Hospital Quality

TASK. As health care director for your company, your job is to select which of two local hospitals you will send your employees to in case of emergency. Mercy Hospital is the larger of the two and a local emergency care facility. It had 2,100 surgery patients last year, many of whom entered the hospital in poor condition. Of its surgery patients, 63 died. Excelsior is smaller. It had 800 surgery patients last year, a smaller percentage entered in poor condition, and 16 of its surgery patients died. The detailed information is given in Table 1.

Table 1: Patient mortality, two hospitals
	Mercy Hospital Excelsior Hospital Patients Deaths Patients Deaths In Good Condition 600 6 600 8 In Poor Condition 1,500 57 200 8 Combined Total 2,100 63 800 16

The director of public relations at Excelsior claims that the overall death rate at Excelsior is smaller than the overall death rate at Mercy and that the intimacy of a small hospital is preferable to the hustle and bustle of a large facility. The director of public relations at Mercy claims that if you look at the death rates more carefully, you will see that they are a better facility--they simply treat a lot of patients who are more seriously ill.

Analyze the given data and make a recommendation to your board of directors. Make the recommendation in the form of a memo in which you clearly justify your decision, knowing that the director of the hospital you do not choose may appeal your decision.

COMMENTARY. This is a decision-making situation that might actually arise in the workplace, but its relevance is much broader. Drawing sound conclusions in such situations requires understanding and careful thinking. In the news and in everyday life, we are inundated with statistics supporting various positions. Thus, it is important that students learn to look for complexities that are often hidden behind the statistics.

Both directors of public relations are correct, despite their seemingly contradictory statements. These data provide an example of an occurrence known in probability as Simpson's paradox; it can also occur in other situations involving "weighted averages." Similar apparent paradoxes arise, for example, in situations where women or minorities in various jobs earn about the same as their male counterparts, but their overall average earnings may be far less. (This can happen if most women are employed in low paying jobs, for example.)

Because of the apparent paradox, this task provides an intriguing context for discussing more fundamental notions, such as probability, rates, and weighted averages. Working through such examples can sensitize students to the need to understand the numbers and trends that give rise to statistics. It will also give them a better sense of what to believe and what to question when confronted with statistical assertions.

MATHEMATICAL ANALYSIS. To check the directors' assertions, one must compute death rates. For example, the death rate for patients in good condition at Mercy is 6/600 or 1%. The other results are shown in Table 2.

Table 2: Patient mortality, two hospitals, with rates
	Mercy Hospital Excelsior Hospital Patients Deaths Rate Patients Deaths Rate In Good Condition 600 6 1% 600 8 1.33% In Poor Condition 1500 57 3.8% 200 8 4% Combined Total 2100 63 3% 800 16 2%

Looking only at the combined death rate, it looks like Excelsior is the better hospital, for a 2% death rate is better than 3%. Looking at the separate death rates, however, the picture is different. For patients in good condition, the death rate is lower at Mercy. Similarly, a patient in poor condition is better off at Mercy. So the public relations director at Mercy is correct: Mercy Hospital has a better success rate both with patients in good health and with those in poor health. The reason Mercy loses more patients overall is that it treats many more seriously ill patients.

Here's an easy way to see how averages based on aggregates can deliver a different message than averages based on components. Suppose a company, in an attempt to recruit women into all positions, pays them more than men in all positions. If it is easy to recruit women for the low-paying positions, and hard to recruit them for the high-paying positions, it is possible that the average salary for women will still be lower than the average salary for men, seemingly contradicting the company's intent to pay women more.

EXTENSIONS. Students might find and analyze employment and salary patterns in various professions. They might look at admissions rates at a university by gender or by race, for the university as a whole, and then separated by college or by department. Such assignments should not be given, however, without allowing for discussion of equity issues that can be raised by such data.

Students might construct data that illustrates analogous paradoxes in contexts that appeal to them. In baseball, for example, it is possible for a batter to have the best batting average before the all-star break and the best average after the all-star break and yet fail to have the best average for the whole season.

Students might also explore other instances of weighted averages, perhaps first as simple ways of computing more familiar averages. For example, if a teacher explains that homework counts 50%, each of three exams count 10%, and the final exam counts 20%, a student can determine his or her average going into the final as follows:

50 * Homework + 10 * Exam₁ + 10 * Exam₂ + 10 * Exam₃ / 50 + 10 + 10 + 10

The arithmetic in this task deserves comment. If one thinks of the death rates as fractions, then one might consider the relationship between the separate death rates and the combined death rate to be like addition. In the case of Mercy hospital, the "addition" looks as follows, where the indicates that this is not the standard addition of fractions.

6 / 600 57 / 1500 = 63 / 2100

Notice that this "addition" is performed by adding the numerators and adding denominators--one of the mistakes that students make when they are supposed to perform the standard addition of fractions. Yet, this "addition" is used in many contexts, from computing batting averages in baseball to computing terms in Farey sequences, an advanced topic in number theory.

Students might be asked, "Why does this 'addition' make sense here?" "What is the difference between this and the standard addition of fractions?" "What is different about the contexts that gives rise to a different kind of addition?" Discussion of such questions can provide for a firmer understanding of the concepts of fraction, rate, and average.

This kind of "addition by component" is reminiscent of addition of vectors, which gives us a geometric model of the situation. The data for patients in good condition at Mercy (600 patients, 6 deaths) can be represented as the vector (600, 6) which can be represented geometrically as an arrow from the origin to the point (600, 6) on a coordinate plane. (See Figure 1.) Then the death rate, 6/600, is precisely the slope of the vector. By similarly representing the data for patients in poor condition as the vector (1500, 57), the sum of the vectors is given by adding the components of the vectors. That is, (600, 6) + (1500, 57) = (2100, 63). Geometrically, the sum of these vectors is the diagonal of the parallelogram formed by the vectors. (See Figure 1.) Note that because the death rate is represented by the slope of the vector, a steeper vector corresponds to a higher death rate. We can similarly represent the data from Excelsior Hospital (Figure 2).

Figure 1: Patient mortality at Mercy Hospital

Figure 2: Patient mortality at Excelsior Hospital

Superimposing the data from Excelsior Hospital upon that from Mercy (Figure 3) shows that the sides of the Excelsior parallelogram are steeper than the corresponding sides of the Mercy parallelogram, but Mercy has a steeper diagonal. To gain a spatial and kinesthetic sense of this paradox, students might use dynamic geometry software to draw such a picture to construct data that exhibit this paradox.

Figure 3: Patient mortality, two hospitals

Observe that the diagonal representing the sum must be between the two vectors, indicating that the slope of the sum must be between the other slopes. This provides a compelling geometric argument for the algebraic fact that a / b c / d , defined as above, is always between the two fractions a/b and c/d, as long as a, b, c, and d are all positive. Proving this algebraically, on the other hand, requires some non-obvious techniques.