The risk of an outcome is the proportion of all at risk individuals who develop disease over some time period, in this case 0.17 (3/18).
Risk is the most basic and most immediate of measurements of disease. It specifies the probability that a given individual will develop disease. It is calculated by dividing the number of new cases of disease by the total number of individuals who are at risk of developing the disease. In this example, out of 20 students 2 contract influenza in the first week. As such, the risk of influenza in the first week is 2/20 or 0.1. In the second week 3 more students get infected, but the denominator is now different. Of the original 20, only 18 students were capable of getting infected in the second week, as two students were already infected with influenza. Thus the risk is 3/18 or 0.17. In the same way, the risk of influenza in the third week is 5/15 or 0.33.
Gordis clarifies what characteristics constrain the values which are in the numerator and denominator of risk calculations. The numerator only includes NEW cases of an event. The event can be whatever the investigator wishes it to be--be it death, disease onset, or successful treatment--so long as it is a new incident of the event. In the case of death this is self evident, but in the case of disease onset, individuals who already have the disease do not count towards the risk. Indeed, they count towards the prevalence. Moreover, the denominator can only include individuals who are capable of experiencing the event. For example, in order to calculate the lifetime risk of uterine cancer the denominator may include all women, but most exclude all men and women without uteri.
Rothman explores the temporal characteristics of risk. As described in the definition, a value assigned to a risk is only valid if the time period has been specified. The risk of death from cardiovascular disease is very different if the time period is the next month or the next 10 years. The risk of death is invariably higher the longer the time period, and for many populations the risk of certain diseases is only non-negligible over extended periods of time. Extended time periods, however, raise their own concerns in regards to risk: the longer the time period the more likely something which precludes the event will occur.
Answer 1: 0.1 (2/20) is the risk of developing flu in the first week, not the second week.
Answer 2: 0.15 (3/20) is the number of new infections in the second week, divided by the total number of students. Because the denominator includes the 2 students who contracted influenza in the first week, it is NOT the risk.
Answer 4: 0.25 ((2+3)/20) is the prevalence of flu during the second week, but not the risk of developing flu in the second week.
Answer 5: 0.5 ((2+3+5)/20) is the risk of developing flu over the entire study, not the second week. Also, assuming that the flu lasts for at least three weeks, 0.5 is also the prevalence of flu by the end of the third week.
Gordis, L. Epidemiology, 4 ed. Saunders Elsevier, 2004.
Rothman KJ. Epidemiology: An Introduction, 1 ed. Oxford University Press, 2002.