Greg Brewster, DePaul University Page 1 LSP 121 Math and Tech Literacy II Greg Brewster DePaul University Risk Topics Probabilities of Adverse Events Impacts of Adverse Events Insurance Test Results Test Accuracy and Errors Probabilities of: Type I Errors: False Type II Errors: False Risk Analysis As we know, every event has multiple outcomes. We ve looked at the probabilities of different outcomes. In the business world, risk analysis is the study of adverse outcomes or unwanted outcomes
Greg Brewster, DePaul University Page 2 Risk Analysis Risk analysis uses a similar formula to the Expected Value formula: Risk Factor = (probability of outcome) * (impact of that outcome) Risk Example You invest $100,000 in a stock. The probability that the stock value loses $50,000 in the next year is 1%. What is the risk factor? Risk Factor = 1% x (-$50,000) = -$500 This is the average amount you expect to lose due to the risk that the stock value drops. Personal Risks We all take risks every day whenever we engage in some activity that might have an adverse outcome. For example, let s look at the risks of traveling
Travel Risks Are cars today safer than those 30 years ago? If you need to travel across country, are you safer flying or driving? The Risk of Driving In 1966, there were 51,000 deaths related to driving, and people drove 9 x 10 11 miles In 2000, there were 42,000 deaths related to driving, and people drove 2.75 x 10 12 miles Was driving safer in 2000? The Risk of Driving 1966 Probability of death per mile = (51,000 deaths) / 9 x 10 11 miles = 5.7 x 10-8 (= 0.0000057%) 2000 Probability of death per mile = (42,000 deaths) / 2.75 x 10 12 miles = 1.5 x 10-8 (= 0.0000015%) Driving has gotten almost 4 times safer! Greg Brewster, DePaul University Page 3
Driving vs. Flying Over the last 20 years, airline travel has averaged 100 deaths per year Airlines have averaged 7 billion miles in the air Probability of death per mile = 100 deaths / 7 billion miles = 1.4 x 10-8 (= 0.0000014%) Flying is a little safer than driving Calculating Risk Factors? So, the Risk Factor associated with death by flying = (probability of death) x (impact of death) = (0.0000014%) * (negative value of dying) BUT how do you calculate a numeric value for the impact of death? Insurance companies do it But, for most of us, this impact is infinite, so risk factor analysis doesn t work so well Insurance Insurance companies can associate a monetary value for death. It s the amount of money they have to pay out on your life insurance policy. How do we calculate the value of the life insurance to us (i.e. our family)? It is somewhat subjective, but includes your expected lifetime income, cost of child care, personal hardship, etc. etc. Greg Brewster, DePaul University Page 4
Greg Brewster, DePaul University Page 5 Spending Money to Reduce Risks Suppose you are buying a new car. For an additional $500 you can add a device that will reduce your chances of death in a highway accident by 20%. Interested? Are you willing to spend more to reduce your risk even further? There are many factors that would influence each person s personal answer to these tradeoffs. What Should We Do? Hide in a cave? Know the data be aware! Now, let s start our first med school lecture Tumors and Cancer Welcome to med school! Most people associate tumors with cancers, but not all tumors are cancerous Tumors caused by cancer are malignant Non-cancerous tumors are benign
s Suppose your patient has a breast tumor. Is it cancerous? Probably not Studies have shown that only about 1 in 100 breast tumors turn out to be malignant Nonetheless, you order a mammogram Suppose the mammogram comes back positive. Now does the patient have cancer? Accuracy Early mammogram screening was 85% accurate 85% would lead you to think that if you tested positive, there is a pretty good chance that you have cancer. But let s do the math! Consider a study in which mammograms are given to 10,000 women with breast tumors Assume that 1% (1 in 100) of the tumors are actually malignant (100 women actually have cancer) while 99% (9900) have benign tumors. Greg Brewster, DePaul University Page 6
Greg Brewster, DePaul University Page 7 Malignant Benign Totals Positive Negative Total 100 9900 10,000 Malignant in 1/100 th of the total 10,000. screening correctly identifies 85% of the 100 malignant tumors as malignant These are called true positives The other 15% had negative results even though they actually have cancer These are called false negatives Malignant Benign Totals Positive Negative 85 True 15 False Total 100 9900 10,000
Greg Brewster, DePaul University Page 8 screening correctly identifies 85% of the 9900 benign tumors as benign Thus it gives negative (benign) results for 85% of 9900, or 8415 These are called true negatives The other 15% of the 9900 (1485) get positive results in which the mammogram incorrectly suggest their tumors are malignant. These are called false positives. Benign Totals Malignant Positive Negative 85 True 15 False 1485 False 8415 True Total 100 9900 10,000 Malignant Benign Totals Positive 85 True 1485 False 1570 Negative 15 False 8415 True 8430 Total 100 9900 10,000 Now compute the row totals.
Results Overall, the mammogram screening gives positive results to 85 women who actually have cancer and to 1485 women who do not have cancer The total number of positive results is 1570 Only 85 of these are true positives, that is 85/1570, or 0.054 = 5.4% are true positives. Accuracy => Error The accuracy of a test determines what percentage of the time the result is correct. If a test is not 100% accurate, then we get errors of two types, false positives and false negatives. Type I error of Errors (false positives) 15% of the 9900 women (1485) who do not have cancer get positive diagnostic test results which incorrectly indicate that they do have cancer (i.e. that their tumors are malignant). Type II error (false negatives) 15% of the 100 women (15) who do have cancer get negative diagnostic test results which incorrectly indicate that they do not have cancer. Greg Brewster, DePaul University Page 9
Greg Brewster, DePaul University Page 10 Results Thus, the chance that a positive result really means cancer is only 5.4% Therefore, when your patient s mammogram comes back positive, you should reassure her that there s still only a small chance that she has cancer In the best case, there should be another follow-up test that can verify the results Another Question Suppose you are a doctor seeing a patient with a breast tumor. Her mammogram comes back negative. Based on the numbers above, what is the chance that she has cancer? Malignant Benign Totals Positive 85 True 1485 False 1570 Negative 15 False 8415 True 8430 Total 100 9900 10,000 15/8430, or 0.0018, or slightly less than 0.2% Now what do you do?