#LyX 1.6.2 created this file. For more info see http://www.lyx.org/ \lyxformat 345 \begin_document \begin_header \textclass article \use_default_options true \language english \inputencoding auto \font_roman default \font_sans default \font_typewriter default \font_default_family default \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 \graphics default \paperfontsize default \spacing single \use_hyperref false \papersize default \use_geometry true \use_amsmath 1 \use_esint 1 \cite_engine basic \use_bibtopic false \paperorientation portrait \leftmargin 1in \topmargin 1in \rightmargin 1in \bottommargin 1in \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \author "" \author "" \end_header \begin_body \begin_layout Standard Variance Functions: \end_layout \begin_layout Standard Constant: \begin_inset Formula $\boldsymbol{1}$ \end_inset \end_layout \begin_layout Standard Power: \begin_inset Formula $\boldsymbol{X}^{2}$ \end_inset \end_layout \begin_layout Standard Binomial: \begin_inset Formula $np(1-p)\text{ where }p=\frac{\mu}{n};\,\, V(\mu)=np(1-p)$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\frac{\partial\mu}{\partial\eta}$ \end_inset \end_layout \begin_layout Standard Links: initialization of base class returns the actual mean vector \begin_inset Formula $\boldsymbol{\mu}$ \end_inset ; \begin_inset Formula $p$ \end_inset in the logit and subclasses; \begin_inset Formula $x$ \end_inset elsewhere. \end_layout \begin_layout Standard \begin_inset Float table placement H wide false sideways false status open \begin_layout Plain Layout \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Link \begin_inset Formula $g(p)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Inverse \begin_inset Formula $g^{-1}(p)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Analytic Derivative \begin_inset Formula $g^{\prime}(p)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Logit \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $z=\log\frac{p}{1-p}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $p=\frac{e^{z}}{1+e^{z}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $g^{\prime}(p)=\frac{1}{p(1-p)}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Power \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $z=x^{\text{pow}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $x=z^{\frac{1}{\text{pow}}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $g^{\prime}(x)=\text{pow}\cdot x^{\text{power}-1}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Inverse \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout same as above with \begin_inset Formula $\text{pow}=-1$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Square Root \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{pow}=0.5$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Identity \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{pow}=1$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Log \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $z=\log x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $g^{-1}(z)=e^{z}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $g^{\prime}(x)=\frac{1}{x}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout CDFLink/Probit \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $z=\Phi^{-1}(p)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $p=\Phi(z)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $g^{\prime}(x)=\frac{1}{\int_{-\infty}^{p}f(t)dt}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Cauchy \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout same as the above with the Cauchy distribution \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout CLogLog \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $z=\log(-\log p)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $p=e^{-e^{z}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $g^{\prime}(p)=-\frac{1}{p\log p}$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption \begin_layout Plain Layout Link Functions \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard Initializing the family sets a link property and a variance based on the link(?) \end_layout \begin_layout Standard \begin_inset Float table placement H wide false sideways false status open \begin_layout Plain Layout \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout Family \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Weights \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Deviance \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout DevResid \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Fitted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Predict \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Base Class \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\frac{1}{(g^{\prime}(\mu))^{2}\cdot V(\mu)}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\frac{\sum_{i}\text{DevResid}^{2}}{\text{scale}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(Y-\mu\right)\cdot\sqrt{\text{weights}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\mu=g^{-1}(\eta)$ \end_inset * \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\eta=g(\mu)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Poisson \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{sign}\left(Y-\mu\right)\sqrt{2Y\log\frac{Y}{\mu}-2(Y-\mu)}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Gaussian \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\frac{\left(Y-\mu\right)}{\text{\sqrt{\text{scale}\cdot V\left(\mu\right)}}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Gamma \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Bug? \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Binomial \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{sign}\left(Y-\mu\right)\sqrt{-2Y\log\frac{\mu}{n}+\left(n-Y\right)\log\left(1-\frac{\mu}{n}\right)}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Inverse Gaussian \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout ? \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption \begin_layout Plain Layout Families \end_layout \end_inset \end_layout \begin_layout Plain Layout * \begin_inset Formula $\eta$ \end_inset is the linear predictor ie., \begin_inset Formula $X\beta$ \end_inset in the generalized linear model \end_layout \end_inset \end_layout \end_body \end_document