1. Sample size and study design
Brian Healy, PhD

2. 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

3. Objectives Type II error How to improve power? Sample size calculation
Study design considerations

4. 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

5. 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?

6. 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

7. 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?

9. 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?

10. Simulation http://onlinestatbook.com/stat_sim/sampling_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

11. 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

12. 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

13. 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 area is still part of the null curve, but it is in the tail of the distribution a

14. 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

15. 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?

16. 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!!!

17. 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)

18. 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

19. Fail to reject H0
Reject Ho
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
This is the cut-off value.
This is the population distribution under the alternative hypothesis m0 m1

20. Fail to reject H0
Reject Ho
a = P(reject H0| H0 is true) m0
Power = P(reject H0| HA is true)
= P(fail to reject H0|
HA is true) m1

21. 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

22. 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?

23. 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

24. 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

25. Using STATA, we find that we have 88
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 z-test
Question: If we were able to enroll more subjects would our power increase or decrease?

26. 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

27. 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?

28. 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

29. Hypothesis testing approach
State null and alternative hypothesis
Null usually pretty easy
Alternative is more difficult, but very important
State standard deviation of outcome
State desired power and alpha level
Power=0.8
Alpha=0.05 for two-sided test
State test
Use statistical package to calculate sample size

30. 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

31. General form of sample size calculation
Here is the general form of the normal sample size
One-sided
Two-sided
Standard deviation
Related to Type I error
Sample size
Mean under null and alternative
Related to Type II error

32. Hypothesis testing approach
State null and alternative hypothesis
H0: m0=80
HA: m1=70
sd=20
State desired power and alpha level
Power=0.8
Alpha=0.05 for two-sided test
State test: z-test
n=31.36  n=32

33. 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

34. Hypothesis testing approach
State null and alternative hypothesis
H0: meanRRMS=meanSPMS=0.54
HA: meanRRMS=0.54, meanSPMS=0.94, Difference between groups=0.4
sdRRMS=0.37, sdSPMS=0.42
State desired power and alpha level
Power=0.8
Alpha=0.05 for two-sided test
State test: t-test

35. Results Use these values in statistical package
17 samples from each group are required
Website:

36. Statistical considerations for grant
“Group sample sizes of 17 and 17 achieve at least 80% power to detect a difference of between the null hypothesis that both group means are and the alternative hypothesis that the mean of group 2 is with estimated group standard deviations of and and with a significance level (alpha) of 0.05 using a two-sided two-sample t-test.”

37. 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.

38. 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

39. Under a normal distribution, the confidence interval for a single sample mean is
We can choose the sample size to provide the specified width of the confidence interval

40. 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

41. Study design

42. 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

43. 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.

44. 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

45. Study design considerations
Null hypothesis
Outcome variable
Explanatory variable
Sources of variability
Experimental unit
Potential correlation
Analysis plan
Sample size

46. 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?

47. Original group
Condition 1
Condition 3
Condition 2

48. 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.

49. 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?

50. Plate 1: 10 patients, 14 controls
Estimate of difference in this plate
Plate 2: 14 patients, 10 controls
Estimate of difference in this plate
Plate 3: 12 patients, 12 controls
Estimate of difference in this plate
We can test if there is a different effect in each plate by investigating the interaction

51. 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?

52. 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

53. 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

54. 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

55. 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