Reply to “Probability of Eradication Using Vancomycin Alone or in Combination for Methicillin-Resistant Staphylococcus aureus Bacteremia”

REPLY

We thank Dr. Chambers for correctly pointing out in his letter (1) that the odds ratio (OR) was misinterpreted as relative risk (RR) in our paper (2). However, the discrepancies between the OR values reported in our study and the corresponding RR values calculated by Chambers are mainly attributed to the decision to model the success rate (i.e., microbiological eradication) rather than the failure rate (i.e., microbiological failure) (1). In the classical cohort (or case-control) study, it is usually evident which level of the outcome variable should be used as the reference category. In other instances, including our study, the choice of whether to model the success rate or the failure rate is rather arbitrary.

The OR, by virtue of possessing the symmetry and invertibility properties, is not affected by the above choice. However, RR lacks these desirable mathematical properties, and its value may vary considerably based on the chosen outcome, as is the case in our study. To illustrate this, let us consider the association between the treatment group (VAN-alone versus Combo) and the outcome (microbiological failure versus microbiological eradication). The probability of microbiological failure in the VAN-alone group is 6/30 (20%), while the corresponding probability in the Combo group is 2/50 (4%). Dividing the former by the latter yields the unadjusted RR for microbiological failure: RR_failure = 5.0. The unadjusted RR of microbiological eradication (RR_eradication) for the combo group (versus the VAN-alone group), as shown by Chambers, is (48/50)/(24/30), or 1.2 (1).

The interpretation of RR_failure is that the VAN-alone group is 5 times more likely to experience microbiological failure than Combo group, while RR_eradication indicates that the Combo group is 1.2 times (or 20%) more likely to achieve microbiological eradication than the Van-alone group. This discordance between the two RRs may lead to a difficulty in assessing the effect of the treatment group on the outcome that may offset the more intuitive interpretation of the RR compared to the OR. Reporting RR_eradication alone, as done by Chambers, may obscure the much stronger association between treatment group and outcome reflected in RR_failure, as illustrated in the example above (1).

As explained above, the OR is symmetrical and invertible, and thus the unadjusted OR_failure, (6/24)/(2/48) (= 6), is equivalent to the unadjusted OR_eradication, (48/2)/(24/6) (= 6), and likewise, Firth's bias-corrected unadjusted OR reported in Table 4, OR_eradication, is equal to OR_failure, 5.15 (2). Therefore, the unadjusted OR_failure of 5.15 approximates an RR_failure of 5, while an OR_eradication of 5.15 greatly overestimates RR_eradication, which is 1.2. In general, the OR approximates the RR when the outcome is rare (e.g., microbiological failure), while the two measures of association will be discrepant when the outcome is common (e.g., microbiological eradication). In the former scenario, the adjusted OR from a logistic regression model may be interpreted as an adjusted RR. However, since reporting the RR leads to very different conclusions depending on whether one chooses the outcome to be microbiological eradication or microbiological failure (as explained above), we believe that OR is the appropriate summary statistic for our study. Therefore, the statement on page 104 in our paper (2) that was highlighted by Chambers (1) “Patients in the Combo group were 11.24 (95% CI, 1.7 to 144.3 times; P = 0.01) times more likely to achieve microbiological eradication than patients in the VAN-alone group” can be amended by correctly interpreting the OR as follows: patients in the Combo group had 11.24 times higher odds of achieving microbiological eradication than patients in the VAN-alone group (95% CI, 1.7 to 144.3 times; P = 0.01).

The above discussion regarding the comparability of the OR and RR when the outcome that is modeled is microbiological failure is also applicable to the relationship between hemodialysis and outcome. There were 12 hemodialysis patients, of which 5 experienced microbiological failure, and 68 nonhemodialysis patients, of which 3 experienced microbiological failure. Therefore, for hemodialysis versus nonhemodialysis, the unadjusted OR_failure, (5/7)/(3/65) (= 15.5), is equivalent to the unadjusted OR_eradication, (65/3)/(7/5) (= 15.5), for nonhemodialysis versus hemodialysis, and the corresponding Firth's bias-corrected unadjusted OR for hemodialysis status is the inverse of the OR reported in Table 4; i.e., OR_eradication = OR_failure = 1/0.073 = 13.7. The unadjusted RR_eradication is (65/68)/(7/12), or 1.6, while the unadjusted RR_failure is (5/12)/(3/68), or 9.4. Therefore, the unadjusted analysis indicates that hemodialysis patients were 9.4 times more likely to experience microbiological failure than nonhemodialysis patients. Reporting RR_eradication as 1.6 markedly obscures the strong association between hemodialysis and microbiological failure, and to avoid these inherent limitations of reporting RR in our study, we will retain our preference for reporting OR as the summary statistic. Regarding the second misstatement noted by Chambers (1) that hemodialysis patients were 24 times less likely to achieve eradication than nonhemodialysis patients, we propose the following amendment: “Additionally, hemodialysis patients had almost 24 times lower odds of achieving microbiological eradication than nonhemodialysis patients (AOR, 0.042; 95% CI, 0.01 to 0.25; P < 0.001), and this significant association persisted in the adjusted analysis (Table 5).”

Lastly, Chambers mentions that the association between treatment group and microbiological eradication/failure may have been confounded by factors related to hemodialysis (1). Out of the 12 hemodialysis patients in our study, 6 received VAN-alone and 6 received Combo. Five of the 12 hemodialysis patients experienced microbiological failure, and all 5 were in the VAN-alone group. A likely explanation is that hemodialysis status acts as an effect modifier, rather than a confounder, for the relationship between treatment group and microbiological eradication of MRSA bacteremia. That is, vancomycin monotherapy for the treatment of MRSA bacteremia is particularly ineffective in hemodialysis patients, while it may be more effective in nonhemodialysis patients. Conversely, combining vancomycin with a β-lactam may offer a distinct advantage over vancomycin monotherapy in the treatment of MRSA bacteremia in hemodialysis patients but not in nonhemodialysis patients. However, while it is difficult to test this effect modification and make a definitive conclusion given the small number of failures in our study, we believe that the apparent effectiveness of combining vancomycin with a β-lactam for the treatment of MRSA bacteremia in hemodialysis patients observed in our study is an interesting finding and worthy of future study.