Regression and correlation
Two-variable vocabulary: fitting a line, reading residuals, and what r and r-squared report.
18 terms
AssociationTwo variables are associated when knowing the value of one changes what you expect for the other, whether the pattern is linear or curved.Coefficient of determination (r²)The coefficient of determination, r-squared, is the fraction of the variation in the response variable that the regression line explains, from 0 to 1.Computer regression outputComputer regression output is the table software prints for a fitted line, listing each coefficient with its standard error plus summaries such as s and R-Sq.CorrelationCorrelation measures the strength and direction of the linear relationship between two quantitative variables on a unit-free scale from -1 to 1.Correlation coefficientThe correlation coefficient r measures the direction and strength of a linear relationship between two quantitative variables, always between -1 and 1.ExtrapolationExtrapolation is using a regression line to predict outside the range of x-values the data covered, where the pattern may no longer hold.High-leverage pointA high-leverage point has an x-value far from the mean of x, which gives it the power to move the regression line, whether or not it actually does.Influential pointAn influential point is an observation that, if removed, would markedly change the regression line's slope, intercept, or correlation.Least-squares criterionThe least-squares criterion picks the line that makes the sum of the squared vertical distances from the points to the line as small as possible.Least-squares regression lineThe least-squares regression line is the straight line through a scatterplot that makes the sum of the squared residuals as small as possible.Linear modelA linear model is a straight-line equation that uses an explanatory variable x to predict a response variable y, written y-hat = a + bx.Predicted value (y-hat)The predicted value y-hat is the response a regression model gives for a chosen x, found by substituting that x into the equation of the line.ResidualA residual is the difference between an observed y-value and the value the regression line predicts for it: observed minus predicted.Residual plotA residual plot graphs the residuals against the explanatory variable or predicted values, used to check whether a line fits the data well.Slope of a regression lineThe slope of a regression line is the predicted change in the response variable for each one-unit increase in the explanatory variable.Standard deviation of the residuals (s)The standard deviation of the residuals, s, is roughly the typical size of a prediction error from the regression line, in the units of y.Transformation to achieve linearityA transformation to achieve linearity applies a function such as a logarithm or square root to x or y so a curved pattern straightens out.Y-intercept of a regression lineThe y-intercept a is the response the line predicts when x equals 0, and it often has no sensible meaning because x = 0 lies outside the data.