136 PART 3 Getting Down and Dirty with Data

Using this method, you first calculate the SE of the proportion using this formula:

SE

p

p

N

(

) /

1

where p stands for proportion. Next, you use the normal-based

formulas in the earlier section “Before you begin: Formulas for confidence limits

in large samples” to calculate the ME and the confidence limits.

Using the numbers from the sample of 100 adult diabetics (of whom 70 have their

diabetes under control), you have p

0 7. andN

100. Using those numbers, the SE

for the proportion is 0 7 1

0 7

100

. (

. ) /

or 0.046. From Table 10-1, k is 1.96 for 95

percent confidence limits. So for the confidence limits, CLL

0 7

1 96

0 046

.

.

.

and

CLU

0 7

1 96

0 046

.

.

.

. If you calculate these out, you get a 95 percent CI of 0.61 to

0.79 (around the original estimate of 0.7). To express these fractions as percent-

ages, you report your result this way: “The percentage of adult diabetics in the

sample whose diabetes was under control was 70 percent (95 percent CI = 61 – 79

percent).”

The confidence interval around

an event count or rate

Suppose that you learned that at a large hospital, there were 36 incidents of

patients having a serious fall resulting in injury in the last three months. If that’s

the only incident report data you have to go on, then your best estimate of the

monthly serious fall rate is simply the observed count (N), divided by the length

of time (T) during which the N counts were observed: 36/3, or 12.0 serious falls per

month. What is the 95 percent CI around that estimate?

There are many approximate formulas for the CIs around an observed event count

or rate, which is also called a Poisson CI. The simplest method to calculate a Pois-

son CI is based on approximating the Poisson distribution by a normal distribu-

tion (see Chapter 24). It should be used only when N is large (at least 50). You first

calculate the SE of the event rate using this formula: SE

N

T

/

. Next, you use

the normal-based formulas in the earlier section “Before you begin: Formulas for

confidence limits in large samples” to calculate the lower and upper confidence

limits.

Using the numbers from hospital falls example, N

36 and T=3, so the SE for the

event rate is 36

3

/ , which is the same as the square root of 2, which is 1.41.

According to Table 10-1, k is 1.96 for 95 percent CLs. So CLL = 12.0 – 1.96 × 1.41 and

CLU = 12.0 + 1.96 × 1.41, which works out to 95 percent confidence limits of 9.24 and

14.76. You report your result this way: “The serious fall rate was 12.0 (95 percent

CI = 9.24 – 14.76) per month.”

To calculate the CI around the event count itself, you estimate the SE of the count

N as SE

N , then calculate the CI around the observed count using the formulas