Week 4 Discussion

Linear Regression Models Simple Linear Regression Model The following is an example of a linear regression model. yi = α0 + (α1xi) + εi yi This letter represents the dependent variable. It is the variable measured, predicted, or otherwise monitored by the researcher. It is expected to be affected by the manipulation of the independent variable. α0 This letter represents the intercept and it is one of the two regression coefficients. Intercept is the value for the linear function when it crosses the Y-axis or the estimate of y when x is 0. α1 This letter represents the slope and it is one of the two regression coefficients. Slope is the change in y for a 1-unit change in x. xi This letter represents the independent variable. It is the variable manipulated by the researcher, thereby causing an effect or change on the dependent variable. εi This letter represents an error term. The error term is needed to account for errors in the measurement of y and those that might be caused due to other variables impacting y. However, it is not possible to observe this error. Therefore, an assumption can be made about the error term being a random variable with a normal distribution having a mean of 0 and a standard deviation of sigma.

Fitted Regression Model The following is an example of a fitted regression model. yi = a0 +a1xi yi Estimated value of yi a0 Estimated intercept

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a1 Estimated slope xi Any given value of x Given a value of x, the fitted regression model can be used to predict a value of y. When the estimated value of yi is subtracted from the observed value of yi, a value known as residual is calculated. The residual can be used to estimate sigma. © 2017 South University

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