Fitting probability models to frequency data

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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
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Confidence interval for a proportion
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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not so weird
weird
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Sample
Null hypothesis
Test statistic
Null distribution
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Binomial test
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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
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Binomial distribution, n 20, p 0.5
x
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Binomial distribution, n 20, p 0.5
Test statistic
x
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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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Binomial distribution, n 20, p 0.5
P 2(Pr16Pr17Pr18 Pr19Pr20)
x
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Sample
Null hypothesis
Test statistic
Null distribution
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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N 20, p0 0.5
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
Null hypothesis
Test statistic
Null distribution
compare
How unusual is this test statistic?
P gt 0.05
P lt 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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c2 Goodness-of-fit test

• Compares counts to a discrete probability
  distribution

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Hypotheses for c2 test
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Test statistic for c2 test
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Hypotheses for day of birth example
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The calculation for Sunday
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The sampling distribution of c2 by simulation
Frequency
c2
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Sampling distribution of c2 by the c2 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 c2 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 a.
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12.59
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Plt0.05, 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 c2 test

• No more than 20 of categories have Expectedlt5
• No category with Expected ? 1

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c2 test as approximation of binomial test

• If the number of data points is large, then a c2
  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
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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?
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World Cup 2002 scores
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Number of goals for a team(World Cup 2002)
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Whats the mean, m?
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Poisson with m 1.26
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Finding the Expected

Too small!
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Calculating c2
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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 value
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Comparing c2 to the critical value
So we cannot reject the null hypothesis. There
is no evidence that the score of a World Cup
Soccer game is not Poisson distributed.