Linear Regression

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Prediction

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The term used to describe what a regression model does.

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Dependent Variable Y

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The variable that a regression model seeks to predict. Also known as the response variable.

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Independent Variable X

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A variable that a regression model seeks to use to predict the dependent variable.

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Least-Squares Method

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The calculational method that minimizes the sum of the squared differences between the actual values of the dependent variable Y and the predicted values of Y.

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Simple Linear Regression

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The regression model that uses a straight line (linear) relationship to predict a numerical dependent variable Y from a single numerical independent variable X.

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Simple Linear Regression Equation

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Slope (b1)

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SSXY/SSX

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SSXY

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The sums of X and Y minus the sum of X multiplied by the sum of Y divided by the sample size.

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SSX

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The sum of each X variable squared minus the total sum of X squared divided by the sample size.

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Intercept (b0)

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mean Y - slope * mean X

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Residual

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The difference between the observed and predicted values of the dependent variable Y for a given value of X.

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Regression Sum of Squares

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The variation that is due to the relationship between X and Y.

SSR = SUM (Predicted Y value - mean Y value)^2

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Error Sum of Squares

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The variation that is due to factors other than the relationship between X and Y.

SSE = SUM (Observed Y value - Predicted Y value)^2

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Total Sum of Squares

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The measure of variation of the Yi values around their mean.

SST = SUM (Observed Y value - Mean Y value)^2

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Coefficient of Determination

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The ratio of the regression sum of squares to the total sum of squares, represented by the symbol r^2.

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SSR

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Equivalent to intercept * sum of Y + slope * sum of XY - minus (sum of Y)^2/n.

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SSE

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Equivalent of sum of Y^2 - intercept * sum of Y - slope * sum of XY.

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SST

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SSE + SSR

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Coefficient of Correlation

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The measure of the strength of the linear relationship between two variables, represented by the symbol r. Basically, the square root of the coefficient of determination.

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Standard Error of the Estimate

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The standard deviation around the fitted line of regression that measures the variability of the actual Y values from the predicted Y, represented by the symbol S_yx.

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Standard Error of the Slope

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Or the standard error of the estimate divided by the square root of the sum of squares X.

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t Test Statistic

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sample slope/standard error of the slope

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Confidence Interval Estimate of the Slope

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b1 +- t_n-2 * Sb1

Multiply the t statistic by the standard error of the slope and then add and subtract the product to the sample slope.

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Multiple Regression Model

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The statistical method that extends the simple linear regression model by assuming a straight-line or linear relationship between each independent variable and the dependent variable.

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Net Regression Coefficients

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The coefficients that measure the change in Y per unit change in a particular X, holding constant the effect of the other X variables. Also known as partial regression coefficients

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Multiple Regression Equation

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Coefficient of Multiple Determination

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The statistic that represents the proportion of the variable in Y that is explained by the set of independent variables included in the multiple regression model.

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The Overall F Test

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The test for the significance of the overall multiple regression model.