statistics for the social sciences psychology 340 fall 2013 correlation and regression

Statistics for the Social Statistics for the Social SciencesSciences

Psychology 340Fall 2013

Correlation and Regression

Homework #12 due11/19Homework #12 due11/19Chapter 16: 1, 2, 7, 8, 10, 22 (Use

SPSS), 24

(HW #13 – the last homework – is due on 11/21)

Last Time:

•We reviewed Pearson’s r, and how to calculate it from raw scores and Z scores, and with SPSS

•We learned about Spearman’s Rho

•We learned how to get a scatter plot using SPSS

•We learned about bivariate regression

•One x (predictor, independent) variable is used to predict one y (outcome, dependent) variable

•Review of the formula for a line (y=mx+b)

•Applied formula for a line to regression (Y=a + bX + error or Y=β0+β1X+error)

This time:

• Clarification and review of some regression concepts

• Multiple regression

• Regression in SPSS

• Scatterplots and simple linear (bivariate) regression in Excel

Regression is all about prediction• If you want to predict (make an educated guess about) an individual person’s score on a variable (Y), what’s the best estimate, based on the information available to you?

• Suppose you only know the mean of the variable (MY) and nothing else. Then the best estimate of any individual’s score is the mean (more scores are close to the mean than far from it).

• Suppose you also know that the variable is correlated with another variable (X), and you know a person’s score on X, but not on Y.

• Regression involves using the person’s score on X, combined with what you know about the relationship between X & Y to get a much more accurate prediction of the person’s score on Y than you can get just using the mean of Y (MY).

• If X and Y are uncorrelated, then the best predictor of any individual score on Y is the mean of Y (Y = MY + Error). The regression line is flat (has no slope). Note that in this scenario, the average error is based on deviations of Y values from the mean of Y. (It’s the standard deviation)

vs. Y vs. µY

Refers to the predicted value of Y

µY Refers to the expected (mean) value of Y. In the context of regression, it refers to the expected value of Y for a given value of X.

Y Refers to an actual (observed) value of Y.

If you know an individual’s score for Y and X, and you know the regression equation, you can calculate a residual score for that individual.

By looking at the residuals for a group of individuals, you can determine how good a “fit” the regression model is (i.e., how much error there is)

Regression• The “best fitting line” is the one that minimizes the

error (differences) between the predicted scores (the line) and the actual scores (the points)

Y

X

1

2

3

4

5

6

1 2 3 4 5 6

• Rather than compare the errors from different lines and picking the best, we will directly compute the equation for the best fitting line

Regression

• The linear model

Y = intercept + slope (X) + error

Beta’s (β) are sometimes called parameters

Come in two types:

• standardized

• unstandardized

Now let’s go through an example computing these things

From when we computed Pearson’s r

6 61 25 6

3 4

3 2

X Y

mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0

4.85.2

2.8

0.0

1.2

5.766.76

1.96

0.36

0.36

4.04.0

4.0

0.0

4.0

14.015.20 16.0

SSYSSX

SP

Computing regression line (with raw scores)

6 61 25 6

3 4

3 2

X Y

14.015.20 16.0

SSYSSX

SPmean 3.6 4.0

Computing regression line(with raw scores)

6 61 25 6

3 4

3 2

X Y

mean 3.6 4.0

Y

X

1

23456

1 2 3 4 5 6

Computing regression line (with raw scores)

6 61 25 6

3 4

3 2

X Y

mean 3.6 4.0

Y

X

1

23456

1 2 3 4 5 6

The two means will be on the line

Computing regression line (with z-scores)

mean

ZY

ZX

-1

1

2

0

1 2

0.0

1.1-1.1

0.0

-1.1

1.1

0.0

1.38-1.49

0.8

- 0.34

- 0.34

-2

-1-2

Regression

• Error– Actual score minus the predicted score

• Measures of error– r2 (r-squared)

– Proportionate reduction in error

• Note: Total squared error when predicting from the mean = SSTotal=SSY

– Squared error using prediction model = Sum of the squared residuals = SSresidual= SSerror

Computing Error around the line

• Compute the difference between the predicted values and the observed values (“residuals”)

• Square the differences

• Add up the squared differences

Y

X

1

23456

1 2 3 4 5 6

• Sum of the squared residuals = SSresidual = SSerror


6 61 25 6

3 4

3 2

X Y

mean 3.6 4.0

Predicted values of Y (points on the line)



6 61 25 6

3 4

3 2

X Y

mean 3.6 4.0

6.2 = (0.92)(6)+0.688

Predicted values of Y (points on the line)



6 61 25 6

3 4

3 2

X Y

mean 3.6 4.0

6.2 = (0.92)(6)+0.688

1.6 = (0.92)(1)+0.688

5.3 = (0.92)(5)+0.688

3.45 = (0.92)(3)+0.688

3.45 = (0.92)(3)+0.688



Y

X

123

45

6

1 2 3 4 5 6


X Y6 61 25 6

3 4

3 2

6.21.6

5.3

3.45

3.45

6.2

1.6

5.3

3.45


6 61 25 6

3 4

3 2

X Y

mean 3.6 4.0

6.2

0.00

-0.200.40

0.70

0.55

-1.45

1.6

5.3

3.45

3.45

residuals• Sum of the squared residuals = SSresidual = SSerror

Quick check

6 - 6.2 =

2 - 1.6 =

6 - 5.3 =

4 - 3.45 =

2 - 3.45 =


6 61 25 6

3 4

3 2

X Y

mean 3.6 4.0

6.2

0.00

0.040.16

0.49

0.30

2.10

3.09

-0.200.40

0.70

0.55

-1.45

1.6

5.3

3.45

3.45

SSERROR



6 61 25 6

3 4

3 2

X Y

mean 3.6 4.0

6.2

0.00

0.040.16

0.49

0.30

2.10

3.09

-0.200.40

0.70

0.55

-1.45

1.6

5.3

3.45

3.45

SSERROR


4.04.0

4.0

0.0

4.0

16.0

SSY


• Also (like r2) represents the percent variance in Y accounted for by X

3.09

SSERROR


16.0

SSY

– Proportionate reduction in error

• In fact, in bivariate regression it is mathematically identical to r2

Regression in SPSS

• Running the analysis in SPSS is pretty easy– Analyze: Regression: Linear– X or predictor variable(s) go into the

‘independent variable’ field– Y or predicted variable goes into the

‘dependent variable’ field

• You get a lot of output

Regression in SPSS• The variables in the model

• r

• r2

• Unstandardized coefficients

• Slope (indep var name)• Intercept (constant)

• Standardized coefficients

• We’ll get back to these numbers in a few weeks

In Excel• With Data Analysis “Tool Pack” you can

perform regression analysis• With standard software package, you

can get bivariate correlation (which is the same as the standardized regression coefficient), you can create a scatterplot, and you can request a trend line, which is a regression line (what is y and what is x in that case?)

Multiple Regression• Multiple regression prediction

models

“fit” “residual”

Prediction in Research Articles

• Bivariate prediction models rarely reported

• Multiple regression results commonly reported

Multiple Regression

• Typically researchers are interested in predicting with more than one explanatory variable

• In multiple regression, an additional predictor variable (or set of variables) is used to predict the residuals left over from the first predictor.

Multiple Regression


• Bi-variate regression prediction models

Multiple Regression

• Multiple regression prediction models

“fit” “residual”


• Bi-variate regression prediction models

Multiple Regression

• Multiple regression prediction models

First

Explanatory

Variable

Second

Explanatory

Variable

Fourth

Explanatory

Variable

whatever variability

is left over

Third

Explanatory

Variable

Multiple Regression• Predict test performance based on:

First

Explanatory

Variable

Second

Explanatory

Variable

Fourth

Explanatory

Variable

whatever variability

is left over

Third

Explanatory

Variable

• Study time • Test time • What you eat for breakfast • Hours of sleep

Multiple Regression• Predict test performance based on:

• Study time • Test time • What you eat for breakfast • Hours of sleep

• Typically your analysis consists of testing multiple regression models to see which “fits” best (comparing R2s of the models)

versus

versus

• For example:

Multiple Regression

Response variableTotal variability it test performance

Total study timer = .6

Model #1: Some co-variance between the two variables

R2 for Model = .36

64% variance unexplained

• If we know the total study time, we can predict 36% of the variance in test performance

Multiple Regression


Test timer = .1

Model #2: Add test time to the model


R2 for Model = .37


• Little co-variance between these test performance and test time• We can explain more the of variance in test performance

Multiple Regression


breakfastr = .0

Model #3: No co-variance between these test performance and breakfast food


Test timer = .1

R2 for Model = .37


• Not related, so we can NOT explain more the of variance in test performance

Multiple Regression


breakfastr = .0

• We can explain more the of variance • But notice what happens with the overlap (covariation between explanatory

variables), can’t just add r’s or r2’s


Test timer = .1

Hrs of sleepr = .45

R2 for Model = .45


Model #4: Some co-variance between these test performance and hours of sleep

Multiple Regression

The “least squares” regression equation when there are multiple intercorrelated predictor (x) variables is found by calculating “partial regression coefficients” for each x

A partial regression coefficient for x1 shows the relationship between y and x1 while statistically controlling for the other x variables (or holding the other x variables constant)

Multiple Regression

The formula for the partial regression coefficient is :

b1= (rY1-rY2r12)/(1-r122)*(sY/s1)

WhererY1=correlation of x1and yrY2=correlation of x2and yr12=correlation of x1 and x2

sY=standard deviation of y, s1=standard deviation of x1

Multiple Regression

• Multiple correlation coefficient (R) is an estimate of the relationship between the dependent variable (y) and the best linear combination of predictor variables (correlation of y and y-pred.)

• R2 tells you the amount of variance in y explained by the particular multiple regression model being tested.

Multiple Regression in SPSS

Setup as before: Variables (explanatory and response) are entered into columns

• A couple of different ways to use SPSS to compare different models

Regression in SPSS• Analyze: Regression, Linear


• Method 1:enter all the explanatory

variables together – Enter:

• All of the predictor variables into the Independent Variable field

• Predicted (criterion) variable into Dependent Variable field


• The variables in the model

• r for the entire model

• r2 for the entire model

• Unstandardized coefficients

• Coefficient for var1 (var name)



• The variables in the model

• r for the entire model

• r2 for the entire model

• Standardized coefficients



Multiple Regression

– Which coefficient to use, standardized or unstandardized?

– Unstandardized b’s are easier to use if you want to predict a raw score based on raw scores (no z-scores needed).

– Standardized β’s are nice to directly compare which variable is most “important” in the equation


• Predicted (criterion) variable into Dependent Variable field

• First Predictor variable into the Independent Variable field

• Click the Next button

• Method 2: enter first model, then add another

variable for second model, etc. – Enter:


• Method 2 cont: – Enter:

• Second Predictor variable into the Independent Variable field

• Click Statistics


– Click the ‘R squared change’ box


• The variables in the first model (math SAT)• Shows the results of two models

• The variables in the second model (math and verbal SAT)


• The variables in the first model (math SAT)

• r2 for the first model

• Coefficients for var1 (var name)

• Shows the results of two models


• Model 1


• The variables in the first model (math SAT)



• Shows the results of two models

• r2 for the second model


• Model 2


• The variables in the first model (math SAT)• Shows the results of two models


• Change statistics: is the change in r2 from Model 1 to Model 2 statistically significant?

Cautions in Multiple Regression

• We can use as many predictors as we wish but we should be careful not to use more predictors than is warranted.– Simpler models are more likely to generalize to other

samples.– If you use as many predictors as you have participants in

your study, you can predict 100% of the variance. Although this may seem like a good thing, it is unlikely that your results would generalize to any other sample and thus they are not valid.

– You probably should have at least 10 participants per predictor variable (and probably should aim for about 30).

statistics for the social sciences psychology 340 fall 2013 correlation and regression

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regression y

mean of y y

line y

variable y

y outcome

variable x

expected value of y

regression equation