upper P left-parenthesis upper B right-parenthesis equals left-parenthesis 1 minus upper P left-parenthesis upper A right-parenthesis right-parenthesis upper P left-parenthesis upper B right-parenthesis 3rd Row 1st Column upper P left-parenthesis not upper A and not upper B right-parenthesis 2nd Column equals upper P left-parenthesis not upper A right-parenthesis upper P left-parenthesis not upper B right-parenthesis equals left-parenthesis 1 minus upper P left-parenthesis upper A right-parenthesis right-parenthesis left-parenthesis 1 minus upper P left-parenthesis upper B right-parenthesis right-parenthesis EndLayout"/>
Example 2.26
Achondroplasia is a genetic disorder related to dwarfism caused by an abnormal gene on one of the 23 chromosome pairs. Twenty percent of the individuals having achondroplasia inherit a mutated gene from their parents. An individual inherits one chromosome from each parent. It takes only one abnormal gene from a parent to cause dwarfism and two abnormal genes can cause death. If only one parent has a mutated gene then, there is a 50% chance that a child will receive this gene and inherit achondroplasia. On the other hand, if both parents have achondroplasia, there is a 50% chance that each parent will pass on the gene. Thus, because the parents’ genes are passed on independently, there is a 25% chance that a child will inherit neither gene, a 50% chance that the child will inherit only one abnormal gene, and a 25% chance that the child will inherit two abnormal genes and be at risk of death. To see this, suppose each parent has achondroplasia and let
and
Then, A and B are independent and P(A)=P(B)=0.5, and the probability that the child inherits
1 no abnormal genes is
2 only one abnormal gene is
3 inherits two abnormal genes is
Independence plays an important role in data collection and the analysis of the observed data. In most statistical applications, it is important that the observed data values are independent of each other. That is, knowing the value of one observation does not influence the value of any other observation.
2.3.4 The Relative Risk and the Odds Ratio
Many biomedical research studies concern the incidence of a particular disease or condition. The absolute risk (AR) of a disease is the probability that an individual develops the disease. Most conditions/diseases are affected by risk factors that increase the incidence of the disease. For example, it is well known that the risk factor smoking cigarettes increases an individual’s chance of developing lung cancer.
An important research question that is often asked in the study of a disease is whether or not the disease is independent of a particular risk factor. When the disease is independent of the risk factor, the risk factor does not increase or decrease the incidence of the disease. On the other hand, when the disease is dependent on the risk factor, the risk factor does affect the chance of having this disease, and in this case, the disease is said to be associated with the risk factor.
In a prospective study where the risk factor is either labeled present or absent, exposed or unexposed, or treated or untreated, an important measure of the strength of the association between the disease and the risk factor is the relative risk (RR), which is also known as the risk ratio. The relative risk is the ratio of the probability of having the disease when the risk factor is present and the probability of having the disease when the risk factor is absent. In particular, the formula for the relative risk is
Note that the relative risk is only appropriate for prospective studies, it takes on values between 0 and ∞, and it is one when the disease is independent of the risk factor. When the relative risk is close to 1, there is only a small difference in the probabilities of having the disease when the risk factor is present or absent, and therefore, the relative risk suggests that the risk factor does not have much influence on the incidence of the disease. A risk ratio larger than 1 suggests there is an increased risk of the disease when the risk factor is present, and a risk ratio smaller than 1, suggests there is decreased risk of the disease when the risk factor is absent.
The larger the relative risk is, the more likely it is for an individual to have the disease when the risk factor is present. For example, if the relative risk is RR = 5.8, then the disease is 5.8 times as likely when the risk factor is present than when it is absent.
Example 2.27
Suppose in a prospective study, the probability of having the disease given a particular risk factor is present is 0.10, and the probability of having the disease when the risk factor is absent is 0.02. Then, the relative risk of the disease for this risk factor is RR=0.100.02=5. Thus, the disease is five times as likely when the risk factor is present.
Finally, in the presence of a significant relative risk, it is also important to look at the absolute risk to assess the practical significance of the risk factor. For example, if the relative risk is RR = 10, but the absolute risk is AR = 0.000001, then the disease is very rare and even with the presence of the risk factor the risk is only 1 in 100,000. Thus, when the absolute risk of the disease is small, large values of relative risk may not truly indicate significant effects of having the risk factor. Also, when the absolute risk of the disease is large, a relative risk close slightly larger than 1 can indicate a significant effect due to the risk factor. Therefore, it is recommended that both the absolute risk and the relative risk be reported with the results of the study.
Because the relative risk can only be used in prospective studies, an alternative measure of the association of the disease and risk factor is required for retrospective studies. The odds ratio (OR) is an alternative measure of association that can be used in both prospective and retrospective studies.
The odds ratio is based on the odds of having the disease rather than the probability of having the disease. The odds of having the disease is ratio of the probability of having the disease to the probability of not having the disease. The formula for computing the odds of having the disease is
The odds of a disease is between 0 and ∞. Furthermore, when the odds = 1, having the disease is just as likely as not having the disease, when the odds < 1, the disease is less likely than not having the disease, and when the odds > 1, the disease is more likely than not having the disease.
For example, if the probability of having the disease, which is the absolute risk of the disease, is 0.2, then the odds of having the disease is odds(Disease)=0.20.8=0.25. Thus, the disease is one-fourth as likely as not having the disease.
In most cases the odds of having a disease will be different for the presence or absence of a particular risk factor. Thus, it is often useful
