Establishing continuous reference intervals for liver function analytes using fractional polynomial regression

Introduction

Reference intervals (RIs) are crucial in contemporary healthcare as they provide critical support for clinical decision-making [1]. The establishment of RIs is conducted in accordance with the protocols set out by the Clinical Laboratory Standards Institute (CLSI) and the International Federation of Clinical Chemistry (IFCC). They are defined as the middle 95 % range of laboratory results obtained from a population of healthy individuals [2]. RIs are used to interpret a patient’s laboratory test results [3], [4], [5], particularly for liver function tests, such as alanine aminotransferase (ALT) and aspartate aminotransferase (AST). These tests are used to diagnose liver diseases and monitor treatment efficacy [6]. Inaccurate RIs can lead to misdiagnosis, ineffective treatment, and compromised patient outcomes. Therefore, improving RI determination methods is significant in contemporary healthcare [7], [8]. Particularly, the non-linear relationship between factors such as age and ALT/AST levels should be considered.

Traditional RI determination methods involve dividing the population into subgroups based on age, sex, and other factors and have several limitations [2], [9], [10], [11]. Particularly, these methods may not consider the non-linear relationship between these factors and ALT and AST levels. For example, categorizing age into discrete age groups may not accurately capture age-related variations in laboratory values. Consequently, lower and upper age limits may exhibit significant disparities when transitioning between two age groups, resulting in inaccurate medical diagnoses. Additionally, categorizing continuous predictors including age causes information loss, limiting RI accuracy [12], [13], [14], [15], [16], [17], [18]. To overcome these limitations and improve the accuracy of RIs for ALT and AST markers, alternative methods should be explored, such as fractional polynomial (FP) regression models [13], [17], [18].

FP regression models are statistical models that address the limitations of traditional RI determination methods [19], [20]. They are useful when variables are non-linear. They preserve the structure of continuous variables, such as age, by capturing non-linear relationships accurately. FP models are flexible and can capture non-linear relationships and thus are suitable for calculating RIs with continuous age variables including ALT and AST markers [21], [22].

This study aimed to (1) introduce and suggest FP models for RI calculation with continuous age and (2) estimate the reference limits based on this method for ALT and AST markers.

Materials and methods

Results

After outlier detection, this study included 4,051 and 2,667 patients for ALT and AST results, respectively. Among the ALT patients, 2,916 (72.0 %) were female, and the mean age ± SD was 35.08 ± 15.06 years. For AST, 1846 (69.2 %) were female, 821 (30.8 %) were male, and the mean age ± SD was 34.20 ± 14.95 years. The parameters had a non-normal distribution. Therefore, a logarithmic or a Box–Cox transformation was performed for each parameter. Then, the values were transformed to normal values. The Harris–Boyd method was used to check whether gender should be separated. The results indicated that ALT and AST measurements should be separated by gender.

Model selection with FP models

The FP models consist of different models according to the RA2 model selection procedures. The analysis procedure begins with an FP model of the maximum permitted complexity, determined by its dimension m₀. The FP functions with m₀=2 (FP2) are sufficient for most applications when beginning the analysis [21]. We studied m₀=2 and m₀=4 to demonstrate the interpretation of complex model structures.

Table 1 presents the FP models, which start at m₀=2, for the AST values of females. First, the null models were compared with the FP2 models (p=0.001), indicating that age affects the female AST. Thus, the FP2 model was selected. Second, the FP2 model was compared with the linear model. The process was stopped because the result was not significant (p=0.673). Age had a linear relationship with AST values in females, and the linear model was selected as a fit model for calculating RIs. Table 2 summarizes the RIs for AST values of females for 10 years apart. Supplemental Data Table S1 presents the RIs for each age.

Table 1:

FP models comparison for AST values of females.

Model	df	Deviance	Deviance difference	p-Valuea	Powers
Null	0	15,089.260	18.201	0.001
Linear	1	15,072.610	1.543	0.673	1
m=1, FP1	2	15,071.280	0.213	0.899	3
m=2, FP2	4	15,071.060	0.000	–	0.5 0.5

1. df, degrees of freedom and FP, fractional polynomial. ^ap-Values are comprising FP2 model, respectively.

Table 2:

Reference intervals of AST values of female individuals calculated by FP models.

	Centiles of ASTb
Age	0.05 (90 % CIa)	0.95 (90 % CIa)
20	11 (11.6–12.1)	24 (23.7–24.7)
30	11 (11.7–12.1)	24 (24.3–25.1)
40	11 (11.8–12.2)	25 (24.8–25.6)
50	11 (11.8–12.4)	25 (25.1–26.3)
60	12 (11.8–12.6)	26 (25.3–27.0)
70	12 (11.7–12.8)	26 (25.6–27.8)
80	12 (11.7–13.0)	26 (25.8–28.6)
90	12 (11.6–13.3)	27 (26.0–29.5)

1. CI, confidence interval; SD, standard deviation. ^aBootstrap confidence interval (5,000 iterations). ^bCalculated by FP models: mean model=16.975+0.025×Age and SD, model=3.522×0.012×Age.

We found that m₀=4 (FP4) is the most complex function only in female ALT values in Table 3. The closed testing procedure begins by comparing the best fitting. The null model, linear model, and the best FP1, FP2, and FP3 models were compared against the more complex reference model FP4, at α=0.05. First, the null model was compared with the best FP4 function, and it was significant. Thus, age was effective (p<0.001) on female ALT values. Then, the FP4 model was compared with the linear model. The test was significant, indicating that age had a non-linear relationship with female ALT values (p<0.001). Otherwise, the linear model would have been selected: compared with the best FP1 model, the FP4 model was selected because the tests were significant (p<0.001). Otherwise, the FP1 model should have been selected. A similar conclusion applies to the FP4 vs. the best FP2 comparison (p=0.006). Finally, FP4 was compared with the best FP3 model, indicating that FP4 was not significant (p=0.098). Thus, FP3 was selected as the best-fit model FP3.

Table 3:

FP models comparison for ALT values of females.

Model	df	Deviance	Deviance difference	p-Valuea	Powers
Null	0	23,869.310	245.601	<0.001
Linear	1	23,662.350	38.633	<0.001	1
m=1, FP1	2	23,653.200	29.490	<0.001	0.5
m=2, FP2	4	23,638.130	14.421	0.006	2 2
m=3, FP3	6	23,628.370	4.660	0.098	0 0.5 0.5
m=4, FP4	8	23,623.710	0.000	–	2 2 2 2

1. df, degrees of freedom and FP, fractional polynomial. ^ap-Values are comprising FP4 model, respectively.

According to the powers of the FP3 model (0, 0.5, 0.5), Table 4 summarizes the RIs for female ALT values for 10 years apart. Supplemental Data Table S2 provides the RIs for each age. Figure 1 illustrates the best-fitting model graphs.

Table 4:

Reference intervals of ALT values of female individuals calculated by FP models.

	Centiles of ALTb
Age	0.05 (90 % CIa)	0.95 (90 % CIa)
20	5 (6.1–6.7)	21 (21.3–23.1)
30	6 (7.1–7.6)	24 (24.5–26.2)
40	6 (7.7–8.3)	26 (26.4–28.3)
50	7 (8.1–8.8)	27 (27.7–29.6)
60	7 (8.4–9.3)	28 (28.0–31.2)
70	7 (8.5–9.8)	29 (27.5–33.0)
80	8 (8.5–10.3)	30 (26.7–34.9)
90	8 (8.4–10.8)	30 (25.6–36.9)

1. CI, confidence interval; SD, standard deviation. ^aBootstrap confidence interval (5,000 iterations). ^bCalculated by FP models: mean model=0.144+10.561×Log(Age)–0.183×Age^0.5–0.183×Age^0.5×ln (Age) and SD, model=1.561+2.600×Log (Age)+0.016×Age^0.5+0.016×Age^0.5×ln (Age).

Figure 1:

The best fitting model for AST and ALT of female individuals. This information is important for understanding the trends and patterns in AST and ALT values among females. One of the problems for the non-statistician is understanding the systematic approach used to compute the predicted mean and standard deviation, components of the Z score equation, which may vary as the independent variable changes over time (e.g., age) [18]. (A) AST tended to a linear increase in age and (B) ALT tended to a polynomial relation with age.

According to the FP analysis results (for m₀=2), Table 5 presents separately the comparison between the null models and FP2 models for male ALT and AST values. The tests were not significant (p=0.679, p=0.507, respectively), and the processes were stopped; the null models were selected. Therefore, age did not affect the male ALT and AST values. Therefore, RIs were calculated using traditional methods. The method is non-parametric because of the non-normality of values. Regardless of age, the lower and upper limits for ALT and AST were 8 (7–9) and 33 (32–33) and 12 (12–13) and 27 (27–28), respectively.

Table 5:

FP models comparison for ALT and AST values of male individuals.

ALT
Model	df	Deviance	Deviance difference	p-Valuea	Powers
Null	0	9,543.251	2.320	0.679
Linear	1	9,541.950	1.019	0.798	1
m=1, FP1	2	9,541.144	0.214	0.899	−1
m=2, FP2	4	9,540.931	0.000	–	0 2
AST
Model	df	Deviance	Deviance difference	p-Value a	Powers
Null	0	6,871.416	3.328	0.507
Linear	1	6,868.902	0.814	0.847	1
m=1, FP1	2	6,868.187	0.099	0.952	−2
m=2, FP2	4	6,868.088	0.000	–	−2 –2

1. df, degrees of freedom and FP, fractional polynomial. ^ap-Values are comprising FP4 model, respectively.

Discussion

We evaluated male and female ALT and AST levels to identify the best-fitting model and thus obtain RIs. Different methods were used to calculate RIs based on the relationship between age and liver function values. Female individuals showed a non-linear relationship between age and ALT values; thus, FP models were used to calculate RIs for each age. A linear model was used for female AST values, whereas the traditional non-parametric method was used for male ALT and AST values. Our findings showed variations in different RI calculation methods.

The reference ranges of 18-year-old females’ AST were 11–24 U/L. As age increases, the lower and upper limits of the range also increase. For the highest age of 92, the reference range was 12–27 U/L. Although ages with close gaps have slight changes, as the age gap increases, the difference widens, especially for the upper limit. The reference range for male AST is 12–27 U/L. Thus, men had higher values than women. For ALT, the reference ranges for women are 5–21 U/L and 8–30 U/L for the minimum (18) and maximum (92) ages, respectively. According to age, the change in the lower and upper limits of the reference range was greater in ALT than in AST. For males, the range is 8–33 U/L. These values are within the accepted reference ranges. Therefore, age and sex-specific RIs are crucial in clinical practice to reduce the error rate of the post-analytical process, the correct grading of aminotransferase elevation, and evaluation of the fold increase or proportional values of abnormal aminotransferases [1], [3], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. By using these intervals, healthcare professionals can better assess whether a patient’s test results fall within the expected range for their specific demographic group [9]. This assessment can aid in the diagnosis and monitoring of various conditions, such as hormonal imbalances or age-related changes in organ function [7], [8], [14].

Although the increase in aminotransferases guides the evaluation of liver diseases, such as chronic hepatitis C, autoimmune hepatitis, and nonalcoholic steatohepatitis, the exact limits on this issue are not completely clear [29]. Therefore, this study contributes to the literature significantly. Although different approaches are available to handle age in reference range calculations, researchers have traditionally divided RIs into age groups based on statistical significance and physiological relevance. Nevertheless, discrete RIs are insufficient to represent accurately the gradual change in analytes, such as ALT and AST, in response to age. The most common approach is to categorize age groups. However, to prevent information loss due to continuous variable categorization, FPs that keep the variable continuous and consider non-linearity have been recently proposed [17], [18], [19], [20], [21], [22], [23, 25], [26].

We compared the continuous and discrete RIs for only ALT and AST values for female because we calculated continuous RIs for them. We compare the results according to the Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) values. Discrete RIs were calculated by Decision Tree (DT) method. The sub-groups according to the Decision Tree are used to calculate the reference intervals. The model with the smallest value for AIC and BIC is the best model for the data. For ALT female, a 39.5-year cut-off value was determined for discrete RIs and AIC and BIC values were 4,592.59 and 4,597.59, respectively. We calculated these values for FP model 4,577.94 and 4,587.94, respectively. According to the results, continuous RIs were better than discrete RIs. For AST female, a cut-off value could not be calculated by DT and this showed that continuous method was more precise.

Our study reveals the importance of calculating using the FP models and providing results for all ages in making successful and accurate interpretations. This type of RI was referred to as continuous RI. Many continuous RI studies have been conducted recently, particularly in the pediatric population [7], [8], [14], [15], with several methodological studies conducted on modeling continuous RIs in addition to FP models [30], [31], [32], [33]. Non-parametric models including quantile regression with smoothing splines, such as natural, restricted cubic, and B-splines, and semi-parametric models including lambda, mu, sigma and generalized additive models for location, scale, and shape are common methods [7], [8]. The use of FPs in the construction of RIs has increasingly become popular. Therefore, we selected FP models as a statistical method in this study. Chitty and Altman published the first study used in clinical science regarding size charts for fetal bones (radius, ulna, humerus, tibia, fibula, femur, and foot) after fitting FPs [27]. Kurmanavicius et al. used this method to produce ranges for cephalic index, head circumference, and occipitofrontal diameter [34]. They also used FPs to estimate the mean abdominal diameter, circumference, and femur length [35]. Zierk et al. calculated pediatric RIs for 15 biochemical analytes using FP models [13]. Our study is the first to calculate the continuous RIs for adults using FP models.

The validity and reliability of the calculated RIs were assessed using bootstrap resampling techniques, which allowed for the estimation of confidence intervals and evaluation of potential sources of bias. The results indicated that the RIs were robust and stable, with minimal variation observed across different resampling iterations. Therefore, the calculated intervals are reliable and can be confidently used for clinical decision-making. However, the calculation process may still have inherent limitations and sources of error, such as measurement errors or variations in the dataset used. These limitations should be considered when interpreting and applying the RIs in practice.

The strengths of our study are as follows. First, continuous RIs represent accurately variations in biomarker concentrations that occur with aging [7], [8], [13]. Second, our sample size is sufficient, though continuous RIs may require a larger sample size or subgroups to ensure statistical power and accuracy than partitioning RIs [7]. It is recommended that laboratories determine their own reference intervals (RIs) and regularly validate them. However, determining the RIs through traditional methods requires resources. For this reason, many laboratories use the reference intervals (RIs) recommended by manufacturers, but the values provided often do not align with population values [36], [37]. For this reason, considering the challenges in traditional RI calculations, the practicality of RI calculations made using hospital data is higher. In this study, the existing AST and ALT reference intervals used were calculated for each age, whereas this study provided more precise reference interval values based on age and sex variables.

Demographic variables such as age and sex require the calculation of reference ranges for the population as specific as possible, reflecting the diversity among individuals. The traditional popRI model, in which population data is sorted from the lowest to the highest value to define RI, does not fully account for the natural biological variation of the measured entity [38]. However, it is emphasized that the genetic effects on RIs need to be investigated in more detail and that genotype-specific RIs should be produced [39]. Based on this information, we believe that the data obtained in this study will contribute to the ongoing transition towards individual reference ranges in clinical applications.

However, several limitations should be acknowledged. First, continuous RIs are not supported by laboratory information systems for clinical laboratory reporting due to mathematical functions. Furthermore, to guarantee accurate interpretation and use in clinical practice, the introduction of new RIs may require healthcare professionals complete training or education. Second, the process recommended by CLSI for determining reference intervals (RIs) is a direct method that involves the prior selection of the reference population [2]. However, due to the challenges encountered in the pre-analytical and analytical processes of this procedure, it is extremely difficult to carry it out properly. At the same time, it is time-consuming and also costly. For this reason, an indirect method is used that leverages routine laboratory data stored in the laboratory information system (LIS) to derive RIs [40], [41]. Therefore, the study was conducted in a single geographic region, which may limit the generalizability of the findings to other populations but the aim of this research is to investigate the applicability of fractional polynomial (FP) regression models as an alternative method for determining the reference range. Addressing these limitations and exploring potential sources of bias can enhance the validity and applicability of the research findings.

The study findings have important implications for clinical practice. By using age and sex-specific RIs, healthcare providers can make accurate diagnoses and treatment decisions, leading to improved patient outcomes. These RIs can also aid in the early detection of diseases or abnormalities, allowing for timely intervention and prevention of complications. Compared with discrete RIs, continuous RIs are more accurate [8]. Future research could explore the application of FP models in different populations, such as pediatric and geriatric patients, to determine whether the method performs equally well across different age groups. Additionally, incorporating additional covariates, such as BMI or medical history, into the regression analysis may further improve the accuracy of the RIs.