Multinomial logistic regression

1. SW388R7
Data Analysis &
Computers II
Slide 1
Multinomial Logistic Regression
Basic Relationships
Multinomial Logistic Regression
Describing Relationships
Classification Accuracy
Sample Problems

2. Compu
ters II
Multinomial logistic regression
Slide 2

Multinomial logistic regression is used to analyze relationships
between a non-metric dependent variable and metric or
dichotomous independent variables.

Multinomial logistic regression compares multiple groups
through a combination of binary logistic regressions.

The group comparisons are equivalent to the comparisons for a
dummy-coded dependent variable, with the group with the
highest numeric score used as the reference group.

For example, if we wanted to study differences in BSW, MSW,
and PhD students using multinomial logistic regression, the
analysis would compare BSW students to PhD students and MSW
students to PhD students. For each independent variable, there
would be two comparisons.

3. Compu
ters II
What multinomial logistic regression predicts
Slide 3

Multinomial logistic regression provides a set of coefficients for
each of the two comparisons. The coefficients for the
reference group are all zeros, similar to the coefficients for the
reference group for a dummy-coded variable.

Thus, there are three equations, one for each of the groups
defined by the dependent variable.

The three equations can be used to compute the probability
that a subject is a member of each of the three groups. A case
is predicted to belong to the group associated with the highest
probability.

Predicted group membership can be compared to actual group
membership to obtain a measure of classification accuracy.

4. Compu
ters II
Level of measurement requirements
Slide 4

Multinomial logistic regression analysis requires that the
dependent variable be non-metric. Dichotomous, nominal, and
ordinal variables satisfy the level of measurement requirement.

Multinomial logistic regression analysis requires that the
independent variables be metric or dichotomous. Since SPSS
will automatically dummy-code nominal level variables, they
can be included since they will be dichotomized in the analysis.

In SPSS, non-metric independent variables are included as
“factors.” SPSS will dummy-code non-metric IVs.

In SPSS, metric independent variables are included as
“covariates.” If an independent variable is ordinal, we will
attach the usual caution.

5. Compu
ters II
Assumptions and outliers
Slide 5

Multinomial logistic regression does not make any assumptions
of normality, linearity, and homogeneity of variance for the
independent variables.

Because it does not impose these requirements, it is preferred
to discriminant analysis when the data does not satisfy these
assumptions.

SPSS does not compute any diagnostic statistics for outliers. To
evaluate outliers, the advice is to run multiple binary logistic
regressions and use those results to test the exclusion of
outliers or influential cases.

6. Compu
ters II
Sample size requirements
Slide 6

The minimum number of cases per independent variable is 10,
using a guideline provided by Hosmer and Lemeshow, authors of
Applied Logistic Regression, one of the main resources for
Logistic Regression.

For preferred case-to-variable ratios, we will use 20 to 1.

7. Compu
ters II
Methods for including variables
Slide 7

The only method for selecting independent variables in SPSS is
simultaneous or direct entry.

8. Compu
ters II
Overall test of relationship - 1
Slide 8

The overall test of relationship among the independent
variables and groups defined by the dependent is based on the
reduction in the likelihood values for a model which does not
contain any independent variables and the model that contains
the independent variables.

This difference in likelihood follows a chi-square distribution,
and is referred to as the model chi-square.

The significance test for the final model chi-square (after the
independent variables have been added) is our statistical
evidence of the presence of a relationship between the
dependent variable and the combination of the independent
variables.

9. Compu
ters II
Slide 9
Overall test of relationship - 2
Model Fitting Information
Model
Intercept Only
Final
-2 Log
Likelihood
284.429
265.972
Chi-Square
18.457
df
Sig.
6
.005
The presence of a relationship between the dependent
variable and combination of independent variables is
based on the statistical significance of the final model
chi-square in the SPSS table titled "Model Fitting
Information".
In this analysis, the probability of the model chi-square
(18.457) was 0.005, less than or equal to the level of
significance of 0.05. The null hypothesis that there was
no difference between the model without independent
variables and the model with independent variables
was rejected. The existence of a relationship between
the independent variables and the dependent variable
was supported.

10. ters II
Strength of multinomial logistic regression
relationship
Slide
10

While multinomial logistic regression does compute correlation
measures to estimate the strength of the relationship (pseudo R
square measures, such as Nagelkerke's R²), these correlations
measures do not really tell us much about the accuracy or
errors associated with the model.

A more useful measure to assess the utility of a multinomial
logistic regression model is classification accuracy, which
compares predicted group membership based on the logistic
model to the actual, known group membership, which is the
value for the dependent variable.

11. ters II
Slide
11
Evaluating usefulness for logistic models

The benchmark that we will use to characterize a multinomial
logistic regression model as useful is a 25% improvement over
the rate of accuracy achievable by chance alone.

Even if the independent variables had no relationship to the
groups defined by the dependent variable, we would still
expect to be correct in our predictions of group membership
some percentage of the time. This is referred to as by chance
accuracy.

The estimate of by chance accuracy that we will use is the
proportional by chance accuracy rate, computed by summing
the squared percentage of cases in each group. The only
difference between by chance accuracy for binary logistic
models and by chance accuracy for multinomial logistic models
is the number of groups defined by the dependent variable.

12. ters II
Slide
12
Computing by chance accuracy
The percentage of cases in each group defined by the dependent
variable is found in the ‘Case Processing Summary’ table.
Case Processing Summary
N
HIGHWAYS
AND BRIDGES
Valid
Missing
Total
Subpopulation
1
2
3
62
93
12
167
103
270
153a
Marginal
Percentage
37.1%
55.7%
7.2%
100.0%
a. The dependent variable has only one value observed
in 146 (95.4%) subpopulations.
The proportional by chance accuracy rate was
computed by calculating the proportion of cases for
each group based on the number of cases in each
group in the 'Case Processing Summary', and then
squaring and summing the proportion of cases in each
group (0.371² + 0.557² + 0.072² = 0.453).
The proportional by chance accuracy criteria is 56.6%
(1.25 x 45.3% = 56.6%).

13. ters II
Slide
13
Comparing accuracy rates

To characterize our model as useful, we compare the overall
percentage accuracy rate produced by SPSS at the last step in which
variables are entered to 25% more than the proportional by chance
accuracy. (Note: SPSS does not compute a cross-validated accuracy
rate for multinomial logistic regression .)
Classification
Predicted
Observed
1
2
3
Overall Percentage
1
15
7
5
16.2%
2
47
86
7
83.8%
3
0
0
0
.0%
The classification accuracy rate was 60.5%
which was greater than or equal to the
proportional by chance accuracy criteria of
56.6% (1.25 x 45.3% = 56.6%).
The criteria for classification accuracy is
satisfied in this example.
Percent
Correct
24.2%
92.5%
.0%
60.5%

14. ters II
Slide
14
Numerical problems




The maximum likelihood method used to calculate multinomial
logistic regression is an iterative fitting process that attempts
to cycle through repetitions to find an answer.
Sometimes, the method will break down and not be able to
converge or find an answer.
Sometimes the method will produce wildly improbable results,
reporting that a one-unit change in an independent variable
increases the odds of the modeled event by hundreds of
thousands or millions. These implausible results can be
produced by multicollinearity, categories of predictors having
no cases or zero cells, and complete separation whereby the
two groups are perfectly separated by the scores on one or
more independent variables.
The clue that we have numerical problems and should not
interpret the results are standard errors for some independent
variables that are larger than 2.0.

15. ters II
Relationship of individual independent
variables and the dependent variable
Slide
15

There are two types of tests for individual independent
variables:
 The likelihood ratio test evaluates the overall relationship
between an independent variable and the dependent
variable
 The Wald test evaluates whether or not the independent
variable is statistically significant in differentiating between
the two groups in each of the embedded binary logistic
comparisons.

If an independent variable has an overall relationship to the
dependent variable, it might or might not be statistically
significant in differentiating between pairs of groups defined by
the dependent variable.

16. ters II
Relationship of individual independent
variables and the dependent variable
Slide
16

The interpretation for an independent variable focuses on its
ability to distinguish between pairs of groups and the
contribution which it makes to changing the odds of being in
one dependent variable group rather than the other.

We should not interpret the significance of an independent
variable’s role in distinguishing between pairs of groups unless
the independent variable also has an overall relationship to the
dependent variable in the likelihood ratio test.

The interpretation of an independent variable’s role in
differentiating dependent variable groups is the same as we
used in binary logistic regression. The difference in
multinomial logistic regression is that we can have multiple
interpretations for an independent variable in relation to
different pairs of groups.

17. ters II
Relationship of individual independent
variables and the dependent variable
Slide
17
Parameter Estimates
HIGHWAYS
a
AND BRIDGES
1
2
Intercept
AGE
EDUC
CONLEGIS
Intercept
AGE
EDUC
CONLEGIS
B
3.240
.019
.071
-1.373
3.639
.003
.172
-1.657
Std. Error
2.478
.020
.108
.620
2.456
.020
.110
.613
95% Confidence Interva
Exp(B)
SPSS identifies the comparisons Exp(B)
it makes for Bound Upper B
Wald
df
Sig.
Lower
groups defined by1the dependent variable in
1.709
.191
the table of ‘Parameter Estimates,’ 1.019 either .980
using
.906
1
.341
the value codes or the value labels, depending
.427
1
.514
1.073
on the options settings for pivot table labeling. .868
4.913
1
.027
.253
.075
The 2.195
reference category is .138
identified in the
1
footnote to the table.
.017
1
.897
1.003
.963
In this analysis, two comparisons will be
2.463
1
.117
1.188
.958
made:
7.298
1
.007
.191
.057
a. The reference category is: 3.
HIGHWAYS
a
AND BRIDGES
TOO LITTLE
ABOUT RIGHT
Intercept
AGE
EDUC
CONLEGIS
Intercept
AGE
EDUC
CONLEGIS
•the TOO LITTLE group (coded 1, shaded
blue) will be compared to the TOO MUCH
Parameter Estimates
group (coded 3, shaded purple)
•the ABOUT RIGHT group (coded 2 ,
shaded orange)) will be compared to the
TOO MUCH group (coded 3, shaded
purple). Wald
Std. Error
df
Sig.
Exp(B)
B
3.240
2.478
1.709
1
.191
The reference category plays the same role in
.019
.020
.906
1
.341
multinomial logistic regression that it plays in
.071
.108
.427
1
.514
the dummy-coding of a nominal variable: it is
the category that4.913
would be coded with .027
zeros
-1.373
.620
1
for all of the dummy-coded variables that all
3.639
2.456
2.195
1
.138
other categories are interpreted against.
.003
.020
.017
1
.897
.172
.110
2.463
1
.117
-1.657
.613
7.298
1
.007
a. The reference category is: TOO MUCH.
1.019
1.073
.253
1.003
1.188
.191
95% C
Lower B

18. ters II
Relationship of individual independent
variables and the dependent variable
Slide
18
Likelihood Ratio Tests
Effect
Intercept
AGE
EDUC
CONLEGIS
-2 Log
Likelihood of
Reduced
Model
268.323
268.625
270.395
275.194
Chi-Square
2.350
2.652
4.423
9.221
df
2
2
2
2
Sig.
.309
.265
.110
.010
In this example, there is a
statistically significant
relationship between the
independent variable
CONLEGIS and the dependent
variable. (0.010 < 0.05)
The chi-square statistic is the difference in -2 log-likelihoods
between the final model and a reduced model. The reduced model is
Parameter Estimates
formed by omitting an effect from the final model. The null hypothesis
is that all parameters of that effect are 0.
HIGHWAYS
a
AND BRIDGES
1
2
B
3.240
.019
.071
-1.373
3.639
.003
.172
-1.657
Intercept
AGE
EDUC
CONLEGIS
Intercept
AGE
EDUC
CONLEGIS
Std. Error
2.478
.020
.108
.620
2.456
.020
.110
.613
Wald
1.709
.906
.427
4.913
2.195
.017
2.463
7.298
df
1
1
1
1
1
1
1
1
As well, the independent
variable CONLEGIS is
significant in distinguishing
both category 1 of 95% Confidence Interval f
the
dependent variable from Exp(B)
category 3 of the dependent
Sig.
Exp(B)
Lower
variable. (0.027 < 0.05) Bound Upper Bou
.191
.341
.514
.027
.138
.897
.117
.007
a. The reference category is: 3.
And the independent variable CONLEGIS is significant in
distinguishing category 2 of the dependent variable from
category 3 of the dependent variable. (0.007 < 0.05)
1.019
1.073
.253
.980
.868
.075
1.0
1.3
.8
1.003
1.188
.191
.963
.958
.057
1.0
1.4
.6

19. ters II
Interpreting relationship of individual independent
variables to the dependent variable
Slide
19
Likelihood Ratio Tests
Effect
Intercept
AGE
EDUC
CONLEGIS
-2 Log
Survey
Likelihood of respondents who had less confidence in congress (higher
values correspond to lower confidence) were less likely to be in the
Reduced
group ofChi-Square
survey respondents who thought we spend too little money
Model
df
Sig.
on highways and bridges (DV category 1), rather than the group of
268.323 respondents who thought we spend too much money on
2.350
2
.309
survey
268.625
2.652
.265
highways and bridges (DV 2
category 3).
270.395
4.423
2
.110
For each unit9.221
increase in confidence in Congress, the odds of being
275.194
2
.010
in the group of survey respondents who thought we spend too little
The chi-square statistic is theon highwayslog-likelihoods decreased by 74.7%. (0.253 – 1.0
money difference in -2 and bridges
between the final model-0.747)
= and a reduced model. The reduced model is
Parameter Estimates
formed by omitting an effect from the final model. The null hypothesis
is that all parameters of that effect are 0.
HIGHWAYS
a
AND BRIDGES
1
2
Intercept
AGE
EDUC
CONLEGIS
Intercept
AGE
EDUC
CONLEGIS
a. The reference category is: 3.
B
3.240
.019
.071
-1.373
3.639
.003
.172
-1.657
Std. Error
2.478
.020
.108
.620
2.456
.020
.110
.613
Wald
1.709
.906
.427
4.913
2.195
.017
2.463
7.298
df
1
1
1
1
1
1
1
1
Sig.
.191
.341
.514
.027
.138
.897
.117
.007
Exp(B)
95% Confidence Interval f
Exp(B)
Lower Bound
Upper Bou
1.019
1.073
.253
.980
.868
.075
1.0
1.3
.8
1.003
1.188
.191
.963
.958
.057
1.0
1.4
.6

20. ters II
Interpreting relationship of individual independent
variables to the dependent variable
Slide
20
Likelihood Ratio Tests
Effect
Intercept
AGE
EDUC
CONLEGIS
-2 Log
Likelihood of
Reduced
Model
268.323
268.625
270.395
275.194
Chi-Square
2.350
2.652
4.423
9.221
df
2
2
2
2
Sig.
.309
.265
.110
.010
Survey respondents who had less confidence in congress (higher
The chi-square statistic is the difference in -2 log-likelihoods confidence) were less likely to be in the
values correspond to lower
group of survey The reduced model is
between the final model and a reduced model. respondents who thought we spend about the right
Parameter Estimates
amount of money The null hypothesis
formed by omitting an effect from the final model.on highways and bridges (DV category 2), rather
than the group of survey respondents who thought we spend too
is that all parameters of that effect are 0.
much money on highways and bridges (DV Category 3).
HIGHWAYS
a
AND BRIDGES
1
2
B
Std. Error
Wald
df
Sig.
Exp(B)
For each unit increase in confidence in Congress, the odds of being
in
Intercept the group of survey respondents who thought we spend about the
3.240
2.478
1.709
1
.191
right amount of money on highways and bridges decreased by
AGE
.019
.020
1
.341
1.019
80.9%. (0.191 – 1.0 = 0.809) .906
EDUC
.071
.108
.427
1
.514
1.073
CONLEGIS
-1.373
.620
4.913
1
.027
.253
Intercept
3.639
2.456
2.195
1
.138
AGE
.003
.020
.017
1
.897
1.003
EDUC
.172
.110
2.463
1
.117
1.188
CONLEGIS
-1.657
.613
7.298
1
.007
.191
a. The reference category is: 3.
95% Confidence Interval f
Exp(B)
Lower Bound
Upper Bou
.980
.868
.075
1.0
1.3
.8
.963
.958
.057
1.0
1.4
.6

21. ters II
Relationship of individual independent
variables and the dependent variable
Slide
21
Likelihood Ratio Tests
Effect
Intercept
AGE
EDUC
POLVIEWS
SEX
-2 Log
Likelihood of
Reduced
Model
327.463a
333.440
329.606
334.636
338.985
Chi-Square
.000
5.976
2.143
7.173
11.521
df
Sig.
0
2
2
2
2
.
.050
.343
.028
.003
The chi-square statistic is the difference in -2 log-likelihoods
Parameter Estimates
between the final model and a reduced model. The reduced model
is formed by omitting an effect from the final model. The null
hypothesis is that all parameters of that effect are 0.
a.
a
NATCHLD
B
Std. Error
Wald
df
This reduced model is equivalent to the final2.233 because
model
TOO LITTLE
Intercept
8.434
14.261
1
omitting the effect does not increase the degrees of freedom.
AGE
-.023
.017
1.756
1
EDUC
-.066
.102
.414
1
POLVIEWS
-.575
.251
5.234
1
[SEX=1]
-2.167
.805
7.242
1
b
[SEX=2]
0
.
.
0
ABOUT RIGHT Intercept
4.485
2.255
3.955
1
AGE
-.001
.018
.003
1
EDUC
.011
.104
.011
1
POLVIEWS
-.397
.257
2.375
1
[SEX=1]
-1.606
.824
3.800
1
b
[SEX=2]
0
.
.
0
a. The reference category is: TOO MUCH.
In this example, there is
a statistically significant
relationship between SEX
and the dependent
variable, spending on
childcare assistance.
As well, SEX plays a
statistically significant role
in differentiating 95% Confidence Interval
the TOO
LITTLE group from the TOO
Exp(B)
MUCH Exp(B)
(reference) group.
Sig.
Lower Bound
Upper Bo
(0.007 < 0.5)
.000
.185
.977
.944
.520
.936
.766
.022
.563
.344
.007
.115
.024
.
.
.
However, SEX does not
.047differentiate the ABOUT
.955RIGHT .999
.965
group from the
TOO MUCH (reference)
.916
1.011
.824
group.(0.51 > 0.5)
.123
.673
.406
.051
.201
.040
.
.
.
1.
1.
.
.
1.
1.
1.
1.

22. ters II
Slide
22
Interpreting relationship of individual independent
variables and the dependent variable
Likelihood Ratio Tests
Effect
Intercept
AGE
EDUC
POLVIEWS
SEX
-2 Log
Likelihood of
Reduced
Model
Chi-Square
df
Sig.
327.463a
.000
0
.
Survey respondents who were2 male (code 1 for sex) were less likely
333.440
5.976
.050
to 329.606
be in the group of survey respondents who thought we spend too
2.143
2
.343
little money on childcare assistance (DV category 1), rather than the
334.636
2
.028
group of survey 7.173
respondents who thought we spend too much
money on childcare assistance (DV category 3).
338.985
11.521
2
.003
The chi-square statistic is the difference in -2 log-likelihoods
Survey respondents who were male were 88.5% less likely (0.115 –
Parameter Estimates
between the final model and a reduced model. The reduced model
1.0 = -0.885) to be in the group of survey respondents who thought
is formed by omittingspend too little final model. The null
we an effect from the money on childcare assistance.
hypothesis is that all parameters of that effect are 0.
a.
a
NATCHLD
B
Std. Error
Wald
df
Sig.
Exp(B)
This reduced model is equivalent to the final2.233 because
model
TOO LITTLE
Intercept
8.434
14.261
1
.000
omitting the effect does not increase the degrees of freedom.
AGE
-.023
.017
1.756
1
.185
.977
EDUC
-.066
.102
.414
1
.520
.936
POLVIEWS
-.575
.251
5.234
1
.022
.563
[SEX=1]
-2.167
.805
7.242
1
.007
.115
b
[SEX=2]
0
.
.
0
.
.
ABOUT RIGHT Intercept
4.485
2.255
3.955
1
.047
AGE
-.001
.018
.003
1
.955
.999
EDUC
.011
.104
.011
1
.916
1.011
POLVIEWS
-.397
.257
2.375
1
.123
.673
[SEX=1]
-1.606
.824
3.800
1
.051
.201
b
[SEX=2]
0
.
.
0
.
.
a. The reference category is: TOO MUCH.
95% Confidence Interval
Exp(B)
Lower Bound
Upper Bo
.944
.766
.344
.024
.
1.
1.
.
.
.965
.824
.406
.040
.
1.
1.
1.
1.

23. ters II
Interpreting relationships for independent
variable in problems
Slide
23

In the multinomial logistic regression problems, the problem
statement will ask about only one of the independent variables.
The answer will be true or false based on only the relationship
between the specified independent variable and the dependent
variable. The individual relationships between other
independent variables are the dependent variable are not used
in determining whether or not the answer is true or false.

24. ters II
Slide
24
Problem 1
11. In the dataset GSS2000, is the following statement true, false, or an incorrect application
of a statistic? Assume that there is no problem with missing data, outliers, or influential cases,
and that the validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "age" [age], "highest year of school completed" [educ] and "confidence in
Congress" [conlegis] were useful predictors for distinguishing between groups based on
responses to "opinion about spending on highways and bridges" [natroad]. These predictors
differentiate survey respondents who thought we spend too little money on highways and
bridges from survey respondents who thought we spend too much money on highways and
bridges and survey respondents who thought we spend about the right amount of money on
highways and bridges from survey respondents who thought we spend too much money on
highways and bridges.
Among this set of predictors, confidence in Congress was helpful in distinguishing among the
groups defined by responses to opinion about spending on highways and bridges. Survey
respondents who had less confidence in congress were less likely to be in the group of survey
respondents who thought we spend too little money on highways and bridges, rather than the
group of survey respondents who thought we spend too much money on highways and bridges.
For each unit increase in confidence in Congress, the odds of being in the group of survey
respondents who thought we spend too little money on highways and bridges decreased by
74.7%. Survey respondents who had less confidence in congress were less likely to be in the
group of survey respondents who thought we spend about the right amount of money on
highways and bridges, rather than the group of survey respondents who thought we spend too
much money on highways and bridges. For each unit increase in confidence in Congress, the
odds of being in the group of survey respondents who thought we spend about the right amount
of money on highways and bridges decreased by 80.9%.
1.
2.
3.
4.
True
True with caution
False
Inappropriate application of a statistic

25. ters II
Slide
25
Dissecting problem 1 - 1
11. In the dataset GSS2000, is the following statement true, false, or an incorrect application
of a statistic? Assume that there is no problem with missing data, outliers, or influential cases,
and that the validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "age" [age], "highest year of school completed" [educ] and "confidence in
Congress" [conlegis] were useful predictors for distinguishing between groups based on
responses to "opinion about spending on highways and bridges" [natroad]. These predictors
differentiate survey respondents who For thesewe spend too little money on highways and
thought problems, we will
bridges from survey respondents who assume that spend is nomuch money on highways and
thought we there too problem
bridges and survey respondents who thought we spend about the right amount of money on
with missing data, outliers, or
highways and bridges from survey respondents who thought wethe
influential cases, and that spend too much money on
highways and bridges.
validation analysis will confirm
the generalizability of the
Among this set of predictors, confidence in Congress was helpful in distinguishing among the
results
groups defined by responses to opinion about spending on highways and bridges. Survey
respondents who had less confidence in congress were less likely to be in the group of survey
In this money we are told and
respondents who thought we spend too littleproblem,on highways to bridges, rather than the
use we spend too much
group of survey respondents who thought 0.05 as alpha for the money on highways and bridges.
For each unit increase in confidence in Congress, logistic regression. in the group of survey
multinomial the odds of being
respondents who thought we spend too little money on highways and bridges decreased by
74.7%. Survey respondents who had less confidence in congress were less likely to be in the
group of survey respondents who thought we spend about the right amount of money on
highways and bridges, rather than the group of survey respondents who thought we spend too
much money on highways and bridges. For each unit increase in confidence in Congress, the
odds of being in the group of survey respondents who thought we spend about the right amount
of money on highways and bridges decreased by 80.9%.
1.
2.
3.
4.
True
True with caution
False
Inappropriate application of a statistic

26. ters II
Slide
26
Dissecting problem 1 - 2
The variables listed first in the problem
statement are the independent variables
(IVs): "age" [age], "highest year of school
11. In the dataset GSS2000,"confidence in
completed" [educ] and is the following statement true, false, or an incorrect application
of a statistic? Assume that there is no problem with missing data, outliers, or influential cases,
Congress" [conlegis].
and that the validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "age" [age], "highest year of school completed" [educ] and "confidence in
Congress" [conlegis] were useful predictors for distinguishing between groups based on
responses to "opinion about spending on highways and bridges" [natroad]. These predictors
differentiate survey respondents who thought we spend too little money on highways and
bridges from survey respondents who thought we spend too much money on highways and
bridges and survey respondents who thought we spend about the right amount of money on
highways and bridges from survey respondents who thought we spend too much money on
The variable used to define
highways and bridges.the dependent
groups is
variable (DV): "opinion about
Among this set of predictors, confidence in Congress was helpful in distinguishing among the
spending on highways and
groups defined by responses to opinion about spending on highways and bridges. Survey
respondents bridges" [natroad].
who had less confidence in congress were less likely to be in the group of survey
respondents who thought we spend too little money on highways and bridges, rather than the
group of survey respondents who thought we spend too much money on highways and bridges.
For each unit increase in confidence in Congress, the odds of being in the group of survey
respondents who thought we spend too little moneySPSS only supports direct or
on highways and bridges decreased by
simultaneous entry of independent in the
74.7%. Survey respondents who had less confidence in congress were less likely to be
group of survey respondents who thought we spend variables in multinomial logistic
about the right amount of money on
regression, so we have no choice of
highways and bridges, rather than the group of survey respondents who thought we spend too
much money on highways and bridges. For each unitmethod for entering variables.
increase in confidence in Congress, the
odds of being in the group of survey respondents who thought we spend about the right amount
of money on highways and bridges decreased by 80.9%.

27. ters II
Slide
27
Dissecting problem 1 - 3
SPSS multinomial logistic regression models the relationship by
comparing each of the groups defined by the dependent variable to the
group with the highest code value.
11. In the dataset GSS2000, opinionfollowing statement true, false, or an incorrect application
The responses to is the about spending on highways and bridges were:
of a statistic? Assume that there is no problem with missing data, outliers, or influential cases,
and that the validation analysis will confirm the= Too much.
generalizability of the results. Use a level of
1= Too little, 2 = About right, and 3
significance of 0.05 for evaluating the statistical relationships.
The variables "age" [age], "highest year of school completed" [educ] and "confidence in
Congress" [conlegis] were useful predictors for distinguishing between groups based on
responses to "opinion about spending on highways and bridges" [natroad]. These predictors
differentiate survey respondents who thought we spend too little money on highways and
bridges from survey respondents who thought we spend too much money on highways and
bridges and survey respondents who thought we spend about the right amount of money on
highways and bridges from survey respondents who thought we spend too much money on
highways and bridges.
Among this set of predictors, confidence in Congress was helpful in distinguishing among the
groups defined by responses to opinion about spending on highways and bridges. Survey
respondents who had less confidence in congress were less likely to be in the group of survey
respondents who The analysis spend too in two money on highways and bridges, rather than the
thought we will result little comparisons:
group of survey respondents who thought we spend too spend too little money
• survey respondents who thought we much money on highways and bridges.
For each unit increase in confidence in Congress, the odds of being in the group of survey
versus survey respondents who thought we spend too much
respondents who thought we spend too and bridges on highways and bridges decreased by
money on highways little money
74.7%. Survey respondents respondents who thought wecongress were less likely to be in the
• survey who had less confidence in spend about the right
group of survey respondentsof money versus survey respondents whoamount of money on
who thought we spend about the right thought we
amount
highways and bridges, rather than the group of survey respondents who thought we spend too
spend too bridges. For on highways and bridges.
much money on highways and much money each unit increase in confidence in Congress, the
odds of being in the group of survey respondents who thought we spend about the right amount
of money on highways and bridges decreased by 80.9%.

28. ters II
Slide
28
Dissecting problem 1 - 4
Each problem includes a statement about the relationship between
one independent variable and the dependent variable. The answer
to the problem is based on the stated relationship, ignoring the
The variablesrelationships between the other independent variables and the
"age" [age], "highest year of school completed" [educ] and "confidence in
dependent variable.
Congress" [conlegis] were useful predictors for distinguishing between groups based on
responses to "opinion about spending on highways and bridges" [natroad]. These predictors
differentiate This problem identifies a difference forspendof the comparisons highways and
survey respondents who thought we both too little money on
bridges from among respondents who thought we spend too much money on highways and
survey groups modeled by the multinomial logistic regression.
bridges and survey respondents who thought we spend about the right amount of money on
highways and bridges from survey respondents who thought we spend too much money on
highways and bridges.
Among this set of predictors, confidence in Congress was helpful in distinguishing among the
groups defined by responses to opinion about spending on highways and bridges. Survey
respondents who had less confidence in congress were less likely to be in the group of
survey respondents who thought we spend too little money on highways and bridges, rather
than the group of survey respondents who thought we spend too much money on highways
and bridges. For each unit increase in confidence in Congress, the odds of being in the
group of survey respondents who thought we spend too little money on highways and
bridges decreased by 74.7%. Survey respondents who had less confidence in congress were
less likely to be in the group of survey respondents who thought we spend about the right
amount of money on highways and bridges, rather than the group of survey respondents
who thought we spend too much money on highways and bridges. For each unit increase in
confidence in Congress, the odds of being in the group of survey respondents who thought
we spend about the right amount of money on highways and bridges decreased by 80.9%.

29. ters II
Slide
29
Dissecting problem 1 - 5
11. In the dataset GSS2000, is the following statement true, false, or an incorrect application
of a statistic? Assume that there is no problem with missing data, outliers, or influential cases,
and that the validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "age" [age], "highest year of school completed" [educ] and "confidence in
Congress" [conlegis] were useful predictors for distinguishing between groups based on
responses to "opinion about spending on highways and bridges" [natroad]. These predictors
differentiate survey respondents who thought we spend too little money on highways and
bridges from survey respondents who thought we spend too much money on highways and
bridges and survey respondents who thought we spend about the right amount of money on
highways and bridges from survey respondents who thought we spend too much money on
highways and bridges.
Among this set of predictors, confidence in Congress was helpful in distinguishing among the
groups defined by responses to opinion about spending on highways and bridges. Survey
respondents who had less confidence in congress were less likely to be in the group of survey
respondents who thought we spend too little money on highways and bridges, rather than the
group of survey respondents who thought we spend too much money on highways and bridges.
In order for the multinomial logistic regression
For each unit increase in confidence in Congress, the odds of being in the group of survey
question to be on highways and bridges decreased
respondents who thought we spend too little money true, the overall relationship must by
be statistically significant, were less be no
74.7%. Survey respondents who had less confidence in congress there mustlikely to be in the
evidence of numerical problems, the classification
group of survey respondents who thought we spend about the right amount of money on
highways and bridges, rather than the accuracy rate must be substantiallythought we spend too
group of survey respondents who better than
much money on highways and bridges.couldeach unit increase in confidence in Congress, the
For be obtained by chance alone, and the
odds of being in the group of survey respondents who thought we spendbe statistically amount
stated individual relationship must about the right
of money on highways and bridges decreased by and interpreted correctly.
significant 80.9%.

30. ters II
Slide
30
Request multinomial logistic regression
Select the Regression |
Multinomial Logistic…
command from the
Analyze menu.

31. ters II
Slide
31
Selecting the dependent variable
First, highlight the
dependent variable
natroad in the list
of variables.
Second, click on the right
arrow button to move the
dependent variable to the
Dependent text box.

32. ters II
Slide
32
Selecting metric independent variables
Metric independent variables are specified as covariates
in multinomial logistic regression. Metric variables can
be either interval or, by convention, ordinal.
Move the metric
independent variables,
age, educ and conlegis to
the Covariate(s) list box.
In this analysis, there are no nonmetric independent variables. Nonmetric independent variables would be
moved to the Factor(s) list box.

33. ters II
Slide
33
Specifying statistics to include in the output
While we will accept most of
the SPSS defaults for the
analysis, we need to specifically
request the classification table.
Click on the Statistics… button
to make a request.

34. ters II
Slide
34
Requesting the classification table
First, keep the SPSS
defaults for Summary
statistics, Likelihood
ratio test, and
Parameter estimates.
Second, mark the
checkbox for the
Classification table.
Third, click
on the
Continue
button to
complete the
request.

35. ters II
Slide
35
Completing the multinomial
logistic regression request
Click on the OK
button to request
the output for the
multinomial logistic
regression.
The multinomial logistic procedure supports
additional commands to specify the model
computed for the relationships (we will use the
default main effects model), additional
specifications for computing the regression,
and saving classification results. We will not
make use of these options.

36. ters II
Slide
36
LEVEL OF MEASUREMENT - 1
11. In the dataset GSS2000, is the following statement true, false, or an incorrect application
of a statistic? Assume that there is no problem with missing data, outliers, or influential cases,
and that the validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "age" [age], "highest year of school completed" [educ] and "confidence in
Congress" [conlegis] were useful predictors for distinguishing between groups based on
responses to "opinion about spending on highways and bridges" [natroad]. These predictors
differentiate survey respondents who thought we spend too little money on highways and
bridges from survey respondents who thought we spend too much money on highways and
bridges and survey respondents who thought we spend about the right amount of money on
highways and bridges from survey respondents who thought we spend too much money on
highways and bridges.
Among this set of predictors, confidence in Congress was helpful in distinguishing among the
groups defined by responses to opinion about spending on highways and bridges. Survey
respondents who had less confidence in congressrequires that the to be in the group of survey
Multinomial logistic regression were less likely
respondents who thought we spend too little money andhighways and bridges, rather than the
dependent variable be non-metric on the
group of survey respondents who thought we spend too much money on highways and bridges.
independent variables be metric or dichotomous.
For each unit increase in confidence in Congress, the odds of being in the group of survey
respondents who thought we spend too little money on highways and bridges decreased by
"Opinion about spending on highways and
bridges" [natroad] is confidence in congress were less likely to be in the
74.7%. Survey respondents who had lessordinal, satisfying the nonmetric level of thought we spend about the the
group of survey respondents who measurement requirement forright amount of money on
dependent variable.
highways and bridges, rather than the group of survey respondents who thought we spend too
much money on highways and bridges. For each unit increase in confidence in Congress, the
It contains three respondents who thought we
odds of being in the group of surveycategories: survey respondents spend about the right amount
who thought we spend too
of money on highways and bridges decreased little money, about
the right amount of money, by 80.9%.
and too much
money on highways and bridges.
1. True
2. True with caution

37. ters II
Slide
37
LEVEL OF MEASUREMENT - 2
"Age" [age] and "highest year of
school completed" [educ] are interval,
11. satisfying the metric or dichotomous
In the dataset GSS2000, is the following statement true, false, or an incorrect application
of alevel of measurement requirement for
statistic? Assume that there is no problem with missing data, outliers, or influential cases,
independent variables.
and that the validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "age" [age], "highest year of school completed" [educ] and "confidence in
Congress" [conlegis] were useful predictors for distinguishing between groups based on
responses to "opinion about spending on highways and bridges" [natroad]. These predictors
differentiate survey respondents who thought we spend too little money on highways and
bridges from survey respondents who thought we spend too much money on highways and
bridges and survey respondents who thought we spend about the right amount of money on
highways and bridges from survey respondents who thought we spend too much money on
"Confidence in Congress" [conlegis] is ordinal,
highways and bridges. satisfying the metric or dichotomous level of
measurement requirement for independent
variables. If we follow the convention of treating
Among this set of predictors, confidence in Congress was helpfulthe distinguishing among the
ordinal level variables as metric variables, in level
groups defined by responses to opinion about spending on highways is bridges. Survey
of measurement requirement for the analysis and
respondents who had less confidence in congress analysts do not agree in the group of survey
satisfied. Since some data were less likely to be
with this convention, a note of caution should be
respondents who thought we spend too little money on highways and bridges, rather than the
included in our interpretation.
group of survey respondents who thought we spend too much money on highways and bridges.
For each unit increase in confidence in Congress, the odds of being in the group of survey
respondents who thought we spend too little money on highways and bridges decreased by
74.7%. Survey respondents who had less confidence in congress were less likely to be in the
group of survey respondents who thought we spend about the right amount of money on
highways and bridges, rather than the group of survey respondents who thought we spend too
much money on highways and bridges. For each unit increase in confidence in Congress, the
odds of being in the group of survey respondents who thought we spend about the right amount
of money on highways and bridges decreased by 80.9%.

38. ters II
Slide
38
Sample size – ratio of cases to variables
Case Processing Summary
N
HIGHWAYS
AND BRIDGES
Valid
Missing
Total
Subpopulation
1
2
3
62
93
12
167
103
270
153a
Marginal
Percentage
37.1%
55.7%
7.2%
100.0%
a. The dependent variable has only one value observed
Multinomial logistic regression requires that the minimum ratio
in 146 (95.4%) subpopulations.
of valid cases to independent variables be at least 10 to 1. The
ratio of valid cases (167) to number of independent variables
(3) was 55.7 to 1, which was equal to or greater than the
minimum ratio. The requirement for a minimum ratio of cases
to independent variables was satisfied.
The preferred ratio of valid cases to independent variables is
20 to 1. The ratio of 55.7 to 1 was equal to or greater than the
preferred ratio. The preferred ratio of cases to independent
variables was satisfied.

39. ters II
Slide
39
OVERALL RELATIONSHIP BETWEEN
INDEPENDENT AND DEPENDENT VARIABLES
Model Fitting Information
Model
Intercept Only
Final
-2 Log
Likelihood
284.429
265.972
Chi-Square
18.457
df
Sig.
6
.005
The presence of a relationship between the dependent
variable and combination of independent variables is
based on the statistical significance of the final model
chi-square in the SPSS table titled "Model Fitting
Information".
In this analysis, the probability of the model chi-square
(18.457) was 0.005, less than or equal to the level of
significance of 0.05. The null hypothesis that there was
no difference between the model without independent
variables and the model with independent variables
was rejected. The existence of a relationship between
the independent variables and the dependent variable
was supported.

40. ters II
Slide
40
NUMERICAL PROBLEMS
Parameter Estimates
HIGHWAYS
a
AND BRIDGES
1
2
Intercept
AGE
EDUC
CONLEGIS
Intercept
AGE
EDUC
CONLEGIS
a. The reference category is: 3.
B
3.240
.019
.071
-1.373
3.639
.003
.172
-1.657
Std. Error
2.478
.020
.108
.620
2.456
.020
.110
.613
Wald
1.709
.906
.427
4.913
2.195
.017
2.463
7.298
95% Confidence Inter
Exp(B)
Multicollinearity in the multinomial
df
Sig.
Exp(B)
logistic regression solution is Lower Bound Upper
1 by examining the standard
.191
detected
errors1for the .341
b coefficients. A
1.019
.980
standard error larger than 2.0
1
.514
1.073
.868
indicates numerical problems, such
1
.027
.253
.075
as multicollinearity among the
1
.138
independent variables, zero cells for
a dummy-coded independent
1
.897
1.003
.963
variable because all of the subjects
1
.117
1.188
.958
have the same value for the
1
.007
.191
variable, and 'complete separation' .057
whereby the two groups in the
dependent event variable can be
perfectly separated by scores on
one of the independent variables.
Analyses that indicate numerical
problems should not be interpreted.
None of the independent variables
in this analysis had a standard error
larger than 2.0. (We are not
interested in the standard errors
associated with the intercept.)

41. ters II
Slide
41
RELATIONSHIP OF INDIVIDUAL INDEPENDENT
VARIABLES TO DEPENDENT VARIABLE - 1
Likelihood Ratio Tests
Effect
Intercept
AGE
EDUC
CONLEGIS
-2 Log
Likelihood of
Reduced
Model
268.323
268.625
270.395
275.194
Chi-Square
2.350
2.652
4.423
9.221
df
2
2
2
2
Sig.
.309
.265
.110
.010
The chi-square statistic is the difference in -2 log-likelihoods
between the final model and a reduced model. The reduced model is
formed by omitting an effect from the final model. The null hypothesis
is that all parameters of that effect are 0.
The statistical significance of the relationship between
confidence in Congress and opinion about spending on
highways and bridges is based on the statistical significance of
the chi-square statistic in the SPSS table titled "Likelihood
Ratio Tests".
For this relationship, the probability of the chi-square statistic
(9.221) was 0.010, less than or equal to the level of
significance of 0.05. The null hypothesis that all of the b
coefficients associated with confidence in Congress were equal
to zero was rejected. The existence of a relationship between
confidence in Congress and opinion about spending on
highways and bridges was supported.

42. ters II
RELATIONSHIP OF INDIVIDUAL INDEPENDENT
VARIABLES TO DEPENDENT VARIABLE - 2
Slide
42
Parameter Estimates
HIGHWAYS
a
AND BRIDGES
1
2
Intercept
AGE
EDUC
CONLEGIS
Intercept
AGE
EDUC
CONLEGIS
B
3.240
.019
.071
-1.373
3.639
.003
.172
-1.657
Std. Error
2.478
.020
.108
.620
2.456
.020
.110
.613
Wald
1.709
.906
.427
4.913
2.195
.017
2.463
7.298
df
1
1
1
1
1
1
1
1
Sig.
.191
.341
.514
.027
.138
.897
.117
.007
a. The reference category is: 3.
In the comparison of survey respondents who thought we spend
too little money on highways and bridges to survey respondents
who thought we spend too much money on highways and
bridges, the probability of the Wald statistic (4.913) for the
variable confidence in Congress [conlegis] was 0.027. Since the
probability was less than or equal to the level of significance of
0.05, the null hypothesis that the b coefficient for confidence in
Congress was equal to zero for this comparison was rejected.
Exp(B)
95% Confiden
Exp
Lower Bound
1.019
1.073
.253
.980
.868
.075
1.003
1.188
.191
.963
.958
.057

43. ters II
RELATIONSHIP OF INDIVIDUAL INDEPENDENT
VARIABLES TO DEPENDENT VARIABLE - 3
Slide
43
Parameter Estimates
HIGHWAYS
a
AND BRIDGES
1
2
Intercept
AGE
EDUC
CONLEGIS
Intercept
AGE
EDUC
CONLEGIS
B
3.240
.019
.071
-1.373
3.639
.003
.172
-1.657
Std. Error
2.478
.020
.108
.620
2.456
.020
.110
.613
Wald
1.709
.906
.427
4.913
2.195
.017
2.463
7.298
df
1
1
1
1
1
1
1
1
Sig.
.191
.341
.514
.027
.138
.897
.117
.007
a. The reference category is: 3.
The value of Exp(B) was 0.253 which implies that for each unit
increase in confidence in Congress the odds decreased by 74.7%
(0.253 - 1.0 = -0.747).
The relationship stated in the problem is supported. Survey
respondents who had less confidence in congress were less likely
to be in the group of survey respondents who thought we spend
too little money on highways and bridges, rather than the group of
survey respondents who thought we spend too much money on
highways and bridges. For each unit increase in confidence in
Congress, the odds of being in the group of survey respondents
who thought we spend too little money on highways and bridges
decreased by 74.7%.
Exp(B)
95% Confiden
Exp
Lower Bound
1.019
1.073
.253
.980
.868
.075
1.003
1.188
.191
.963
.958
.057

44. ters II
RELATIONSHIP OF INDIVIDUAL INDEPENDENT
VARIABLES TO DEPENDENT VARIABLE - 4
Slide
44
Parameter Estimates
HIGHWAYS
a
AND BRIDGES
1
2
Intercept
AGE
EDUC
CONLEGIS
Intercept
AGE
EDUC
CONLEGIS
B
3.240
.019
.071
-1.373
3.639
.003
.172
-1.657
Std. Error
2.478
.020
.108
.620
2.456
.020
.110
.613
Wald
1.709
.906
.427
4.913
2.195
.017
2.463
7.298
df
1
1
1
1
1
1
1
1
Sig.
.191
.341
.514
.027
.138
.897
.117
.007
a. The reference category is: 3.
In the comparison of survey respondents who thought we spend
about the right amount of money on highways and bridges to
survey respondents who thought we spend too much money on
highways and bridges, the probability of the Wald statistic
(7.298) for the variable confidence in Congress [conlegis] was
0.007. Since the probability was less than or equal to the level
of significance of 0.05, the null hypothesis that the b coefficient
for confidence in Congress was equal to zero for this comparison
was rejected.
Exp(B)
95% Confiden
Exp
Lower Bound
1.019
1.073
.253
.980
.868
.075
1.003
1.188
.191
.963
.958
.057

45. ters II
Slide
45
RELATIONSHIP OF INDIVIDUAL INDEPENDENT
VARIABLES TO DEPENDENT VARIABLE - 5
Parameter Estimates
95% Con
HIGHWAYS
a
AND BRIDGES
1
2
Intercept
AGE
EDUC
CONLEGIS
Intercept
AGE
EDUC
CONLEGIS
B
3.240
.019
.071
-1.373
3.639
.003
.172
-1.657
Std. Error
2.478
.020
.108
.620
2.456
.020
.110
.613
Wald
1.709
.906
.427
4.913
2.195
.017
2.463
7.298
df
1
1
1
1
1
1
1
1
Sig.
.191
.341
.514
.027
.138
.897
.117
.007
a. The reference category is: 3.
The value of Exp(B) was 0.191 which implies that for each unit increase in
confidence in Congress the odds decreased by 80.9% (0.191-1.0=-0.809).
The relationship stated in the problem is supported. Survey respondents
who had less confidence in congress were less likely to be in the group of
survey respondents who thought we spend about the right amount of
money on highways and bridges, rather than the group of survey
respondents who thought we spend too much money on highways and
bridges. For each unit increase in confidence in Congress, the odds of
being in the group of survey respondents who thought we spend about the
right amount of money on highways and bridges decreased by 80.9%.
Exp(B)
Lower Bou
1.019
1.073
.253
.9
.8
.0
1.003
1.188
.191
.9
.9
.0

46. ters II
Slide
46
CLASSIFICATION USING THE MULTINOMIAL LOGISTIC
REGRESSION MODEL: BY CHANCE ACCURACY RATE
The independent variables could be characterized as useful
predictors distinguishing survey respondents who thought we
spend too little money on highways and bridges, survey
respondents who thought we spend about the right amount
of money on highways and bridges and survey respondents
who thought we spend too much money on highways and
bridges if the classification accuracy rate was substantially
higher than the accuracy attainable by chance alone.
Operationally, the classification accuracy rate should be 25%
or more higher than the proportional by chance accuracy
rate.
Case Processing Summary
N
HIGHWAYS
AND BRIDGES
1
2
3
Marginal
Percentage
37.1%
55.7%
7.2%
100.0%
62
93
12
Valid
167
Missing
103
Total
270
The proportional by chance accuracy rate was computed by
Subpopulation
153
calculating the proportion of cases for eachagroup based on
the number of cases in each group in the 'Case Processing
a.
Summary',The dependent variable has only one value the proportion of
and then squaring and summing observed
in 146 (95.4%) subpopulations.
cases in each group (0.371² + 0.557² + 0.072² = 0.453).

47. ters II
Slide
47
CLASSIFICATION USING THE MULTINOMIAL LOGISTIC
REGRESSION MODEL: CLASSIFICATION ACCURACY
Classification
Predicted
Observed
1
2
3
Overall Percentage
1
15
7
5
16.2%
2
47
86
7
83.8%
3
0
0
0
.0%
The classification accuracy rate was 60.5%
which was greater than or equal to the
proportional by chance accuracy criteria of
56.6% (1.25 x 45.3% = 56.6%).
The criteria for classification accuracy is
satisfied.
Percent
Correct
24.2%
92.5%
.0%
60.5%

48. ters II
Slide
48
Answering the question in problem 1 - 1
11. In the dataset GSS2000, is the following statement true, false, or an incorrect application
of a statistic? Assume that there is no problem with missing data, outliers, or influential cases,
and that the validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "age" [age], "highest year of school completed" [educ] and "confidence in
Congress" [conlegis] were useful predictors for distinguishing between groups based on
responses to "opinion about spending on highways and bridges" [natroad]. These predictors
differentiate survey respondents who thought we spend too little money on highways and
bridges from survey respondents who thought we spend too much money on highways and
bridges and survey respondents who thought we spend about the right amount of money on
highways and bridges from survey respondents who thought we spend too much money on
highways and bridges.
Among this set of predictors, confidence in Congress was helpful in distinguishing among the
groups defined by responses to opinion about spending on highways and bridges. Survey
We found a statistically significant be in
respondents who had less confidence in congress were less likely tooverallthe group of survey
relationship between highways and bridges, rather than the
respondents who thought we spend too little money onthe combination of
independent variables and the dependent
group of survey respondents who thought we spend too much money on highways and bridges.
variable.
For each unit increase in confidence in Congress, the odds of being in the group of survey
respondents who thought we spend too little money on highways and bridges decreased by
74.7%. Survey respondents who had less was no evidence of numerical less likelyin be in the
There confidence in congress were problems to
group of survey respondents who thought we spend about the right amount of money on
the solution.
highways and bridges, rather than the group of survey respondents who thought we spend too
much money on highways and bridges. For each unit increaseaccuracy surpassed
Moreover, the classification in confidence in Congress, the
odds of being in the group of survey respondents whochance accuracy criteria, the right amount
the proportional by thought we spend about
of money on highways and bridges supporting the 80.9%.of the model.
decreased by utility
1. True
2. True with caution
3. False

49. ters II
Slide
49
Answering the question in problem 1 - 2
We verified that each statement about the [educ] and
The variables "age" [age], "highest year of school completed" relationship "confidence in
Congress" [conlegis]between an independent for distinguishingdependent groups based on
were useful predictors variable and the between
variable was correct in both direction of the relationship These predictors
responses to "opinion about spending on highways and bridges" [natroad].
differentiate surveyand the change in likelihoodwe spend too little money on highways and
respondents who thought associated with a one-unit
bridges from survey change of the who thought variable, for both of the
respondents independent we spend too much money on highways and
bridges and survey respondents who thought we stated in the problem. amount of money on
comparisons between groups spend about the right
highways and bridges from survey respondents who thought we spend too much money on
highways and bridges.
Among this set of predictors, confidence in Congress was helpful in distinguishing among the
groups defined by responses to opinion about spending on highways and bridges. Survey
respondents who had less confidence in congress were less likely to be in the group of survey
respondents who thought we spend too little money on highways and bridges, rather than the
group of survey respondents who thought we spend too much money on highways and bridges.
For each unit increase in confidence in Congress, the odds of being in the group of survey
respondents who thought we spend too little money on highways and bridges decreased by
74.7%. Survey respondents who had less confidence in congress were less likely to be in the
group of survey respondents who thought we spend about the right amount of money on
highways and bridges, rather than the group of survey respondents who thought we spend too
much money on highways and bridges. For each unit increase in confidence in Congress, the
odds of being in the group of survey respondents who thought we spend about the right amount
of money on highways and bridges decreased by 80.9%.
1.
2.
3.
4.
True
True with caution
False
Inappropriate application of a statistic
The answer to the question is true
with caution.
A caution is added because of the
inclusion of ordinal level variables.

50. ters II
Slide
50
Problem 2
1. In the dataset GSS2000, is the following statement true, false, or an incorrect application of
a statistic? Assume that there is no problem with missing data, outliers, or influential cases,
and that the validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "highest year of school completed" [educ], "sex" [sex] and "total family income"
[income98] were useful predictors for distinguishing between groups based on responses to
"opinion about spending on space exploration" [natspac]. These predictors differentiate survey
respondents who thought we spend too little money on space exploration from survey
respondents who thought we spend too much money on space exploration and survey
respondents who thought we spend about the right amount of money on space exploration from
survey respondents who thought we spend too much money on space exploration.
Among this set of predictors, total family income was helpful in distinguishing among the
groups defined by responses to opinion about spending on space exploration. Survey
respondents who had higher total family incomes were more likely to be in the group of survey
respondents who thought we spend about the right amount of money on space exploration,
rather than the group of survey respondents who thought we spend too much money on space
exploration. For each unit increase in total family income, the odds of being in the group of
survey respondents who thought we spend about the right amount of money on space
exploration increased by 6.0%.
1.
2.
3.
4.
True
True with caution
False
Inappropriate application of a statistic

51. ters II
Slide
51
Dissecting problem 2 - 1
1. In the dataset GSS2000, is the following statement true, false, or an incorrect
application of a statistic? Assume that there is no problem with missing data, outliers, or
influential cases, and that the validation analysis will confirm the generalizability of the
results. Use a level of significance of 0.05 for evaluating the statistical relationships.
The variables "highest year of school completed" [educ], "sex" [sex] and "total family income"
[income98] were useful predictors for distinguishing between groups based on responses to
"opinion about spending on space exploration" [natspac]. we will predictors differentiate survey
For these problems, These
respondents who thought we spend too little money on is no problem
assume that there space exploration from survey
respondents who thought we spend too much money on outliers, or
with missing data, space exploration and survey
respondents who thought we spend about the right amount of money on space exploration from
influential cases, and that the
survey respondents who thought we spend too much moneyconfirm exploration.
on space
validation analysis will
the generalizability of the
Among this set of predictors, total family income was helpful in distinguishing among the
results
groups defined by responses to opinion about spending on space exploration. Survey
respondents who had higher total familythis problem, we are told to to be in the group of survey
In incomes were more likely
respondents who thought we spend about0.05 right amount of money on space exploration,
use the as alpha for the
rather than the group of survey respondents who logistic regression. too much money on space
multinomial thought we spend
exploration. For each unit increase in total family income, the odds of being in the group of
survey respondents who thought we spend about the right amount of money on space
exploration increased by 6.0%.
1.
2.
3.
4.
True
True with caution
False
Inappropriate application of a statistic

52. ters II
Slide
52
Dissecting problem 2 - 2
The variables listed first in the problem
statement are the independent variables
1. In (IVs): "highest year of is the following statement true, false, or an incorrect application of
the dataset GSS2000, school completed"
a statistic? Assume [sex] there is nofamily
[educ], "sex" that and "total problem with missing data, outliers, or influential cases,
and that the validation analysis will confirm the generalizability of the results. Use a level of
income" [income98].
significance of 0.05 for evaluating the statistical relationships.
The variables "highest year of school completed" [educ], "sex" [sex] and "total family
income" [income98] were useful predictors for distinguishing between groups based on
responses to "opinion about spending on space exploration" [natspac]. These predictors
differentiate survey respondents who thought we spend too little money on space exploration
from survey respondents who thought we spend too much money on space exploration and
survey respondents who thought we spend about the right amount of money on space
exploration from survey respondents who thought we spend too much money on space
The variable
exploration. used to define
groups is the dependent
variable (DV): "opinion about
Among this on space
spending set of predictors, total family income was helpful in distinguishing among the
groups defined by responses to opinion about spending on space exploration. Survey
exploration" [natspac].
respondents who had higher total family incomes were more likely to be in the group of survey
respondents who thought we spend about the right amount of money on space exploration,
rather than the group of survey respondents who thought we spend too much money on space
SPSS only odds of direct in
exploration. For each unit increase in total family income, thesupports being or the group of
simultaneous entry of independent
survey respondents who thought we spend about the right amount of money on space
variables in multinomial logistic
exploration increased by 6.0%.
1. True
2. True with caution
3. False
regression, so we have no choice of
method for entering variables.

53. ters II
Slide
53
Dissecting problem 2 - 3
SPSS multinomial logistic regression models the relationship
by comparing each of the groups defined by the dependent
variable to the group with the highest code value.
1. In the dataset GSS2000,to opinion about spending ontrue, false, or an incorrect application of
The responses is the following statement the space
a statistic? Assume that there is no problem with missing data, outliers, or influential cases,
program were:
and that the1= Too little, 2 = About right, and 3 = Too much.
validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "highest year of school completed" [educ], "sex" [sex] and "total family income"
[income98] were useful predictors for distinguishing between groups based on responses to
"opinion about spending on space exploration" [natspac]. These predictors differentiate
survey respondents who thought we spend too little money on space exploration from
survey respondents who thought we spend too much money on space exploration and
survey respondents who thought we spend about the right amount of money on space
exploration from survey respondents who thought we spend too much money on space
exploration.
Among this set of predictors, total family income was helpful in distinguishing among the
The analysis will result about spending on
groups defined by responses to opinion in two comparisons:space exploration. Survey
respondents who • survey respondents who thought we spend likely to be in the group of survey
had higher total family incomes were more too little money
versus survey respondents who amount of money on space
respondents who thought we spend about the rightthought we spend too much exploration,
money on space exploration
rather than the group of survey respondents who thought we spend too much money on space
• survey increase in total family income, the odds the being in the group of
exploration. For each unit respondents who thought we spend about of right
amount of money versus survey respondents who money on
survey respondents who thought we spend about the right amount ofthought we space
exploration increased by 6.0%.
spend too much money on space exploration.
1. True

54. ters II
Slide
54
Dissecting problem 2 - 4
Each problem includes a statement about the
The variables "highest year of school completed" [educ], "sex" [sex] and "total family income"
[income98]relationship between onefor distinguishing between groups based on responses to
were useful predictors independent variable and
the dependenton space exploration" [natspac]. These predictors differentiate survey
"opinion about spending variable. The answer to the
problem is based on the stated relationship,
respondents who thought we spend too little money on space exploration from survey
respondents who thought we spend too much money on space exploration and survey
ignoring the relationships between the other
respondents who thought we spend about the right variable. of money on space exploration from
independent variables and the dependent amount
survey respondents who thought we spend too much money on space exploration.
Among this set of predictors, total family income was helpful in distinguishing among the
groups defined by responses to opinion about spending on space exploration. Survey
respondents who had higher total family incomes were more likely to be in the group of
survey respondents who thought we spend about the right amount of money on space
exploration, rather than the group of survey respondents who thought we spend too much
money on space exploration. For each unit increase in total family income, the odds of
being in the group of survey respondents who thought we spend about the right amount of
money on space exploration increased by 6.0%.
1.
2.
3.
4.
True
True with caution
This problem identifies a difference for only one
of the two comparisons based on the three values
False
Inappropriate application of a of the dependent variable.
statistic
Other problems will specify both of the possible
comparisons.

55. ters II
Slide
55
Dissecting problem 2 - 5
The variables "highest year of school completed" [educ], "sex" [sex] and "total family income"
[income98] were useful predictors for distinguishing between groups based on responses to
"opinion about spending on space exploration" [natspac]. These predictors differentiate survey
respondents who thought we spend too little money on space exploration from survey
respondents who thought we spend too much money on space exploration and survey
respondents who thought we spend about the right amount of money on space exploration from
survey respondents who thought we spend too much money on space exploration.
Among this set of predictors, total family income was helpful in distinguishing among the
groups defined by responses to opinion about spending on space exploration. Survey
respondents who had higher total family incomes were more likely to be in the group of survey
respondents who thought we spend about the right amount of money on space exploration,
rather than the group of survey respondents who thought we spend too much money on space
exploration. For each unit increase in total family income, the odds of being in the group of
survey respondents who thought we spend about the right amount of money on space
exploration increased by 6.0%.
1.
2.
3.
4.
True
In order for the multinomial logistic regression
question to be true, the overall relationship must
True with caution
be statistically significant, there must be no
False
evidence of numerical problems, the classification
Inappropriate application of a statistic
accuracy rate must be substantially better than
could be obtained by chance alone, and the
stated individual relationship must be statistically
significant and interpreted correctly.

56. ters II
Slide
56
LEVEL OF MEASUREMENT - 1
1. In the dataset GSS2000, is the following statement true, false, or an incorrect application of
a statistic? Assume that there is no problem with missing data, outliers, or influential cases,
and that the validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "highest year of school completed" [educ], "sex" [sex] and "total family income"
[income98] were useful predictors for distinguishing between groups based on responses to
"opinion about spending on space exploration" [natspac]. These predictors differentiate
survey respondents who thought we spend too little money on space exploration from
survey respondents who thought we spend too much money on space exploration and
survey respondents who thought we spend about the right amount of money on space
exploration from survey respondents who thought we spend too much money on space
exploration.
Among this set of predictors, total family income was helpful in distinguishing among the
Multinomial opinion about spending on space
groups defined by responses tologistic regression requires that the exploration. Survey
dependent variable be non-metric and the
respondents who had higher total family incomes were more likely to be in the group of survey
independent variables be metric or dichotomous.
respondents who thought we spend about the right amount of money on space exploration,
rather than the group of survey respondents who thought we spend too much money on space
"Opinion about spending on space exploration"
exploration. For each unit increase in total family income, the odds of being in the group of
[natspac] is ordinal, satisfying the non-metric
survey respondentslevel of measurement requirement for the
who thought we spend about the right amount of money on space
exploration increased by 6.0%.
dependent variable.
1.
2.
3.
4.
It contains three categories: survey respondents
True
who thought we spend too little money, about
True with cautionright amount of money, and too much
the
money on space exploration.
False
Inappropriate application of a statistic

57. ters II
Slide
57
LEVEL OF MEASUREMENT - 2
"Highest year of school
"Sex" [sex] is dichotomous,
completed" [educ] is interval,
satisfying the metric or
satisfying the metric or
dichotomous level of measurement
1. In the dataset GSS2000, is the following statement true, false, or an incorrect application of
dichotomous level of
requirement for independent
measurement Assume that there is no problem with missing data, outliers, or influential cases,
a statistic? requirement for
variables.
independent variables.
and that the validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "highest year of school completed" [educ], "sex" [sex] and "total family
income" [income98] were useful predictors for distinguishing between groups based on
responses to "opinion about spending on space exploration" [natspac]. These predictors
differentiate survey respondents who thought we spend too little money on space exploration
from survey respondents who thought we spend too much money on space exploration and
survey respondents who thought we spend about the right amount of money on space
exploration from survey family income" [income98] we spend too much money on space
"Total respondents who thought is ordinal,
exploration.
satisfying the metric or dichotomous level of
measurement requirement for independent
variables. If we follow the convention of treating
Among this set of ordinal level total family incomevariables, the in distinguishing among the
predictors, variables as metric was helpful level
groups defined byof measurement requirementspending on space exploration. Survey
responses to opinion about for the analysis is
respondents who had higher total family incomes were not agree to be in the group of survey
satisfied. Since some data analysts do more likely
with this convention, a note of caution should money on space exploration,
respondents who thought we spend about the right amount of be
included in our interpretation.
rather than the group of survey respondents who thought we spend about the right amount of
money on space exploration. For each unit increase in total family income, the odds of being in
the group of survey respondents who thought we spend about the right amount of money on
space exploration increased by 6.0%.
1. True
2. True with caution

58. ters II
Slide
58
Request multinomial logistic regression
Select the Regression |
Multinomial Logistic…
command from the
Analyze menu.

59. ters II
Slide
59
Selecting the dependent variable
First, highlight the
dependent variable
natspac in the list
of variables.
Second, click on the right
arrow button to move the
dependent variable to the
Dependent text box.

60. ters II
Slide
60
Selecting non-metric independent variables
Non-metric independent variables are specified as
factors in multinomial logistic regression. Non-metric
variables can be either dichotomous, nominal, or
ordinal.
These variables will be dummy coded as needed and
each value will be listed separately in the output.
Select the
dichotomous
variable sex.
Move the non-metric
independent variables
listed in the problem to
the Factor(s) list box.

61. ters II
Slide
61
Selecting metric independent variables
Metric independent variables are specified as covariates
in multinomial logistic regression. Metric variables can
be either interval or, by convention, ordinal.
Move the metric
independent variables,
educ and income98, to
the Covariate(s) list box.

62. ters II
Slide
62
Specifying statistics to include in the output
While we will accept most of
the SPSS defaults for the
analysis, we need to specifically
request the classification table.
Click on the Statistics… button
to make a request.

63. ters II
Slide
63
Requesting the classification table
First, keep the SPSS
defaults for Summary
statistics, Likelihood
ratio test, and
Parameter estimates.
Second, mark the
checkbox for the
Classification table.
Third, click
on the
Continue
button to
complete the
request.

64. ters II
Slide
64
Completing the multinomial
logistic regression request
Click on the OK
button to request
the output for the
multinomial logistic
regression.
The multinomial logistic procedure supports
additional commands to specify the model
computed for the relationships (we will use the
default main effects model), additional
specifications for computing the regression,
and saving classification results. We will not
make use of these options.

65. ters II
Slide
65
Sample size – ratio of cases to variables
Case Processing Summary
N
SPACE EXPLORATION
PROGRAM
RESPONDENTS SEX
Valid
Missing
Total
Subpopulation
1
2
3
1
2
33
90
85
94
114
208
62
270
138a
Marginal
Percentage
15.9%
43.3%
40.9%
45.2%
54.8%
100.0%
a. The dependent variable has only one value observed in 112
Multinomial logistic regression requires that the minimum ratio
(81.2%) subpopulations.
of valid cases to independent variables be at least 10 to 1. The
ratio of valid cases (208) to number of independent
variables( 3) was 69.3 to 1, which was equal to or greater than
the minimum ratio. The requirement for a minimum ratio of
cases to independent variables was satisfied.
The preferred ratio of valid cases to independent variables is
20 to 1. The ratio of 69.3 to 1 was equal to or greater than the
preferred ratio. The preferred ratio of cases to independent
variables was satisfied.

66. ters II
Slide
66
OVERALL RELATIONSHIP BETWEEN
INDEPENDENT AND DEPENDENT VARIABLES
Model Fitting Information
Model
Intercept Only
Final
-2 Log
Likelihood
354.268
334.967
Chi-Square
19.301
df
Sig.
6
.004
The presence of a relationship between the dependent
variable and combination of independent variables is
based on the statistical significance of the final model
chi-square in the SPSS table titled "Model Fitting
Information".
In this analysis, the probability of the model chi-square
(19.301) was 0.004, less than or equal to the level of
significance of 0.05. The null hypothesis that there was
no difference between the model without independent
variables and the model with independent variables
was rejected. The existence of a relationship between
the independent variables and the dependent variable
was supported.

67. ters II
Slide
67
NUMERICAL PROBLEMS
Parameter Estimates
SPACE EXPLORATION
a
PROGRAM
1
2
Intercept
EDUC
INCOME98
[SEX=1]
[SEX=2]
Intercept
EDUC
INCOME98
[SEX=1]
[SEX=2]
B
Std. Error
-4.136
1.157
.101
.089
.097
.050
.672
.426
b
0
.
-2.487
.840
.108
.068
.058
.034
.501
.317
b
0
.
a. The reference category is: 3.
b. This parameter is set to zero because it is redundant.
Wald
12.779
1.276
3.701
2.488
.
8.774
2.521
2.932
2.492
.
df
95% Confidence
Exp(B)
Lower Bound
U
Sig.
Exp(B)
1
Multicollinearity .000
in the multinomial
logistic regression solution is
1
.259
1.106
detected by examining the
1
.054
1.102
standard errors for the b
1
.115
1.959
coefficients. A standard error
larger than 2.0 indicates numerical
0
.
.
problems, such .003
as multicollinearity
1
among the independent variables,
1
.112
1.114
zero cells for a dummy-coded
independent variable because all of
1
.087
1.060
the subjects have the same value
1
.114
1.650
for the variable, and 'complete
0
.
separation' whereby the two .
groups in the dependent event
variable can be perfectly separated
by scores on one of the
independent variables. Analyses
that indicate numerical problems
should not be interpreted.
None of the independent variables
in this analysis had a standard
error larger than 2.0.
.929
.998
.850
.
.975
.992
.886
.

68. ters II
Slide
68
RELATIONSHIP OF INDIVIDUAL INDEPENDENT
VARIABLES TO DEPENDENT VARIABLE - 1
Likelihood Ratio Tests
Effect
Intercept
EDUC
INCOME98
SEX
-2 Log
Likelihood of
Reduced
Model
334.967a
337.788
340.154
338.511
Chi-Square
.000
2.821
5.187
3.544
df
Sig.
0
2
2
2
.
.244
.075
.170
The chi-square statistic is the difference in -2 log-likelihoods
between the final model and a reduced model. The reduced model
is formed by omitting an effect from the final model. The null
hypothesis is that all parameters of that effect are 0.
a.
The statistical significance of the relationship between
This reduced model spending on space
total family income and opinion aboutis equivalent to the final model because
exploration is based on the statistical significance of the
omitting the effect does not increase the degrees of freedom.
chi-square statistic in the SPSS table titled "Likelihood
Ratio Tests".
For this relationship, the probability of the chi-square
statistic (5.187) was 0.075, greater than the level of
significance of 0.05. The null hypothesis that all of the b
coefficients associated with total family income were
equal to zero was not rejected. The existence of a
relationship between total family income and opinion
about spending on space exploration was not supported.

69. ters II
Slide
69
Answering the question in problem 2
1. In the dataset GSS2000, is the following statement true, false, or an incorrect application of
a statistic? Assume that there is no problem with missing data, outliers, or influential cases,
and that the validation analysis will confirm the generalizability of the results. Use a level of
significance of 0.05 for evaluating the statistical relationships.
The variables "highest year of school completed" [educ], "sex" [sex] and "total family income"
[income98] were useful predictors for distinguishing between groups based on responses to
"opinion about spending on space exploration" [natspac]. These predictors differentiate survey
respondents who thought we spend too little money on space exploration from survey
respondents who thought we spend too much money on space exploration and survey
respondents who thought we spend about the right amount of money on space exploration from
survey respondents who thought we spend too much money on space exploration.
We found a statistically significant overall
relationship between the combination of
Among this set of predictors, totalindependent variables and the dependent
family income was helpful in distinguishing among the
groups defined by responses to opinion about spending on space exploration. Survey
variable.
respondents who had higher total family incomes were more likely to be in the group of survey
respondents who thought we spend about the right amount numerical problems in
There was no evidence of of money on space exploration,
rather than the group of survey respondents who thought we spend too much money on space
the solution.
exploration. For each unit increase in total family income, the odds of being in the group of
survey respondents who thought we spend about the right amount of money on space
However, the individual relationship between
exploration increased by 6.0%.
1.
2.
3.
4.
total family income and spending on space was
not statistically significant.
True
True with caution
The answer to the question is false.
False
Inappropriate application of a statistic

70. ters II
Slide
70
Steps in multinomial logistic regression:
level of measurement and initial sample size
The following is a guide to the decision process for answering
problems about the basic relationships in multinomial logistic
regression:
Dependent non-metric?
Independent variables
metric or dichotomous?
No
Inappropriate
application of
a statistic
Yes
Ratio of cases to
independent variables at
least 10 to 1?
Yes
Run multinomial logistic regression
No
Inappropriate
application of
a statistic

71. ters II
Slide
71
Steps in multinomial logistic regression:
overall relationship and numerical problems
Overall relationship
statistically significant?
(model chi-square test)
No
False
Yes
Standard errors of
coefficients indicate no
numerical problems (s.e.
<= 2.0)?
Yes
No
False

72. ters II
Slide
72
Steps in multinomial logistic regression:
relationships between IV's and DV
Overall relationship
between specific IV and DV
is statistically significant?
(likelihood ratio test)
No
False
Yes
Role of specific IV and DV
groups statistically significant
and interpreted correctly?
(Wald test and Exp(B))
Yes
No
False

73. ters II
Slide
73
Steps in multinomial logistic regression:
classification accuracy and adding cautions
Overall accuracy rate is
25% > than proportional
by chance accuracy rate?
No
False
Yes
Satisfies preferred ratio of
cases to IV's of 20 to 1
No
True with caution
Yes
One or more IV's are
ordinal level treated as
metric?
No
True
Yes
True with caution