STAT 3120
Statistical Methods I
Lecture 8
Chi-Square
STAT3120 – Chi Square
Dependent
Variable
Independent
(predictor)
Variable
Statistical
Test
Comments
Quantitative
Categorical
T-TEST (one,
two or
paired
sample)
Determines if categorical
variable (factor) affects
dependent variable; typically
used for experimental or
planned change studies
Quantitative
Quantitative
Correlation
/Regression
Analysis
Test establishes a regression
model; used to explain, predict
or control dependent variable
Categorical
Categorical
Chi-Square
Tests if variables are statistically
independent (i.e. are they
related or not?)
STAT3120 – Chi Square
When presented with categorical data, one common
method of analysis is the “Contingency Table” or “Cross
Tab”. This is a great way to display frequencies For example, lets say that a firm has the following data:
120 male and 80 female employees
40 males and 10 females have been promoted
STAT3120 – Chi Square
Using this data, we could create the following 2x2
matrix:
Promoted
Not Promoted
Total
Male
40
80
120
Female
10
70
80
Total
50
150
200
STAT3120 – Chi Square
Now, a few questions…
1) From the data, what is the probability of being
promoted?
2) Given that you are MALE, what is the probability of
being promoted?
3) Given that you are promoted, what is the probability
that you are MALE?
4) Given that you are FEMALE, what is the probability of
being promoted?
5) Given that you are promoted, what is the probability
that you are female?
STAT3120 – Chi Square
The answers to these questions help us start to understand
if promotion status and gender are related.
Specifically, we could test this relationship using a ChiSquare. This is the test used to determine if two variables
are related.
The relevant hypothesis statements for a Chi-Square test
are:
H0: Variable 1 and Variable 2 are NOT Related
Ha: Variable 1 and Variable 2 ARE Related
Develop the appropriate hypothesis statements and
testing matrix for the gender/promotion data.
STAT3120 – Chi Square
The Chi-Square Test uses the Χ2 test statistic, which has a
distribution that is skewed to the right (it approaches
normality as the number of obs increases). You can see an
example of the distribution on pg 641.
The Χ2 test statistic calculation can be found on page 640.
The observed counts are provided in the dataset.
The expected counts are the counts which would be
expected if there was NO relationship between the two
variables.
STAT3120 – Chi Square
Going back to our example, the data provided is
“observed”:
Promoted
Not Promoted
Total
Male
40
80
120
Female
10
70
80
Total
50
150
200
What would the matrix look like if there was no relationship
between promotion status and gender? The resulting
matrix would be “expected”…
STAT3120 – Chi Square
From the data, 25% of all employees were promoted.
Therefore, if gender plays no role, then we should see 25%
of the males promoted (75% not promoted) and 25% of the
females promoted…
Promoted
Male
Female
Total
Not Promoted
Total
120*.25 = 30
120*.75 = 90
120
80*.25 = 20
50
80*.75 = 60
150
80
200
Notice that the marginal values did not change…only the
interior values changed.
STAT3120 – Chi Square
Now, calculate the X2 statistic using the observed
and the expected matrices:
((40-30)2/30)+((80-90)2/90)+((10-20)2/20)+((7060)2/60) =
3.33+1.11+5+1.67 = 11.11
This is conceptually equivalent to a t-statistic or a
z-score.
STAT3120 – Chi Square
To determine if this is in the rejection region, we
must determine the df and then use the table on
page 732.
Df = (r-1)*(c-1)…
In the current example, we have two rows and
two columns. So the df = 1*1 = 1.
At alpha = .05 and 1df, the critical value is
3.84…our value of 11.11 is clearly in the reject
region…so what does this mean?
STAT3120 – Chi Square
From the book Outliers, Malcolm Glidewell makes
the point that the month in which a boy is born
will determine his probability of playing in the
NHL.
The months of birth for players in the NHL are on
the next page…
(data taken from
http://sports.espn.go.com/espn/page2/story?pa
ge=merron/081208)
STAT3120 – Chi Square
January
February
March
April
May
June
July
August
September
October
November
December
51
46
61
49
46
49
36
41
36
34
33
30
Now, if there is NO relationship
between birth month and playing
hockey, what SHOULD the
distribution of months look like?
Lets do this one in EXCEL…
Note that this is technically
referred to as a “goodness of fit”
test – where we are assessing if
the actual distribution “fits” what
would be expected.
STAT3120 – Chi Square
Practice Problems for Chi-Square:
15.55
15.56
15.57
15.58
For all of these, identify the hypothesis
statements, the testing matrix, and the decision.
Categorical Example
Using credit data.
Credit
• Sample Data Set
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–
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–
Purchase: $: 1=$250+, 0=<$250
Age: Customer Age
Gender: male,female
Income: Low, Medium, High
What do we have?
• Predictors
• Gender
• Income
• Age
• Outcome
• GT $250
• LT $250
Determine ‘Scale’
• Nominal variables:
– Values with no logical ordering.
» Gender
• Ordinal variables:
– Variables have values with a logical ordering.
» Income
Lets Examine!?
• Determine distribution of categorical values
• Recognize possible associations among variables
• Association ?
– Two variables when one level or value of the
other changes.
– No changes? Distribution of the variable is the
same regardless of the level of the other
variable
Determine Association
• No Association?
– Statistic professor temperament changes with
golf.
Great golf
Sunshine
Raining
Bad Golf
65%
35%
65%
35%
Watch Out!
• Association?
– Statistic professor temperament changes with
golf.
Great golf
Sunshine
Raining
Bad Golf
95%
5%
30%
70%
Crosstabulation Table
• Table shows the number of observations for each combination of the
row and the column variables
Column 1
Column 1
…
Column 1
Row 1
Row 2
Cell11
Cell12
…
Cell1c
Cell12
Cell22
…
Cell2c
…
Row r
…
…
…
…
Cellr1
Cellr2
…
Cellrc
-
Frequency: nbr of observations falling into a category formed by row variable and column variable
Percent: nbr of observations in each cell as a percentage of the total nbr of observations
Row percent: nbr of observations in each cell as a percentage of the total nbr of observations in that row
col percent: nbr of observations in each cell as a percentage of the total nbr of observations in that column
Distributions
• SAS Freq procedure
– Examine distributions
– Ordering values
SAS Proc Freq Distributions
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libname JLLP 'E:\JenniferPriestly\Chi_Square';
%let outpath=E:\JenniferPriestly\Chi_Square;
%let libpath=E:\JenniferPriestly\Chi_Square;
options nodate nonumber ls=95 ps=80;
run;
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Proc format;
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ods rtf close;
ods listing;
value purfmt 1 = "$ 100 +"
0 = "< $100"
;
Run;
ods graphics on;
ods listing close;
ods Rtf path="&outpath"
style=journal
file='freq.rtf';
proc freq data=JLLP.Online;
tables purchase gender income
gender*purchase income*purchase /
plots(only)=(freqplot);
format purchase purfmt.;
run;
ods select histogram probplot;
proc univariate data=JLLP.Online;
var age;
histogram age / normal (mu=est sigma=est);
probplot age / normal (mu=est sigma=est);
run;
SAS Ordering Values
• Change Income
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ods graphics on;
ods listing;
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data JLLP.Online_inc;
set JLLP.Online;
if income='Low' then IncLevel=1;
else if income='Medium' then IncLevel=2;
else if income='High' then IncLevel=3;
run;
proc format;
value incfmt 1='Low Income'
2='Medium Income'
3='High Income';
run;
ods graphics on;
ods rtf path="&outpath"
style=statistical
file='freq2.rtf';
proc freq data=JLLP.Online_inc;
tables IncLevel*Purchase;
format IncLevel incfmt. Purchase purfmt.;
title1 'Change Variable IncLevel to Correct Income';
run;
ods rtf close;
Tests for Association
• Determine
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Chi-square test for association
Examine strength of the association
Calculate exact p-value
Cramer’s V
Chi-Square Test
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ods graphics on;
ods rtf path="&outpath"
style=statistical
file='freq3.rtf';
proc freq data=JLLP.Online_inc;
tables Gender*purchase
/ chisq expected cellchi2 nocol nopercent
relrisk;
format purchase purfmt.;
Title1 'Association Between Gender and Purchase';
run;
ods rtf close;
Gender by Purchase
Table of Gender by Purchase
Gender
Purchase
Frequenc
y
Percent
Row Pct
$ 100
Col Pct
< $100
+
Total
Female
139
101
240
32.25 23.43 55.68
57.92 42.08
51.67 62.35
Male
130
61
191
30.16 14.15 44.32
68.06 31.94
48.33 37.65
Total
269
162
431
62.41 37.59 100.00
Chi-Square Test
• No association
• Observed frequencies=expected frequencies
– Null Hypothesis:
• No association between Gender and Purchase
• Probability of purchasing items more than $100 is the same for both sexes.
• Association
• Observed frequencies≠expected frequencies
– Alternative Hypothesis:
• There is an association between Gender and Purchase
• Probability of purchasing items more than $100 is the same for both sexes.
Pearson Chi-square Test
• Commonly used test to determine whether there is
association between 2 categorical values
• Test measure the difference between the observed cell
frequencies and the cell frequencies that are expected if
there is no association between the variables
• Significant test statistic, strong evidence an association
exists
Frequencies Calculation
• Expected frequencies are calculated by:
» (row total * column total) / sample size
No association between Row and Column variable the expected
percentage in any R*C will be equal to the percentage in that cell rows
(R/T) times the percentage in the cell column (C/T). The expected
percentage times the total sample size.
Expected count=(R/T)*(C/T)*T=(R*C)/T
Chi-square tests
• Measures of association
– P-value tests only indicates how confident you can be
that the null hypothesis if no association exists.
– Cramer’s V statistics: measures association between
two nominal variables. Range from -1 to 1 for a 2-by2 table. 0 to 1 for larger tables. Values further from 0
indicate the presence of a relativity strong association.
– Odds Ratios indicates how much more likely, with the
respect to odds a certain event occurs in one group
relative to its occurrence in another group.
Odds Ratio
Probability of odds of an outcome
No
Yes
Total
Group A
20
60
80
Group B
10
90
100
Total
30
150
180
Prob of Yes outcome in
Group B = 90/100 (.90)
Prob of a No Outcome in
Group B = 10/100 (.10)
Odds Ratio
• Odds of outcome in Group B
» .90 / .10 = 9
• Odds of outcome in Group A
» .75 / .25 = 3
• Odds Ratio of Group B to Group A
» 9/3=3
Odds ratio of Group B to Group A is 3 times
.
Properties of the Odds Ratio, B to A
• Odds ratio shows strength of association.
– If odds ration is 1 then there is no association
– If odds ratio is greater than 1then Grp B is
more likely to have the outcome.
– If odds ratio is less than 1 then Grp A is more
likely to have the outcome
Example
• Determine association between Gender and
purchase.
• Generate expected cell frequencies and the cell’s
contribution to the total chi-square statistic
Results
Table of Gender by Purchase
Gender
Purchase
Frequency
Expected
Cell Chi-Square
Row Pct
< $100 $ 100 +
Total
Female
139
101
240
149.79 90.209
0.7774 1.2909
57.92
42.08
Male
130
119.21
0.9769
68.06
61
71.791
1.6221
31.94
191
Total
269
162
431
Calculate cell Chi-square
Results
Statistic
Chi-Square
Likelihood Ratio Chi-Square
Continuity Adj. Chi-Square
Mantel-Haenszel Chi-Square
Phi Coefficient
Contingency Coefficient
Cramer's V
DF
1
1
1
1
Value
4.6672
4.6978
4.2447
4.6564
-0.1041
0.1035
-0.1041
Prob
0.0307
0.0302
0.0394
0.0309
Fisher's Exact Test
Cell (1,1) Frequency (F)
139
Left-sided Pr <= F
0.0195
Right-sided Pr >= F
0.9883
Table Probability (P)
Two-sided Pr <= P
0.0078
0.0355
Estimates of the Relative Risk (Row1/Row2)
Type of Study
Value
95% Confidence Limits
Case-Control (Odds Ratio)
0.6458
0.4339
0.9612
Cohort (Col1 Risk)
0.8509
0.7360
0.9839
Cohort (Col2 Risk)
1.3177
1.0214
1.7000
P-value is 0.0307<.05 , reject the Null hypothesis
Appendix A.5: .05<p-value<.025
Cramer’s V indicates association is relatively weak.
Relative Risk at 95% CI that Males in the right column (+100)
compared to Females has value of .6458. Males has a 65% odds of
purchasing more then $100
Odds ratio (OR-1)*100, (0.6458-1)*100=-35.42%, males have a
35.42% lower odds than females.
Gender by Purchase