The Chi-Squared Test of Independence – An Example in Both R and SAS

Introduction

The chi-squared test of independence is one of the most basic and common hypothesis tests in the statistical analysis of categorical data.  Given 2 categorical random variables, $X$ and $Y$, the chi-squared test of independence determines whether or not there exists a statistical dependence between them.  Formally, it is a hypothesis test with the following null and alternative hypotheses:

$H_0: X \perp Y \ \ \ \ \ \text{vs.} \ \ \ \ \ H_a: X \not \perp Y$

If you’re not familiar with probabilistic independence and how it manifests in categorical random variables, watch my video on calculating expected counts in contingency tables using joint and marginal probabilities.  For your convenience, here is another video that gives a gentler and more practical understanding of calculating expected counts using marginal proportions and marginal totals.

Today, I will continue from those 2 videos and illustrate how the chi-squared test of independence can be implemented in both R and SAS with the same example.

Applied Statistics Lesson of the Day – The Coefficient of Variation

In my statistics classes, I learned to use the variance or the standard deviation to measure the variability or dispersion of a data set.  However, consider the following 2 hypothetical cases:

1. the standard deviation for the incomes of households in Canada is $2,000 2. the standard deviation for the incomes of the 5 major banks in Canada is$2,000

Even though this measure of dispersion has the same value for both sets of income data, $2,000 is a significant amount for a household, whereas$2,000 is not a lot of money for one of the “Big Five” banks.  Thus, the standard deviation alone does not give a fully accurate sense of the relative variability between the 2 data sets.  One way to overcome this limitation is to take the mean of the data sets into account.

A useful statistic for measuring the variability of a data set while scaling by the mean is the sample coefficient of variation:

$\text{Sample Coefficient of Variation (} \bar{c_v} \text{)} \ = \ s \ \div \ \bar{x},$

where $s$ is the sample standard deviation and $\bar{x}$ is the sample mean.

Analogously, the coefficient of variation for a random variable is

$\text{Coefficient of Variation} \ (c_v) \ = \ \sigma \div \ \mu,$

where $\sigma$ is the random variable’s standard deviation and $\mu$ is the random variable’s expected value.

The coefficient of variation is a very useful statistic that I, unfortunately, never learned in my introductory statistics classes.  I hope that all new statistics students get to learn this alternative measure of dispersion.

Organic and Inorganic Chemistry Lesson of the Day – Diastereomers

I previously introduced the concept of chirality and how it is a property of any molecule with only 1 stereogenic centre.  (A molecule with $n$ stereogenic centres may or may not be chiral, depending on its stereochemistry.)  I also defined 2 stereoisomers as enantiomers if they are non-superimposable mirror images of each other.  (Recall that chirality in inorganic chemistry can arise in 2 different ways.)

It is possible for 2 stereoisomers to NOT be enantiomers; in fact, such stereoisomers are called diastereomers.  Yes, I recognize that defining something as the negation of something else is unusual.  If you have learned set theory or probability (as I did in my mathematical statistics classes) then consider the set of all pairs of the stereoisomers of one compound – this is the sample space.  The enantiomers form a set within this sample space, and the diastereomers are the complement of the enantiomers.

It is important to note that, while diastereomers are not mirror images of each other, they are still non-superimposable.  Diastereomers often (but not always) arise from stereoisomers with 2 or more stereogenic centres; here is an example of how they can arise.

1) Consider a stereoisomer with 2 tetrahedral stereogenic centres and no meso isomers*.  This isomer has $2^{n = 2}$ stereoisomers, where $n = 2$ denotes the number of stereogenic centres.

2) Find one pair of enantiomers based on one of the stereogenic centres.

3) Find the other pair enantiomers based on the other stereogenic centre.

4) Take any one molecule from Step #2 and any one molecule from Step #3.  These cannot be mirror images of each other.  (One molecule cannot have 2 different mirror images of itself.)  These 2 molecules are diastereomers.

Think back to my above description of enantiomers as a proper subset within the sample space of the pairs of one set of stereoisomers.  You can now see why I emphasized that the sample space consists of pairs, since multiple different pairs of stereoisomers can form enantiomers.  In my example above, Steps #2 and #3 produced 2 subsets of enantiomers.  It should be clear by now that enantiomers and diastereomers are defined as pairs.  To further illustrate this point,

a) call the 2 molecules in Step#2 A and B.

b) call the 2 molecules in Step #3 C and D.

A and B are enantiomers.  A and C are diastereomers.  Thus, it is entirely possible for one molecule to be an enantiomer with a second molecule and a diastereomer with a third molecule.

Here is an example of 2 diastereomers.  Notice that they have the same chemical formula but different 3-dimensional orientations – i.e. they are stereoisomers.  These stereoisomers are not mirror images of each other, but they are non-superimposable – i.e. they are diastereomers.

D-Threose

D-Erythrose

Images courtesy of Popnose, DMacks and Edgar181 on Wikimedia.  For brevity, I direct you to the Wikipedia entry for diastereomers showing these 4 images in one panel.

*I will discuss meso isomers in a separate lesson.