Categorical Data

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Categorical Data
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Lesson Objective

• Understand basic rules of probability.
• Calculate marginal and conditional
  probabilities.
• Determine if two categorical variables are
  independent.

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Recall Rule of Thumb
C
Quantitative variables averages or
differences have meaning.

• Ex weight, height, income, age

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Recall Rule of Thumb
C
Categorical variables classify people or
things.

• Ex gender, race, occupation, political
  affiliation, country of origin

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Note Sometimes quantitative variables are
expressed as categorical.
Income (Family Economic Income)
Class Definition 1. Less than
30,000 2. 30,000 but less than 100,000
3. 100,000 or more.
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Relationships between variables
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Relationship between two quantitative
variables?
Is relationship linear (scatterplot)?

• J Use Correlation Least Squares
  Regression.
• L Data transformations.

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Recall Boxplots

• Best graphical tool for examining the
  relationship between a quantitative variable and
  a categorical variable,(i.e., comparing
  distributions).

Example Weight vs. Country of Origin
Boxplot can be used to answer
Do the distributions of weights vary for
different countries of origin?
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Relationship between two categorical
variables?
Use two-way frequency tables

• Look at marginal probabilities and
  conditional probabilities.

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STATISTICS

• is the science of
• transforming data
• into informationto make decisionsin the face of
  uncertainty.

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Probability
How do we measure "uncertainty"?

• A numerical measure of the likelihood that an
  outcome or an event occurs.
• P(A) probability of event A

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• Three Methods for Assessing Probability
• Classical
• Relative Frequency
• Subjective

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Probability requirements fordiscrete variables
_
_

• 1. 0 lt P(A) lt 1

2. Sum of the probabilities of all possible
outcomes must equal 1. (Binomial, Poisson)
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Conditional probability The chance one event
happens,given that another event will occur.
P(A B)
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• Problem Credit Card Manager
• New credit test to determine credit worthiness.
• Credit test checked against500 previous
  customers.

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Credit Test A
Credit History
Failed (F)
Passed (P)
Good (G)
400
350
50
Default (D)
20
80
100
370
130
500
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What is the probability of a customer defaulting?
P(Defaults)
What is the probability of a customer defaulting
given that he fails test A?
P(Defaults given failed test A)
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General Rules

• P(A and B) P(A) P(BA)
• P(B) P(AB)
• P(A or B) P(A) P(B) - P(A and B)

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P(Fails AND Defaults)
P(F) P(DF)
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P(Fails OR Defaults)
P(F) P(D) - P(D AND F)
Note The overlap group would be counted twice
if no subtraction.
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Does knowledge of test A resulthelp you make a
better decision?

P
(
D
)
Do you want to know the test A results before
you give the loan?
Credit test A results and defaulting are
____________ on each other.
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A Newer Credit Test.Is it even better?
A different sample of 500 credit records
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Credit Test B
Credit History
Failed (F)
Passed (P)
Good (G)
400
340
60
Default (D)
85
15
100
425
75
500
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What is the probability of a customer defaulting?
P(Defaults)
What is the probability of a customer defaulting
given that he fails test B?
P(Defaults given failed test B)
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Does knowledge of test B resulthelp you make a
better decision?

P
(
D
)
Test B tells me .
Credit test B results and defaulting are
of each other.
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Independence
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Two events are independent if the occurrence, or
non-occurrence, of one does not affect the
chances of the other occurring, or not occurring.

• Otherwise, we say the events are dependent.

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If A and B independent, then

• P(A and B) P(A) P(B)
• P(A or B) P(A) P(B) - P(A) P(B)
• P(AB) P(A)
• P(BA) P(B)

Note The condition does NOT changethe
probability.
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Survey of randomly selectedpeople voters in Jan.
2001
Q1 Did you vote in the 2000 election? Q2 Do
you favor an amendment to require a
balanced budget? Q3 To which political party
do you belong ?
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Do you favor amendmentfor a balancedbudget?
Political Party
Yes No Total
Republican Democrat OtherTotal
90 44 48
172148 80
82 104 32
182 218 400
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Sample size
Marginal totals for Party.
Marginal totalsfor opinion.
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What proportionfavor the amend.?
What proportionclaim to be Rep?
What proportionfavor the amend.and are Other?
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What proportionfavor the amend,given those that
claim to be Rep?
Considering onlythose opposed, what
proportionare not Republican?
Of those that claim to be Democrat,what
proportionfavor the amend.
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Conditional Distribution
Restrict subjects to only those that meet a
condition. Within this restricted group, what is
the distribution of some other var.?
Distribution of opinion given those that claim
to be Republican
P( Yes Rep. ) .523P( No Rep. )
.477
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Is there a relationship betweenthe party and
the opinion on the amendment?

• What would you expect to happen if no
  relationship existed?

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Three Conditional Distributions
Marginal Distribution P( Yes ) .455,
P( No ) .545
Is there a relationship?Why? or Why not?
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If there is NO relationship(i.e.,
independence)between the party andthe opinion,
then
the three conditional probabilitiesshould be
the close to each other and close to the
marginal probability.
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Three Conditional Probabilities
P( Yes Rep.) .523
Are these close toeach other?
P( Yes Demo) .297
P( Yes Other) .600
Marginal Probability P( Yes ) .455
AND close to the marginal?
Not close therefore, party and the opinion
are ____________.
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Create with Pivot Tablesin Excel.
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Barchart- Clustered
Yes
Rep.
Demo.
Other
Frequency
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Barchart- Stacked
Rep.
Yes
Demo.
Other
Frequency
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Barchart- Percents
Yes
Rep.
Demo.
Other
Percent
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SummaryFor two categorical variables

• Must use conditional probabilities to
  determine if a relationship exists.
• Cannot use correlation.
• Visual display Stacked percentage
  bar charts

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Associations between TWO Variables
numerical graphical
Variables
Quant. vs. Quant
LS regression line, r, r-sq, std error
Scatterplot,residual plots
Quant. vs. Cat.
X-bar and sfor each category
Side-by-side box plots
Two-way table, conditional marginal
distributions
Bar chart stacked, percent.
Cat. vs. Cat.
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The End