16.2: The Mathematics
- Page ID
- 57787
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Remember from the chapter on GLMs that performing generalized linear modeling requires that we specify three things about our model:
- the linear predictor;
- the conditional distribution of the dependent variable; and
- the function that links the two.
Let us go through these with the Binomial distribution.
Linear Predictor
As usual, the linear predictor is the function that relates the independent variable(s) with the dependent variable. For \(k\) predictor (independent) variables, the linear predictor is
\[ \eta = \beta_0 + \beta_1 x_1 + \cdots + \beta_{k} x_{k} \]Frequently, this is what the researcher cares most about. This is the start of where we can test if certain variables can help in better understanding the data-generating process.
Conditional Distribution
The second need is the conditional distribution of the dependent variable, the distribution of \(Y\), given the values of the x-variables. For the Binomial distribution, the probability mass function (pmf) is
\[ f(y,\ \pi) = \binom{n}{y} \pi^y (1-\pi)^{n-y} \qquad y \in \{0, 1, 2, \ldots, n\} \]I leave it as an exercise for you to show that the Binomial distribution is a member of the Exponential Class of distributions. In other words, you will need to show that the above probability mass function can be written as
\[ f(y,\ \pi) = \exp \left[ \frac{y \mathrm{logit } (\pi) + n \log(1-\pi)}{1} + \log \binom{n}{y} \right] \label{eq:bin-link} \]With this, we can calculate \(\mathrm{E}[Y]\) and \(\mathrm{V}[Y]\). Note that equation \ref{eq:bin-link} above shows us that the canonical link is the logit function,
\begin{equation}
g(\pi) = \mathrm{logit}(\pi)
\end{equation}
As always, the canonical link offers some mathematical cleanness but little else. If the situation calls for a different link function, you should use it.
Link Function
From above, we know that the canonical link function is the logit function. However, as discussed in the section on GLM mathematics, many alternative link functions are available. If the model is sound, then predictions based on those alternatives will tend to be similar. Let me emphasize that here:
It is extremely rare that the link function can be determined from the scientific theory — extremely rare. Thus, if the model significantly depends on the choice of link, then the model is weak. You should improve the model.
This also suggests another model test. Fit the model using several link functions. That the results are substantively the same across the link functions supports the goodness of your model.
Assumptions/Requirements
As you have read through this chapter, what assumptions were made? Those are the requirements you need to check. Allow me to repeat this extremely important point:
As you have read through this chapter, what assumptions were made? Those are the requirements you need to check.