Regression Analysis with SPSS

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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 [phone]
• E-mail yaffee_at_nyu.edu
• February 04

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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

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Conceptualization of Regression Analysis

• Hypothesis testing
• Path Analytical Decomposition of effects

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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.

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Regression AnalysisHave a clear notion of what
you can and cannot do with regression analysis

• Conceptualization
• A Path Model of a Regression Analysis

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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
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Interaction coefficient C X1 and X2 must be in
model for interaction to be properly specified.
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A Precursor to Modeling with Regression

• Data Exploration Run a scatterplot matrix and
  search for linear relationships with the
  dependent variable.

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Click on graphs and then on scatter
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When the scatterplot dialog box appears, select
Matrix
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A Matrix of Scatterplots will appear
Search for distinct linear relationships
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Decomposition of the Sums of Squares
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Graphical Decomposition of Effects
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Decomposition of the sum of squares
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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

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F test for significance and R2 for magnitude of
effect

• R2 Model var/total var

• F test for model significance
• Model Var/Error Var

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ANOVA tests the significance of the Regression
Model
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The Multiple Regression Equation

• We proceed to the derivation of its components
  
• The intercept a
• The regression parameters, b1 and b2

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Derivation of the Intercept
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Derivation of the Regression Coefficient
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• If we recall that the formula for the
  correlation coefficient can be expressed as
  follows

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Extending the bivariate case To the Multiple
linear regression case
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It is also easy to extend the bivariate
intercept to the multivariate case as follows.
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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.

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F and T tests for significance for overall model
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Significance tests

• If we are using a type II sum of squares, we
  are dealing with the ballantine. DV Variance
  explained a b

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Significance tests

• T tests for statistical significance

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Significance tests

• Standard Error of intercept

Standard error of regression coefficient
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Programming Protocol
After invoking SPSS, procede to File, Open, Data
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Select a Data Set (we choose employee.sav) and
click on open
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We open the data set
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To inspect the variable formats, click on
variable view on the lower left
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Because gender is a string variable, we need to
recode gender into a numeric format
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We autorecode gender by clicking on transform and
then autorecode
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We select gender and move it into the variable
box on the right
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Give the variable a new name and click on add new
name
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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.
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To invoke Regression analysis,Click on Analyze
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Click on Regression and then linear
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Select the dependent variable Current Salary
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Enter it in the dependent variable box
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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.

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After entering the covariates, we click on next
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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)

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After entering these variables, click on
statistics
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We select the following statistics from the
dialog box and click on continue
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Click on plots to obtain the plots dialog box
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We click on OK to run the regression analysis
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Navigation window (left) and output window(right)
This shows that SPSS is reading the variables
correctly
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Variables Entered and Model Summary
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Omnibus ANOVA
Significance Tests for the Model at each stage of
the analysis
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Full ModelCoefficients
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We omit insignificant variables and rerun the
analysis to obtain trimmed model coefficients
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Beta weights

• These are standardized regression coefficients
  used to compare the contribution to the
  explanation of the variance of the dependent
  variable within the model.

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T tests and signif.

• These are the tests of significance for each
  parameter estimate.
• The significance levels have to be less than .05
  for the parameter to be statistically significant.

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Assumptions of the Linear Regression Model

• Linear Functional form
• Fixed independent variables
• Independent observations
• Representative sample and proper specification of
  the model (no omitted variables)
• Normality of the residuals or errors
• Equality of variance of the errors (homogeneity
  of residual variance)
• No multicollinearity
• No autocorrelation of the errors
• No outlier distortion

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Explanation of the Assumptions

• 1. Linear Functional form
• Does not detect curvilinear relationships
• Independent observations
• Representative samples
• Autocorrelation inflates the t and r and f
  statistics and warps the significance tests
• Normality of the residuals
• Permits proper significance testing
• Equality of variance
• Heteroskedasticity precludes generalization and
  external validity
• This also warps the significance tests
• Multicollinearity prevents proper parameter
  estimation. It may also preclude computation of
  the parameter estimates completely if it is
  serious enough.
• Outlier distortion may bias the results If
  outliers have high influence and the sample is
  not large enough, then they may serious bias the
  parameter estimates

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Diagnostic Tests for the Regression Assumptions

• Linearity tests Regression curve fitting
• No level shifts One regime
• Independence of observations Runs test
• Normality of the residuals Shapiro-Wilks or
  Kolmogorov-Smirnov Test
• Homogeneity of variance if the residuals
  Whites General Specification test
• No autocorrelation of residuals Durbin Watson or
  ACF or PACF of residuals
• Multicollinearity Correlation matrix of
  independent variables.. Condition index or
  condition number
• No serious outlier influence tests of additive
  outliers Pulse dummies.
• Plot residuals and look for high leverage of
  residuals
• Lists of Standardized residuals
• Lists of Studentized residuals
• Cooks distance or leverage statistics

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Explanation of Diagnostics

• Plots show linearity or nonlinearity of
  relationship
• Correlation matrix shows whether the independent
  variables are collinear and correlated.
• Representative sample is done with probability
  sampling

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Explanation of Diagnostics

• Tests for Normality of the residuals. The
  residuals are saved and then subjected to either
  of
• Kolmogorov-Smirnov Test Tests the limit of the
  theoretical cumulative normal distribution
  against your residual distribution.
• Nonparametric Tests
• 1 sample K-S test

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Collinearity Diagnostics
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More Collinearity Diagnostics

• condition numbers
• maximum eigenvalue/minimum eigenvalue.
• If condition numbers are between 100 and 1000,
  there is moderate to strong collinearity

If Condition index gt 30 then there is strong
collinearity
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Outlier Diagnostics

• Residuals.
• The predicted value minus the actual value. This
  is otherwise known as the error.
• Studentized Residuals
• the residuals divided by their standard errors
  without the ith observation
• Leverage, called the Hat diag
• This is the measure of influence of each
  observation
• Cooks Distance
• the change in the statistics that results from
  deleting the observation. Watch this if it is
  much greater than 1.0.

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Outlier detection

• Outlier detection involves the determination
  whether the residual (error predicted actual)
  is an extreme negative or positive value.
• We may plot the residual versus the fitted plot
  to determine which errors are large, after
  running the regression.

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Create Standardized Residuals

• A standardized residual is one divided by its
  standard deviation.

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Limits of Standardized Residuals

• If the standardized residuals have values in
  excess of 3.5
• and -3.5, they are outliers.
• If the absolute values are less than 3.5, as
  these are, then there are no outliers
• While outliers by themselves only distort mean
  prediction when the sample size is small enough,
  it is important to gauge the influence of
  outliers.

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Outlier Influence

• Suppose we had a different data set with two
  outliers.
• We tabulate the standardized residuals and obtain
  the following output

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Outlier a does not distort and outlier b does.
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Studentized Residuals

• Alternatively, we could form studentized
  residuals. These are distributed as a t
  distribution with dfn-p-1, though they are not
  quite independent. Therefore, we can
  approximately determine if they are statistically
  significant or not.
• Belsley et al. (1980) recommended the use of
  studentized residuals.

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Studentized Residual
These are useful in estimating the statistical
significance of a particular observation, of
which a dummy variable indicator is formed. The
t value of the studentized residual will indicate
whether or not that observation is a
significant outlier. The command to generate
studentized residuals, called rstudt is predict
rstudt, rstudent
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Influence of Outliers

• Leverage is measured by the diagonal components
  of the hat matrix.
• The hat matrix comes from the formula for the
  regression of Y.

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Leverage and the Hat matrix

• The hat matrix transforms Y into the predicted
  scores.
• The diagonals of the hat matrix indicate which
  values will be outliers or not.
• The diagonals are therefore measures of leverage.
• Leverage is bounded by two limits 1/n and 1.
  The closer the leverage is to unity, the more
  leverage the value has.
• The trace of the hat matrix the number of
  variables in the model.
• When the leverage gt 2p/n then there is high
  leverage according to Belsley et al. (1980) cited
  in Long, J.F. Modern Methods of Data Analysis
  (p.262). For smaller samples, Vellman and Welsch
  (1981) suggested that 3p/n is the criterion.

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Cooks D

• Another measure of influence.
• This is a popular one. The formula for it is

Cook and Weisberg(1982) suggested that values of
D that exceeded 50 of the F distribution (df
p, n-p) are large.
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Using Cooks D in SPSS

• Cook is the option /R
• Finding the influential outliers
• List cook, if cook gt 4/n
• Belsley suggests 4/(n-k-1) as a cutoff

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DFbeta

• One can use the DFbetas to ascertain the
  magnitude of influence that an observation has on
  a particular parameter estimate if that
  observation is deleted.

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Programming Diagnostic TestsTesting
homoskedasiticitySelect histogram, normal
probability plot, and insert zresid in Yand
zpred in X
Then click on continue
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Click on Save to obtain the Save dialog box
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We select the following
Then we click on continue, go back to the Main
Regression Menu and click on OK
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Check for linear Functional Form

• Run a matrix plot of the dependent variable
  against each independent variable to be sure that
  the relationship is linear.

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Move the variables to be graphed into the box on
the upper right, and click on OK
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Residual Autocorrelation check
See significance tables for this statistic
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Run the autocorrelation function from the Trends
Module for a better analysis
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Testing for Homogeneity of variance
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Normality of residuals can be visually inspected
from the histogram with the superimposed normal
curve. Here we check the skewness for symmetry
and the kurtosis for peakedness
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Kolmogorov Smirnov Test An objective test of
normality
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Multicollinearity test with the correlation
matrix
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Alternatives to Violations of Assumptions

• 1. Nonlinearity Transform to linearity if
  there is nonlinearity or run a nonlinear
  regression
• 2. Nonnormality Run a least absolute
  deviations regression or a median regression
  (available in other packages or generalized
  linear models SPLUS glm, STATA glm, or SAS
  Proc MODEL or PROC GENMOD).
• 3. Heteroskedasticity weighted least squares
  regression (SPSS) or white estimator (SAS,
  Stata, SPLUS). One can use a robust regression
  procedure (SAS, STATA, or SPLUS) to obtain
  downweighted outlier effect in the estimation.
• 4. Autocorrelation Run AREG in SPSS Trends
  module or either Prais or Newey-West procedure
  in STATA.
• 4. Multicollinearity components regression or
  ridge regression or proxy variables. 2sls in
  SPSS or ivreg in stata or SAS proc model or proc
  syslin.

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Model Building Strategies

• Specific to General Cohen and Cohen
• General to Specific Hendry and Richard
• Extreme Bounds analysis E. Leamer.

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Nonparametric Alternatives

• If there is nonlinearity, transform to linearity
  first.
• If there is heteroskedasticity, use robust
  standard errors with STATA or SAS or SPLUS.
• If there is non-normality, use quantile
  regression with bootstrapped standard errors in
  STATA or SPLUS.
• If there is autocorrelation of residuals, use
  Newey-West autoregression or First order
  autocorrelation correction with Areg. If there
  is higher order autocorrelation, use Box Jenkins
  ARIMA modeling.