FI=LAB9908
Lab:
PART 1 Taking Care of Outliers
PART 2: Examining Curvilinear Prediction
PART 3: Examine Indicators of Multicollinearity
PART 1
OBJECTIVES. In this part of the lab you will
1) look at some simple regression results
2) look at residual output to decide if you have come cases that are strongly influencing the regression
3) re-run the regression leaving that case/those cases out, to see if it makes a difference
EXAMPLE:
Predicting 1985 State-Level Reported Property Crime Rates Using AVGPAY
In this example we go ahead to do something about problems revealed by our analysis of residuals.
Our independent variable is average yearly pay in a state in 1985. We will use the square root transform of this variable to make it somewhat more normal We are going to use AVGPAY even though it has high skewness (2.76) and an outlier (AK) because the purpose of this exercise is to show you how a single case influences regression results, and how we can use regression diagnostics to fix things.
Original Simple Regression
First, let's run the regression, and save the the following regression results:
the predicted score
the residual
the leverage
the studentized residual
Here are the results. Let's interpret the b weight and the R squared, and the beta weight.
Block Number 1. Method: Enter AVGPAY
Variable(s) Entered on Step Number
1.. AVGPAY AVERAGE ANNUAL PAY PER WORKER 1983
Multiple R .46669
R Square .21780
Adjusted R Square .20150
Standard Error 911.49191
Analysis of Variance
DF Sum of Squares Mean Square
Regression 1 11104017.41392 11104017.41392
Residual 48 39879239.86608 830817.49721
F = 13.36517 Signif F = .0006
------------------ Variables in the Equation ------------------
Variable B SE B Beta T Sig T
AVGPAY .206358 .056446 .466688 3.656 .0006
(Constant) 876.530150 955.057691 .918 .3633
Residuals Statistics:
Min Max Mean Std Dev N
*PRED 3597.9778 6803.1284 4336.1200 476.0384 50
*ZPRED -1.5506 5.1824 .0000 1.0000 50
*SEPRED 128.9094 687.0144 164.2911 79.8029 50
*ADJPRED 3684.7815 8642.1709 4373.4438 687.4895 50
*RESID -2109.1367 1870.0493 .0000 902.1430 50
*ZRESID -2.3139 2.0516 .0000 .9897 50
*SRESID -2.3375 2.0785 -.0163 1.0335 50
*DRESID -3237.1704 1919.4299 -37.3238 1014.7210 50
*SDRESID -2.4571 2.1561 -.0161 1.0565 50
*MAHAL .0001 26.8570 .9800 3.7710 50
*COOK D .0000 3.5828 .0846 .5051 50
*LEVER .0000 .5481 .0200 .0770 50
Name Contents
---- --------
PRE_1 Predicted Value
RES_1 Residual
SRE_1 Studentized Residual
LEV_1 Leverage
We can see that the maximum leverage of a case is .54. Hamilton tells us that above .2 is possibly risky, and above .5 we should avoid. Let's find this case. Sort your file by the leverage statistic (LEV_1 or LEV_N where N is the number you see in your output), make it a Descending sort, and look at the first case. We see it is ALASKA.
Why does Alaska have such huge leverage? Look at the scatterplot and see if you can see why
OK, so let's drop Alaska from the file. Just put your blinker on farthest left part of row, click to highlight and delete. Be sure to save the file as something with a different name, so you do NOT confuse your 49 state file with your 50 state file. Also, let's manually drop the variables just added from the regression results.
After you have done this, be sure to save the file with a new name and an identifier that let's you clearly know there are only 49 cases in it.
An alternate procedure (not demonstrated here) would be to:
* calculate a new variable, called WEIGHT
* compute it so that is scores 1 for all cases
* go to the offending case, then manually change WEIGHT=1 to WEIGHT=0
* before you do the regression, WEIGHT the data file
If you run the regression results again, they look like this:
Listwise Deletion of Missing Data
Equation Number 1 Dependent Variable.. PROPRA85 PROPERTY CRIME RATE 1985
Block Number 1. Method: Enter AVGPAY
Variable(s) Entered on Step Number
1.. AVGPAY AVERAGE ANNUAL PAY PER WORKER 1983
Multiple R .53884
R Square .29034
Adjusted R Square .27524
Standard Error 867.29292
Analysis of Variance
DF Sum of Squares Mean Square
Regression 1 14464176.63749 14464176.63749
Residual 47 35353259.77068 752197.01640
F = 19.22924 Signif F = .0001
------------------ Variables in the Equation ------------------
Variable B SE B Beta T Sig T
AVGPAY .354773 .080904 .538835 4.385 .0001
(Constant) -1546.900075 1342.344011 -1.152 .2550
End Block Number 1 All requested variables entered.
* * * * M U L T I P L E R E G R E S S I O N * * * *
Equation Number 1 Dependent Variable.. PROPRA85 PROPERTY CRIME RATE 1985
Check your residual output? Do you see any high leverage cases?
* COMMANDS
* YOUR DRIVE DESIGNATIONS WILL BE DIFFERENT
* .
GET
FILE='C:\PCW\GRSTAT99\USDANU11.SAV'.
EXECUTE .
******************************
* NOW DOING THE FIRST REGRESSION
* .
REGRESSION
/MISSING LISTWISE
/STATISTICS COEFF OUTS R ANOVA
/CRITERIA=PIN(.05) POUT(.10)
/NOORIGIN
/DEPENDENT propra85
/METHOD=ENTER avgpay
/SAVE PRED LEVER RESID SRESID .
*********************************
* SORTING CASES BY LEVERAGE
* .
SORT CASES BY
lev_1 (D) .
************************************
* GRAPHING THE REGRESSION
* .
GRAPH
/SCATTERPLOT(BIVAR)=avgpay WITH propra85
/MISSING=LISTWISE .
* Here we manually dropped Alaska, and
* then re-saved the file with different name.
* Also suggest you drop the four variables at the
* right hand side of the file, that were just added
* SAVE FILE W/ A DIFFERENT NAME
* SEE ABOVE ALTERNATE PROCEDURE
* .
SAVE OUTFILE='C:\PCW\GRSTAT99\US49.SAV'
/COMPRESSED.
****************************************
* NOW RUN THE REGRESSION AGAIN
* .
REGRESSION
/MISSING LISTWISE
/STATISTICS COEFF OUTS R ANOVA
/CRITERIA=PIN(.05) POUT(.10)
/NOORIGIN
/DEPENDENT propra85
/METHOD=ENTER avgpay
/SAVE PRED LEVER RESID SRESID .
****************************************
* LOOK AT RESULTS AGAIN
* .
GRAPH
/SCATTERPLOT(BIVAR)=avgpay WITH propra85
/MISSING=LISTWISE .
PART 2
Examining Curvilinear Regression
OBJECTIVES
* To see if a variable has a curvilinear impact. What does this mean? It means the impacts of X on Y EITHER "flatten out" at high values of X or they "accelerate" at high values of X.
* To use the F test of R squared change to see if the curvilinear impact is significant.
BACKGROUND
The outcome is the 1990 imprisonment rate in states per 100K population. The predictor is the proportion of the population aged 18 to 24. Although this is a bit "young" for imprisonment prone ages, we are going to use it anyway.
Here is the basic plot of X and Y:
STEPS OVERVIEW
1. Run the simple regression
2. Create the age squared term using compute statements
3. Run a regression with both the age and the age squared term. Tell SPSS to enter the predictors separately - the age term in the first block, the age squared term in the second block.
4. Inspect the R squared change and the F test of the R squared change
* ACTUAL SYNTAX TO USE
* FILE=LAB08PT2
* FOR LAB CARRIED OUT ON 04-08-99
* USES ECOLOGICAL DATA FILE TO CREATE A SQUARED TERM FOR THE
* PROPORTION 18-24 IN THE POPULATION IN 1990
* AND THEN EXAMINES THE IMPACT OF THE LINEAR AND THE SQUARED TERM ON THE
* OUTCOME, WHICH IS THE 1990 IMPRISONMENT RATES
**************************************************
* STEP 1: GET THE FILE AND DO THE FIRST REGRESSION
* NOTE: YOU WILL NEED TO CHANGE WHERE YOU TELL IT TO GET THE DATA FILE
* AS AN INTERPRETIVE AID, YOU ALSO SHOULD RUN THE SCATTERPLOT OF Y*X
* ALTHOUGH I DO NOT SHOW YOU THE SYNTAX FOR DOING THAT HERE
**************************************************
* .
GET
FILE='G:\gradstat\USDANU12.SAV)'.
REGRESSION
EXECUTE .
/MISSING LISTWISE
/STATISTICS COEFF OUTS R ANOVA
/CRITERIA=PIN(.05) POUT(.10)
/NOORIGIN
/DEPENDENT prisra90
/METHOD=ENTER ag182490 .
**************************************************
* OF PARTICULAR INTEREST IN THE ABOVE RESULTS IS JUST THE R SQUARED, AND THE
* F TEST OF THE R SQUARED
* NOW YOU WANT TO COMPUTE THE NEW CURVILINEAR TERM
**************************************************
* .
COMPUTE SQ1824 = ag182490*ag182490 .
EXECUTE .
**************************************************
* IF YOU WANTED, JUST FOR DUCKS AND TO BETTER UNDERSTAND THIS
* NEW VARIABLE, YOU COULD PLOT IT AGAINST AG182490
* NOW WE WANT TO TEST IF THERE IS A CURVILINEAR RELATIONSHIP
* IN PARTICULAR YOU WANT TO BE SURE TO REQUEST THE STATISTIC FOR
* R SQUARED CHANGE
* YOU ALSO WANT TO BE SURE TO ENTER THE VARIABLES IN TWO DIFFERENT SETS
* THIS IS NOT SOMETHING WE HAVE DONE BEFORE
* IN BLOCK ONE IS THE LINEAR TERM
* IN BLOCK 2 IS THE QUADRATIC TERM
* HERE IS THE SYNTAX FOR SPSSW 6
* MAY DIFFER IN LATER VERSIONS
* IT IS ACTUALLY RATHER A CHORE TO GET SPSS TO GIVE YOU THE R SQUARED CHANGE
* THE EXACT SYNTAX BELOW WILL DO IT
* I TRIED USING A DIFFERENT SYNTAX THAT WAS SUPPOSED TO DO IT, BUT IT DID NOT
* WORK - RECOMMEND THAT YOU TYPE THESE COMMANDS INTO THE SYNTAX BOX
* ON YOUR OWN, AND RUN IN SPSS V 6
* YOU WANT TO LOOK IN PARTICULAR AT:
* R SQUARE CHANGE
* F CHANGE
************************
* .
REGRESSION
/MISSING MEANSUB
/STATISTICS COEFF OUTS R ANOVA CHA LINE COLLIN TOL
/CRITERIA=PIN(.10) POUT(.15)
/NOORIGIN
/DEPENDENT prisra90
/METHOD=ENTER ag182490 /METHOD=ENTER sq1824 .
PART 3
Collinearity Diagnostics
PURPOSE
When predictors correlate too highly with one another, it is difficult for multiple regression to pull things apart. B weights may change sign unexpectedly. See the note file for more details. Here you are going to do a series of regressions, adding in more predictors, and examining both the tolerance at each step, and the changes in b weights and standard errors of b weights.
In the regression the outcome is VIOLRA90. The predictor we are interested in, to see how it shifts, is PROPRA85.
In the regressions, be sure to ask for collinearity statistics.
***************
* predicting violra90
* with just violra85 and imprisonment rate
***************
* .
REGRESSION
/DESCRIPTIVES MEAN STDDEV CORR SIG N
/MISSING MEANSUB
/STATISTICS COEFF OUTS R ANOVA COLLIN TOL
/CRITERIA=PIN(.05) POUT(.10)
/NOORIGIN
/DEPENDENT violra90
/METHOD=ENTER violra85 prisra85 .
******************************
* adding property crime rate and average pay
******************************
* .
REGRESSION
/DESCRIPTIVES MEAN STDDEV CORR SIG N
/MISSING MEANSUB
/STATISTICS COEFF OUTS R ANOVA COLLIN TOL
/CRITERIA=PIN(.05) POUT(.10)
/NOORIGIN
/DEPENDENT violra90
/METHOD=ENTER violra85 prisra85 propra85 avgpay .
*******************************
* now adding bunches more
* all theoretically driven, of course
*******************************
* .
REGRESSION
/DESCRIPTIVES MEAN STDDEV CORR SIG N
/MISSING MEANSUB
/STATISTICS COEFF OUTS R ANOVA COLLIN TOL
/CRITERIA=PIN(.05) POUT(.10)
/NOORIGIN
/DEPENDENT violra90
/METHOD=ENTER violra85 prisra85 propra85 avgpay allbooz hsrate82 fdstrat2
divrate2 mwest south.