{"id":1017,"date":"2026-10-13T22:16:00","date_gmt":"2026-10-13T22:16:00","guid":{"rendered":"https:\/\/freestatisticaltools.com\/blog\/?p=1017"},"modified":"2026-10-13T22:16:00","modified_gmt":"2026-10-13T22:16:00","slug":"multiple-regression-interpretation","status":"publish","type":"post","link":"https:\/\/freestatisticaltools.com\/blog\/multiple-regression-interpretation\/","title":{"rendered":"How to Interpret Multiple Regression Output (with an APA Example)"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><strong>Multiple linear regression<\/strong> predicts a continuous outcome from two or more predictors and tells you how much each predictor contributes while holding the others constant. It is the workhorse of quantitative research, and also the analysis students most often struggle to interpret.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This guide goes through a typical regression output line by line, explains what each number means, lists the assumptions you need to check, and ends with an APA-style write-up.<\/p>\n\n\n<div class=\"fst-tool\"><span class=\"ico\"><svg viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.8\" stroke-linecap=\"round\" stroke-linejoin=\"round\" aria-hidden=\"true\"><path d=\"M3 21 21 3\"\/><circle cx=\"6\" cy=\"16\" r=\"1.5\"\/><circle cx=\"11\" cy=\"12\" r=\"1.5\"\/><circle cx=\"16\" cy=\"9\" r=\"1.5\"\/><\/svg><\/span><span><span class=\"t-eyebrow\">Free tool \u00b7 runs in your browser<\/span><a class=\"t-name\" href=\"https:\/\/freestatisticaltools.com\/multiple-regression-calculator\/\" style=\"text-decoration:none;color:inherit\">Multiple Linear Regression Calculator<\/a><span class=\"t-desc\">Free multiple regression calculator: B, SE, standardized beta, t, p, 95% CI and VIF for each predictor, with R\u00b2, ANOVA and automatic dummy coding.<\/span><\/span><a class=\"btn accent\" href=\"https:\/\/freestatisticaltools.com\/multiple-regression-calculator\/\" data-tool=\"multiple-regression-calculator\">Run a multiple regression \u2192<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">The example<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A lecturer wants to predict final exam scores (0\u2013100) for 150 students from three predictors: weekly <strong>study hours<\/strong>, <strong>prior GPA<\/strong> and average nightly <strong>sleep hours<\/strong>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Part 1: Model summary (how well does the model fit?)<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>R<\/th><th>R\u00b2<\/th><th>Adjusted R\u00b2<\/th><th>SE of the estimate<\/th><\/tr><\/thead><tbody><tr><td>.64<\/td><td>.41<\/td><td>.40<\/td><td>8.92<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>R\u00b2<\/strong> = .41: together, the three predictors explain 41% of the variance in exam scores.<\/li>\n\n<li><strong>Adjusted R\u00b2<\/strong> = .40: R\u00b2 corrected for the number of predictors. R\u00b2 always goes up when you add a predictor, even a useless one; adjusted R\u00b2 does not. Report it when you compare models with different numbers of predictors.<\/li>\n\n<li><strong>SE of the estimate<\/strong> = 8.92: the typical size of a prediction error, in exam points.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">What counts as a &quot;good&quot; R\u00b2 depends on the field. In the social sciences, .10\u2013.30 is common for individual behaviour; in the physical sciences, values above .90 are expected. Cohen&#39;s f\u00b2 = R\u00b2 \/ (1 \u2212 R\u00b2) is an effect size for the whole model; here f\u00b2 = 0.69, a large effect (benchmarks: .02 small, .15 medium, .35 large).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Part 2: The ANOVA table (is the model significant?)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">F(3, 146) = 33.82, p &lt; .001.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This tests whether the model as a whole predicts the outcome better than just using the mean. The degrees of freedom are k (number of predictors) and n \u2212 k \u2212 1. A significant F means at least one predictor is useful; it does not tell you which.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Part 3: The coefficients table (what does each predictor do?)<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Predictor<\/th><th>B<\/th><th>SE<\/th><th>\u03b2<\/th><th>t<\/th><th>p<\/th><th>95% CI for B<\/th><th>VIF<\/th><\/tr><\/thead><tbody><tr><td>(Constant)<\/td><td>21.40<\/td><td>5.10<\/td><td><\/td><td>4.20<\/td><td>&lt; .001<\/td><td>[11.32, 31.48]<\/td><td><\/td><\/tr><tr><td>Study hours<\/td><td>1.85<\/td><td>0.32<\/td><td>.39<\/td><td>5.78<\/td><td>&lt; .001<\/td><td>[1.22, 2.48]<\/td><td>1.21<\/td><\/tr><tr><td>Prior GPA<\/td><td>6.20<\/td><td>1.45<\/td><td>.29<\/td><td>4.28<\/td><td>&lt; .001<\/td><td>[3.33, 9.07]<\/td><td>1.28<\/td><\/tr><tr><td>Sleep hours<\/td><td>0.95<\/td><td>0.60<\/td><td>.10<\/td><td>1.58<\/td><td>.116<\/td><td>[\u22120.24, 2.14]<\/td><td>1.09<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">B (unstandardized coefficient)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">B is the expected change in the outcome for a <strong>one-unit increase<\/strong> in the predictor, <strong>holding the other predictors constant<\/strong>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Each extra weekly study hour is associated with 1.85 more exam points, for students with the same GPA and sleep.<\/li>\n\n<li>Each extra GPA point is associated with 6.20 more exam points.<\/li>\n\n<li>The <strong>constant<\/strong> (21.40) is the predicted score when all predictors are 0. It is often not meaningful (nobody has a GPA of 0), and that is fine.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">\u03b2 (standardized coefficient, &quot;beta&quot;)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u03b2 expresses the effect in <strong>standard-deviation units<\/strong>: a 1-SD increase in study hours predicts a 0.39-SD increase in exam score. Because all \u03b2s are on the same scale, you can use them to compare the relative importance of predictors measured in different units. Here, study hours (\u03b2 = .39) matter most, followed by GPA (\u03b2 = .29).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">t, p and the confidence interval<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Each predictor&#39;s t-test checks whether its B differs from zero. Study hours and GPA are significant; sleep is not (p = .116), and its 95% CI [\u22120.24, 2.14] includes zero. That does not prove sleep is irrelevant. It means this sample gives no clear evidence of a unique effect once study hours and GPA are taken into account.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">VIF (multicollinearity)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The <strong>variance inflation factor<\/strong> shows how much a predictor overlaps with the others. VIF = 1 means no overlap. Values above 5 (some say 10) signal problematic multicollinearity, which inflates standard errors and makes coefficients unstable. Here, all VIFs are below 1.3, so multicollinearity is not a concern.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Assumptions to check<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Linearity:<\/strong> each predictor relates linearly to the outcome. Check scatter plots or a residuals-vs-fitted plot.<\/li>\n\n<li><strong>Independence of residuals:<\/strong> usually guaranteed by design. The Durbin-Watson statistic (around 2 is good) checks this for time-ordered data.<\/li>\n\n<li><strong>Homoscedasticity:<\/strong> residuals have constant spread across fitted values. A funnel shape signals trouble. Use <a href=\"\/robust-standard-errors-regression-diagnostics\/\">robust standard errors<\/a> if needed.<\/li>\n\n<li><strong>Normality of residuals:<\/strong> check a Q-Q plot of the residuals. This matters most in small samples.<\/li>\n\n<li><strong>No multicollinearity:<\/strong> VIF below 5.<\/li>\n\n<li><strong>No overly influential cases:<\/strong> check Cook&#39;s distance (values above 1, or clearly larger than the rest).<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">Categorical predictors<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The calculator dummy-codes categorical predictors automatically. With a variable such as &quot;faculty&quot; (Arts, Science, Business), one category becomes the <strong>reference group<\/strong>, and each dummy&#39;s B is the difference from that reference, holding other predictors constant.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">How to report multiple regression in APA 7<\/h2>\n\n\n\n<div class=\"fst-report\"><b>How to report it (APA 7)<button type=\"button\" data-copy-report>Copy<\/button><\/b><div><p>A multiple linear regression was conducted to predict final exam scores from weekly study hours, prior GPA and nightly sleep hours. The model was significant, <i>F<\/i>(3, 146) = 33.82, <i>p<\/i> &lt; .001, and explained 41% of the variance in exam scores (<i>R<\/i>\u00b2 = .41, adjusted <i>R<\/i>\u00b2 = .40). Study hours (<i>B<\/i> = 1.85, 95% CI [1.22, 2.48], \u03b2 = .39, <i>p<\/i> &lt; .001) and prior GPA (<i>B<\/i> = 6.20, 95% CI [3.33, 9.07], \u03b2 = .29, <i>p<\/i> &lt; .001) were significant positive predictors, whereas sleep hours were not (<i>B<\/i> = 0.95, 95% CI [\u22120.24, 2.14], \u03b2 = .10, <i>p<\/i> = .116). Variance inflation factors were below 1.3.<\/p><\/div><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Put the full coefficients table in your results, formatted as an APA table. The calculator produces one you can copy straight into Word.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">When multiple regression is not the right tool<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Binary outcome<\/strong> (yes\/no): use <a href=\"\/blog\/logistic-regression-odds-ratios\/\">logistic regression<\/a>.<\/li>\n\n<li><strong>Count outcome<\/strong> (number of visits): use <a href=\"\/poisson-negative-binomial-regression\/\">Poisson or negative binomial regression<\/a>.<\/li>\n\n<li><strong>Repeated observations over time<\/strong> for the same units: use <a href=\"\/panel-data-regression-fixed-random-effects\/\">panel data regression<\/a>.<\/li>\n\n<li><strong>Testing indirect or conditional effects:<\/strong> see our guides to <a href=\"\/blog\/mediation-analysis-bootstrapping\/\">mediation<\/a> and <a href=\"\/blog\/moderation-analysis-simple-slopes\/\">moderation<\/a>.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Frequently asked questions<\/h2>\n\n\n\n<div class=\"fst-faq\"><details><summary>What is the difference between B and beta in regression?<\/summary><div class=\"a\"><p>B is the unstandardized coefficient: the change in the outcome, in its own units, for a one-unit change in the predictor. Beta (\u03b2) is standardized: the change in standard deviations for a one-SD change in the predictor, which lets you compare predictors measured in different units.<\/p><\/div><\/details><details><summary>What is a good R-squared value?<\/summary><div class=\"a\"><p>It depends on the field and the outcome. In social and behavioural research, R\u00b2 between .10 and .30 is common and can be meaningful; in tightly controlled physical systems, much higher values are expected.<\/p><\/div><\/details><details><summary>What VIF value indicates multicollinearity?<\/summary><div class=\"a\"><p>VIF values above 5 are a common warning sign, and above 10 indicate serious multicollinearity. Values close to 1 indicate little overlap between predictors.<\/p><\/div><\/details><details><summary>How many participants do I need for multiple regression?<\/summary><div class=\"a\"><p>A common rule of thumb is at least 10 to 15 cases per predictor, but a power analysis is better. To detect a medium effect (f\u00b2 = .15) with three predictors, 80% power and \u03b1 = .05, you need about 77 participants.<\/p><\/div><\/details><\/div>\n\n\n<div class=\"fst-report\"><b>Cite the tool (APA 7)<button type=\"button\" data-copy-report>Copy<\/button><\/b><div>Xavier, B. (2026). <i>Free Statistical Tools<\/i> [Computer software]. Zenodo. https:\/\/doi.org\/10.5281\/zenodo.23372149 (web app: https:\/\/freestatisticaltools.com\/multiple-regression-calculator\/)<\/div><small>Name the tool in your methods: \u201cThe analysis was carried out with the Multiple Linear Regression Calculator in Free Statistical Tools.\u201d<\/small><\/div>\n","protected":false},"excerpt":{"rendered":"<p>A line-by-line guide to multiple regression output, from R\u00b2 and the F-test to unstandardized and standardized coefficients, multicollinearity checks and an APA results paragraph.<\/p>\n","protected":false},"author":1,"featured_media":1018,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_fst_seo_title":"How to Interpret Multiple Regression Output (R\u00b2, Beta, VIF)","_fst_meta_desc":"Read multiple regression output line by line: R\u00b2, adjusted R\u00b2, the F-test, B vs beta, t and p, confidence intervals and VIF, with an APA reporting example.","_fst_primary_tool":"multiple-regression-calculator","_fst_canonical":"","_fst_noindex":false,"_fst_no_toc":false,"_fst_no_autolink":false,"footnotes":""},"categories":[7],"tags":[19,16,41,38],"class_list":["post-1017","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-correlation-regression","tag-apa-style","tag-beginners","tag-multicollinearity","tag-regression"],"_links":{"self":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/posts\/1017","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/comments?post=1017"}],"version-history":[{"count":0,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/posts\/1017\/revisions"}],"wp:attachment":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/media?parent=1017"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/categories?post=1017"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/tags?post=1017"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}