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Chapter 7: Problem 116

A study was conducted in Sweden to relate the age at surgery for undescended testis to the subsequent risk of testicular cancer [10]. Twelve events were reported in 22,884 person-years of follow-up among men who were \(13-15\) years at age of surgery. Provide a 95\% Cl for the number of events among men who were \(13-15\) years at age of surgery.

Short Answer

Expert verified
The 95% CI for the number of events is approximately 5.54 to 18.44 events.

Step by step solution

01

Determine the Point Estimate

First, calculate the incidence rate. The point estimate for the incidence rate is the number of events (12) divided by the total number of person-years (22,884). \[\text{Point Estimate} = \frac{12}{22,884} = 0.000524\]
02

Calculate the Variance

The variance of the incidence rate can be calculated as follows:\[\text{Variance} = \frac{12}{22,884^2}\]
03

Determine Standard Error

Calculate the standard error (SE) using the square root of the variance from Step 2.\[SE = \sqrt{\frac{12}{22,884^2}} = \sqrt{\frac{12}{523,627,456}} \approx 0.000144\]
04

Find the 95% Confidence Interval

Use the standard error and the point estimate to find the 95% CI. For the normal distribution, the critical z-value for 95% confidence is approximately 1.96.\[CI = \text{Point Estimate} \pm (1.96 \times SE)\]\[CI = 0.000524 \pm (1.96 \times 0.000144)\]Calculate the margin of error:\[MOE = 1.96 \times 0.000144 \approx 0.000282\]So, the confidence interval is:\[CI = (0.000524 - 0.000282, 0.000524 + 0.000282)\]\[CI = (0.000242, 0.000806)\]
05

Convert to Number of Events

To convert the confidence interval of the incidence rate into a confidence interval for the number of events, multiply each end of the interval by the total person-years (22,884).\[CI = (0.000242 \times 22,884, 0.000806 \times 22,884)\]This results in:\[CI = (5.54, 18.44)\]So the 95% confidence interval for the number of events is approximately between 5.54 and 18.44 events.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Incidence Rate
The incidence rate is a measure used in epidemiology to describe the frequency at which a certain event, such as a disease or health outcome, occurs in a specific population during a specified period of time. In the original exercise, the event of interest was the occurrence of testicular cancer after surgery for undescended testis.

To calculate this rate, you divide the number of events by the number of person-years investigated, which here was 12 events over 22,884 person-years. This gives us a point estimate or the observed rate.
  • **Steps to calculate**
  • Count the total number of observed events.
  • Divide by the amount of person-years.
  • Example: 12 / 22,884 = 0.000524
This calculation provides a baseline to understand how often testicular cancer might develop in the studied group, helping in further statistical analyses and interpretations like estimating confidence intervals.
Variance
The variance is a statistic that measures the spread or dispersion of a set of numbers. In the context of incidence rates, it helps to understand the variability of the observed rate, giving us an idea of how much the rate can fluctuate.

For our incidence rate of testicular cancer, we calculated the variance by taking the number of events and dividing it by the square of the total person-years.
The variance formula used was:
  • **Variance** = (Number of Events / (Person-Years)^2)
  • Example: 12 / 22,8842
A smaller variance indicates that the observed incidence rate is likely to be closer to the true population rate.
Standard Error
The standard error (SE) tells us how accurate our estimate of the incidence rate is likely to be. It gives us an idea of the likely "sampling error."

To find the standard error, we take the square root of the variance calculated earlier. This involves using the variance formula and simplifying it through a square root:
  • **Formula**: SE = \( \sqrt{\text{Variance}} \)
  • Example: \( \sqrt{\frac{12}{22,884^2}} \approx 0.000144 \)
The smaller the standard error, the more confident we can be that our incidence rate reflects the real population situation.By using the standard error, we prepare ourselves to create confidence intervals which tell us the extent within which we expect to find the true incidence rate.
Person-Years
Person-years are a useful unit of measurement that accounts for both the number of individuals in a study and the amount of time each individual spends in the study.
This measurement is crucial in calculating incidence rates and for understanding the exposure time individuals have concerning the event in question.
  • **Calculating person-years**
  • Track how long each participant is observed.
  • Add these times together for all individuals.
  • Example: If 1 person is observed for 1 year, that equals 1 person-year.
  • If 22,884 individuals are observed for one year each, that equals 22,884 person-years.
Person-years help to adjust for varying follow-up times across participants, offering a weighted account of the exposure to risk over the collected data, and is essential for accurately measuring incidence rates in cohort studies.

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Most popular questions from this chapter

An investigator wants to test a new eye drop that is supposed to prevent ocular itching during allergy season. To study the drug, she uses a contralateral design whereby for each participant one eye is randomized (using a random-number table) to get active drug (A) while the other eye gets placebo (P). The participants use the eye drops three times a day for a 1-week period and then report their degree of itching in each eye on a 4-point scale \((1=\text { none }, 2=\text { mild, } 3=\text { moderate, } 4=\) severe) without knowing which eye drop is used in which eye. Ten participants are randomized into the study. What is the principal advantage of the contra-lateral design?

Another type of outcome reported was fetal deaths. Suppose the investigators expect, given general population pregnancy statistics at the time of the survey, that \(1.5 \%\) of pregnancies will result in a fetal death. They found that among 586 pregnancies where an exposed worker was the father, 11 resulted in a fetal death. Provide a \(95 \%\) Cl for the underlying fetal death rate among offspring of exposed men. Given the \(\mathrm{Cl}\), how do you interpret the results of the study?

A screening program for neuroblastoma (a type of cancer) was undertaken in Germany among children born between November \(1,1993,\) and June \(30,2000,\) who were 9 to 18 months of age between May 1995 and April 2000 [6]. A total of 1,475,773 children participated in the screening program, of whom 204 were diagnosed between 12 and 60 months of age. The researchers expected the incidence rate of neuroblastoma to be 7.3 per 100,000 children during this period in the absence of screening. Provide a \(95 \%\) Cl for the incidence rate of neuroblastoma in the screened population. Express the \(95 \%\) Cl as \(\left(p_{1}, p_{2}\right),\) where \(p_{1}\) and \(p_{2}\) are in the units of number of cases per 100,000 children. Is \(p_{0}(7.3 \text { cases per } 100,000\) children) in this interval?

Osteoporosis is an important cause of morbidity in middle-aged and elderly women. Several drugs are currently used to prevent fractures in postmenopausal women. Suppose the incidence rate of fractures over a 4 -year period is known to be \(5 \%\) among untreated postmenopausal women with no previous fractures. A pilot study conducted among 100 women without previous fractures aims to determine whether a new drug can prevent fractures. It is found that two of the women have developed fractures over a 4-year period. Suppose the new drug is hypothesized to yield a fracture rate of \(2.5 \%\) over a 4 -year period. How many subjects need to be studied to have an \(80 \%\) chance of detecting a significant difference between the incidence rate of fractures in treated women and the incidence rate of fractures in untreated women (assumed to be \(5 \%\) from Problem 7.105 )?

A study was conducted in Sweden to relate the age at surgery for undescended testis to the subsequent risk of testicular cancer [10]. Twelve events were reported in 22,884 person-years of follow-up among men who were \(13-15\) years at age of surgery. What is the estimated incidence rate of testicular cancer among this group of men? Express the rate per 100,000 person-years.
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