To review probability concepts, you should read Chapter 3 of your text. This handout will focus on Section 3.6 and also some elements of Section 13.3.

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1 To review probability concepts, you should read Chapter 3 of your text. This handout will focus on Section 3.6 and also some elements of Section EXAMPLE: The table below contains the results of treatment for patients with different types of Hodgkin s disease. The data was collected by taking a random sample of 538 patients diagnosed with some form of Hodgkin s disease; therefore, both Type of Hodgkin s Disease and Response to Treatment are random variables. Type of Hodgkin s Disease Response to Treatment None Partial Positive ROW TOTALS Lymphocyte depletion Lymphocyte predominant Mixed cellularity Nodular schlerosis COLUMN TOTALS n 538 Questions: For a patient selected at random from these 538 Hodgkin s patients, find the probability that the patient 1. had a positive response. 2. had at least some response to the treatment. 3. had lymphocyte predominant Hodgkin s disease and had a positive response to treatment. 4. Had lymphocyte predominant or nodular schlerosis Hodgkin s disease. 1

2 Conditional Probability The probability of event A occurring given that event B has already occurred is usually denoted by P(A B). Recall the following formula: P(A B) P(A B) P(B) Once again, consider the Hodgkin s data. Type of Hodgkin s Disease Response to Treatment None Partial Positive ROW TOTALS Lymphocyte depletion Lymphocyte predominant Mixed cellularity Nodular schlerosis Questions: Find the following conditional probabilities: 1. P(Positive Response LD) 2. P(Partial Response LD) 3. P(No Response LD) 2

3 These conditional probabilities are easily seen using JMP. First, enter the data from the contingency table into JMP as shown in the data table below. Note: The frequencies/counts in the Freq column are interpreted by JMP as frequencies associated with the histological type/response categories. To obtain the contingency table and mosaic plot in JMP select Analyze > Fit Y by X and place Histological type in the X box and Response in the Y box. 3

4 EXAMPLE: A study was conducted in 1991 by the University of Wisconsin and the Wisconsin Department of Transportation in which linked police reports and hospital discharge records were used to assess, among other things, the risk for head injury for motorcyclists in motor-vehicle crashes. The data shown below can be used to examine the relationship between helmet use and whether brain injury was sustained in the accident. Brain Injury No Brain Injury Row Totals Helmet Worn No Helmet Column Totals Questions: 1. What is the probability that a motorcycle accident victim in Wisconsin suffered brain injury? 2. What is the probability that a motorcyclist involved in an accident was wearing a helmet? Can this be used to estimate the probability that a randomly sampled motorcyclist in WI wears a helmet? 3. What is the probability that a motorcyclist suffered brain injury given that they were wearing a helmet? 4. What is the probability that a motorcyclist not wearing a helmet suffered brain injury? 5. How many times more likely is a motorcyclist not wearing a helmet to sustain a brain injury? 4

5 RELATIVE RISK AND ODDS RATIOS Relative Risk: This is a measure of how much a particular risk factor influences the risk of a specified outcome. For the motorcycle helmet data, we calculate the relative risk as follows: P(Brain Injury NoHelmet Worn) Relative Risk P(Brain Injury Helmet Worn) Comments: 1. We interpret this number by saying that the risk of brain injury is times as large for those that do not wear helmets as for those that do wear helmets. 2. A relative risk value of 1.0 is the reference value for making comparisons. That is, a relative risk of 1.0 implies that there is no difference in the two probabilities (or risks). 3. The relative risk ratio is easily displayed in the mosaic plot: 4. If we had alternatively calculated the relative risk ratio as P(Brain Injury Helmet Worn) Relative Risk P(Brain Injury NoHelmet Worn) then the interpretation changes. Now, we say the risk of brain injury for those that do wear helmets is times as large as the risk of brain injury for those that do not wear helmets. 5

6 Odds Ratios The relative risk ratio is frequently used when investigating the relationship between two categorical variables. Although this quantity is relatively easy to calculate and interpret, statisticians often use another quantity known as an odds ratio in this situation. Before computing an odds ratio, we need to compute the odds: Odds: With counts given for two distinct response categories (Helmet Use or Non-use), the odds of a Yes versus a No is computed as the number of Yes events versus the number of No events for each group. You can also think of this as the probability that something is true divided by the probability that something is not true. Once again, consider our data. Brain Injury No Brain Injury Row Totals Helmet Worn No Helmet Column Totals Find the odds of sustaining a brain injury for both groups: Odds of Brain Injury for Wearing Helmet Number with Brain Injury in Helmet group Number with No Brain Injury in Helmet group Odds of Brain Injury for No Helmet Number with Brain Injury in No Helmet group Number with No Brain Injury in No Helmet group The odds ratio is simply the ratio of the odds for the two groups: Odds Ratio Odds of Brain Injury for No Helmet Odds of Brain Injury for Helmet The interpretation is that the odds of brain injury in the No Helmet group are times as high as the odds of brain injury in the Helmet group. 6

7 We could also have calculated the odds ratio as follows: Odds Ratio Odds of Brain Injury for Helmet Odds of Brain Injury for No Helmet The interpretation is that the odds of brain injury in the Helmet group are as large as those in the No Helmet group. Comments: 1. An odds ratio of 1.0 implies that there is no observable difference between the two odds. 2. The odds can also be visualized in the mosaic plot: 7

8 EXAMPLE: A case-control study was conducted to determine whether there was an increased risk of cervical cancer amongst women who had their first child before age 25. A sample of 49 women with cervical cancer was taken of which 42 had their first child before the age of 25. From a sample of 317 similar women without cervical cancer it was found that 203 of them had their first child before age 25. Do these data suggest that having a child at or before age 25 increases risk of cervical cancer? Cancer (Case) No cancer (Control) Row Totals Age Age > Column Totals Questions: 1. Why can t we meaningfully calculate P(disease risk factor status)? 2. Even though it is not appropriate to do so, calculate P(disease risk factor status) for both risk factor groups and compute the relative risk ratio. 3. Find the odds for disease for both risk factor groups, and find the odds ratio for having the disease associated with the risk factor being present. 8

9 Now, suppose that the data from the study were slightly different from what was observed in the above table. Note that we have simply doubled the number of controls, but kept the percent of controls falling in each age group constant. Cancer (Case) No cancer (Control) Row Totals Age Age > Column Totals Questions: 4. Even though it is not appropriate to do so, calculate P(disease risk factor status) for both risk factor groups and compute the relative risk ratio using the revised data. 5. With the revised data, find the odds for disease for both risk factor groups, and find the odds ratio for having the disease associated with the risk factor being present. 6. What do you notice? Why do you suppose the odds ratio is much more commonly used that the relative risk? 9

10 Comments on Odds Ratios and Relative Risk 1. When the disease is rare in the population being studied then there is little difference between the RR and OR, with the difference getting smaller the rarer the disease is. Thus, for many diseases RR OR, which makes it easier to discuss and interpret odds ratios because we can state the results in terms of how many times more likely something is rather than using a multiplicative statement in terms of the odds. 2. The most commonly cited advantage of the relative risk over the odds ratio is that the former has the more natural interpretation. The relative risk comes closer to what most people think of when they compare the relative likelihood of events. 10