1. Regression lecture 2 1. Review: deterministic and random components
2. The coefficient of determination
3. Using the regression line
4. Estimation of the mean value of Y for some X
5. Prediction of an individual value of Y for some X
6. Estimation and prediction contrasted
7. Estimation and prediction formulas
8. Examples
Regression Lecture 2

2. 1. Deterministic & random components
Our basic question is whether there is a relationship between two variables, X and Y.
To answer this question, we compare the deterministic part of the relationship to the random part.
the deterministic part is the part that would look the same if we sampled and measured again – it’s there for a reason (if it’s there at all).
Regression Lecture 2

3. 1. Deterministic & random components
Deterministic part – the least squares line (which determines a Y for each value of X).
Random part – deviations of observed Y scores from least squares line
(Note similarity here to t, F, and Z ratios, where we compare the numerator (treatment + error) to the denominator (error).)
Regression Lecture 2

4. 2. Coefficient of determination
Once we have regression line, we can assess its usefulness as a numerical “model” of the X-Y relationship.
We can do this by testing a hypothesis about the slope β1 or the correlation ρ, as last week.
We can also square the correlation coefficient, to get the coefficient of determination, r2.
Regression Lecture 2

5. The sum of the line (---) lengths gives the total error when we compute the Yi using the mean, YX
SSYY = Σ(Yi – Y)2
Regression Lecture 2

6. Regression line Y
The sum of the line (---) lengths gives the total error when we compute the Yi using the regression line. X ^
SSE = Σ(Yi – Yi)2
Regression Lecture 2

7. 2. The coefficient of determination
If knowing X reduces our uncertainty about Y, then SSE << SSYY. In that case, r2 – the coefficient of determination – tells us something useful:
SSYY – SSE
SSYY
r2 = explained sample variability in Y
total sample variability in Y
When thinking about r2, keep the graphs on the two previous slides in mind. The sums of line lengths on those slides give you SSYY and SSE.
Regression Lecture 2

8. 3. Using the regression line
So far, we’ve learned how to decide whether our regression line is useful. Suppose it is useful.
What can we do with it?
We’ll consider two alternative uses: estimation and prediction.
Regression Lecture 2

9. 3. Using the regression line
Estimation:
gives the average value of Y (Y) for all cases that have a given value of X
Prediction:
gives an individual Y score for one case that has a given value of X
Regression Lecture 2

10. 4. Estimation of the mean value of Y for some X
We can estimate the mean value of Y for a specific value of X.
e.g., we can estimate Y for ALL people whose blood contains a 4% concentration of some drug
here, Y would be some variable of interest such as (for example) reaction time (RT) to perform some task
we could estimate mean RT for all people who have the 4% drug concentration in their blood
Regression Lecture 2

11. 5. Prediction of an individual value of Y for some X
We can predict an individual value of Y for a given value of X.
e.g., we could predict RT for a specific person whose blood contains a 4% concentration of the drug
Regression Lecture 2

12. 6. Estimation and prediction contrasted
Recall from last week the two sources of error when using X to calculate an expected Y:
1. In the population, Y is not uniquely determined by X. As a result, for each value of X, there is a distribution of possible Y values.
if we knew the line Y = β0 + β1X + ε, we would still have this source of error
Regression Lecture 2

13. 6. Estimation and prediction contrasted
Two sources of error when using X to calculate an expected value of Y
2. The line we do have, Y = β0 + β1X, is not precisely correct
it does not capture the relationship between X and Y very precisely, because it is based on sample data. ^ ^ ^
Regression Lecture 2

14. 6. Estimation and prediction contrasted
only the second source of error is at work
things other than X that influence Y in the population are random effects, so on average across all cases they cancel out
Predicting
both sources of error are at work
Regression Lecture 2

15. 7. Estimation and prediction formulas
Estimation interval:
Y ± (tα/2)(s) (XP – X)2
n SSXX
tα/2 is based on d.f. = n – 2 ^ √
Regression Lecture 2

16. 7. Estimation and prediction formulas
Prediction interval:
Y ± (tα/2)(s) (XP – X)2
n SSXX
tα/2 is based on d.f. = n – 2 ^ √
The extra 1 under the square root sign makes the interval bigger than the corresponding estimation interval, because conclusions about individual cases (predictions) are less certain (more variable) than conclusions about means across large numbers of cases (estimates).
Regression Lecture 2

17. Examples – Emotional intelligence
First, we find X and Y:
X = ΣX = 74 = n
Y = ΣY = 82 =
n 7
Regression Lecture 2

18. Examples – Emotional intelligence
From last week:
SSXY =
SSXX =
Thus, β1 = = .781
139.71 ^
Regression Lecture 2

19. Examples – Emotional intelligence^ ^
β0 = Y – β1X
= – .781(10.571)
= 3.46
SSE = SSYY – β1(SSXY)
= – .781 ( ) = ^
Regression Lecture 2

20. Examples – Emotional intelligence
S = SSE
n – 2 = 5 = √ √
Regression Lecture 2

21. Examples – Emotional intelligence
The question says: “Use the data to form a 95% prediction interval for the Openness score of someone with an EI score of 13.”
Y = β0 + β1(X) = (13) =
tcrit = t(5, α/2 = .025) =
^ ^ ^
Regression Lecture 2

22. Examples – Emotional intelligence
Interval is:
± (2.571) (2.457) (13 – )2
13.613± 6.877 √
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23. Examples – Laughing First, we find X and Y: X = ΣX = 4.2 = .60 n 7
Y = ΣY = 32 =
n 7
Regression Lecture 2

24. Examples – Laughing From last week: SSXY = 2.15 SSXX = .34
Thus, β1 = =
.34 ^
Regression Lecture 2

25. Examples – Laughing ^ ^ β0 = Y – β1X = 4.5714 – 6.3235(.60) = .7773=
SSE = SSYY – β1(SSXY)
= – (2.15) = ^
Regression Lecture 2

26. √ √ Examples – Laughing S = SSE n – 2 = 1.6188 5 = .569= 5 = √ √
Regression Lecture 2

27. Examples – Laughing
The question says: “Regardless of your answer to part (a), form the 95% confidence interval for the predicted y value for a delay of .5 seconds (i.e., for all instances of .5).”
Y = β0 + β1(X) = (.5) =
tcrit = t(5, α/2 = .025) =
^ ^ ^
Regression Lecture 2

28. √ Examples – Laughing Interval is:
3.939 ± (2.571) (.569) (.5 – .6)2
3.939 ± (2.571) 9.569) (.4151)
3.939 ± .607 √
Regression Lecture 2