multinomial logisticregression basicrelationships

SW388R7Data Analysis

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Slide 1

Multinomial Logistic RegressionBasic Relationships

Multinomial Logistic Regression

Describing Relationships

Classification Accuracy

Sample Problems


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Slide 2

Multinomial logistic regression

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.


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Slide 3

What multinomial logistic regression predicts

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.


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Slide 4

Level of measurement requirements

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.


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Slide 5

Assumptions and outliers

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.


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Slide 6

Sample size requirements

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.


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Slide 7

Methods for including variables

The only method for selecting independent variables in SPSS is simultaneous or direct entry.


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Slide 8

Overall test of relationship - 1

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.


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Slide 9

Overall test of relationship - 2

Model Fitting Information

284.429

265.972 18.457 6 .005

ModelIntercept Only

Final

-2 LogLikelihood Chi-Square df Sig.

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.


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Slide 10

Strength of multinomial logistic regression relationship

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.


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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.


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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

62 37.1%

93 55.7%

12 7.2%

167 100.0%

103

270

153a

1

2

3

HIGHWAYSAND BRIDGES

Valid

Missing

Total

Subpopulation

NMarginal

Percentage

The dependent variable has only one value observedin 146 (95.4%) subpopulations.

a.

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%).


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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

15 47 0 24.2%

7 86 0 92.5%

5 7 0 .0%

16.2% 83.8% .0% 60.5%

Observed1

2

3

Overall Percentage

1 2 3PercentCorrect

Predicted

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.


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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.


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Slide 15

Relationship of individual independent variables and the dependent variable

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.


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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.


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Slide 17


Parameter Estimates

3.240 2.478 1.709 1 .191

.019 .020 .906 1 .341 1.019 .980

.071 .108 .427 1 .514 1.073 .868

-1.373 .620 4.913 1 .027 .253 .075

3.639 2.456 2.195 1 .138

.003 .020 .017 1 .897 1.003 .963

.172 .110 2.463 1 .117 1.188 .958

-1.657 .613 7.298 1 .007 .191 .057

Intercept

AGE

EDUC

CONLEGIS

Intercept

AGE

EDUC

CONLEGIS

HIGHWAYSAND BRIDGES

a

TOO LITTLE

ABOUT RIGHT

B Std. Error Wald df Sig. Exp(B) Lower Bound

95% Confidence Interval forExp(B)

The reference category is: TOO MUCH.a.

Parameter Estimates

3.240 2.478 1.709 1 .191

.019 .020 .906 1 .341 1.019 .980

.071 .108 .427 1 .514 1.073 .868

-1.373 .620 4.913 1 .027 .253 .075

3.639 2.456 2.195 1 .138

.003 .020 .017 1 .897 1.003 .963

.172 .110 2.463 1 .117 1.188 .958

-1.657 .613 7.298 1 .007 .191 .057

Intercept

AGE

EDUC

CONLEGIS

Intercept

AGE

EDUC

CONLEGIS

HIGHWAYSAND BRIDGES

a

1

2



The reference category is: 3.a.

SPSS identifies the comparisons it makes for groups defined by the dependent variable in the table of ‘Parameter Estimates,’ using either the value codes or the value labels, depending on the options settings for pivot table labeling.

The reference category is identified in the footnote to the table.

In this analysis, two comparisons will be made: •the TOO LITTLE group (coded 1, shaded blue) will be compared to the TOO MUCH 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).

The reference category plays the same role in multinomial logistic regression that it plays in the dummy-coding of a nominal variable: it is the category that would be coded with zeros for all of the dummy-coded variables that all other categories are interpreted against.


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Slide 18


Likelihood Ratio Tests

268.323 2.350 2 .309

268.625 2.652 2 .265

270.395 4.423 2 .110

275.194 9.221 2 .010

EffectIntercept

AGE

EDUC

CONLEGIS

-2 LogLikelihood of

ReducedModel Chi-Square df Sig.

The chi-square statistic is the difference in -2 log-likelihoodsbetween the final model and a reduced model. The reduced model isformed by omitting an effect from the final model. The null hypothesisis that all parameters of that effect are 0.

Parameter Estimates

3.240 2.478 1.709 1 .191

.019 .020 .906 1 .341 1.019 .980

.071 .108 .427 1 .514 1.073 .868

-1.373 .620 4.913 1 .027 .253 .075

3.639 2.456 2.195 1 .138

.003 .020 .017 1 .897 1.003 .963

.172 .110 2.463 1 .117 1.188 .958

-1.657 .613 7.298 1 .007 .191 .057

Intercept

AGE

EDUC

CONLEGIS

Intercept

AGE

EDUC

CONLEGIS

HIGHWAYSAND BRIDGES

a

1

2




In this example, there is a statistically significant relationship between the independent variable CONLEGIS and the dependent variable. (0.010 < 0.05)

As well, the independent variable CONLEGIS is significant in distinguishing both category 1 of the dependent variable from category 3 of the dependent variable. (0.027 < 0.05)

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)


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Slide 19

Interpreting relationship of individual independent variables to the dependent

variableLikelihood Ratio Tests

268.323 2.350 2 .309

268.625 2.652 2 .265

270.395 4.423 2 .110

275.194 9.221 2 .010

EffectIntercept

AGE

EDUC

CONLEGIS

-2 LogLikelihood of



Parameter Estimates

3.240 2.478 1.709 1 .191

.019 .020 .906 1 .341 1.019 .980

.071 .108 .427 1 .514 1.073 .868

-1.373 .620 4.913 1 .027 .253 .075

3.639 2.456 2.195 1 .138

.003 .020 .017 1 .897 1.003 .963

.172 .110 2.463 1 .117 1.188 .958

-1.657 .613 7.298 1 .007 .191 .057

Intercept

AGE

EDUC

CONLEGIS

Intercept

AGE

EDUC

CONLEGIS

HIGHWAYSAND BRIDGES

a

1

2




Survey respondents who had less confidence in congress (higher values correspond to lower confidence) were less likely to be in the group of survey respondents who thought we spend too little money on highways and bridges (DV category 1), rather than the group of survey respondents who thought we spend too much money on highways and bridges (DV category 3).

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%. (0.253 – 1.0 = -0.747)


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Slide 20

Interpreting relationship of individual independent variables to the dependent


268.323 2.350 2 .309

268.625 2.652 2 .265

270.395 4.423 2 .110

275.194 9.221 2 .010

EffectIntercept

AGE

EDUC

CONLEGIS

-2 LogLikelihood of



Parameter Estimates

3.240 2.478 1.709 1 .191

.019 .020 .906 1 .341 1.019 .980

.071 .108 .427 1 .514 1.073 .868

-1.373 .620 4.913 1 .027 .253 .075

3.639 2.456 2.195 1 .138

.003 .020 .017 1 .897 1.003 .963

.172 .110 2.463 1 .117 1.188 .958

-1.657 .613 7.298 1 .007 .191 .057

Intercept

AGE

EDUC

CONLEGIS

Intercept

AGE

EDUC

CONLEGIS

HIGHWAYSAND BRIDGES

a

1

2




Survey respondents who had less confidence in congress (higher values correspond to lower confidence) 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 (DV category 2), rather than the group of survey respondents who thought we spend too much money on highways and bridges (DV Category 3).

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%. (0.191 – 1.0 = 0.809)


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Slide 21



327.463a .000 0 .

333.440 5.976 2 .050

329.606 2.143 2 .343

334.636 7.173 2 .028

338.985 11.521 2 .003

EffectIntercept

AGE

EDUC

POLVIEWS

SEX

-2 LogLikelihood of


The chi-square statistic is the difference in -2 log-likelihoodsbetween the final model and a reduced model. The reduced modelis formed by omitting an effect from the final model. The nullhypothesis is that all parameters of that effect are 0.

This reduced model is equivalent to the final model becauseomitting the effect does not increase the degrees of freedom.

a.

Parameter Estimates

8.434 2.233 14.261 1 .000

-.023 .017 1.756 1 .185 .977 .944

-.066 .102 .414 1 .520 .936 .766

-.575 .251 5.234 1 .022 .563 .344

-2.167 .805 7.242 1 .007 .115 .024

0b . . 0 . . .

4.485 2.255 3.955 1 .047

-.001 .018 .003 1 .955 .999 .965

.011 .104 .011 1 .916 1.011 .824

-.397 .257 2.375 1 .123 .673 .406

-1.606 .824 3.800 1 .051 .201 .040

0b . . 0 . . .

Intercept

AGE

EDUC

POLVIEWS

[SEX=1]

[SEX=2]

Intercept

AGE

EDUC

POLVIEWS

[SEX=1]

[SEX=2]

NATCHLDa

TOO LITTLE

ABOUT RIGHT




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 the TOO LITTLE group from the TOO MUCH (reference) group.(0.007 < 0.5)

However, SEX does not differentiate the ABOUT RIGHT group from the TOO MUCH (reference) group.(0.51 > 0.5)


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Slide 22

Interpreting relationship of individual independent variables and the dependent


327.463a .000 0 .

333.440 5.976 2 .050

329.606 2.143 2 .343

334.636 7.173 2 .028

338.985 11.521 2 .003

EffectIntercept

AGE

EDUC

POLVIEWS

SEX

-2 LogLikelihood of




a.

Parameter Estimates

8.434 2.233 14.261 1 .000

-.023 .017 1.756 1 .185 .977 .944

-.066 .102 .414 1 .520 .936 .766

-.575 .251 5.234 1 .022 .563 .344

-2.167 .805 7.242 1 .007 .115 .024

0b . . 0 . . .

4.485 2.255 3.955 1 .047

-.001 .018 .003 1 .955 .999 .965

.011 .104 .011 1 .916 1.011 .824

-.397 .257 2.375 1 .123 .673 .406

-1.606 .824 3.800 1 .051 .201 .040

0b . . 0 . . .

Intercept

AGE

EDUC

POLVIEWS

[SEX=1]

[SEX=2]

Intercept

AGE

EDUC

POLVIEWS

[SEX=1]

[SEX=2]

NATCHLDa

TOO LITTLE

ABOUT RIGHT




Survey respondents who were male (code 1 for sex) were less likely to be in the group of survey respondents who thought we spend too little money on childcare assistance (DV category 1), rather than the group of survey respondents who thought we spend too much money on childcare assistance (DV category 3).

Survey respondents who were male were 88.5% less likely (0.115 – 1.0 = -0.885) to be in the group of survey respondents who thought we spend too little money on childcare assistance.


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Slide 23

Interpreting relationships for independent variable in problems

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.


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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. True 2. True with caution 3. False 4. Inappropriate application of a statistic


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Slide 25

Dissecting problem 1 - 1





For these problems, we will 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

In this problem, we are told to use 0.05 as alpha for the multinomial logistic regression.


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Slide 26





SPSS only supports direct or simultaneous entry of independent variables in multinomial logistic regression, so we have no choice of method for entering variables.

The variables listed first in the problem statement are the independent variables (IVs): "age" [age], "highest year of school completed" [educ] and "confidence in Congress" [conlegis].

The variable used to define groups is the dependent variable (DV): "opinion about spending on highways and bridges" [natroad].


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Slide 27





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.

The responses to opinion about spending on highways and bridges were: 1= Too little, 2 = About right, and 3 = Too much.

The analysis will result in two comparisons:• survey respondents who thought we spend too little money

versus survey respondents who thought we spend too much money on highways and bridges

• survey respondents who thought we spend about the right amount of money versus survey respondents who thought we spend too much money on highways and bridges.


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Slide 28




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 relationships between the other independent variables and the dependent variable.

This problem identifies a difference for both of the comparisons among groups modeled by the multinomial logistic regression.


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Slide 29





In order for the multinomial logistic regression question to be true, the overall relationship must be statistically significant, there must be no evidence of numerical problems, the classification 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.


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Slide 30

Request multinomial logistic regression

Select the Regression | Multinomial Logistic… command from the Analyze menu.


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Slide 31

Selecting the dependent variable

Second, click on the right arrow button to move the dependent variable to the Dependent text box.

First, highlight the dependent variable natroad in the list of variables.


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Slide 32

Selecting metric independent variables

Move the metric independent variables, age, educ and conlegis to the Covariate(s) list box.

Metric independent variables are specified as covariates in multinomial logistic regression. Metric variables can be either interval or, by convention, ordinal.

In this analysis, there are no non-metric independent variables. Non-metric independent variables would be moved to the Factor(s) list box.


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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.


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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.


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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.


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Slide 36

LEVEL OF MEASUREMENT - 1




1. True 2. True with caution

Multinomial logistic regression requires that the dependent variable be non-metric and the independent variables be metric or dichotomous.

"Opinion about spending on highways and bridges" [natroad] is ordinal, satisfying the non-metric level of measurement requirement for the dependent variable.

It contains three categories: survey respondents who thought we spend too little money, about the right amount of money, and too much money on highways and bridges.


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Slide 37





"Age" [age] and "highest year of school completed" [educ] are interval, satisfying the metric or dichotomous level of measurement requirement for independent variables.

"Confidence in Congress" [conlegis] is ordinal, satisfying the metric or dichotomous level of measurement requirement for independent variables. If we follow the convention of treating ordinal level variables as metric variables, the level of measurement requirement for the analysis is satisfied. Since some data analysts do not agree with this convention, a note of caution should be included in our interpretation.


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Slide 38

Sample size – ratio of cases to variables


62 37.1%

93 55.7%

12 7.2%

167 100.0%

103

270

153a

1

2

3

HIGHWAYSAND BRIDGES

Valid

Missing

Total

Subpopulation

NMarginal

Percentage


a. Multinomial logistic regression requires that the minimum ratio 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.


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Slide 39

OVERALL RELATIONSHIP BETWEEN INDEPENDENT AND DEPENDENT

VARIABLES


284.429

265.972 18.457 6 .005

ModelIntercept Only

Final





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Slide 40

Parameter Estimates

3.240 2.478 1.709 1 .191

.019 .020 .906 1 .341 1.019 .980

.071 .108 .427 1 .514 1.073 .868

-1.373 .620 4.913 1 .027 .253 .075

3.639 2.456 2.195 1 .138

.003 .020 .017 1 .897 1.003 .963

.172 .110 2.463 1 .117 1.188 .958

-1.657 .613 7.298 1 .007 .191 .057

Intercept

AGE

EDUC

CONLEGIS

Intercept

AGE

EDUC

CONLEGIS

HIGHWAYSAND BRIDGES

a

1

2




NUMERICAL PROBLEMS

Multicollinearity in the multinomial logistic regression solution is detected by examining the standard errors for the b coefficients. A standard error larger than 2.0 indicates numerical problems, such as multicollinearity among the independent variables, zero cells for a dummy-coded independent variable because all of the subjects have the same value for the variable, and 'complete separation' whereby the two groups in the dependent event variable can be perfectly separated by scores on one 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.)


& Computers II

Slide 41


268.323 2.350 2 .309

268.625 2.652 2 .265

270.395 4.423 2 .110

275.194 9.221 2 .010

EffectIntercept

AGE

EDUC

CONLEGIS

-2 LogLikelihood of



RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT

VARIABLE - 1

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.


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Slide 42

Parameter Estimates

3.240 2.478 1.709 1 .191

.019 .020 .906 1 .341 1.019 .980

.071 .108 .427 1 .514 1.073 .868

-1.373 .620 4.913 1 .027 .253 .075

3.639 2.456 2.195 1 .138

.003 .020 .017 1 .897 1.003 .963

.172 .110 2.463 1 .117 1.188 .958

-1.657 .613 7.298 1 .007 .191 .057

Intercept

AGE

EDUC

CONLEGIS

Intercept

AGE

EDUC

CONLEGIS

HIGHWAYSAND BRIDGES

a

1

2





VARIABLE - 2

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.


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Slide 43

Parameter Estimates

3.240 2.478 1.709 1 .191

.019 .020 .906 1 .341 1.019 .980

.071 .108 .427 1 .514 1.073 .868

-1.373 .620 4.913 1 .027 .253 .075

3.639 2.456 2.195 1 .138

.003 .020 .017 1 .897 1.003 .963

.172 .110 2.463 1 .117 1.188 .958

-1.657 .613 7.298 1 .007 .191 .057

Intercept

AGE

EDUC

CONLEGIS

Intercept

AGE

EDUC

CONLEGIS

HIGHWAYSAND BRIDGES

a

1

2





VARIABLE - 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%.


& Computers II

Slide 44

Parameter Estimates

3.240 2.478 1.709 1 .191

.019 .020 .906 1 .341 1.019 .980

.071 .108 .427 1 .514 1.073 .868

-1.373 .620 4.913 1 .027 .253 .075

3.639 2.456 2.195 1 .138

.003 .020 .017 1 .897 1.003 .963

.172 .110 2.463 1 .117 1.188 .958

-1.657 .613 7.298 1 .007 .191 .057

Intercept

AGE

EDUC

CONLEGIS

Intercept

AGE

EDUC

CONLEGIS

HIGHWAYSAND BRIDGES

a

1

2





VARIABLE - 4

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.


& Computers II

Slide 45

Parameter Estimates

3.240 2.478 1.709 1 .191

.019 .020 .906 1 .341 1.019 .980

.071 .108 .427 1 .514 1.073 .868

-1.373 .620 4.913 1 .027 .253 .075

3.639 2.456 2.195 1 .138

.003 .020 .017 1 .897 1.003 .963

.172 .110 2.463 1 .117 1.188 .958

-1.657 .613 7.298 1 .007 .191 .057

Intercept

AGE

EDUC

CONLEGIS

Intercept

AGE

EDUC

CONLEGIS

HIGHWAYSAND BRIDGES

a

1

2





VARIABLE - 5

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%.


& Computers II

Slide 46


62 37.1%

93 55.7%

12 7.2%

167 100.0%

103

270

153a

1

2

3

HIGHWAYSAND BRIDGES

Valid

Missing

Total

Subpopulation

NMarginal

Percentage


a.

CLASSIFICATION USING THE MULTINOMIAL LOGISTIC REGRESSION MODEL: BY CHANCE ACCURACY RATE

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 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.


& Computers II

Slide 47

Classification

15 47 0 24.2%

7 86 0 92.5%

5 7 0 .0%

16.2% 83.8% .0% 60.5%

Observed1

2

3

Overall Percentage

1 2 3PercentCorrect

Predicted

CLASSIFICATION USING THE MULTINOMIAL LOGISTIC REGRESSION MODEL: CLASSIFICATION ACCURACY

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.


& Computers II

Slide 48

Answering the question in problem 1 - 1





We found a statistically significant overall relationship between the combination of independent variables and the dependent variable.

There was no evidence of numerical problems in the solution.

Moreover, the classification accuracy surpassed the proportional by chance accuracy criteria, supporting the utility of the model.


& Computers II

Slide 49

Answering the question in problem 1 - 2




We verified that each statement about the relationship between an independent variable and the dependent variable was correct in both direction of the relationship and the change in likelihood associated with a one-unit change of the independent variable, for both of the comparisons between groups stated in the problem.

The answer to the question is true with caution.

A caution is added because of the inclusion of ordinal level variables.


& Computers II

Slide 50

Problem 2


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%.



& Computers II

Slide 51






For these problems, we will 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

In this problem, we are told to use 0.05 as alpha for the multinomial logistic regression.


& Computers II

Slide 52






SPSS only supports direct or simultaneous entry of independent variables in multinomial logistic regression, so we have no choice of method for entering variables.

The variables listed first in the problem statement are the independent variables (IVs): "highest year of school completed" [educ], "sex" [sex] and "total family income" [income98].

The variable used to define groups is the dependent variable (DV): "opinion about spending on space exploration" [natspac].


& Computers II

Slide 53





1. True 2. True with caution 3. False

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.

The responses to opinion about spending on the space program were: 1= Too little, 2 = About right, and 3 = Too much.

The analysis will result in two comparisons:• survey respondents who thought we spend too little money

versus survey respondents who thought we spend too much money on space exploration

• survey respondents who thought we spend about the right amount of money versus survey respondents who thought we spend too much money on space exploration.


& Computers II

Slide 54





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 relationships between the other independent variables and the dependent variable.

This problem identifies a difference for only one of the two comparisons based on the three values of the dependent variable.

Other problems will specify both of the possible comparisons.


& Computers II

Slide 55





In order for the multinomial logistic regression question to be true, the overall relationship must be statistically significant, there must be no evidence of numerical problems, the classification 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.


& Computers II

Slide 56






Multinomial logistic regression requires that the dependent variable be non-metric and the independent variables be metric or dichotomous.

"Opinion about spending on space exploration" [natspac] is ordinal, satisfying the non-metric level of measurement requirement for the dependent variable.

It contains three categories: survey respondents who thought we spend too little money, about the right amount of money, and too much money on space exploration.


& Computers II

Slide 57




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 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

"Sex" [sex] is dichotomous, satisfying the metric or dichotomous level of measurement requirement for independent variables.

"Highest year of school completed" [educ] is interval, satisfying the metric or dichotomous level of measurement requirement for independent variables.

"Total family income" [income98] is ordinal, satisfying the metric or dichotomous level of measurement requirement for independent variables. If we follow the convention of treating ordinal level variables as metric variables, the level of measurement requirement for the analysis is satisfied. Since some data analysts do not agree with this convention, a note of caution should be included in our interpretation.


& Computers II

Slide 58

Request multinomial logistic regression

Select the Regression | Multinomial Logistic… command from the Analyze menu.


& Computers II

Slide 59

Selecting the dependent variable

Second, click on the right arrow button to move the dependent variable to the Dependent text box.

First, highlight the dependent variable natspac in the list of variables.


& Computers II

Slide 60

Selecting non-metric independent variables

Move the non-metric independent variables listed in the problem to the Factor(s) list box.

Select the dichotomous variable sex.

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.


& Computers II

Slide 61

Selecting metric independent variables

Move the metric independent variables, educ and income98, to the Covariate(s) list box.

Metric independent variables are specified as covariates in multinomial logistic regression. Metric variables can be either interval or, by convention, ordinal.


& Computers 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.


& Computers 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.


& Computers 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.


& Computers II

Slide 65


33 15.9%

90 43.3%

85 40.9%

94 45.2%

114 54.8%

208 100.0%

62

270

138a

1

2

3

SPACE EXPLORATIONPROGRAM

1

2

RESPONDENTS SEX

Valid

Missing

Total

Subpopulation

NMarginal

Percentage

The dependent variable has only one value observed in 112(81.2%) subpopulations.

a.

Sample size – ratio of cases to variables

Multinomial logistic regression requires that the minimum ratio 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.


& Computers II

Slide 66


354.268

334.967 19.301 6 .004

ModelIntercept Only

Final


OVERALL RELATIONSHIP BETWEEN INDEPENDENT AND DEPENDENT

VARIABLES




& Computers II

Slide 67

Parameter Estimates

-4.136 1.157 12.779 1 .000

.101 .089 1.276 1 .259 1.106 .929

.097 .050 3.701 1 .054 1.102 .998

.672 .426 2.488 1 .115 1.959 .850

0b . . 0 . . .

-2.487 .840 8.774 1 .003

.108 .068 2.521 1 .112 1.114 .975

.058 .034 2.932 1 .087 1.060 .992

.501 .317 2.492 1 .114 1.650 .886

0b . . 0 . . .

Intercept

EDUC

INCOME98

[SEX=1]

[SEX=2]

Intercept

EDUC

INCOME98

[SEX=1]

[SEX=2]

SPACE EXPLORATIONPROGRAM

a

1

2




This parameter is set to zero because it is redundant.b.

NUMERICAL PROBLEMS

Multicollinearity in the multinomial logistic regression solution is detected by examining the standard errors for the b coefficients. A standard error larger than 2.0 indicates numerical problems, such as multicollinearity among the independent variables, zero cells for a dummy-coded independent variable because all of the subjects have the same value for the variable, and 'complete separation' whereby the two groups in the dependent event variable can be perfectly separated by scores on one 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.


& Computers II

Slide 68


VARIABLE - 1Likelihood Ratio Tests

334.967a .000 0 .

337.788 2.821 2 .244

340.154 5.187 2 .075

338.511 3.544 2 .170

EffectIntercept

EDUC

INCOME98

SEX

-2 LogLikelihood of




a. The statistical significance of the relationship between total family income and opinion about spending on space exploration 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 (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.


& Computers II

Slide 69

Answering the question in problem 2





We found a statistically significant overall relationship between the combination of independent variables and the dependent variable.

There was no evidence of numerical problems in the solution.

However, the individual relationship between total family income and spending on space was not statistically significant.

The answer to the question is false.


& Computers 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:

Inappropriate application of a statistic

Yes

NoDependent non-metric?Independent variables metric or dichotomous?

Yes

Ratio of cases to independent variables at least 10 to 1?

Yes

No Inappropriate application of a statistic

Run multinomial logistic regression


& Computers II

Slide 71

Steps in multinomial logistic regression: overall relationship and numerical

problems

Yes

Yes

Standard errors of coefficients indicate no numerical problems (s.e. <= 2.0)?

NoFalse

Overall relationship statistically significant?(model chi-square test)

NoFalse


& Computers 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)

Yes

Role of specific IV and DV groups statistically significant and interpreted correctly?(Wald test and Exp(B))

No

Yes

False

NoFalse


& Computers II

Slide 73

Steps in multinomial logistic regression: classification accuracy and adding

cautions

Yes

Overall accuracy rate is 25% > than proportional by chance accuracy rate?

Yes

NoFalse

One or more IV's are ordinal level treated as metric?

No

Yes

True

Satisfies preferred ratio of cases to IV's of 20 to 1

No

YesYes

True with caution

True with caution

multinomial logisticregression basicrelationships

Business

ordinal variables

nonmetric dependent

final model chisquare

dummycoded dependent

variable ratios

compu ters iiassumptions

compu ters iimethods

group comparisons