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Statistics for the Social SciencesPsychology 340
Spring 2005
Prediction cont.
Statistics for the Social Sciences
Outline (for week)
• Simple bi-variate regression, least-squares fit line– The general linear model
– Residual plots
– Using SPSS
• Multiple regression– Comparing models, (?? Delta r2)
– Using SPSS
Statistics for the Social Sciences
From last time
• Review of last time
Y = intercept + slope(X) + errorY
X123456
1 2 3 4 5 6
Y
residuals are = Y −Y( )
Statistics for the Social Sciences
From last time
Y
X123456
1 2 3 4 5 6
residuals are = Y −Y( )
Y • The sum of the residuals should always equal 0.– The least squares regression line splits the data
in half
• Additionally, the residuals to be randomly distributed. – There should be no pattern to the residuals. – If there is a pattern, it may suggest that there is
more than a simple linear relationship between the two variables.
Statistics for the Social Sciences
Seeing patterns in the error
– Useful tools to examine the relationship even further. • These are basically scatterplots of the Residuals (often
transformed into z-scores) against the Explanatory (X) variable (or sometimes against the Response variable)
• Residual plots
Statistics for the Social Sciences
Seeing patterns in the error
• The residual plot shows that the residuals fall randomly above and below the line. Critically there doesn't seem to be a discernable pattern to the residuals.
Residual plotScatter plot
• The scatter plot shows a nice linear relationship.
Statistics for the Social Sciences
Seeing patterns in the error
Residual plot
• The scatter plot also shows a nice linear relationship.
• The residual plot shows that the residuals get larger as X increases.
• This suggests that the variability around the line is not constant across values of X.
• This is referred to as a violation of homogeniety of variance.
Scatter plot
Statistics for the Social Sciences
Seeing patterns in the error
• The residual plot suggests that a non-linear relationship may be more appropriate (see how a curved pattern appears in the residual plot).
Residual plotScatter plot
• The scatter plot shows what may be a linear relationship.
Statistics for the Social Sciences
Regression in SPSS
– Variables (explanatory and response) are entered into columns
– Each row is an unit of analysis (e.g., a person)
• Using SPSS
Statistics for the Social Sciences
Regression in SPSS
• Analyze: Regression, Linear
Statistics for the Social Sciences
Regression in SPSS
– Predicted (criterion) variable into Dependent Variable field
– Predictor variable into the Independent Variable field
• Enter:
Statistics for the Social Sciences
Regression in SPSS
• The variables in the model
• r
• r2
• We’ll get back to these numbers in a few weeks
• Slope (indep var name)• Intercept (constant)
• Unstandardized coefficients
Statistics for the Social Sciences
Regression in SPSS
(indep var name)
• Standardized coefficient
• Recall that r = standardized in
bi-variate regression
Statistics for the Social Sciences
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.
Statistics for the Social Sciences
Multiple Regression
Y = intercept + slope (X) + error
• Bi-variate regression prediction models
Statistics for the Social Sciences
Multiple Regression
• Multiple regression prediction models
μY = β0 + β1X1 + β2 X2 + β 3X3 + β 4 X4 + ε
“fit” “residual”
Y = intercept + slope (X) + error
μY = β0 + β1X + ε
• Bi-variate regression prediction models
Statistics for the Social Sciences
Multiple Regression
• Multiple regression prediction models
First
Explanatory
Variable
Second
Explanatory
Variable
Fourth
Explanatory
Variable
whatever variability
is left overμY = β0 + β1X1 + β2 X2 + β 3X3 + β 4 X4 + ε
Third
Explanatory
Variable
Statistics for the Social Sciences
Multiple Regression
First
Explanatory
Variable
Second
Explanatory
Variable
Fourth
Explanatory
Variable
whatever variability
is left overμY = β0 + β1X1 + β2 X2 + β 3X3 + β 4 X4 + ε
Third
Explanatory
Variable
• Predict test performance based on: • Study time • Test time • What you eat for breakfast • Hours of sleep
Statistics for the Social Sciences
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)
μY = β0 + β1X1 + β2 X2 + ε
μY = β0 + β1X1 + β2 X2 + β 4 X4 + εversus
μY = β0 + β1X1 + β2 X2 + β 3X3 + β 4 X4 + εversus
• For example:
Statistics for the Social Sciences
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
μY = β0 + β1X1 + ε
Statistics for the Social Sciences
Multiple Regression
Response variableTotal variability it test performance
Test timer = .1
Model #2: Add test time to the model
Total study timer = .6
R2 for Model = .49
51% variance unexplained
• Little co-variance between these test performance and test time• We can explain more the of variance in test performance
μY = β0 + β1X1 + β2 X2 + ε
Statistics for the Social Sciences
Multiple Regression
Response variableTotal variability it test performance
breakfastr = .0
Model #3: No co-variance between these test performance and breakfast food
Total study timer = .6
Test timer = .1
R2 for Model = .49
51% variance unexplained
μY = β0 + β1X1 + β2 X2 + β 3X3 + ε
• Not related, so we can NOT explain more the of variance in test performance
Statistics for the Social Sciences
Multiple Regression
Response variableTotal variability it test performance
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
Total study timer = .6
Test timer = .1
Hrs of sleepr = .45
R2 for Model = .60
40% variance unexplained
μY = β0 + β1X1 + β2 X2 + β 3X3 + β 4 X4 + ε
Model #4: Some co-variance between these test performance and hours of sleep
Statistics for the Social Sciences
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
Statistics for the Social Sciences
Regression in SPSS
• Analyze: Regression, Linear
Statistics for the Social Sciences
Multiple Regression in SPSS
• 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
Statistics for the Social Sciences
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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)
Statistics for the Social Sciences
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Multiple Regression in SPSS
• The variables in the model
• r for the entire model
• r2 for the entire model
• Standardized coefficients
• Coefficient for var1 (var name)
• Coefficient for var2 (var name)
Statistics for the Social Sciences
Multiple Regression
– Which to use, standardized or unstandardized?
– Unstandardized ’s are easier to use if you want to predict a raw score based on raw scores (no z-scores needed).
– Standardized ’s are nice to directly compare which variable is most “important” in the equation
Statistics for the Social Sciences
Multiple Regression in SPSS
• 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:
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Multiple Regression in SPSS
• Method 2 cont: – Enter:
• Second Predictor variable into the Independent Variable field
• Click Statistics
Statistics for the Social Sciences
Multiple Regression in SPSS
– Click the ‘R squared change’ box
Statistics for the Social Sciences
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Multiple Regression in SPSS
• The variables in the first model (math SAT)• Shows the results of two models
• The variables in the second model (math and verbal SAT)
Statistics for the Social Sciences
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Multiple Regression in SPSS
• The variables in the first model (math SAT)
• r2 for the first model
• Coefficients for var1 (var name)
• Shows the results of two models
• The variables in the second model (math and verbal SAT)
• Model 1
Statistics for the Social Sciences
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Multiple Regression in SPSS
• The variables in the first model (math SAT)
• Coefficients for var1 (var name)
• Coefficients for var2 (var name)
• Shows the results of two models
• r2 for the second model
• The variables in the second model (math and verbal SAT)
• Model 2
Statistics for the Social Sciences
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Multiple Regression in SPSS
• The variables in the first model (math SAT)• Shows the results of two models
• 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?
Statistics for the Social Sciences
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).