Transcript Chapter 4 - the Department of Psychology at Illinois State

Reminders
 HW2 due today
 Exam 1 next Tues (9/27) – Ch 1-5
– 3 sections:
• Short answers (concepts, definitions)
• Calculations (you’ll be given the formulas)
• SPSS output interpretation
Chapter 4
Prediction, Part 3
Sept 20, 2005
The Regression Line
 Relation between predictor variable and
predicted values of the criterion variable
 Slope of regression line
– Equals b, the raw-score regression
coefficient
 Intercept of the regression line (where
line crosses y axis)
– Equals a, the regression constant
Drawing the Regression Line
1. Draw and label the axes for a scatter diagram
2. Figure predicted value on criterion for a low
value on predictor variable
You can randomly choose what value to plug in..
Y hat = -.271 + .4 (x)
Y hat = -.271 + .4 (20) = 7.73
3. Repeat step 2. with a high value on predictor
Y hat = -.271 + .4 (80) = 31.73
4. Draw a line passing through the two marks
5. Hint: you can also use (Mx, My) to save time
as one of your 2 points. Reg line always
passes through the means of x and y.
Drawing the Regression Line
Regression Error
 Now that you have a regression line or
equation, can find predicted y scores…
– Then, assume that you later collect a new sample
of x & y scores
• You can compare how the accuracy of predicted ŷ to the
actual y scores
• Sometimes you’ll overestimate, sometimes
underestimate…this is ERROR.
– Can we get a measure of error? How much is OK?
Error and Proportionate
Reduction in Error
 Error
– Actual score minus the predicted score
2
ˆ
Error  (Y  Y )
2
 Proportionate reduction in error
– Squared error using prediction (reg) model =
SSError =  (y - ŷ)2
– Compare this to amount of error w/o this
prediction (reg) model. If no other model, best
guess would be the mean.
– Total squared error when predicting from the
mean is SSTotal = (y – My)2
Error and Proportionate
Reduction in Error
 Formula for proportionate reduction in
error: compares reg model to mean
baseline
SS Total  SS Error
Proportion ate reduction in error 
SS Total
Want reg model to be much better than
mean (baseline) – fewer prediction
errors
Example – Hrs. Slept & Mood
See Tables 4-5 and 4-6
 Reg model was ŷ = -6.57 + 1.33(x)
 Use mean model to find error (y-My)2 for
each person & sum up that column  SStot
 Find prediction using reg model:
– plug in x values into reg model to get ŷ
– Find (y-ŷ)2 for each person, sum up that column
 SSerror
 Find PRE
Error and Proportionate
Reduction in Error (cont.)
 If our reg model no better than mean, SSerror =
SStotal, so (0/ SStot) = 0.
– Using this regression model, we reduce error over the
mean model by 0%….not good prediction.
 If reg model has 0 error (perfect), SStot-0/SStot = 1, or
100% reduction of error.
 Proportionate reduction in error = r2
 aka “Proportion of variance in y accounted for by
x”, ranges between 0-100%.
Multiple Regression
 Bivariate prediction – 1 predictor, 1 criterion
 Multiple regression – use multiple predictors
– Reg model/equations are same, just use
separate reg coefficients () for each predictor
– Ex) Z-score multiple regression formula with
three predictor variables
ZˆY  (1 )( Z X1 )  (2 )( Z X 2 )  (3 )( Z X 3 )
– Note that here,  does not equal r due to
overlap among predictors.
Mult Reg (cont.)
 How to judge the relative importance of
each predictor variable in predicting the
criterion?
 Consider both the rs and the βs
– Not necessarily the same rank order of
magnitude for rs and βs, so check both.
– βs indicate unique relationship betw a predictor
and criterion, controlling for other predictors
– r’s indicate general relationship betw x & y
(includes effects of other predictors)
Prediction in Research Articles
 Bivariate prediction models rarely reported
 Multiple regression results commonly reported
– Note example table in book, reports r’s and βs
for each predictor; reports R2 in note at bottom.
SPSS Reg Example
– Analyze Regression  Linear
– Note that terms used in SPSS are
“Independent Variable”…this is x (predictor)
– “Dependent Variable”…this is y (criterion)
– Climate data, IV = exclusion experiences
• DV = likelihood of choosing ISU again
• What to look for:
– “Model Summary” section - shows r2
– ANOVA section – 1st line gives ‘sig value’, if < .05  signif
– Coefficients section – 1st line gives ‘constant’ = a
» 2nd line gives ‘standardized coefficients’ = b or beta
Group Activity
 Use climate data to find regression
model using views of ISU climate (as
IV) to predict likelihood of attending ISU
again (as DV).
 1) No need to print the output, just write
out the regression model on your paper.
 2) What is the r2 value you get? What
does it mean here?
Group Activity…

Finishing the patient satisfaction / therapist empathy
problem from Thurs. (remember, r = .9)
Pair
Ther.
Empathy(x)
1
2
3
4
70
94
36
48
M = 62
SD = 22.14
Patient
Satisf (y)
Predicted
Y
Y - Predicted Y…..(see board).
4
5
2
1
M=3
SD = 1.58
Reg Equation = ………
a) Figure error & squared error for each prediction, then find
proportion of reduction in error over SStotal
b) Does it match r2?