Regression What is regression to the mean? Suppose the mean temperature in November is 5 degrees What’s your best guess for tomorrow’s temperature? exactly 5? warmer than 5? colder than 5?

Regression What is regression to the mean? Suppose the mean temperature in November is 5 degrees and today the temperature is 15 What’s your best guess for tomorrow’s temperature? exactly 15 again? exactly 5? warmer than 15? something between 5 and 15?

Regression What is regression to the mean? Regression to the mean is the fact that scores tend to be closer to the mean than the values they are paired with e.g. Daughters tend to be shorter than mothers if the mothers are taller than the mean and taller than mothers if the mothers are shorter than the mean e.g. Parents with high IQs tend to have kids with lower IQs, parents with low IQs tend to have kids with higher IQs

Regression What is regression to the mean? The strength of the correlation between two variables tells you the degree to which regression to the mean affects scores strong correlation means little regression to the mean weak correlation means strong regression to the mean no correlation means that one variable has no influence on values of the other - the mean is always your best guess

Regression Suppose you measured workload and credit hours for 8 students Could you predict the number of homework hours from credit hours?

Regression Suppose you measured workload and credit hours for 8 students Your first guess might be to pick the mean number of homework hours which is 12.9

Regression Sum of Squares Adding up the squared deviation scores gives you a measure of the total error of your estimate

Regression Sum of Squares ideally you would pick an equation that minimized the sum of the squared deviations You would need a line is as close as possible to each point

Regression The regression line That line is called the regression line The sum of squared deviations from it is called the sum of squared error or SSE

Regression The regression line That line is called the regression line its equation is:

Regression remember: y = ax + b ax + b predicted y

Regression What happens if you had transformed all the scores to z scores and were trying to predict a z score?

Regression What happens if you had transformed all the scores to z scores and were trying to predict a z score? but… Sy = Sx = 1 So….

The Regression Line The regression line is a linear function that generates a y for a given x

The Regression Line The regression line is a linear function that generates a y for a given x What should its slope and y-intercept be to be the best predictor?

The Regression Line The regression line is a linear function that generates a y for a given x What should its slope and y-intercept be to be the best predictor? What does best predictor mean? It means least distance between the predicted y and an actual y for a given x

The Regression Line The regression line is a linear function that generates a y for a given x What should its slope and y-intercept be to be the best predictor? What does best predictor mean? It means least distance between the predicted y and an actual y for a given x in other words, how much variability is residual after using the correlation to explain the y scores

Mean Square Residual Recall that

Mean Square Residual The variance of Zy is the average squared distance of each point from the x axis (note that the mean of Zy = 0)

Mean Square Residual Some of the variance in the Zy scores is due to the correlation with x Some of the variance in the Zy scores is due to other (probably random) factors

Mean Square Residual the variance due to other factors is called “residual” because it is “leftover” after fitting a regression line The best predictor should minimize this residual variance

Mean Square Residual MSres is the average squared deviation of the actual scores from the regression line

Minimizing MSres the regression line (the best predictor of y) is the line with a slope and y intercept such that MSres is minimized

Minimizing MSres What will be its y intercept? if there was no correlation at all, your best guess for y at any x would be the mean of y if there was a strong correlation between x and y, your best guess for the y that matches the mean x would be the mean y the mean of Zx is zero so the best guess for the Zy that goes with it will be zero (the mean of the Zy’s)

Minimizing MSres In other words, the regression line will predict zero when Zx is zero so the y intercept of the regression line will be zero (only so for Z scores !)

Minimizing MSres y intercept is zero

Minimizing MSres what is the slope?

Minimizing MSres what is the slope? consider the extremes: Do the slopes look familiar? Zy = Zx Zy’=Zx slope = 1 Zy=-Zx Zy’=-Zx slope = -1 Zy is random with respect to Zx Zy’=mean Zy=0 slope = 0

Minimizing MSres a line (regression of Zy on Zx) that has a slope of rxy and a y intercept of zero minimizes MSres

Predicting raw scores we have a regression line in z scores: can we predict a raw-score y from a raw-score x?

Predicting raw scores recall that: and

Predicting raw scores by substituting we get:

Predicting raw scores + b a y = ax + b by substituting we get: note that this is still of the form: note that the slope still depends on r and the intercept still depends on the mean of y + b a y = ax + b

Interpreting rxy in terms of variance Recall that rxy is the slope of the regression line that minimizes MSres

Interpreting rxy in terms of variance Recall that rxy is the slope of the regression line that minimizes MSres

Interpreting rxy in terms of variance MSres can be simplified to:

Interpreting rxy in terms of variance Thus:

Interpreting rxy in terms of variance Thus: So can be thought of as the proportion of original variance accounted for by the regression line

Interpreting rxy in terms of variance Observed y Subtract this distance What % of this distance Regression Line is this distance Predicted y Mean of y

Interpreting rxy in terms of variance it follows that 1 - is the proportion of variance not accounted for by the regression line - this is the residual variance

Interpreting rxy in terms of variance this can be thought of as a partitioning of variance into the variance accounted for by the regression and the variance unaccounted for

Interpreting rxy in terms of variance this can be thought of as a partitioning of variance into the variance accounted for by the regression and the variance unaccounted for

Interpreting rxy in terms of variance often written in terms of sums of squares: or simply SStotal = SSregression + SSresidual