}}\right)$, where the +$\bm{\Theta}$ argument is the matrix of true model parameters and +$\widehat{\bm{\Theta}}$ is the matrix of estimated model parameters. In essence, +the performance measure is a scalar value computed by comparing the two sets of +model parameters in $\bm{\Theta}$ and $\widehat{\bm{\Theta}}$. Together with the +performance measure, the researcher also needs to specify a target value +$\delta$ indicating the desired value for the performance measure. Therefore, +what constitutes an optimal sample size largely depends on what performance +measure the researcher is interested in, which, in turn, is driven by the +research question at hand. + +### Statistic +The **_statistic_** is conceptualized as a definition for power that captures +how the target value $\delta$ for the performance measure should be reached. It +takes the form of a function $g(\bm{\xi})$, where the argument $\bm{\xi}$ is +vector of performance measures of size $R$ indexed by $r(r = 1, . . . , R)$, +with each element representing a replicated performance measure computed for a +particular sample size. + +Most commonly, this statistic may be defined in terms of the probability of +observing a target value $\delta$ for the performance measure (i.e., analogous +to a power computation) as: +$$ +\begin{equation} + g \left( \bm{\xi} \right) = \frac{1}{R} \sum_{r = 1}^{R} \left\lbrack \xi_{r} \geq \delta \right\rbrack\,, +\end{equation} +$$ +where the notation $\left\lbrack \cdot \right\rbrack$ represents the Iverson +brackets ([Knuth, +1992](https://www.tandfonline.com/doi/abs/10.1080/00029890.1992.11995869)), with +$\left\lbrack\xi_{r}\geq\delta\right\rbrack$ defined to be $1$ if the statement +$\xi_{r}\geq\delta$ is true and $0$ otherwise. Therefore, the optimal sample +size needs to satisfy $g\left(\bm{\xi}\right)\geq\tau$, where $\tau$ is an +arbitrary threshold indicating a certain probability of interest, i.e., the +target value for the statistic. + +## Question +Taken together, the true model matrix $\bm{\Theta}$, the performance measure +$f\left(\bm{\Theta},\widehat{\bm{\Theta}}\right)$ with its corresponding target +value $\delta$, and the statistic $g(\bm{\xi})$ with its target value $\tau$ +allow one to formulate the following question: + +
+ +Given the hypothesized parameters in $\bm{\Theta}$, what sample size does one +need to observe $f\left(\bm{\Theta},\widehat{\bm{\Theta}}\right)\geq\delta$ with +probability $\tau$ as defined by $g(\bm{\xi})$? + +
+ +`powerly` strives for flexibility, allowing researchers to provide custom +implementations for performance measures and statistics that best suit their +study goals. However, it also provides out of the box support for several models +and common related performance measures. Check out the [Reference +Section](/reference/powerly) for the [`powerly`](/reference/powerly) function +for an overview of the currently supported models and performance measures. + +::: tip +Coming in `powerly` version `2.0.0` we are introducing an `API` that allows +researchers to build upon and easily extend the current method to new models and +performance measures. Check out the [Developer Section](/developer/) for updates +on the `API` design and examples on how to extend `powerly`. +::: + +## Steps +Given the inputs presented above, the search for the optimal sample size is +conducted in three steps, i.e., a **_Monte Carlo (MC) simulation step_** for +computing the performance measure and statistic at various sample sizes, a +**_curve-fitting step_** for interpolating the statistic, and a **_bootstrapping +step_** to quantify the uncertainty around the fitted curve. + +### Step 1 + +
+ +The goal of the first step is to get a rough understanding of how the +performance measure $f\left(\bm{\Theta},\widehat{\bm{\Theta}}\right)$ changes as +a function of sample size. + +
+ +To achieve this we perform several MC simulations for different sample sizes +selected from a candidate sample size range denoted $\mathbb{N}_{s}$, for +example, $\mathbb{N}_{s}=[400..1400]$. More specifically, we select $T$ indexed +by $t(t = 1,\ldots,T)$ equidistant samples as +$S=\{s_{1},\ldots,s_{T}\}\subseteq\mathbb{N}_{s}$. Then, for each $s_{t} \in S$, +we perform $R$ MC replications as follows: + +1. Generate data with $s_{t}$ number of observations using the true model + parameters in $\bm{\Theta}$. +2. Estimate the model parameters matrix $\widehat{\bm{\Theta}}^{(rt)}$ using + the generated data. +3. Compute the performance measure by applying function + $f\left(\bm{\Theta},\widehat{\bm{\Theta}}^{(rt)}\right)$. + +This procedure gives us an $R \times T$ matrix $\bm{\Xi}$, where each $\xi_{rt}$ +entry is a performance measure computed for the $t$-th sample size, during the +$r$-th MC replication. Each column of the $\bm{\Xi}$ matrix is, therefore, a +vector $\bm{\xi}_{t}$ that holds the replicated performance measures associated +with the $t$-th sample size (i.e., see the plot below). + +
+ Example performance measures values for powerly package +
+ +Finally, we compute the statistic of choice (e.g., Equation $1$) by applying the +function $g(\bm{\xi})$ to each column of the matrix of performance measures +$\bm{\Xi}$. In essence, we collapse the $R \times T$ matrix $\bm{\Xi}$ to a $T +\times 1$ vector of statistics +$\bm{g}^{\top}=[g\left(\bm{\xi}_{1}\right),\ldots,g\left(\bm{\xi}_{T}\right)]$. +The plot below shows the statistics computed according to Equation $1$, where +each diamond represents the probability of observing a performance measure value +$\delta \ge 0.6$. We can see that the target value for the statistic is, in this +case, $\tau = 0.8$. + +
+ Example statistic values for powerly package +
+ +### Step 2 + +
+ +The goal of the second step is to obtain a smooth power function and interpolate +the statistic across all sample sizes in the candidate range $\mathbb{N}_{s}$. + +
+ +To achieve this, we fit a spline to the statistic values $\bm{g}$ obtain in the +previous step and use the resulting spline coefficients to interpolate across +the entire range $\mathbb{N}_{s}$. Depending on the choice of performance +measure and statistic, it may be appropriate to impose monotonicity constraints +when fitting the spline. For the example above, we assume a monotone +non-decreasing trend and use cubic *I-Splines* ([de Leeuw, +2017](http://dx.doi.org/10.13140/RG.2.2.36758.96327); [Ramsay, +1988](https://doi.org/10.1214/ss/1177012761)) via the `R` package +[`splines2`](https://CRAN.R-project.org/package=splines2) ([Wang & Yan, +2021](https://doi.org/10.6339/21-JDS1020)) with the number of inner knots +selected based on a cross-validation procedure (e.g., leave-one-out). This +assumption implies that the statistic increases as a function of sample size, as +depicted in the plot below. + +
+ Example of fitted spline for powerly package +
+ +::: tip +More information about the second step (i.e., basis functions, spline +coefficients, and information about the cross-validation) can be obtained by +running `plot(results, step = 2)`, where the `results` object represents the +output provided by the `powerly` function). +::: + +### Step 3 + +
+ +The goal of the third step is to quantify the Monte Carlo error around the +estimated spline. + +
+ +To achieve this, we use nonparametric bootstrapping to capture the variability +in the replicated performance measures associated with each sample size +$s_{t} \in S$. More specifically, we perform $B$ bootstraps runs, with the index +$b(b = 1,\ldots, B)$ indicating the current bootstrap run, as follows: + +1. Efficiently emulate the MC procedure in *Step 1* by sampling with replacement + $R$ elements from each vector $\bm{\xi}_{t}$ containing the replicated + performance measures for the sample size $s_{t}\in S$. +2. Compute a bootstrapped version of the vector of statistics + $\bm{g}^{(b)} = \left\lbrack g \left( \bm{\xi}_{1}^{(b)} \right), \ldots, g \left( \bm{\xi}_{T}^{(b)} \right) \right\rbrack^{\top}$, + as discussed in *Step 1* (i.e., see the animation below). + +
+ +
+ +3. Repeat the spline fitting procedure in *Step 2*, using as input the the + bootstrapped vector of statistics $\bm{g}^{(b)}$ instead of $\bm{g}$. Then, + use the resulting bootstrapped spline coefficients to interpolate across the + entire range $\mathbb{N}_{s}$. + +Performing the procedure above gives us a bootstrap distribution of statistic +values for each sample size in the candidate range $\mathbb{N}_{s}$. In +addition, we obtain this information using relatively few computational +resources since we resample the performance measures directly and thus bypass +the model estimation and data generation steps. Therefore, using this +information we can compute Confidence Intervals (CI) around the observed spline +we fit in *Step 2*. The plot below shows the observed spline (i.e, the thick +dark line) and the resulting CI (i.e., the gray and blue bands) for the example +above. Furthermore, the distribution depicted below shows the bootstrapped +statics associated with a sample size (i.e., $854$ in this case). + +
+ Example of confidence intervals for powerly package +
+ +### Iterations + +At this point, we can decide whether subsequent method iterations are needed by +zooming in on intersection of the CI and the target value $\tau$ for the +statistic (i.e., see the plot below). + +
+ Example of confidence intervals intersection for powerly package +
+ +More specifically, we let $N_l$ and $N_u$ be the first sample sizes for which +the upper and lower bounds of the $95\%$ CI reach the target value $\tau$. +Deciding whether subsequent method iterations are needed boils down to computing +the distance between $N_l$ and $N_u$ and comparing it to a scalar $\varepsilon$ +that indicates the tolerance for the uncertainty around the recommended sample +size. If $N_u - N_l \ge \varepsilon$, the initial candidate range +$\mathbb{N}_{s}$ is updated by setting its lower and upper bounds to $N_l$ and +$N_u$, respectively. Then, we repeat the tree steps of method discussed above, +this time concentrating the new set of MC replications on the updated and +narrowed-down range of sample sizes, i.e., $\mathbb{N}_{s}=[N_l..N_u]$. The +algorithm continues to iterate until either $N_u − N_l \geq \varepsilon$, or a +certain number of iterations has been elapsed. + +## Implementation + +As discussed in the introduction of the [Tutorial Section](/developer/), the +main function [`powerly`](/reference/powerly) implements the sample size +calculation method described above. When using the +[`powerly`](/reference/powerly) function to run a sample size analysis, several +arguments can be provided as input. For example, the function signature for +consists of the following arguments: + +```r +# Arguments supported by `powerly`. +powerly( + range_lower, + range_upper, + samples = 30, + replications = 30, + model = "ggm", + ..., + model_matrix = NULL, + measure = "sen", + statistic = "power", + measure_value = 0.6, + statistic_value = 0.8, + monotone = TRUE, + increasing = TRUE, + spline_df = NULL, + solver_type = "quadprog", + boots = 10000, + lower_ci = 0.025, + upper_ci = 0.975, + tolerance = 50, + iterations = 10, + cores = NULL, + backend_type = NULL, + save_memory = FALSE, + verbose = TRUE +) +``` + +::: warning +Please note that the function signature of [`powerly`](/reference/powerly) will +change (i.e., be simplified) with the release of the version `2.0.0`. +::: + +These arguments can be grouped in three categories: + +1. Researcher **_input_**. +2. Method **_parameter_**. +3. **_Miscellaneous_**. + +The table below provides an overview of the mapping between the notation used in +Constantin et al. (2021) and the `R` function arguments. The order of the rows +in the table is indicative of the order in which the arguments appear in the +method steps. + +
+ +| Notation | Argument | Category | Description | +| :-----------------------------------------------: | :-----------------------------: | :------: | :----------------------------------------------------------------------------------------------- | +| $\mathbb{N}_{s}$ | `range_lower` and `range_upper` | param | The initial candidate sample size range to search for the optimal sample size. | +| $T$ | `samples` | param | The number of sample sizes to select from the candidate sample size range. | +| $R$ | `replications` | param | The number of MC replications to perform for each sample size selected from the candidate range. | +| $\bm{\Theta}$ | `model_matrix` | input | The matrix of true parameter values. | +| - | `model` | input | The type or family of true model. | +| $f\left(\bm{\Theta},\widehat{\bm{\Theta}}\right)$ | `measure` | input | The performance measure. | +| $\delta$ | `measure_value` | input | The target value for the performance measure. | +| $g(\bm{\xi})$ | `statistic` | input | The statistic (i.e., the definition for power). | +| $\tau$ | `measure_value` | input | The target value for the statistic. | +| - | `monotone` | param | Whether to impose monotonicity constraints on the fitted spline. | +| - | `increasing` | param | Whether the spline is assumed to be monotone non-decreasing or increasing. | +| - | `spline_df` | param | The degrees of freedom to consider for constructing the basis matrix. | +| - | `solver_type` | param | The type of solver for estimating the spline coefficients. | +| $B$ | `boots` | param | The number of bootstrap runs. | +| - | `lower_ci` and `upper_ci` | param | The lower and upper bounds of the CI around the fitted spline. | +| $\varepsilon$ | `tolerance` | param | The tolerance for the uncertainty around the recommended sample size. | +| - | `iterations` | param | The number of method iterations allowed. | +| - | `cores` | misc | The number of cores to use for running the algorithm in parallel. | +| - | `backend_type` | misc | The type of cluster to create for running the algorithm in parallel. | +| - | `save_memory` | misc | Whether to save memory by limiting the amount results stored. | +| - | `verbose` | misc | Whether information should be printed while the method is running. | + +
+ +::: tip +For more information about the data types and default values for the arguments +listed above, consult the [Reference Section](/reference/) for the +[`powerly`](/reference/powerly) function, or the documentation in `R` by via +`?powerly`. +::: + +## Validation + +Aside for the method steps discussed earlier, we can also perform a validation +check to determine whether the sample size recommendation consistently recovers +the desired value $\delta$ for the performance measure according to the +statistic of interest. To achieve this, we repeat the MC procedure described in +*Step 1*, with only one selected sample size, namely, the recommendation +provided by the method. This results in a vector of replicated performance +measures for which compute the statistic of interest. In order to trust the +sample size recommendation, the value of the statistic obtained during the +validation procedure should be close to the target value specified by the +researcher. + +::: tip +Check out the [Reference Section](/reference/) for the +[`validate`](/reference/validate) function for more details on how validate a +sample size recommendation. +::: + +## References + +
+ +Constantin, M., Schuurman, N. K., & Vermunt, J. (2021). A General Monte Carlo Method for Sample Size Analysis in the Context of Network Models. https://doi.org/10.31234/osf.io/j5v7u + +de Leeuw, J. (2017). Computing and Fitting Monotone Splines. http://dx.doi.org/10.13140/RG.2.2.36758.96327 + +Knuth, D. E. (1992). Two Notes on Notation. The American Mathematical Monthly, 99(5), 403–422. https://doi.org/10.1080/00029890.1992.11995869 + +Ramsay, J. O. (1988). Monotone Regression Splines in Action. Statistical Science, 3(4), 425–441. https://doi.org/10.1214/ss/1177012761 + +Wang, W., & Yan, J. (2021). Shape-restricted regression splines with R package splines2. *Journal of Data Science*, 19(3), 498–517. https://doi.org/10.6339/21-JDS1020 + +
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