# Chapter 12.3: What Determines the Width of the Confidence Interval?

Clinicians sometimes equate the size of a study or the number of participants in a study with the width of the *confidence interval* (CI) and thus with its precision. This chapter deals with issues of precision and the resulting CIs associated with *treatment effects* on *dichotomous* (yes/no) *outcomes,* such as death, stroke, or myocardial infarction. As it turns out, for the relative measures of *effect sizes* (eg, *relative risk* [RR] or *relative risk reduction* [RRR]), the number of patients in a study is a secondary determinant of the width of a CI, with the primary determinant being the absolute number of events*.*

Consider 2 hypothetical studies. Both show an RRR of 33% of some adverse outcome with an intervention A vs control. Study 1 has enrolled 100 patients in each of the experimental and *control groups*, and study 2 has enrolled 1000 patients in each group. Which of the 2 studies will generate a more precise estimate of treatment effect, represented by a narrower CI? The apparently obvious answer is study 2, with its sample size an order of magnitude larger than that of study 1.

Suppose, however, that study 2—the study with the larger sample size—generated its RRR of 33% on the basis of 2 outcomes among 1000 people receiving intervention A vs 3 outcomes among 1000 people in the control group. Study 1 produced an RRR of 33% on the basis of 20 outcomes among 100 people receiving intervention A vs 30 outcomes in the control group.

Which RRR of 33% do you trust more? Which one is more precise? Which has the narrower associated CI? As shown in Table 12.3-1, study 1 yields the narrower CI because it is not the number of participants but rather the number of outcome events that matters most.

**Sample Size, Event Rate, and the Width of the Confidence Interval**

**Sample Size, Event Rate, and the Width of the Confidence Interval**

Study No. | No. of Events in Control Group | Total No. in Control Group | No. of Events in Experimental Group | Total No. in Experimental Group | RRR (95% CI) |
---|---|---|---|---|---|

1 | 30 | 100 | 20 | 100 | 33% (–8% to 59%) |

2 | 3 | 1000 | 2 | 1000 | 33% (–233% to 87%) |

In the following figures, we explore the association among sample size, number of events, and the precision of the study results by calculating CIs around the RRR from a set of hypothetical studies. The starting point is 100 patients per group, with 8 patients having an event ...

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