week-07-equations.tex

\documentclass{tufte-handout}

\usepackage{xcolor}
\usepackage{graphicx}

% set hyperlink attributes
\hypersetup{colorlinks}

\usepackage{amsmath}

% set image attributes:
\usepackage{graphicx}
\graphicspath{ {images/} }

% create environment for bottom paragraph:
\newenvironment{bottompar}{\par\vspace*{\fill}}{\clearpage}

% ============================================================

% define the title
\title{SOC 4930/5050: Week 07 Equations Quick \\Reference}
\author{Christopher Prener, Ph.D.}
\date{October 9\textsuperscript{th}, 2017}
% ============================================================
\begin{document}
% ============================================================
\maketitle % generates the title
% ============================================================

\vspace{5mm}
\section{One-sample T-test}\marginnote{Degrees of freedom ($v$) is defined as $v=n-1$.}
\begin{equation}
\scalebox{2} {$ t=\frac { \bar { x } - \mu }{ \frac{s}{\sqrt{n}} }  $}
\end{equation}

\vspace{5mm}
\section{Independent T-test, Homogeneous Variance}
\begin{subequations}
\paragraph{Independent T-test}\marginnote{Degrees of freedom ($v$) is defined as $v={n}_{a}+{n}_{b}-2$.}
\begin{equation}
\scalebox{2} {$ t=\frac { { \bar{X}  }_{ a }-{ \bar{X}  }_{ b } }{ \sqrt{\frac{{ s }_{ p }^{ 2 }}{{ n }_{ a }}+\frac{{ s }_{ p }^{ 2 }}{{ n }_{ b }}} } $}
\end{equation}

\vspace{3mm}
\paragraph{Pooled Variance}
\begin{equation}
\scalebox{2} {$ { s }_{ p }^{ 2 }=\frac{\left({n}_{a}-1\right){s}_{a}^{2}+\left({n}_{b}-1\right){s}_{b}^{2}}{{n}_{a}+{n}_{b}-2} $}
\end{equation}
\end{subequations}

\vspace{5mm}
\section{Independent T-test, Heterogeneous Variance}
\begin{subequations}
\paragraph{Independent T-test}
\begin{equation}
\scalebox{2} {$ t=\frac { { \bar{X}  }_{ a }-{ \bar{X}  }_{ b } }{ \sqrt{\frac{{ s }_{ a }^{ 2 }}{{ n }_{ a }}+\frac{{ s }_{ b }^{ 2 }}{{ n }_{ b }}} } $}
\end{equation}

\vspace{3mm}
\paragraph{Welch's Corrected Degrees of Freedom ($v$)}
\begin{equation}
\scalebox{2} {$ v\approx \frac { \left( \frac { { s }_{ a }^{ 2 } }{ { n }_{ a } } +\frac { { s }_{ b }^{ 2 } }{ { n }_{ b } }  \right) ^{ 2 } }{ \frac { { s }_{ a }^{ 4 } }{ \left( { n }_{ a }^{2} \right)\left( { n }_{ a }-1 \right) } +\frac { { s }_{ b }^{ 4 } }{ \left( { n }_{ b }^{2} \right)\left( { n }_{ b }-1 \right) }  } $}
\end{equation}
\end{subequations}

\vspace{5mm}
\section{Dependent T-test}
\begin{equation}
\scalebox{2} {$ t=\frac{\bar{d}}{\sqrt{\frac{{s}_{d}^{2}}{n}}} $}
\end{equation}

\vspace{5mm}
\section{Cohen's $D$}\marginnote{Note that groups $t$ and $c$ are defined for controlled experiments where $t=treatment$ and $c=control$. This can be applied to the above equations by defining $t=a$ and $c=b$.}
\begin{subequations}
\paragraph{General Equation}
\begin{equation}
\scalebox{2} {$ d=\frac{{M}_{t}-{M}_{c}}{\sqrt{\frac{\left({n}_{t}-1\right){s}_{t}^{2}+\left({n}_{c}-1\right){s}_{c}^{2}}{{n}_{t}+{n}_{c}-2}}} $}
\end{equation}

\vspace{3mm}
\paragraph{Cohen's $D$ after T-test, ${n}_{a}={n}_{b}$}
\begin{equation}
\scalebox{2} {$ d=\frac{2t}{\sqrt{v}} $}
\end{equation}

\vspace{3mm}
\paragraph{Cohen's $D$ after T-test, ${n}_{a}\neq{n}_{b}$}
\begin{equation}
\scalebox{2} {$ d=\frac{t\left({n}_{t}+{n}_{c}\right)}{\sqrt{v}\left(\sqrt{{n}_{t}+{n}_{c}} \right)} $}
\end{equation}
\end{subequations}

% ============================================================
\end{document}