Transcript Power and sample size - MGH Biostatistics Center

Sample size and study
design
Brian Healy, PhD
Comments from last time


We did not cover confounding
Too much in one class/Not enough
examples/Superficial level
– I wanted to show one example for each type of
analysis so that you can determine what your data
matches. This way you can speak to a statistician
knowing the basic ideas.
– My hope was for you to feel confident enough to
learn more about the topics relevant to you
– Worked example lectures


This is not basic biostatistics
I did Teach for America
Objectives
Type II error
 How to improve power?
 Sample size calculation
 Study design considerations

Review

Previous classes we have focused on data
analysis
– AFTER data collection

Hypothesis testing allowed us to
determine whether there was a
statistically significant:
– Difference between groups
– Association between two continuous factors
– Association between two dichotomous factors
Example
We know that the heart rate for healthy
adult is 80 beats per minute and this has
an approximately normal distribution
(according to my wife)
 Some elite athletes, like Lance Armstrong,
have lower heart rate, but it is not known
if this is true on average
 How could we address this question?

Experimental design

One way to do this is to collect a sample
of normal controls and a sample of elite
athletes and compare their mean
– What test would you use?

Another way is to collect a sample of elite
athletes and compare their mean to the
known population mean
– This is a one sample test
– Null hypothesis: meanelite=80
Question
How large a sample of elite athletes should I
collect?
 What is the benefit of having a large sample
size?

– More information
– More accurate estimate of the population mean

What is the disadvantage of a large sample size?
– Cost
– Effort required to collect

What is the “correct” sample size?
Effect of sample size

Let’s say we wanted to estimate the blood
pressure of people at MGH
– If we sampled 3 people, would we have a good
estimate of the population mean?
 How much will sample mean vary from sample to sample?
– Does our estimate of the improve if we sampled 30
people?
 Would the sample mean to vary more or less from sample to
sample?
– What about 300 people?
Simulation
http://onlinestatbook.com/stat_sim/sampli
ng_dist/index.html
 What is the shape of the distribution of
sample means?
 Where is the curve centered?
 What happens to curve as sample size
increases?
 Technical: Central limit theorem

Standard error of the mean

There are two measures of spread in the data
– Standard deviation: measure of spread of the
individual observations
 The estimate of this is the standard deviation of the
observations: 
– Standard error: standard deviation of the sample
mean
 The estimate of this is the standard deviation of the
observations divided by the sample size 
n
Technical: Distribution of sample
mean under the null

If we took
repeated samples
and calculated the
sample mean, the
distribution of the
sample means
would have a
distribution
Spread in distribution is
based on standard error
Mean of
distribution=80
Type I error
We could plot the distribution of the sample
means under the null before collecting data
 Type I error is the probability that you reject
the null given that the null is true
a = P(reject H0 | H0 is true) Notice that the shaded

a
area is still part of the
null curve, but it is in
the tail of the
distribution
Hypothesis test-review
After data collection, we can calculate the
p-value
 If the p-value is less than the pre-specified
a-level, we reject the null hypothesis

As the sample size increases, the standard error
decreases
 p-value is based on the standard error

– As you sample size increases, the p-value decreases if
the mean and standard deviation do not change
– With an extremely large sample, a very small
departure from the null is statistically significant

What would you think if you found the sample
mean heart rate of three elite athletes was 70
beats per minute?
– Do your thoughts change if you sampled 300 athletes
and found the same sample mean?
How much data should we collect?

Depends on several factors:
– Type I error
– Type II error (power)
– Difference we are trying to detect (null and
alternative hypotheses)
– Standard deviation

Remember this is decided BEFORE the
study!!!
Type II error
Definition: when you fail to reject the
null hypothesis when the alternative is in
fact true (type II error)
 This type of error is based on a specific
alternative
b= P(fail to reject the H0 | HA is true)

Power

Definition: the probability that you reject
the null hypothesis given that the
alternative hypothesis is true. This is what
we want to happen.
Power = P(reject Ho | HA is true) = 1 - b

Since this is a good thing, we want this to
be high
Fail to reject H0
This is the
population
distribution
under the null
hypothesis
The location of
the curve is m0
and the spread
in the curve is
the standard
error
Reject Ho
This is the cut-off value.
This is the population
distribution under the
alternative hypothesis
m0
m1
Fail to reject H0
Reject Ho
a = P(reject H0| H0 is true)
m0
b= P(fail to reject H0|
Power = P(reject H0| HA is
true)
HA is true)
m1
Life is a trade off

These two errors are related
– We usually assume that the type I error is
0.05 and calculate the type II error for a
specific alternative
– If you are want to be more strict and falsely
reject the null only 1% of the time (a=0.01),
the chance of a type II error increases

Sensitivity/specificity or false positive/false
negative
Changing the power
Note how the power
(green) increases as
you increase the
difference between
the null and
alternative
hypotheses
 How else do you think
we could increase the
power?

Another way to increase power is to
increase type I error rate
 Two other ways to increase power involve
changing the shape of the distribution

– Increasing the sample size
 When the sample size increases, the curve for the
sample means tightens
– Decreasing the variability in the population
 When there is less variability, the curve for the
sample means also tightens
Example
For our study, we know that we can enroll 40
elite athletes.
 We also know that the population mean is 80
beats per minute and the standard deviation is
20
 We believe the elite athletes will have a mean of
70 beats per minute
 How much power would we have to detect this
difference at the two-sided 0.05 level?

– All this information fully defined our curves
• Using STATA, we find that we have 88.5% power to
detect the difference of 10 beats per minute between the
groups at the two-sided 0.05 level using a one sample ztest
• Question: If we were able to enroll more subjects would
our power increase or decrease?
Conclusions
For a specific sample size, standard
deviation, difference between the means
and type I error, we can calculate the
power
 Changing any of the four parameters
above will change the power

– Some under the control of the investigator,
but others are not
Sample size
Up to now we have shown how to find the
power given a specific sample size, difference
between the means, standard deviation and
alpha level.
 We can vary any four of these five factors and
find the fifth.

– Usually the alpha level is required to be two-sided
0.05
– How can we calculate the sample size for specific
values of the remaining parameters?
Two approaches to sample size

Hypothesis testing
– When you have a specific null AND alternative
hypothesis in mind

Confidence interval
– When you want to place an interval around an
estimate
Hypothesis testing approach
1)
State null and alternative hypothesis
–
–
Null usually pretty easy
Alternative is more difficult, but very important
State standard deviation of outcome
3) State desired power and alpha level
2)
–
–
Power=0.8
Alpha=0.05 for two-sided test
State test
5) Use statistical package to calculate sample size
4)

We know the location
of the null and
alternative curves, but
we do not know the
shape because the
sample size
determines the shape.
We need to find the
sample size that will
give the curves the
shape so that the a
level and power equal
the specified values.
Alpha=0.025
Power=0.8
Beta=0.2
General form of sample size
calculation

Here is the general form of the normal sample
Standard deviation Related to Type
size
I error
– One-sided
  z1a  z1 b     z1a  z1 b  
 = 

n = 


 m0  m1  
2
Sample size
– Two-sided
Mean under null and
alternative
  z
n=


1a / 2
 z1 b
m0  m1
Related to Type II
error
 


2
2
Hypothesis testing approach
1)
State null and alternative hypothesis
–
–
H0: m0=80
HA: m1=70
sd=20
3) State desired power and alpha level
2)
–
–
Power=0.8
Alpha=0.05 for two-sided test
State test: z-test
5) n=31.36  n=32
4)
Example-more complex
In a recently submitted grant, we
investigated the sample size required to
detect a difference between RRMS and
SPMS patients in terms of levels of a
marker
 Preliminary data:

– RRMS: mean level=0.54 +/- 0.37
– SPMS: mean level=0.94 +/- 0.42
Hypothesis testing approach
1)
State null and alternative hypothesis
– H0: meanRRMS=meanSPMS=0.54
– HA: meanRRMS=0.54, meanSPMS=0.94,
Difference between groups=0.4
2)
3)
sdRRMS=0.37, sdSPMS=0.42
State desired power and alpha level
– Power=0.8
– Alpha=0.05 for two-sided test
4)
State test: t-test
Results

Use these values in statistical package
– 17 samples from each group are required

Website:
http://hedwig.mgh.harvard.edu/sample_si
ze/size.html
Statistical considerations for grant
“Group sample sizes of 17 and 17 achieve at
least 80% power to detect a difference of
-0.400 between the null hypothesis that
both group means are 0.540 and the
alternative hypothesis that the mean of
group 2 is 0.940 with estimated group
standard deviations of 0.370 and 0.420
and with a significance level (alpha) of
0.05 using a two-sided two-sample t-test.”
Technical remarks
So we have shown that we can calculate the
power for a given sample size and sample size
for a given power. We can also change the
clinically meaningful difference if we set the
sample size and power.
 In many grant applications, we show the power
for a variety of sample sizes and differences in
the means in a table so that the grant reviewer
can see that there is sufficient power to detect a
range of differences with the proposed sample
size.

Confidence interval approach
If we do not have a set alternative, we
can choose the sample size based on how
close to the truth we want to get
 In particular we choose the sample size so
that the confidence interval is of a certain
width


Under a normal distribution, the
confidence interval for a single sample
mean is


 
 mean 1.96 *
, mean 1.96 *

n
n


We can choose the sample size to provide
the specified width of the confidence
interval
Conclusions
Sample size can be calculated if the
power, alpha level, difference between the
groups and standard deviation are
specified
 For more complex setting than those
presented here, statisticians have worked
out the sample size calculations, but still
need estimates of the hypothesized
difference and variability in the data

Study design
Reasons for differences between
groups
Actual effect-when there is a difference
between the two groups (ex. the
treatment has an effect)
 Chance
 Bias
 Confounding

Chance

When we run a study, we can only take a
sample of the population. Our conclusions
are based on the sample we have drawn.
Just by chance, sometimes we can draw
an extreme sample from the population. If
we had taken a different sample, we may
have drawn different conclusions. We call
this sampling variability.
Note on variability

Even though your experiments are well
controlled, not all subjects will behave exactly
the same
– This is true for almost all experiments
– If all animals acted EXACTLY the same, we would only
need one animal

Since one is not enough, we observe a group of
mice
– We call this our sample

Based on our sample, we draw a conclusion
regarding the entire population
Study design considerations
Null hypothesis
 Outcome variable
 Explanatory variable
 Sources of variability
 Experimental unit
 Potential correlation
 Analysis plan
 Sample size

Example
We start with a single group (ex. Genetically
identical mice)
 The group are broken into 3 groups that are
treated with 3 different interventions
 An outcome is measured in each individual
 Questions:

– What analysis should we do?
– What is the effect of starting from the same
population?
– Do we need to account for repeated measures?
Original
group
Condition 3
Condition 1
Condition 2
Generalizability
Assume that we have found a difference
between our exposure and control group
and we have shown that this result is not
likely due to chance, bias or confounding.
 What does this mean for the general
population? Specifically, to which group
can we apply our results?

– This is often based on how the sample was
originally collected.
Example 2
We want to compare the expression of a
marker in patients vs. controls
 Full sample size is 288 samples
 Can only run 24 samples (1 plate) per day
 Questions:

– What types of analysis should we do?
– Can we combine across the plates?
– Could other confounders be important to
collect?
Plate 1: 10 patients, 14
controls
Plate 2: 14 patients, 10
controls
Plate 3: 12 patients, 12
controls
Estimate of difference
in this plate
Estimate of difference
in this plate
Estimate of difference
in this plate
We can test if there is a different effect in each plate by
investigating the interaction
Example 3
We want to compare the expression of 6
markers
 We measure the six markers in 5 mice
 Questions:

– What types of analysis should we do?
– How many independent groups do we have?
– What is the null hypothesis?
Example 4
“In our experiments, we collect 3
measurements. If it is significant, we call
it a day. If it is close to significant, we
measure 1 more animal”
 Question:

– Is this valid?

Always more statistically valid if the
number is specified BEFORE the
experiment
Spreadsheet formation

What to collect
– Everything that might be important for the
analysis
 Plate
 Batch
 Technician
 All potential sources of variability
 All potential confounders
– Most accurate version of this you can
 If it is continuous, collect it as such. Can always
dichotomize later
Spreadsheet formation

Easiest to move to a statistical package if
– One row per measurement
– One column for the outcome, each predictor
and potential confounders
– No open space
Conclusions
Sample size for experiment must be
considered BEFORE collecting data
 Can improve power by reducing standard
deviation, increasing sample size or
increasing difference between groups
 Important to consider study design as you
develop your analysis plan
