# Negative binomial regression

## Predict for count outcomes with extensive variance

Negative binomial regression is used to test for associations between predictor and confounding variables on a count outcome variable

**when the variance of the count is higher than the mean of the count**. Negative binomial regression is interpreted in a similar fashion to logistic regression with the use of odds ratios with 95% confidence intervals. Just like with other forms of regression, the assumptions of linearity, homoscedasticity, and normality have to be met for negative binomial regression.### The steps for conducting negative binomial regression in SPSS

1. The data is entered in a multivariate fashion.

2. Click

3. Drag the cursor over the

4. Click

5. In the

6. Click on the

7. Click on the count outcome variable in the

8. Click on the arrow to move the variable into the

9. Click on the

10. Click on a categorical or ordinal predictor variable in the

11. Click on the arrow to move the variable into the

12. Repeat Steps 10 and 11 until all of the categorical predictor variables are in the

13. Click on a continuous predictor variable in the

14. Click on the arrow to move the variable into the

15. Repeat Steps 13 and 14 until all of the continuous predictor variables are in the

16. Click on the

17. Look in the

18. Click on the arrow to move the variable into the

19. Repeat Steps 17 and 18 until all of the predictor variables are in the

20. Click on the

21. In the

22. Click on the arrow to move the variable into the

23. Repeat Steps 21 and 22 until all of the predictor variables are in the

24. Click on the

25. Click on the

26. Click

2. Click

**.**__A__nalyze3. Drag the cursor over the

**Generali**drop-down.__z__ed Linear Models4. Click

**.**__G__eneralized Linear Model5. In the

**Type of Model**tab, under the**Counts**header, click on the**marker to select it.**__N__egative binomial with log link6. Click on the

**Response**tab.7. Click on the count outcome variable in the

**box to highlight it.**__V__ariables:8. Click on the arrow to move the variable into the

**box.**__D__ependent Variable:9. Click on the

**Predictors**tab.10. Click on a categorical or ordinal predictor variable in the

**box to highlight it.**__V__ariables:11. Click on the arrow to move the variable into the

**box.**__F__actors:12. Repeat Steps 10 and 11 until all of the categorical predictor variables are in the

**box.**__V__ariables:13. Click on a continuous predictor variable in the

**box to highlight it.**__V__ariables:14. Click on the arrow to move the variable into the

**box.**__C__ovariates:15. Repeat Steps 13 and 14 until all of the continuous predictor variables are in the

**box.**__C__ovariates:16. Click on the

**Model**tab.17. Look in the

**box. Click on the first predictor variable to highlight it.**__F__actors and Covariates:18. Click on the arrow to move the variable into the

**box.**__M__odel:19. Repeat Steps 17 and 18 until all of the predictor variables are in the

**box.**__M__odel:20. Click on the

**EM Means**tab.21. In the

**table, click on the first predictor variable to highlight it.**__F__actors and Interactions:22. Click on the arrow to move the variable into the

**box.**__D__isplay Means for:23. Repeat Steps 21 and 22 until all of the predictor variables are in the

**box.**__D__isplay Means for:24. Click on the

**Save**tab.25. Click on the

**Predicted value of mean of response**,**Standardized Pearson residual**, and**Standardized deviance residual**boxes to select them.26. Click

**OK**.With more complex statistics such as a negative binomial regression, a little bit more

1. Go to the

2. Scroll up to the very top of the output where the

3. Click in the area of the syntax code and it will become highlighted.

4. Use the cursor to

5. Look in the upper left hand part of the computer screen for the

6. Click on the

7. Drag the cursor over the

8. Click

9.

10. The last line of the syntax code should end with

11. Type the last line of code like this:

12. Use the cursor to highlight all of the

13. Click the

**complexity**is necessitated to run the analysis. Researchers are going to have to use**syntax**to get the adjusted odds ratios and 95% confidence intervals for the model.**SPSS does not have a point-and-click button**for these important values. However, do not fret! It is very simple to do. Here are the steps:1. Go to the

**output file**.2. Scroll up to the very top of the output where the

**syntax code**for the analysis is located.3. Click in the area of the syntax code and it will become highlighted.

4. Use the cursor to

**highlight all of the syntax code and COPY it by right-clicking your mouse and selecting Copy or press Ctrl + c**.5. Look in the upper left hand part of the computer screen for the

**File**drop down menu.6. Click on the

**File**drop-down menu.7. Drag the cursor over the

**New**drop-down menu.8. Click

**Syntax**.9.

**Paste**the syntax code into the syntax editor.10. The last line of the syntax code should end with

**SOLUTION.**11. Type the last line of code like this:

**SOLUTION (EXPONENTIATED).**12. Use the cursor to highlight all of the

**new syntax code**.13. Click the

**Green triangle**to run the code. (Looks like a green "play" button)### The steps for interpreting the SPSS output for negative binomial regression

1. Look in the

If the value is

If the value is

2. Look in the

If the

If the

3. Look in the

For

The last category of the categorical or ordinal variable is going to serve as the reference group for interpretation purposes.

If the

If the

If the

For

If the

If the

**Goodness of Fit**table, at the**Value/df**column for the**Pearson Chi-Square**row.If the value is

**LESS THAN .05**, then the model does not fit the data well and other analyses should be considered.If the value is

**MORE THAN .05**, then the model does the fit the data well and researchers can continue with interpreting the results.2. Look in the

**Omnibus Test**table, under the**Sig.**column. This is the*p*-value that is interpreted.If the

*p*-value is**LESS THAN .05**, then researchers have statistically significant model and should continue interpreting the results.If the

*p*-value is**MORE THAN .05**, then researchers do not have a significant model. Report the*p*-values as needed.3. Look in the

**Tests of Model Effects**table, under the**Sig.**,**Exp(B)**,**Lower**, and**Upper**columns.For

**categorical or ordinal predictors**:The last category of the categorical or ordinal variable is going to serve as the reference group for interpretation purposes.

If the

*p*-value is**LESS THAN .05**and the adjusted odds ratio with its 95% CI is**above 1.0**, the**risk of the outcome occurring increases**that many more times versus the reference category.If the

*p*-value is**LESS THAN .05**and the adjusted odds ratio with its 95% CI is**below 1.0**, then the**risk of the outcome occurring decreases**that many times versus the reference category.If the

*p*-value is**MORE THAN .05**, then the 95% CI for the adjusted odds ratio crosses over 1.0 and the association is non-significant.For

**continuous predictors**:If the

*p*-value is**LESS THAN .05**and the adjusted odds ratio with its 95% CI is**above 1.0**,**for every one-unit increase**in the continuous variable, the**risk of the outcome occurring increases**that many more times versus the reference category.If the

*p*-value is**LESS THAN .05**and the adjusted odds ratio with its 95% CI is**below 1.0**,**for every one-unit increase**in the continuous variable, the**risk of the outcome occurring decreases**that many times versus the reference category.### Residuals

At this point, researchers need to construct and interpret several plots of the raw and standardized residuals to fully assess model fit. Residuals can be thought of as

**the error associated with predicting or estimating outcomes using predictor variables**. Residual analysis is**extremely important**for meeting the linearity, normality, and homogeneity of variance assumptions of negative binomial regression.Here is how to conduct the analysis in SPSS:

1. Go back to the

The first is the

The second variable contains the

The third variable has

2. Click

3. Drag the cursor over the

4. Click

5. Click

6. Click

7. Click on the

8. Click on the

9. Click on the

10. Click on the

11. Click

1. Go back to the

**Data View**. There are three new variables that have been created.The first is the

**predicted value of the mean of response**of that observation and is given the variable name**MeanPredicted**.The second variable contains the

**standardized Pearson residual**and is given the variable name of**StdPearsonResidual**.The third variable has

**standardized Deviance residuals**and will be given the variable name of as**StdDevianceResidual**.2. Click

**.**__G__raphs3. Drag the cursor over the

**drop-down menu.**__L__egacy Dialogs4. Click

**.**__S__catter/Dot5. Click

**Simple Scatter**to select it.6. Click

**Define**.7. Click on the

**StdDevianceResidual**variable to highlight it.8. Click on the

**arrow**to move the variable into the**Y Axis:**box.9. Click on the

**MeanPredicted**variable to highlight it.10. Click on the

**arrow**to move the variable into the**X Axis:**box.11. Click

**OK**.### The steps for interpreting the SPSS scatterplot output

1. If there are not significant deviations away from 0 and 95% of the residuals are under absolute value of 2.0, then the model is thought to fit the data.

### Outliers

Normality and equal variance assumptions apply to negative binomial regression analyses. Here is how to assess outliers in the dataset:

1. Click

2. Drag the cursor over the

3. Click

4. Click on the

5. Click on the

6. Click

**.**__A__nalyze2. Drag the cursor over the

**D**drop-down menu.__e__scriptive Statistics3. Click

**.**__F__requencies4. Click on the

**StdPearsonResidual**variable to highlight it.5. Click on the

**arrow**to move the variable into the**Variable(s):**box.6. Click

**OK**.### The steps for interpreting the SPSS output for outliers

1. Look in the

2. Scroll through the entirety of the table.

3. If there are values that are

**Standardized Pearson Residual**table, under the**first column**. (It has the word "Valid" in it).2. Scroll through the entirety of the table.

3. If there are values that are

**above an absolute value of 2.0**, then are outliers.Click on the

**Download Database**and**Download Data Dictionary**buttons for a configured database and data dictionary for negative binomial regression.**Click on the****Validation of Statistical Findings**button to learn more about bootstrap, split-group, and jack-knife validation methods.## Statistician Services for Professionals

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