# Quick sample size calculations

## 2001-01-01

Dear Professor Mean, I’m reading a research paper. I suspect that the sample size is way too small. I don’t like the findings of the study anyway, so I’m hoping that you will help me discredit this study. Is there a quick sample size calculation that I can use?

– Cynical Chris

Dear Cynical,

That’s not a paper that I wrote, is it? I was very tired, and the dog ate my homework, and the Year 2000 bug just struck (my computer procrastinates worse than I do).

The simplest way to see if your sample size is too small is to look at the confidence intervals. Look for a confidence interval so wide that you can drive a truck through it. Look for an interval that contains both clinically large and clinically insignificant values.

What? The paper doesn’t have any confidence intervals? That’s terrible. Many journal editors require the use of confidence intervals instead of p-values in their publications. Write a letter to the editor and complain.

A second way to assess the sample size is to do a quick power calculation. If your research paper compares two groups (e.g., a treatment group and a control group), then you can use the rule of 50 or the rule 16. The first rule applies for categorical outcomes; the second applies for continuous outcomes.

### Assumptions

These rules presume that you are comparing two groups. Some examples would be comparing a control group to a treatment group, a standard therapy to a new therapy, or an exposed group to an unexposed group.

They also presume that you are performing

• a two-sided test,
• the alpha level is .05, and
• the power is 80%.

Finally, keep in mind that both rules are approximations, which means that you need to consult with a statistician for an official sample size. Still, these quick rules help you get a feel for whether you need dozens versus hundreds versus thousands of research subjects. These quick rules are also helpful when you are reading someone else’s research and you want to get a rough idea about what an appropriate sample size might be.

By the way, I have to give credit where credit is due. Both of these rules came from a website with the title “STRUTS: Statistical Rules of Thumb” which is no longer on the web. There is a book out with the same title (see further reading) as well as a new website. Unfortunately, the details about the rule of 50 seem to have disappeared in the update.

### The rule of 50

The rule of 50 applies when your outcome measure is a discrete event such as morbidity or mortality. The rule works well if that event is relatively rare. If you want enough subjects to be able to detect a halving of risk from your control group, be sure to collect enough data so that you will have at least 50 events in your control group. Then sample the same number of subjects in your treatment group.

For example, patients using a control medication will have a risk over five years for a heart attack of roughly 8%. You want to try a new drug to see if it can reduce the risk to 4%. You would need a large enough sample in each group to ensure that at least 50 patients in the control group will have a heart attack. It seems a bit morbid to plan for such a thing. Still, it makes sense. If hardly anyone in either group has a heart attack, you will have a hard time deciding whether one medication is better than another.

A control group of 625 subjects would suffice (8% of 625 is 50). With the same number of treated subjects, you would have a total of 1,250 patients in your study. This is just an approximation. The sample size that provides 80% power for detecting a halving of risk is actually 553 per group, not 625.

### The rule of 16

The rule of 16 applies when your outcome measure is continuous, such as birth weight. For the continuous outcome, you need to define how much of a difference you would consider to be clinically significant, and then compute the ratio of this clinically significant difference to the standard deviation of the outcome measure. This ratio is called the effect size or the standardized effect size.Your sample size per group is 16 divided by the square of the effect size.

For example, you are measuring the duration of breast feeding in a sample of newborn infants. Let’s presume that any intervention that can increase the average duration of breast feeding by at least two weeks is considered clinically significant. Furthermore, in this population, you expect the standard deviation of breast feeding duration to be 10 weeks. The effect size is 0.2, and the required sample size per group is 400 (=16/0.04).

Again, this is just an approximation. The sample size that provides 80% for detecting a two week shift in breast feeding duration is actually 394, not 400.

Disclaimer: I am not a medical expert, so I cannot say with any authority whether two weeks of additional breast feeding is clinically significant. All my knowledge of breast feeding comes from experiences more than forty years ago.

### Example of a hypothetical research study (using the rule of 50)

An article by Schwartz et al proposes an interesting scenario for a research study (N Engl J Med. 1998;338:1709-1714). These authors noticed an association between prolonged QT interval and Sudden Infant Death Syndrome. In the discussion of these findings, the authors raise the possibility of screening all newborn infants using electrocardiography and the placing those infants with prolonged QT intervals on a beta blocker. The authors discuss the complexity of the cost benefit issues, which is beyond the scope of this web page. It is interesting, however, to speculate on how to test whether beta blockers would be effective as a therapy to prevent SIDS in those infants with long QT intervals.

The paper provides much interesting data to help you calculate an appropriate sample size for this study. The risk of SIDS in infants with prolonged QT intervals is 1.5%. Suppose that a beta blocker could cut this risk in half (to 0.75%). What sample size would you have to collect in order to have adequate power?

The rule of 50 tells us that we would need 50 SIDS events in the placebo arm of the trial. At a rate of 1.5% that translates into recruiting 3,333 infants with prolonged QT interval for the placebo arm. You would recruit a similar number of infants for the beta-blocker arm of the study.

Not every infant, however, will have a prolonged QT interval. The cutoff used in this paper for a prolonged interval represented the 97.5 percentile. So only 2.5% of the infants screened could qualify to be in the study. In order to recruit 6,666 infants who qualify for the study, you would have to screen 266,640 normal infants.

Disclaimer: I am not a medical expert, so I cannot comment intelligently on cost benefit issues, the amount of improvement that a beta blocker might have, and other related issues. This example should only serve as an illustration of how difficult it would be to prospectively examine a therapy in a group of children where both the adverse event and the qualifying condition for therapy are both rare.

### Assessing the sample size in an existing publication (using the rule of 16)

An article by Adkinson et al presents a negative finding on the use of immunotherapy for asthma in allergic children (N Engl J Med. 1997;336:324-331). You may want to examine whether this negative finding is due to an inadequate sample size.

One of the outcome measures is change in medication score, which has a standard deviation of roughly 2.0. Suppose you felt that a difference of one unit on the medication score represented a clinically significant effect. This represents a standardized effect size of 0.5 +1.0/2.0). Using the rule of 16, you would want 64 (=16/0.25) subjects in each group to have adequate power.

Another outcome measure is change in symptom scores, which has a standard deviation of roughly 0.4. If you believed that a 0.25 unit change in the symptom score, this would represent a standardized effect size of 0.625. Using the rule of 16, you would need 41 patients in each group.

The actual study had 60 and 61 patients, so the sample size appears adequate (presuming that the differences of 1.0 and 0.25 are reasonable values).

Disclaimer: I am not a medical expert, so I can only speculate on what a clinically significant difference would be. Also, there are some tricky issues in the paper involving compliance and the quality of care received. My discussion of this example may oversimplify some of the issues, but I hope you still find the example interesting and informative.

### Mathematical details, rule of 50

The rule of 50 starts with the a standard formula for sample size.

$n_1=\frac{\Big(z_{1-\alpha / 2} \sqrt{2 \bar p \bar q} + z_{1-\beta}\sqrt{p_1q_1 + p_2q_2}\Big)^2}{(p_1-p_2)^2}$

If the event in question is rare, then

$q_1 \approx 1;\ q_2 \approx 1;\ \bar q \approx 1$

Suppose we want to detect a 50% decline from group 1 to group 2. This implies

$p_2 = 0.5 p_1;\ \bar p = 0.75 p_1$

Let’s also assume that k=1. Set $\alpha=0.05$ and $\beta=0.2$. Then the sample size formula simplifies to

$n_1 \approx \frac{\big(1.96 \sqrt{2(0.75 p_1)} + 0.84 \sqrt{p_1 + 0.5 p_1}\big)^2}{(p_1-0.5 p_1)^2}$

or

$n_1 p_1 \approx 47.04$

Round this up to 50. Then

$n_2 p_2 \approx 25$

### Mathematical details, rule of 16

The formula for sample size for a two-sample t-test is

$n_1=\frac{(z_{1-\alpha/2} + z_{1-\beta})^2 2 \sigma^2}{(\mu_1 - \mu_2)^2}$

where D is the difference in population means. Set $\alpha=0.05$ and $\beta=0.2$ as before to get

$n_1 = \frac{(1.96 + 0.84)^2 2 \sigma^2}{(\mu_1 - \mu_2)^2}$

or

$n_1 = \frac{15.68}{\frac{(\mu_1 - \mu_2)^2}{\sigma^2}}$

The denominator is the effect size squared. Round the numerator up to 16.