FrontPage Outliers

1. Outliers e.g.,

This is further reading for detecting outliers, adopted from http://www.ats.ucla.edu/stat/spss/webbooks/reg/chapter2/spssreg2.htm .

@crime.sav (3.91 KB)
@outlierCheck.sps (2.53 KB)


get file = "DirectoryOfYourComputer\crime.sav".

descriptives
  /var=crime murder pctmetro pctwhite pcths poverty single.

		Descriptive Statistics
			N	Minimum	Maximum	Mean	Std. Deviation
violent crime rate	51	82	2922	612.84	441.100
murder rate		51	1.60	78.50	8.7275	10.71758
pct metropolitan	51	24.00	100.00	67.3902	21.95713
pct white		51	31.80	98.50	84.1157	13.25839
pct hs graduates	51	64.30	86.60	76.2235	5.59209
pct poverty		51	8.00	26.40	14.2588	4.58424
pct single parent	51	8.40	22.10	11.3255	2.12149
Valid N (listwise)	51				

graph
  /scatterplot(matrix)=crime murder pctmetro pctwhite pcths poverty single .

r.crime.scatterplot.for.all.variables.jpg
scatterplot for all variables [JPG image (141.43 KB)]


GRAPH /SCATTERPLOT(BIVAR)=pctmetro WITH crime BY state(name) .
r.crime.scatterplot.for.crime.by.state.jpg
scatterplot of pcmetro by crime by state [JPG image (24.93 KB)]


GRAPH /SCATTERPLOT(BIVAR)=poverty WITH crime BY state(name) .
r.crime.scatterplot.for.poverty.by.state.jpg
scatterplot of poverty by state [JPG image (23.89 KB)]


GRAPH /SCATTERPLOT(BIVAR)=single WITH crime BY state(name) .
r.crime.scatterplot.for.single.by.state.jpg
scatterplot of single by state [JPG image (26.6 KB)]


regression
  /dependent crime
  /method=enter pctmetro poverty single.

		Model Summary
Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	.916a	.840	.830	182.068
a. Predictors: (Constant), pct single parent, pct metropolitan, pct poverty

			ANOVA(b)
Model			Sum of Squares	df	Mean Square	F	Sig.
1	Regression	8170480.211	3	2723493.404	82.160	.000a
	Residual	1557994.534	47	33148.820		
	Total		9728474.745	50			
a. Predictors: (Constant), pct single parent, pct metropolitan, pct poverty
b. Dependent Variable: violent crime rate

			Coefficients(a)
		Unstandardized Coefficients		Standardized Coefficients
Model				B		Std. Error	Beta	t	Sig.
1	(Constant)		-1666.436	147.852			-11.271	.000
	pct metropolitan	7.829		1.255		.390	6.240	.000
	pct poverty		17.680		6.941		.184	2.547	.014
	pct single parent	132.408		15.503		.637	8.541	.000
a. Dependent Variable: violent crime rate



regression
  /dependent crime
  /method=enter pctmetro poverty single
  /residuals=histogram.

		Model Summary(b)
Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	.916a	.840	.830	182.068
a. Predictors: (Constant), pct single parent, pct metropolitan, pct poverty
b. Dependent Variable: violent crime rate

			ANOVA(b)
Model			Sum of Squares	df	Mean Square	F	Sig.
1	Regression	8170480.211	3	2723493.404	82.160	.000a
	Residual	1557994.534	47	33148.820		
	Total		9728474.745	50			
a. Predictors: (Constant), pct single parent, pct metropolitan, pct poverty
b. Dependent Variable: violent crime rate

			Coefficients(a)
		Unstandardized Coefficients		Standardized Coefficients
Model				B		Std. Error	Beta	t	Sig.
1	(Constant)		-1666.436	147.852		-11.271	.000
	pct metropolitan	7.829		1.255		.390	6.240	.000
	pct poverty		17.680		6.941		.184	2.547	.014
	pct single parent	132.408		15.503		.637	8.541	.000
a. Dependent Variable: violent crime rate

		Residuals Statistics(a)
			Minimum		Maximum		Mean	Std.Deviation	N
Predicted Value	-30.51		2509.43		612.84	404.240		51
Residual		-523.013	426.111		.000	176.522		51
Std. Predicted Value	-1.592		4.692		.000	1.000		51
Std. Residual		-2.873		2.340		.000	.970		51
a. Dependent Variable: violent crime rate
r.crime.residual.histogram.jpg
histogram [JPG image (26.38 KB)]


regression
  /dependent crime
  /method=enter pctmetro poverty single
  /residuals=histogram(sdresid).
r.crime.residual.histogram.sdresidual.jpg
histogram sdresid [JPG image (25.2 KB)]


regression
  /dependent crime
  /method=enter pctmetro poverty single
  /residuals=histogram(sdresid) id(state) outliers(sdresid).
see at http://www2.bc.edu/~stevenw/MB875/mb875_Analyzing Residuals.htm for sdresid (studentized deleted residuals).
		Residuals Statistics(a)
					Minimum		Maximum		Mean	Std. Deviation	N
Predicted Value				-30.51		2509.43		612.84	404.240		51
Std. Predicted Value			-1.592		4.692		.000	1.000		51
Standard Error of Predicted Value	25.788		133.343		47.561	18.563		51
Adjusted Predicted Value		-39.26		2032.11		605.66	369.075		51
Residual				-523.013	426.111		.000	176.522		51
Std. Residual				-2.873		2.340		.000	.970		51
Stud. Residual				-3.194		3.328		.015	1.072		51
Deleted Residual			-646.503	889.885		7.183	223.668		51
Stud. Deleted Residual		-3.571		3.766		.018	1.133		51
Mahal. Distance			.023		25.839		2.941	4.014		51
Cook's Distance			.000		3.203		.089	.454		51
Centered Leverage Value		.000		.517		.059	.080		51
a. Dependent Variable: violent crime rate


regression
  /dependent crime
  /method=enter pctmetro poverty single
  /residuals=histogram(sdresid) id(state) outliers(sdresid)
  /casewise=plot(sdresid) outliers(2)  .

		Casewise Diagnostics(a)
Case Number	state	Stud. Deleted 	violent crime 	Predicted 	Residual
			Residual	rate		Value
9		fl	2.620		1206		779.89		426.111
25		ms	-3.571		434		957.01		-523.013
51		dc	3.766		2922		2509.43		412.566
a. Dependent Variable: violent crime rate

regression
  /dependent crime
  /method=enter pctmetro poverty single
  /residuals=histogram(sdresid lever) id(state) outliers(sdresid lever)
  /casewise=plot(sdresid) outliers(2).
		Outlier Statistics(a)
				Case 	state	Statistic
				Number
Stud. Deleted Residual	1	51	dc 	3.766
			2	25	ms 	-3.571
			3	9	fl 	2.620
			4	18	la 	-1.839
			5	39	ri 	-1.686
			6	12	ia 	1.590
			7	47	wa 	-1.304
			8	13	id 	1.293
			9	14	il 	1.152
			10	35	oh 	-1.148
Centered Leverage Value	1	51	dc 	.517
			2	1	ak 	.241
			3	25	ms 	.171
			4	49	wv 	.161
			5	18	la 	.146
			6	46	vt 	.117
			7	9	fl 	.083
			8	26	mt 	.080
			9	31	nj 	.075
			10	17	ky 	.072
a. Dependent Variable: violent crime rate

r.crime.residual.histogram.sdresidual.jpg
histogram sdresid [JPG image (25.2 KB)]

r.crime.residual.histogram.leverage.outlierl.jpg
histogram leverage [JPG image (20.5 KB)]


regression
  /dependent crime
  /method=enter pctmetro poverty single
  /residuals=histogram(sdresid lever) id(state) outliers(sdresid, lever)
  /casewise=plot(sdresid)  outliers(2)
  /scatterplot(*lever, *sdresid).
r.crime.residual.scatterplot.leverage.sdresid.jpg
histogram sdresid [JPG image (26.07 KB)]


regression
  /dependent crime
  /method=enter pctmetro poverty single
  /residuals=histogram(sdresid lever) id(state) outliers(sdresid, lever, cook)
  /casewise=plot(sdresid)  outliers(2) cook dffit
  /scatterplot(*lever, *sdresid).
		Casewise Diagnostics(a)
Case Number	state	Stud. 		violent 	Cook's		DFFIT
			Deleted		crime		Distance	
			Residual	rate
9	fl		2.620		1206		.174		48.507
25	ms		-3.571		434		.602		-123.490
51	dc		3.766		2922		3.203		477.319
a. Dependent Variable: violent crime rate

		Outlier Statistics(a)
		Case Number	state	Statis	Sig. F
Stud.  		1	51	dc 	3.766	
Deleted		2	25	ms 	-3.571	
Residual	3	9	fl 	2.620	
		4	18	la 	-1.839	
		5	39	ri 	-1.686	
		6	12	ia 	1.590	
		7	47	wa 	-1.304	
		8	13	id 	1.293	
		9	14	il 	1.152	
		10	35	oh 	-1.148	
Cook's 		1	51	dc 	3.203	.021
Distance	2	25	ms 	.602	.663
		3	9	fl 	.174	.951
		4	18	la 	.159	.958
		5	39	ri 	.041	.997
		6	12	ia 	.041	.997
		7	13	id 	.037	.997
		8	20	md 	.020	.999
		9	6	co 	.018	.999
		10	49	wv 	.016	.999
Centered  	1	51	dc 	.517	
Leverage	2	1	ak 	.241	
Value		3	25	ms 	.171	
		4	49	wv 	.161	
		5	18	la 	.146	
		6	46	vt 	.117	
		7	9	fl 	.083	
		8	26	mt 	.080	
		9	31	nj 	.075	
		10	17	ky 	.072	
a. Dependent Variable: violent crime rate


regression
  /dependent crime
  /method=enter pctmetro poverty single
  /residuals=histogram(sdresid lever) id(state) outliers(sdresid, lever, cook)
  /casewise=plot(sdresid)  outliers(2) cook dffit
  /scatterplot(*lever, *sdresid)
  /save sdbeta(sdfb).
list
  /variables state sdfb1 sdfb2 sdfb3
  /cases from 1 to 10.
state       sdfb1       sdfb2       sdfb3

ak        -.10618     -.13134      .14518
al         .01243      .05529     -.02751
ar        -.06875      .17535     -.10526
az        -.09476     -.03088      .00124
ca         .01264      .00880     -.00364
co        -.03705      .19393     -.13846
ct        -.12016      .07446      .03017
de         .00558     -.01143      .00519
fl         .64175      .59593     -.56060
ga         .03171      .06426     -.09120


Number of cases read:  10    Number of cases listed:  10


VARIABLE LABLES sdfb1 "Sdfbeta pctmetro"
                              /sdfb2 "Sdfbeta poverty"
                              /sdfb3 "Sdfbeta single" .

GRAPH
  /SCATTERPLOT(OVERLAY)=sid sid sid  WITH sdfb1 sdfb2 sdfb3 (PAIR) BY state(name)
  /MISSING=LISTWISE .
r.crime.residual.scatterplot.dbfBeta.jpg
dbfBeta value [JPG image (33.11 KB)]

Note
MeasureValue
leverage >(2k+2)/n
abs(rstu) > 2
Cook's D > 4/n
abs(DFBETA) > 2/sqrt(n)

PRED
Unstandardized predicted values.
RESID
Unstandardized residuals.
DRESID
Deleted residuals.
ADJPRED
Adjusted predicted values.
ZPRED
Standardized predicted values.
ZRESID
Standardized residuals.
SRESID
Studentized residuals.
SDRESID
Studentized deleted residuals.
SEPRED
Standard errors of the predicted values.
MAHAL
Mahalanobis distances.
COOK
Cook’s distances.
LEVER
Centered leverage values.
DFBETA
Change in the regression coefficient that results from the deletion of the ith case. A DFBETA value is computed for each case for each regression coefficient generated by a model.
SDBETA
Standardized DFBETA. An SDBETA value is computed for each case for each regression coefficient generated by a model.
DFFIT
Change in the predicted value when the ith case is deleted.
SDFIT
Standardized DFFIT.
COVRATIO
Ratio of the determinant of the covariance matrix with the ith case deleted to the determinant of the covariance matrix with all cases included.
MCIN
Lower and upper bounds for the prediction interval of the mean predicted response. A lowerbound LMCIN and an upperbound UMCIN are generated. The default confidence interval is 95%. The confidence interval can be reset with the CIN subcommand. (See Dillon & Goldstein
ICIN
Lower and upper bounds for the prediction interval for a single observation. A lowerbound LICIN and an upperbound UICIN are generated. The default confidence interval is 95%. The confidence interval can be reset with the CIN subcommand. (See Dillon & Goldstein



regression
  /dependent crime
  /method=enter pctmetro poverty single
  /residuals=histogram(sdresid lever) id(state) outliers(sdresid, lever, cook)
  /casewise=plot(sdresid)  outliers(2) cook dffit
  /scatterplot(*lever, *sdresid)
  /partialplot.  
r.crime.regression.outlier.01.jpg
[JPG image (28.2 KB)]

r.crime.regression.outlier.02.jpg
[JPG image (28.53 KB)]

r.crime.regression.outlier.03.jpg
[JPG image (27.35 KB)]


regression
  /dependent crime
  /method=enter pctmetro poverty single.

			Coefficients(a)
		Unstandardized Coefficients		Standardized Coefficients
Model		B	Std. Error	Beta	t	Sig.
1	(Constant)	-1666.436	147.852		-11.271	.000
	pct metropolitan	7.829	1.255	.390	6.240	.000
	pct poverty	17.680	6.941	.184	2.547	.014
	pct single parent	132.408	15.503	.637	8.541	.000
a. Dependent Variable: violent crime rate

compute filtvar = (state NE "dc").
filter by filtvar.
regression
  /dependent crime
  /method=enter pctmetro poverty single . 


			Coefficients(a)
		Unstandardized Coefficients		Standardized Coefficients
Model		B	Std. Error	Beta	t	Sig.
1	(Constant)	-1197.538	180.487		-6.635	.000
	pct metropolitan	7.712	1.109	.565	6.953	.000
	pct poverty	18.283	6.136	.265	2.980	.005
	pct single parent	89.401	17.836	.446	5.012	.000
a. Dependent Variable: violent crime rate

2. e.g., 2

redirected from . . . multiple regression.
@elemapi2.sav (28.49 KB)
@r.api00.OutlierDetection.sps (1.24 KB)

2.1. inspection

descriptives /var= ALL .

Descriptive Statistics
N Minimum Maximum Mean Std. Deviation
api 2000 400 369 940 647.62 142.249
english language learners 400 0 91 31.45 24.839
avg class size k-3 398 14 25 19.16 1.369
avg parent ed 381 1.00 4.62 2.6685 .76379
pct free meals 400 0 100 60.32 31.912
Valid N (listwise) 379

graph
  /scatterplot(matrix)=api00 ell acs_k3 avg_ed meals .
r.graph.whole.jpg
[JPG image (167.94 KB)]

This graph does not give any suspicious cases.
GRAPH /SCATTERPLOT(BIVAR)=ell with api00 .
GRAPH /SCATTERPLOT(BIVAR)=acs_k3 with api00  .
GRAPH /SCATTERPLOT(BIVAR)=avg_ed with api00 .
GRAPH /SCATTERPLOT(BIVAR)=meals with api00  .
r.01.jpg
ell [JPG image (129.45 KB)]
r.02.jpg
acsk3 [JPG image (77.27 KB)]
r.03.jpg
ave_ed [JPG image (117.17 KB)]
r.04.jpg
meals [JPG image (119.43 KB)]

We speculate that the second IV (average class size) is not quite related to DV (api00). And, there seems no particular suspicious data.


REGRESSION
  /DEPENDENT api00
  /METHOD=ENTER ell acs_k3 avg_ed meals 
   /residuals=histogram(sdresid lever) id(snum) outliers(sdresid, lever, cook)
  /casewise=plot(sdresid)  outliers(2) cook dffit
  /scatterplot(*lever, *sdresid)
  /save sdbeta(sdfb) 
   /partialplot.  

Model Summary
Model R R Square Adjusted
R Square
Std. Error
of the Estimate
1 .912a .833 .831 58.633
a. Predictors: (Constant), pct free meals, avg class size k-3, english language learners, avg parent ed

ANOVA(b)
Model Sum of Squares df Mean Square F Sig.
1 Regression 6393719.254 4 1598429.813 464.956 .000a
Residual 1285740.498 374 3437.809
Total 7679459.752 378
a. Predictors: (Constant), pct free meals, avg class size k-3, english language learners, avg parent ed
b. Dependent Variable: api 2000

Coefficients(a)
Unstandardized
Coefficients
Standardized
Coefficients
Model B Std. Error Beta t Sig.
1 (Constant) 709.639 56.240 12.618 .000
english language learners -.843 .196 -.147 -4.307 .000
avg class size k-3 3.388 2.333 .032 1.452 .147
avg parent ed 29.072 6.924 .156 4.199 .000
pct free meals -2.937 .195 -.655 -15.081 .000
a. Dependent Variable: api 2000

Casewise Diagnostics(a)
Case Number school number Stud. Deleted
Residual
api 2000 Cook's
Distance
DFFIT
93 1497 2.170 604 .010 1.292
97 1539 2.230 700 .006 .826
100 1515 2.222 667 .005 .661
105 1516 2.128 597 .010 1.380
135 1633 2.072 584 .044 6.085
188 1731 2.121 719 .015 2.126
203 1621 2.034 717 .006 .831
226 211 -3.241 386 .015 -1.325
227 182 -2.653 411 .005 -.581
228 167 2.903 774 .010 .987
232 210 -2.369 432 .018 -2.263
234 165 -2.734 449 .019 -1.997
252 3700 2.036 717 .013 1.878
259 3537 -2.425 694 .012 -1.436
271 3758 3.012 690 .022 2.108
272 3794 2.083 610 .010 1.400
274 3759 -2.290 585 .069 -8.646
304 4507 2.011 751 .013 1.917
327 4737 2.470 808 .012 1.447
334 4744 2.160 700 .005 .645
346 5362 -2.138 487 .010 -1.359
a. Dependent Variable: api 2000

Residuals Statistics(a)
Minimum Maximum Mean Std. Deviation N
Predicted Value 449.17 910.04 647.64 130.056 379
Std. Predicted Value -1.526 2.018 .000 1.000 379
Standard Error of Predicted Value 3.218 14.681 6.496 1.780 379
Adjusted Predicted Value 449.44 909.36 647.65 130.056 379
Residual -187.020 173.697 .000 58.322 379
Std. Residual -3.190 2.962 .000 .995 379
Stud. Residual -3.201 2.980 .000 1.002 379
Deleted Residual -188.345 175.805 -.016 59.138 379
Stud. Deleted Residual -3.241 3.012 .000 1.005 379
Mahal. Distance .141 22.702 3.989 3.030 379
Cook's Distance .000 .069 .003 .006 379
Centered Leverage Value .000 .060 .011 .008 379
a. Dependent Variable: api 2000

Outlier Statistics(a)
Case Number school number Statistic Sig. F
Stud. Deleted Residual 1 226 211 -3.241
2 271 3758 3.012
3 228 167 2.903
4 234 165 -2.734
5 227 182 -2.653
6 327 4737 2.470
7 259 3537 -2.425
8 232 210 -2.369
9 274 3759 -2.290
10 97 1539 2.230
Cook's Distance 1 274 3759 .069 .997
2 135 1633 .044 .999
3 26 4299 .030 1.000
4 193 1952 .025 1.000
5 271 3758 .022 1.000
6 234 165 .019 1.000
7 232 210 .018 1.000
8 200 1872 .018 1.000
9 108 1606 .018 1.000
10 388 4878 .017 1.000
Centered Leverage Value 1 274 3759 .060
2 37 4308 .058
3 209 1795 .050
4 135 1633 .046
5 26 4299 .040
6 69 3000 .037
7 372 6068 .036
8 30 4317 .035
9 147 1709 .035
10 193 1952 .033
a. Dependent Variable: api 2000

r.api.histogram.sdresid.jpg
sdresidual check [JPG image (29.53 KB)]

r.api.histogram.leverage.jpg
leverage check [JPG image (22.84 KB)]


r.api.regression.predbyresi.01.jpg
"plot spred by sresid [JPG image (62.22 KB)]

2.2. Outlier dection

Let's say, we decide to opt out cases whose studentized deleted residual value exceed normal. We set the criterion as ABS(sdresid) > 2. These cases which meet this criterion will filtered out.

We need to save some residual statistics first, with regression method. Saved values include:
PRED
ZPRED
MAHAL
COOK
LEVER
RESID
ZRESID
SDRESID
DFBETA
Among them, we take a look at SDRESID, whose variable name will be SDR_1 in spss data set.

For the referece,

Note: outlier detection
MeasureValue
leverage >(2k+2)/n 0.021108179
abs(rstu) > 2 2
Cook's D > 4/n 0.01055409
abs(DFBETA) > 2/sqrt(n) 0.102733099



REGRESSION
  /DESCRIPTIVES MEAN STDDEV CORR SIG N
  /MISSING LISTWISE
  /STATISTICS COEFF OUTS R ANOVA COLLIN TOL CHANGE ZPP
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN 
  /DEPENDENT api00
  /METHOD=ENTER meals ell acs_k3 avg_ed
  /residuals=histogram(sdresid lever) id(snum) outliers(sdresid, lever, cook) Durbin
  /casewise=plot(sdresid)  outliers(2) cook dffit
  /SCATTERPLOT=(*ZRESID ,*ZPRED)
  /SAVE PRED ZPRED MAHAL COOK LEVER RESID ZRESID SDRESID DFBETA.
Then, we need to filter out cases whose SDR_1 value exceed:
abs(SDR_1) > 2
with the below command.
USE ALL.
COMPUTE filterVar=(abs(SDR)_1 < 2).
FILTER BY filterVar.
EXECUTE.

Then, we do regression again, excluding the suspicious cases. But, this time we do not save the residuals.
REGRESSION
  /DESCRIPTIVES MEAN STDDEV CORR SIG N
  /MISSING LISTWISE
  /STATISTICS COEFF OUTS R ANOVA CHANGE ZPP
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN 
  /DEPENDENT api00
  /METHOD=ENTER ell avg_ed acs_k3 meals
  /SCATTERPLOT=(*ZRESID ,*ZPRED) .

Compare the ouptput between the previous and this regression.

Model Summaryb
Model R R
Square
Adjusted
R Square
Std. Error of
the Estimate
Change
Statistics
R Square Change F Change df1 df2 Sig. F Change
1 .938a .880 .879 49.914 .880 649.458 4 353 .000

ANOVAb
Model Sum of
Squares
df Mean
Square
F Sig.
1 Regression 6472284.822 4 1618071.206 649.458 .000a
Residual 879470.664 353 2491.418
Total 7351755.486 357

Coefficientsa
Model Unstandardized
Coefficients
Standardized
Coefficients
t Sig. Correlations
B Std. Error Beta Zero-order Partial Part
1 (Constant) 705.495 51.072 13.814 .000
ell -.915 .170 -.160 -5.374 .000 -.789 -.275 -.099
avg_ed 25.661 6.061 .138 4.234 .000 .809 .220 .078
acs_k3 4.452 2.127 .040 2.093 .037 .204 .111 .039
meals -3.056 .171 -.683 -17.868 .000 -.928 -.689 -.329




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