Septic arthritis is a bacterial infection in a joint. Patients with septic arthritis of the knee present with a painful, swollen, warm knee, but other conditions such as gout or pseudogout can cause a similar presentation. One test for septic arthritis is to insert a needle into the joint space, withdraw fluid, and send it to the lab for a white blood cell (WBC) count. Septic arthritis tends to cause higher WBC counts than the non-septic arthritis conditions.
You study 15 consecutive patients who presented to the emergency department with a painful, swollen, warm knee, and who had joint fluid WBC counts. On all 15 patients, a final diagnosis was established by an independent, valid gold standard. Five had septic arthritis, ten had something else. Here are the joint fluid WBC counts:
| Septic Arthritis | Not Septic Arthitis |
|---|---|
| 30 | 0 |
| 37 | 6 |
| 64 | 7 |
| 112 | 8 |
| 128 | 12 |
| 12 | |
| 23 | |
| 37 | |
| 48 | |
| 71 |
We are going to ask you to draw the ROC curve, so we are doing you the favor of sorting the test results from most abnormal to least abnormal:
| Septic Arthritis | No Septic Arthritis |
|---|---|
| 128 | |
| 112 | |
| 71 | |
| 64 | |
| 48 | |
| 37 | 37 |
| 30 | |
| 23 | |
| 12,12 | |
| 8 | |
| 7 | |
| 6 | |
| 0 | |
a. Draw an ROC curve for this test.
b. Estimate the area under the ROC curve. (Hint: Count boxes and divide by 5 ×10 =
50.)
c. Now assign ranks to each distinct result. The highest result gets Rank = 1. Assign the
average rank to ties. For example, if the same result appears twice after the result ranked
5, assign both occurrences the average rank (6 + 7)/2 = 6.5. If it occurs 5 times, assign all
5 occurrences rank 8 (the average of 6, 7, 8, 9, and 10). You can write the ranks next to
the values in the sorted list above. (Hint: you can check your answer by remembering
that the sum of all of the ranks should = N × (N+1)/2, where N is the total number of
subjects).
d. Now calculate the RANK SUM, S, as well as Smin and Smax.
e. Now use the formula given in Chapter 3 to determine the area under the ROC curve from these ranks. You should get the same answer you got for part b above. Isn’t that satisfying?
For math/puzzle nerds only:
f. If we measured WBC count out to enough significant digits, we could eliminate ties. If there are no ties, how many different possible ROC curves are there? This is the number of different 15-step paths from the lower left-hand corner to the upper right-hand corner, when the only allowed steps are up one or over one (to the right).
g. Again assuming no ties, how many different ROC Curve areas are there?
Back to all Chapter 3 Problems
Back to all Chapters