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Limitations of Linear Models
 difficult to assert normality and constant var for resp variable (can transform like ln(x) to satisfy)
 values from resp var may be restricted to be > 0 (violates assumption of normality)
 if resp var strictly > 0 then σ^{2} → 0 as μ → 0 ⇒σ^{2} is a fctn of μ
 additivity effect not realistic for some applications

Generalized Linear Model assumptions
 (GLM1) random component: each cpnt of Y is independent and is from one of the exponential family of distribution
 (GLM2) systematic component: the p covariates are combined to give the linear predictor η = X β
 (GLM3) link fctn: relationship btwn rdm & syst cpnts is specified via link fctn g that is differentiable & monotonic such that E[Y] = μ = g^{1}(η)

What changed from LM to GLM
 no additivity assumption
 no assumption that the response var has constant var
 Var(Y_{i}) = φVar(μ_{i}) / ω_{i}
 reponse variable is not assumed to be normal, but rather from a member of the exponential family
 Y depends on X first & then g^{1}(Σβ_{i}X_{i}) + ε

Advantages of exponential family
 (+) each dist is fully specified in terms of μ and σ^{2}
 (+) σ^{2} is a function of its μ: Var(Y_{i}) = φV(μ_{i}) / ω_{i}
 (+) incl normal, poisson, gamma, binomial, inv gaussian

Canonical link function
 Distribution  g(x)  g^{1}(x)
 Normal  x  x
 Poisson  ln(x)  e^{x}
 Gamma  1/x  1/x
 Binomial  ln(x(1x))  e^{x}(1+e^{x})
 Inverse Gaussian  1/x^{2}  1/√x

GLM Aliasing
 solving routine to remove as many param as necessary to make the model uniquely defined
 occurs when there is a linear dependency among covariates
 intrinsic: dependencies inherent in the definition of covariates
 extrinsinc: from the nature of the data (eg: if X = . Y is .)
 choice of alias does not modify fitted values
 near aliasing: occurs when 2 var are almost 100% correlated. Convergence problems may occur, so exclude, delete or reclassify

GLM Model Diagnostics
 std error: speed w which loglikelihood falls from the maximum given a change in parameter
 deviance test: measures how much fitted values diff from obs. Adjusts for V(x) giving more weight to deviance if V(x) is small. Helps assess theoretical significance of a particular factor.

