Inference for Proportions

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Inference for Proportions

• Cord Heuer
• EpiCentre, Massey University

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counts proportions
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Herd level mastitis data
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Normal, so why bother??
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(No Transcript)
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Plot this
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Better create intervals
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and plot again
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Proportions

• disease prevalence
• test 200 sheep
• 76 positive
• 76/200 0.38 or 38
• Bernoulli process 2 possible outcomes
• follows a binomial distribution

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Binomial Distribution
P of x positive in a sample of 18 when p0.178
P
x
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Binomial Distribution n100
P 0.3
n 100
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Binomial Distribution n10
P 0.3
n 10
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Distributions for statistics

• 3 Parameters
• population ?,
• sample p,
• sampling distribution ?,

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Normal approximations to binomial distribution

• provided sample size is large enough

or
t
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Normal approximation to binomial distribution

• how big is big enough?
• If np ? 9 or 10
• AND
• n(1 - p) ? 9 or 10

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Confidence interval

• confidence interval for mean proportion

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n Z2 variance / L2
L 0.2P
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L may overlap zero
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Source J.L. Fleiss, 1981 Statistical methods
for rates and proportions. 2nd ed. John Wiley
Sons Inc., New York
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Single proportion hypothesis test

• Assume population proportion p0
• Collect a single sample (n) and measure
  proportion p-hat

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Look up P-value for z (or t)
1-sided or 2-sided
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Two proportions

• use normal approximation

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Two proportions

• H0 p1 p2 HA p1 ? p2
• need a pooled variance
• first find

assumption of equal variances
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Two proportions

• then get SEM for the difference p1 - p2
• then estimate a z-statistic

assumption of equal variances
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2 proportions unequal variances
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Small sample binomial procedures

• when sample size is small, normal approximations
  dont hold
• have to use the exact binomial distribution -
  awkward to deal with

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Binomial distribution - exact

• B(n,p) n number of trials
• p proportion of success in each trial

Probability of exactly f successes
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Miscarriage in chip workers

• Population miscarriage rate 0.178
• H0 p ? 0.178
• HA p gt 0.178
• n18 number of miscarriages7
• p 0.389

how likely is this to occur if the true rate is
0.178?
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• need to use
• complicated formula
• and sum possible
• outcomes
• p lt 0.05

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2 x 2 table

• Alternative to normal approximation
• Very widely used
• Example

P 10/500 0.02
Ho 2 proportions are equal
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2 x 2 table

• Want to compare the observed 2x2 table with a
  table expected by chance alone
• Difference is chi-square distributed
• D ?(obs exp)2 / exp

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D is Chi2 distributed
df3
?2df1 Z2
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Review
or
One sample tests
DF n-1
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Two sample t-tests
Equal variances
DF n1 n2 - 2
Pooled variance
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2-Sample t-test
Unequal variances
or use smaller of n1-1 and n2-1
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1 proportion inference
Proportion
Confidence interval
One sample inference
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2 proportion inference
Pooled variance estimate assuming equal variances
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2 proportion inference
Unequal variances denominator made up of
individual variances
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Binomial distribution - exact

• B(n,p) n number of trials
• p proportion of success
• in each trial

Probability of exactly f successes