P.Mean: Sample chapter: The first three steps in selecting an appropriate sample size (created 2010-07-24).

As I mentioned in an earlier webpage, I am talking to some publishers about writing a second book. The working title is "Jumpstart Statistics: How to Restart Your Stalled Research Project." Here's a tentative chapter from that book. It is an early draft and I do not have all the references in place yet. It should be enough, though, to give you a sense of what I want to write about.

One of your most critical choices in designing a research study is selecting an appropriate sample size. A sample size that is either too small or too large will be wasteful of resources and will raise ethical concerns.

Here's a story (not true, but it should be) that I tell to people who come asking about sample size issues. A researcher is finishing up a six year, ten million dollar NIH grant and writes up in the final report "This is a new and innovative surgical procedure and we are 95% confident that the cure rate is somewhere between 3% and 96%."

This story illustrates that the heart and soul of all sample size calculations is economic. You want to insure that your research dollars are well spent, that you are getting something of value for your investment. Spending ten million dollars and only having a 3% to 96% confidence interval to show for it is a poor use of limited economic resources.

There's also an ethical component to most sample size calculations. People volunteer for research studies for three reasons. Some are in it for the money, and others are curious about the research process. The biggest reason, though, for many people to volunteer is that they want to help other people. They are hoping that the research will help and if your sample size is too small, your failure to produce a reasonable level of precision has betrayed their hopes.

Too large a sample size is also an ethical problem. Research volunteers often suffer during a clinical trial. They may experience pain. They may endure a risky procedure. They may forgo an appropriate medical treatment (if there is a placebo arm) or endure an inferior treatment (if there is an active control). You do all that you can to minimize these problems, of course, but most research requires some type of sacrifice to the volunteers. An excessive sample size, a sample size far beyond the needs of the research creates needless suffering among research volunteers.

You commonly justify the sample size of a study through a power calculation. Power is the probability that you will successfully reject the null hypothesis when there is a clinically important difference between your treatment and control group. Power an be defined for more complex data analyses, such as comparisons involving multiple treatment groups, or assessing the strength of association between two variables. Calculations in these settings are a bit more complex than the example discussed below, but the general steps are similar.

Selecting an appropriate sample size is one of the most important choices you will have to make in planning your research study. There are three basic steps in determining an appropriate sample size: specifying a research hypothesis, identifying the variation in your outcome, and determining the minimum clinically important difference

Step 1: Specify a research hypothesis.

Not all research can or should have a research hypothesis. But for those studies that do have a research hypothesis, this needs to be shared with your consulting statistician. This will help him/her identify the appropriate research design, and test statistic

As mentioned in the previous chapter, I like to use the PICO format (Patient, Intervention, Control, Outcome) described in Evidence-Based Medicine to help people formulate a good research hypothesis. The PICO format is also helpful in understanding the steps you need to select an appropriate sample size.

When I find the O (outcome) in the research hypothesis, I can begin to visualize the statistical approach to analyzing the data. You can't justify a sample size, of course, if you haven't settled on a specific statistical approach. If the outcome is continuous, for example, you might consider t-tests, ANOVA or linear regression models. If the outcome is categorical, you might plan for a logistic regression model instead.

The C (control) is also important in visualizing the statistical approach. Are controls selected through randomization? Are they matched one-to-one with patients in the treatment group? These will also help in deciding your statistical approach?

So why didn't I just be direct about it and ask you what statistical approach you were planning? Well that's something that you might not have considered just yet or maybe you were considering several different approaches. Maybe the thought of specifying a statistical approach terrifies you.

Most people are not afraid of telling me what their research hypothesis is. If they don't have a research hypothesis yet, I can usually work them through the process (see the previous chapter). Finally, all the details of the statistical approach don't need to be nailed down in order for you to start working on justifying the sample size. Often, just knowing the O (outcome) is enough to start making some progress.

Example: In a study, I helped with at Children's Mercy Hospital, the researchers were interested in the following hypothesis.

• P: infants with burns
• I: treated with an artificial skin protection membrane
• C: similar infants treated with an alternative membrane
• O: pain as measured by the Oucher scale

Additional outcome measures included healing time and total costs of treatment.

Step 2: Identify the variation in your outcome measure.

You've already done a literature review haven't you? If so, search through the papers in your review that used the same outcome measure that you are proposing in your study (the O in PICO). Ideally, the outcome measure will be examined in a group of patients that is close to the types of patients that you are studying (the P in PICO, or possibly the C in PICO). This is not always easy, and you will sometimes be forced to use a study where the patients are quite different from your patients. Don't fret too much about this, but make a good faith effort to find the most representative population that you can.

Some clients will raise an objection here and say that their research is unique, so it is impossible to find a comparable paper. It is true that most research is unique (otherwise it wouldn't be research). But what these people are worried about is that their intervention (the I in PICO) is unique. In these situations, the remainder of the hypothesis is usually quite mundane: the patients, the comparison group, and the outcome (P, C, and O in PICO) are all well studied. If you find a study where the P, C, and O match reasonably well, but the I doesn't, then you are probably going to get a good estimate of variation.

If there are major dissimilarities because this patient population (P)  is very different than any previously studied patient population, or because the outcome measure (O) is newly developed by the researcher, then perhaps a pilot study would be needed to establish a reasonable estimate of variation.

Sometimes you can infer a standard deviation through general principles. If a variable is constrained to be between 0 and 100, it would be impossible, for example, for the standard deviation to be five thousand. There are approximate formulas relating the range of a distribution to the standard deviation. Divide the range by four or six to get an approximate standard deviation. There are also formulas that allow you calculate a standard deviation from a coefficient of variation, a confidence interval, or a standard error. Just about any measure of variation can be converted into a standard deviation.

If your outcome measure is a proportion, then the variation is related to the estimated proportion. Similarly, the variation in a count variable is related to the mean of the counts. Find a paper that establishes a proportion or average count in a control group similar to your control group and any competent statistician will be able to get an estimate of variation.

In some situations, the amount of variation in a proportion or count is larger than would be expected by the statistical distributions (binomial and Poisson)  traditionally associated with these measures. Still, a calculation based on binomial or Poisson assumptions is a reasonable starting point for further calculations.

If you have multiple outcome measures, pick the one that is most critical. If you are indecisive, pick two or three. But don't try to justify your sample size for ten or twenty different outcome measures (but do adjust for multiple comparisons). There's a general presumption in the research community that if you have an adequate sample size for one of your outcome measures that the sample size is probably reasonable for any other closely related outcome measure. In my experience, this is generally true, but do include a separate sample size justification for an outcome that is substantially different in nature. So, for example, if most of your outcome measures involve quality of life measures but one of them is mortality, then perform a separate sample size justification for mortality because it is discrete rather than continuous and because it uses a substantially different form of data analysis.

Example: The researchers examining infants with burns could easily find a standard deviation for the Oucher scale, 1.5, from previous literature. This number seemed a bit high to me, because the range of the Oucher scale they were using was 1 to 5. Typically, the standard deviation is 1/4 to 1/6 the range, so I would have been happier with a standard deviation of 0.67 to 1.0. But 1.5 wasn't outrageously too large.

Healing time is a more difficult endpoint to assess. Medical textbooks cite that the healing time for second degree burns has a range of 4 days (minimum 10, maximum 14). A study of healing times for a glove made from one of the skin barriers showed a healing time range of 6 (minimum 2 and maximum 8 days). Note that the average healing time is quite different from the two sources, with the minimum healing time in the first study being 2 days longer than the maximum healing time in the second study. But the ranges are quite similar, and this is reassuring.

Since the standard deviation is approximately 1/4 to 1/6 of the range, it's possible that the standard deviation for healing time could be as small as 0.5 or as large as 1.5.

For one type of skin barrier, a study of costs showed a range of \$4.00 (\$5.50 to \$9.50). Thus, a standard deviation of 0.67 to 1 would be reasonable.

Step 3: Determine the minimum clinically important difference

Determining the minimum clincally (or scientifically) important difference is the most difficult step so far, but you need to do this if you want any hope of determining an appropriate sample size. The minimum clinically significant difference (MCID) is the boundary between two important regions. The first region is the land of yawns. This region is all the differences so small that all your colleagues say "so what?" These are trivial differences, differences that no one would adopt the new intervention on the basis of such a meager change. The second region is the land of wow. This region is all the differences large enough that people sit up and take notice. These are large changes, changes large enough to justify changing how you might act.

Establishing the MCID is a tricky task, but it is something that should be done prior to any research study. You might start by asking yourself "How much of an improvement would I have to see before I would adopt a new treatment?" or "How severe would the side effects have to be before I would abanadon a treatment."

For binary outcomes, the choice is not too difficult in theory. Suppose that an intervention "costs" X dollars in the sense that it produces that much pain, discomfort, and inconvenience, in addition to any direct monetary costs. Suppose the value of a cure is kX where k is a number greater than 1. A number less than 1, of course, means that even if you could cure everyone, the costs outweigh the benefits of the cure.

For k>1, the minimum clinically significant difference in proportions is 1/k. So if the cure is 10 times more valuable than the costs, then you need to show at least a 10% better cure rate (in absolute terms) than no treatment or the current standard of treatment. Otherwise, the cure is worse than the disease.

It helps to visualize this with certain types of alternative medicine. If your treatment is aromatherapy, there is almost no cost involved, so even a very slight probability of improvement might be worth it. But Gerson therapy, which involves, among other things, coffee enemas, is a different story. An enema is reasonably safe, but is not totally risk free. And it involves a substantially greater level of inconvenience than aromatherapy. So you'd only adopt Gerson therapy if it helped a substantial fraction of patients. Exactly how many depends on the dollar value that you place on having to endure a coffee enema, a task that I will leave for someone else to quantify.

For continuous variables, the minimum clinically significant difference could be defined as above. Define a threshold that represents "better" versus "not better" and then try to shift the entire distribution so that the fraction "better" under the new treatment is at least 1/k. There have also been efforts to elucidate, through experiments, interviews, and other approaches, what the average person considers an important shift to be. For the visual analog scale of pain, for example, a shift of at least 15 mm is considered the smallest value that is noticeable to the average patient.

There are some informal rules of thumb. Generally, a change that represents a doubling of a halving is pretty important. So if you cut the length of stay in a hospital in half, from 4 days on average to 2, that's pretty big. A side effect that occurs 8% of the time rather than 4% of the time is pretty large. Rules of thumb are not perfect, though. A 25% shortening in length of stay, from 4 days on average to 3 would probably also be clinically important. And, depending on the type of side effect, we might not get too worried unless we saw a tripling of side effect rates, from 4% to 12%. So use this rule of thumb to establish a starting point for further discussion.

If you're totally stumped, try talking about what's clinically important with some of your colleagues. In a pinch, you can also look at the size of improvements for other successful treatments. This is an example, though, of the lemming school of research (If all you your friends jumped off a cliff would you jump off also?). As a last resort, you can try inverting the calculations. Specify the largest sample size that you could collect without killing yourself in the process and then back calculate what the minimum clinically important difference might be.

I often get told "you tell me what the minimum clinically important difference is." I can't do it, because of that adjective "clinically." I do not exercise good clinical judgment, as I do not work in a clinic. I'd also have trouble if it were the minimum scientifically important difference, as my scientific judgment stopped developing when I skipped all those high school biology labs involving dissection (it was more a weak stomach than a strong ethical objection). I'm sometimes willing to venture an opinion, but mostly just to start the discussion and get a reaction. If pressed, I will often state a number that I know they will say is way too big or way too small. Once I get them to commit to such a judgment, then it is only a few short steps to arriving at a reasonable number for the MCID.

Example: The researchers said that a shift of 1 unit in the Oucher scale was the smallest value that would be important from a clinical perspective. That seemed reasonable to me. It would be hard to argue that a change much smaller than the finest unit of measurement for the scale would be important from a clinical perspective. An average shift of one day in healing time was also considered clinically significant. Finally, a difference in average costs of \$0.50 would be considered clinically significant.

Here's an example of how the sample size calculations worked out, using a sample size calculation package, PiFace, that is freely available on the web. The steps shown here would be similar if you used a different program.

With the Oucher scale, a sample of 36 patients per group would provide 80% power for detecting a difference of 1 unit, assuming the standard deviation of the Oucher is 1.5 units. This was well within the researchers budget, so this was welcome news. Also reassuring was that I had thought that the standard deviation was a bit big. You can check easily that a smaller standard deviation would leave to a smaller sample size.

For the healing time, a standard deviation of 0.5 leads to a ridiculously small sample size (5 or 6 per group).

A standard deviation of 1.5 leads to the exact same sample size, which is not surprising.

For total costs, a standard deviation of 0.67 and a MCID of \$0.50 leads to a sample size of 29 per group. That's reassuring, but the standard deviation could possibly be as large as 1.0.

In this case, the sample size would be 64 per group, which would bust the budget. I asked if they could live with a study that could detect a \$1.00 difference in costs. That seemed reasonable to them.

A study that would try to detect a difference of \$1.00 would need 17 patients per group, assuming the standard deviation that was also \$1.00. Looking at all the calculations, it appears that a sample of 36 patients per group is a reasonable choice. It fits within the research budget. It provides 80% for detecting a shift of one unit in the Oucher scale. The same sample size provides 80% power for healing time using the worst case scenario of a standard deviation of 1.5. It's not quite adequate for detecting a shift of \$0.50 in costs, depending on what the standard deviation is, but more than adequate for detecting a shift of \$1.00 in costs.

The fly in the ointment: research without a research hypothesis.

What do you do if you don't have a research hypothesis? This is a situation where you need to discuss things in more detail with your statistical consultant.

In some research studies, the goal is exploratory. You don't have a formal hypothesis at the start of the study, but rather you are hoping that the data you collect will generate hypotheses for future studies. The path to selecting a sample size in these settings is quite different. Often you want to establish that the confidence intervals for some of the key descriptive statistics in these studies has a reasonable amount of precision.

Pilot studies also do not normally have a research hypothesis. It is tricky to determine the appropriate sample size for a pilot study. This will be dealt with in the next chapter.

If your study involves assessing validity or reliability, then you could force your research goal into a formal hypothesis, but I don't recommend it. Most efforts to establish reliability and/or validitiy involve estimation of a correlation (for example, a correlation between two different observers or a correlation of your measure and a gold standard). If this is the case, simply calculate how wide you would expect your confidence interval for the correlation to be. Specify a sample size and a target for your correlation. Then your sample size is adequate if the confidence interval is sufficiently narrow.