Variables Entered/Removed(b)
Model Variables Entered Removed Variables Method
1 height(a) . Enter
a All requested variables entered. b Dependent Variable: fev1
Model Summary(b)
Model R R Square Adjusted R Square the Estimate Std. Error of
1 .562(a) .315 .314 .55337
a Predictors: (Constant), height b Dependent Variable: fev1
ANOVA(b)
Model Squares Sum of df Square Mean F Sig.
1 Regression 112.380 1 112.380 366.997 .000(a)
Residual 244.054 797 .306
Total 356.434 798
a Predictors: (Constant), height b Dependent Variable: fev1
Information on the independent
variables and dependent variable in the regression model, and the method of entering the independent variables into the regression model.
R-Square = proportion of the total variation in the dependent variable explained by the independent variable(s) = .315 or 31.5% R is square root of R Square Adjusted R Square – “adjusts” the R square for the number of
variables in the model Std. error of the estimate = standard deviation of the error or residuals. Not usually reported, but used in estimating the standard error of the regression
coefficients.
ANOVA = analysis of variance table. Not needed when there is only 1 independent variable in the model. The F test is
equivalent to the t test for testing if the slope is equal to zero in the output that follows. (F = t2)
Coefficients(a)
Model Unstandardized Coefficients Standardized Coefficients t Sig. 95% Confidence Interval for B
B Error Std. Beta Lower Bound Upper Bound
1 (Constant) -4.330 .335 -12.943 .000 -4.987 -3.673
height .039 .002 .562 19.157 .000 .035 .043
a Dependent Variable: fev1
Charts
1.0 0.8 0.6 0.4 0.2 0.0Observed Cum Prob
1.0 0.8 0.6 0.4 0.2 0.0 Ex p e c te d C u m Pr o b
Normal P-P Plot of Regression Standardized Residual
Dependent Variable: fev1
Unstandardized coefficients B = regression coefficient
In this example B = 0.039 is the slope and B = -4.330 the intercept Std. Error = standard error of the regression coefficient.
Standardized coefficients Beta = standardized regression coefficient
t = t statistic for testing if the regression coefficient is equal to zero (versus not equal to zero)
Sig. = p – value for testing if the regression coefficient is equal to zero (versus not equal to zero).
95% confidence interval for B = 95% confidence interval for the regression coefficient In this example, you would report the slope (.039), standard error of the slope (.002) and the p-value (<.001), or the slope (.039) and 95% confidence interval (.035 to 0.043).
Normal probability plot of the residuals. The points fall along a straight line, indicating the residuals have, at least
approximately, a Normal distribution.
Linear Regression Example with three independent variables
Statistics… options
By default, Estimates and Model fit are selected.
In this example, part and partial
correlations and collinearity diagnostics are also selected.
Plots… options
Normal probability plot (of the standardized residuals) and partial (residual) plots are selected.
The dependent variable is forced expiratory volume (fev1).
The independent variables are height, age and enter.
The Enter method means all 3 independent variables will be included in the regression model.
Regression
Variables Entered/Removed(b)
Model Variables Entered Removed Variables Method
1 gender,
age,
height(a) .
Enter
a All requested variables entered. b Dependent Variable: fev1
Model Summary(b) Model R R Square Adjusted R Square Std. Error of the Estimate 1 .601(a) .361 .358 .53531
a Predictors: (Constant), gender, age, height b Dependent Variable: fev1
ANOVA(b) Model Sum of Squares df Mean Square F Sig. 1 Regression 128.623 3 42.874 149.621 .000(a) Residual 227.811 795 .287 Total 356.434 798
a Predictors: (Constant), gender, age, height b Dependent Variable: fev1
Coefficients(a)
Unstandardized Coefficients
Standardized
Coefficients t Sig. Correlations
Collinearity Statistics
B Error Std. Beta Zero-order Partial Part Tolerance VIF
(Constant) -.780 .593 -1.315 .189
height .028 .003 .399 9.143 .000 .562 .308 .259 .423 2.364
age -.025 .004 -.200 -6.857 .000 -.206 -.236 -.194 .944 1.059
gender .273 .059 .201 4.591 .000 .478 .161 .130 .420 2.379
a Dependent Variable: fev1
Height, age, and gender are all statistically significant (P < .001), i.e., the regression coefficients are different from zero.
The partial correlations (and partial R-squares, .3082=.095, -.2362=.056, and .1612=.026) indicate the correlation with the dependent variable adjusted for the other variables in the regression model.
A low tolerance value (say, <.20) or a high variance inflation factor (VIF) (say, > 5 or 10) may indicate a multicollinearity problem.
Information on the independent
variables, method of variable entry, and dependent variable.
R-square is .361 or 36.1% (adjusted R-square is 35.8%). About 36% of the variation in the dependent variables can be explained by the 3 independent variables.
The overall F test, indicates 1 or more the independent variables is significant (P < .001). Degrees of freedom of the F test are 3 and 795.
1.0 0.8 0.6 0.4 0.2 0.0
Observed Cum Prob
1.0 0.8 0.6 0.4 0.2 0.0 Ex p e c te d C u m Pr o b
Normal P-P Plot of Regression Standardized Residual
Dependent Variable: fev1
30.00 20.00 10.00 0.00 -10.00 -20.00 -30.00 height 2.00 0.00 -2.00 fev1
Partial Regression Plot
Dependent Variable: fev1
20.00 15.00 10.00 5.00 0.00 -5.00 -10.00 -15.00 age 2.00 0.00 -2.00 fev1
Partial Regression Plot
Dependent Variable: fev1
Note that SPSS will also produce a partial residual plot for gender. In general, the partial residuals plots for categorical/nominal variables are not very useful. Boxplots of the residuals for each category of a categorical/nominal variable are useful for regression diagnostics. To produce the boxplots you could use the Save… options to save the residuals from a regression and then the Boxplot commands to plot the residuals.
Normal probability plot of the residuals. The points fall approximately along a straight line, indicating the residuals have (approximately) a Normal
distribution.
Partial regression plots for height and age with lowess smooths.
The plot for height is assessing the relationship between height and fev1 after adjusting for age and gender (e.g., is the relationship linear).
Similarly, the plot for age is assessing the relationship between age and fev1 adjusting for height and gender.