Generalized Linear Models

Linear regression is the only one where variance is independent from the mean

--> thus, we don't rely on residuals.

Exponential family : any equation that can fit in the following equation format:

GLMs are members of the exponential family, which can be expressed by a certain form (complex looking equation)

Meaning of coefficients?

• Additive models: linear, logit, log hazard, exponentiation of binomial
• Multiplicative models: logistic, hazard, GLMs

Hypothesis testing = when the nuance parameters come in to play. For determining the relationships when adjusting the variables, they do not influence.

Binomial Distribution

Values: between 0 and n (n= number of times the event occurs; p = probability of each event)

Mean: np

Variance: np (1-p)

Poisson distribution

Values: integer values 0 or greater

mean and variance are the same. Useful if the data is not particularly right skewed. -- if it is right skewed, then negative binomial is the best approach. -- contrast with negative binomial - variance is allowed to be something else: better for right skewed data.

Use: categorical outcome where mean and variance are the same

Exponential Distribution

Lambda is a rate

Values: continuous values greater than 0

mean: 1/lambda

variance 1/lambda^2

Negative binomial

values: integers greater than or equal to 0 mean: pr / 1-p variance: pr / (1-p)^2

use: categorical outcome where mean and variance are not the same

Gamma distribution

utility - mean and variance are related, but assumptions of normality can be violated - thus, this can be used with with continuous outcomes when normality assumptions don't hold .

To do the actual molding, use the log link (not inverse, even though inverse is it's natural link) - if log link used, it's similar to poisson or negative binomial interpretation. -- this gets used often with things like laboratory values: skew, but don't need negatives.

values: continuous values over 0

omega = the scale mean: komega variance: komega^2

##Model fit:

• 1/deviance, 1/pearson = theoretically you want to be close to 1 if you choose the right model.
• AIC and BIC (can compare this between models to see which one fits best). Note: if difference is greater than 2, it's significant. BIC favors more parsimonious option than AIC, but you can use either.
• Lastly, you can look at the amount of variance in the y and see if which is best

The tests for poisson, negative binomial (analogous to Hosmer Lemshow) aren't all that good in practice, so not used.

Catorigcal predictor: use testparm to see: is this generally important? (Any categories?)