Student Name:____________________________________ Week 2 Assignment - Kill- EZ example Please show your calculations. If there are in your work, the calculations can show me where the errors were made, and I can point you in the right direction. The easiest way to show your work would be to use an Excel spreadsheet. Submit it with your assignment. If you prefer to show your work in other ways, you may. Table: Hypothetical Data for Kil-EZ Carcinogenicity Test Number of test rats Numbe r of rats with tumors Rat test dose in mg Kil-EZ per kg rat body weight (mg/kg/day) 50 1 0 50 5 20 50 10 40 50 20 80 50 7* Probability of a rat's developing tumors Standard Deviation Lower 95% confidence level Upper 95% confidence level 160 * Acute effects were observed in the 160 mg/kg/day group, leading to significant premature deaths Part A 1. Fill in the fourth column of the Table - the probability of tumor, which is the number of rats with tumors divided by the number of rats in the sample 2. Explain why the probability is lower at a dose of 160 mg/kg.day than at a dose of 80 mg/kg.day 3. Plot the points on a graph with probability on the vertical axis (y-axis) and dose on the horizontal axis (x-axis). 4. Draw a straight line through your points to the origin (x = 0, y = 0) and calculate its slope the way you did in high school algebra. That is the cancer potency in kg-day/mg. (If you don't remember how to calculate the slope of a straight line, ask) Should you include the observed probabilities of tumor at the zero dose and the highest dose? Why or why not? 5. Using equation (1), calculate the dose that would correspond to cancer probabilities of 1E-6 and 1E-4. (That is the range EPA considers in setting regulatory standards.) Part B 6. Using the equation for standard deviation (s) in the module, calculate the standard deviation at each dose level and enter these values in the fifth column. The standard deviation needs to be calculated separately for each dose level. 7. Calculate the 95% confidence intervals for each of the relevant probabilities in your Table, and insert into the sixth and seventh columns of the Table. 8. Using the upper 95% confidence level for the 20 mg/kg-day dose, calculate the cancer potency by drawing a straight line from the 20 mg/kg-day dose level to the origin. 9. Do the same using the lower 95% confidence level. 10. Calculate the range of doses that would correspond to a one in a million risk (p = 1E-6) using your answers to Questions 8 and 9. 11. What conclusions do you draw from your answer? Assignment Part C 12. If you were the EPA administrator, would you ban Kil-EZ to avoid human cancers? Explain your answer. Number of Experiments Normal Distribution p0 - 2s p0 - s p0 p0 + s p0 + 2s Probability WHAT IS ALL THIS STATISTICS STUFF ABOUT? This is not a statistics course. I gave you all the statistics you needed to solve the problems in the assignments. This paper explains the basis for the tools I gave you. Everything we did prior to Crystal Ball is based on the normal distribution. The normal distribution, as shown in the above graph, is a bell-shaped curve. It is symmetrical around the mean. The mean or average is the highest point of the curve. If you cut the curve out and folded it on the mean line, the two sides would match exactly. The curve is completely determined by two parameters - the mean and the standard deviation. If the standard deviation is small, the curve will be tall and skinny. If the standard deviation is large, the curve will be short and fat. Let's go back to the Kil-EZ assignment. Suppose the experiment at the 20 mg/kg-day exposure level were repeated many times with different groups of rats. Each experiment might report different values for the probability of tumors. Some of the probabilities would be lower than that in the data set I gave you. Some would be higher. Looking at all the experimental results, you could determine how many resulted in a probability of p0, p1, p2, p3, etc. You could then plot those points on a graph with number of experiments on the vertical axis and probabilities on the horizontal axis. In the Kil-EZ example, we assumed that all the points would fall on a normal distribution curve like the one above. A normal distribution curve has some remarkable properties. In the above graph, I have drawn vertical lines at one and two standard deviations above and below the mean. Calling the mean p0, one standard deviation above the mean is p0 + s. One standard deviation below the mean is p0 - s. The area of the curve between those two vertical lines is 68% of the area under the total curve. What that means in the Kil-EZ example is that 68% of the probabilities obtained in all the experiments would lie between p0 - s and p0 + s. Two standard deviations above the mean is p0 + 2s. Two standard deviations below the mean is p0 - 2s. The area under the curve between the vertical lines drawn from those two points is 95.4% of the area under the total curve. That means that 95.4% of the probabilities obtained in all the experiments would lie between p0 - 2s and p0 + 2s What we call the lower and upper 95% confidence limits are the probabilities that would correspond to 95% of the area under the curve. Those probability values are p0 - 1.96s and p0 + 1.96s. Note that 2.5% of the probabilities observed in the multiple experiments would be less than the lower 95% confidence limit; and 2.5% would be greater than the upper 95% confidence limit. The following table shows you how to derive the 90%, 95%, and 99% confidence intervals. Those are all we will need in this class. Confidence LimitRange of Probabilities 90% p0 - 1.65s to p0 + 1.65s 95% p0 - 1.96s to p0 + 1.96s 99% p0 - 2.58s to p0 + 2.58s I gave you the formula for standard deviation. You will just have to take my word for it. As I said at the beginning, this is not a statistics class. What about the Z stuff? In the above discussion, I plotted measured probabilities on the horizontal axis. In the cholera problem next week, we will work with a new variable, Z. Everything goes back to the normal distribution curve and Z is defined as follows: Z = (p - p0)/s Let's look at the 95% confidence limits again. When p = p0 , the mean, Z = 0 When p = p0 - 1.96s, Z = -1.96 When p = p0 + 1.96s, Z = +1.96 Note that the shape of the curve would not change at all. In the cholera problem, I will ask you whether the different probabilities observed in Lambeth and the rest of London could have occurred by chance. I will give you a standard statistical formula for getting the answer. Another way of approaching the problem is to determine whether the 99% confidence intervals overlap, and some students prefer to do it that way. If they do overlap, you can be 99% confident that the results occurred by chance. You would find the 99% confidence intervals for Lambeth and the rest of London just as we did above