Box-Cox Transformation Research Articles

Joint Optimization Strategy of Condition-Based Maintenance and Spare Parts Ordering for Nonlinear Degraded Equipment under Imperfect Maintenance

When existing methods consider the impact of the imperfect maintenance activities on random degraded equipment, they usually assume that the degraded equipment is linear, and set the number of imperfect maintenance activities in advance. However, in engineering practice, most of the degraded equipment is nonlinear, and the number of imperfect maintenance activities has an impact on the development of maintenance strategies. Therefore, this paper proposes a new joint maintenance strategy considering the uncertainty of the number of imperfect maintenance activities for nonlinear degraded equipment. First, nonlinear degradation data are linearized based on Box-Cox transformation (BCT), and the degradation model under the influence of imperfect activities is constructed by the random coefficient regression model. Accordingly, the remaining useful life (RUL) probability distribution can be derived. Secondly, the number of maintenances is calculated by imperfect maintenance level and success probability. Then, the detection cycle, preventive maintenance threshold and maintenance times are taken as decision variables. The optimization objective is to minimize the expected average cost. However, this is restricted by availability and probability of success for imperfect maintenance activities. A multi-objective joint optimization model of condition-based maintenance and spare parts ordering is constructed. Finally, the analysis results based on numerical examples verify the feasibility of the proposed joint optimization strategy.

Changing Box–Cox transformation parameter as an early warning signal for abrupt climate change

There are some critical thresholds in many complex dynamic systems. When a system approaches its critical threshold, the probability density distribution of the system might be changed. In view of this, we presented a new early warning indicator of an abrupt climate change based on the Box–Cox transformation, namely, the changing Box–Cox transformation parameter (BCTP). A significant increasing or decreasing trend in BCTP can always be found before the onset of an abrupt change in different folding models. Based on the Greenland paleotemperature data reconstructed through Greenland Ice Sheet Project II, we found that BCTP showed an increasing trend about 2800 years before the onset of Bølling–Allerød warming. The trends of BCTP are statistically significant at a significant level of of α = 0.05 for both artificial abrupt changes and Greenland paleotemperature. Compared with kurtosis coefficient and skewness coefficient, BCTP can present an effectively early warning in some cases of abrupt change where kurtosis coefficient or skewness coefficient showed a poor warning performance. In general, the variation magnitudes of BCTP are greater than that of kurtosis coefficient and skewness coefficient, and BCTP shows a better warning performance in most of cases. These results indicate that changing BCTP could be a good new indicator for warning upcoming abrupt changes.

Forecasting Indonesian Oil, Non-Oil and Gas Import Export with Fuzzy Time Series

Indonesia is active in export and import activities. Some of the commodities traded are oil and gas, as well as food and other industrial materials. Export and import activities play a role in determining the stability of the country's economy seen from its trade balance. According to the Central Statistics Agency, Indonesia experienced a deficit of USD 864 million due to a decline in exports at the beginning of 2020. Based on the state of the trade balance, the government needs to make policies in order to maintain the stability of the Indonesian economy. The right decision-making must be supported by accurate information, therefore, through this research, the value of Indonesia's exports and imports will be forecasted in the oil and gas and non-oil and gas sectors for the next period using the Fuzzy Time Series (FTS). FTS was chosen as the forecasting method because it is able to predict free real time data with arbitrary patterns. The data used is data on the value of exports and imports of oil and gas and non-oil and gas sectors for 2011-2020. To overcome the problem of stationary data variance and reduce the error value, a Box Cox transformation will be applied. The research stages include data transformation with Box Cox, forming universe and linguistic sets, determining interval length, fuzzification, forming FLR and FLR, defuzzification and forecasting. The final forecast results estimate that exports and imports in the oil and gas sector in 2021 will decline, while for the non-oil and gas sector will fluctuate and increase from the previous year. Forecasting with Box Cox transform data is more accurate with MAPE 19.56% and RMSE 121.52 compared to forecasting with original data with MAPE 74.89% and RMSE 132.09.

Data mining for prothrombin time and international normalized ratio reference intervals in children.

Reference intervals (RIs) help physicians in differentiating healthy from sick individuals. The prothrombin time (PT) and International normalized ratio (INR) fluctuate in coagulation pathway defects and have interlaboratory variability due to the instrument/reagent used. As direct method is difficult in children, we chose an indirect data mining method for the determining PT/INR RIs. The indirect method overcomes the substantial financial and logistic challenges, and ethical restrictions in children, moreover, allows partitioning in more fine-grained age groups. Prothrombin Time/INR measurements performed in patients aged birth-18 years between January 2013 and December 2020, were retrieved from laboratory management system of the Aga Khan Hospital. Reference intervals were computed using an indirect KOSMIC algorithm. The KOSMIC package function on the assumption that the non-pathologic samples follow a Gaussian distribution (after Box-Cox transformation of the data), following an elaborate statistical process to isolate distribution of physiological samples from mixed dataset. A total of 56,712 and 52,245 values were retrieved for PT and INR respectively. After the exclusion of patients with multiple specimens obtained during the study period, RIs were calculated for 37,356 (PT) and 37,192 (INR) children with stratification into 9 age groups. A comparison of 2.5th and 97.5th percentile results with those of established RIs from SickKids Handbook of Pediatric Thrombosis and Hemostasis demonstrated good agreement in between different age groups. This study supports data mining as an alternate approach for establishing PT/INR RIs, specifically in resource-limited settings. The results obtained are specific to studied population and instrument/reagent used. The study also allows understanding of fluctuations in coagulation pathways with increasing age and hence better clinical decision-making based on PT and INR results.

Improving the normalization procedure of the simplified standardized precipitation index (SSPI) using Box–Cox transformation

Improving the normalization procedure of the simplified standardized precipitation index (SSPI) using Box–Cox transformation

A Retrospective Analysis of NRG/RTOG 0522: Low Myofibroblast Content in the Tumor-Associated Stroma may Predict Response to Epidermal Growth Factor Receptor (EGFR) Inhibitors

<h3>Purpose/Objective(s)</h3> Most head and neck cancers (HNC) overexpress EGFR, a feature associated with poor clinical outcome. Preclinical and clinical data suggest that peritumoral myofibroblasts (MF) express EGFR ligands and the presence of a high MF density in the tumor stroma is associated with statistically worse survival in patients with locally advanced HNC. We evaluated whether peritumoral MF density was prognostic of outcome in patients treated on RTOG 0522, a negative phase III trial evaluating the addition of C225 to cisplatin-based chemoradiation (CRT). We also evaluated whether MF density was predictive of benefit to C225. <h3>Materials/Methods</h3> Specimens from HPV-negative patients treated on RTOG 0522 were obtained from the NRG Oncology Biospecimen Repository and evaluated by a blinded pathologist (JW). MF density was determined as a percentage and the skewed data were normalized using Box-Cox transformation, from which the mean was calculated and used to stratify patients into low- and high-density groups. Overall survival (OS), progression-free survival (PFS), and distant metastasis (DM) rates were estimated using the Kaplan-Meier method and a multivariable analysis (MVA) by Cox proportional hazards model was used to test for the prognostic effect of this biomarker for patients with low vs. high MF density. To test for predictive effect of MF density on clinical outcomes in patients treated with CRT vs. CRT+C225, a Cox model with a treatment-by-biomarker interaction term was used. <h3>Results</h3> A total of 186 patients were evaluated, 94 were treated with CRT and 92 were treated with CRT+C225. On MVA, including primary site, age, PS, smoking history, T-stage, N-stage, and EGFR level, there was no significant difference in OS (Hazard ratio [HR], 0.65; 95% CI, 0.41-1.02; P=0.06), PFS (HR, 0.75; 95% CI, 0.50-1.12; P=0.16), or DM (HR, 0.76; 95% CI, 0.38-1.54; P=0.44) based on MF density. However, MVA Cox model including biomarker, treatment arm and biomarker-treatment interaction demonstrated that this interaction was significant (P=0.01). Within the low MF density subgroup, patients treated with CRT+C225 had significantly higher PFS (HR, 0.42; 95% CI, 0.21 to 0.82; P=0.01) and lower incidence of DM (HR, 0.18; 95% CI, 0.04 to 0.82; P=0.01), as well as numerically higher OS (HR, 0.48; 95% CI, 0.22 to 1.05; P=0.06) compared to patients treated with CRT. Patients with high MF density demonstrated no difference in OS, PFS, or DM based on treatment arm. <h3>Conclusion</h3> Peritumoral MF density was not prognostic of OS, PFS, or DM among patients treated on RTOG 0522. However, patients with low peritumoral MF density who received CRT+C225 had significantly higher PFS and lower incidence of DM compared to patients who received CRT alone. These results suggest that MF density is not a prognostic biomarker but may be predictive of benefit to C225. Further investigations are needed to validate these exploratory findings.

APRO: Secret Key Based Data Encryption and Mutual Authentication with Privacy Preservation Model in IoT Using Autoregressive Poor and Rich Optimization

An essential security component for the Internet of Things (IoT) is device authentication. IoT devices are widely dispersed in public and open areas, making them vulnerable to attacks. To detect the security violations, an authentication with a privacy preservation model is developed in this research using devised Autoregressive Poor and Rich Optimization (APRO)-based secret key. Here, the secret key used to perform the encryption process is generated by the proposed optimization algorithm in such a way that the data encryption mechanism is carried out by considering the Box-Cox transformation. However, the authentication process is achieved between the server and the Authorization center (AC) through the generation of authentication and verification messages. The original data is retrieved back by employing the decryption process along with the usage of the retrieval key. The proposed method achieved a minimum computation time of 0.062 sec, higher encryption quality of 92.56%, and minimum memory usage of 2.86MB.

ARIMA MODEL FOR FORECASTING THE BITCOIN EXCHANGE RATE AGAINST THE USD

This study analysis forecasting the bitcoin exchange rate against the USD. The dataset selected for this study starts from January 2015 to June 2022. This study's methodology uses autoregressive integrated moving average forecasting (ARIMA). The overall outcomes of this study were gathered from the statistical software Minitab 21.1. The Box Jenkins approaches are also used to predict the best model. To determine the ARIMA model parameter, this study did autocorrelation function (ACF) and partial autocorrelation function (PACF) analyses. According to the Box-Cox transformation method, log transformation was selected. The outcome demonstrates that the seasonal with the regular difference in the Bitcoin exchange rate against the USD is a stationary data series. The forecasting model used in this study is ARIMA (1,1,0) (2,1,1)12. This predicted model is identified through the Mean squared error by comparing the other guessing ARIMA models. After the prediction, 5 Month bitcoin exchange rate against the USD. Investors will be able to estimate the bitcoin exchange rate against the USD with the use of this information, but volatility must also be properly watched. This will aid investors in making better investment decisions and increase profits. In future studies, better consider another exchange rate of BTC and software experts will develop such type of software based on ARIMA models for prediction.

NONLINEAR REGRESSION MODELS FOR ESTIMATING THE DURATION OF SOFTWARE DEVELOPMENT IN JAVA FOR PC BASED ON THE 2021 ISBSG DATA

Context. The problem of estimating the duration of software development in Java for personal computers (PC) is important because, first, failed duration estimating is often the main contributor to failed software projects, second, Java is a popular language, and, third, a personal computer is a widespread multi-purpose computer. The object of the study is the process of estimating the duration of software development in Java for PC. The subject of the study is the nonlinear regression models to estimate the duration of software development in Java for PC.&#x0D; Objective. The goal of the work is to build nonlinear regression models for estimating the duration of software development in Java for PC based on the normalizing transformations and deleting outliers in data to increase the confidence of the estimation in comparison to the ISBSG model for the PC platform.&#x0D; Method. The models, confidence, and prediction intervals of nonlinear regressions to estimate the duration of software development in Java for PC are constructed based on the normalizing transformations for non-Gaussian data with the help of appropriate techniques. The techniques to build the models, confidence, and prediction intervals of nonlinear regressions are based on normalizing transformations. Also, we apply outlier removal for model construction. In general, the above leads to a reduction of the mean magnitude of relative error, the widths of the confidence, and prediction intervals in comparison to nonlinear models constructed without outlier removal application in the model construction process.&#x0D; Results. A comparison of the model based on the decimal logarithm transformation with the nonlinear regression models based on the Johnson (for the SB family) and Box-Cox transformations as both univariate and bivariate ones has been performed.&#x0D; Conclusions. The nonlinear regression model to estimate the duration of software development in Java for PC is constructed based on the decimal logarithm transformation. This model, in comparison with other nonlinear regression models, has smaller widths of the confidence and prediction intervals for effort values that are bigger than 900 person-hours. The prospects for further research may include the application of bivariate normalizing transformations and data sets to construct the nonlinear regression models for estimating the duration of software development in other languages for PC and other platforms, for example, mainframe.

Corporate Probability of Default: A Single-Index Hazard Model Approach

Corporate probability of default (PD) prediction is vitally important for risk management and asset pricing. In search of accurate PD prediction, we propose a flexible yet easy-to-interpret default-prediction single-index hazard model (DSI). By applying it to a comprehensive U.S. corporate bankruptcy database we constructed, we discover an interesting V-shaped relationship, indicating a violation of the common linear hazard specification. Most importantly, the single-index hazard model passes the Hosmer-Lemeshow goodness-of-fit calibration test while neither does a state-of-the-art linear hazard model in finance nor a parametric class of Box-Cox transformation survival models. In an economic value analysis, we find that this may translate to as much as three times of profit compared to the linear hazard model. In model estimation, we adopt a penalized-spline approximation for the unknown function and propose an efficient algorithm. With a diverging number of spline knots, we establish consistency and asymptotic theories for the penalized-spline likelihood estimators. Furthermore, we reexamine the distress risk anomaly, that is, higher financially distressed stocks deliver anomalously lower excess returns. Based on the PDs from the proposed single-index hazard model, we find that the distress risk anomaly has weakened or even disappeared during the extended period.

Equivalence tests for the difference of two survival functions under the class of Box–Cox transformation model

Equivalence tests for the difference of two survival functions under the class of Box–Cox transformation model

Early size estimation of web apps created using codeigniter framework by nonlinear regression models

Subject matter: Early software size estimation is one of the project managers' significant problems in evaluating app development efforts because software size is the major determinant of software project effort. Function points (FPs) and lines of code (LOC) are most commonly used as measures of size in existing software effort estimation methods and models. As is known, both these metrics have their advantages and disadvantages when used for software effort estimation. Although the FPs-based measure has the advantage over the LOC in that it does not depend on the technologies used, however, the assessment of efforts requires considering such factors (environmental factors). Considering the above factors can be ensured by appropriate models for estimating the LOC-based effort. Nowadays, many Web apps are created using PHP frameworks making the app development faster. CodeIgniter is one such powerful framework. However, there are no regression models for estimating the software size of Web apps created using the CodeIgniter framework. This requires the construction of the appropriate models. The task of this paper is to develop a nonlinear regression model for estimating the software size (in KLOC, kilo lines of code) of Web apps created using the CodeIgniter framework. Method: We apply the technique for constructing nonlinear regression models based on the multivariate normalizing transformations and prediction intervals. The result is three nonlinear regression models with three predictors: the total number of classes, the average number of methods per class, and the DIT (Depth of Inheritance Tree) average per class. To build these models for estimating the size of Web apps created using the CodeIgniter framework, we used three well-known normalizing transformations: two univariate transformations (the decimal logarithm and the Box-Cox transformation) and the Box-Cox four-variate transformation. Conclusions. The nonlinear regression model constructed by the Box-Cox four-variate transformation has better size prediction results than other regression models based on the univariate transformations.

Polycystin-2 (TRPP2) regulates primary cilium length in LLC-PK1 renal epithelial cells.

Polycystin-2 (PC2, TRPP2) is a Ca2+ permeable nonselective cation channel whose dysfunction generates autosomal dominant polycystic kidney disease (ADPKD). PC2 is present in different cell locations, including the primary cilium of renal epithelial cells. However, little is known as to whether PC2 contributes to the primary cilium structure. Here, we explored the effect(s) of external Ca2+, PC2 channel blockers, and PKD2 gene silencing on the length of primary cilia in wild-type LLC-PK1 renal epithelial cells. Confluent cell monolayers were fixed and immuno-labeled with an anti-acetylated α-tubulin antibody to identify primary cilia and measure their length. Although primary cilia length measurements did not follow a Normal distribution, the data were normalized by Box-Cox transformation rendering statistical differences under all experimental conditions. Cells exposed to high external Ca2+ (6.2 mM) decreased a 13.5% (p < 0.001) primary cilia length as compared to controls (1.2 mM Ca2+). In contrast, the PC2 inhibitors amiloride (200 μM) and LiCl (10 mM), both increased primary ciliary length by 33.2% (p < 0.001), and 17.4% (p < 0.001), respectively. PKD2 gene silencing by siRNA elicited a statistically significant, 10.3% (p < 0.001) increase in primary cilia length compared to their respective scrambled RNA transfected cells. The data indicate that conditions that regulate PC2 function or gene expression modify the length of primary cilia in renal epithelial cells. Blocking of PC2 mitigates the effects of elevated external Ca2+ concentration on primary cilia length. Proper regulation of PC2 function in the primary cilium may be essential in the onset of mechanisms that trigger cyst formation in ADPKD.

Automatic differentiation of Box-Cox transformations with application to random effects models for continuous non-normal response

Automatic differentiation of Box-Cox transformations with application to random effects models for continuous non-normal response

Traffic Volatility Forecasting Using an Omnibus Family GARCH Modeling Framework

Traffic volatility modeling has been highly valued in recent years because of its advantages in describing the uncertainty of traffic flow during the short-term forecasting process. A few generalized autoregressive conditional heteroscedastic (GARCH) models have been developed to capture and hence forecast the volatility of traffic flow. Although these models have been confirmed to be capable of producing more reliable forecasts than traditional point forecasting models, the more or less imposed restrictions on parameter estimations may make the asymmetric property of traffic volatility be not or insufficiently considered. Furthermore, the performance of the models has not been fully evaluated and compared in the traffic forecasting context, rendering the choice of the models dilemmatic for traffic volatility modeling. In this study, an omnibus traffic volatility forecasting framework is proposed, where various traffic volatility models with symmetric and asymmetric properties can be developed in a unifying way by fixing or flexibly estimating three key parameters, namely the Box-Cox transformation coefficient , the shift factor , and the rotation factor . Extensive traffic speed datasets collected from urban roads of Kunshan city, China, and from freeway segments of the San Diego Region, USA, were used to evaluate the proposed framework and develop traffic volatility forecasting models in a number of case studies. The models include the standard GARCH, the threshold GARCH (TGARCH), the nonlinear ARCH (NGARCH), the nonlinear-asymmetric GARCH (NAGARCH), the Glosten–Jagannathan–Runkle GARCH (GJR-GARCH), and the family GARCH (FGARCH). The mean forecasting performance of the models was measured with mean absolute error (MAE) and mean absolute percentage error (MAPE), while the volatility forecasting performance of the models was measured with volatility mean absolute error (VMAE), directional accuracy (DA), kickoff percentage (KP), and average confidence length (ACL). Experimental results demonstrate the effectiveness and flexibility of the proposed framework and provide insights into how to develop and select proper traffic volatility forecasting models in different situations.

Ensemble Based Box-Cox Transformation via Meta Analysis

Normal distribution has a vital role for the most of statistical methods. Box-Cox power transformation is the most usually applied method when the distribution of data is not normal. In this study, a novel algorithm is proposed assembling different Box-Cox transformation estimates of the well performed six techniques through random effect model in meta analysis. These techniques include the use of goodness-of-fit tests for normality; Anderson–Darling, Lilliefors, Cramer-von Mises, Shapiro–Wilk, Jarque–Bera and Shapiro–Francia tests. For the estimation of Box-Cox parameter, we assemble all possible combinations (63 combinations) of estimates calculated by these six methods. A Monte-Carlo simulation study is implemented to investigate which combination performs better compared to the rest. The simulation study states that the combination of Shapiro–Wilk, Jarque–Bera and Ander-son–Darling tests performs well in most of the simulation scenarios constructed under different transformation parameters and sample sizes. In this study, this combination is proposed as ensemble based Box-Cox transformation via meta analysis. The proposed approach is implemented on white blood count data of leukaemia patients which are not normally distributed. Also, the proposed methodology is provided in AID R package with “box-coxmeta” function for public use.

Forecasting admissions in psychiatric hospitals before and during Covid-19: a retrospective study with routine data

The COVID-19 pandemic has strong effects on most health care systems. Forecasting of admissions can help for the efficient organisation of hospital care. We aimed to forecast the number of admissions to psychiatric hospitals before and during the COVID-19 pandemic and we compared the performance of machine learning models and time series models. This would eventually allow to support timely resource allocation for optimal treatment of patients. We used admission data from 9 psychiatric hospitals in Germany between 2017 and 2020. We compared machine learning models with time series models in weekly, monthly and yearly forecasting before and during the COVID-19 pandemic. A total of 90,686 admissions were analysed. The models explained up to 90% of variance in hospital admissions in 2019 and 75% in 2020 with the effects of the COVID-19 pandemic. The best models substantially outperformed a one-step seasonal naïve forecast (seasonal mean absolute scaled error (sMASE) 2019: 0.59, 2020: 0.76). The best model in 2019 was a machine learning model (elastic net, mean absolute error (MAE): 7.25). The best model in 2020 was a time series model (exponential smoothing state space model with Box-Cox transformation, ARMA errors and trend and seasonal components, MAE: 10.44). Models forecasting admissions one week in advance did not perform better than monthly and yearly models in 2019 but they did in 2020. The most important features for the machine learning models were calendrical variables. Model performance did not vary much between different modelling approaches before the COVID-19 pandemic and established forecasts were substantially better than one-step seasonal naïve forecasts. However, weekly time series models adjusted quicker to the COVID-19 related shock effects. In practice, multiple individual forecast horizons could be used simultaneously, such as a yearly model to achieve early forecasts for a long planning period and weekly models to adjust quicker to sudden changes.

Predictive kappa (κ) models for Turkey: Regional effects and uncertainty analysis

High-frequency attenuation of ground motions is commonly characterized by the κ parameter. κ and its value at zero distance, [Formula: see text], are critical parameters for site characterization, ground motion simulations, and adjustment of ground motion models for different site conditions. In this study, we derive regional κ models for Eastern, Western, and Northwestern Turkey, using a ground motion data set of 3212 strong ground motion records with 9636 components from 1053 earthquakes, recorded at 178 stations with varying site conditions. Initially, models with linear functional forms are derived with distance and site conditions as predictor variables. Residual analyses showed that these models have non-normal residuals with heterogeneous variances (heteroscedasticity). Box–Cox transformations on κ values are performed to address these two issues. Models using these transformations resulted in normal residuals with homogeneous variances (homoscedasticity). Effects of earthquake magnitude on the κ value are also investigated, and a linear adjustment for magnitudes is proposed. A constant vertical-to-horizontal ratio of 0.78 is proposed to estimate the vertical κ values instead of deriving separate models for the vertical component. This approach does not result in significant decreases in accuracy and is preferred due to its simplicity. Next, the physical implications of logarithmic transformations for the κ values are discussed. Finally, regional differences in κ models are highlighted.

An Objective Approach to Deriving the Clinical Performance of Autoverification Limits.

This study describes an objective approach to deriving the clinical performance of autoverification rules to inform laboratory practice when implementing them. Anonymized historical laboratory data for 12 biochemistry measurands were collected and Box-Cox-transformed to approximate a Gaussian distribution. The historical laboratory data were assumed to be error-free. Using the probability theory, the clinical specificity of a set of autoverification limits can be derived by calculating the percentile values of the overall distribution of a measurand. The 5th and 95th percentile values of the laboratory data were calculated to achieve a 90% clinical specificity. Next, a predefined tolerable total error adopted from the Royal College of Pathologists of Australasia Quality Assurance Program was applied to the extracted data before subjecting to Box-Cox transformation. Using a standard normal distribution, the clinical sensitivity can be derived from the probability of the Z-value to the right of the autoverification limit for a one-tailed probability and multiplied by two for a two-tailed probability. The clinical sensitivity showed an inverse relationship with between-subject biological variation. The laboratory can set and assess the clinical performance of its autoverification rules that conforms to its desired risk profile.

Indirect Method for Estimation of Reference Intervals of Inflammatory Markers.

The direct method for reference interval (RI) estimating is limited due to the requirement of resources, difficulties in defining a non-diseased population, or ethical problems in obtaining samples. We estimated the RI for inflammatory biomarkers using an indirect method (RII). C-reactive protein (CRP), erythrocyte sedimentation rate (ESR) and presepsin (PSEP) data of patients visiting a single hospital were retrieved from April 2009 to April 2021. Right-skewed data were transformed using the Box-Cox transformation method. A mixed population of non-diseased and diseased distributions was assumed, followed by latent profile analysis for the two classes. The intersection point of the distribution curve was estimated as the RI. The influence of measurement size was evaluated as the ratio of abnormal values and adjustment (n×bandwidth) of the distribution curve. The RIs estimated by the proposed RII method (existing method) were as follows: CRP, 0-4.1 (0-4.7) mg/L; ESR, 0-10.2 (0-15) mm/hr and PSEP, 0-411 (0-300) pg/mL. Measurement sizes ≥2,500 showed stable results. An abnormal-to-normal value ratio of 0.5 showed the most accurate result for CRP. Adjustment values ≤5 or >5 were applicable for a measurement size <25,000 or ≥25,000, respectively. The proposed RII method could provide additional information for RI verification or estimation with some limitations.