Download Hypothesis testing

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Statistical Inference: Brief Overview
 Statistics: Learning from Samples about Populations
 Inference 1: Confidence Intervals
 What does the 95% CI really mean?
 Inference 2: Hypothesis Tests
 What does a p-value really mean?
 When to use which test?
Examples of hypothesis testing in medical research
 In epidemiological studies: Is there a relationship between a
variable of interest and an outcome of interest?
 In example: smoking and lung cancer

stress and thyroid cancer
 In clinical trails: Is experimental therapy more effective than
standard therapy or placebo?
Hypothesis testing = testing of
statistical hypothesis
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Statistical hypothesis
Statements about population parameter values.
 Null hypothesis (H0) says a parameter is unchanged from a
default, pre-specified value;
and
 Alternative hypothesis (H1) says parameter has a value
incompatible with H0
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Population
Sample
?
Parameters
Statistics
_
 . . . . . . . . Mean . . . . . . . x
 . . . Standard Deviation. . s
Size . . . . . . . n
Postulated (unknown)
Seen (known)
…
…
Example: Hypertension and Cholesterol
Make appropriate statistical hypotheses:
Assumption: Mean cholesterol in hypertensive men is
equal to mean cholesterol in male general population
(20-74 years old).
We estimated: In the 20-74 year old male population
the mean serum cholesterol is 211 mg/ml with a
standard deviation of 46 mg/ml
Example: Hypertension and Cholesterol
Null hypothesis => no difference between treatments
 H0: μhypertensive = μgeneral population
 H0: μhypertensive = 211 mg/ml
• μ - population mean of serum cholesterol
• Mean cholesterol for hypertensive men = mean for general male
population
Alternative hypothesis
 HA: μhypertensive ≠ μ general population
 HA: μ hypertensive ≠ 211 mg/ml
Null and alternative hypothesis
Two-sided tests
One-sided tests
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How to choose one or the other?
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1.
Assume H0 is true i.e. believe results are a matter of chance
2.
Quantify how far away are data from being consistent with H0
by evaluating quantity called a test statistic
3.
Assess probability of results at least this extreme - call this the
p-value of the test
4.
Reject H0 (believe H1) if this p-value is small or keep H0 (do not
believe H1) otherwise
Interpretation of P-value (0.05)
P>=0.05
No difference between the treatments
(observed difference having happened by
chance)
Null hypothesis is accepted
P<0.05
5%
Significant difference between the
treatments
Null hypothesis is rejected, alternative is
accepted
P-value
The P value gives the probability of observed and more
extreme difference having happened by chance.
 P = 0.500 means that the probability of the difference
having happened by chance is 0.5=50% in 1 ~ 1 in 2.
 P = 0.05 means that the probability of the difference
having happened by chance is 0.05=5% in 1 ~ 1 in 20.
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P-value
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P-value
 The lower the P value, the less likely it is that the
difference happened by chance and so the higher the
significance of the finding.
 P = 0.01 is often considered to be “highly significant”. It
means that the difference will only have happened by
chance 1 in 100 times. This is unlikely, but still possible.
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Example 1
 Out of 50 new babies on average 25 will be girls,
sometimes more, sometimes less.
 Say there is a new fertility treatment and we want to
know whether it affects the chance of having a boy or a
girl.
 Null hypothesis –the treatment does not alter the chance
of having a girl.
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Example 1
 Null hypothesis –the treatment does not alter the chance
of having a girl.
 Out of the first 50 babies resulting from the treatment, 15
are girls.
 We need to know the probability that this just happened
by chance, i.e. did this happen by chance or has the
treatment had an effect on the sex of the babies?
 P=0.007
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Example 1
 The P value in this example is 0.007.
 This means the result would only have happened by
chance in 0.007 in 1 (or 1 in 140) times if the treatment
did not actually affect the sex of the baby.
 This is highly unlikely, so we can reject our hypothesis and
conclude that the treatment probably does alter the
chance of having a girl.
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Example 2
 Patients with minor illnesses were randomized to
see either Dr Smith or Dr Jones. Dr Smith ended up
seeing 176 patients in the study whereas Dr Jones saw
200 patients.
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Example 2
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How to choose the appropriate
statistical test?
1. Type of data (type of variable)?
2. Number of groups?
3. Related or independent groups?
4. Normal or asymmetric distribution?
Numerical
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Example: Hypertension and Cholesterol
Make appropriate statistical hypotheses:
Mean cholesterol in hypertensive men is 220 mg/ml
with a standard deviation of 39 mg/ml.
In the 20-74 year old male population the mean
serum cholesterol is estimated to 211 mg.
Hypothesis vs Statictical Hypothesis
 Alcohol intake increases
 Mean reaction time in
driver’s reaction time.
examinees drinking
alcohol is greater than in
nondrinking controls.
Research hypothesis
Statistical hypothesis