Download Power & Sample Size Calculations

Power & Sample Size Calculations
Presented by:
Ms. Kamla Kumari D Maharaj
MSc Nursing
CON JPMC Karachi
Course facilitator:
Ms. Rabia Riaz
Dated: 28th Sep, 2010
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Sample size
• Sample size refer to the number of subjects or
participants studied in a trial, including the treatment
and control group, where applicable.
• Sample size refers to the specific size of the group or
groups being studied in research.
• Four criteria are used to estimate the appropriate
sample size for a study. Sometime called power
analysis.
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Criteria for estimating sample size
Level of Significance (alpha)
Statistical power (beta-1)
Expected difference (effect size).
Standard deviation
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Level of significance (alpha)
This is the threshold for finding statistical
significance. Normally this is set out at .05, a 5%
chance of rejecting null hypothesis when there is
no fact no significance difference or relationship
between underling population. As alpha get
smaller, sample size requirement increase.
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Power (beta-1)
Power is 1-Beta and is defined as the
probability of correctly finding statistical
significance. A common value for power is
.80.
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Expected difference (effect size)
This is the expected difference or relationship
between two independent samples. Also known as
the effect size. the effect size is determined by
literature review, logical assertion, and conjecture.
Formula:
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Basic Terms and Concepts involved in
Sample Size and Power Estimation
Null Hypothesis:
is the supposition that the effect which we are
checking for does not exist.
comparing treatment to no-treatment:
H0 = treatment has no effect
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Basic Terms and Concepts involved in
Sample Size and Power Estimation
Alternative Hypothesis:
is the supposition that the effect which we are
checking
for does exist.
comparing treatment to no-treatment:
H1 = treatment has effect
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Hypothesis Testing in Sample Size or Power
Estimation
Hypothesis tests are inferential procedures, they
concern the inferences from sample to population.
It involves the calculation of some test statistics.
The most commonly used test statistics are
Z (normal), χ2 (Chi-square), and t.
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Two Possible Errors of Hypothesis Testing
in Sample Size or Power Estimation
 The Type I Error
occurs when we conclude from an experiment that a
difference between groups exists when in truth it does
not.
Rejecting H0 when H0 is in Fact True
Probability of making type I error is denoted by “α” ,
Investigators reject H0 and declare that a real effect
exists. when the chance of this decision being wrong
is less than 5%. This is what is meant when it is
claimed that the result is statistically significant at
p<.05
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Two Possible Errors of Hypothesis Testing
in Sample Size or Power Estimation cont…
 The Type II Error
occurs when we conclude that there is no difference
between treatments when in truth there is a difference
fail to reject H0 when H0 is in Fact False
probability of making type II error is denoted by β
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Errors and Probabilities in Hypothesis
Testing
Not Reject H0
Reject H0
H0 True
Type-I Error
Power
H0 False
Power
Type-II Error
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Basic Terms and Concepts involved in
Sample Size and Power Estimation cont…
Two-tailed test
When the investigator is interested in determining
whether treatment A is different from treatment B
(either better or worse) a 2 tailed test is indicated.
Usually a 2 tailed test is performed with the risk of
making a
Type-1 error set at α / 2 in each tail.
For a 2 tailed test at α = .05 and equal allocation of
type-1 error to each tail Zα = 1.96
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Basic Terms and Concepts involved in
Sample Size and Power Estimation cont…
One-tailed test
Sometimes an investigator is only interested in a
difference between treatments in one direction.
This is appropriate when either
1. the scientific reasoning behind the experiment leads
to a prediction in one direction or
2. a new treatment will be used if it is better than the
standard but abandoned if it is worse or the same
For a 1 tailed test at α = .05 Zα = 1.65
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SAMPLE SIZE CALCULATIONS :
SYMBOLS
Z α is the “Standard Normal Deviate” corresponding to
the probability α
Z β is the “Standard Normal Deviate” corresponding to
the probability β
Common α, β= .2
.1
.05
.025
.01 .005
value Zα β= .84 1.28 1.65 1.96
2.33 2.58
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Mathematical Symbols used to denote
some common summary Statistics
Population
Parameters
Mean Variance
Standard Proportion
deviation
Correlation
Greek
letters
μ
σ2
σ
π
ϒ
Roman
letters
_
×
s2
s
p
r
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Sample Size and Power
Calculations
The calculation of sample size depends on
the summary statistics chosen. The most common
choices are
Treatment mean
e.g. average blood pressure, average cholesterol,
average days in hospital
Treatment proportion
e.g. % of patients who die, recover, achieve some
therapeutic goal or any defined state
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Most Common Sample Size Calculations
 Comparing 2 independent groups- means
 Comparing 2 related groups- means
 Comparing 2 independent groups- proportions
 Comparing 2 related groups- proportions
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Sample size estimation for tests between two
independent sample proportions
Formula:
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Sample size estimation for tests between two
independent sample proportions cont…
Where as
N= the sample size estimate
Zcv=Z critical value for alpha (.05 alpha has a Zcv of
1.96)
Z power=Z value for 1-beta (.80 power has a Z of
0.842)
P1=expected proportion for sample 1
P2=expected proportion for sample 2
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Sample size estimation for tests between two
independent sample proportions cont…
Proportion Example
Alpha=.05
Power=.80
P1=.70
P2=.80
p= .75
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Sample size estimation for tests between two
independent sample means
where
N= the sample size estimate
Zcv=Z critical value for alpha (.05 alpha has a
Zcv of 1.96)
Zpower=Z value for 1-beta (.80 power has a Z
of 0.842)
s=standard deviation
D=the expected difference between the two
means.
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Sample size estimation for tests between two
independent sample means cont…
Mean Example
Alpha=.05
Power=.80
D=10
S=20
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Reference
• Germann, E. (2003). Sample Size and Statistical
Power in the Planning of Experiments retrieved from
www.nihtraining.com/cc/ippcr/current/.../Johnson111
505bw.ppt on dated 20/08/2010.
• Johnson,L,L.(2005). Sample Size and Power
retrieved from
http://www.nihtraining.com/cc/ippcr/current/downloa
ds/Johnson111505bw.ppt on dated 22/08/2010.
• Thalheimer, W. Cook, S.(2002). How to calculate
effect sizes from published research articles: A
simplified methodology retrieved from www.worklearning.com/effect_sizes.htm. 0n dated 21/08/2010
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