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Cox proportional hazard model (Cox) with a binary covariate is used widely to assess the effect of intervention (I) on cardiorenal event risks. As Kaplan-Meier (KM) curves of I and control (C) approximate proportional hazards (PH), quality of data analyzed using Cox becomes high. Quantitative methods for assessing PH are unknown. If PH can be quantitatively estimated from published KM curves of I and C, data reliability is easier to assess. Thus, we conducted a simulation study using real-world data for quantitative evaluation of PH.
We used glucose levels (GL) measured at 15-minute intervals using continuous glucose monitoring (CGM: FreeStyle Libre Pro) as time-to-event data. This data was selected for two reasons. Firstly, there is no censoring; secondly, observation periods and timepoints are unified. We analyzed GL for 100 “outpatients with Type 2 diabetes” (pT2D), measured by CGM over 24-h for 13 days (from 0 AM on Day 2 to 0 AM on Day 15 [CGM attachment: Day 1]). The pT2D did not change their treatments throughout the CGM observation period. We analyzed cumulative survival rates [S(t)] using Cox [S(t)Cox] and KM method [S(t)KM] for 13 endpoints where time to event was expected to be delayed by I, compared to C (Figures). Observations were done at all GL measurement. The GL value at each of these timepoints is estimated to be identical to the GL values in the preceding 1–14 minutes for calculating metrics proposed later. We defined “–Loge(S(t)KM for I) [Log‘e’, ‘Napier's constant’] ÷ – Loge(S(t)KM for control [C])” as “e hazard ratio” (eHR). We proposed a quantitative metric, “absolute value of difference between eHR and HR” (|eHR–HR|) to estimate PH for time-to-event outcomes. “ΔLoge(–Loge(S(t)Cox)) [C – I]” (Δlog–logS(t)Cox) is constant at every observation time because Cox has a fundamental PH. Hence, we proposed a metric, absolute value of coefficient of variation (CV) of 18720 (60×24×13) "ΔLoge(–Loge(S(t)KM)) [C – I] at one-minute intervals [Δlog–logS(t)KM]" (|CV|Δlog–logKM) for estimating PH. Since we thought lower CV of 18720 “Δ‘1 – S(t)KM’ [C – I] at one-minute intervals [ΔKM]” (CVΔKM) better satisfy PH, we proposed a metric, CVΔKM, to visually assess PH from KM curves. Corresponding with the concept of Restricted Mean Survival Time (RMST), we propose a metric, RMST related index (RMSTrI), as sum of 18720 ΔKM.
Using |eHR–HR| as a quantitative metric, PH may be estimated from published KM curves. Catching variability of difference in S(t)KM at the same timepoint between I and C visually, PH may be estimated. Area between KM curves of I and C may reflect |1–HR|.