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Index to Module 5 Notes
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Our last module for the
course (did I hear loud applause again?) presents descriptive and inferential
techniques for the analysis of categorical (also called qualitative) data. We
already examined categorical data in the multiple regression material of Module
3 - recall we incorporated a "dummy" variable to represent gender
(male/female), season of the year (in season/out of season), output
(defective/not defective) and so forth. But in that case, the categorical
variable just served to stratify the data in the same multiple regression
model.
Now we want to learn about techniques for analyzing data that is all categorical.
For example, a consumer products company was hired to survey 1000 shoppers at
four stores in Fort Myers several years ago. Worksheet 5.1.1 presents the
results of their survey.
Worksheet 5.1.1.
|
Row 1 |
Col B |
C |
D |
E |
F |
|
2 |
Excel |
Good |
Poor |
Total |
|
|
3 |
Kmart |
272 |
477 |
251 |
|
|
4 |
Sears |
315 |
457 |
228 |
|
|
5 |
JCP |
323 |
470 |
207 |
|
|
6 |
Wards |
391 |
404 |
205 |
|
|
7 |
Total |
||||
This table of
cross-classification (also known as cross-tabulation or contingency table)
presents two categorical variables. One of the variables is Store, and there
are four "values" for store - Kmart, Sears, JCP and Wards. This is a
categorical variable - its values are categories or names - we can't average
them, or find their standard deviation, or their median - those type of
descriptive statistics for numerical variables do not apply for categorical
variables.
But there are simple descriptive statistics for categorical variables and we
cover them in this module. There are also inferential statistics for single
categorical variables - those are covered in Module 5.2. We wrap up this module
by studying descriptive and inferential statistics for multiple samples of
categorical variables in Module 5.3.
Before we get to work, let me also note that there is a second categorical
variable in Worksheet 5.1.1 - rating of the quality of the shopping experience
by shoppers participating in the customer survey. The Rating variable has three
"values": Excellent, Good and Poor. I need to point out that by
tradition, we sometimes do assign a value to this type of categorical
variable, such as 3 = Excellent, 2 = Good, and 1 = Poor. When we do that, we
are treating the variable as if it were numerical (quantitative) and
it's data were measured on an interval scale. Sometimes we even compute
descriptive statistics such as the average rating. Of course, when we do so, we
have to recognize that the numbers assigned are arbitrary and zero is
meaningless to interval scaled data (we could use the scores 10 = Excellent; 5
= Good; and 1 = Poor).
However, for this module, we are not going to assume we can convert the
categorical variable into a quantitative variable. We are going to analyze it
as a categorical variable. The tools we introduce in this Module are used
frequently in business - especially with customer surveys that contain many
categorical variables, from demographic characteristics to attitudes to
behaviors.
The cross-classification table is to categorical variables, as simple linear
regression is to quantitative variables. The cross-classification table
provides a way of looking at the relationship between two categorical
variables - a very powerful tool when one wants to study the relationship between
categorical variables that model demographic, attitude and behavioral
characteristics.
Descriptive
Statistics for Categorical Variables
Counting
So, if we can't find
the average, or standard deviation, or median, or interquartile range of
categorical variables, how do we measure them. We simply count their
occurrences in a way that provides useful information. Worksheet 5.1.1 already
illustrated counts in cross-tabulation classes. That is, we know 477 shoppers
rated their Kmart shopping experience as Good.
Worksheet 5.1.2 provides some more ways of counting the survey information.
Worksheet 5.1.2
|
Row 1 |
Col B |
C |
D |
E |
F |
|
2 |
Excel |
Good |
Poor |
Total |
|
|
3 |
Kmart |
272 |
477 |
251 |
1000 |
|
4 |
Sears |
315 |
457 |
228 |
1000 |
|
5 |
JCP |
323 |
470 |
207 |
1000 |
|
6 |
Wards |
391 |
404 |
205 |
1000 |
|
7 |
Total |
1301 |
1808 |
891 |
4000 |
Note that I added
marginal totals to the worksheet by entering, for example, =SUM(C3:E3) in cell
F3; and =SUM(C3:C6) in cell C7. Now I know that more shoppers rated the four
stores as Good, followed by Excellent (Excel), followed by poor. I also know
that the same number of shoppers were surveyed at each store, and that the
sample size was very large (much larger than political opinion poles conducted
by the major news organizations - we will look at that later).
Is that all there is to descriptive statistics for categorical variables? No -
there is a little more. We can also convert a count into a probability (also
called long term relative frequency or proportion or percent or
chance).
Before we do this, let's take a moment to review some simple counting rules
from mathematics and statistics (Mason, 1999). You may remember these from math
courses you took a long time ago.
Multiplication
Rule
If there are m
ways of doing one thing, and n ways of doing another, there are mn
possible arrangements. So, in the cross-classification table, shoppers can
select between four stores and choose between three
possible ratings, giving 12 total combinations or arrangements as shown in the
body of Worksheet 5.1.1. This can be expanded. If there are m ways
of doing one thing, n ways of doing another, and o
ways of doing yet another; then there are mno possible
arrangements. If the shoppers in our example can choose between paying cash or
using credit card, then there would be 4 times 3 times 2 or 24 possible
arrangements.
The Permutation Formula
The multiplication
rule applies to finding the number of arrangements when there are two or more
groups. The permutation formula applies to arrangements when there is only one
group. The scenario for this counting rule might be something like: how many
different ways can shoppers visit the four stores if order matters. For
example, one arrangement would be go to Kmart first, then Sears, then JCP, then
Wards. Another arrangement might be Sears, JCP, Wards, then Kmart. These
arrangements are called permutations.
A permutation is any arrangement of r objects selected from
a group of n objects, where order matters. The formula for a
permutation is:
Eq. 5.1.1: nPr = n! / (n - r)! where ! means factorial, the product of
n(n-1)(n-2)...(1). By definition, 0! = 1
So, if n = 4, r = 4,
Eq. 5.1.2: 4P4 = 4! / (4 - 4)! = 4! / 0! = 4! / 1 = (4 * 3 * 2 * 1) = 24
Another scenario might be: how many different ways can shoppers visit just two of the four stores if order matters.
Eq. 5.1.3: 4P2 = 4! / (4 - 2)! = 4! / 2! = (4 * 3 * 2 * 1) / (2 * 1) = 12
The arrangements here are
Kmart/Sears; Sears/Kmart; Kmart/JCP; JCP/Kmart; Kmart/Wards; Wards/Kmart;
Sears/JCP; JCP/Sears; Sears/Wards; Wards/Sears; JCP/Wards; and Wards/JCP.
The final counting rule is for combinations.
The Combination Formula
This is similar to
permutations, but now order is not important. The equation for the combination
rule is:
Eq. 5.1.4: nCr = n! / [ r! (n - r!) ]
How many different arrangements can shoppers follow to visit two of four stores, if order is not important?
Eq. 5.1.5: nCr = 4! / [ 2! (4 - 2)!] = (4 * 3 * 2 * 1) / [(2 * 1) * (2 * 1)] = 6
The combinations are Kmart/Sears; Kmart/JCP; Kmart/Wards; Sears/JCP; Sears/Wards; and JCP/Wards.
Simple Probability
The simple probability
of an event of interest is the count of observations for that particular
event divided by all observations for all possible events in the sample
space. Let's not get to technical for this simple concept. The probability
that shoppers give an Excellent rating, when we consider all of the shoppers,
is 1301 divided by 4000, or 0.325 gave an Excellent rating. We can convert
0.325 into a percent by multiplying by 100. So, there is a 32.5% chance that
shoppers give an Excellent rating. We follow generally accepted practice by
writing the probability of Excellent as P(Excellent).
Eq. 5.1.6: Simple Probability of Event Excellent =
P(Excellent)
= Nbr of Excellent Ratings/Total Shoppers
P(Excellent) = 1301/4000 = 0.325 x 100 = 32.5%
This probability is called a simple probability when I am just looking
at one categorical variable. It is called a marginal probability when we
are looking at any of the marginal sums divided by the grand total in a
cross-classification table. All of the marginal probabilities are shown in
Worksheet 5.1.3. Worksheet 5.1.3 is a copy of Worksheet 5.1.2 in rows 12 to 18
of the same Excel Worksheet. To compute the marginal probability in Cell C13,
using the data in Worksheet 5.1.2, I enter the formula =C6/F7 in Cell C13.
Worksheet 5.1.3
|
PERCENT OF TOTAL |
|||||
|
Row 12 |
Col B |
C |
D |
E |
F |
|
13 |
Excel |
Good |
Poor |
Total |
|
|
14 |
Kmart |
6.8% |
11.9% |
6.3% |
25% |
|
15 |
Sears |
7.9% |
11.4% |
5.7% |
25% |
|
16 |
JCP |
8.1% |
11.8% |
5.2% |
25% |
|
17 |
Wards |
9.8% |
10.1% |
5.1% |
25% |
|
18 |
Total |
32.5% |
45.2% |
22.3% |
100% |
Note there are some other
percents or probabilities shown in Worksheet 5.1.3. These are called joint
probabilities in a cross-classification table.
Joint Probability
The joint probabilities occurs in the body of the cross-classification table at
the intersection of two events for each categorical variable. In Worksheet
5.1.1 we see that there are 457 shoppers who rated Sears as Good. The joint
probability of Sears and Good is 457 divided by 4,000 or 11.4%.
Eq. 5.1.7: Joint Probability of Sears and Good events =
P(Sears and Good) = (Number of Sears Shoppers and Good Ratings)/Total Shoppers
To compute this probability
in cell C14 of the Worksheet, I enter =C3/F7 in cell C14.
Probabilities, such as these simple and joint probabilities, have no dimensions
and enable us to make relative comparisons. That is, we generally get more
relative information by comparing 32.5% for Excellent Rating to 45.2% for Good
to 22.3% for poor than by comparing the count data 1301 to 1808 to 891. Same is
true for the joint probabilities.
Assumptions
The only assumptions that we need for computing these probabilities is that
they be considered long term relative frequencies and that the events
within a categorical variable are mutually exclusive and exhaustive.
We consider probabilities to be long term relative frequencies for making
inferences. We are not talking about one shopper going to Kmart tomorrow and
finding the experience excellent, since the probability of an excellent event
vs. an not excellent event for that shopper is 50%. Rather, we are
talking about probabilities that occur over the longer time period dictated by
our sample. These long term relative frequencies are expressed as any number
between 0 and 1. When the resulting fraction is multiplied by 100 we convert
the long term relative frequency into a percent.
Side note: I don't think people who gamble believe in long term relative
frequencies. For example, a roulette wheel has 18 red slots, 18 black slots,
one 0 slot, and one 00 slot. If a gambler bets that a ball will fall in a
"red" slot during a spin of the roulette wheel, the long term
probability of winning is 18 red/(18 red+ 18 black + 1 zero + 1 double zero) =
18/38 = 0.474 or 47.4%. The long term chance of the house winning is 100% -
47.4% or 52.6%. The casino cannot (and does not) loose in the long run. Does
that matter to the gambler? Of course not. Their chance of winning is 50% in
the short term (they win or lose on the next spin) (or they enjoy the free food
and ambiance).
Back to the notes. Mutually exclusive means that if you rate Kmart as
Excellent, you cannot also rate it as Good - an observation has to fall in one
event classification. Exhaustive means that all events within a categorical
variable are presented. There cannot be an event "no opinion" unless
it is represented with its counts in the cross-classification table. Given that
the mutually exclusive and exhaustive conditions are met, then all of the probabilities
for all of the events within a categorical variable event space must sum to
100%.
General Addition Rule
Having covered simple, marginal and joint probabilities, we can present the
addition rule:
Eq. 5.1.8: P(A or B) = P(A) + P(B) - P(A and B)
Note the fine distinction
between P(A or B), the addition of two simple probabilities, and P(A and
B), the joint probability of events A and B.
For example: what is P(JCP or Excellent)?
Eq. 5.1.9: P(JCP or Excellent) = P(JCP) + P(Excellent) -
P(JCP and Excellent) = 25% + 32.5% - 8.1% = 49.4%.
Another example: what is P(Good or Poor)?
Eq. 5.1.10: P(Good or Poor) = P(Good) + P(Poor) -
P(Good and Poor) = 45.2% + 22.3% - 0% = 67.5%
I hope this last example did
not seem tricky. Note that there cannot be a joint probability of Good and Poor
since the events good and poor are marginal events for the same category.
Recall that events have to be mutually exclusive, so if a shopper scored a
"Good," they cannot also score a "Poor." The only joint
events are those that represent the combination of events from two
different variables.
Complementary Events and Their Probabilities
In the last example, Equation 5.1.10, I gave P(Good and Poor) as 67.5%.
What is P(Excellent)? Because of the mutually exclusive and exhaustive
assumptions, all probabilities for all events within the categorical event
space must sum to 100%. Since the only other event that can occur besides Good
and Poor, is Excellent, P(Excellent) must be:
Eq. 5.1.11: P(Excellent) = 100% - P(Good and Poor) = 32.5%.
There is one more classification of probability that we need to complete our
study of descriptive statistics for categorical variables. This is called the conditional
probability.
Conditional Probability
The last probability can occur whenever we are using cross-classification
tables. A conditional probability conditions the total event space
(denominator of the relative frequency equation) to some desired subset. For
example, we may want to ask, what is the probability that a shopper rates their
experience as excellent given that we are only interested in Wards
shoppers? Mathematically, the formula is:
Eq. 5.1.12: P(Excel|Wards) = P(Excel and Wards)/P(Wards) =
9.8%/25% = 39.1%
The vertical bar,
"|" in equation 5.1.6 represents the word "given" which
provides the subset of the event space of interest. In other words, we are not
interested in the total sample space of 4,000 shoppers shown in Worksheet
5.1.1, we are only interested in the subset of 1,000 shoppers who shopped at
Wards. So, a direct way of computing this conditional probability would be to
just divide the number of shoppers who rated the Wards experience as Excellent
by the total shoppers at Wards which gives 391/1000 or 39.1%.
Worksheet 5.1.4 presents this and the other row conditional probabilities. That
is, the probabilities for the various levels of ratings given store
variable. To compute the conditional probability for cell C24, I enter =C3/F7
in cell C24.
Worksheet 5.1.4
|
PERCENT OF ROW TOTALS: |
|||||
|
Row 23 |
Col B |
C |
D |
E |
F |
|
24 |
Excel |
Good |
Poor |
Total |
|
|
25 |
Kmart |
27.2% |
47.7% |
25.1% |
100.0% |
|
26 |
Sears |
31.5% |
45.7% |
22.8% |
100.0% |
|
27 |
JCP |
32.3% |
47.0% |
20.7% |
100.0% |
|
28 |
Wards |
39.1% |
40.4% |
20.5% |
100.0% |
|
29 |
Total |
32.5% |
45.2% |
22.3% |
100.0% |
Let's look at another
example. What is the probability that a shopper is a Sears shopper given that
the rating was Good?
Eq. 5.1.13: P(Sears| Good) = P(Sears and Good)/P(Good) =
11.4%/45.2% = 25.2%
Worksheet 5.1.5 gives this
and the other column conditional probabilities. That is, the probability of one
of the four stores given the rating. To compute the conditional
probability in cell C25, I enter =C3/C7 in cell C25.
Worksheet 5.1.5
|
PERCENT OF COLUMN TOTALS: |
|||||
|
Row 1 |
Col B |
C |
D |
E |
F |
|
2 |
Excel |
Good |
Poor |
Total |
|
|
3 |
Kmart |
20.9% |
26.4% |
28.2% |
25.0% |
|
4 |
Sears |
24.2% |
25.3% |
25.6% |
25.0% |
|
5 |
JCP |
24.8% |
26.0% |
23.2% |
25.0% |
|
6 |
Wards |
30.1% |
22.3% |
23.0% |
25.0% |
|
7 |
Total |
100.0% |
100.0% |
100.0% |
100.0% |
That's it for
descriptive statistics for categorical variables. You should be able to answer question
5 of the assignment given in Main Module 5 Overview in the course Web site.
The references show another application of simple and conditional
probabilities. The application is in decision trees. That material is covered
in the quantitative methods course so I will not duplicate it here. Other
material covered in reference texts includes probability distributions for
discrete random variables which are special applications of categorical
variables. We will cover one of these, the binomial distribution, in Module 5.2
Notes. The Poisson Distribution is covered in the waiting line (queuing)
material in the quantitative class.
The next subject is inferential statistics. You remember, confidence intervals
and test of hypothesis - this time for a proportion. That is the subject of
Module Notes 5.2.
References:
Anderson,
D., Sweeney, D., & Williams, T. (2010). Essential of Modern Business
Statistics with Microsoft Excel. Cincinnati, OH: South-Western, Chapter 4 and
Chapter 5.
Ken Black. Business Statistics for
Contemporary Decision Making. Fourth Edition, Wiley. Chapter 4
& 12
D.
Groebner, P. Shannon, P. Fry & K. Smith.
Business Statistics: A Decision Making Approach, Fifth Edition,
Prentice Hall,
Chapter
4 and 14
Levine, D., Berenson,
M. & Stephan, D. (1999). Statistics for Managers Using Microsoft Excel (2nd.
ed.). Upper Saddle River, NJ: Prentice-Hall, Chapter 4.
Mason, R., Lind, D. & Marchal, W. (1999). Statistical Techniques in
Business and Economics (10th. ed.). Boston: Irwin McGraw Hill, Chapter
5.