In mac-theobio/effects: isolated confidence intervals and bias-corrected predictor effects

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library(shellpipes)
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library(vareffects); varefftheme()

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\newcommand{\nset}[1]{#1_{{n}}} \newcommand{\yref}{y_{\textrm{ref}}} \newcommand{\cdist}{{D(\nset{x}|x_f)}} \newcommand{\cdistprime}{{D(\nset{x}|x_{f'})}} \newcommand{\xfprime}{x_{f'}}

\newcommand{\bX}{{\mathbf X}} \newcommand{\bbeta}{{\boldsymbol \beta}} \newcommand{\boldeta}{{\boldsymbol \eta}}

Notation :

• $x_f$: a value of the focal predictor
• $\nset{x}$: a vector of values of the non-focal predictors for a particular observation
• $\eta(x_f, \nset{x}) = \beta_f x_f + \sum \nset{\beta} \nset{x}$ = linear predictor (e.g. prediction on the log-odds scale)
• $g^{-1}()$: inverse-link function (e.g. logistic)
• $\cdist$: distribution of the non-focal predictors conditional on a particular value of the focal predictor
• $\beta_{fi}$: the coefficient describing the interaction(s) of the focal and non-focal parameters

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\section{Population-based approach for bias correction}

\subsection{Binned non-focal linear predictor}

To implement this: \begin{itemize} \item compute linear predictor of the non-focal predictors, $\nset{\eta} = \sum \nset{\beta} \nset{x}$ \item find a list of vectors of observations of $\nset{\eta}$ associated with each value (bin) of the focal predictor, $\nset{{\eta_j}}$, $j = 1, 2, \cdots$ \item for each $\nset{{\eta_j}}$: \begin{itemize} \item compute $\hat{y}j = \textrm{mean} ~ g^{-1} \left(\beta_f x{j_f} + \nset{{\eta_j}}\right)$ \end{itemize} \end{itemize}

If we compute the individual back-transformed predictions for a poorly sampled/finely spaced set of focal values, we will get a noisy prediction line as the values of the non-focal predictors shift across the focal values. Simple example: suppose everyone below the median age has wealth index $w_1$, everyone above the median has $w_2$. Then the predicted value will have a discontinuity at the median age. We can deal with this by taking bigger bins (a form of smoothing), or by post-smoothing the results (by loess, for example). The principled form of this would be to assume/recognize that our uneven distribution of observed non-focal predictors actually represents a sample of a distribution that will vary \emph{smoothly} as a function of the focal predictor.

\subsection{Whole population non-focal linear predictor}

Instead of binning non-focal associated with particular level of focal predictor, we add the overall contribution of the non-focal predictor to the corresponding value of the focal linear predictor. In particular:

\begin{itemize} \item compute linear predictor of the non-focal predictors, $\nset{\eta} = \sum \nset{\beta} \nset{x}$ \item for every value of the focal predictor, $x_{j_f}$: \begin{itemize} \item compute $\hat{y}j = \textrm{mean} ~ g^{-1} \left(\beta_f x{j_f} + \nset{\eta}\right)$ \end{itemize} \end{itemize}

\section{Delta method}

Suppose that we have a response $Y$ and the transformed response on the link scale, $\eta$ and an inverse-link function $h(.) = g^{-1}(.)$. We consider $\eta$ as a one-dimensional random variable with mean ($\mu_{\eta}$) and standard deviation ($\sigma_{\eta}$). We are interested in the expected value of $h(\eta)$, i.e., the back-transformed values so that $Y = h(\eta)$. More specifically, based on a second-order Taylor expansion

\begin{align} Y \approx h(\mu_{\eta}) + h'(\eta)(\eta - \mu_{\eta}) + \frac{1}{2}h''(\eta)(\eta - \mu_{\eta})^2 \end{align}

so that

\begin{align} E(Y) = h(\eta) + \frac{1}{2}h''(\mu_{\eta})\sigma^2_{\eta} \end{align}

\section{McCulloch and Searle (2001)} [ E(Y) \approx h\left(\eta-\frac{0.5\sigma^2\tanh(\eta(1 + 2\exp(-0.5\sigma^2)))}{6}\right) ]

\section{Diggle et al. (2004)} [ E(Y) \approx h\left(\frac{\eta}{\sqrt{1 + \left(\frac{16\sqrt{3}}{15\pi}\right)^2\sigma^2}}\right) ]

mac-theobio/effects documentation built on July 6, 2023, 4:19 a.m.