Investigation 1.4

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Background

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 in the last 11 cases (since Dec. 1999), 8 had died within 30 days of the transplant, "a death rate massively out of kilter with the average or other transplant units, which are all supposed to achieve an 80% survival rate within a year of opening." Could this just be an unlucky run?

https://www.theguardian.com/uk/2001/sep/13/sarahboseley

Research Question: Is the long-run probability of survival at this hopsital lower than 0.80?

Goals: In this investigation, you will

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

Describing the study
(a1) Identify the observational units
(a2) Identify the variable of interest
(a3) Is the variable quantitative or categorical?
In fact this variable is binary, so we let's define 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.” But let's start with success = survival for at least 30 days.
(b1) Considering survival for at least 30 days as a success, define the parameter of interest in this study (in words)
Consider the following symbols:
- p-hat p̂
- pi π
- x-bar x̅
- mu μ
(b2) Which symbol could we use to refer to the unknown parameter value?
(c) Is it reasonable to model this heart transplantation process as a binomial process? You have already defined success and failure, so evaluate the independence and constant probability conditions in this context.
(d1) If there is nothing unusual about the survival rate for heart transplantations at this hospital (compared to transplant units at other U.K. hospitals, government targets), what does this imply about the value of the probability of “success”? (Enter a number)
(d2) If the patients at this hospital are indeed surviving at a lower rate than the target, what does this imply about the value of the probability of “success”? (Enter a comparison to the previous number)
(e) Translate your answers to (d) to null and alternative hypothesis statements. Keep in mind, the null hypothesis claims the observed result is “just by chance,” whereas the alternative hypothesis translates the research conjecture
Null hypothesis Ho: parameter (=, <, >) enter a number
Alternative hypothesis Ha: same parameter (=, >, <)same number

Sample data
Of the hospital’s 11 most recent transplantations at the time of the study, there had been eight deaths within the first 30 days following surgery.
(f) Compute the sample proportion (remember to be consistent with how we have defined success)

Statistical Inference
Calculate the binomial p-value: Use either the One Proportion applet (or R or JMP )
(g) Enter the p-value that you found.
Submit a screen capture showing your inputs and output.

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Interpreting the results
Example JMP output
(h1) Provide a detailed interpretation of this p-value: the p-value is the probability of obtaining or successes in trials from a random process assuming .
(h2) What conclusion do you draw from this p-value?
(h2) What do you conclude about the probability of survival for 30 days after a heart transplantation at this hospital?

Convincing evidence the probability is smaller than 0.80Not convincing evidence that the probability is smaller than 0.80Proof that the probability is smaller than 0.80Evidence that the probability is equal to 0.80

Does it matter which outcome I choose as success?
Suppose that we had focused on death within 30 days rather than survival for 30 days as a “success” in this study.
(i) Describe how the hypotheses would change and how the calculation of the binomial p-value would change.
After describing your plan in (i), check this button to continue.
Recalculate the binomial p-value
(j) Enter the p-value that you found.
Include a screen capture showing your inputs and results.

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Does the sample size matter?

Following up on the suspicion that the sample of size 11 aroused, Poloneicki, Sismanidis, Bland, and Jones (2004) 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.

(k) Compute the new sample proportion for these data
(l) PREDICT whether this sample provides more or less convincing evidence that this hospital’s death rate exceeds 0.20. Explain your reasoning.
Submit your prediction(s) to reveal the next question.
(m) Use the applet to determine the binomial probability of finding at least 71 deaths in a sample of 361 if the underlying probability of death is 0.20.
Report your new p-value here.
Submit a screen capture showing your inputs and output.

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Does the hypothesized probability matter?
In the Poloneicki, Sismanidis, Bland, and Jones (2004) study, they state that "The national average mortality was not available and a rate of 15% was used instead as the benchmark."
(n) PREDICTION: The sample data (71 deaths in 361 operations) will provide stronger evidence against which null hypothesis?
Submit prediction to reveal the next question.
(o) Use the applet to determine the binomial probability of finding at least 71 deaths in a sample of 361 if the underlying probability of death is 0.15.
Report your new p-value here.
Submit a screen capture showing your inputs and output.

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What factors impact the p-value?

Here are the p-values you said you found

3 out of 11 vs. .80	{gEnter106}
8 out of 11 vs. .20	{jEnter}
71 deaths out of 361 vs. .20	{reportYour}
71 deaths out of 361 vs. .15	{Report}

(p) Compare your p-values and summarize what you learned:
- If all else stays the same, changing which outcome is success increase, decrease, no change the p-value.
- If all else stays the same, a larger sample size will (increase, decrease, not change) the p-value.
- If all else stays the same, a larger difference between the observed proportion and the hypothesized probability will (increase, decrease, not change) the p-value.
(Optional): If you ran any additional analyses or don't have an appropriate comparison, explain here.
Reflection
(q) In your opinion, which data set do you think is more valid to use – the larger sample size or the more recent data? Explain how you are deciding.
Should be Empty:

Background

Describing the study

Sample data

Statistical Inference

Interpreting the results

Does it matter which outcome I choose as success?

Does the sample size matter?

Does the hypothesized probability matter?

What factors impact the p-value?

Reflection