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Chapter 6

### Introduction

Our clinical questions have correct answers that correspond to an underlying reality or truth. For instance, there is a true underlying magnitude of the impact of β-blockers on mortality in patients with heart failure, the impact of inhaled corticosteroids on exacerbations in patients with asthma, the impact of reamed vs unreamed nailing of tibial fractures, the prognosis of patients with hip osteoarthritis, and the diagnostic properties of a pregnancy test. Research studies attempt to estimate that underlying truth. Unfortunately, however, we will never know the exact truth. Studies may be flawed in their design or conduct and introduce systematic error (or bias). Even if a study could be perfectly designed and executed, the estimated treatment effect may miss the mark because of random error. The next section explains why.

### Random Error

Consider a perfectly balanced coin. Every time we flip the coin, the probability of it landing with its head up or tail up is equal—50%. Assume, however, that we as investigators do not know that the coin is perfectly balanced—in fact, we have no idea how well balanced it is, and we would like to find out. We can state our question formally: What is the true underlying probability of a resulting head or tail on any given coin flip? Our first experiment addressing this question is a series of 10 coin flips; the result: 8 heads and 2 tails. What are we to conclude? Taking our result at face value, we infer that the coin is very unbalanced (ie, biased in such a way that it yields heads more often than tails) and that the probability of heads on any given flip is 80%.

Few would be happy with this conclusion. The reason for our discomfort is that we know that the world is not constructed so that a perfectly balanced coin will always yield 5 heads and 5 tails in any given set of 10 coin flips. Rather, the result is subject to the play of chance, otherwise known as random error. Some of the time, 10 flips of a perfectly balanced coin will yield 8 heads. On occasion, 9 of 10 flips will turn up heads. On rare occasions, we will find heads on all 10 flips. Figure 6-1 shows the actual distribution of heads and tails in repeated series of coin flips.

What if the 10 coin flips yield 5 heads and 5 tails? Our awareness of the play of chance leaves us uncertain that the coin is balanced: a series of 10 coin flips of a very biased coin (a true probability of heads of 0.8, for instance) could, by chance, yield 5 heads and 5 tails.

Let us say that a funding agency, intrigued by the results of our first small experiment, provides us with resources to conduct a larger study. This time, we increase the sample size of our experiment markedly, conducting a series of ...

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