Since the mean error term is zero, the outcome variable y can be approximately estimated as follow: This is one the metrics used to evaluate the overall quality of the fitted regression model. The average variation of points around the fitted regression line is called the Residual Standard Error ( RSE). The sum of the squares of the residual errors are called the Residual Sum of Squares or RSS. Some of the points are above the blue curve and some are below it overall, the residual errors (e) have approximately mean zero. the error terms (e) are represented by vertical red linesįrom the scatter plot above, it can be seen that not all the data points fall exactly on the fitted regression line.the intercept (b0) and the slope (b1) are shown in green.the best-fit regression line is in blue.The figure below illustrates the linear regression model, where: e is the error term (also known as the residual errors), the part of y that can be explained by the regression model.b1 is the slope of the regression line.b0 is the intercept of the regression line that is the predicted value when x = 0.b0 and b1 are known as the regression beta coefficients or parameters:.As the x values in your chart are the same, MAX and MIN functions could be used interchangeably.The mathematical formula of the linear regression can be written as y = b0 + b1*x + e, where: The MEDIAN function is used in case there are multiple instances of the same x ( Number) in your sample. SELECT ( Alpha + ( Beta * (SELECT MEDIAN( Number)))) We can now calculate our linear regression estimate using α, β, and the x value ( Number). Metric 10 - hi Linear Regression Estimate The two mean metrics are carried over from the topic Covariance and Correlation and R-Squared. SELECT ( Avg Claim Value (Mean Y ) - ( Beta * Avg Claim Number (Mean X))) BY ALL OTHER Then, we are able to calculate α using β from above, the mean of x, and the mean of y: The BY ALL OTHER clause is used to prevent the amount from being sliced by anything present in the report. SELECT ((SELECT Pearson Correlation (r))*(SELECT (SELECT STDEV( Value))/(SELECT STDEV( Number)))) BY ALL OTHER Metric 8 - Beta Regression Coefficientįirst, we calculate β using Pearson Correlation (r), the standard deviation of x ( Number) and the standard deviation of y ( Value): Let's assume the same scenario as the insurance company in the topic Covariance and Correlation and R-Squared.Īfter we have generated Metric 6: Pearson Correlation (r) defined in the above topic, you can immediately calculate metrics for β, α and our linear estimate hi. The above metrics enable us to solve for our linear regression equation h: Scenario The result yields the following two equalities for β and α: The standard deviation of the dependent variable s y.The standard deviation of the explanatory variable s x.
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