Introduction to Biostatistics (BIO/EPI 540) Lecture 11: Hypothesis Testing

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Introduction to Biostatistics(BIO/EPI 540)
Lecture 11 Hypothesis Testing
Acknowledgement Thanks to Professor Pagano
(Harvard School of Public Health) for lecture
material
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Testing
No human investigation can be called true
science without passing through mathematical
tests. Leonardo da Vinci (1452-1519)
(in Treatise on Painting)
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Sampling Paradigm
Inference
µ,
s
Population
,S
Sample
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Inference

• Sample mean is an estimate of
• Sample variance (S) is an estimate
• of
• Confidence intervals and
• hypothesis tests are equivalent
• techniques to quantify uncertainty
• in sample derived inferences
• regarding population parameters

µ
s2
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Confidence Interval - Illustration
We know that cholesterol levels in US men 20-24
yrs are normally distributed with sX ? 46
mg/100ml. We obtain a sample of n25 and want
to infer µ.
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Use of C.I. to infer value
value of µ
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Population mean 211?
IF 95 C.I.
? 200 mg/100ml (182,218)


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Population mean 211?
IF 95 C.I.
? 200 mg/100ml (182,218)
? 190 mg/100ml (172,208)

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Population mean 211?
IF 95 C.I.
? 200 mg/100ml (182,218)
? 190 mg/100ml (172,208)
? 175 mg/100ml (157,193)
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If true

• Alternatively IF
• 211 and ? 46 and we take a
• sample of size n25 from this pop.,
• then the Central Limit Theorem
• says that the sample mean is
• approx. normal with mean ? 211
• and std. dev. 46/5 i.e.

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Hypothesis Testing

• Hypothesis Testing Trial by jury

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Hypothesis Testing Trial by jury
Individual on trial. Is he/she innocent?
Evidence
Trial
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Hypothesis Testing Trial by jury
Individual on trial. Is he/she innocent?
Evidence
Trial
Person Person
Innocent Guilty


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Hypothesis Testing Trial by jury
Individual on trial. Is he/she innocent?
Evidence
Trial
Jury Person Person
Jury Innocent Guilty
Not Guilty
Guilty
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Hypothesis Testing Trial by jury
Individual on trial. Is he/she innocent?
Evidence
Trial
Jury Person Person
Jury Innocent Guilty
Not Guilty ?
Guilty ?
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Hypothesis Testing Trial by jury
Individual on trial. Is he/she
innocent?
Evidence
Trial
Jury Person Person
Jury Innocent Guilty
Not Guilty ? x
Guilty x ?
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Hypothesis Testing
Test of Hypothesis that ? ?0?


Evidence
Trial
Evidence
Trial
Jury Person Person
Jury Innocent Guilty
Not Guilty ? x
Guilty x ?
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Hypothesis Testing
Test of Hypothesis that ? ?0?


Sample
Trial
Trial
Jury Person Person
Jury Innocent Guilty
Not Guilty ? x
Guilty x ?
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Hypothesis Testing
Test of Hypothesis that ? ?0?


Sample
Analysis
Jury Person Person
Jury Innocent Guilty
Not Guilty ? x
Guilty x ?
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Hypothesis Testing
Test of Hypothesis that ? ?0?


Sample
Analysis
Jury Population Population
Jury Innocent Guilty
Not Guilty ? x
Guilty x ?
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Hypothesis Testing
Test of Hypothesis that ? ?0?


Sample
Analysis
Jury Population Population
Jury ? ?0 Guilty
Not Guilty ? x
Guilty x ?
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Hypothesis Testing
Test of Hypothesis that ? ?0?


Sample
Analysis
Jury Population Population
Jury ? ?0 ? ? ?0
Not Guilty ? x
Guilty x ?
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Hypothesis Testing
Test of Hypothesis that ? ?0?


Sample
Analysis
Us Population Population
Us ? ?0 ? ? ?0
Not Guilty ? x
Guilty x ?
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Hypothesis Testing
Test of Hypothesis that ? ?0?


Sample
Analysis
Us Population Population
Us ? ?0 ? ? ?0
Not reject ? x
Guilty x ?
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Hypothesis Testing
Test of Hypothesis that ? ?0?


Sample
Analysis
Us Population Population
Us ? ?0 ? ? ?0
Not reject ? x
Reject x ?
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Hypothesis Testing
Test of Hypothesis that ? ?0?


Sample
Analysis
Us Population Population
Us ? ?0 ? ? ?0
Not reject ? Type II
Reject Type I ?
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Possible errors in analysis results
Probability of Type I error is ? i.e. the
probability of rejecting the null hypothesis
when it is true. Probability of Type II error
is ? i.e the probability of not rejecting the
null hypothesis when it is false. 1-? is the
power of the test.
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Hypothesis testing about ?
1o Hypothesize a value (?0)
2o Take a random sample (n).
3o Is it likely that the sample came from a population with mean ?0 (? 0.05) ?
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2 sided hypothesis test -Illustration
We know that cholesterol levels in US men 20-74
yrs are normally distributed with sX ? 46
mg/100ml and µ 211. We obtain a random sample
of 12 hypertensive smokers and obtain a sample
mean of 217 mg/100ml. We want to test whether
their population mean is the same as that of the
general population?
H0 ? 211
HA ? ? 211
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2 sided hypothesis test -Illustration
H0 ? 211
HA ? ? 211
? 46 mg/100ml
12 hypertensive smokers have
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P-value
Some prefer to quote the p-value. The p-value
answers the question, What is the probability of
get- ting as large, or larger, a Discrepancy
given the null hypothesis is true?
Question Do hypertensive smokers have the same
mean as the general population?
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Rejecting the null hypothesis

• Assume a specific threshold of Type I error, a
• Typically a 0.05
• If p value lt a ? Reject null

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P-value
Some prefer to quote the p-value. The p-value
answers the question, What is the probability of
get- ting as large, or larger, a Discrepancy
given the null hypothesis is true?
Answer Do not reject the null hypothesis. No
evidence that hypertensive smokers have a
different mean than general population
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Summary
Decide on statistic
Determine which values of
are
consonant with the hypothesis that ? ?0 and
which ones are not.
Look at
and decide.
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Alternative hypothesis
Need to set up 2 hypotheses to cover all
possibilities for ?. Choice of 3 possibilities
1. Two-sided H0 ? ?0
1. Two-sided HA ? ? ?0




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Example - One-sided alternative
Blood glucose level of healthy persons has ?
9.7 mmol/L and ? 2.0 mmol/L
H0 ? ? 9.7
HA ? gt 9.7
Sample of 64 diabetics yields
Do diabetics have blood glucose levels that are
higher on average when compared to the general
population?
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Example - One-sided alternative
Blood glucose level of healthy persons has ?
9.7 mmol/L and ? 2.0 mmol/L
H0 ? ? 9.7
HA ? gt 9.7
n 64
p-value ltlt 0.001
Answer Reject the null hypothesis. Significant
evidence that diabetics have a higher mean level
of glucose when compared to the general population
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Alternative hypothesis
Need to set up 2 hypotheses to cover all
possibilities for ?. Choice of 3 possibilities
Two-sided H0 ? ?0
Two-sided HA ? ? ?0
One-sided H0 ? ? ?0
One-sided HA ? lt ?0
One-sided H0 ? ? ?0
One-sided HA ? gt ?0
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Summary

• Hypothesis testing
• Type I and II errors
• Power
• Two sided hypothesis test
• One sided hypothesis test