A.06.Anderson et al.
Home > Flashcards > Print Preview
The flashcards below were created by user
Exam8
on
FreezingBlue Flashcards. What would you like to do?

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.