Abstract
In Chap. 10 we examined nonlinear models with normally-distributed errors. Given these conditions, minimizing the residual sum of squares maximizes the likelihood function. Not all variables of interest to scientists are normally distributed, however. Instead of being continuous and unbounded, many variables are discrete (e.g., number of aphids on a leaf), categorical (e.g., number of men and women who buy or do not buy life insurance in a given year), binary (e.g., employed or unemployed), or restricted to having only non-negative values (e.g., rainfall). Because these variables are not normally-distributed, minimizing the residual sum of squares does not produce maximum likelihood estimates.
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- 1.
When the shape parameter is an integer, the gamma distribution is called an Erlang distribution.
- 2.
The form of the likelihood for the Gaussian distribution is intended to highlight its connections with the other three. This connection will be explained momentarily.
- 3.
The sign of z is reversed for the gamma distribution.
- 4.
Our examples will also use only a single predictor, as extensions to multiple predictors are straightforward.
- 5.
To qualify as a true Poisson process the events must be independent. Yawning can be contagious, so this is not necessarily the best example. But insofar as the data are fabricated anyway, we are also going to stipulate that students are not able to see one another yawn during my lectures.
- 6.
Using dummy-coded vectors, Poisson GLMs can also analyze categorical data. Such an analysis is commonly referred to as a log linear analysis, and readers interested in details can consult Agresti (2013).
- 7.
Predictions are approximate and should be made only for observed values of the predictor.
- 8.
With a very large sample and a single predictor, the significance of the regression coefficient will match the significance of the goodness of fit test.
- 9.
Later in this chapter we will see that the Pearson residuals can also be used to perform a goodness of fit test.
- 10.
Under dispersion can also occur, although it is less common.
- 11.
The logit is the canonical link for the binomial family, but a probit link can also be used. The two links ordinarily produce very similar estimates.
- 12.
To create a smoother curve, the figures use interpolated values of y.
- 13.
Equivalently, Var = μ2/α.
- 14.
The dispersion parameter \( \widehat{\phi} \) can also be estimated from the sum of the squared Pearson residuals, divided by (n − p).
References
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Brown, J.D. (2018). Generalized Linear Models. In: Advanced Statistics for the Behavioral Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-93549-2_11
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