Chapter 6 EXERCISES Last update: 11/15/2008      

6.1 - 6.9 See text pp. 136 - 139.

6.10 Simulating rates. This activity demonstrates rates and population dynamics in an open and closed population.

(A) Stationary open population. The instructor puts these four signs in the corners of the room" "healthy", "ill", "dead," and "not yet born." Half the class congregates in the "healthy" corner while the other half congregates in the "not yet born" corner. One person starts in the ill corner. The instructor counts off years. Each year, on June 30, one person becomes ill and one is born. The ill case moves to the ill corner and newborn to the healthy corner. Ill cases live for one year after which they expire and go to the "dead" corner. On the New Years day each year, data are tallied for the prior year in this table:

(B) Cohort. All members of the class congregate in the "healthy corner." The instructor counts off years. Each year, on June 30, one person becomes ill and moves to the "ill corner." (Comment: This scenario would represent an increasing risk over time since the number of cases remains constant but the denominator is decreasing.) Such cases stay in the ill corner for 2 years, after which they expire and are relegated to the "dead corner." At the beginning of each year, the instructor tallies counts and rates using a table similar to the one shown above. 

6.11 Mortality and life expectancy. This schematics illustrates the survival experience of two cohorts. Each line represents an individual. The symbol D represents death or disease incidence.

(A) Calculate the mortality rate in cohort 1. Calculate the mortality rate in cohort 2. 
(B) Calculate life expectancy in each of the cohorts. 
(C) What is the mathematical relation between the mortality rate and life expectancy? (p. 130, para 1)

Cohort 1 person 2 |-----------------D           | person 1 |-----------------D          |          |-----|-----|-----|-----|          0     25    50    75    100                   years

 

Cohort 2 person 2 |-----------------D          | person 1 |-----------------------D          |          |-----|-----|-----|-----|          0     25    50    75    100                    years

6.12 Person-time and % surviving.

(A) Determine the incidence  rate. [See exercise 6.11 for an explanation of the symbols.]

person 4 |-----D          |      person 3 |-----------D          |             person 2 |-----------------------          | person 1 |-----------------------          |          |-----|-----|-----|-----|          0     1     2     3     4             follow-up (years)

(B) The drawing has been modified to emulate a Kaplan-Meier survival curve. Seventy-five percent of the cohort survived one year. What percentage survived two years?  What percentage survived four years?  

person 4 |-----D          |     | person 3 |     +-----D          |           |  person 2 |           +-----------          | person 1 |          |          |-----|-----|-----|-----|          0     1     2     3     4            follow-up (years)

(C) The area under the "curve" of the Kaplan-Meier curve is equal to the person-time in the cohort. Area under the above Kaplan-Meier curve can be derived by geometry. During the interval 0 -1, we have a rectangle with base 1 and height 4 and, thus, an area 4 × 1 = 4. During the second year we have a rectangle with a base of 1 (from 1 to 2) and height of 3 and thus an areas of 1 × 3 =3. How many person-years are there between years 2 and 4 of follow-up? 

6.13 Breast cancer. A study starts with 10,000 women. Of these, 500 had already experienced breast cancer. The remaining 9,500 are followed for five-years. Two-hundred fifty breast cancer incidents occurred during the 5 years of following. 

(A) What is the five-year cumulative incidence (proportion) of breast cancer? Report the incidence per 1000 people.
(B) What is the incidence rate of breast cancer? No life-table adjustment of the denominator is necessary. Report the rate "per 1000 person-years".

6.14 Cohort study of CHD. One-thousand people are recruited for a study. Eight-hundred-fifty agree to participate. Fifty have coronary heart disease upon examination. Over the next ten years, 100 develop coronary heart disease. 

(A) What is the 10-year incidence proportion (average risk) of coronary heart disease in this cohort? 
(B) What is  the rate?

6.15 Rates of driving errors. "Every two miles, the average driver makes four hundred observations, forty decisions, and one mistake. Once every five hundred miles, one of those mistakes leads to a near collision, and once every sixty-one thousand miles one of those mistakes leads to a crash." (New Yorker, Gladwell, June 11, 2001, pp. 50-61.)

(A) What is the rate of mistakes per mile?
(B) What proportion of observations are mistaken?
(C) Why is the answer to Part A a rate and the answer to Part B a proportion?
(D) What is the odds of near collisions to actual collisions?

6.16 Open population. Using the schematic in the figure below, determine the

(A) Total number of people in the population (N)
(B) Average number of people in the population at any one point (Ñ). [This is equal to the total person-time divided by the total time of observation.]
(C) Mortality rate

Symbols: o = entry or exit from population, D = death

o---------------D
                o----------------o
        o-------D
            o-----------D
o-----------------------------------o    
                o-----------------------D
|---|---|---|---|---|---|---|---|---|---|
0   1   2   3   4   5   6   7   8   9   10
                 Year   

6.17 Another cohort. A cohort of 150 people begins with 10 cases. The cohort is followed for five years, during which time 16 new cases arise. 

(A) What is the prevalence of disease at the start of the study? 
(B) Assume all cases survive. What is the prevalence at the end of the study? 
(C) What is the incidence proportion over the interval? 
(D) What is the incidence rate over the interval? 

6.18 Just like Exercise 6.1. The figure below represents a cohort of 7 individuals observed for a year. In this figure, periods of disease are represented with a D and dashes represents disease-free periods. There are  no losses to follow-up in the cohort, and we assume that recovery will confer life-long immunity.  

(A) What is the prevalence on Jan 1? 
(B) What is the prevalence on Dec 31? 
(C) Estimate the one-year risk of developing illness.

Person 1 -|---------------------|--
Person 2 D|DDDDDDD--------------|--
Person 3 -|-----------------DDDD|DD
Person 4 -|---------------------|--
Person 5 -----DDDDDDDDDDDDDDDDDD|DD
Person 6 -|---------------------|--
Person 7 -|---------------------|--
          |                     |
         Jan1                 Dec31

6.19 Just like Exercise 6.3.  A population demonstrates the following vital statistics: 

Total midyear population                     25,000
Population size, 65-years of age or older     2,500
Number of live births                           300
Total deaths (all cause)                        250
Deaths in under 1-year olds                       3
Deaths in persons 65 and over                    75

(A) Calculate the crude birth rate per 1,000.
(B) Calculate the crude mortality rate per 1,000.
(C) Calculate the infant mortality rate per 1000.
(D) Calculate the age-specific mortality rate for those over 65 (per 1000).

6.20 In the schematic below, dashed lines (--) represent disease-free person-time and D indicates a disease onset. There are three (3) people in the cohort, labeled A, B, and C. 

A|--------D
B|------------------------
C|------------------------
 |---|----|----|----|----|
 0   1    2    3    4    5
          Year

(A) What is the 5-year risk of disease in the cohort?
(B) What is the rate of the disease in the cohort? 

6.21 Twice the prevalence with the same risk. Suppose the prevalence of disease in population A is 20 per 100,000. The prevalence in population B is 10 per 100,000. The groups have identical age distributions. Can we conclude that population A has twice the risk compared to group B? Explain.