HI560 Unit 5 Hypothesis Test Writeup Example

Hypotheses First, Then the Numbers: A Unit 5 Test of Two Proportions for Incomplete MRI Exams at Two Sites

[Student Name]

Purdue University Global

HI560: Health Care Data Analytics

Unit 5 Hypothesis Test Writeup

[Instructor Name]

[Date]

Thistlewood Health, its sites and every count are composite; the sources cited are real.

Question and Hypotheses

The question, carried from Unit 1, is whether adult outpatient MRI exams end incomplete more often at the Fennimore center than at the hospital suite. This unit tests the crude difference, before adjustment for exam group or age.

Let p sub F be the proportion of exams ending incomplete at Fennimore and p sub H the proportion at the hospital.

1. Null hypothesis: p sub F equals p sub H. The two sites have the same proportion of incomplete exams.

2. Alternative hypothesis: p sub F does not equal p sub H.

The test is two-sided, because a lower rate at Fennimore would also matter to the imaging director. Alpha was fixed at 0.05 in the analysis plan, written before the repaired file was opened.

Choice of Method

The outcome is binary, complete or incomplete, and the two sites' exams form independent groups. That points to a comparison of two proportions. A two-proportion z test is used, which gives the same result as a chi-square test on the two-by-two table of site by outcome. Two alternatives were rejected. A t test on outcomes coded zero and one would treat a binary measure as if it were continuous and would misstate the variability. A paired test would require each hospital exam to be matched with a particular Fennimore exam, and no such pairing exists.

Conditions

Independence comes first. Each exam appears once in the analytic file, since repeat attempts were removed in Unit 2. About 6 percent of patients contribute more than one exam on different dates, which introduces mild clustering; a sensitivity check keeping only each patient's first exam is reported below.

Sample size is the second condition. Under the null, the pooled proportion is 734 of 17,840, or 4.11 percent. The smallest expected cell count, incomplete exams at Fennimore, is about 329, far above the usual minimum of five or ten.

Results

Counts and rates by site:

1. Fennimore: 375 incomplete of 7,990 exams, 4.69 percent.

2. Hospital: 359 incomplete of 9,850 exams, 3.64 percent.

Effect size, reported first:

1. Risk difference: 1.05 incomplete exams per 100 more at Fennimore, 95 percent confidence interval 0.46 to 1.64.

2. Relative risk: 1.29.

Test statistic, reported second: z equals 3.51, two-sided p below 0.001.

The interval is computed with the unpooled standard error of the difference. With counts this large, the Wald interval and the score-based methods compared by Newcombe (1998) agree to within a few hundredths of a point, and adding two successes and two failures to each group, as Agresti and Caffo (2000) suggest for small samples, changes nothing at the reported precision.

Fennimore ends about one more MRI per hundred incomplete than the hospital, and the interval excludes zero, but it says nothing yet about why.

Sensitivity Check

Keeping only each patient's first exam in the period leaves 7,520 Fennimore and 9,250 hospital exams. The risk difference is 1.02 points, interval 0.41 to 1.63, essentially unchanged. Clustering by patient does not explain the result.

What the Result Would Mean at Each Site

In exams, the difference is easier to grasp. Fennimore performs about 4,000 adult outpatient MRIs a year. At the hospital's rate it would expect about 146 incomplete exams a year; at its own rate it has about 188. The excess is therefore roughly 42 exams a year, with the interval placing it between about 18 and 66. Each of those exams means a patient who must return, a radiologist reading a partial study or a referring clinician waiting longer for an answer. The figure is crude, so some of those 42 belong to case mix rather than to anything the site does, which is the subject of Unit 6.

Figure

A dot plot shows each site's rate as a dot with its 95 percent interval on an axis running from 0 to 6 percent, with the count of exams printed beside each dot. The intervals do not overlap, though non-overlap is not itself the test.

Interpretation

Three claims are kept separate. First, the gap is larger than chance would readily produce if the sites had the same underlying rate. Second, its size is uncertain within the interval: the true difference could be as small as about half an exam per hundred or as large as about one and a half. Third, nothing in this test says why the gap exists. The sites differ in the exams they perform and the patients they serve, and Unit 3 showed that Fennimore performs proportionally more head-first neuro exams, which are the most likely to stop. Epidemiology texts call a comparison like this one crude for exactly that reason: it compares populations, not like with like (Rothman et al., 2008).

Next Test

Unit 6 adjusts the comparison for exam group and age by direct standardization and takes up the question of whether scheduled slot length belongs in the adjustment.

References

Agresti, A., & Caffo, B. (2000). Simple and effective confidence intervals for proportions and differences of proportions result from adding two successes and two failures. The American Statistician, 54(4), 280-288. https://doi.org/10.1080/00031305.2000.10474560

Newcombe, R. G. (1998). Interval estimation for the difference between independent proportions: Comparison of eleven methods. Statistics in Medicine, 17(8), 873-890. https://doi.org/10.1002/(SICI)1097-0258(19980430)17:8<873::AID-SIM779>3.0.CO;2-I

Rothman, K. J., Greenland, S., & Lash, T. L. (2008). Modern epidemiology (3rd ed.). Lippincott Williams & Wilkins.