{"id":45729,"date":"2023-08-18T11:06:19","date_gmt":"2023-08-18T11:06:19","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=45729"},"modified":"2026-03-05T17:55:28","modified_gmt":"2026-03-05T17:55:28","slug":"analysis-of-variance-anova-2","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/analysis-of-variance-anova-2\/","title":{"rendered":"Analysis of Variance (ANOVA)"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Simple Linear Regression \u2013 Part II (2025 CFA\u00ae Level I Exam \u2013 Quantitative Methods \u2013 Learning Module 10)\",\n  \"description\": \"CFA\u00ae Level I Quantitative Methods video lesson from AnalystPrep covering Simple Linear Regression \u2013 Part II (Learning Module 10). This session builds on Part I by explaining hypothesis testing of regression coefficients using t-tests and F-tests, analysis of variance (ANOVA), the relationship between correlation and regression significance, the use of dummy variables, construction of prediction and confidence intervals, and common functional forms including log-linear and semi-log models, with exam-focused examples.\",\n  \"uploadDate\": \"2024-03-24\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/FmVBqrMYreM\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=FmVBqrMYreM\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/FmVBqrMYreM\",\n  \"duration\": \"PT49M58S\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"FAQPage\",\n  \"mainEntity\": [\n    {\n      \"@type\": \"Question\",\n      \"name\": \"How are R\u00b2 and the F-statistic calculated from an ANOVA table?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"R\u00b2 is calculated as the sum of squares for regression divided by the total sum of squares. Using the ANOVA table, R\u00b2 equals 1,701,563 divided by 1,808,363, which is approximately 0.94 or 94%. The F-statistic is calculated as the mean regression sum of squares divided by the mean squared error. This equals 1,701,563 divided by 13,350, resulting in an F-statistic of approximately 127.\"\n      }\n    },\n    {\n      \"@type\": \"Question\",\n      \"name\": \"What does an R\u00b2 value of 94% indicate in a regression model?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"An R\u00b2 value of 94% indicates that 94% of the total variation in the dependent variable is explained by the regression model. This suggests a strong explanatory power and a good overall fit of the model to the observed data.\"\n      }\n    },\n    {\n      \"@type\": \"Question\",\n      \"name\": \"What does a large F-statistic imply about the regression model?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"A large F-statistic indicates that the regression model explains a significant portion of the variation in the dependent variable relative to unexplained variation. This provides strong evidence that the regression relationship is statistically significant.\"\n      }\n    }\n  ]\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/images\/anova-lm10-img11\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG11.jpg\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG11.jpg\",\n  \"caption\": \"Analysis of Variance (ANOVA)\",\n  \"width\": 404,\n  \"height\": 276,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  },\n  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/analysis-of-variance-anova-2\/\"\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/FmVBqrMYreM\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The sum of squares of a regression model is usually represented in the Analysis of Variance (ANOVA) table. The ANOVA table contains the sum of squares (SST, SSE, and SSR), the degrees of freedom, the mean squares (MSR and MSE), and F-statistics.<\/p>\n<p>The typical format of ANOVA is as shown below:<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c}<br \/>\n\\textbf{Source} &amp; \\textbf{Sum of Squares} &amp; \\textbf{Degrees} &amp; \\textbf{Mean} &amp; \\textbf{F-statistic} \\\\<br \/>\n&amp; &amp; \\textbf{of} &amp; \\textbf{square} &amp; \\\\<br \/>\n&amp; &amp; \\textbf{Freedom} &amp; &amp; \\\\ \\hline<br \/>\n{ \\text{Regression} \\\\ \\text{(Explained)} } &amp; SSR=\\sum_{i=1}^{n}\\left(\\widehat{Y_i}-\\bar{Y}\\right)^2 &amp;<br \/>\n1 &amp; MSR=\\frac{SSR}{1} &amp; F=\\frac{MSR}{MSE} \\\\ \\hline<br \/>\n{ \\text{Residual} \\\\ \\text{(explained)} } &amp; SSE=\\sum_{i=1}^{n}\\left(Y_i-\\widehat{Y_i}\\right)^2 &amp; n-2 &amp; MSE=\\frac{SSE}{n-2} &amp; \\\\ \\hline<br \/>\n\\text{Total} &amp; SST=\\sum_{i=1}^{n}\\left(Y_i-\\bar{Y}\\right)^2 &amp; n-1 &amp; &amp;<br \/>\n\\end{array} $$<\/p>\n<p>Standard Error of EstimateStandard Error of Estimate, \\(S_e\\) or \\(SEE\\), is referred to as the root mean square error or standard error of the regression. It measures the distance between the observed and dependent variables predicted by the regression model. The Standard Error of Estimate is easily calculated from the ANOVA table using the following formula:<\/p>\n<p>$${\\text{Standard Error of Estimate }}(S_e)=\\sqrt{MSE}=\\sqrt{\\frac{\\sum_{i = 1}^{n}\\left(Y_i-\\hat{Y}\\right)^2}{n-2}}$$<\/p>\n<p>The standard error of estimate, coefficient of determination, and F-statistic are the measures that can be used to gauge the goodness of fit of a regression model. In other words, these measures are used to tell the extent to which a regression model syncs with data.<\/p>\n<p>The smaller the Standard Error of Estimate is, the better the fit of the regression line. However, the Standard Error of Estimate does not tell us how well the independent variable explains the variation in the dependent variable.<\/p>\n<p><strong>Example: Calculating and Interpreting F-Statistic<\/strong><\/p>\n<p>The completed ANOVA table for the regression model of the inflation rate against the unemployment rate over 10 years is given below:<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c}<br \/>\n\\textbf{Source} &amp; \\textbf{Sum of} &amp; \\textbf{Degrees of} &amp; \\textbf{Mean Sum} &amp; \\textbf{F-Statistic} \\\\<br \/>\n&amp; \\textbf{Squares} &amp; \\textbf{Freedom} &amp; \\textbf{of Squares} &amp; \\\\ \\hline<br \/>\n\\text{Regression} &amp; 10.568 &amp; 1 &amp; 10.568 &amp; ? \\\\ \\hline<br \/>\n\\text{Error} &amp; 3.136 &amp; 8 &amp; 0.392 \\\\ \\hline<br \/>\n\\text{Total} &amp; 13.704 &amp; 9<br \/>\n\\end{array} $$<\/p>\n<ol type=\"a\">\n<li>Use the above ANOVA table to calculate the F-statistic.<\/li>\n<li>Test the hypothesis that the slope coefficient equals a 5% significance level.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<ol type=\"a\">\n<li>\\( =\\frac{{\\text{Mean Regression Sum of Squares (MSR)}}}{{\\text{Mean Squared Error(MSE)}}}=\\frac{10.568}{0.392}=26.960\\)<\/li>\n<li>We are testing the null hypothesis \\(H_0:b_1=0\\) against the alternative hypothesis \\(H_1: b_1 \\neq 0\\). The critical F-value for \\(k = 1\\) and \\(n-2 = 8\\) degrees of freedom at a 5% significance level is roughly 5.32. Note that this is a one-tail test, so we use the 5% F-table.<img loading=\"lazy\" decoding=\"async\" width=\"404\" height=\"276\" class=\"aligncenter size-full wp-image-45745\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG11.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG11.jpg 404w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG11-300x205.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG11-400x273.jpg 400w\" sizes=\"auto, (max-width: 404px) 100vw, 404px\" \/> Remember that the null hypothesis is rejected if the calculated value of the F-statistic is greater than the critical value of F. Since \\(26.960 \\gt 5.32\\), we reject the null hypothesis and conclude that the slope coefficient is significantly different from zero. Notice that we also rejected the null hypothesis in the previous examples. We did so because the 95% confidence interval did not include zero.An F-test duplicates the t-test in regard to the slope coefficient significance for a linear regression model with one independent variable. In this case, \\(t^2={2.306}^2\\approx 5.32\\). Since the F-statistic is the square of the t-statistic for the slope coefficient, its inferences are the same as the t-test. However, this is not the case for multiple regressions.<\/li>\n<\/ol>\n<blockquote>\n<h2>Question<\/h2>\n<p>Consider the following analysis of variance (ANOVA) table:<\/p>\n<p>$$<br \/>\n\\begin{array}{c|c|c|c}<br \/>\n\\textbf {Source} &amp; \\textbf {Degrees of} &amp; \\textbf { Sum of } &amp; \\textbf {Mean Sum} \\\\<br \/>\n&amp; \\textbf{Freedom} &amp; \\textbf {Squares} &amp; \\textbf{of Squares} \\\\ \\hline<br \/>\n\\text {Regression} &amp; 1 &amp; 1,701,563 &amp; 1,701,563 \\\\ \\hline<br \/>\n\\text {Error} &amp; 3 &amp; 106,800 &amp; 13,350 \\\\<br \/>\n\\text {(Unexplained)} &amp; &amp; &amp; \\\\ \\hline<br \/>\n\\text {Total} &amp; 4 &amp; 1,808,363 &amp; \\\\<br \/>\n\\end{array}<br \/>\n$$<\/p>\n<p>The value of \\(R^2\\) and the F-statistic for the test of fit of the regression model are <em>closest to<\/em>:<\/p>\n<ol type=\"A\">\n<li>6% and 16.<\/li>\n<li>94% and 127.<\/li>\n<li>99% and 127.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>The correct answer is B<\/strong>.<\/p>\n<p>$$R^2=\\frac{\\text{Sum of Squares Regression (SSR)}}{\\text{Sum of Squares Total (SST)}}=\\frac{1,701,563}{1,808,363}=0.94=94\\%$$<\/p>\n<p>$$ \\begin{align*} F &amp; =\\frac{\\text{Mean Regression Sum of Squares (MSR)}}{\\text{Mean Squared Error (MSE)}} \\\\ &amp; =\\frac{1,701,563}{13,350} =127.46\\approx 127 \\end{align*}$$<\/p>\n<\/blockquote>","protected":false},"excerpt":{"rendered":"<p>The sum of squares of a regression model is usually represented in the Analysis of Variance (ANOVA) table. The ANOVA table contains the sum of squares (SST, SSE, and SSR), the degrees of freedom, the mean squares (MSR and MSE),&#8230;<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-45729","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Analysis of Variance (ANOVA) | CFA Level I Notes<\/title>\n<meta name=\"description\" content=\"Learn how ANOVA tests differences between group means, including the ANOVA table, F-statistic interpretation, and standard error of estimate.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, 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