\(\LaTeX\) Guide

Typesetting Mathematical Equations in Quarto

An Introduction to \(\LaTeX\) in Quarto

\(\LaTeX\) is a markup language and document preparation system that allows for (easier) typesetting of mathematical symbols, notation, and equations. Quarto allows for integration of \(\LaTeX\) code alongside text and R code!

Getting Started

To indicate to the document compiler that we want to start typesetting math equations, we offset \(\LaTeX\) code using dollar signs: $your LaTeX code here$. For example, the code $y=mx+b$ produces \(y=mx+b\).

Common Math Commands

Most LaTeX math commands are relatively intuitive and obey common mathematical order. We’ll illustrate some of the most common symbols you might use in an SDS course below!

• Superscripts and Subscripts: we can use underscores to obtain subscripts (e.g., $x_i$ yields \(x_i\)) and carets to obtain superscripts (e.g., $e_i^2$ yields \(e_i^2\)). If your superscript or subscript contains more than one character (e.g., if we wished to write down notation for a \(t\) distribution with \(n-2\) degrees of freedom), we can wrap our superscripts and subscripts in braces, { }. As a concrete example, the code $t_n-2$ produces the output \(t_n-2\), while the code $t_{n-2}$ produces the output \(t_{n-2}\).

• Hats for Estimated and Predicted Values: we can use the \widehat{} command to place hats on top of values that are estimated or predicted from the observed data (e.g., $\widehat{y}_i$ yields \(\widehat{y}_i\)).

• Overbars for the Sample Mean: we can use the command $\overline{}$ to denote the sample average (e.g., $\overline{x}$ yields \(\overline{x}\)).

• Fractions: we can use the command \frac{}{} to typeset fractions; anything in the first set of brackets will go in the numerator, and anything in the second set of brackets will go in the denominator (e.g., $\frac{1}{2}$ yields \(\frac{1}{2}\)).

• Summations: we can use the command $\sum_{}^{}$ to typeset the summation notation we’ve been using in class; the first set of brackets sets up the indexing that goes below/on the bottom of the summation sign while the second set of brackets specifies the upper limit on the sum, which goes above/on top of the summation sign (e.g., $\sum_{i=1}^{n}x_i$ yields \(\sum_{i=1}^{n}x_i\)).

• Multiplication: by convention, we typically do not write a multiplication symbol at all (such as when writing \(y=mx+b\)); alternatives include indicating multiplication using parentheses (such as when writing \((4)(3)=12\)) or using the $\cdot$ or $\times$ commands (which yield \(\cdot\) and \(\times\), respectively).

• Inequalities: we can use the commands $<$, $\leq$, $>$, and $\geq$ to typeset \(<\), \(\leq\), \(>\), and \(\geq\), respectively.

Note

The Spinelli Center maintains an extensive guide to LaTeX and list of LaTeX-related resouces, accessible here should you be interested in learning more!

Typesetting Population Parameters and Population Regression Lines

In statistics, the convention is to represent all population parameters by lowercase Greek letters; here is a list of the most common letters used and the corresponding LaTeX code:

• Alpha: \(\alpha\) (\alpha)
• Beta: \(\beta\) (\beta)
• Gamma: \(\gamma\) (\gamma)
• Delta: \(\delta\) (\delta)
• Epsilon: \(\epsilon\) (\epsilon) or \(\varepsilon\) (\varepsilon)
• Theta: \(\theta\) (\theta)
• Lambda: \(\lambda\) (\lambda)
• Mu: \(\mu\) (\mu)
• Pi: \(\pi\) (\pi)
• Rho: \(\rho\) (\rho)
• Sigma: \(\sigma\) (\sigma)
• Chi: \(\chi\) (\chi)

Equations and Formulas

If we want to typeset full equations that are important enough (or long enough) to be offset from the main text in their own centered paragraph (like the equations for the population or fitted linear regression line), we can do so using display math mode. Display mode is achieved by either (a) wrapping the LaTeX code in two sets of double dollar signs $$code for your offset equation here$$ or (b) wrapping the LaTeX code in \begin{equation} and \end{equation}.

Here we demonstrate both of those approaches; the following code (when rendered) should produce the population model for the mean as a centered equation on its own line in the document twice:

$$
  E[Y_i | x_i] = \beta_0 + \beta_1 x_i
$$

\[ E[Y_i | x_i] = \beta_0 + \beta_1 x_i \]

\begin{equation}
E[Y|x_i] = \beta_0 + \beta_1 x_i.
\end{equation}

\[\begin{equation} E[Y|x_i] = \beta_0 + \beta_1 x_i. \end{equation}\]

The equivalent representation of the population regression line (in data = model + error format), alongside the regression assumptions, can be typeset as

\begin{equation}
Y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \quad \epsilon_i \sim N(0, \sigma^2),
\end{equation}

\[\begin{equation} Y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \quad \epsilon_i \sim N(0, \sigma^2), \end{equation}\] where the code $\sim$ produces \(\sim\), our notational shorthand for the phrase “follows the same distribution is” or “is distributed according to”. So we read the notation \(\epsilon_i \sim N(0, \sigma^2)\) as “the errors (i.e., the way in which the population deviates from the model) follow a Normal distribution with mean 0 and variance \(\sigma^2\)”.

The fitted regression line based on the observed values of \(y_i\) and \(x_i\) in our sample is given by \[\widehat{y}_i = \widehat{\beta}_0 + \widehat{\beta}_1 x_i.\]

If you would like to create a multi-line display math environment where each line of your formula/text is vertically aligned at a particular location, you can alternatively use the align environment, which wraps the LaTeX code in \begin{align} and \end{align}. To insert a line break in the aligned formula, we use two backslashes, \\; the content before the two backslashes forms the first line of the formula and the content after the two backslashes forms the second line. To indicate how these two (or more) lines should be positioned on top of each other, we use the “and” sign, &; \(\LaTeX\) formats the multiple lines so that the locations of the & sign are vertically aligned. For example, the code

\begin{align}
  a &= b + c \\ 
  d + e  f &= \\ 
  g + h &= i + j
\end{align}

produces the output \[\begin{align} a &= b + c \\ d + e + f&= \\ g + h &= i + j \end{align}\] If you would like to turn off the equation numbering, you may do so by replacing \begin{align} \end{align} with \begin{align*} \end{align*}.