A.06.Anderson et al.

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1. 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 σ2 → 0 as μ → 0 ⇒σ2 is a fctn of μ
• additivity effect not realistic for some applications
2. 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)
3. What changed from LM to GLM
• no additivity assumption
• no assumption that the response var has constant var
• Var(Yi) = φVar(μi) / ωi
• 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(ΣβiXi) + ε
4. Advantages of exponential family
• (+) each dist is fully specified in terms of μ and σ2
• (+) σ2 is a function of its μ: Var(Yi) = φV(μi) / ωi
• (+) incl normal, poisson, gamma, binomial, inv gaussian
5. Canonical link function
• Distribution         | g(x)              | g-1(x)
• Normal                 | x                   | x
• Poisson                 | ln(x)             | ex
• Gamma                 |  1/x               | 1/x
• Binomial                | ln(x(1-x))     | ex(1+ex)
• Inverse Gaussian | 1/x2              | 1/√x
6. 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
7. GLM Model Diagnostics
• std error: speed w which log-likelihood 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.
 Author: Exam8 ID: 161157 Card Set: A.06.Anderson et al. Updated: 2012-08-14 02:31:06 Tags: GLM link base linear model generalized offset Folders: Description: A practitioner's Guide to Generalized Linear Models Show Answers: