Solving stepwise regression problems

Stepwise Multiple Regression

Different Methods for Entering Variables in Multiple Regression
 Different types of multiple regression are distinguished by the method for entering the
independent variables into the analysis.
 In standard (or simultaneous) multiple regression, all of the independent variables are
entered into the analysis at the same.
 In hierarchical (or sequential) multiple regression, the independent variables are entered in
an order prescribed by the analyst.
 In stepwise (or statistical) multiple regression, the independent variables are entered
according to their statistical contribution in explaining the variance in the dependent
variable.
 No matter what method of entry is chosen, a multiple regression that includes the same
independent variables and the same dependent variables will produce the same multiple
regression equation.
 The number of cases required for stepwise regression is greater than the number for the
other forms. We will use the norm of 40 cases for each independent variable.

Purpose of Stepwise Multiple Regression
 Stepwise regression is designed to find the most parsimonious set of predictors that are
most effective in predicting the dependent variable.
 Variables are added to the regression equation one at a time, using the statistical criterion
of maximizing the R² of the included variables.
 After each variable is entered, each of the included variables are tested to see if the model
would be better off it were excluded. This does not happen often.
 The process of adding more variables stops when all of the available variables have been
included or when it is not possible to make a statistically significant improvement in R²
using any of the variables not yet included.
 Since variables will not be added to the regression equation unless they make a statistically
significant addition to the analysis, all of the independent variable selected for inclusion
will have a statistically significant relationship to the dependent variable.
 An example of how SPSS does stepwise regression is shown below.

Stepwise Multiple Regression in SPSS
 Each time SPSS includes or removes a variable from the analysis, SPSS considers it a new
step or model, i.e. there will be one model and result for each variable included in the
analysis.
 SPSS provides a table of variables included in the analysis and a table of variables
excluded from the analysis. It is possible that none of the variables will be included. It is
possible that all of the variables will be included.
 The order of entry of the variables can be used as a measure of relative importance.
 Once a variable is included, its interpretation in stepwise regression is the same as it would
be using other methods for including regression variables.

Pros and Cons of Stepwise Regression
 Stepwise multiple regression can be used when the goal is to produce a predictive model
that is parsimonious and accurate because it excludes variables that do not contribute to
explaining differences in the dependent variable.
 Stepwise multiple regression is less useful for testing hypotheses about statistical
relationships. It is widely regarded as atheoretical and its usage is not recommended.
 Stepwise multiple regression can be useful in finding relationships that have not been
tested before. Its findings invite one to speculate on why an unusual relationship makes
sense.
 It is not legitimate to do a stepwise multiple regression and present the results as though
one were testing a hypothesis that included the variables found to be significant in the
stepwise regression.
 Using statistical criteria to determine relationships is vulnerable to over-fitting the data set
used to develop the model at the expense of generalizability.
 When stepwise regression is used, some form of validation analysis is a necessity. We will
use 75/25% cross-validation.

75/25% Cross-validation
 To do cross validation, we randomly split the data set into a 75% training sample and a
25% validation sample. We will use the training sample to develop the model, and we test
its effectiveness on the validation sample to test the applicability of the model to cases not
used to develop it.
 In order to be successful, the follow two questions must be answers affirmatively:
 Did the stepwise regression of the training sample produce the same subset of
predictors produced by the regression model of the full data set?
 If yes, compare the R2
for the 25% validation sample to the R2
for the 75% training
sample. If the shrinkage (R2
for the 75% training sample - R2
for the 25% validation
sample) is 2% (0.02) or less, we conclude that validation was successful.
 Note: shrinkage may be a negative value, indicating that the accuracy rate for the
validation sample is larger than the accuracy rate for the training sample. Negative
shrinkage (increase in accuracy) is evidence of a successful validation analysis.
 If the validation is successful, we base our interpretation on the model that included all
cases.

DV
IV1
DV
IV2
Correlations between dependent variable and independent variables
DV and IV1 are
correlated at r = .
70. The area of
overlap is r² = .49.
We have two independent variables, IV1
and IV2, which each have a relationship
to the dependent variable. The areas of
IV1 and IV2 which overlap with DV are
r² values, i.e. the proportion of the dv
that is explained by the iv.
DV and IV2 are
correlated at r = .
40. The area of
overlap is r² = .16.

Correlations between independent variables
IV1 IV2
The two independent
variables, IV1 and IV2,
are correlated at r = .20.
This correlation
represents redundant
information in the
independent variables.

Variance in the dependent variable explained by the independent variables
The variance explained in DV
is divided into three areas.
The total variance explained
is the sum of the three areas.
DV
IV1 IV2
The brown area is
the variance in DV
that is explained by
both IV1 and IV2.
The green area is the
variance in DV uniquely
explained by IV1.
The orange area is the
variance in DV uniquely
explained by IV2.

Correlations at step 1 of the stepwise regression
Since IV1 had the stronger relationship with
DV (.70 versus .40), it will be the variable
entered first in the stepwise regression.
As the only variable in the regression
equation, it is given full credit (.70) for its
relationship to DV.
The partial correlation and the part
correlation have the same value as the
zero-order correlation at .70.
DV
IV1

Change in variance explained when a second variable in included
At step 2, IV2 enters the model, increasing the total
variance explained from .49 to .56, an increase 0f .07.
By itself, IV2 explained .16 of the variance in DV, but
since it was itself correlated with IV1, a portion of
what it could explain had already been attributed to
IV1.

Differences in correlations when a second variable is entered
While the zero-order correlations do not change, both the
partial and the part correlations decrease.
Partial correlation represents the relationship between the
dependent variable and an independent variable when the
relationship between the dependent variable and other
independent variables has been removed from the
variance of both the dependent and the independent
variable.
Part (or semi-partial) correlation is the portion of the total
variance in the dependent variable that is by only that
independent variable. The square of part correlation is the
amount of change in R² by including this variable.

Zero-order, partial, and part correlations
DV
IV1 IV2
DV
IV1
The zero-order correlation is
based on the relationship between
the independent variable and the
dependent variable, ignoring all
other independent variables.
The partial correlation for
IV1 is the green area divided
by the area in DV and IV1
that is not part of IV2, i.e.
green divided by green +
yellow.
Part correlation for IV1 is
the green area divided by
all parts of DV, i.e. including
areas associated with IV2.
NOTE:
diagrams are
scaled to r2
rather than r.

DV
IV2
Zero-order, partial, and part correlations
DV
IV1 IV2
The zero-order correlation
is based on the
relationship between the
independent variable and
the dependent variable,
ignoring all other
independent variables.
The partial correlation for IV2
is the green area divided by
the area in DV and IV2 that
is not part of IV2, i.e. orange
divided by orange + yellow.
Part correlation for IV2 is
the orange area divided by
all parts of DV, i.e. including
areas associated with IV1.

How SPSS Stepwise Regression Chooses Variables - 1
The table of Correlations shows
the the variable with the
strongest individual relationship
with the dependent variable is
RACE OF HOUSEHOLD=WHITE,
with a correlation of -.247.
Provided that the relationship
between this variable and the
dependent variable is
statistically significant, this will
be the variable that enters first.
We can use the table of
correlations to identify which
variable will be entered at the
first step of the stepwise
regression.

The correlation between RACE OF
HOUSEHOLD=WHITE and
importance of ethnic group to R is
statistically significant at p < .001.
It will be the first variable entered
into the regression equation.

Model 1 contains the variable RACE
OF HOUSEHOLD=WHITE, with a
Multiple R of .247, producing an R²
of .061 (.247²), which is
We cannot use the table of correlations
to show which variable will be entered
second, since the variable entered
second must take into account its
correlation to the independent variable
entered first.

Partial correlation is a measure of the
relationship of the dependent variable to an
independent variable, where the variance
explained by previously entered independent
variables has been removed from both.
The table of Excluded
Variables, however,
shows the Partial
Correlation between
each candidate for entry
and the dependent
variable.
In this example, RACE
OF
HOUSEHOLD=BLACK
has the largest Partial
Correlation (.252) and
is statistically significant
at p < .001, so it will be
entered on the next
step

As expected, Model 2 contains the
variable RACE OF
HOUSEHOLD=WHITE and RACE OF
HOUSEHOLD=BLACK. The R² for
Model 2 increased by 0.059 to a total
of .120. The increase in R² was

The increase in R² of .059 is the square
of the Part Correlation for RACE OF
HOUSEHOLD=BLACK (.244² = 0.059).
Part correlation, also referred to as
semi-partial correlation, is the unique
relationship between this independent
variable and the dependent variable.

In the table of Excluded
Variables for model 2,
the next largest partial
correlation is HOW
OFTEN R ATTENDS
RELIGIOUS SERVICES
at .149.
This is the variable that will be
added in Model 3 because the
relationships is statistically
significant at p = 0.32.
Partial
Correlatio
n
Column
Sig.
Colum
n

As expected, Model 3 contains the
variable RACE OF
HOUSEHOLD=WHITE and RACE OF
HOUSEHOLD=BLACK, and HOW OFTEN
R ATTENDS RELIGIOUS SERVICES .
The R² for Model 3 increased by 0.019
to a total of .140. The increase in R²
was statistically significant at p = .
032.

Partial
Correlatio
n
Column
Sig.
Colum
n
However, the partial
correlation is not significant
(p=.203), so no additional
variables will be added to
the model.
In the table of Excluded
Variables for model 3,
the next largest partial
correlation is THINK OF
SELF AS LIBERAL OR
CONSERVATIVE at .
089.

What SPSS Displays when Nothing is Significant
If none of the independent
variables has a statistically
significant relationship to the
dependent variable, SPSS
displays an empty table for
Variables Entered/Removed.

The Problem in BlackBoard - 1
The introductory problem statement tells us:
• the data set to use:
GSS2002_PrejudiceAndAltruism.SAV
• the method for including variables in the regression
• The dependent variable for the analysis
• the list of independent variables that stepwise
regression will select from

This Week’s Problems
 The problems this week take the 13 questions on prejudice from the general social survey
and explore the relationship of each to the demographic characteristics of age, education,
income, political views (conservative versus liberal), religiosity (attendance at church),
socioeconomic index, gender, and race.
 I had no specific hypothesis about which demographic factors would be related to which
question on prejudice, beyond an expectation that race would be a significant contributor
to explaining differences on each of the questions.
 My analyses were exploratory (to identify what demographic characteristics were
associated with different aspects of prejudice) and, thus, appropriate for stepwise
regression.

In these problems, we will assume that our data
satisfies the assumptions required by multiple
regression without explicitly testing for it.
We should recognize that failing to use a
needed transformation could preclude a variable
from being selected as a predictor.
In your analyses, you would, of course, want to
test for conformity to all of the assumptions.

The next sequence of specific instructions tell us
whether each variable should be treated as
metric or non-metric, along with the reference
category to use when dummy-coding non-
metric variables.
Though we will not use the script to test for
assumptions, we can use it to do the dummy
coding that we need for the problem.

The next pair of instructions tell us the
probability values to use for alpha for
both the tests of statistical
relationships and for the diagnostic
tests.

The final instruction tells us the
random number seed to use in
the validation analysis.
If you do not use this number
for the seed, it is likely that
you will get different results
from those shown in the
feedback.

The Statement about Level of Measurement
The first statement in the problem asks about
level of measurement. Stepwise multiple
regression requires the dependent variable and
the metric independent variables be interval
level, and the non-metric independent variables
be dummy-coded if they are not dichotomous.
The only way we would violate the level of
measurement would be to use a nominal
variable as the dependent variable, or to
attempt to dummy-code an interval level
variable that was not grouped.

Marking the Statement about Level of Measurement - 1
Mark the check box as a correct statement because:
• "Importance of ethnic identity" [ethimp] is ordinal level, but the
problem calls for treating it as metric, applying the common
convention of treating ordinal variables as interval level.
• The metric independent variable "age" [age] was interval level,
satisfying the requirement for independent variables.
• The metric independent variable "highest year of school
completed" [educ] was interval level, satisfying the requirement
for independent variables.
• "Income" [rincom98] is ordinal level, but the problem calls for
treating it as metric, applying the common convention of
treating ordinal variables as interval level.
Stepwise multiple regression requires the dependent
variable and the metric independent variables be interval
level, and the non-metric independent variables be
dummy-coded if they are not dichotomous.

Marking the Statement about Level of Measurement - 2
In addition:
• "Description of political views" [polviews] is ordinal level, but the
problem calls for treating it as metric, applying the common
convention of treating ordinal variables as interval level.
• "Frequency of attendance at religious services" [attend] is
ordinal level, but the problem calls for treating it as metric,
applying the common convention of treating ordinal variables as
interval level.
• The metric independent variable "socioeconomic index" [sei]
was interval level, satisfying the requirement for independent
variables.
• The non-metric independent variable "sex" [sex] was
dichotomous level, satisfying the requirement for independent
variables.
• The non-metric independent variable "race of the household"
[hhrace] was nominal level, but will satisfy the requirement for
independent variables when dummy coded.

The Statement for Sample Size
The statement for sample
size indicates that the
available data satisfies the
requirement.
Because of the tendency for
stepwise regression to over-fit
the data, we have a larger
sample size requirement, i.e. 40
cases per independent variable
(Tabachnick and Fidell, p. 117)
To obtain the number of cases
available for this analysis, we run
the stepwise regression.

Using the Script to Create Dummy-coded Variables - 1
Before we can run the
stepwise regression, we need
to dummy code sex and race.
We will use the script to
create the dummy-coded
variables.
Select the Run
Script command
from the Utilities
menu.

Navigate to the
My Documents
folder, if
necessary.
Highlight the script file
SatisfyingRegressionAssumptionsWit
h
MetricAndNonMetricVariables.SBS.
Click on the Run button
to open the script.

Move the non-metric
variable "sex" [sex] to the
list box for Non-metric
independent variables list
box.
With the variable highlighted,
select the reference category,
2=FEMALE from the
Reference category drop
down menu.

Move the non-metric variable
"race of the household"
[hhrace] to the list box for Non-
metric independent variables list
box.
With the variable highlighted,
select the reference category,
3=OTHER from the Reference
category drop down menu.
The OK button to run the
regression is deactivated until
we select a dependent
variable.

We select the dependent variable
"importance of ethnic identity"
[ethimp], though since we are
not going to interpret the output,
we could select any variable.
To have the script save the dummy-
coded variables, clear the check box
Delete variables created in this analysis.

Click on the OK button to run the
regression, creating the dummy-
coded variables as a by-product.

The Dummy-Coded Variables in the Data Editor
If we scroll the variable list to
the right, we see that the
three dummy-coded variables
have been added to the data
set.

Run the Stepwise Regression - 1
To run the regression, select
Regression > Linear from
the Analyze menu.

Move the dependent variable
•"importance of ethnic identity"
[ethimp]
to the Dependent text box.
Move the independent variables:
•"age" [age]
•"highest year of school completed" [educ],
•"income" [rincom98],
•"description of political views" [polviews],
•"frequency of attendance at religious services"
[attend],
•"socioeconomic index" [sei],
•“survey respondents were male" [sex_1],
•"survey respondents who were white" [hhrace_1],
•"survey respondents who were black" [hhrace_2]
to the Independent(s) list box.

Select Stepwise from the
Method drop down menu.
The critical step to produce a
stepwise regression is the selection
of the method for entering
variables.

Click on the Statistics
button to specify additional
output.

We mark the check boxes for
optional statistics:
• R squared change,
• Descriptives,
• Part and partial
correlations,
• Collinearity diagnostics, and
• Durbin-Watson.
Click on the
Continue button to
close the dialog box.

Click on the OK
button to produce
the output.

Answering the Sample Size Question
The analysis included 9 independent variables (6 metric
independent variables plus 3 dummy-coded variables). The
number of cases available for the analysis was 209, not
satisfying the requirement for 360 cases based on the rule
of thumb that the required number of cases for stepwise
multiple regression should be 40 x the number of
independent variables recommended by Tabachnick and
Fidell (p. 117).
We should consider mentioning the sample size issue as a
limitation of the analysis.

Marking the Statement for Sample Size
The check box is not marked
because we did not satisfy
the sample size
requirement.

Statements about Variables Included in Stepwise Regression
Three statements in the problem list
different combinations of the
variables included in the stepwise
regression.
To determine which is correct, we
look at the table of Variables Entered
and Removed in the SPSS output.

Answering the Question about Variables Included in Stepwise Regression - 1
Three independent variables satisfied
the statistical criteria for entry into the
model. The variable "survey
respondents who were white"
[hhrace_1] had the largest individual
impact on the dependent variable
"importance of ethnic identity" [ethimp].
The second variable included in the
model was "survey respondents who
were black" [hhrace_2]. The third
variable included in the model was
"frequency of attendance at religious
services" [attend].
The column for Variables
Removed is empty, telling us
that no variables were
removed after being
entered.

Marking the Statement about Variables Included in Stepwise Regression
Three independent variables satisfied the
statistical criteria for entry into the model.
The variable "survey respondents who
were white" [hhrace_1] had the largest
individual impact on the dependent
variable "importance of ethnic identity"
[ethimp]. The second variable included in
the model was "survey respondents who
were black" [hhrace_2]. The third variable
included in the model was "frequency of
attendance at religious services" [attend].
We mark the check box for the first of the
three statements.

Statement about the Strength of the Relationship
The next two statements focus on the
strength of the overall relationship
between the dependent variable and
the set of predictors that are selected
in the stepwise entry of variables. The
statement assumes that the overall
relationship will be statistically
significant, which will be true if any
variables are selected for the model.
We will use Cohen’s scale for
assigning an adjective to the
strength of the relationship:
•less than .10 = trivial
•.10 up to 0.30 = weak
•.30 up to .50 = moderately strong
•.50 or greater = strong

Statement about the Strength of the Relationship
The overall relationship was
statistically significant (F(3, 205)
= 11.11, p < .001. The null
hypothesis that "all of the partial
slopes (b coefficients) = 0" is
rejected, supporting the
research hypothesis that "at
least one of the partial slopes (b
coefficients) is not equal to 0".
Applying Cohen's criteria for effect size,
the relationship was correctly characterized
as moderately strong (Multiple R = .374).
Three independent variables satisfied the statistical
criteria for inclusion in the model. We interpret the
results for the last step for all of the questions about
statistical relationships (Model 3 in this example).

Marking the Statement about the Strength of the Relationship
The Multiple R of .374 translates to a
moderately strong relationship, so we
mark the check box for the second
statement on strength of the
relationship.

Statements about Relationships to the Dependent Variable for Individual Predictors
The next set of statements focus on
individual relationships between
predictors and the dependent
variable. In order for a statement to
be true, it must have a statistically
significant individual relationship (i.e.
it entered into the model), and the
direction of the relationship must be
interpreted correctly.

Answering Question about Relationship of RACE OF HOUSEHOLD=WHITE
Again, we base our interpretation about
statistical relationships on the last model for
variables entered, i.e. Model 3 for this problem.
We reject the null hypothesis that the partial slope (b
coefficient) for the variable "survey respondents who
were white" = 0 and conclude that the partial slope (b
coefficient) for the variable "survey respondents who
were white" is not equal to 0. The negative sign of the b
coefficient (-0.518) means that survey respondents who
were white attached less importance to ethnic identity
compared to the average for all survey respondents.
The statement that "survey respondents who were white
attached less importance to ethnic identity compared to the
average for all survey respondents" is correct. The
individual relationship between the independent variable
"survey respondents who were white" [hhrace_1] and the
dependent variable "importance of ethnic identity" [ethimp]
was statistically significant, ß = -.290, t(199) = -4.38, p < .
001.

Marking the Statement about Relationship of RACE OF HOUSEHOLD=WHITE
Since the statement “survey respondents
who were white attached less importance
to ethnic identity compared to the
average for all survey respondents” is
supported by our statistical results, we
mark the check box.

Answering Question about Relationship of RACE OF HOUSEHOLD=BLACK
We reject the null hypothesis that the partial slope (b coefficient)
for the variable "survey respondents who were black" = 0 and
conclude that the partial slope (b coefficient) for the variable
"survey respondents who were black" is not equal to 0. The
positive sign of the b coefficient (0.524) means that survey
respondents who were black attached greater importance to ethnic
identity compared to the average for all survey respondents.
The statement that "survey respondents who were black
attached greater importance to ethnic identity compared to
the average for all survey respondents" is correct. The
individual relationship between the independent variable
"survey respondents who were black" [hhrace_2] and the
dependent variable "importance of ethnic identity" [ethimp]
was statistically significant, ß = .225, t(199) = 3.37, p < .001.

Marking the Statement about Relationship of RACE OF HOUSEHOLD=BLACK
who were black attached greater
importance to ethnic identity compared
to the average for all survey
respondents" is supported by our
statistical results, we mark the check
box.
Since the previous statement was
correct, this statement cannot be
true, so the check box is not
marked.

Answering Question about Relationship of ATTEND RELIGIOUS SERVICES
We reject the null hypothesis that the partial slope (b coefficient) for the
variable "frequency of attendance at religious services" = 0 and conclude that
the partial slope (b coefficient) for the variable "frequency of attendance at
religious services" is not equal to 0. The positive sign of the b coefficient
(0.062) means that higher values of frequency of attendance at religious
services were associated with higher values of "importance of ethnic
identity".
The statement that "survey respondents who attended
religious services more often attached greater importance to
ethnic identity" is correct. The individual relationship between
the independent variable "frequency of attendance at religious
services" [attend] and the dependent variable "importance of
ethnic identity" [ethimp] was statistically significant, ß = .141,
t(199) = 2.16, p = .032.

Marking the Statement about Relationship of ATTEND RELIGIOUS SERVICES
who attended religious services more
often attached greater importance to
ethnic identity" is supported by our
statistical results, we mark the check
box.
The following check box is not marked
because the statement contradicts the
finding we have just made.

Answering Question about Relationship of AGE
The statement that "survey respondents
who were older attached greater importance
to ethnic identity" is not correct. The
variable "age" [age] was not among the list
of variables included in the stepwise model.

Marking the Statement for Age
The check box for the
statement for age is not
marked because the variable
did not enter the model in
the stepwise regression.

Statement about Cross-validation
The final statement concerns the
generalizability of our findings to
the larger population. To answer
this question, we will do a
75/25% cross-validation.
The findings from our analysis are
generalizable to the extent that they are
applicable to cases not included in the
analysis. Since we cannot collect new
cases, we will divide our sample into two
subsets, using one subset to create the
model and test the findings on the second
subset of cases which were not included in
the analysis that created the model.

Creating the Training Sample and the Validation Sample - 1
The 75/25% cross-validation requires
that we randomly divide the cases for
this analysis into two parts:
75% of the cases will be used to run
the stepwise regression (the training
sample), which will be tested for
accuracy on the remaining 25% of the
cases (the validation sample).
To set the seed for the random
number generator, select
Random Number Generator
from the Transform menu.
NOTE: you must use the random number
seed that is stated in the problem in order
to produce the same results that I found.
Any other seed will generate a different
random sequence that can produce results
that are very different from mine.

Third, type the seed
number provided in the
problem directions: 726201.
First, mark the check
for Set Starting Point.
Second, select
the option button
for a Fixed
Value.
Fourth, click on the
OK button to complete
the action.
NOTE: SPSS does not provide any
feedback that the seed has been set or
changed. If you are in doubt, you can
reopen the dialog box and see what it
indicates.

We will create a variable that will
contain the information about whether a
case is in the training sample or the
validation sample. We will name this
variable “split” and use a value of 1 to
indicate the training sample and a value
of 0 to indicate the validation sample.
To create the new
variable, select Compute
from the Transform
menu.

Type the name of
the new variable,
split, in the Target
Variable text box.
Type the formula as
shown in the
Numeric Expression
text box.
Click on the OK
button to create
the variable.
The formula uses the SPSS UNIFORM
function to create a uniform distribution
of decimal numbers between 0 and 1. If
the generated number for a case is less
than or equal to 0.75, the statement in
the text box is True and the split
variable will be assigned a 1 for that
case. If the generated number is larger
than 0.75, the statement is false and
the case will be assigned a 0 for split.

If we scroll the data editor
window to the right, we see
the split variable in a new
column.

If we created a frequency distribution for
the split variable, we see that the
breakdown is approximately, not exactly,
correct. This is a consequence of
generating random numbers – you have
no control over the sequence that it
generates beyond setting an initial seed.
Though I have done it to create
specific results for homework
problems, it is not acceptable to
run repeated series of random
numbers until one gets a
sequence that has desirable
properties.

An Additional Task before Running the Stepwise Regression on the Training Sample
 Before we run the regression on the training sample, we need an additional step that will
enable us to compare the accuracy of the model for the training sample to the accuracy of
the model for the validation sample, using the R2
for each as our measure of accuracy.
 We need to exclude from the analysis cases that are missing data for any of the variables
that we have designated as candidates for inclusion. If we don’t specifically do this, SPSS
may include different cases in predicting values for the dependent variable than it does in
determining which variables to include in the model.
 In model building, SPSS does listwise exclusion of missing data and omits any cases that
have missing data for any variable. In predicting scores on the dependent variable, it
excludes cases that are missing data for only the variables included in the stepwise model.
Thus, when selecting variables, SPSS assumes that only respondents who answer all
questions are valid cases; in predicting scores, it assumes that failing to answer a question
on a variable that is not included has no importance in the analysis.

Selecting Cases with Valid Data for All Variables in the Analysis - 1
To include only those
cases that have valid
data for all variables in
the analysis, choose the
Select Cases command
from the Data menu.

First, mark the
option button for If
condition is
satisfied.
Second, click on
the If button to
add the condition.

Type
NMISS(ethimp,age,educ,rincom98,polviews,
attend,sei,sex_1,hhrace_1,hhrace_2) = 0
in the condition textbox. In the parentheses,
we type the names of the dependent variable
and all of the independent variables.
The SPSS NMISS function counts the number
of variables in the list that have missing data.
Telling SPSS to include cases for which this
calculation results in 0 indicates that the case
was not missing data for any of the variables.

Click on the
Continue button to

Click on the OK
button to
execute the
command.

The excluded cases
have a slash
through the case
number.

Run the Stepwise Regression on the Training Sample - 1
To run the regression, select
Regression > Linear from
the Analyze menu.

Move the dependent variable
•"importance of ethnic identity"
[ethimp]
to the Dependent text box.
Move the independent variables:
•"age" [age]
•"highest year of school completed" [educ],
•"income" [rincom98],
•"description of political views" [polviews],
•"frequency of attendance at religious services"
[attend],
•"socioeconomic index" [sei],
•“survey respondents were male" [sex_1],
•"survey respondents who were white" [hhrace_1],
•"survey respondents who were black" [hhrace_2]
to the Independent(s) list box.

Select Stepwise from the
Method drop down menu.
The critical steps to produce a
stepwise regression on the training
sample are the selection of the
stepwise method for entering
variables and the inclusion of the
training sample cases.

First, highlight
the split variable.
To select the training sample, we
move the split variable to the
Selection Variable text box.
Second, click on the
right arrow button to the
left of the Selection
Variable text box..

Click on the Rule button
to specify the value that
we want split to use to
select cases.

First, type 1 in
the Value text
box. Recall that
this is the value
of split indicating
training cases.
Second, click on the
Continue button to

Click on the Statistics
button to specify additional
output.

• Descriptives,
correlations,
• Durbin-Watson.
Click on the
Continue button to

Click on the OK
button to produce
the output.

Validating the Model - 1
The first step in our validation is to make
certain that the model based on the
training sample reasonably approximates
the model based on the full sample.
Here we see that both models included 3
variables.
If the number of models
(steps) were different, the
validation would fail.

Second, we verify that the model
based on the training sample
included the same three variables as
the model based on the full data set.
We do not require that the variables
be entered in the same order, as the
difference in samples can easily
result in small shifts.
The same variables entered into the
stepwise regression of the training sample
that entered into the stepwise regression
using the full sample ("frequency of
attendance at religious services" [attend],
"survey respondents who were black"
[hhrace_2] and "survey respondents who
were white" [hhrace_1]).

Third, we compare the accuracy of the model
for the validation sample to the accuracy of
the model for the training sample.
We have to calculate the R² for the
validation sample (split ~= 1.0) by hand
from the Multiple R: .402² = .162.
The R² for the 75% training sample was
0.131 and the R² for the 25% validation
sample was 0.162, resulting in a value of
.131 – 162 = -.031 for shrinkage. Since
-.031 is <= .02, the validation is
successful.
If the shrinkage were greater
than .02 (2%), the validation
fails.

Marking the Check Box for the Cross-validation Statement
The validation analysis supported
the generalizability of the findings
of the analysis to the population
represented by the sample in the
data set.
We mark the check box for the
validation.

The Question Graded in Blackboard
When the problem was
submitted, BlackBoard
confirmed that all marked
answers were correct.

Logic Diagram for Solving Homework Problems: Level of Measurement
No
No
Ordinal level variable
treated as metric?
• Do not mark check box
• Mark: Inappropriate
application of the
statistic
• Stop
Yes
Yes
Level of
measurement ok?
Consider limitation in
discussion of findings
Run script to dummy-code
non-metric variables, if needed
Run stepwise regression

Logic Diagram for Solving Homework Problems: Sample Size and Overall Relationship
• Do not mark check box
• Consider limitation in
discussion of findings
Yes
Sample size ok
(number of Iv’s x 40)?
No
Mark check box
for correct sample size
1+ variables entered
in model?
No
Yes
Model is not trivial
(Multiple R >= .10)
No
Yes
Stop (no significant
predictors)
Stop (model is
not usable)
Model will be
statistically
significant if
any
variables
entered

Logic Diagram for Solving Homework Problems: Strength of Overall Relationship
Do not mark check box
Yes
Strength of model
correctly characterized No
Mark check box
for correct strength
Do not mark check box for
correct subset
Yes
Subset of entered
variables correctly
identified?
No
Mark check box
for correct subset

Yes
Variable entered
and not removed? No
Mark check box
for individual relationship
Correct interpretation of
direction of relationship?
Yes
individual relationship
No
Logic Diagram for Solving Homework Problems: Individual Relationships
Additional variables
entered?
No
Yes

Logic Diagram for Solving Homework Problems: Cross-validation
Create split variable
using specified seed
Select cases with no missing
values for all variables
Run stepwise regression
on training sample
Same variables entered
in full model?
Yes
supporting validation
No
Shrinkage
< or = 2%?
Yes
Mark check box for
No

Solving stepwise regression problems

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