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.