Dear Sepideh Jahandideh

Certainly! When working with variables that are not normally distributed in the context of generalized estimating equations (GEE), it's important to select appropriate families and link functions that suit the distributional characteristics of your data. Here are some guidelines and resources that can help you understand and choose suitable options:

1. General Guidelines: Depending on the nature of your data, you can consider different families and link functions. Here are some common combinations:

- For continuous data with a symmetric distribution, the Gaussian family with the identity link function is often appropriate.

- For binary or dichotomous data, the binomial family with the logit or probit link function is commonly used.

- For count data, the Poisson or negative binomial family with the log link function is frequently employed.

- For ordinal data, the cumulative logit or proportional odds model can be applied.

2. Statistical Literature: Several statistical textbooks and research papers provide detailed information on the selection of appropriate families and link functions for GEE models. Some recommended sources include:

- "Generalized Estimating Equations" by Hardin and Hilbe.

- "Generalized Linear Models for Categorical and Continuous Limited Dependent Variables" by Long and Freese.

- "Generalized Linear Models" by McCullagh and Nelder.

- "Analysis of Longitudinal Data" by Diggle, Heagerty, Liang, and Zeger.

3. Software Documentation: If you are using statistical software packages like R, SAS, or Stata to fit GEE models, their documentation often provides guidance on the available families and link functions. For example, the documentation for the "geepack" package in R or the "PROC GENMOD" procedure in SAS can be helpful in understanding the options and making appropriate choices.

4. Consultation and Collaboration: If you have access to a statistician or a research collaborator with expertise in GEE modeling or generalized linear models (GLMs), it would be beneficial to consult with them. They can provide personalized guidance based on the specific characteristics of your data and research objectives.

Remember that the choice of the family and link functions in GEE depends on the distributional assumptions of your data and the specific research question you are addressing. It's important to evaluate the goodness of fit and interpret the estimated coefficients accordingly.

The family and link functions are:

When selecting appropriate family and link functions for Generalized Estimating Equations (GEE), it is essential to consider the distributional characteristics of your data. Here are some guidelines to help you make suitable choices:

1. Gaussian Family with Identity Link:

- Appropriate for continuous data with a symmetric distribution, assuming the response variable follows a normal distribution.

- It is often used when the response variable is measured on an interval or ratio scale and shows a relatively symmetric distribution.

2. Binomial Family with Logit or Probit Link:

- Suitable for binary or dichotomous data where the response variable takes only two possible outcomes.

- The logit or probit link functions are commonly used to model the relationship between the linear predictor and the probability of success.

- The logit function is the default choice and is often preferred due to its simplicity and interpretability.

3. Poisson Family with Log Link:

- Appropriate for count data where the response variable represents the number of occurrences within a given interval.

- The log link function is typically used to ensure the estimated rates remain positive.

- If there is evidence of overdispersion (variance exceeds the mean), the negative binomial family with log link can be considered as an alternative.

4. Gamma Family with Inverse Link:

- Suitable for continuous data with a positively skewed distribution, such as durations or response times.

- The inverse link function is often used to model the reciprocal of the response variable.

- The gamma distribution assumes that the response variable follows a gamma distribution with a specific shape and rate parameter.

5. Other Families and Link Functions:

- There are additional families and link functions available for specific data types and distributions. For example, the ordinal family with a logit or probit link can handle ordered categorical responses.

- Different software packages may offer additional options, such as the quasi-likelihood family for accommodating data with non-standard distributions.

It is important to note that the choice of family and link functions should align with the characteristics and assumptions of your data. Consider the scale, distributional properties, and nature of the response variable when making your selection.

Additionally, consulting statistical textbooks, research papers, and software documentation specific to your chosen statistical package can provide further guidance and examples in selecting appropriate families and link functions for GEE models.