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Fitting probability models to frequency
data
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Review - proportions
• Data: discrete nominal variable with two states (“success” and “failure”)
• You can do two things:– Estimate a parameter with confidence interval
– Test a hypothesis
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Estimating a proportion
€
ˆ p =x
n
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Confidence interval for a proportion*
€
′ p =X + 2
n + 4
€
′ p − Z′ p 1− ′ p ( )
n + 4
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟≤ p ≤ ′ p + Z
′ p 1− ′ p ( )
n + 4
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
where Z = 1.96 for a 95% confidence interval
* The Agresti-Couli method
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Hypothesis testing
• Want to know something about a population
• Take a sample from that population• Measure the sample• What would you expect the sample to look like under the null hypothesis?
• Compare the actual sample to this expectation
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weird
not so weird
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Sample
Test statistic
Null hypothesis
Null distributioncompare
How unusual is this test statistic?
P < 0.05 P > 0.05
Reject Ho Fail to reject Ho
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Binomial test
The binomial test uses data to test whether a population
proportion p matches a null expectation for the proportion.
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Test statistic
• For the binomial test, the test statistic is the number of successes
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Binomial test
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The binomial distribution
€
P(x) =n
x
⎛
⎝ ⎜
⎞
⎠ ⎟px 1− p( )
n−x
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0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of correct choices
Frequency
Binomial distribution, n = 20, p = 0.5
€
P(x) =n
x
⎛
⎝ ⎜
⎞
⎠ ⎟px 1− p( )
n−x
x
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0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of correct choices
Frequency
Binomial distribution, n = 20, p = 0.5
x
Test statistic
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P-value
• P-value - the probability of obtaining the data* if the null hypothesis were true
*as great or greater difference from the null hypothesis
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P-value
• Add up the probabilities from the null distribution
• Start at the test statistic, and go towards the tail
• Multiply by 2 = two tailed test
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0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of correct choices
Frequency
Binomial distribution, n = 20, p = 0.5
x
P = 2*(Pr[16]+Pr[17]+Pr[18]
+Pr[19]+Pr[20])
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Sample
Test statistic
Null hypothesis
Null distributioncompare
How unusual is this test statistic?
P < 0.05 P > 0.05
Reject Ho Fail to reject Ho
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N =20, p0 =0.5
€
P = 2 *(Pr[16] + Pr[17] + Pr[19] + Pr[19] + Pr[20])
= 2
20
16
⎛
⎝ ⎜
⎞
⎠ ⎟ 0.5( )
160.5( )
4+
20
17
⎛
⎝ ⎜
⎞
⎠ ⎟ 0.5( )
170.5( )
3+
20
18
⎛
⎝ ⎜
⎞
⎠ ⎟ 0.5( )
180.5( )
2
+20
19
⎛
⎝ ⎜
⎞
⎠ ⎟ 0.5( )
190.5( )
1+
20
20
⎛
⎝ ⎜
⎞
⎠ ⎟ 0.5( )
200.5( )
0
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟
= 0.0026
This is a pain….
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Calculating P-values
• By hand• Use computer software like jmp, excel
• Use tables
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Sample
Test statistic
Null hypothesis
Null distributioncompare
How unusual is this test statistic?
P < 0.05 P > 0.05
Reject Ho Fail to reject Ho
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Discrete distribution
• A probability distribution describing a discrete numerical random variable
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Discrete distribution
• A probability distribution describing a discrete numerical random variable
• Examples: – Number of heads from 10 flips of a coin
– Number of flowers in a square meter – Number of disease outbreaks in a year
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2 Goodness-of-fit test
Compares counts to a discrete probability
distribution
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H0: The data come from a particular discrete probability
distribution.
HA: The data do not come from that distribution.
Hypotheses for 2 test
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Test statistic for 2 test
€
2 =Observedi − Expectedi( )
2
Expectediall classes
∑
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The day of birth for 350 births in the USin 1999.
Day Sun. Mon. Tues. Wed. Thu. Fri. Sat.Num.of
Births33 41 63 63 47 56 47
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H0: The probability of a birth occurring on any given day of
the week is 1/7.
HA: The probability of a birth occurring on any given day of
the week is NOT 1/7.
Hypotheses for day of birth example
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Day Sun.
Mon. Tues.
Wed Thu.
Fri.
Sat.
Total
Obs. 33 41 63 63 47 56 47 350
Exp. 50 50 50 50 50 50 50 350
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The calculation for Sunday
€
Observed − Expected( )2
Expected=
33− 50( )2
50=
289
50
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€
2 =Observedi − Expectedi( )
2
Expectediall classes
∑
=289
50+
81
50+
169
50+
169
50+
9
50+
36
50+
9
50
= 15.24
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The sampling distribution of 2 by
simulation
Frequency
2
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Sampling distribution of 2 by the 2 distribution
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Degrees of freedom
• The number of degrees of freedom specifies which of a family of distributions to use as the sampling distribution
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Degrees of freedom for 2 test
df = Number of categories - 1 - (Number of parameters estimated from the data)
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Degrees of freedom for day of birth
df = 7 - 1 - 0 = 6
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Finding the P-value
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Critical value
The value of the test statistic where P = .
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12.59
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P<0.05, so we can reject the null hypothesis
Babies in the US are not born randomly with respect to the day of the week.
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Assumptions of 2 test
• No more than 20% of categories have Expected<5
• No category with Expected 1
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2 test as approximation of binomial test
• If the number of data points is large, then a 2 goodness-of-fit test can be used in place of a binomial test.
• See text for an example.
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The Poisson distribution
• Another discrete probability distribution
• Describes the number of successes in blocks of time or space, when successes happen independently of each other and occur with equal probability at every point in time or space
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Poisson distribution
€
Pr X[ ] =e−μ μ X
X!
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Example: Number of goals per side
in World Cup Soccer
Q: Is the outcome of a soccer game (at this level) random?
In other words, is the number of goals per team distributed as expected by pure chance?
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
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World Cup 2002 scores
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Number of goals for a team
(World Cup 2002)Number of goals Frequency
0 371 472 273 134 25 16 07 08 1
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What’s the mean, ?
€
x =37 0( ) + 47 1( ) + 27 2( ) +13 3( ) + 2 4( ) +1 5( ) +1 8( )
128
=161
128
=1.26
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Poisson with = 1.26X Pr[X]
0 0.284
1 0.357
2 0.225
3 0.095
4 0.030
5 0.008
6 0.002
7 0
8 0
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Finding the ExpectedX Pr[X] Expect
ed
0 0.284 36.3
1 0.357 45.7
2 0.225 28.8
3 0.095 12.1
4 0.030 3.8
5 0.008 1.0
6 0.002 0.2
7 0 0.04
8 0 0.007
} Too small!
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Calculating 2
X Expected
Observed
0 36.3 37 0.013
1 45.7 47 0.037
2 28.8 27 0.113
3 12.1 13 0.067
4 5.0 4 0.200
€
Observedi − Expectedi( )2
Expectedi
€
2 =Observedi − Expectedi( )
2
Expectediall classes
∑ = 0.429
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Degrees of freedom for poisson
df = Number of categories - 1 - (Number of parameters estimated from the data)
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Degrees of freedom for poisson
df = Number of categories - 1 - (Number of parameters estimated from the data)
Estimated one parameter,
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Degrees of freedom for poisson
df = Number of categories - 1 - (Number of parameters estimated from the data)
= 5 - 1 - 1 = 3
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Critical valueTable 8.2. An excerpt from the 2 critical value tabl ,e fromAppendi x 3.1
dfX 0.999 0.995 0.99 0.975 0.95 0.05 0.025 0.01 0.005 0.001
1 1.6E-6
3.9E-5 0.00016 0.00098 0.00393 3.84 5.02 6.63 7.88 10.83
2 0 0.01 0.02 0.05 0.1 5.99 7.38 9.21 10.6 13.823 0.02 0.07 0.11 0.22 0.35 7.81 9.35 11.34 12.84 16.274 0.09 0.21 0.3 0.48 0.71 9.49 11.14 13.28 14.86 18.475 0.21 0.41 0.55 0.83 1.15 11.07 12.83 15.09 16.75 20.526 0.38 0.68 0.87 1.24 1.64 12.59 14.45 16.81 18.55 22.467 0.6 0.99 1.24 1.69 2.17 14.07 16.01 18.48 20.28 24.32
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Comparing 2 to the critical value
€
2 = 0.429
χ 32 = 7.81
0.49 < 7.81
So we cannot reject the null hypothesis.
There is no evidence that the score of a World Cup Soccer game is not Poisson distributed.