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# Psychology 340 Spring 2010 - PowerPoint PPT Presentation

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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#### Presentation Transcript

Statistics for the Social Sciences

Prediction with multiple variables

Psychology 340

Spring 2010

### Outline

• Multiple regression

• Comparing models, Delta r2

• Using SPSS

### 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

Y = intercept + slope (X) + error

“residual”

“fit”

### Multiple Regression

• Multiple regression prediction models

• Bi-variate regression prediction models

Y = intercept + slope (X) + error

whatever variability

is left over

First

Explanatory

Variable

Second

Explanatory

Variable

Third

Explanatory

Variable

Fourth

Explanatory

Variable

### Multiple Regression

• Multiple regression prediction models

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

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:

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 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 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 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

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

• 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 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)

• 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 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

• 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

• Second Predictor variable into the Independent Variable field

• Click Statistics

### Multiple Regression in SPSS

• Method 2 cont:

• Enter:

### Multiple Regression in SPSS

• Click the ‘R squared change’ box

### 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 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)

• 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 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?

“residual”

“fit”

### Hypothesis testing with Regression

• Multiple Regression

• We can test hypotheses about the overall model

### 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

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

• 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 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 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)

• 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

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

• Bivariate prediction models rarely reported

• Multiple regression results commonly reported

### 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).