Psychology 340 spring 2010
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Statistics for the Social Sciences. Prediction with multiple variables. Psychology 340 Spring 2010. Outline. Multiple regression Comparing models, Delta r 2 Using SPSS. Multiple Regression. Typically researchers are interested in predicting with more than one explanatory variable

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Psychology 340 Spring 2010

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Psychology 340 spring 2010

Statistics for the Social Sciences

Prediction with multiple variables

Psychology 340

Spring 2010


Outline

Outline

  • Multiple regression

    • Comparing models, Delta r2

    • Using SPSS


Multiple regression

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 regression1

Multiple Regression

  • Bi-variate regression prediction models

Y = intercept + slope (X) + error


Multiple regression2

“residual”

“fit”

Multiple Regression

  • Multiple regression prediction models

  • Bi-variate regression prediction models

Y = intercept + slope (X) + error


Multiple regression3

whatever variability

is left over

First

Explanatory

Variable

Second

Explanatory

Variable

Third

Explanatory

Variable

Fourth

Explanatory

Variable

Multiple Regression

  • Multiple regression prediction models


Multiple regression4

whatever variability

is left over

First

Explanatory

Variable

Second

Explanatory

Variable

Third

Explanatory

Variable

Fourth

Explanatory

Variable

Multiple Regression

  • Predict test performance based on:

  • Study time

  • Test time

  • What you eat for breakfast

  • Hours of sleep


Multiple regression5

versus

versus

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)

  • For example:


Multiple regression6

Response variable

Total variability it test performance

Total study time

r = .6

Multiple Regression

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

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

R2 for Model = .36

64% variance unexplained


Multiple regression7

Multiple Regression

Model #2: Add test time to the model

  • Little co-variance between these test performance and test time

  • We can explain more the of variance in test performance

R2 for Model = .49

Response variable

Total variability it test performance

Total study time

r = .6

51% variance unexplained

Test time

r = .1


Multiple regression8

Multiple Regression

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

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

R2 for Model = .49

Response variable

Total variability it test performance

breakfast

r = .0

Total study time

r = .6

51% variance unexplained

Test time

r = .1


Multiple regression9

Multiple Regression

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

  • 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

R2 for Model = .60

Response variable

Total variability it test performance

breakfast

r = .0

Total study time

r = .6

40% variance unexplained

Hrs of sleep

r = .45

Test time

r = .1


Multiple regression in spss

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

Regression in SPSS

  • Analyze: Regression, Linear


Multiple regression in spss1

  • Predicted (criterion) variable into Dependent Variable field

  • All of the predictor variables into the Independent Variable field

Multiple Regression in SPSS

  • Method 1:enter all the explanatory variables together

    • Enter:


Multiple regression in spss2

Multiple Regression in SPSS

  • The variables in the model

  • r for the entire model

  • r2 for the entire model

  • Unstandardized coefficients

  • Coefficient for var1 (var name)

  • Coefficient for var2 (var name)


Multiple regression in spss3

  • Coefficient for var1 (var name)

  • Coefficient for var2 (var name)

Multiple Regression in SPSS

  • The variables in the model

  • r for the entire model

  • r2 for the entire model

  • Standardized coefficients


Multiple regression10

Multiple Regression

  • Which β to use, standardized or unstandardized?

  • Unstandardized β’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


Multiple regression in spss4

  • First Predictor variable into the Independent Variable field

  • Click the Next button

Multiple Regression in SPSS

  • Method 2: enter first model, then add another variable for second model, etc.

    • Enter:

  • Predicted (criterion) variable into Dependent Variable field


Multiple regression in spss5

  • Second Predictor variable into the Independent Variable field

  • Click Statistics

Multiple Regression in SPSS

  • Method 2 cont:

    • Enter:


Multiple regression in spss6

Multiple Regression in SPSS

  • Click the ‘R squared change’ box


Multiple regression in spss7

Multiple Regression in SPSS

  • Shows the results of two models

  • The variables in the first model (math SAT)

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


Multiple regression in spss8

Multiple Regression in SPSS

  • Shows the results of two models

  • The variables in the first model (math SAT)

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

  • r2 for the first model

  • Model 1

  • Coefficients for var1 (var name)


Multiple regression in spss9

  • Coefficients for var1 (var name)

  • Coefficients for var2 (var name)

Multiple Regression in SPSS

  • Shows the results of two models

  • The variables in the first model (math SAT)

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

  • r2 for the second model

  • Model 2


Multiple regression in spss10

Multiple Regression in SPSS

  • Shows the results of two models

  • The variables in the first model (math SAT)

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

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


Hypothesis testing with regression

“residual”

“fit”

Hypothesis testing with Regression

  • Multiple Regression

  • We can test hypotheses about the overall model


Multiple regression in spss11

Multiple Regression in SPSS

  • Null Hypotheses

  • H0: University GPA is not predicted by SAT verbal or SAT Math scores

  • p < 0.05, so reject H0, SAT math and verbal predict University GPA


Hypothesis testing with regression1

First

Explanatory

Variable

Second

Explanatory

Variable

Third

Explanatory

Variable

Fourth

Explanatory

Variable

Hypothesis testing with Regression

  • Multiple Regression

  • We can test hypotheses about each of these explanatory hypotheses within a regression model

    • So it’ll tell us whether that variable is explaining a “significant”amount of the variance in the response variable

  • We can test hypotheses about the overall model


Multiple regression in spss12

  • H0: Coefficient for var1 = 0

  • p < 0.05, so reject H0, var1 is a significant predictor

  • H0: Coefficient for var2 = 0

  • p > 0.05, so fail to reject H0, var2 is a not a significant predictor

Multiple Regression in SPSS

  • Null Hypotheses


Hypothesis testing with regression2

Hypothesis testing with Regression

  • Multiple Regression

  • We can test hypotheses about each of these explanatory hypotheses within a regression model

    • So it’ll tell us whether that variable is explaining a “significant”amount of the variance in the response variable

  • We can test hypotheses about the overall model

  • We can also use hypothesis testing to examine if the change in r2 is statistically significant


Hypothesis testing with regression3

Hypothesis testing with Regression

  • Shows the results of two models

  • The variables in the first model (math SAT)

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

  • r2 for the first model

  • Model 1

  • Coefficients for var1 (var name)


Hypothesis testing with regression4

  • Coefficients for var1 (var name)

  • Coefficients for var2 (var name)

Hypothesis testing with Regression

  • Shows the results of two models

  • The variables in the first model (math SAT)

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

  • r2 for the second model

  • Model 2


Hypothesis testing with regression5

The 0.002 change in r2

is not statistically

significant (p = 0.46)

Hypothesis testing with Regression

  • Shows the results of two models

  • The variables in the first model (math SAT)

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

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


Regression in research articles

Regression in Research Articles

  • Bivariate prediction models rarely reported

  • Multiple regression results commonly reported


Cautions in multiple regression

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).


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