![]() Regression analysis is a simple and statistical method to understand and quantify the relationship between two variables or more. It also empowers decision-makers with data-driven insights. Aiding in forecasting, risk assessment, and identifying trends, regression analysis plays an important role in diverse fields. \(H_a\): The addition of the term \(\beta_1\) (and hence the predictor \(X\)) helps improve the prediction of \(Y\) over the simple mean model.Regression analysis, a powerful tool for data analysis, helps businesses and researchers make informed decisions by predicting outcomes based on historical data. \(H_o\): The addition of the term \(\beta_1\) (and hence the predictor \(X\)) does not improve the prediction of \(Y\) over the simple mean model. To answer this question, we must setup a hypothesis test. The question then becomes ‘how significant is the observed ratio?’. The larger the \(F\) ratio, the greater the difference between the bivariate model and mean model. In this working example, the percent error explained by the bivariate model is, The following table summarizes some of these key error terms:Ģ.1 Testing if the bivariate model is an improvement over the mean model 2.1.1 The coefficient of determination, \(R^2\) The take away point here is that \(SSR\) is the error that is explained by the new bivariate model. Had we shown each squared residual, red boxes would have been drawn as in earlier figures (the boxes were not drawn here to prevent clutter). ![]() The linear models are shown in blue lines, the square root of each residual element making up \(SSR\) (the difference between the two models) is shown in dashed red lines. It’s no surprise that R has a built in function, lm(), that will estimate these regression coefficients for us. Our final linear equation looks like this: So we can substitute \(X\) and \(\hat Y\) in the equation with their respective means: 0.1. It just so happens that the slope passes through \(\bar Y\) and \(\bar X\). Knowing \(\hat \beta_1\) we can solve for the intercept \(b_0\): \[ The bottom of the last two columns show the sums of the numerator and denominator (3553.60875 and 0.0715438 respectively). In the process, a dataframe called dat is created that stores both \(X\) and \(Y\). The following block of code defines the new variable \(X\) (i.e. fraction of Mainers having attained a bachelor’s degree or greater within each county) and visualizes the relationship between income and education attainment using a scatter plot. This is the equation of a slope where \(\beta_0\) tells us where on the y-axis the slope intersects the axis and \(\beta_1\) is the slope of the line (a positive \(\beta_1\) value indicates that the slope is upward, a negative \(\beta_1\) value indicates that the slope is downward).Ĭontinuing with our example, we will look at the percentage of Mainers having attained a Bachelor’s degree or greater by county (the data are provided by the US Census ACS 2007 - 2011 dataset). ![]() In other words, we are using another variable (aka, an independent or predictor variable) to estimate \(Y\). The bivariate model augments the simple (mean) model by adding a variable, \(X\), that is believed to co-vary with \(Y\). A more common approach in predicting \(Y\) is to use a predictor variable (i.e. a covariate). This simple model (i.e. one where we are using the mean value to predict \(Y\) for each county) obviously has its limits. Note too that the unit used to measure the area of each square (the squared error) is the one associated with the Y-axis (recall that the error is associated with the prediction of \(Y\) and not of \(X\)). Yet, the area for index 3 appears almost 9 times bigger than that associated with index 5 (as it should since we are squaring the difference). Note how the larger error at index 3 has an error line length about 3 times that associated with index 5 (these correspond to the records number three and five in the above table). It is more natural to visualize a squared amount using a geometric primitive whose area changes proportionately to the squared error changes.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |