Assignment #3 - Conditional Probabilities, Relative Risk, Odds Ratios, Mosaic Plots

Assignment #3 - Conditional Probabilities, Relative Risk, Odds Ratios, Mosaic Plots, Correspondence Analysis, Bayes Rule& Medical Screening Tests, and the Binomial Distribution

1. Folic Acid and Spina Bifida

A large-scale study is conducted to determine whether regular ingestion of folic acid (vitamin B) by women of child bearing age who are at elevated risk for having babies affected by spina bifida will lower the risk of this disease. To this end, n = 23573 expecting women were randomly assigned to either the folic acid encouraged group or a control group where there was no such encouragement. The live birth outcomes for both groups are presented in the table below.

a) Find the estimated P(spina bifida| treatment group) for both groups. Discuss.
(3 pts.)

b) Take the ratio of P(spina bifida|encouraged)/P(spina bifida|control). What does this ratio tell you in terms of reduction in risk of spina bifida associated with being encouraged to take a folic acid supplement? (3 pts.)

2. Low Birth Weight Risk Factors (Lowbirth.JMP)

The purpose of this study was to identify potential risk factors for low birth weight. The following categorical variables were measured: previous history of premature labor (Prev?), hypertension during pregnancy (Hyper), smoking (Smoke), uterine irritability during pregnancy (Uterine), and minority status (Minority).

a) For each risk factor calculate P(Low|risk factor present) and P(Low|risk factor absent) for each of the four potential risk factors. What do these tell you about each of the potential risk factors? (5 pts.)

b) Use your answers in part (a) to calculate the relative risk (RR) associated with each factor and interpret. (4 pts.)

c) Calculate the odds ratio (OR) associated with each characteristic. Discuss. (4 pts.)
d) Which factor do you think poses the greatest risk of having a child with low birth weight? the least? Explain your answers. (2 pts.)

3.HIV ELISA Tests

The enzyme-linked immuosorbent assay (ELISA) test was the main test used to screen blood samples for antibodies to the HIV virus (rather than the virus itself) in 1985. It gives a measured mean absorbance ratio for HIV (previously called HTLV) antibodies. The table on the following page gives the absorbance ratio values for 297 healthy blood donors and 88 HIV patients. Healthy donors tend to give low ratios, but some are quite high, partly because the test also responds to some other types of antibody, such a human leucocyte antigen or HLA. HIV patients tend to give high ratios, but a few give lower values because they have not been able to mount a strong immune reaction.

To test this in practice we need a cutoff value so that those who fall below the value are deemed to have tested negatively and those above to have tested positively. Any such cutoff will naturally involve misclassifying some people without HIV as having a positive HIV test (which will be a huge emotional shock), and some people with HIV as having a negative HIV test (with consequences to their own health, the health of people around them, and the integrity of the blood bank, etc.) The table on the following page shows the results for a variety of MAR cut-off values.

a) If we regard a MAR value 3 as a positive test result for having the HIV virus what are the sensitivity, specificity, false-positive rate, and false-negative rate of the ELISA test? You will need to set up a 2 X 2 table to do the necessary computations. (4 pts.)

The Economist (July 4, 1992) told a story of a young American who committed suicide on learning that he had tested positive for HIV. Consider this in answering part (c) in particular.

b) In 1992 the number of Americans who were HIV positive was estimated to be 218,301 out of a population of 252.7 million. Using this information what is the estimated probability that a randomly selected American is HIV positive? Use this as your prior probability of being infected with HIV. (1 pt.)

c) Using the estimate from part (b) what is probability that they actually had HIV given a positive ELISA test result? (2 pts.)

d) If the ELISA test for a blood sample is negative what is the probability that the blood sample is actually HIV free? Again use your answer from part (b) for doing this calculation. (2 pts.)

e) If we changed the MAR cutoff value to 4 explain what would happen to the following probabilities in terms of an increase or decrease? You do not necessarily have to calculate each of these to answer this question. (3 pts.)

SensitivityFalse-positive

SpecificityPositive-predictive value

False-negativeNegative-predictive value

4. Renal Blockage

A study is run to determine the effects of removing a renal blockage in patients whose renal function is impaired because of advanced metastatic malignancy of nonurologic cause. The arterial blood pressure in each patient is measured before (X) and after (Y) surgery. The results are in the table below.

In parts (a) – (c) you conduct analysis to answer the question:
Is there evidence to suggest that the surgery tends to lower arterial blood pressure?
To do this consider the sign (+ or - ) of the difference in the arterial blood pressure before minus after, i.e. X – Y by first filling out the table above.
a) If surgery has no effect on arterial blood pressure what is the
P(X - Y > 0) = P(+) or P(X – Y < 0) = P(-)? Explain your answer.(1 pts.)
b) Count how many positive differences there are and compute the probability of getting that many or more using the probability of a positive difference from part (a). (2 pts.)
c) Use the probability found in part (b) to answer the question of interest to the researchers. Explain your reasoning. (3 pts.)