**Coefficients(a)**
Model		Unstandardized Coefficients	Standardized Coefficients	t	Sig.	95% Confidence Interval for B	Correlations	Collinearity Statistics
B	Std. Error	Beta	Lower Bound	Upper Bound	Zero-order	Partial	Part	Tolerance	VIF	B	Std. Error
1	(Constant)	3.949	1.014		3.893	.000	1.958	5.939
AGE WHEN FIRST MARRIED	.001	.022	.001	.043	.966	-.043	.045	.102	.001	.001	.899	1.113
AGE OF RESPONDENT	.014	.008	.054	1.638	.102	-.003	.030	.012	.057	.053	.967	1.034
HIGHEST YEAR OF SCHOOL COMPLETED	.407	.076	.379	5.330	.000	.257	.556	.344	.182	.173	.209	4.777
RS HIGHEST DEGREE	-.085	.183	-.033	-.464	.643	-.443	.274	.301	-.016	-.015	.209	4.793
a Dependent Variable: RESPONDENTS INCOME

**Collinearity Diagnostics(a)**
Model	Dimension	Eigenvalue	Condition Index	Variance Proportions
(Constant)	AGE WHEN FIRST MARRIED	AGE OF RESPONDENT	HIGHEST YEAR OF SCHOOL COMPLETED	RS HIGHEST DEGREE	(Constant)	AGE WHEN FIRST MARRIED
1	1	4.638	1.000	.00	.00	.00	.00	.00
2	.279	4.074	.00	.00	.04	.00	.19
3	.055	9.223	.01	.15	.80	.01	.05
4	.024	13.998	.08	.80	.05	.09	.04
5	.005	31.365	.91	.05	.11	.90	.72
a Dependent Variable: RESPONDENTS INCOME

**Casewise Diagnostics(a)**
Case Number	Std. Residual	RESPONDENTS INCOME	Predicted Value	Residual
110	-3.225	1	10.20	-9.199
411	-3.278	1	10.35	-9.350
767	-3.310	1	10.44	-9.442
789	-3.112	1	9.88	-8.878
878	-3.415	2	11.74	-9.741
933	-3.510	1	11.01	-10.013
1115	-3.250	1	10.27	-9.272
1174	-3.104	1	9.85	-8.853
1269	-3.713	1	11.59	-10.591
a Dependent Variable: RESPONDENTS INCOME

**Residuals Statistics(a)**
	Minimum	Maximum	Mean	Std. Deviation	N
Predicted Value	4.67	12.67	9.94	1.056	837
Std. Predicted Value	-4.988	2.592	.000	1.000	837
Standard Error of Predicted Value	.113	.795	.208	.072	837
Adjusted Predicted Value	4.81	12.68	9.94	1.057	837
Residual	-10.591	6.356	.000	2.846	837
Std. Residual	-3.713	2.228	.000	.998	837
Stud. Residual	-3.762	2.286	.000	1.001	837
Deleted Residual	-10.875	6.691	-.002	2.865	837
Stud. Deleted Residual	-3.792	2.292	-.001	1.003	837
Mahal. Distance	.310	63.998	3.995	4.443	837
Cook's Distance	.000	.110	.001	.006	837
Centered Leverage Value	.000	.077	.005	.005	837
a Dependent Variable: RESPONDENTS INCOME

SPSS Regression Output

Notes This example is from SPSS 15 for the file "gss93.sav". The dependent is "rincome91" (respondent's income), The independents are age, agewed, degree, and educ.

To obtain this output:

File, Open, point to gss93.sav.
Analyze, Regression, Linear
In the Regression dialog box, select "rincome91" as the "dependent", and as independents select age, agewed, degree, and educ. In the Statistics and Plots dialog boxes, check all output options and ask for plots of SDRESID by ZPRED,and ZPRED by DEPENDENT.

Comments in blue are by the instructor and are not part of SPSS output.

Regression

REGRESSION
  /DESCRIPTIVES MEAN STDDEV CORR SIG N
  /MISSING LISTWISE
  /STATISTICS COEFF OUTS CI BCOV R ANOVA COLLIN TOL CHANGE ZPP
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN
  /DEPENDENT rincome
  /METHOD=ENTER agewed age educ degree
  /PARTIALPLOT ALL
  /SCATTERPLOT=(*SDRESID ,*ZPRED ) (*ZPRED ,rincome )
  /RESIDUALS DURBIN HIST(ZRESID) NORM(ZRESID)
  /CASEWISE PLOT(ZRESID) OUTLIERS(3) .

Regression

**Notes**
Output Created		29-JAN-2007 16:51:32
Comments
Input	Data	I:\PC\DATASETS\GSS\gss93\GSS93.SAV
	Active Dataset	DataSet1
	Filter	<none>
	Weight	<none>
	Split File	<none>
	N of Rows in Working Data File	1606
Missing Value Handling	Definition of Missing	User-defined missing values are treated as missing.
Missing Value Handling	Cases Used	Statistics are based on cases with no missing values for any variable used.
Syntax		REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF OUTS CI BCOV R ANOVA COLLIN TOL CHANGE ZPP /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT rincome /METHOD=ENTER agewed age educ degree /PARTIALPLOT ALL /SCATTERPLOT=(SDRESID ,ZPRED ) (*ZPRED ,rincome ) /RESIDUALS DURBIN HIST(ZRESID) NORM(ZRESID) /CASEWISE PLOT(ZRESID) OUTLIERS(3) .
Resources	Elapsed Time	0:00:06.61
	Memory Required	15276 bytes
	Additional Memory Required for Residual Plots	2768 bytes
	Processor Time	0:00:04.88

[DataSet1] I:\PC\DATASETS\GSS\gss93\GSS93.SAV

**Descriptive Statistics**
	Mean	Std. Deviation	N
RESPONDENTS INCOME	9.94	3.035	837
AGE WHEN FIRST MARRIED	22.79	4.660	837
AGE OF RESPONDENT	43.13	12.028	837
HIGHEST YEAR OF SCHOOL COMPLETED	13.56	2.827	837
RS HIGHEST DEGREE	1.61	1.182	837

The correlation matrix below shows the Pearsonian r's, the significance of each r, and the sample size (n) for each r. All correlations are significant at the .05 level or better, except age with respondent's income, age when first married, and respondent's highest degree. At r = .886, there may be multicollinearity between highest year of education completed and highest degree.

Correlations

RESPONDENTS INCOME AGE WHEN FIRST MARRIED AGE OF RESPONDENT HIGHEST YEAR OF SCHOOL COMPLETED RS HIGHEST DEGREE

Pearson Correlation RESPONDENTS INCOME 1.000 .102 .012 .344 .301

AGE WHEN FIRST MARRIED .102 1.000 .039 .286 .312

AGE OF RESPONDENT .012 .039 1.000 -.114 -.042

HIGHEST YEAR OF SCHOOL COMPLETED .344 .286 -.114 1.000 .886

RS HIGHEST DEGREE .301 .312 -.042 .886 1.000

Sig. (1-tailed) RESPONDENTS INCOME . .002 .362 .000 .000

AGE WHEN FIRST MARRIED .002 . .127 .000 .000

AGE OF RESPONDENT .362 .127 . .000 .113

HIGHEST YEAR OF SCHOOL COMPLETED .000 .000 .000 . .000

RS HIGHEST DEGREE .000 .000 .113 .000 .

N RESPONDENTS INCOME 837 837 837 837 837

AGE WHEN FIRST MARRIED 837 837 837 837 837

AGE OF RESPONDENT 837 837 837 837 837

HIGHEST YEAR OF SCHOOL COMPLETED 837 837 837 837 837

RS HIGHEST DEGREE 837 837 837 837 837

**Correlations**
		RESPONDENTS INCOME	AGE WHEN FIRST MARRIED	AGE OF RESPONDENT	HIGHEST YEAR OF SCHOOL COMPLETED	RS HIGHEST DEGREE
Pearson Correlation	RESPONDENTS INCOME	1.000	.102	.012	.344	.301
AGE WHEN FIRST MARRIED	.102	1.000	.039	.286	.312
AGE OF RESPONDENT	.012	.039	1.000	-.114	-.042
HIGHEST YEAR OF SCHOOL COMPLETED	.344	.286	-.114	1.000	.886
RS HIGHEST DEGREE	.301	.312	-.042	.886	1.000
Sig. (1-tailed)	RESPONDENTS INCOME	.	.002	.362	.000	.000
AGE WHEN FIRST MARRIED	.002	.	.127	.000	.000
AGE OF RESPONDENT	.362	.127	.	.000	.113
HIGHEST YEAR OF SCHOOL COMPLETED	.000	.000	.000	.	.000
RS HIGHEST DEGREE	.000	.000	.113	.000	.
N	RESPONDENTS INCOME	837	837	837	837	837
AGE WHEN FIRST MARRIED	837	837	837	837	837
AGE OF RESPONDENT	837	837	837	837	837
HIGHEST YEAR OF SCHOOL COMPLETED	837	837	837	837	837
RS HIGHEST DEGREE	837	837	837	837	837

Below: because stepwise regression was not requested, SPSS entered all independent variables in a single step. Income is shown as the dependent variable.

Variables Entered/Removed(b)
Model Variables Entered Variables Removed Method

1 RS HIGHEST DEGREE, AGE OF RESPONDENT, AGE WHEN FIRST MARRIED, HIGHEST YEAR OF SCHOOL COMPLETED(a) . Enter

a All requested variables entered.
b Dependent Variable: RESPONDENTS INCOME

**Variables Entered/Removed(b)**
Model	Variables Entered	Variables Removed	Method
1	RS HIGHEST DEGREE, AGE OF RESPONDENT, AGE WHEN FIRST MARRIED, HIGHEST YEAR OF SCHOOL COMPLETED(a)	.	Enter
a All requested variables entered.
b Dependent Variable: RESPONDENTS INCOME

The table below is the "bottom line."

R-squared is the percent of the dependent explained by the independents. In this case, the two-independent model (age and highest year of school explaining respondent's income) explains only 12.1% of the variance. This suggests the model was misspecified. Other variables, not suggested by the researcher, explain the bulk of the variance. Moreover, age and highest year may share common variance with these unmeasured variables and in fact part or even all of the observed 12.1% might disappear if these unmeasured variables were entered first in the equation.
Adjusted R-squared is a standard, arbitrary downward adjustment to penalize for the possibility that, with many independents, some of the variance may be due to chance. The more independents, the more the adjustment penalty. Since there are only four independents here, the penalty is minor.
Standard error of estimate is 2.85 units. From the earlier table of descriptive statistics, we can note the the mean respondent's income is 9.94. If on a given case the prediction happened to be 9.94, we could be 95% confident that the actual value would be within plus or minus 1.96*2.85 = 5.59 units -- that is, between about 4.4 and about 15.5. In noting this we are looking to see if two standard errors are the bulk of the entire range of the dependent, since then any predictions using this model will be very poor. Here, since the lower confidence limit dips almost halfway from the mean to zero, we would expect predictions would not be terribly reliable, but not completely worthless either.
The F value for the "Change Statistics" shows the significance level associated with adding the variable or set of variables for that step in stepwise or hierarchical regression, not pertinent here.
The Durbin-Watson statistic tests for serial correlation of error terms for adjacent cases. This test is mainly used in relation to time series data, where adjacent cases are sequential years.

Model Summary(b)
Model R R Square Adjusted R Square Std. Error of the Estimate Change Statistics Durbin-Watson

R Square Change F Change df1 df2 Sig. F Change R Square Change F Change df1 df2 Sig. F Change

1 .348(a) .121 .117 2.852 .121 28.634 4 832 .000 1.980

a Predictors: (Constant), RS HIGHEST DEGREE, AGE OF RESPONDENT, AGE WHEN FIRST MARRIED, HIGHEST YEAR OF SCHOOL COMPLETED
b Dependent Variable: RESPONDENTS INCOME

**Model Summary(b)**
Model	R	R Square	Adjusted R Square	Std. Error of the Estimate	Change Statistics	Durbin-Watson
R Square Change	F Change	df1	df2	Sig. F Change	R Square Change	F Change	df1	df2	Sig. F Change
1	.348(a)	.121	.117	2.852	.121	28.634	4	832	.000	1.980
a Predictors: (Constant), RS HIGHEST DEGREE, AGE OF RESPONDENT, AGE WHEN FIRST MARRIED, HIGHEST YEAR OF SCHOOL COMPLETED
b Dependent Variable: RESPONDENTS INCOME

The ANOVA table below tests the overall significance of the model (that is, of the regression equation). If we had been doing stepwise regression, significance for each step would be computed. Here the significance of the F value is below .05, so the model is significant.

ANOVA(b)
Model
Sum of Squares df Mean Square F Sig.

1 Regression 931.945 4 232.986 28.634 .000(a)

Residual 6769.699 832 8.137

Total 7701.644 836

a Predictors: (Constant), RS HIGHEST DEGREE, AGE OF RESPONDENT, AGE WHEN FIRST MARRIED, HIGHEST YEAR OF SCHOOL COMPLETED
b Dependent Variable: RESPONDENTS INCOME

**ANOVA(b)**
Model		Sum of Squares	df	Mean Square	F	Sig.
1	Regression	931.945	4	232.986	28.634	.000(a)
Residual	6769.699	832	8.137
Total	7701.644	836
a Predictors: (Constant), RS HIGHEST DEGREE, AGE OF RESPONDENT, AGE WHEN FIRST MARRIED, HIGHEST YEAR OF SCHOOL COMPLETED
b Dependent Variable: RESPONDENTS INCOME

The table below gives the b and beta coefficients and other coefficients for the model.

The b coefficients and the constant are used to create the prediction (regression) equation. For this model, predicted rincome91 = ..001*(age when first married) + .014*(age of respondent) + .407*(highest school year) - .085*(respondent's highest degree) + 3.949. However, as noted above in the summary table, the standard error of estimate is 2.85. This means that at the .05 significance level, the estimate is the one from this formula plus or minus 1.96*2.85.
The beta coefficients are the standardized regression coefficients. Their relative absolute magnitudes reflect their relative importance in predicting respondent's income. Betas are only compared within a model, not between. Moreover, betas are highly influenced by misspecification of the model. Adding or subtracting variables in the equation will affect the size of the betas. Highest year of school completed is shown to be much more important than the other independent variables.
The t-test tests the significance of each b coefficient. It is possible to have a regression model which is significant overall by the F test, but where a particular coefficient is not significant. Here. only highest year of school completed is significant. This would lead the researcher to re-run the model, dropping one variable at a time, starting with the least significant (her, age when first married).
The confidence interval on a b coefficient are the b's which could be placed in the prediction equation to get the high and low estimates, though this is rarely done. Recall that b is the number of units the dependent changes when the independent changes one unit.

The zero-order correlation is simply the raw correlation from the correlation matrix given at the top of this output. The partial correlation is the correlation of the given variable with respondent's income, controlling for other independent variables in the equation. Partial correlation "removes" the effect of the control variable(s) on both the dependent and the independent variables. Part correlation, in contrast, "removes" the effect of the control variable(s) on the independent variable alone. Part correlation is used when one hypothesizes that the control variable affects the independent variable but not the dependent variable and when one wants to assess the unique effect of the independent variable on the dependent variable.

Coefficients(a)
Model
Unstandardized Coefficients Standardized Coefficients t Sig. 95% Confidence Interval for B Correlations Collinearity Statistics

B Std. Error Beta Lower Bound Upper Bound Zero-order Partial Part Tolerance VIF B Std. Error

1 (Constant) 3.949 1.014
3.893 .000 1.958 5.939

AGE WHEN FIRST MARRIED .001 .022 .001 .043 .966 -.043 .045 .102 .001 .001 .899 1.113

AGE OF RESPONDENT .014 .008 .054 1.638 .102 -.003 .030 .012 .057 .053 .967 1.034

HIGHEST YEAR OF SCHOOL COMPLETED .407 .076 .379 5.330 .000 .257 .556 .344 .182 .173 .209 4.777

RS HIGHEST DEGREE -.085 .183 -.033 -.464 .643 -.443 .274 .301 -.016 -.015 .209 4.793

a Dependent Variable: RESPONDENTS INCOME

**Coefficient Correlations(a)**
Model			RS HIGHEST DEGREE	AGE OF RESPONDENT	AGE WHEN FIRST MARRIED	HIGHEST YEAR OF SCHOOL COMPLETED
1	Correlations	RS HIGHEST DEGREE	1.000	-.120	-.125	-.876
		AGE OF RESPONDENT	-.120	1.000	-.060	.168
		AGE WHEN FIRST MARRIED	-.125	-.060	1.000	-.031
		HIGHEST YEAR OF SCHOOL COMPLETED	-.876	.168	-.031	1.000
	Covariances	RS HIGHEST DEGREE	.033	.000	-.001	-.012
		AGE OF RESPONDENT	.000	6.96E-005	-1.11E-005	.000
		AGE WHEN FIRST MARRIED	-.001	-1.11E-005	.000	-5.20E-005
		HIGHEST YEAR OF SCHOOL COMPLETED	-.012	.000	-5.20E-005	.006
a Dependent Variable: RESPONDENTS INCOME

The collinearity statistics need scrutiny when the independents are highly intercorrelated.
Above: The tolerance for a variable is 1 - R-squared for the regression of that variable on all the other independents, ignoring the dependent. When tolerance is close to 0 there is high multicollinearity of that variable with other independents and the b and beta coefficients will be unstable. VIF is the variance inflation factor, which is simply the reciprocal of tolerance. Therefore, when VIF is high there is high multicollinearity and instability of the b and beta coefficients. However, the tolerance and VIF reported in the "Correlations" table above are not the coefficients needed.
Below: The table below is another way of assessing if there is too much multicollinearity in the model. To simplify, crossproducts of the independent variables are factored. High eigenvalues indicate dimensions (factors) which account for a lot of the variance in the crossproduct matrix. Eigenvalues close to 0 indicate dimensions which explain little variance. Multiple eigenvalues close to 0 indicate an ill-conditioned crossproduct matrix, meaning there is a problem with multicollinearity. The condition index summarizes the findings, and a common rule of thumb is that a condition index over 15 indicates a possible multicollinearity problem and a condition index over 30 suggests a serious multicollinearity problem. If a factor has a high condition index, one looks in the variance proportions column to see if it accounts for a sizable proportion of variance in other variables. If it does, multicollinearity is a problem. Here we see that the fifth dimension has a high condition index. The fifth dimension corresponds to the 4th variable entered (with the constant makes 5), which is highest degree. It has a high variance proportion with the third dimension, which is highest year of education. The researcher will wish to drop a variable, combine variables, or pursue some other multicollinearity strategy.

Collinearity Diagnostics(a)
Model Dimension Eigenvalue Condition Index Variance Proportions

(Constant) AGE WHEN FIRST MARRIED AGE OF RESPONDENT HIGHEST YEAR OF SCHOOL COMPLETED RS HIGHEST DEGREE (Constant) AGE WHEN FIRST MARRIED

1 1 4.638 1.000 .00 .00 .00 .00 .00

2 .279 4.074 .00 .00 .04 .00 .19

3 .055 9.223 .01 .15 .80 .01 .05

4 .024 13.998 .08 .80 .05 .09 .04

5 .005 31.365 .91 .05 .11 .90 .72

a Dependent Variable: RESPONDENTS INCOME

The table below is a listing of outliers: cases where the prediction is 3 standard deviations or more from the mean value of the dependent. The researcher looks at these cases to consider if they merit a separate model, or if they reflect measurement errors. Either way, the researcher may decide to drop these cases from analysis.

Casewise Diagnostics(a)
Case Number Std. Residual RESPONDENTS INCOME Predicted Value Residual

110 -3.225 1 10.20 -9.199

411 -3.278 1 10.35 -9.350

767 -3.310 1 10.44 -9.442

789 -3.112 1 9.88 -8.878

878 -3.415 2 11.74 -9.741

933 -3.510 1 11.01 -10.013

1115 -3.250 1 10.27 -9.272

1174 -3.104 1 9.85 -8.853

1269 -3.713 1 11.59 -10.591

a Dependent Variable: RESPONDENTS INCOME

The table below contains summary data regarding the residuals (the difference between predicted and actual values). Std. residual, for instance, is the standardized residual (raw residual divided by the standard deviation of residuals). Since the minimum standardized residual is -3.71, at least one prediction is more than 3 standard deviations below the mean residual. Studentized residuals are very similar to standardized residuals and follow the t distribution. These are used in plots of standardized or studentized predicted values vs. observed values. The "deleted residual" rows have to do with coefficients when the model is recomputed over and over, dropping one case from the analysis each time. The bottom three rows are measures of the influence of the minimum, maximum, and mean case on the model. Mahalanobis distance is (n-1) times leverage (the bottom row), which is a measure of case influence. Cases with leverage values less than .2 are not a problem, but cases with leverage values of .5 or higher may be unduly influential in the model and should be examined. Cook's distance measures how much the b coefficients change when a case is dropped. In this example, it does not appear there are problem cases since the maximum leverage is only .077.

Residuals Statistics(a)

Minimum Maximum Mean Std. Deviation N

Predicted Value 4.67 12.67 9.94 1.056 837

Std. Predicted Value -4.988 2.592 .000 1.000 837

Standard Error of Predicted Value .113 .795 .208 .072 837

Adjusted Predicted Value 4.81 12.68 9.94 1.057 837

Residual -10.591 6.356 .000 2.846 837

Std. Residual -3.713 2.228 .000 .998 837

Stud. Residual -3.762 2.286 .000 1.001 837

Deleted Residual -10.875 6.691 -.002 2.865 837

Stud. Deleted Residual -3.792 2.292 -.001 1.003 837

Mahal. Distance .310 63.998 3.995 4.443 837

Cook's Distance .000 .110 .001 .006 837

Centered Leverage Value .000 .077 .005 .005 837

a Dependent Variable: RESPONDENTS INCOME

Charts

The zresid histogram below provides a visual way of assessing if the assumption of normally distributed residual error is met. Regression is robust in the face of some deviation from this assumption, and for this example the small skewness to the right should not affect substantive conclusions.

The normal probability plot (zresid normal p-p plot) below is another test of normally distributed residual error. Under perfect normality, the plot will be a 45-degree line. For this example, it is imperfect but close enough for exploratory conclusions.

We would like residuals to be randomly related to the value of the dependent, but here we see there is a downward-sloping trend.

In the plot of stnardized predicted values vs. observed values if 100% of the variance is explained in a linear relationship, the points will form a straight line. The lower the percent of variance explained, the more the points will form a cloud with no trend. The more the points are dispersed around the trend (a lot, in this case), the higher the standard error of estimate and the poorer the model. The plot below reflects the fact that the model in this case only explains a small percentage of the variance.

Partial regression plots, like those below, simply show the plot of one independent (such as agewed, for instance) on the dependent (respondent income). A partial residual plot (not shown), also called an "added variable plot," is plotted when there are two or more predictors. You get one partial residual plot for each predictor. In a partial residual plot, the dependent (rincome91) is regressed on all predictors except the one of current interest, and likewise that predictor is also regressed on all other predictors. When the two sets of residuals are plotted, the extent to which the points fall on a line shows the correlation of the dependent with the given independent, controlling for all other predictors. Thus the partial residual plot is a visual form of the t-test of the b coefficient for the given variable.