An Illustrative Example of Logistic Regression

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An Illustrative Example of Logistic Regression
This is the sample problem presented in the text
on pages 314 to 321.  Consistent with the
authors strategy for presenting the problem, we
will divide the data set into a learning sample
and a validation sample, after a brief overview
of logistic regression.
An Illustrative Example of Logistic Regression
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Overview of Logistic Regression - 1
Multiple regression requires that the dependent
variable be a metric variable. There are,
however, many problems in which the dependent
variable is a non-metric class or category and
the goal of our analysis is to produce a model
that predicts group membership or classification.
For example, we might be interested in predicting
whether individuals will succeed or fail in some
treatment, i.e. the likelihood that they will be
a member of a particular outcome group. We will
look at two strategies for addressing this type
of problem discriminant analysis and logistic
regression. Discriminant analysis can be used for
any number of groups. Logistic regression is
commonly used with two groups, i.e. the dependent
variable is dichotomous. Discriminant analysis
requires that our data meet the assumptions of
multivariate normality and equality of
variance-covariance across groups. Logistic
regression does not require these assumptions
"In logistic regression, the predictors do not
have to be normally distributed, linearly
related, or of equal variance within each group."
(Tabachnick and Fidell, page 575) Logistic
regression predicts the probability that the
dependent variable event will occur given a
subject's scores on the independent variables.
The predicted values of the dependent variable
can range from 0 to 1. If the probability for an
individual case is equal to or above some
threshold, typically 0.50, then our prediction is
that the event will occur. Similarly, if the
probability for an individual case is less than
0.50, then our prediction is that the event will
not occur. One of the criticisms of logistic
regression is that its group prediction does not
take into account the relative position of a case
within the distribution, a case that has a
probability of .51 is classified in the same
group as a case that has a probability of .99,
since both are above the .50 cutoff. The
dependent variable plotted against the
independent variables follows an s-shaped curve,
like that shown in the text on page 277. The
relationship between the dependent and
independent variable is not linear.
An Illustrative Example of Logistic Regression
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Overview of Logistic Regression - 2
As with multiple regression, we are concerned
about the overall fit, or strength of the
relationship between the dependent variable and
the independent variables, but the statistical
measures of the fit are different than those
employed in multiple regression. Instead of the
F-test for overall significance of the
relationship, we will interpret the Model
Chi-Square statistic which is the test of a model
which has no independent variables versus a model
that has independent variables. There is a
"pseudo R square" statistic that can be computed
and interpreted as an R square value. We can also
examine a classification table of predicted
versus actual group membership and use the
accuracy of this table in evaluating the utility
of the statistical model. The coefficients for
the predictor variables measure the change in the
probability of the occurrence of the dependent
variable event in log units. Since the B
coefficients are in log units, we cannot directly
interpret their meaning as a measure of change in
the dependent variable.  However, when the B
coefficient is used as a power to which the
natural log (2.71828) is raised, the result
represents an odds ratio, or the probability that
an event will occur divided by the probability
that the event will not occur. If a coefficient
is positive, its transformed log value will be
greater than one, meaning that the event is more
likely to occur. If a coefficient is negative,
its transformed log value will be less than one,
and the odds of the event occurring decrease. A
coefficient of zero (0) has a transformed log
value of 1.0, meaning that this coefficient does
not change the odds of the event one way or the
other. We can state the information in the odds
ratio for dichotomous independent variables as
subjects having or being the independent variable
are more likely to have or be the dependent
variable, assuming the that a code of 1
represents the presence both the independent and
the dependent variable. For metric independent
variables, we can state that subjects having more
of the independent variable are more likely to
have or be the dependent variable, assuming that
the independent variable and the dependent
variable are both coded in this direction.
An Illustrative Example of Logistic Regression
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Overview of Logistic Regression - 3
There are several numerical problems that can
occur in logistic regression that are not
detected by SPSS or other statistical packages
multicollinearity among the independent
variables, zero cells for a dummy-coded
independent variable because all of the subjects
have the same value for the variable, and
"complete separation" whereby the two groups in
the dependent event variable can be perfectly
separated by scores on one or a combination of
the independent variables.  All of these problems
produce large standard errors (I recall the
threshold as being over 2.0, but I am unable to
find a reference for this number) for the
variables included in the analysis and very often
produce very large B coefficients as well.  If we
encounter large standard errors for the predictor
variables, we should examine frequency tables,
one-way ANOVAs, and correlations for the
variables involved to try to identify the source
of the problem. Like multiple regression and
discriminant analysis, we are concerned with
detecting outliers and influential cases and the
effect that they may be having on the model. 
Finally, we can use diagnostic plots to evaluate
the fit of the model to the data and to identify
strategies for improving the relationship
expressed in the model. Sample size, power, and
the ratio of cases to variables are important
issues in logistic regression, though the
specific information is less readily available.
In the absence of any additional information, we
may employ the standards required for multiple
regression.
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Preliminary Division of the Data Set
The data for this problem is the Hatco.Sav data
set. Instead of conducting the analysis with the
entire data set, and then splitting the data for
the validation analysis, the authors opt to
divide the sample prior to doing the analysis. 
They use the estimation or learning sample of 60
cases to build the discriminant model and the
other 40 cases for a holdout sample to validate
the model. To replicate the author's analysis,
we will create a randomly generated variable,
randz, to split the sample.  We will use the
cases where randz 0 to create the logistic
regression model and apply that model to the
validation sample to estimate the model's true
accuracy rate. Note the results produced in the
chapter example were obtained by using the same
random seed and compute statement as the
two-group discriminant analysis, not the SPSS
syntax commands specified in the text on page 707.
An Illustrative Example of Logistic Regression
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Specify the Random Number Seed
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Compute the random selection variable
An Illustrative Example of Logistic Regression
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Stage One Define the Research Problem

• In this stage, the following issues are
  addressed
• Relationship to be analyzed
• Specifying the dependent and independent
  variables
• Method for including independent variables

Relationship to be analyzed
The research problem is still to determine if
differences in perception of HATCO can
distinguish between customers using specification
buying versus total value analysis (text, page
314).
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Method for including independent variables
Specifying the dependent and independent variables

• The dependent variable is
• X11 'Purchasing Approach',
• a dichotomous variable, where 1 indicates "Total
  Value Analysis" and 0 indicates "Specification
  Buying."
• The independent variables are
• X1 'Delivery Speed'
• X2 'Price Level'
• X3 'Price Flexibility'
• X4 'Manufacturer Image'
• X5 'Service'
• X6 'Salesforce Image'
• X7 'Product Quality

Since the authors are interested in the best
subset of predictors, they use the forward
stepwise method for selecting independent
variables.
An Illustrative Example of Logistic Regression
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Stage 2 Develop the Analysis Plan Sample Size
Issues

• In this stage, the following issues are
  addressed
• Missing data analysis
• Minimum sample size requirement 15-20 cases per
  independent variable

Missing data analysis
There is no missing data in this problem.
Minimum sample size requirement15-20 cases per
independent variable
The data set has 60 cases and 7 independent
variables for a ratio of 9 to 1, short of the
requirement that we have 15-20 cases per
independent variable.
An Illustrative Example of Logistic Regression
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Stage 2 Develop the Analysis Plan Measurement
Issues

• In this stage, the following issues are
  addressed
• Incorporating nonmetric data with dummy variables
• Representing Curvilinear Effects with Polynomials
• Representing Interaction or Moderator Effects

Incorporating Nonmetric Data with Dummy Variables
All of the independent variables are metric.
Representing Curvilinear Effects with Polynomials
We do not have any evidence of curvilinear
effects at this point in the analysis.
Representing Interaction or Moderator Effects
We do not have any evidence at this point in the
analysis that we should add interaction or
moderator variables.
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Stage 3 Evaluate Underlying Assumptions

• In this stage, the following issues are
  addressed
• Nonmetric dependent variable with two groups
• Metric or dummy-coded independent variables

Nonmetric dependent variable having two groups
The dependent variable is X11 'Purchasing
Approach', a dichotomous variable, where 1
indicates "Total Value Analysis" and 0 indicates
"Specification Buying."
Metric or dummy-coded independent variables
All of the independent variables in the analysis
are metric X1 'Delivery Speed',  X2 'Price
Level',  X3 'Price Flexibility',  X4
'Manufacturer Image',  X5 'Service',  X6
'Salesforce Image', and  X7 'Product Quality
An Illustrative Example of Logistic Regression
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Stage 4 Estimation of Logistic Regression and
Assessing Overall Fit Model Estimation

• In this stage, the following issues are
  addressed
• Compute logistic regression model

Compute the logistic regression
The steps to obtain a logistic regression
analysis are detailed on the following screens.
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Requesting a Logistic Regression
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Specifying the Dependent Variable
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Specifying the Independent Variables
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Specify the method for entering variables
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Specifying Options to Include in the Output
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Specifying the New Variables to Save
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Specifying the cases to include in the analysis
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Complete the Logistic Regression Request
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Stage 4 Estimation of Logistic Regression and
Assessing Overall Fit  Assessing Model Fit

• In this stage, the following issues are addressed
  for the stepwise inclusion of variables
• Significance test of the model log likelihood
  (Change in -2LL)
• Measures Analogous to R² Cox and Snell R² and
  Nagelkerke R²
• Classification matrices
• Check for Numerical Problems
• Once we have decided on the number of variables
  to be included in the equation, we will examine
  other issues of fit and compare model accuracy to
  the by chance accuracy rates.
• Hosmer-Lemeshow Goodness-of-fit
• By chance accuracy rates
• Presence of outliers

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Step 1 of the Stepwise Logistic Regression Model
In this section, we will examine the results
obtained at the first step of the analysis.
Initial statistics before independent variables
are included
The Initial Log Likelihood Function, (-2 Log
Likelihood or -2LL) is a statistical measure like
total sums of squares in regression. If our
independent variables have a relationship to the
dependent variable, we will improve our ability
to predict the dependent variable accurately, and
the log likelihood value will decrease.  The
initial 2LL value is 78.859 on step 0, before
any variables have been added to the model.
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Significance test of the model log likelihood
At step 1, the variable X7 'Product Quality' is
added to the logistic regression equation.  The
addition of this variable reduces the initial log
likelihood value (-2 Log Likelihood) of 78.859 to
37.524. 
The difference between these two measures is the
model child-square value (41.335 78.859 -
37.524) that is tested for statistical
significance. This test is analogous to the
F-test for R² or change in R² value in multiple
regression which tests whether or not the
improvement in the model associated with the
additional variables is statistically
significant.  
In this problem the Model Chi-Square value of
41.335 has a significance of less than 0.0001,
less than 0.05, so we conclude that there is a
significant relationship between the dependent
variable and the set of independent variables,
which includes a single variable at this step.
An Illustrative Example of Logistic Regression
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Measures Analogous to R²
The next SPSS outputs indicate the strength of
the relationship between the dependent variable
and the independent variables, analogous to the
R² measures in multiple regression.
The Cox and Snell R² measure operates like R²,
with higher values indicating greater model fit. 
However, this measure is limited in that it
cannot reach the maximum value of 1, so
Nagelkerke proposed a modification that does
range from 0 to 1.  We will rely upon
Nagelkerke's measure as indicating the strength
of the relationship. If we applied our
interpretive criteria to the Nagelkerke R² of
0.681, we would characterize the relationship as
very strong.
An Illustrative Example of Logistic Regression
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The Classification Matrices
The classification matrices in logistic
regression serve the same function as the
classification matrices in discriminant analysis,
i.e. evaluating the accuracy of the model. If
the predicted and actual group memberships are
the same, i.e. 1 and 1 or 0 and 0, then the
prediction is accurate for that case. If
predicted group membership and actual group
membership are different, the model "misses" for
that case. The overall percentage of accurate
predictions (85.00 in this case) is the measure
of the model that we rely on most heavily in
logistic regression because it has a meaning that
is readily communicated, i.e. the percentage of
cases for which our model predicts accurately.
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Correspondence of Actual and Predicted Values of
the Dependent Variable
The final measure of model fit is the Hosmer and
Lemeshow goodness-of-fit statistic, which
measures the correspondence between the actual
and predicted values of the dependent variable. 
In this case, better model fit is indicated by a
smaller difference in the observed and predicted
classification.  A good model fit is indicated by
a nonsignificant chi-square value.
At step 1, the Hosmer and Lemshow Test is not
statistically significant, indicating predicted
group memberships correspond closely to the
actual group memberships, indicating good model
fit.
An Illustrative Example of Logistic Regression
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Check for Numerical Problems
There are several numerical problems that can
occur in logistic regression that are not
detected by SPSS or other statistical packages
multicollinearity among the independent
variables, zero cells for a dummy-coded
independent variable because all of the subjects
have the same value for the variable, and
"complete separation" whereby the two groups in
the dependent event variable can be perfectly
separated by scores on one or a combination of
the independent variables.
All of these problems produce large standard
errors (over 2) for the variables included in the
analysis and very often produce very large B
coefficients as well.  If we encounter large
standard errors for the predictor variables, we
should examine frequency tables, one-way ANOVAs,
and correlations for the variables involved to
try to identify the source of the problem. Our
final step, in assessing the fit of the derived
model is to check the coefficients and standard
errors of the variables included in the model.
For the single variable included on the first
step, neither the standard error nor the B
coefficient are large enough to suggest any
problem.
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Significance test of the model log likelihood
Step 2 of the Stepwise Logistic Regression Model
In this section, we will examine the results
obtained at the second step of the analysis.
At step 2, the variable X3 'Price Flexibility' is
added to the logistic regression equation.  The
addition of this variable reduces the initial log
likelihood value (-2 Log Likelihood) of 78.858931
to 20.258. 
The difference between these two measures is the
model child-square value (58.601 78.859 -
20.258) that is tested for statistical
significance. This test is analogous to the
F-test for R² or change in R² value in multiple
regression which tests whether or not the
improvement in the model associated with the
additional variables is statistically
significant. In this problem the Model
Chi-Square value of 58.601 has a significance of
less than 0.0001, less than 0.05, so we conclude
that there is a significant relationship between
the dependent variable and the set of independent
variables, which now includes two independent
variables at this step.
An Illustrative Example of Logistic Regression
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Measures Analogous to R²
The next SPSS outputs indicate the strength of
the relationship between the dependent variable
and the independent variables, analogous to the
R² measures in multiple regression.
If we applied our interpretive criteria to the
Nagelkerke R² of 0.852 (up from 0.681 at the
first step), we would characterize the
relationship as very strong.
An Illustrative Example of Logistic Regression
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The Classification Matrices
The classification matrices in logistic
regression serve the same function as the
classification matrices in discriminant analysis,
i.e. evaluating the accuracy of the model.
The overall percentage of accurate predictions
now increases to 98.33 in this case.  Only one
case is classified incorrectly.
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Correspondence of Actual and Predicted Values of
the Dependent Variable
The final measure of model fit is the Hosmer and
Lemeshow goodness-of-fit statistic, which
measures the correspondence between the actual
and predicted values of the dependent variable. 
In this case, better model fit is indicated by a
smaller difference in the observed and predicted
classification.  A good model fit is indicated by
a nonsignificant chi-square value.
At step 2, the Hosmer and Lemshow Test is not
statistically significant, indicating predicted
group memberships correspond closely to the
actual group memberships, indicating good model
fit.
An Illustrative Example of Logistic Regression
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Check for Numerical Problems
Our check for numerical problems is a check for
standard errors larger than 2 or unusually large
B coefficients.
We do not identify any problems from the table of
variables in the equation.
An Illustrative Example of Logistic Regression
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Significance test of the model log likelihood
Step 3 of the Stepwise Logistic Regression Model
In this section, we will examine the results
obtained at the third step of the analysis.
At step 3, the variable X5 'Service' is added to
the logistic regression equation.  The addition
of this variable reduces the initial log
likelihood value (-2 Log Likelihood) of 78.859 to
6.254. 
The difference between these two measures is the
model child-square value (72.605 78.859 -
6.254) that is tested for statistical
significance. This test is analogous to the
F-test for R² or change in R² value in multiple
regression which tests whether or not the
improvement in the model associated with the
additional variables is statistically
significant. In this problem the Model
Chi-Square value of 72.605 has a significance of
less than 0.0001, less than 0.05, so we conclude
that there is a significant relationship between
the dependent variable and the set of independent
variables, which now includes three independent
variables at this step.
An Illustrative Example of Logistic Regression
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Measures Analogous to R²
The next SPSS outputs indicate the strength of
the relationship between the dependent variable
and the independent variables, analogous to the
R² measures in multiple regression.
If we applied our interpretive criteria to the
Nagelkerke R² of 0.960 (up from 0.852 at the
previous step), we would characterize the
relationship as very strong.
An Illustrative Example of Logistic Regression
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Correspondence of Actual and Predicted Values of
the Dependent Variable
The final measure of model fit is the Hosmer and
Lemeshow goodness-of-fit statistic, which
measures the correspondence between the actual
and predicted values of the dependent variable. 
In this case, better model fit is indicated by a
smaller difference in the observed and predicted
classification.  A good model fit is indicated by
a nonsignificant chi-square value.
At step 3, the Hosmer and Lemshow Test is not
statistically significant, indicating predicted
group memberships correspond closely to the
actual group memberships, indicating good model
fit.
An Illustrative Example of Logistic Regression
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The Classification Matrices
The classification matrices in logistic
regression serve the same function as the
classification matrices in discriminant analysis,
i.e. evaluating the accuracy of the model.
The overall percentage of accurate predictions
for the three variable model is 98.33.  Only two
cases are classified incorrectly.
An Illustrative Example of Logistic Regression
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Check for Numerical Problems
Our check for numerical problems is a check for
standard errors larger than 2 or unusually large
B coefficients.
The standard errors for all of the variables in
the model are substantially larger than 2,
indicating a serious numerical problem.  In
addition, the B coefficients have become very
large (remember that these are log values, so the
corresponding decimal value would appear much
larger). This model should not be used, and we
should interpret the model obtained at the
previous step. In hindsight, we may have gotten
a notion that a problem would occur in this step
from the classification table at the previous
step.  Recall that we had only one
misclassification on the previous step, so there
was almost no overlap remaining between the
groups of the dependent variable.
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Returning to the two-variable model
The residual and Cook's distance measures which
we have available are for the three variable
model which SPSS was working with at the time it
concluded the stepwise selection of variables. 
Since I do not know of a way to force SPSS to
stop at step 2, I will repeat the analysis using
direct entry for the two independent variables
which were found to be significant with stepwise
selection  X7 'Product Quality' and X3 'Price
Flexibility.'
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Re-run the Logistic Regression
An Illustrative Example of Logistic Regression
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Complete the specification for the new analysis
An Illustrative Example of Logistic Regression
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The Two-Variable Model
To sum up evidence of model fit presented
previously, the Model Chi-Square value of 58.601
has a significance of less than 0.0001, less than
0.05, so we conclude that there is a significant
relationship between the dependent variable and
the two independent variables.  The Nagelkerke R²
of 0.852 would indicate that the relationship is
very strong.  The Hosmer and Lemeshow
goodness-of-fit measure has a value of 10.334
which has the desirable outcome of
nonsignificance.
An Illustrative Example of Logistic Regression
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The Classification Matrices
The classification matrices in logistic
regression serve the same function as the
classification matrices in discriminant analysis,
i.e. evaluating the accuracy of the model.
The overall percentage of accurate predictions
(98.33 in this case) is very high, with only one
case being misclassified. To evaluate the
accuracy of the model, we compute the
proportional by chance accuracy rate and the
maximum by chance accuracy rates, if appropriate.
The proportional by chance accuracy rate is
equal to 0.536 (0.6332 0.3672). A 25
increase over the proportional by chance accuracy
rate would equal 0.669. Our model accuracy race
of 98.3 exceeds this criterion. Since one of
our groups contains 63.3 of the cases, we might
also apply the maximum by chance criterion. A
25 increase over the largest groups would equal
0.792. Our model accuracy race of 98.3 also
exceeds this criterion. In addition, the
accuracy rates for the unselected validation
sample, 87.50, surpasses both the proportional
by chance accuracy rate and the maximum by chance
accuracy rate.
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The Classification Matrices
SPSS provides a visual image of the
classification accuracy in the stacked histogram
as shown below. To the extent to which the
cases in one group cluster on the left and the
other group clusters on the right, the predictive
accuracy of the model will be higher.
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Presence of outliers
There are two outputs to alert us to outliers
that we might consider excluding from the
analysis listing of residuals and saving Cook's
distance scores to the data set.  SPSS provides
a casewise list of residuals that identify cases
whose residual is above or below a certain number
of standard deviation units.  Like multiple
regression there are a variety of ways to compute
the residual.  In logistic regression, the
residual is the difference between the observed
probability of the dependent variable event and
the predicted probability based on the model. 
The standardized residual is the residual divided
by an estimate of its standard deviation.  The
deviance is calculated by taking the square root
of -2 x the log of the predicted probability for
the observed group and attaching a negative sign
if the event did not occur for that case. Large
values for deviance indicate that the model does
not fit the case well.  The studentized residual
for a case is the change in the model deviance if
the case is excluded.  Discrepancies between the
deviance and the studentized residual may
identify unusual cases. (See the SPSS chapter on
Logistic Regression Analysis for additional
details). In the output for our problem, SPSS
listed one case that may be considered an outlier
with a studentized residuals greater than 2, case
13
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Cooks Distance
SPSS has an option to compute Cook's distance as
a measure of influential cases and add the score
to the data editor.  I am not aware of a precise
formula for determining what cutoff value should
be used, so we will rely on the more traditional
method for interpreting Cook's distance which is
to identify cases that either have a score of 1.0
or higher, or cases which have a Cook's distance
substantially different from the other.  The
traditional method for detecting unusually large
Cook's distance scores is to create a scatterplot
of Cook's distance scores versus case id or case
number.
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Request the Scatterplot of Cook's Distances
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Specifying the Variables for the Scatterplot
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The Scatterplot of Cook's Distances
On the plot of Cook's distances, we see a case
that exceeds the 1.0 rule of thumb for
influential cases and has a distance value much
different than the other cases.  This is actually
the same case that was identified as an outlier
on the casewise plot, though it is difficult to
track down because SPSS uses the case number in
the learning sample for the casewise plot. This
case is the only case in the two variable model
that was misclassified.  We cannot omit it
because we would again be faced with no overlap
between the groups, producing the problematic
numeric results that we found with the three
variable model.
An Illustrative Example of Logistic Regression
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Stage 5 Interpret the Results

• In this section, we address the following issues
• Identifying the statistically significant
  predictor variables
• Direction of relationship and contribution to
  dependent variable

Identifying the statistically significant
predictor variables
The coefficients are found in the column labeled
B, and the test that the coefficient is not zero,
i.e. changes the odds of the dependent variable
event is tested with the Wald statistic, instead
of the t-test as was done for the individual B
coefficients in the multiple regression equation.
The Wald tests for the two independent variables
X7 'Product Quality' and X3 'Price Flexibility'
are both statistically significant (p lt 0.05), as
we knew they would be from the first two steps of
the stepwise procedure.
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Direction of relationship and contribution to
dependent variable
The negative sign of X7 'Product Quality'
indicates an inverse relationship with the
dependent variable.  As the rating for X7
'Product Quality' increases, there is a higher
likelihood that the respondent favored the
'Specification Buying' approach to purchasing. 
In contrast  the positive coefficient for X3
'Price Flexibility' indicates a direct
relationship to the dependent variable.  As the
rating for X3 'Price Flexibility', there is a
higher likelihood that the respondent favored the
'Total Value Analysis' approach to purchasing.
Since the B coefficient is expressed in log
units, we cannot directly interpret the magnitude
of the change associated with a one unit change
in the independent variable.  However, if we
convert the B coefficient from log to decimal
units, the result represents the change in the
odds of having the dependent variable event for a
one unit of change in the independent variable. 
For a one-unit change in X3 'Price Flexibility',
the odds of being in the 'Total Value Analysis'
approach to purchasing increase by 6 times.  A
one unit change in X7 'Product Quality' reduces
the odds of being in the 'Total Value Analysis'
approach to purchasing by 1/20th (i.e. you have a
much lower probability of being in the 'Total
Value Analysis' approach to purchasing). Should
we want to talk about the odds ratio for a
different level of change, e.g. a 2-unit change
in the independent variable, we cannot simply
multiply the odds-ratio for a one unit change by
2.  Instead we would multiply the B coefficient
by 2 and take the antilog EXP(B) of that value. 
For example, in this problem, EXP(1x1.8035)
6.2371, while the odds ratio for a two unit
change in X3 would be EXP(2x1.8035) 38.900.
An Illustrative Example of Logistic Regression
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Stage 6 Validate the Model
By holding out 40 of the subjects as a
validation sample, the authors created the
context for validation analysis. As stated
above, the accuracy rate for the validation
sample (87.50) is in the same range as the
accuracy rate for the learning sample (98.33).
Though the difference is larger than 10, the
very high accuracy rates for both samples would
incline us to support a conclusion that the
findings are generalizable.