Basis of statistical Inference 2

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Basis of statistical Inference 2

• BMLS 202.251
• Dr. Cord Heuer
• EpiCentreMassey University

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Confidence Interval

• sample mean
• ? is not known, but estimated by the sample
  standard deviation (s) if n reasonably large (gt
  30)
• n sample size
• z standard normal score

Confidence interval
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Confidence interval

• informal
• 95 confident that the true mean lies in the
  interval
• formal
• if we repeated sampling an infinite number of
  times, 95 of the intervals would overlap the
  true mean

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Sample size to estimate a mean
becomes
becomes
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Example 95CI Estimation ???Unknown

• A random sample of n 10 has 37.2 and
• S 7.13. Set up a 95 confidence interval
  estimate for ?.
• from to

?
?
?
32.1
42.3
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Confidence Interval
If ? 0.05, the 95 CI of about 5 from 100
samples will not overlap the true population mean
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Hypothesis testing

• generation of hypothesis testing theory depends
  on assumption of random sampling
• begins with null hypothesis
• and alternate hypothesis

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Hypothesis testing

• we cannot prove the alternate hypothesis
• we can only become increasingly confident that
  the null hypothesis may not be true

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Cannot prove the alternative hypothesis
Ho all swans are white H1 at least one swan
is not white
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The concept of reasoning
Induction all swans are white
sample
population
Deduction there were no black swans
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Cannot prove the alternative hypothesis
Ho all swans are white H1 at least one swan
is not white
Oilpest ? Birth defect ? More of those ?
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Inference for Meanst-test One sample
Example Ho sample new value z 1.35
What is the area under the normal curve outside
1.35 and 1.35 i.e. the probability of obtaining
an absolute z-value greater or smaller than
1.35? P 0.18
Not enough evidence that the new value comes
from a different population
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P Ho new value mean1
P 0.089
P 0.089
area outside 2 0.089 0.178 ie. 18
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2-sample t-test
page 16
H0 x-bar1 - xbar2 0 or xbar1 xbar2 Ha
x-bar1 ? xbar2
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P-value

• Answers the question
• Assuming the null hypothesis is true,
• what is the probability of obtaining a
  test-statistic this large or larger.
• the smaller the P-value, the less confidence we
  have in the null hypothesis

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P-value

• Then compare the calculated P-value to a
  pre-defined threshold 0.05
• If Plt 0.05
• if the null hypothesis is true, there is lt5
  probability of getting a difference this large.
• so - reject the null hypothesis !?

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One Sample

• Examples
• Height of a student at Massey
• Ho height of an individual mean height
• Difference between paired body weights
• Weight gain Ho difference 0
• Slope coefficient age vs. WBC
• Ho slope 0
• Size of an association between gender and
  respiratory disease
• Ho OR 1

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Power

• Error
• ? probability of accepting Ho when the sample
  populations are in fact different.
• Commonly set to 0.2
• Power 1- ?
• Commonly set to 0.8
• 80 chance of detecting a difference (effect) if
  it exists

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Power analysis
page 22

• Steps
• Define Ho and Ha
• Find z for Pgt(1-?) ? lack of confidence
• Find x-bar at that point
• Compute the probability that Ha will be concluded

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P(ower) of accepting Ha given X1
1) Define Ho and Ha Ho X0 vs. Ha X1 2) Find
z for Pgt(1-?) 1.67 (one sided, ?0.05) 3) Find
x-bar at that point 01.67SE 0.658 4) Compute
the probability that Ha will be
concluded P(xgt0.658Xbar1) Pzgt(0.658-1)/SE
0.80
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EXAMPLE

• Training effect on race dogs Xdifference
• Ho X0 sec Ha X2 sec
• n20 sd3
• For P gt 0.951-sided tdf19, 1-sided1.729
• find X-bar for the upper end of the conf. int.
  under Ho
• z(1.16) under Ha
• P(Zgtz) 0.89

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Use of power analysis

• Determine sample size while designing a study
• Post hoc after completing a study
• estimate power
• estimate required sample size ? cost
• estimate the effect size that would have been
  significant by the study

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Ways to increase power

• increase sample size
• consider a 1-sided test
• allow a larger type-I error
• increase precision of measurements
• use paired or matched subjects
• use unequal group sizes

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Odds Ratio

• 95 CI
• From elnOR 1.96SE(lnOR)
• to elnOR 1.96SE(lnOR)

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Relative Risk

• RR has approximately normal distribution at the
  log-scale (use ln)
• 95 CI
• From elnRR 1.96SE(lnRR)
• To elnRR 1.96SE(lnRR)