After a multivariate test, it is often desired to know more about the specific groups to find out if they are significantly different or similar. This step after analysis is referred to as 'post-hoc analysis' and is a major step in hypothesis testing. One common and popular method of post-hoc analysis is Tukey's Test. The test is known by several different names. Tukey's test compares the means of all treatments to the mean of every other treatment and is considered the best available method in cases when confidence intervals are desired or if sample sizes are unequal.

## Kruskal-Wallis One-Way Analysis of Variance of Ranks

The Kruskal-Wallis test extends the Mann-Whitney-Wilcoxon Rank Sum test for more than two groups. The test is nonparametric similar to the Mann-Whitney test and as such does not assume the data are normally distributed and can, therefore, be used when the assumption of normality is violated. This example will employ the Kruskal-Wallis test on the

`PlantGrowth`

dataset as used in previous examples. Although the data appear to be approximately normally distributed as seen before, the Kruskal-Wallis test performs just as well as a parametric test.## Quadratic Discriminant Analysis of Several Groups

Quadratic discriminant analysis for classification is a modification of linear discriminant analysis that does not assume equal covariance matrices amongst the groups (\(\Sigma_1, \Sigma_2, \cdots, \Sigma_k\)). Similar to LDA for several groups, quadratic discriminant analysis for several groups classification seeks to find the group that maximizes the quadratic classification function and assign the observation vector \(y\) to that group.

## Quadratic Discriminant Analysis of Two Groups

LDA assumes the groups in question have equal covariance matrices (\(\Sigma_1 = \Sigma_2 = \cdots = \Sigma_k\)). Therefore, when the groups do not have equal covariance matrices, observations are frequently assigned to groups with large variances on the diagonal of its corresponding covariance matrix (Rencher, n.d., pp. 321). Quadratic discriminant analysis is a modification of LDA that does not assume equal covariance matrices amongst the groups. In quadratic discriminant analysis, the respective covariance matrix \(S_i\) of the \(i^{th}\) group is employed in predicting the group membership of an observation, rather than the pooled covariance matrix \(S_{p1}\) in linear discriminant analysis.

## Linear Discriminant Analysis for the Classification of Several Groups

Similar to the two-group linear discriminant analysis for classification case, LDA for classification into several groups seeks to find the mean vector that the new observation \(y\) is closest to and assign \(y\) accordingly using a distance function. The several group case also assumes equal covariance matrices amongst the groups (\(\Sigma_1 = \Sigma_2 = \cdots = \Sigma_k\)).

## Linear Discriminant Analysis for the Classification of Two Groups

In this post, we will use the discriminant functions found in the first post to classify the observations. We will also employ cross-validation on the predicted groups to get a realistic sense of how the model would perform in practice on new observations. Linear classification analysis assumes the populations have equal covariance matrices (\(\Sigma_1 = \Sigma_2\)) but does not assume the data are normally distributed.

## Discriminant Analysis for Group Separation

Discriminant analysis assumes the two samples or populations being compared have the same covariance matrix \(\Sigma\) but distinct mean vectors \(\mu_1\) and \(\mu_2\) with \(p\) variables. The discriminant function that maximizes the separation of the groups is the linear combination of the \(p\) variables. The linear combination denoted \(z = a′y\) transforms the observation vectors to a scalar. The discriminant functions thus take the form:

## Discriminant Analysis of Several Groups

Discriminant analysis is also applicable in the case of more than two groups. In the first post on discriminant analysis, there was only one linear discriminant function as the number of linear discriminant functions is \(s = min(p, k − 1)\), where \(p\) is the number of dependent variables and \(k\) is the number of groups. In the case of more than two groups, there will be more than one linear discriminant function, which allows us to examine the groups' separation in more than one dimension.

## Introduction to Rpoet

The Rpoet package is a wrapper of the PoetryDB API, which enables developers and other users to extract a vast amount of English-language poetry from nearly 130 authors. The package provides a simple R interface for interacting and accessing the PoetryDB database. This vignette will introduce the basic functionality of Rpoet and some example usages of the package.

## Calculating and Performing One-way Multivariate Analysis of Variance (MANOVA)

MANOVA, or Multiple Analysis of Variance, is an extension of Analysis of Variance (ANOVA) to several dependent variables. The approach to MANOVA is similar to ANOVA in many regards and requires the same assumptions (normally distributed dependent variables with equal covariance matrices).

## Calculating and Performing One-way Analysis of Variance (ANOVA)

ANOVA, or Analysis of Variance, is a commonly used approach to testing a hypothesis when dealing with two or more groups. One-way ANOVA, which is what will be explored in this post, can be considered an extension of the t-test when more than two groups are being tested. The factor, or categorical variable, is often referred to as the 'treatment' in the ANOVA setting. ANOVA involves partitioning the data's total variation into variation between and within groups. This procedure is thus known as Analysis of Variance as sources of variation are examined separately.

## Computing Working-Hotelling and Bonferroni Simultaneous Confidence Intervals

There are two procedures for forming simultaneous confidence intervals, the Working-Hotelling and Bonferroni procedures. Each estimates intervals of the mean response using a family confidence coefficient. The Working-Hotelling coefficient is defined by \(W\) and Bonferroni \(B\). In practice, it is recommended to perform both procedures to determine which results in a tighter interval. The Bonferroni method will be explored first.

## Predicting Cat Genders with Logistic Regression

Consider a data set of 144 observations of household cats. The data contains the cats' gender, body weight and height. Can we model and accurately predict the gender of a cat based on previously observed values using logistic regression?

## PetfindeR, R Wrapper for the Petfinder API, Introduction Part Two

The first post introduced and explored the basic usage of the PetfindeR library. In this post, we take a quick look at some of the additional uses of the library and its methods to extract data from the Petfinder database.

## PetfindeR, R Wrapper for the Petfinder API, Introduction Part One

The goal of the PetfindeR package is to provide a simple and straightforward interface for interacting with the Petfinder API through R. The Petfinder database contains approximately 300,000 adoptable pet records and 11,000 animal welfare organization records, which makes it a handy and valuable source of data for those in the animal welfare community. However, the outputs from the Petfinder API are in messy JSON format and thus it makes it more time-consuming and often frustrating to coerce the output data into a form that is workable with R.

## Analyzing Nationwide Utility Rates with R, SQL and Plotly

R and SQL make excellent complements for analyzing data due to their respective strengths. The sqldf package provides an interface for working with SQL in R by querying data from a database into an R data.frame. This post will demonstrate how to query and analyze data using the sqldf package in conjunction with the graphing libraries plotly and ggplot2 as well as some other packages that provide useful statistical tests and other functions.

## Cartesian Product and Ordered and Unordered Pairs

A pair set is a set with two members, for example, {2, 3}, which can also be thought of as an unordered pair, in that {2, 3}={3, 2}. However, we seek a more a strict and rich object that tells us more about two sets and how their elements are ordered.

## Algebra of Sets with R

The set operations, union and intersection, the relative complement − and the inclusion relation (subsets) are known as the algebra of sets. The algebra of sets can be used to find many identities related to set relations.

## N-Union and Intersection Set Operations

Set unions and intersections can be extended to any number of sets. This post introduces notation to simplify the expression of n-sets and the set union and intersection operations themselves with R.

## Combined Linear Congruential Generator for Pseudo-random Number Generation

Combined linear congruential generators, as the name implies, are a type of PRNG (pseudorandom number generator) that combine two or more LCGs (linear congruential generators). The combination of two or more LCGs into one random number generator can result in a marked increase in the period length of the generator which makes them better suited for simulating more complex systems.

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