## Arnab Chakraborty on The Monty Hall Problem and Bayes’ Theorem – The Central Equilibrium – Episode 6

I am pleased to welcome Arnab Chakraborty back to my talk show, “The Central Equilibrium“, to talk about the Monty Hall Problem and Bayes’ theorem.  In this episode, he shows 2 solutions to this classic puzzle in probability, and invokes Bayes’ Theorem for the second solution.

If you have not watched Arnab’s first episode on Bayes’ theorem, then I encourage you to do that first.

Marilyn Vos Savant provided a solution to this problem in PARADE Magazine in 1990-1991.  Thousands of readers disagreed with her solution and criticized her vehemently (and incorrectly) for her error.  Some of these critics were mathematicians!  She included some of those replies and provided alternative perspectives that led to the same conclusion.  Although I am dismayed by the disrespect that some people showed in their letters to her, I am glad that a magazine column on probability was able to attract so much readership and interest.  Arnab and I referred to one of her solutions in our episode.  Thank you, Marilyn!

Enjoy this episode of “The Central Equilibrium“!

## Layne Newhouse on representing neural networks – The Central Equilibrium – Episode 4

I am excited to present the first of a multi-episode series on neural networks on my talk show, “The Central Equilibrium”.  My guest in this series in Layne Newhouse, and he talked about how to represent neural networks. We talked about the biological motivations behind neural networks, how to represent them in diagrams and mathematical equations, and a few of the common activation functions for neural networks.

Check it out!

## A macro to execute PROC TTEST for multiple binary grouping variables in SAS (and sorting t-test statistics by their absolute values)

In SAS, you can perform PROC TTEST for multiple numeric variables in the same procedure.  Here is an example using the built-in data set SASHELP.BASEBALL; I will compare the number of at-bats and number of walks between the American League and the National League.

proc ttest
data = sashelp.baseball;
class League;
var nAtBat nBB;
ods select ttests;
run;

Here are the resulting tables.

Method Variances DF t Value Pr > |t|
Pooled Equal 320 2.05 0.0410
Satterthwaite Unequal 313.66 2.06 0.04

Method Variances DF t Value Pr > |t|
Pooled Equal 320 0.85 0.3940
Satterthwaite Unequal 319.53 0.86 0.3884

What if you want to perform PROC TTEST for multiple grouping (a.k.a. classification) variables?  You cannot put more than one variable in the CLASS statement, so you would have to run PROC TTEST separately for each binary grouping variable.  If you do put LEAGUE and DIVISION in the same CLASS statement, here is the resulting log.

1303 proc ttest
1304 data = sashelp.baseball;
1305 class league division;
--------
22
202
ERROR 22-322: Expecting ;.
ERROR 202-322: The option or parameter is not recognized and will be ignored.
1306 var natbat;
1307 ods select ttests;
1308 run;

There is no syntax in PROC TTEST to use multiple grouping variables at the same time, so this tutorial provides a macro to do so.  There are several nice features about my macro:

1. It allows you to use multiple grouping variables at the same time.
2. It sorts the t-test statistics by their absolute values within each grouping variable.
3. It shows the name of each continuous variable in the output table, unlike the above output.

Here is its basic skeleton.

## A macro to automate the creation of indicator variables in SAS

In a recent blog post, I introduced an easy and efficient way to create indicator variables from categorical variables in SAS.  This method pretends to run logistic regression, but it really is using PROC LOGISTIC to get the design matrix based on dummy-variable coding.  I shared SAS code for how to do so, step-by-step.

I write this follow-up post to provide a macro that you can use to execute all of those steps in one line.  If you have not read my previous post on this topic, then I strongly encourage you to do that first.  Don’t use this macro blindly.

Here is the macro.  The key steps are

1. Run PROC LOGISTIC to get the design matrix (which has the indicator variables)
2. Merge the original data with the newly created indicator variables
3. Delete the “INDICATORS” data set, which was created in an intermediate step
%macro create_indicators(input_data, target, covariates, output_data);

proc logistic
data = &input_data
noprint
outdesign = indicators;
class &covariates / param = glm;
model &target = &covariates;
run;

data &output_data;
merge    &input_data
indicators (drop = Intercept &target);
run;

proc datasets
library = work
noprint;
delete indicators;
run;

%mend;

I will use the built-in data set SASHELP.CARS to illustrate the use of my macro.  As you can see, my macro can accept multiple categorical variables as inputs for creating indicator variables.  I will do that here for the variables TYPE, MAKE, and ORIGIN.

## An easy and efficient way to create indicator variables (a.k.a. dummy variables) from a categorical variable in SAS

#### Introduction

In statistics and biostatistics, the creation of binary indicators is a very useful practice.

• They can be useful predictor variables in statistical models.
• They can reduce the amount of memory required to store the data set.
• They can treat a categorical covariate as a continuous covariate in regression, which has certain mathematical conveniences.

However, the creation of indicator variables can be a long, tedious, and error-prone process.  This is especially true if there are many categorical variables, or if a categorical variable has many categories.  In this tutorial, I will show an easy and efficient way to create indicator variables in SAS.  I learned this technique from SAS usage note #23217: Saving the coded design matrix of a model to a data set.

#### The Example Data Set

Let’s consider the PRDSAL2 data set that is built into the SASHELP library.  Here are the first 5 observations; due to a width constraint, I will show the first 5 columns and the last 6 columns separately.  (I encourage you to view this data set using PROC PRINT in SAS by yourself.)

COUNTRY STATE COUNTY ACTUAL PREDICT
U.S.A. California $987.36$692.24
U.S.A. California $1,782.96$568.48
U.S.A. California $32.64$16.32
U.S.A. California $1,825.12$756.16
U.S.A. California $750.72$723.52

PRODTYPE PRODUCT YEAR QUARTER MONTH MONYR
FURNITURE SOFA 1995 1 Jan JAN95
FURNITURE SOFA 1995 1 Feb FEB95
FURNITURE SOFA 1995 1 Mar MAR95
FURNITURE SOFA 1995 2 Apr APR95
FURNITURE SOFA 1995 2 May MAY95

## Arnab Chakraborty on Bayes’ Theorem – The Central Equilibrium – Episode 3

Arnab Chakraborty kindly came to my new talk show, “The Central Equilibrium”, to talk about Bayes’ theorem.  He introduced the concept of conditional probability, stated Bayes’ theorem in its simple and general forms, and showed an example of how to use it in a calculation.

Check it out!

## Christopher Salahub on Markov Chains – The Central Equilibrium – Episode 2

It was a great pleasure to talk to Christopher Salahub about Markov chains in the second episode of my new talk show, The Central Equilibrium!  Chris graduated from the University of Waterloo with a Bachelor of Mathematics degree in statistics.  He just finished an internship in data development at Environics Analytics, and he is starting a Master’s program in statistics at ETH Zurich in Switzerland.

Chris recommends “Introduction to Probability Models” by Sheldon Ross to learn more about probability theory and Markov chains.

The Central Equilibrium is my new talk show about math, science, and economics. It focuses on technical topics that involve explanations with formulas, equations, graphs, and diagrams.  Stay tuned for more episodes in the coming weeks!

You can watch all of my videos on my YouTube channel!

Please watch the video on this blog.  You can also watch it directly on YouTube.

## Store multiple strings of text as a macro variable in SAS with PROC SQL and the INTO statement

I often need to work with many variables at a time in SAS, but I don’t like to type all of their names manually – not only is it messy to read, it also induces errors in transcription, even when copying and pasting.  I recently learned of an elegant and efficient way to store multiple variable names into a macro variable that overcomes those problems.  This technique uses the INTO statement in PROC SQL.

To illustrate how this storage method can be applied in a practical context, suppose that we want to determine the factors that contribute to a baseball player’s salary in the built-in SASHELP.BASEBALL data setI will consider all continuous variables other than “Salary” and “logSalary”, but I don’t want to write them explicitly in any programming statements.  To do this, I first obtain the variable names and types of a data set using PROC CONTENTS.

* create a data set of the variable names;
proc contents
data = sashelp.baseball
noprint
out = bvars (keep = name type);
run;

## Sorting correlation coefficients by their magnitudes in a SAS macro

#### Theoretical Background

Many statisticians and data scientists use the correlation coefficient to study the relationship between 2 variables.  For 2 random variables, $X$ and $Y$, the correlation coefficient between them is defined as their covariance scaled by the product of their standard deviations.  Algebraically, this can be expressed as $\rho_{X, Y} = \frac{Cov(X, Y)}{\sigma_X \sigma_Y} = \frac{E[(X - \mu_X)(Y - \mu_Y)]}{\sigma_X \sigma_Y}$.

In real life, you can never know what the true correlation coefficient is, but you can estimate it from data.  The most common estimator for $\rho$ is the Pearson correlation coefficient, which is defined as the sample covariance between $X$ and $Y$ divided by the product of their sample standard deviations.  Since there is a common factor of $\frac{1}{n - 1}$

in the numerator and the denominator, they cancel out each other, so the formula simplifies to $r_P = \frac{\sum_{i = 1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i - \bar{x})^2 \sum_{i = 1}^{n}(y_i - \bar{y})^2}}$.

In predictive modelling, you may want to find the covariates that are most correlated with the response variable before building a regression model.  You can do this by

1. computing the correlation coefficients
2. obtaining their absolute values
3. sorting them by their absolute values.

## Potato Chips and ANOVA, Part 2: Using Analysis of Variance to Improve Sample Preparation in Analytical Chemistry

In this second article of a 2-part series on the official JMP blog, I use analysis of variance (ANOVA) to assess a sample-preparation scheme for quantifying sodium in potato chips.  I illustrate the use of the “Fit Y by X” platform in JMP to implement ANOVA, and I propose an alternative sample-preparation scheme to obtain a sample with a smaller variance.  This article is entitled “Potato Chips and ANOVA, Part 2: Using Analysis of Variance to Improve Sample Preparation in Analytical Chemistry“.

If you haven’t read my first blog post in this series on preparing the data in JMP and using the “Stack Columns” function to transpose data from wide format to long format, check it out!  I presented this topic at the last Vancouver SAS User Group (VanSUG) meeting on Wednesday, November 4, 2015.

My thanks to Arati Mejdal, Louis Valente, and Mark Bailey at JMP for their guidance in writing this 2-part series!  It is a pleasure to be a guest blogger for JMP! ## Potato Chips and ANOVA in Analytical Chemistry – Part 1: Formatting Data in JMP

I am very excited to write again for the official JMP blog as a guest blogger!  Today, the first article of a 2-part series has been published, and it is called “Potato Chips and ANOVA in Analytical Chemistry – Part 1: Formatting Data in JMP“.  This series of blog posts will talk about analysis of variance (ANOVA), sampling, and analytical chemistry, and it uses the quantification of sodium in potato chips as an example to illustrate these concepts.

The first part of this series discusses how to import the data into the JMP and prepare them for ANOVA.  Specifically, it illustrates how the “Stack Columns” function is used to transpose the data from wide format to long format.

I will present this at the Vancouver SAS User Group (VanSUG) meeting later today. ## Odds and Probability: Commonly Misused Terms in Statistics – An Illustrative Example in Baseball

Yesterday, all 15 home teams in Major League Baseball won on the same day – the first such occurrence in history.  CTV News published an article written by Mike Fitzpatrick from The Associated Press that reported on this event.  The article states, “Viewing every game as a 50-50 proposition independent of all others, STATS figured the odds of a home sweep on a night with a full major league schedule was 1 in 32,768.”  (Emphases added)

Out of curiosity, I wanted to reproduce this result.  This event is an intersection of 15 independent Bernoulli random variables, all with the probability of the home team winning being 0.5. $P[(\text{Winner}_1 = \text{Home Team}_1) \cap (\text{Winner}_2 = \text{Home Team}_2) \cap \ldots \cap (\text{Winner}_{15}= \text{Home Team}_{15})]$

Since all 15 games are assumed to be mutually independent, the probability of all 15 home teams winning is just $P(\text{All 15 Home Teams Win}) = \prod_{n = 1}^{15} P(\text{Winner}_i = \text{Home Team}_i)$ $P(\text{All 15 Home Teams Win}) = 0.5^{15} = 0.00003051757$

Now, let’s connect this probability to odds.

It is important to note that

• odds is only applicable to Bernoulli random variables (i.e. binary events)
• odds is the ratio of the probability of success to the probability of failure

For our example, $\text{Odds}(\text{All 15 Home Teams Win}) = P(\text{All 15 Home Teams Win}) \ \div \ P(\text{At least 1 Home Team Loses})$ $\text{Odds}(\text{All 15 Home Teams Win}) = 0.00003051757 \div (1 - 0.00003051757)$ $\text{Odds}(\text{All 15 Home Teams Win}) = 0.0000305185$

The above article states that the odds is 1 in 32,768.  The fraction 1/32768 is equal to 0.00003051757, which is NOT the odds as I just calculated.  Instead, 0.00003051757 is the probability of all 15 home teams winning.  Thus, the article incorrectly states 0.00003051757 as the odds rather than the probability.

This is an example of a common confusion between probability and odds that the media and the general public often make.  Probability and odds are two different concepts and are calculated differently, and my calculations above illustrate their differences.  Thus, exercise caution when reading statements about probability and odds, and make sure that the communicator of such statements knows exactly how they are calculated and which one is more applicable.

## Mathematics and Applied Statistics Lesson of the Day – Contrasts

A contrast is a linear combination of a set of variables such that the sum of the coefficients is equal to zero.  Notationally, consider a set of variables $\mu_1, \mu_2, ..., \mu_n$.

Then the linear combination $c_1 \mu_1 + c_2 \mu_2 + ... + c_n \mu_n$

is a contrast if $c_1 + c_2 + ... + c_n = 0$.

There is a reason for why I chose to use $\mu$ as the symbol for the variables in the above notation – in statistics, contrasts provide a very useful framework for comparing multiple population means in hypothesis testing.  In a later Statistics Lesson of the Day, I will illustrate some examples of contrasts, especially in the context of experimental design.

## The advantages of using count() to get N-way frequency tables as data frames in R

#### Introduction

I recently introduced how to use the count() function in the “plyr” package in R to produce 1-way frequency tables in R.  Several commenters provided alternative ways of doing so, and they are all appreciated.  Today, I want to extend that tutorial by demonstrating how count() can be used to produce N-way frequency tables in the list format – this will magnify the superiority of this function over other functions like table() and xtabs().

#### 2-Way Frequencies: The Cross-Tabulated Format vs. The List-Format

To get a 2-way frequency table (i.e. a frequency table of the counts of a data set as divided by 2 categorical variables), you can display it in a cross-tabulated format or in a list format.

In R, the xtabs() function is good for cross-tabulation.  Let’s use the “mtcars” data set again; recall that it is a built-in data set in Base R.

> y = xtabs(~ cyl + gear, mtcars)
> y
gear
cyl      3     4     5
4        1     8     2
6        2     4     1
8        12    0     2

## How to Get the Frequency Table of a Categorical Variable as a Data Frame in R

#### Introduction

One feature that I like about R is the ability to access and manipulate the outputs of many functions.  For example, you can extract the kernel density estimates from density() and scale them to ensure that the resulting density integrates to 1 over its support set.

I recently needed to get a frequency table of a categorical variable in R, and I wanted the output as a data table that I can access and manipulate.  This is a fairly simple and common task in statistics and data analysis, so I thought that there must be a function in Base R that can easily generate this.  Sadly, I could not find such a function.  In this post, I will explain why the seemingly obvious table() function does not work, and I will demonstrate how the count() function in the ‘plyr’ package can achieve this goal.

#### The Example Data Set – mtcars

Let’s use the mtcars data set that is built into R as an example.  The categorical variable that I want to explore is “gear” – this denotes the number of forward gears in the car – so let’s view the first 6 observations of just the car model and the gear.  We can use the subset() function to restrict the data set to show just the row names and “gear”.

> head(subset(mtcars, select = 'gear'))
gear
Mazda RX4            4
Mazda RX4 Wag        4
Datsun 710           4
Hornet 4 Drive       3
Valiant              3

## Exploratory Data Analysis – All Blog Posts on The Chemical Statistician

This series of posts introduced various methods of exploratory data analysis, providing theoretical backgrounds and practical examples.  Fully commented and readily usable R scripts are available for all topics for you to copy and paste for your own analysis!  Most of these posts involve data visualization and plotting, and I include a lot of detail and comments on how to invoke specific plotting commands in R in these examples.

Useful R Functions for Exploring a Data Frame

The 5-Number Summary – Two Different Methods in R

Combining Histograms and Density Plots to Examine the Distribution of the Ozone Pollution Data from New York in R

Conceptual Foundations of Histograms – Illustrated with New York’s Ozone Pollution Data

Quantile-Quantile Plots for New York’s Ozone Pollution Data

Kernel Density Estimation and Rug Plots in R on Ozone Data in New York and Ozonopolis

2 Ways of Plotting Empirical Cumulative Distribution Functions in R

Conceptual Foundations of Empirical Cumulative Distribution Functions

Combining Box Plots and Kernel Density Plots into Violin Plots for Ozone Pollution Data

Kernel Density Estimation – Conceptual Foundations

Variations of Box Plots in R for Ozone Concentrations in New York City and Ozonopolis

Computing Descriptive Statistics in R for Data on Ozone Pollution in New York City

How to Get the Frequency Table of a Categorical Variable as a Data Frame in R

The advantages of using count() to get N-way frequency tables as data frames in R

## Performing Logistic Regression in R and SAS

#### Introduction

My statistics education focused a lot on normal linear least-squares regression, and I was even told by a professor in an introductory statistics class that 95% of statistical consulting can be done with knowledge learned up to and including a course in linear regression.  Unfortunately, that advice has turned out to vastly underestimate the variety and depth of problems that I have encountered in statistical consulting, and the emphasis on linear regression has not paid dividends in my statistics career so far.  Wisdom from veteran statisticians and my own experience combine to suggest that logistic regression is actually much more commonly used in industry than linear regression.  I have already started a series of short lessons on binary classification in my Statistics Lesson of the Day and Machine Learning Lesson of the Day.    In this post, I will show how to perform logistic regression in both R and SAS.  I will discuss how to interpret the results in a later post.

#### The Data Set

The data set that I will use is slightly modified from Michael Brannick’s web page that explains logistic regression.  I copied and pasted the data from his web page into Excel, modified the data to create a new data set, then saved it as an Excel spreadsheet called heart attack.xlsx.

This data set has 3 variables (I have renamed them for convenience in my R programming).

1. ha2  – Whether or not a patient had a second heart attack.  If ha2 = 1, then the patient had a second heart attack; otherwise, if ha2 = 0, then the patient did not have a second heart attack.  This is the response variable.
2. treatment – Whether or not the patient completed an anger control treatment program.
3. anxiety – A continuous variable that scores the patient’s anxiety level.  A higher score denotes higher anxiety.

Read the rest of this post to get the full scripts and view the full outputs of this logistic regression model in both R and SAS!

## Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Chebyshev’s Inequality

In 2 recent Statistics Lessons of the Day, I

Chebyshev’s inequality is just a special version of Markov’s inequality; thus, their motivations and intuitions are similar. $P[|X - \mu| \geq k \sigma] \leq 1 \div k^2$

Markov’s inequality roughly says that a random variable $X$ is most frequently observed near its expected value, $\mu$.  Remarkably, it quantifies just how often $X$ is far away from $\mu$.  Chebyshev’s inequality goes one step further and quantifies that distance between $X$ and $\mu$ in terms of the number of standard deviations away from $\mu$.  It roughly says that the probability of $X$ being $k$ standard deviations away from $\mu$ is at most $k^{-2}$.  Notice that this upper bound decreases as $k$ increases – confirming our intuition that it is highly improbable for $X$ to be far away from $\mu$.

As with Markov’s inequality, Chebyshev’s inequality applies to any random variable $X$, as long as $E(X)$ and $V(X)$ are finite.  (Markov’s inequality requires only $E(X)$ to be finite.)  This is quite a marvelous result!

## Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Markov’s Inequality

Markov’s inequality may seem like a rather arbitrary pair of mathematical expressions that are coincidentally related to each other by an inequality sign: $P(X \geq c) \leq E(X) \div c,$ where $c > 0$.

However, there is a practical motivation behind Markov’s inequality, and it can be posed in the form of a simple question: How often is the random variable $X$ “far” away from its “centre” or “central value”?

Intuitively, the “central value” of $X$ is the value that of $X$ that is most commonly (or most frequently) observed.  Thus, as $X$ deviates farther and farther from its “central value”, we would expect those distant-from-the-centre values to be less frequently observed.

Recall that the expected value, $E(X)$, is a measure of the “centre” of $X$.  Thus, we would expect that the probability of $X$ being very far away from $E(X)$ is very low.  Indeed, Markov’s inequality rigorously confirms this intuition; here is its rough translation:

As $c$ becomes really far away from $E(X)$, the event $X \geq c$ becomes less probable.

You can confirm this by substituting several key values of $c$.

• If $c = E(X)$, then $P[X \geq E(X)] \leq 1$; this is the highest upper bound that $P(X \geq c)$ can get.  This makes intuitive sense; $X$ is going to be frequently observed near its own expected value.

• If $c \rightarrow \infty$, then $P(X \geq \infty) \leq 0$.  By Kolmogorov’s axioms of probability, any probability must be inclusively between $0$ and $1$, so $P(X \geq \infty) = 0$.  This makes intuitive sense; there is no possible way that $X$ can be bigger than positive infinity.

## 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.