STAT 250
Dr. Kari Lock Morgan
Testing Goodness-ofFit for a Single
Categorical Variable
SECTION 7.1
• Testing the distribution of a single
categorical variable : 2 goodness of fit (7.1)
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Statistics!
 Statistics might be the most important class you
take in college
 http://college.usatoday.com/2015/04/08/voices-
statistics-might-be-the-most-important-class-youtake-in-college/
 (4/8/15)
 Why you need to study statistics
 https://www.youtube.com/watch?v=wV0Ks7aS7YI
 (4/2/15)
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Multiple Categories
• So far, we’ve learned how to do inference
for categorical variables with only two
categories
• Today, we’ll learn how to do hypothesis
tests for categorical variables with multiple
categories
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Genetic Variants for Fast-Twitch Muscles
 A gene called ACTN3 encodes a protein which functions
in fast twitch muscles
 Three different variants of the gene: RR, RX, and XX
 In a sample, we observe 130 RR, 226 RX, and 80 XX.
 If both R and X are equally likely, then by the Hardy-
Weinberg principle about 50% of the population should
be heterozygotes (RX) and about 25% should be each of
the homozygotes (25% RR, 25% XX)
 Do our data contradict these hypothesized
proportions?
Yang, N. et. al. (2003). “ACTN3 genotype is associated with human elite athletic
performance,” American Journal of Human Genetics, 73: 627-631.
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Hypothesis Testing
1. State Hypotheses
2. Calculate a statistic, based on your sample data
1. Create a distribution of this statistic, as it would
be observed if the null hypothesis were true
2. Measure how extreme your test statistic from (2)
is, as compared to the distribution generated in (3)
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Hypotheses
Define the null hypothesized proportions in
each category:
H0 : pRR = 0.25, pRX = 0.5, pxx = 0.25
Ha : At least one pi is not as specified in H0
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Observed Counts
• The observed counts are the actual
counts observed in the study
Observed
RR
130
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RX
226
XX
80
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Test Statistic
Why can’t we use the familiar formula
sample statistic  null value
SE
to get the test statistic?
We need something a bit more complicated…
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Expected Counts
• The expected counts are the expected
counts if the null hypothesis were true
• For each cell, the expected count is the
sample size (n) times the null proportion, pi
expected = npi
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Expected Counts
n = 436
Null Proportion
Expected
RR
0.25
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RX
0.5
XX
0.25
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Chi-Square Statistic
Observed
Expected
RR
130
109
RX
226
218
XX
80
109
 Need a way to measure how far the observed
counts are from the expected counts…
 Use the chi-square statistic :
 
2
 observed - expected 
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2
expected
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Chi-Square Statistic
Observed
Expected
RR
130
109
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RX
226
218
XX
80
109
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What Next?
We have a test statistic. What else do we need
to perform the hypothesis test?
How do we get this?
Two options:
1) Simulation
2) Distributional Theory
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Upper-Tail p-value
 To calculate the p-value for a chi-square test,
we always look in the upper tail
 Why?
 Values
of the χ2 are always positive
 The
higher the χ2 statistic is, the farther the
observed counts are from the expected counts, and
the stronger the evidence against the null
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Simulation
p-value
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Chi-Square (χ2) Distribution
• If each of the expected counts are at least 5,
AND if the null hypothesis is true, then the χ2
statistic follows a χ2 –distribution, with
degrees of freedom equal to
df = number of categories – 1
•
Gene variants:
df = 3 – 1 = 2
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Chi-Square Distribution
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p-value using χ2 distribution
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Conclusion
Do our data provide evidence that the
population proportions differ from 25% RR,
50% RX, and 25% XX?
a) Yes
b) No
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Chi-Square Test for Goodness of Fit
1. State null hypothesized proportions for each category, pi.
Alternative is that at least one of the proportions is different
than specified in the null.
2. Calculate the expected counts for each cell as npi .
3. Calculate the
χ2
statistic:
observed - expected 

2
 
expected
2
4. Compute the p-value as the proportion above the χ2 statistic
for either a randomization distribution or a χ2 distribution with
df = (# of categories – 1) if expected counts all > 5
5. Interpret the p-value in context.
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Mendel’s Pea Experiment
In 1866, Gregor Mendel, the “father of genetics”
published the results of his experiments on peas
•
• He found that his experimental distribution of
peas closely matched the theoretical distribution
predicted by his theory of genetics (involving
alleles, and dominant and recessive genes)
Source: Mendel, Gregor. (1866). Versuche über Pflanzen-Hybriden. Verh.
Naturforsch. Ver. Brünn 4: 3–47 (in English in 1901, Experiments in Plant
Hybridization, J. R. Hortic. Soc. 26: 1–32)
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Mendel’s Pea Experiment
Mate SSYY with ssyy:
 1st Generation: all Ss Yy
Mate 1st Generation:
=> 2nd Generation
 Second Generation
S, Y: Dominant
s, y: Recessive
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Phenotype
Theoretical
Proportion
Round, Yellow
9/16
Round, Green
3/16
Wrinkled, Yellow
3/16
Wrinkled, Green
1/16
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Mendel’s Pea Experiment
Phenotype
Theoretical Observed
Proportion
Counts
Round, Yellow
Round, Green
Wrinkled, Yellow
9/16
3/16
3/16
315
101
108
Wrinkled, Green
1/16
32
Let’s test this data against the null hypothesis of each
pi equal to the theoretical value, based on genetics
H 0 : p1  9 /16, p2  3 /16, p3  3 /16, p4  1/16
H a : At least one pi is not as specified in H 0
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Mendel’s Pea Experiment
Phenotype
Round, Yellow
Null pi Observed
Counts
Wrinkled, Yellow
9/16
3/16
3/16
315
101
108
Wrinkled, Green
1/16
32
Round, Green
Expected Counts
The expected count for the round, yellow phenotype is
a)177.2
b)310.5
c) 312.75
d)318.25
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Mendel’s Pea Experiment
Phenotype
Round, Yellow
Null pi Observed
Counts
Expected
Counts
Contribution
to χ2
Wrinkled, Yellow
9/16
3/16
3/16
315
101
108
312.75
104.25
104.25
0.101
0.135
Wrinkled, Green
1/16
32
34.75
0.1218
Round, Green
The contribution to the χ2 statistic for the round, yellow
phenotype is
a)0.012
b)0.014
c) 0.016
d)0.018
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Mendel’s Pea Experiment
• χ2 = 0.47
• Two options:
o
Simulate a randomization distribution
o
Compare to a χ2 distribution with 4 – 1 = 3 df
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Mendel’s Pea Experiment
p-value = 0.925
Does this prove Mendel’s theory of genetics?
Or at least prove that his theoretical
proportions for pea phenotypes were correct?
a) Yes
b) No
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To Do
 Read Section 7.1
 Do HW 7.1 (due Friday, 4/17)
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