estimated regression equation, in statistics, an equation constructed to model the relationship between dependent and independent variables.
Either a simple or multiple regression model is initially posed as a hypothesis concerning the relationship among the dependent and independent variables. The least squares method is the most widely used procedure for developing estimates of the model parameters. For simple linear regression, the least squares estimates of the model parameters β0 and β1 are denoted b0 and b1. Using these estimates, an estimated regression equation is constructed: ŷ = b0 + b1x . The graph of the estimated regression equation for simple linear regression is a straight line approximation to the relationship between y and x.
As an illustration of regression analysis and the least squares method, suppose a university medical centre is investigating the relationship between stress and blood pressure. Assume that both a stress test score and a blood pressure reading have been recorded for a sample of 20 patients. The data are shown graphically in , called a scatter diagram. Values of the independent variable, stress test score, are given on the horizontal axis, and values of the dependent variable, blood pressure, are shown on the vertical axis. The line passing through the data points is the graph of the estimated regression equation: ŷ = 42.3 + 0.49x. The parameter estimates, b0 = 42.3 and b1 = 0.49, were obtained using the least squares method.
A primary use of the estimated regression equation is to predict the value of the dependent variable when values for the independent variables are given. For instance, given a patient with a stress test score of 60, the predicted blood pressure is 42.3 + 0.49(60) = 71.7. The values predicted by the estimated regression equation are the points on the line in , and the actual blood pressure readings are represented by the points scattered about the line. The difference between the observed value of y and the value of y predicted by the estimated regression equation is called a residual. The least squares method chooses the parameter estimates such that the sum of the squared residuals is minimized.