14.3: Assumptions of GLMs
- Page ID
- 57770
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This section distills the assumptions of Generalized Linear Models (GLMs) into three clear requirements. While the Classical Linear Model rested primarily on the assumption of Normally-distributed errors, a GLM makes assumptions about each of its three core components: the linear predictor must be correctly specified, the conditional distribution of the response variable must be appropriate for the data, and the chosen link function must correctly relate the linear predictor to the expected value. The section emphasizes that violations of any of these assumptions mean the model is missing systematic information present in the data. It also notes that testing these assumptions is more complex than in OLS, as the correctness of the distribution involves understanding its theoretical properties (such as its range and mean-variance relationship) and because heteroskedasticity may be an inherent feature of certain distributions rather than a violation.
By the end of this section, you will be able to:
- State the three core assumptions of a Generalized Linear Model: that the linear predictor (\(\eta = \mathbf{XB}\)) is correctly specified, that the conditional distribution of the response variable is correctly chosen, and that the link function (\(g(\mu)=\eta\)) is correct.
- Recognize that if any of these assumptions are violated, the model is misspecified and fails to capture all the information in the data, potentially leading to biased or inefficient estimates.
- Explain why testing the assumptions in a GLM is more complex than in OLS, particularly because the correctness of the conditional distribution involves understanding its theoretical properties (range, mean-variance relationship) and because heteroskedasticity may be an inherent feature of the distribution rather than a violation.
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When we were creating ordinary least squares (OLS) regression, we made one assumption: \(\varepsilon \stackrel{\text{iid}}{\sim} N(0, \sigma^2)\). After learning the mathematics of fitting the models, we went back and figured out how to test these assumptions. The same will be true here.
When performing generalized linear modeling, you make at least three assumptions:
- you assume the linear predictor is correct;
- you assume the conditional distribution of the dependent variable is correct; and
- you assume the link function is correct.
If these assumptions are not met by the data and model, then there is information in the data that you are ignoring.
Testing these is usually not as easy as in the case of OLS regression. The linear predictor and the link function, together, determine the functional form. It can sometimes be tested using a runs test. That is the easy part. Testing the correctness of the conditional distribution is much more involved. It requires that one understands the hypothesized distribution, especially in terms of range, expected values, and variances. Note that tests of heteroskedasticity may not be useful here; many distributions are heteroskedastic by definition.
The testing must be done, however.
As you read through this part of the book, always keep in mind what we are assuming. That will help you determine the requirements and how to test them.