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Previously, I’ve written about how to interpret regression coefficients and their individual P values. I’ve also written about how to interpret R-squared to assess the strength of the relationship between your model and the response variable. Recently I've been asked, how does the F-test of the overall significance and its P value fit in with these other statistics? That’s the topic of this post! In general, an F-test in regression compares the fits of different linear models. Unlike t-tests that can assess only one regression coefficient at a time, the F-test can assess multiple coefficients simultaneously. The F-test of the overall significance is a specific form of the F-test. It compares a model with no predictors to the model that you specify. A regression model that contains no predictors is also known as an intercept-only model. The hypotheses for the F-test of the overall significance are as follows:
In Minitab statistical software, you'll find the F-test for overall significance in the Analysis of Variance table.  If the P value for the F-test of overall significance test is less than your significance level, you can reject the null-hypothesis and conclude that your model provides a better fit than the intercept-only model. Great! That set of terms you included in your model improved the fit! Typically, if you don't have any significant P values for the individual coefficients in your model, the overall F-test won't be significant either. However, in a few cases, the tests could yield different results. For example, a significant overall F-test could determine that the coefficients are jointly not all equal to zero while the tests for individual coefficients could determine that all of them are individually equal to zero. There are a couple of additional conclusions you can draw from a significant overall F-test. In the intercept-only model, all of the fitted values equal the mean of the response variable. Therefore, if the P value of the overall F-test is significant, your regression model predicts the response variable better than the mean of the response. While R-squared provides an estimate of the strength of the relationship between your model and the response variable, it does not provide a formal hypothesis test for this relationship. The overall F-test determines whether this relationship is statistically significant. If the P value for the overall F-test is less than your significance level, you can conclude that the R-squared value is significantly different from zero. To see how the F-test works using concepts and graphs, see my post about understanding the F-test. If your entire model is statistically significant, that's great news! However, be sure to check the residual plots so you can trust the results! If you're learning about regression, read my regression tutorial!
Read my blog post about how F-tests work in ANOVA. To calculate the F-test of overall significance, your statistical software just needs to include the proper terms in the two models that it compares. The overall F-test compares the model that you specify to the model with no independent variables. This type of model is also known as an intercept-only model. The F-test for overall significance has the following two hypotheses:
In statistical output, you can find the overall F-test in the ANOVA table. An example is below. Related Post: What are Independent and Dependent Variables? Interpreting the Overall F-test of SignificanceCompare the p-value for the F-test to your significance level. If the p-value is less than the significance level, your sample data provide sufficient evidence to conclude that your regression model fits the data better than the model with no independent variables. This finding is good news because it means that the independent variables in your model improve the fit! Generally speaking, if none of your independent variables are statistically significant, the overall F-test is also not statistically significant. Occasionally, the tests can produce conflicting results. This disagreement can occur because the F-test of overall significance assesses all of the coefficients jointly whereas the t-test for each coefficient examines them individually. For example, the overall F-test can find that the coefficients are significant jointly while the t-tests can fail to find significance individually. These conflicting test results can be hard to understand, but think about it this way. The F-test sums the predictive power of all independent variables and determines that it is unlikely that all of the coefficients equal zero. However, it’s possible that each variable isn’t predictive enough on its own to be statistically significant. In other words, your sample provides sufficient evidence to conclude that your model is significant, but not enough to conclude that any individual variable is significant. Related post: How to Interpret Regression Coefficients and their P-values. If you have a statistically significant overall F-test, you can draw several other conclusions. For the model with no independent variables, the intercept-only model, all of the model’s predictions equal the mean of the dependent variable. Consequently, if the overall F-test is statistically significant, your model’s predictions are an improvement over using the mean. R-squared measures the strength of the relationship between your model and the dependent variable. However, it is not a formal test for the relationship. The F-test of overall significance is the hypothesis test for this relationship. If the overall F-test is significant, you can conclude that R-squared does not equal zero, and the correlation between the model and dependent variable is statistically significant. It’s fabulous if your regression model is statistically significant! However, check your residual plots to determine whether the results are trustworthy! And, learn how to choose the correct regression model! If you’re learning regression and like the approach I use in my blog, check out my Intuitive Guide to Regression Analysis book! You can find it on Amazon and other retailers. Note: I wrote a different version of this post that appeared elsewhere. I’ve completely rewritten and updated it for my blog site. What is the general form of the regression equation?A linear regression line has an equation of the form Y = a + bX, where X is the explanatory variable and Y is the dependent variable.
When expressed as a percentage What is the range of values for multiple R?R-squared values range from 0 to 1 and are commonly stated as percentages from 0% to 100%. An R-squared of 100% means that all movements of a security (or another dependent variable) are completely explained by movements in the index (or the independent variable(s) you are interested in).
What is the null hypothesis to test the significance of the slope in a regression equation?The null hypothesis states that the slope is equal to zero, and the alternative hypothesis states that the slope is not equal to zero.
What is a residual for a multiple regression model and the data that is used to create it?A residual is a measure of how far away a point is vertically from the regression line. Simply, it is the error between a predicted value and the observed actual value.
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Which statistic is used to test a global hypothesis about a multiple regression equation
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