2. 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?

3. 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?

4. 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

5. 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

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

7. 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

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

9. 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

10. 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

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

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

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

14. 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….

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

16. 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?

17. 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

18. 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

19. Mean Square Residual
Recall that

20. 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)

21. 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

22. 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

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

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

25. 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)

26. 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 !)

27. Minimizing MSres
y intercept is zero

28. Minimizing MSres
what is the slope?

29. 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

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

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

32. Predicting raw scores
recall that:
and

33. Predicting raw scores
by substituting we get:

34. 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

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

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

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

38. Interpreting rxy in terms of variance
Thus:

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

40. 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

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

42. 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

43. 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

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