Fitting Linear Models • Generalized Linear Models • Overview of Generalized Linear Models
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Overview of Generalized Linear Models
While traditional linear models are used extensively in statistical data analysis, there are types of problems for which they are not appropriate.
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It may not be reasonable to assume that data are normally distributed. For example, the normal distribution (which is continuous) may not be adequate for modeling counts or measured proportions.
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If the mean of the data is naturally restricted to a range of values, the traditional linear model may not be appropriate, since the linear predictor can take on any value. For example, the mean of a measured proportion is between 0 and 1, but the linear predictor of the mean in a traditional linear model is not restricted to this range.
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It may not be realistic to assume that the variance of the data is constant for all observations. For example, it is not unusual to observe data where the variance increases with the mean of the data.
A generalized linear model extends the traditional linear model and is, therefore, applicable to a wider range of data analysis problems. See the section Examples of Generalized Linear Models for the form of a probability distribution from the exponential family of distributions.
As in the case of traditional linear models, fitted generalized linear models can be summarized through statistics such as parameter estimates, their standard errors, and goodness-of-fit statistics. You can also make statistical inference about the parameters using confidence intervals and hypothesis tests. However, specific inference procedures are usually based on asymptotic considerations, since exact distribution theory is not available or is not practical for all generalized linear models.