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Sample Size And Power Warren Browner and Stephen Hulley

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Hypothesis. Null and alternative. One-sided vs two-sided. Statistical test. Type of variables ... If The Null Hypothesis Is True ... Null hypothesis: ... – PowerPoint PPT presentation

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Title: Sample Size And Power Warren Browner and Stephen Hulley


1
Sample Size And PowerWarren Browner and Stephen
Hulley
  • The ingredients for sample size planning, and how
    to design them
  • An example, with strategies for minimizing sample
    size

2
Sampling and Inference
  • A sample is designed to represent a larger
    population
  • Therefore, findings in the sample allow
    inferences about events in the population
  • Problem what if the inferences are wrong?
  • Finding something in the sample that isnt real
    in the population
  • Missing something that is real

3
Preventing Wrong Inferences
  • Difficult when caused by systematic error (bias)
  • Easy when caused by Random error (chance)
  • Solution increase sample size
  • Problem cost, feasibility
  • Goldilocks solution a sample size that is big
    enough but not too big

4
Ingredients For Planning Sample Size in an
Analytic Study or RCT
  • Hypothesis
  • Null and alternative
  • One-sided vs two-sided
  • Statistical test
  • Type of variables
  • Effect size (and its variance)
  • Power and alpha

5
Research Hypothesis
  • A clear statement of what you are studying
  • Simple one predictor, one outcome
  • Specific who, what, when, where
  • Stated in advance

6
Research Hypothesis
  • In patients with early ALS seen at UCSF in 2007,
    those randomly assigned to be treated with newmol
    will have a lower 1-year mortality than those
    randomly assigned to placebo.

7
The Null Hypothesis
  • Theres nothing going on.
  • Purpose in life to be rejected in favor of its
    alternative.
  • In patients with early ALS seen at UCSF in 2007,
    those randomly assigned to be treated with newmol
    will have the same 1-year mortality as those
    randomly assigned to placebo.

8
Whats This All About?
  • A long time ago, statisticians figured out the
    probability that a sample of a given size would
    find something even if there were nothing going
    on in the population.

9
This means that...
  • After a study, we can determine the likelihood
    that whatever we found in our sample could have
    occurred by chance...
  • Even if nothing was going on in the population
    (i.e., the null hypothesis was true)--a Type I
    error
  • If this is very unlikely (say lt 1 in 20) we
    reject the null hypothesis in favor of the
    alternative hypothesis we call the finding
    statistically significant (P lt .05)

10
Two-sided Alternative Hypothesis
  • In patients with early ALS seen at UCSF in 2007,
    those randomly assigned to be treated with newmol
    will have a different 1-year mortality than those
    randomly assigned to placebo.

11
Two One-sided Alternative Hypotheses
  • Side A In patients with early ALS seen at UCSF
    in 2007, those randomly assigned to be treated
    with newmol will have a higher 1-year mortality
    than those randomly assigned to placebo.
  • Side B In patients with early ALS seen at UCSF
    in 2007, those randomly assigned to be treated
    with newmol will have a lower 1-year mortality
    than those randomly assigned to placebo.

12
If The Null Hypothesis Is True
  • By chance alone, each of the two one-sided
    alternative hypotheses is...
  • Possible
  • Equally likely
  • Wrong
  • Thus a two-sided alternative hypothesis has twice
    the likelihood of happening by chance alone

13
Next Ingredient Statistical Test (Types of
Variable)
  • The statistical test determines how the sample
    size will be calculated
  • The type of predictor and outcome variable
    determine which statistical test will be used to
    analyze the data
  • Both dichotomous Chi square
  • One dichotomous, one continuous t test
  • Both continuous correlation coeff or t test

14
Statistical Test (Types of Variable)
  • ALS study
  • Predictor newmol vs placebo
  • Outcome dead
  • Both are dichotomous
  • Chi square test

15
Next IngredientEffect Sizes (dichotomous
variables)
  • How big an effect you anticipate seeing
  • Newmol halves mortality
  • newmol 5, Placebo 10

16
Penultimate Ingredient Power
  • The chance of finding something in your sample if
    its really going on in the population (avoiding
    a Type II error)
  • Something the effect size (or greater)
  • Usually set at 80 or 90
  • (1 - beta)

17
and the Final Ingredient Alpha
  • The chance of finding something in your sample if
    theres nothing going on in the population.

18
Alpha Explained
  • The level of statistical significance (ie, the
    p-value that will be considered significant)
  • The pre-set maximum chance of finding something,
    if it really isnt there.
  • Usually set at 0.05.
  • May be one-sided or two-sided.

19
Sidedness Of Alpha
  • With a two-sided alternative hypothesis, you have
    two chances of finding something that isnt
    really there
  • One (equal) chance for each side.
  • So a one-sided alpha of 0.05 corresponds to a
    two-sided alpha of 0.10.

20
SAMPLE SIZE AN EXAMPLE
  • Null hypothesis
  • In patients with early ALS seen at UCSF in 2007,
    those randomly assigned to be treated with newmol
    will have the same 1-year mortality as those
    randomly assigned to placebo.
  • Two-sided alternative hypothesis
  • Dichotomous predictor and outcome
  • Effect size 10 mortality 5
  • Power, alpha 90, 0.05 (two-sided)

21
THE SAMPLE SIZE IS
  • Appendix 6.B
  • Smaller of P1 and P2 0.05 power of 90 alpha
    of 0.05 (two-sided)
  • Difference 0.05
  • 381
  • 473
  • 620
  • This is per group

22
Sample Size Reduction Strategy 1Statistical
Manipulation
  • Use a lower power
  • Use a one-sided alpha
  • Power of 80
  • One-sided alpha of 0.05

23
The New Sample Size Is
  • Appendix 6.B
  • Smaller of P1 and P2 0.05 power of 80 alpha
    of 0.05 (one-sided)
  • Difference 0.05
  • 381
  • 473
  • 620
  • This is also per group

24
SS Reduction Strategy 2 Use A More Common
Outcome
  • Change from 1-year mortality to 2-year mortality
    or loss of independent living
  • Placebo 40
  • Newmol 20

25
The New Sample Size Is
  • Appendix 6.B
  • Smaller of P1 and P2 0.20 power of 80 alpha
    of 0.05 (two-sided)
  • Difference 0.20
  • 74
  • 91
  • 118

26
SS Reduction Strategy 3 Use A Continuous
Outcome
  • Change mortality or loss of independent living
    to muscle strength
  • NOTE Big change in research question and
    research hypothesis.
  • New null hypothesis
  • In patients with early ALS seen at UCSF in 2007,
    those randomly assigned to be treated with newmol
    will have the same grip strength at the end of
    six months as those treated with placebo.
  • Two-sided alternative hypothesis

27
Estimate The Mean And Variability Of Grip Strength
  • Patients with untreated ALS have a (mean SD)
    grip strength of 20 10 kg after 6 months of
    disease
  • Newmol may improve that by 25

28
Then
  • Grip strength
  • Placebo 20 kg
  • Newmol 25 kg (25 more)
  • Effect size 5 kg SD 10 kg
  • Standardized effect size E/S 5/10 0.5

29
The New Sample Size Is ...
  • Appendix 6.A
  • E/S 0.5
  • ß 0.20, Alpha (two-sided) 0.05
  • N 64 per group

30
Ss Reduction Strategy 4 Use A More Precise
Outcome
  • Buy a better instrument to measure grip strength
  • Use a well-defined protocol
  • Repeat measurements on two consecutive days
  • Reduce SD from 10 kg to 8 kg

31
The New Sample Size Is ...
  • New E/S 5 kg/8 kg 0.625
  • ß 0.20, Alpha (two-sided) 0.05
  • N about 45 per group
  • This helped quite a bit.

32
SS Reduction Strategy 5 Use Paired Measurements
  • Most of the variability in grip strength at the
    end of the study is likely to be due to
    differences between subjects in grip strength at
    the beginning of the study.
  • Switch the outcome to change in grip strength
    from the beginning to the end of the study.

33
Paired Measurements
  • Each subject contributes a pair of measurements
    (before, after)
  • The outcome variable is the difference between
    that pair for each subject.
  • The SD of the change in a measurement is usually
    lt than the SD of the measurement
  • SD of change in grip strength is 5 kg
  • New standardized effect size 5/5 1.0

34
The New Sample Size Is...
  • E/S 1.0
  • ß 0.20, Alpha (two-sided) 0.05
  • N 17 per group
  • We now have a potentially do-able study, albeit
    one that is very different from the original aim.

35
The Bottom Line
  • Sample size estimation is an integral part of
    study planning
  • Almost never the last thing you do
  • More often, one of your first tasks

36
SAMPLE SIZE PLANNING REVIEW OF INGREDIENTS
  • Looking for something in a sample
  • Hypotheses (null and alternative)
  • Will you be able to ...
  • Know its there in the population if you find it
    in your sample (avoid a Type I error)
  • Test of significance, alpha
  • Find it in your sample if its there in the
    population (avoid a type II error)?
  • Effect size, power
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