# Generalized Linear Models

**Generalized linear models** are an extension, or generalization, of the linear modeling process which allows for non-normal distributions. Common non-normal distributions are Poisson, Binomial, and Multinomial. Related linear models include ANOVA, ANCOVA, MANOVA, and MANCOVA, as well as the regression models. In SPSS, generalized linear models can be performed by selecting "Generalized Linear Models" from the analyze of menu, and then selecting the type of model to analyze from the Generalized Linear Models options list.

Generalized Estimating Equations extends Generalized Linear Models further by involving dependent data such as, repeated measures, logistic regression and other various models involving correlated data. In SPSS, Generalized Estimating Equations can be done by selecting "Generalized Linear Models" from the analyze menu, and then selecting the "Generalized Estimating Equations" from the Generalized Linear Models options list.

### Questions Answered:

- Do differences exist in IQ and test scores by management position and tenure?

- Do differences exist in IQ and test scores by management position and time?

- What effect did the intervention program have on attitudes and self-esteem among the different races and genders on students from pre-test to post-test?

The dependent variable in the implementation of Generalized Estimating Equations and Generalized Linear Models are distributed in the following distributions:

- Normal distribution when the dependent variable is continuous (numeric).
- Multinomial distribution when the dependent variable is ordinal (numeric or string).
- Binomial distribution when the dependent variable is binary (e.g., yes or no).
- Poisson distribution when the dependent variable is count in nature or when the events are rare in nature.
- The dependent or criterion variables in Generalized Estimating Equations and Generalized Linear Models are not distributed as free variables.

There are certain assumptions in Generalized Estimating Equations and Generalized Linear Models. These assumptions are as follows:

- Generalized Estimating Equations and Generalized Linear Models do not assume that the dependent/independent variables are not normally distributed.
- Generalized Estimating Equations and Generalized Linear Models neither assume linearity between the predictors and the dependent variables, nor homogeneity of variance for the range of the dependent variable.
- There must be linearity in the link function.
- Multicollinearity is absent.
- The data must be centered in order to reduce multicollinearity.
- The dependent data in Generalized Estimating Equations and Generalized Linear Models are continuous, ordinal, or binary.

**Generalized Linear Model Resources**

Ballinger, G. A. (2004). Using generalized estimating equations for longitudinal data analysis. *Organizational Research Methods, 7*(2), 127-150.

Beretvas, S. N., & Williams, N. J. (2004). The use of hierarchical generalized linear model for item dimensionality assessment. *Journal of Educational Measurement, 41*(4), 379-395.

Cardot, H., & Sarda, P. (2005). Estimation in generalized linear models for functional data via penalized likelihood. *Journal of Multivariate Analysis, 92*(1), 24-41.

Fox, J. (2008). *Applied regression analysis and generalized linear models* (2nd ed.). Thousand Oaks, CA: Sage Publications.

Hardin, J. W., & Hilbe, J. M. (2007). *Generalized linear models and extensions *(2nd ed.). College Station, TX: StataCorp LP.

Hoffman, J. P. (2003). *Generalized linear models: An applied approach*. Boston: Pearson, Allyn, & Bacon.

Hwang, H., & Takane, Y. (2005). Estimation of growth curve models with structured error covariances by generalized estimation equations. *Behaviormetrika, 32*(2), 155-163.

Johnson, T. R. (2006). Generalized linear models with ordinally-observed covariates. *British Journal of Mathematical and Statistical Psychology, 59*(2), 275-300.

Johnson, T. R., & Kim, J. -S. (2004). A generalized estimating equations approach to mixed-effects ordinal probit models. *British Journal of Mathematical and Statistical Psychology, 57*(2), 295-310.

McCullagh, P., & Nelder, J. A. (1989). *Generalized linear models* (2nd ed.)*.* London: Chapman & Hall.

Mukherjee, B., & Liu, I. (2009). A note on bias due to fitting prospective multivariate generalized linear models to categorical outcomes ignoring retrospective sampling. *Journal of Multivariate Analysis, 100*(3), 459-472.

Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized linear models. *Journal of the Royal Statistical Society, 135*(3), 370-384.

Rogers, W. H. (1993). Comparison of nbreg and glm for negative binomial. *Stata Technical Bulletin, 3*(16), 1-32.

Schluchter, M. D. (2008). Flexible approaches to computing mediated effects in generalized linear models: Generalized estimating equations and bootstrapping. *Multivariate Behavioral Research, 43*(2), 268-288.