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Regression Analysis with SPSS . Robert A. Yaffee, Ph.D. Statistics, Mapping and Social Science Group Academic Computing Services Information Technology Services New York University Office: 75 Third Ave Level C3 Tel: 212.998.3402 E-mail: firstname.lastname@example.org February 04. Outline .
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Regression Analysiswith SPSS Robert A. Yaffee, Ph.D. Statistics, Mapping and Social Science Group Academic Computing Services Information Technology Services New York University Office: 75 Third Ave Level C3 Tel: 212.998.3402 E-mail: email@example.com February 04
Outline • Conceptualization • Schematic Diagrams of Linear Regression processes • Using SPSS, we plot and test relationships for linearity • Nonlinear relationships are transformed to linear ones • General Linear Model • Derivation of Sums of Squares and ANOVADerivation of intercept and regression coefficients • The Prediction Interval and its derivation • Model Assumptions • Explanation • Testing • Assessment • Alternatives when assumptions are unfulfilled
Conceptualization of Regression Analysis • Hypothesis testing • Path Analytical Decomposition of effects
Hypothesis Testing • For example: hypothesis 1 : X is statistically significantly related to Y. • The relationship is positive (as X increases, Y increases) or negative (as X decreases, Y increases). • The magnitude of the relationship is small, medium, or large. If the magnitude is small, then a unit change in x is associated with a small change in Y.
Regression AnalysisHave a clear notion of what you can and cannot do with regression analysis • Conceptualization • A Path Model of a Regression Analysis
In a path analysis, Yi is endogenous. It is the outcome of several paths. Direct effects on Y3: C,E, F Indirect effects on Y3: BF, BDF Total Effects= Direct + Indirect effects
Interaction coefficient: C X1 and X2 must be in model for interaction to be properly specified.
A Precursor to Modeling with Regression • Data Exploration: Run a scatterplot matrix and search for linear relationships with the dependent variable.
A Matrix of Scatterplots will appear Search for distinct linear relationships
Decomposition of the sum of squares • Total SS = model SS + error SS and if we divide by df • This yields the Variance Decomposition: We have the total variance= model variance + error variance
F test for significance and R2 for magnitude of effect • R2 = Model var/total var • F test for model significance • = Model Var/Error Var
The Multiple Regression Equation • We proceed to the derivation of its components: • The intercept: a • The regression parameters, b1 and b2
If we recall that the formula for the correlation coefficient can be expressed as follows:
Extending the bivariate case To the Multiple linear regression case
It is also easy to extend the bivariate intercept to the multivariate case as follows.
Significance Tests for the Regression Coefficients • We find the significance of the parameter estimates by using the F or t test. • The R2 is the proportion of variance explained.
Significance tests • If we are using a type II sum of squares, we are dealing with the ballantine. DV Variance explained = a + b
Significance tests T tests for statistical significance
Significance tests Standard Error of intercept Standard error of regression coefficient
Programming Protocol After invoking SPSS, procede to File, Open, Data
To inspect the variable formats, click on variable view on the lower left
Because gender is a string variable, we need to recode gender into a numeric format
Click on ok and the numeric variable sex is created It has values 1 for female and 2 for male and those values labels are inserted.
Entering independent variables • These variables are entered in blocks. First the potentially confounding covariates that have to entered. • We enter time on job, beginning salary, and previous experience.
We now enter the hypotheses we wish to test • We are testing for minority or sex differences in salary after controlling for the time on job, previous experience, and beginning salary. • We enter minority and numeric gender (sex)
We select the following statistics from the dialog box and click on continue