Reply to the Letter to the Editor: What Injury Mechanism and Patterns of Ligament Status Are Associated With Isolated Coronoid, Isolated Radial Head, and Combined Fractures?

To the Editor,

My colleagues and I would like to thank Drs. Ayubi and Safiri for their comments regarding our manuscript [3]. After reviewing their letter, we agree that our methods for calculating estimated odds ratios (ORs) and confidential internals (CI) were influenced by sparse-data bias, and, as the authors of the letter suggested, the data could be made more exact by performing a statistical compensation called penalization [1, 2].

We revised our estimated ORs and CIs using the penalization method, which we believe make our data more accurate, although our conclusions did not change. The ORs and CIs from chi square tests were reanalyzed using R statistical software, based on Bayesian logistic regression with penalization [1, 4]. We assumed prior 95% limits for all odds ratios to be 1/39 to 39, derived from a F(2,2) prior for the OR, as suggested by Greenland and Mansournia [1].

Next, we added the penalized ORs to Tables 1-3.

Table 1.

Number of patients in isolated coronoid and isolated radial head fractures groups based on MRI findings and the results of chi square test, odds ratio, and penalized odds ratio.

Table 3.

Number of patients in isolated radial head fractures and combined coronoid and radial head fractures groups based on MRI findings and the results of chi square test and odds ratios.

After reading your letter, and discussing your concerns with CORR’s Senior Editor, we agreed that the OR evaluations in Tables 1 and 3 showed something strange; for example, since complete rupture of the medial collateral ligament had an OR over 1, the OR for the lateral ulnar collateral ligament rupture (Table 1) should be below 1. We have modified and corrected the data with proper ORs using penalization as shown now in the revised Tables 1 and 3. We also found large sparse-data bias for the anteromedial facet fracture in isolated coronoid fracture and combined fracture. Accordingly, we have adjusted the ORs in Table 2 by using the penalization method.

Table 2.

Number of patients in coronoid process fracture of isolated coronoid process and combined fractures and results of chi square and Fisher’s exact tests.

Drs. Safiri and Ayubi also suggested that we apply the penalization approach to the analysis on medial bone contusion (Table 3). We tried again, but found that the chi square test did not work because there was no occurrence of isolated radial-head fractures in patients with medial bone contusions. Furthermore, we did not perform the penalization method for the lateral bone contusion (Table 3) because that OR was correct.