Homework 6

Due: 2022-10-02, 11:59pm

All homework must be submitted via Blackboard. Your answers must be in a MS Word (DOCX) or PDF format. Your submitted document should have sections corresponding to those in this homework.

Please make sure that you have watched the videos and have done the readings. Everyone should do this independently; you can discuss the process, but the answers are expected to be different.

Include graphs as images in your document. Use the lecture notes as a guide.

1. Diagnostic tests (50%)

In this exercise we will see the interplay between the characteristics of a diagnostic test, disease prevalence (prior probability) and positive/negative predictive values.

  • Consider a disease for which there is a test that is 90% sensitive and 80% specific. Calculate the positive and negative predictive values for this test if the disease prevalence is 10%.
  • If you have a high-risk individual, for whom you assess the prior probability for having the disease is 40%, what would be the positive and negative predictive values for this test?
  • A patient comes to the clinic, and based on your initial assessment they have a 50-50 chance of bacterial vs a viral infection. You want to decide if you give antibiotics for the bacterial infection. You order test for the bacterial infection, which comes out positive. What is your updated assessment of the odds of bacterial vs viral infection?
  • Let’s say you a want to be at least 90% certain that it is a bacterial infection before prescribing antibiotics. You order a second test that is 95% sensitive but 60% specific. It is positive. Will you prescribe antibiotics?
  • Suppose you are in the biotech industry and want to design a test such that if the disease prevalence is 10%, the positive predictive value is at least 90%. What is the minimum likelihood ratio your test has to achieve? (You can use algebra and solve an equation.)

2. Likelihoods and likelihood ratio (50%)

Use the coin spins data for this exercise.

  • Recall your coin flips (all 20 spins). How many heads and tails you got in 20 spins? Calculate the likelihood as a function of q, the probability of heads. Assume that the spins are independent, and have the same chance of turning heads for both sets of 10 spins (heads facing you and tails facing you). The answer will be a formula in terms of q.
  • Plot the likelihood function. Based on this data, what is the range of plausible values for q?
  • Now use the data for the whole class. Count the total number of heads and tails in all spins for the whole class. Calculate the likelihood function, and plot it. Compare the plot with the plot for just your data. What is the difference?
  • Now assume that there are different values for the probability of heads when the coin is spun with heads facing the spinner (call is qH) vs tails facing the spinner (call it qT). Using data for the whole class, plot the likelihood functions for qH and qT on the same graph. (Calculate the likelihoods as two different columns of data.)
  • Based on the plots of the likelihood functions, what is your conclusion regarding plausible values for qH and qT? Are they same or different?

3. Acknowledgements

Please acknowledge individuals who helped you or resources thay were helpful in completing the homework.