Poloneicki, Sismanidis, Bland, and Jones (2004) reported that in September 2000 heart transplantation at St. George’s Hospital in London was suspended because of concern that more patients were dying than previously. Newspapers reported that the 80% mortality rate in the last 10 cases was of particular concern because it was over five times the national average. The variable measured was whether or not the patient died within 30 days of the transplant. Although there was not an officially reported national mortality rate (probability of death within 30 days for patients undergoing this procedure), the researchers determined that 15% was a reasonable benchmark for comparison

Research Question: Is the true death rate at this hopsital larger than the national benchmark?

Goals: In this lab, you will

• Investigate a research question through a binomal test of significance
• Investigate factors that impact the strength of evidence against the null hypothesis

We need to consider which outcome we will consider “success” and which we will consider “failure.” The choice is often arbitrary, though sometimes we may want to focus on the more unusual outcome as success. In fact, in many epidemiology studies, “death” is typically the outcome of interest or “success.”

Consider the following symbols:

• p-hat
• pi 
• x-bar
• mu μ

Of the hospital’s ten most recent transplantations at the time of the study, there had been eight deaths within the first 30 days following surgery.

To conduct a Binomial Test in JMP

Choose Analyze > Distribution  With raw data, drag the variable to the Y, columns slot. Press OK.  With summarized data, move the column with the category names to the Y, columns slot and move the column with the counts to the Freq slot. Press OK. From the variable’s hot spot, select Test Probabilities. Specify the hypothesized probability of success for the category you want to define as success (only).
Then pick a one-sided alternative hypothesis (greater than or less than). Press Done.
JMP assumes “success” to be the first category alphabetically unless you specify otherwise with Column Properties > Value Ordering. Note: “not equal to” alternatives will be discussed in Investigation 1.5.

Example JMP output

Suppose that we had focused on survival for 30 days rather than death within 30 days as a “success” in this study. 

Following up on the suspicion that the sample of size 10 aroused, these researchers proceeded to gather data on the previous 361 patients who received a heart transplant at this hospital dating back to 1986. They found 71 deaths within 30 days among heart transplantations.

The following graphs display the two theoretical probability distributions (for sample sizes n = 10 and n = 361), both assuming the null hypothesis ( = 0.15) is true. These graphs show just how far the observed values (8 and 71) are from the expected value of the number of deaths (0.15 × 10 = 1.5 and 0.15 × 361 = 54.15) in each case. You should also note that the shape, center, and variability of the probability distribution for number of successes are all affected by the sample size n.

 

Keep in mind that of interest to us is the observed statistic’s relative location in the null distribution. Thus, we are most interested in how variable the possible outcomes are from the “expected” outcome. The center of the distribution isn’t all that interesting to us in answering the research question because we determine what the center of the distribution will be by how we specify the null hypothesis. Even the shape isn’t all that interesting on its own in answering our research question.