{"id":29169,"date":"2022-12-16T12:20:17","date_gmt":"2022-12-16T12:20:17","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=29169"},"modified":"2026-05-01T13:51:49","modified_gmt":"2026-05-01T13:51:49","slug":"anova-table-and-measures-of-goodness-of-fit","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/quantitative-method\/anova-table-and-measures-of-goodness-of-fit\/","title":{"rendered":"ANOVA Table and Measures of Goodness of Fit"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Which of the following is most appropriate for adjusted R\u00b2?\",\n    \"text\": \"Which of the following is most appropriate for adjusted R\u00b2?\\n\\nA. It is always positive.\\nB. It may or may not increase when one adds an independent variable.\\nC. It is non-decreasing in the number of independent variables.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B.\\n\\nThe value of adjusted R\u00b2 increases only when the added independent variables improve the fit of the regression model. It decreases when the added variables do not improve the model fit sufficiently.\\n\\nChoice A is incorrect: Adjusted R\u00b2 can be negative if R\u00b2 is low enough, although multiple R\u00b2 is always positive.\\n\\nChoice C is incorrect: Adjusted R\u00b2 can decrease when additional variables do not sufficiently improve model fit, whereas multiple R\u00b2 is non-decreasing as variables are added.\"\n    }\n  }\n}\n<\/script><\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/Dk9KuKkhtM8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><strong>R-squared<\/strong> \\(\\bf{(R^2)}\\) measures how well an estimated regression fits the data. It is also known as the <strong>coefficient of determination<\/strong> and can be formulated as:<\/p>\n<p>$$ R^2=\\frac{\\text{Sum of regression squares}}{\\text{Sum of squares total}}=\\frac{{\\sum_{i=1}^{n}{(\\widehat{Y_i}-\\bar{Y})}}^2}{{\\sum_{i=1}^{n}{(Y_i-\\bar{Y})}}^2} $$<\/p>\n<p>Where:<\/p>\n<p>\\(n\\) = Number of observations.<\/p>\n<p>\\(Y_i\\) = Dependent variable observations.<\/p>\n<p>\\(\\widehat{Y_i}\\) = Dependent variables predicted value to the independent variable.<\/p>\n<p>\\(\\bar{Y}\\)= Dependent variable mean.<\/p>\n<p>In the presence of independent variables, \\(R^2\\) will either increase or remain constant. However, \\(R^2\\) cannot be used to measure the goodness of fit of a model as it will not decrease with the addition of independent variables.<\/p>\n<div style=\"text-align: center; margin: 25px 0;\"><a style=\"display: inline-block; padding: 12px 20px; border: 2px solid #2f5cff; border-radius: 999px; color: #2f5cff; text-decoration: none; background: #f7f9fc; white-space: nowrap;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Understand goodness of fit with our free trial. <\/a><\/div>\n<h3>Limitations of R<sup>2<\/sup><\/h3>\n<ul>\n<li>It is impossible to determine the statistical significance of the coefficients from \\(R^2\\).<\/li>\n<li>A bias in the predicted coefficients or estimates cannot be determined with \\(R^2\\).<\/li>\n<li>When a model is good, it has a high \\(R^2\\); when it is bad, it has a low \\(R^2\\), usually due to overfitting and biases in the model.<\/li>\n<\/ul>\n<p>An <strong>overfitted regression model<\/strong> is one with too many independent variables to the number of observations in a sample. Overfitting may produce coefficients that do not reflect the true relationship between the independent and dependent variables.<\/p>\n<p>Multiple regression software packages usually produce an <strong>adjusted<\/strong> \\(\\bf{R^2} (\\bar{R}^2)\\) as an alternative measure of goodness of fit. Using adjusted \\(R^2\\) in regression is beneficial since it does not automatically increase when more independent variables are included, given that it adjusts for degrees of freedom.<\/p>\n<p>$$ \\bar{R^2}=1-\\left[\\cfrac{\\frac{\\text{Sum of squares error} }{n-k-1}}{\\frac{\\text{Sum of squares total}}{n-1}}\\right] $$<\/p>\n<p>Therefore, the relationship between \\(\\bar{R^2}\\) and \\(R^2\\) can mathematically be derived as follows:<\/p>\n<p>$$ \\bar{R^2}=1-\\left[\\left(\\frac{n-1}{n-k-1}\\right)\\ \\left(1-R^2\\right)\\right] $$<\/p>\n<p>Note that:<\/p>\n<ul>\n<li>If \\(k \\geq 1\\) then \\(R^2 &gt; \\text{adjusted } R^2\\) the result is that adjusted \\(R^2\\) can be negative while \\(R^2\\) is zero at minimum.<\/li>\n<\/ul>\n<p>When including a new variable in the regression, the following should be taken into consideration:<\/p>\n<ul>\n<li>\\(\\bar{R^2}\\) increases when the coefficient t-statistic is \\(&gt; \\left|1.0\\right|\\).<\/li>\n<li>\\(\\bar{R^2}\\) decreases when the coefficient t-statistic is \\(&lt; \\left|1.0\\right|\\).<\/li>\n<li>At typical significance levels, 5% and 1%, a t-statistic with an absolute value of 1.0 does not indicate that the independent variable is different from zero. Therefore, the adjusted \\(R^2\\) doesn&#8217;t demonstrate that it will increase significantly.<\/li>\n<\/ul>\n<h2>ANOVA Table<\/h2>\n<p>One of the outputs of multiple regression is the ANOVA table. The following shows the general structure of an Anova table.<\/p>\n<p>$$ \\begin{array}{c|c|c|c} \\textbf{ANOVA} &amp; \\textbf{Df (degrees} &amp; \\textbf{SS (Sum of squares)} &amp; \\textbf{MSS (Mean sum} \\\\ &amp; \\textbf{of freedom)} &amp; &amp; \\textbf{of squares)}\\\\ \\hline \\text{Regression} &amp; k &amp; \\text{RSS} &amp; MSR \\\\ &amp; &amp; \\text{(Explained variation)} &amp; \\\\ \\hline \\text{Residual} &amp; n-(k+1) &amp; \\text{SSE} &amp; MSE \\\\ &amp; &amp; \\text{(Unexplained variation)} &amp; \\\\ \\hline \\text{Total} &amp; n-1 &amp; \\text{SST} &amp; \\\\ &amp; &amp; \\text{(Total variation) } &amp; \\end{array} $$<\/p>\n<p>We can use the information in an ANOVA table to determine \\(R^2\\), the F-statistic, and the standard error estimates (SEE) as expressed below:<\/p>\n<p>$$ R^2=\\frac{RSS}{SST} $$<\/p>\n<p>$$ F=\\frac{MSR}{MSE} $$<\/p>\n<p>$$ SEE=\\sqrt{MSE} $$<\/p>\n<p>Where:<\/p>\n<p>$$ \\begin{align*} MSR &amp; =\\frac{RSS}{k} \\\\ MSE &amp; =\\frac{SSE}{n-k-1} \\end{align*} $$<\/p>\n<h4>Example: Interpreting Regression Output<\/h4>\n<p>Consider the following regression results generated from multiple regression analysis of the price of the US Dollar index on the inflation rate and real interest rate.<\/p>\n<p>$$ \\begin{array}{cccc} \\text{ANOVA} &amp; &amp; &amp; \\\\ \\hline &amp; \\text{df} &amp; \\text{SS} &amp; \\text{Significance F} \\\\ \\hline \\text{Regression} &amp; 2 &amp; 432.2520 &amp; 0.0179 \\\\ \\text{Residual} &amp; 7 &amp; 200.6349 &amp; \\\\ \\text{Total} &amp; 9 &amp; 632.8869 &amp; \\\\ \\hline \\\\ &amp; \\text{Coefficients} &amp; \\text{Standard Error} &amp; \\\\ \\hline \\text{Intercept} &amp; 81 &amp; 7.9659 &amp; \\\\ \\text{Inflation rates} &amp; -276 &amp; 233.0748 &amp; \\\\ \\text{Real interest Rates} &amp; 902 &amp; 279.6949 &amp; \\\\ \\hline \\end{array} $$<\/p>\n<p>Given the above information, the regression equation can be expressed as:<\/p>\n<p>$$ P=81-276INF+902IR $$<\/p>\n<p>Where:<\/p>\n<p>\\(P\\) = Price of USDX.<\/p>\n<p>\\(INF\\) = Inflation rate.<\/p>\n<p>\\(IR\\) = Real interest rate.<\/p>\n<p>\\(R^2\\) and adjusted \\(R^2\\) can also be calculated as follows:<\/p>\n<p>$$ \\begin{align*} R^2 &amp; =\\frac{RSS}{SST}=\\frac{432.2520}{632.8869}=0.6830=68.30\\% \\\\ \\\\ \\text{Adjusted } R^2 &amp; =1-\\left(\\frac{n-1}{n-k-1}\\right)\\left(1-R^2\\right)=1-\\frac{10-1}{10-2-1}\\left(1-0.6830\\right) \\\\\u00a0 \u00a0&amp; =0.5924 = 59.24\\% \\end{align*} $$<\/p>\n<p>It&#8217;s important to note the following:<\/p>\n<ul>\n<li>Multiple regression does not provide a straightforward explanation of adjusted \\(R^2\\) in terms of the variance explained by the dependent variable, as is the case in simple regression.<\/li>\n<li>Adjusted \\(R^2\\) does not indicate whether a regression coefficient&#8217;s predictions are true or biased. Residual plots and other statistics are required to determine whether or not the predictions are accurate.<\/li>\n<li>To assess the significance of the model&#8217;s fit, we use the F-Statistic and other goodness-of-fit metrics from the ANOVA rather than \\(R^2\\) and adjusted \\(R^2\\).<\/li>\n<\/ul>\n<blockquote>\n<h2>Question<\/h2>\n<p>Which of the following is <em>most appropriate<\/em> for adjusted \\(R^2\\)?<\/p>\n<ol type=\"A\">\n<li>It is always positive.<\/li>\n<li>It may or may not increase when one adds an independent variable.<\/li>\n<li>It is non-decreasing in the number of independent variables.<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>The value of the adjusted \\(R^2\\) increases only when the added independent variables improve the fit of the regression model. Moreover, it decreases when the added variables do not improve the model fit sufficiently.<\/p>\n<p><strong>A is incorrect<\/strong>: The adjusted \\(R^2\\) can be negative if \\(R^2\\) is low enough. However, multiple \\(R^2\\) is always positive.<\/p>\n<p><strong>C is incorrect<\/strong>: The adjusted \\(R^2\\) <strong>can decrease<\/strong> when the added variables do not improve the model fit by a good enough amount. However, multiple \\(R^2\\) is non-decreasing in the number of independent variables. For this reason, it is less reliable as a measure of goodness of fit in regression with more than one independent variable than in a one-independent variable regression.<\/p>\n<\/blockquote>\n<div style=\"text-align: center; margin: 40px 0;\"><a style=\"display: inline-block; padding: 14px 26px; background: #4a76d1; color: #fff; border-radius: 999px; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Start Free Trial \u2192 <\/a><\/p>\n<p style=\"margin-top: 10px;\">Learn R\u00b2, model limitations, and regression evaluation with clear study tools.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>R-squared \\(\\bf{(R^2)}\\) measures how well an estimated regression fits the data. It is also known as the coefficient of determination and can be formulated as: $$ R^2=\\frac{\\text{Sum of regression squares}}{\\text{Sum of squares total}}=\\frac{{\\sum_{i=1}^{n}{(\\widehat{Y_i}-\\bar{Y})}}^2}{{\\sum_{i=1}^{n}{(Y_i-\\bar{Y})}}^2} $$ Where: \\(n\\) = Number of observations&#8230;.<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,229],"tags":[],"class_list":["post-29169","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>ANOVA Table and Goodness of Fit | CFA Level 2<\/title>\n<meta name=\"description\" content=\"Learn how ANOVA tables assess 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