Copula Dependent Censoring Models for Survival Prognosis: Application to Lactylation-Related Genes

Abstract

Survival for cancer patients is predictable by gene expressions obtained from DNA microarrays for tumor samples. For analyzing survival data with gene expressions, traditional survival analysis methods have been employed. However, these methods rely on the independent censoring model. In real survival data, dependent censoring arises, which violates the fundamental assumption of independent censorship. In addition, how to handle dependent censoring has not been clearly demonstrated for scientists working on molecular genetics. In this article, we review copula-based methods to handle dependent censoring, including the copula-graphic estimator and significance test. We illustrate the copula-based method by the prognostic analysis of the lactylation-related genes from 327 breast cancer tumor tissues. To justify the correctness of the copula-based significance test, we examine the performance of the copula-based methods using a simulation study. The results of our analysis indicate that the copula-based analyses may reverse the conclusions derived from the traditional independent censoring model.

1. Introduction

Survival for cancer patients is predictable by gene expressions obtained from DNA microarrays for tumor samples [1,2,3,4,5,6]. In the gene expression data, one of the main goals is to assess the prognostic abilities of genes. For survival data with gene expressions, traditional survival analysis methods, such as the Kaplan–Meier (KM) survival curve and Cox regression, have been employed. Using these methods, one can assess genes via significance tests [5,6,7,8,9]. The resultant genes become useful biomarkers for predicting survival in breast cancer [10,11,12,13,14], lung cancer [6,7,8,15,16], gastric cancer [17,18,19], ovarian cancer [20,21,22,23,24,25], liver cancer [26,27], bladder cancer [28], head and neck cancer [29,30], myeloproliferative neoplasms [31], and cancers of mixed types [32,33].

These previous studies demonstrated the abilities of relevant genes to predict survival. However, a primary concern for these studies is the occurrence of dependent censoring (see Section 2), which may violate the fundamental assumptions needed for the traditional survival analysis methods. However, how to handle dependent censoring has not been clearly demonstrated for scientists working on molecular genetics. In this article, we review copula-based methods to handle dependent censoring with illustrations though the assessments of the prognostic values of the lactylation-related genes in cancer [34,35,36,37] using a breast cancer dataset of Kao et al. [10].

Traditional survival analysis methods critically rely on the “independent censoring model”, where survival time and censoring time are assumed to be statistically independent. This simplistic model may be invalid in poorly controlled clinical settings, where patients drop out from a follow-up study due to the worsening of his/her health condition or patients are removed due to organ transplantations [38,39]. Such a phenomenon is termed “dependent censoring” [39]. Moreover, dependent censoring naturally arises in clustering [40,41] and the significance analyses based on univariate gene selection [39,42,43,44]. In the presence of dependent censoring, traditional survival analysis methods may provide biased results [39,45], and, hence, fail to yield effective prognostic models based on gene expressions.

Copula-based survival models are novel approaches for gene selection [43,44], which can effectively adjust for the dependent censoring phenomenon. These models work if the underlying dependence structure is correctly specified by a copula function. In particular, the copula-graphic (CG) estimator unbiasedly estimates survival probabilities, serving as an alternative to the traditional KM estimates for survival [45]. Along with the mathematical integrity of copulas [46], these copula models have gained popularity in recent years, especially for dealing with survival data with dependent censoring [43,44,45,47,48,49,50,51,52,53,54,55]. However, these novel copula methods have not been widely utilized in genomic analyses. Therefore, there is an urgent need to explain how to use the copula models to scientists working on molecular genetics.

In this article, we demonstrate the usefulness of these copula-based approaches via an application to the prognostic evaluation of lactylation-related genes [37] using a publicly available breast cancer dataset [10]. We conclude that the copula-based analyses may reverse the conclusions derived from the independent censoring model. Specifically, genes declared to be significantly predictive of survival using the traditional methods are no longer significant under the copula-based models. This indicates that the traditional methods may overestimate the effect of genes when dependent censoring actually exists.

The article is constructed as follows: Section 2 describes the dataset and reviews the copula-based models for dependent censoring. Section 3 reports the results of analyzing the lactylation-related genes using the breast cancer dataset and the copula-based models. Section 4 provides a simulation study to validate the copula-based significance test. Section 5 concludes with the discussion. The Supplementary Materials section provides the code to reproduce all the numerical results of the article, as well as additional results of the data analysis.

2. Materials and Methods

In this section, we shall first describe the gene expression dataset to be analyzed. We then review copula-based methods [39,43,44,45] for analyzing the dataset.

2.1. Dataset

Lactylation-related genes are recently identified as key players in cancer biology [34,35,36,37]. Lactylation-related gene expressions (abundance of mRNA transcripts) measured in DNA microarray analyses are considered to be potential biomarkers for survival prognosis in various cancers. By regulating the mRNA transcription and the protein functions, the lactylation genes facilitate metabolic reprogramming. Large-scale sequencing has confirmed the widespread occurrence of lactylation sites across the tumor proteome.

In this context, we used the same breast cancer dataset as previously analyzed by Kao et al. [10]. This dataset was also recently reanalyzed in Jiao et al. [37] for understanding the role of lactylation-related genes. The dataset consists of $n=327$ breast cancer tumor tissues [10] publicly available on the GEO (https://www.ncbi.nlm.nih.gov/geo/ (accessed on 20 September 2025)) under the accession number GSE20685. This data was chosen because it had a large number of tissues in the dataset and included survival information of patients.

We denote the ID of tumors as $i (=1,2,\dots ,n)$. The dataset provides follow-up time $t_i$ (either death time or censoring time), survival status ${\delta }_i$ (1 for death, or 0 for censor), and a ${\mathrm{l}\mathrm{o}\mathrm{g}}_2$-transformed gene expression value $x_i$ for each tumor $i$. For gene expressions, we used five lactylation-related genes for the analysis of gene expressions as considered in Jiao et al. [37], which are sufficiently independent: ARID3A, PKM2, HBB, CCNA2, and G6PD. Thus, the dataset consists of triplets $(t_i,{\delta }_i,x_i)$, $i=1,2,\dots ,n$, where $n=327.$

Figure 1 displays scatter plots for the five lactylation-related gene expressions based on the 327 tumor tissues. We observe that the five gene expressions are only weakly dependent (pairwise Kendall’s tau correlations < 0.20). This implies that the five genes are reasonably independent predictors with little concern for multicollinearity.

2.2. Copula Model for Dependent Censoring

We first define basic notations and terms. Let $T$ be survival time and $U$ be censoring time. Let $x$ denote a ${\mathrm{l}\mathrm{o}\mathrm{g}}_2$-transformed expression value. The conditional survival functions given $x$ are denoted by $S\left(t|x\right)=P\left(T>t|x\right)$, and $G\left(u|x\right)=P\left(U>u|x\right)$, respectively. In the traditional survival analysis models, the independent censoring assumption is imposed, namely,

$$P\left(T>t,U>u|x\right)=S\left(t|x\right)G\left(u|x\right), t\ge 0, u\ge 0.$$

Copulas can relax the assumption of independent censoring. Copula-based dependent censoring models have gained popularity in recent years [39,43,44,45,46,47,48,49,50,51,52,53,54,55] along with other copula models in different fields [56,57,58,59]. For the purpose of measuring the prognostic ability of the five lactylation-related genes, we take the following copula model:

$$P\left(T>t,\hspace{0.33em}U>u|x\right)=C_{\theta }\left(S\left(t|x\right),G\left(u|x\right)\right), t\ge 0, u\ge 0,$$

(1)

where $C_{\theta }$ is a copula [46] with a parameter $\theta$ that specifies the strength of dependence.

The copula model (1) accommodates a variety of dependent censoring structures. For implementation, a copula with a simple form, such as the Clayton, Gumbel, or Frank copula, is suggested. As an instance, the Clayton copula takes the following expression:

$$C_{\theta }(v,w)=(v^{-\theta }+w^{-\theta }-1{)}^{-{\displaystyle \frac{1}{\theta }}},\hspace{1em}0<v,w<1,\hspace{1em}\theta >0.$$

The parameter $\theta$ specifies the strength of dependence. The degenerate case gives the independent censoring model $C_{\theta }\left(v,w\right)\to vw$ for $\theta \to 0$. To measure the degree of dependence, one can use Kendall’s tau, defined as $\tau =\theta /(\theta +2)$ under the Clayton copula. The range of $\tau$ varies from independence ($\tau =0$) to perfect positive dependence ($\tau =1$).

2.3. Copula-Graphic Estimator

Copula-graphic (CG) estimators provide unbiased estimates of survival curves when the copula in model (1) is correctly specified [45]. The virtue of the CG estimator is the ability to handle dependent censoring in a rigorous and efficient way. Therefore, the CG estimators are alternative methods to the traditional KM estimator.

To define the CG estimator, we introduce the structure of the observed dataset. Define $T_i$ as survival time, $U_i$ as censoring time, and $x_i$ as gene expression from tumors $i=1,2,\dots ,n$, where $n$ is the sample size. The dataset consists of triplets $(t_i,{\delta }_i,x_i)$, $i=1,2,\dots ,n$, where $t_i=\mathrm{m}\mathrm{i}\mathrm{n}(T_i,U_i)$ and ${\delta }_i=1\{T_i\le U_i\}$. We divide the dataset into the two groups via the gene expression ($x_i>c$ or $x_i\le c$) for a cut-off value $c$. The value $c$ may be chosen to be the 50th percentile (median) of the gene expressions to achieve robust groupings by eliminating the instability due to unevenly allocated sample sizes. Additionally, this rule has been commonly employed in the analysis of survival prognostic prediction with gene expressions [5,20,26,44].

The dataset yields two subsets: over-expressed group $\{i;x_i>c\}$ and under-expressed group $\{i;x_i\le c\}$. Let $n_1={\sum }_{i=1}^{n}1\{x_i>c\}$ and $n_2={\sum }_{i=1}^{n}1\{x_i\le c\}$ be the sample sizes of the two groups. For each group, we assume Archimedean copula models:

$$P\left(T_i>t,U_i>u|x_i>c\right)={\varphi }_{\theta }^{-1}\left[{\varphi }_{\theta }\left\{S\left(t\right|x_i>c)\right\}+{\varphi }_{\theta }\left\{G\left(u\right|x_i>c)\right\}\right],$$

(2)

$$P\left(T_i>t,U_i>u|x_i\le c\right)={\varphi }_{\theta }^{-1}\left[{\varphi }_{\theta }\left\{S\left(t\right|x_i\le c)\right\}+{\varphi }_{\theta }\left\{G\left(u\right|x_i\le c)\right\}\right],$$

(3)

where ${\varphi }_{\theta }$ is a generator function satisfying ${\varphi }_{\theta }(0)=\infty$ and ${\varphi }_{\theta }(1)=0$ [46].

If the models (2) and (3) are assumed, one can estimate survival curves for the two groups. For instance, one can estimate the survival function for the over-expressed group $S\left(t\right|x_i>c)$ by the CG estimator:

$${\widehat{S}}^{\mathrm{C}\mathrm{G}}(t|x_i>c)={\varphi }_{\theta }^{-1}\left[\sum _{j:t_j\le t, {\delta }_j=1, x_j>c}\left\{{\varphi }_{\theta }\left({\displaystyle \frac{n_{1j}-1}{n_1}}\right)-{\varphi }_{\theta }\left({\displaystyle \frac{n_{1j}}{n_1}}\right)\right\}\right],$$

where $n_{1j}={\sum }_{i=1}^{n}1\left\{t_i\ge t_j,x_i>c\right\}$ is the number of individuals at risk at time $t_j$. The survival function $S\left(t\right|x_i\le c)$ for the under-expressed group is also estimated in a similar formula. Note that the CG estimator reduces to the KM estimator by ${\varphi }_0\left(t\right)=-\log (t)$. For the Clayton copula, one has ${\varphi }_{\theta }(t)=(t^{-\theta }-1)/\theta \to -\log (t)$ with $\theta \to 0$. The copula generated by ${\varphi }_0\left(t\right)=-\log (t)$ is the independence copula: ${\varphi }_0^{-1}\left[{\varphi }_0(v)+{\varphi }_0(w)\right]=vw$. This means that the KM estimator is obtained as the limiting case of the Clayton CG estimator. Appendix A gives the formulas of the CG estimator under the Gumbel copula.

To compute the CG estimators, the parameter $\theta$ must be specified. However, it is difficult to be estimated by observed data due to their modest information for dependence [45,47,48,51,52]. Thus, a sensitivity analysis may be carried out for selected values of $\theta$ [39,44,45,48,49,50,51,52,53,54,55]. We suggest three values of $\theta$ yielding the low, medium, and high positive correlations (τ = 0.2, 0.5, and 0.8) and three values yielding the low, medium, and high negative correlations (τ = −0.2, −0.5, and −0.8). The computation can be carried out by the R functions, CG.Clayton(.), CG.Gumbel(.), and CG.Frank(.), available in the R package compound.Cox (version 3.33) [8].

2.4. Testing Significance of Survival Difference

The statistical significance of the gene can be visualized by plotting the CG estimators for a given copula parameter $\theta$. If ${\widehat{S}}^{\mathrm{C}\mathrm{G}}(t|x_i>c)$ and ${\widehat{S}}^{\mathrm{C}\mathrm{G}}(t|x_i\le c)$ are clearly separated, we conclude that the prognostic ability for the gene $x_i$ is significant. An objective metric of “separation” is based on a large value of $\left|D\right|$, where the distance $D$ is defined as

$$D={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }\{{\widehat{S}}^{\mathrm{C}\mathrm{G}}(t|x_i\le c)-{\widehat{S}}^{\mathrm{C}\mathrm{G}}(t|x_i>c)\}dt,$$

where $\tau =\mathrm{m}\mathrm{i}\mathrm{n}\{{\mathrm{m}\mathrm{a}\mathrm{x}}_{{ x}_i\le c}\left(t_i\right),{\mathrm{m}\mathrm{a}\mathrm{x}}_{{ x}_i>c}\left(t_i\right)\}$ as previously suggested [44]. A p-value is computed by a permutation method [44]. The computation can be carried out by the R function CG.test(.) available in the R package compound.Cox [8].

As this testing method has not been validated by means of a simulation study in the literature, we carried out an experiment under various conditions (see Section 4).

2.5. Hazard Ratio

Copula models can be used to estimate the hazard ratio (HR), a metric for the effect of gene expressions on survival. Although the metric $D$ from the CG estimator can be used to measure the effect, a more interpretable metric in survival analysis is the HR, defined as $\mathrm{e}\mathrm{x}\mathrm{p}\left(\beta \right)$, where $\beta$ is a coefficient in the Cox model [60] for a gene $x_i$. That is,

$$\lambda \left(t|x_i\right)={\lambda }_0\left(t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(\beta x_i\right),$$

where $\lambda \left(t|x_i\right)=-d\log \{P\left(T_i>t|x_i\right)\}/dt$ is the hazard function and ${\lambda }_0(.)$ is the baseline hazard function. Fitting survival data to the copula-based dependent censoring model, the estimator of the HR (and confidence interval) is obtained by the method of [44], which can also test $H_0:\beta =0$ vs. $H_1:\beta \ne 0$. The computation is carried out by the R function, dependCox.reg(.), available in the R package compound.Cox [8].

2.6. Prognostic Index

Once genes are selected, one can combine them into a predictor, called Prognostic Index (PI). The PI is a weighted sum of gene expressions $(x_1,\dots ,x_q)$. For weights $\beta$s computed by fitting suitable models, the PI is defined as

$$\mathrm{P}\mathrm{I}={\beta }_1{x}_1+\hspace{0.33em}\cdots +\hspace{0.33em}{\beta }_q{x}_q.$$

A convenient option for computing $\beta$s is fitting univariate Cox models for each gene. In this case, the PI is known as the compound covariate predictor [7,8,22,61]. The computation of $\beta$s can be implemented by the R function uni.score(.) available in the R package compound.Cox [8], or by a sequence of univariate Cox regression analyses.

By these weights, a high (low) value of the PI gives a low (high) survival probability. Thus, a patient is assigned to a low-risk ($\mathrm{P}\mathrm{I}<c$) or a high-risk ($\mathrm{P}\mathrm{I}>c$) group using $c$, the median of the PI values. The classification by the PI is supposed to be more accurate than the classifications by single genes since the PI combines the prognostic abilities of all genes.

An alternative method for computing $\beta$s is under the Clayton copula model as suggested by [44]. This may enhance the prediction ability under dependent censoring if the dependence parameter $\theta$ is estimated accurately. However, as the estimation of the parameter $\theta$ is time-consuming, we do not use this approach in this article.

3. Results

We report the results of the prognostic abilities of five lactylation-related genes, HBB, ARID3A, PKM2, CCNA2, and G6PD, using the breast cancer dataset and copula models described in Section 2. Our reports focus on the results under the Clayton copula model while those under the Gumbel and Frank copulas are given in Supplementary Materials. The copula-based models are compared to the traditional independent censoring model.

3.1. HBB Gene

The KM estimates (under the independent censoring model) of survival curves did not show a clear difference between the high- and low-HBB-expression groups, as presented in the top-left panel of Figure 2 (the independence case of $\theta =0$) and

Table 1. Test results for the difference in survival curves predicted by the five genes and PI (using ARID3A, CCNA2, and G6PD) under copula models for dependent censoring.

Model	$\mathbf{Parameter} \mathit{\theta }$	Difference Metric D (p-Value)
		HBB	ARID3A	PKM2	CCNA2	G6PD	PI
Clayton copula	$0.0$ (τ = 0.0)	0.043 (0.254)	−0.097 (0.007)	−0.018 (0.615)	−0.087 (0.018)	−0.107 (0.004)	−0.111 (0.003)
	$0.5$ (τ = 0.2)	0.044 (0.293)	−0.111 (0.006)	−0.027 (0.506)	−0.095 (0.020)	−0.11 (0.008)	−0.113 (0.006)
	$2.0$ (τ = 0.5)	0.040 (0.443)	−0.136 (0.010)	−0.041 (0.425)	−0.101 (0.054)	−0.097 (0.063)	−0.105 (0.046)
	$8$.0 (τ = 0.8)	0.028 (0.496)	−0.086 (0.047)	0.009 (0.824)	−0.072 (0.082)	−0.058 (0.151)	−0.058 (0.160)
Gumbel copula	$0$.00 (τ = 0.0)	0.043 (0.254)	−0.097 (0.007)	−0.018 (0.615)	−0.087 (0.018)	−0.107 (0.004)	−0.111 (0.003)
	$0.25$(τ = 0.2)	0.039 (0.305)	−0.099 (0.008)	−0.022 (0.543)	−0.081 (0.031)	−0.101 (0.007)	−0.102 (0.006)
	$1.00$ (τ = 0.5)	0.033 (0.422)	−0.098 (0.013)	−0.028 (0.475)	−0.073 (0.068)	−0.086 (0.029)	−0.085 (0.034)
	$4$.00 (τ = 0.8)	0.025 (0.514)	−0.083 (0.038)	−0.009 (0.821)	−0.07 (0.074)	−0.058 (0.128)	−0.059 (0.128)
Frank copula	$0$.00 (τ = 0.0)	0.043 (0.254)	−0.097 (0.007)	−0.018 (0.615)	−0.087 (0.018)	−0.107 (0.004)	−0.111 (0.003)
	$1.86$ (τ = 0.2)	0.041 (0.298)	−0.101 (0.007)	−0.024 (0.534)	−0.084 (0.029)	−0.104 (0.007)	−0.105 (0.007)
	$5.74$ (τ = 0.5)	0.035 (0.399)	−0.099 (0.016)	−0.027 (0.508)	−0.076 (0.065)	−0.088 (0.033)	−0.085 (0.041)
	$18.19$ (τ = 0.8)	0.027 (0.465)	−0.077 (0.044)	0.002 (0.968)	−0.07 (0.063)	−0.056 (0.123)	−0.057 (0.124)
	$-1.86$ (τ = −0.2)	0.043 (0.229)	−0.093 (0.009)	−0.015 (0.680)	−0.089 (0.013)	−0.107 (0.003)	−0.114 (0.001)
	$-5.74$ (τ = −0.5)	0.043 (0.217)	−0.088 (0.011)	−0.011 (0.747)	−0.089 (0.010)	−0.106 (0.003)	−0.114 (0.001)
	$-18.2$(τ = −0.8)	0.043 (0.220)	−0.085 (0.013)	−0.010 (0.778)	−0.088 (0.011)	−0.105 (0.003)	−0.114 (0.000)

(τ = 0). Likewise, in the CG estimators for survival curves accounting for various degrees of dependent censoring, we did not observe any notable effect of HBB (Figure 2, Table 1). The conclusion remains consistent under the Frank and Gumbel copulas (figures in Supplement). The Cox regression analysis also failed to reach significance (HR: 0.964, 95% CI: 0.78–1.2). The HR adjusted for various degrees of dependent censoring under various copula were also non-significant (

Table 2. Hazard ratio (HR) and p-values (in parenthesis) for the five gene expressions under the Clayton copula model with various degrees of dependence.

Model	$\mathbf{Parameter} \mathit{\theta }$	Hazard Ratio: exp(β) (p-Value)
		HBB	ARID3A	PKM2	CCNA2	G6PD	PI
Univariate Cox	None	0.964 (0.743)	2.103 (0.002)	1.229 (0.645)	1.222 (0.042)	1.611 (0.000)	1.995 (0.000)
Clayton copula	$0.1$ (τ = 0.05)	0.964 (0.736)	2.118 (0.001)	1.256 (0.288)	1.222 (0.033)	1.619 (0.007)	1.987 (0.000)
	$0.5$ (τ = 0.2)	0.969 (0.781)	2.095 (0.002)	1.135 (0.616)	1.222 (0.035)	1.599 (0.001)	1.959 (0.000)
	$2.0$ (τ = 0.5)	0.991 (0.934)	1.951 (0.002)	1.049 (0.701)	1.218 (0.030)	1.327 (0.011)	1.981 (n.a.)
	$8.0$ (τ = 0.8)	1.022 (0.843)	1.344 (0.468)	1.056 (n.a.)	1.216 (0.041)	1.303 (0.014)	1.122 (0.574)

n.a.: not available due to convergence issues.

). In summary, the HBB gene does not yield a significant influence on survival outcomes under varying dependent censoring scenarios.

3.2. ARID3A Gene

The ARID3A gene is significantly associated with survival under the independent censoring model. Specifically, KM estimators for survival curves demonstrate a clear separation between high- and low-expression groups as presented in Figure 3 (the case of $\theta =0$) and Table 1 (τ = 0). The Cox regression analysis yielded the HR of 2.1 (95% CI: 1.3–3.4), indicating that the unit increase in ARID3A expression increases the risk of death twice. However, for the CG estimators, the significance is reduced under the strongly dependent censoring scenario (τ $=0.8$). Indeed, the survival curves of the low- and high-expression groups cross at 10 years (τ $=0.8$ in Figure 3). Moreover, the HR approaches the null value when the dependence becomes stronger (Table 2). In summary, the significance declared using the traditional methods is reverted under the strongly dependent censoring scenario.

3.3. PKM2 Gene

The KM estimators for survival probabilities indicate no significant difference between high- and low-expression groups (Figure 4 (the case of $\theta =0$) and Table 1 (τ = 0)). Even using the CG estimators accounting for dependent censoring, no difference is found (Figure 4 and Table 1). Furthermore, Cox regression produced non-significant HRs under both the independent and dependent censoring scenarios (Table 2), suggesting no effective impact on survival under varying levels of dependent censoring. This result indicates that the PKM2 gene has no influence on survival under various dependent censoring scenarios.

3.4. CCNA2 Gene

The survival probabilities estimated by the KM estimator indicated a significant difference between high- and low-expression groups (Figure 5 (the case of $\theta =0$) and Table 1 (τ = 0)). After accounting for dependent censoring using the CG estimators, the significance was somewhat reduced to non-significant levels due to the crossing of two survival curves. Especially, the p-values become higher than 0.05 under Kendall’s tau τ = 0.5 and 0.8. On the other hand, estimates for HR for the high- and low-expression groups are significant across various dependent censoring scenarios. This implies that the CCNA2 gene expression has a potential ability to predict survival; yet, it alone may be not be strong enough to justify the predictive ability. Thus, this gene will be considered as a component of the PI.

3.5. G6PD Gene

The high and low gene expression levels give significantly different KM estimators for survival (Figure 6 for the case of $\theta =0$). The difference remains clears under the weak dependent censoring scenario (Figure 6 for the case of $\theta =0.2$); yet, it reduces to non-significant levels under moderate and strong dependent censoring scenarios. We also find statistically significant HRs for survival under the independent censoring model ($\theta$ = 0, τ = 0) (HR = 1.61, p < 0.001). The significance of HRs persists even after accounting for low-to-moderate dependent censoring. This observation demonstrates the predictive power of the G6PD gene expression under various dependent censoring scenarios.

3.6. Combining Five Genes

To combine the predictive abilities of individual genes, we created a PI by including significant genes and excluding non-significant genes under the independent censoring model (Table 2). The three genes yielded significance (ARID3A: HR = 2.10, p = 0.0023; CCNA2: HR = 1.22, p = 0.039; G6PD: HR = 1.55, p < 0.001). Using estimates $\beta$s from univariate Cox models using the uni.score(.) function, the resultant PI is the weighted sum:

$$\mathrm{P}\mathrm{I}=0.744\times x_{ARID3A}+0.194\times x_{CCNA2}+0.504\times x_{G6PD}$$

The PI exhibited quite a strong prognostic performance under independent censoring (τ = 0.00, HR = 1.995, p < 0.001). Moreover, the significance is stronger compared to the significance found for individual genes (Table 2). This means that the predictive power is enhanced by synthesizing the predictive power of the three genes. Moreover, the predictive power remains, even under weak or moderate degrees of dependent censoring (Table 2, Figure 7). However, the significance reduces under the strong dependent censoring at τ = 0.80: see Figure 7 (p = 0.1605) and Table 2 (HR = 1.122, p = 0.574).

In conclusion, although the PI yields a higher predictive ability than individual genes, its ability reduces to the non-significant level under the strong dependent censoring.

4. Simulation Study

We conducted a simulation study to validate the performance of the significance test based on a distance statistic $D$ (defined in Section 2.4). We examined if the type I error rate is well-controlled, and if the power is reasonably high. Such a performance study has not been conducted in the literature since the method was proposed by [44] and implemented in the R function CG.test(.) in the R package compound.Cox [8].

We simulated survival time $T_i$ and censoring time $U_i$ from exponential survival models $S\left(t|x_i\right)=\exp \left(-\exp \left(\beta x_i\right)t\right)$ and $G\left(u|x_i\right)=\exp \left(-u\right)$, where $x_i$ is a gene expression generated from the standard normal distribution. The value $\beta$ was set in the range $-1\le \beta \le 1$. We adopted the Clayton copula with $\theta =2$ for the dependence, specified as

$$P(T_i>t,\hspace{0.33em}{U}_i>u|x)={\left({S\left(t|x\right)}^{-\theta }+{G\left(u|x\right)}^{-\theta }-1\right)}^{-{\displaystyle \frac{1}{\theta }}}, t,u\ge 0,$$

(4)

Setting $t_i=\mathrm{m}\mathrm{i}\mathrm{n}(T_i,U_i)$ and ${\delta }_i=1\{T_i\le U_i\}$, we formed an observed dataset of size $n$, denoted as $(t_i,{\delta }_i,x_i)$, $i=1,2,\dots ,n$, with $n=50 \mathrm{a}\mathrm{n}\mathrm{d} 100$. A significance test was caried out by applying the dataset to the CG.test(.) function at the significance level of $\alpha$, 0.01, 0.05, and 0.10. We analyzed the power (rejection rates defined as the proportion of p-values less than $\alpha$) for the test based on 1000 repetitions. We plotted the power as a function of $\beta$ in the range $-1\le \beta \le 1$.

Figure 8 displays the power functions of $\beta$ for the CG.test(.). We observe that the power functions at $\beta =0$ are close to the significance level of 0.01, 0.05, and 0.10. Hence, the type I error rates are well-controlled. We also observe that the power functions become higher as $\beta$ deviates from zero or $n$ increases. This indicates the desired ability to detect the effect of gene expression on survival.

5. Conclusions and Discussion

In this article, we have introduced copula-based tools to cope with dependent censoring phenomena when developing survival prognostic models using gene expressions. These statistical tools consist of the CG estimator [45] for estimating survival probabilities and the Clayton–Cox model [43] for estimating HRs, both of which are valid under dependent censoring. These copula methods can effectively adjust for the impact of dependent censoring on survival data. We have also reviewed the R software (version 4.5.1) functions to implement these tools. While these copula methods are known in the statistical literature, they have not been widely used in the field of molecular genetics. Therefore, the major contribution of this article is to introduce these copula approaches to researchers of molecular genetics via its empirical application to the prognostic analyses of lactylation-related genes. Another methodological contribution is the simulation-based validation of the significance test based on a distance statistic $D$ (Section 4).

The dependent censoring phenomenon is an important concern among medical statisticians in a variety of survival analysis methods [38,39,59]. While the problem of dependent censoring is not avoidable in real datasets, the copula-based methods can assess the influence of dependent censoring. This article illustrates how to analyze the influence of dependent censoring by copula-based models for dependent censoring. An important finding from our data analysis is that the significance of the three genes found in the independent censoring model is reversed in the presence of strongly dependent censoring (Section 3). This implies that some prognostic genes previously declared “significant” should be re-analyzed using the dependent censoring models. In addition, a strong impact of dependent censoring on the analysis results will be found in future genomic studies.

When developing a multi-gene predictor, a common practice is to re-fit a multivariate Cox regression model based on the selected genes. However, we have reservations about this strategy due to the poor predictive performance observed in many papers [62]. Alternatively, we have suggested using the PI based on a compound covariate predictor (Section 2.6) that combines the results from univariate Cox models without going through a multivariate analysis. This approach is commonly used to build a prognostic model based on DNA microarray data in medical studies [22,43,44,61,63,64,65,66,67]. This approach has the advantage of directly incorporating the significance analysis of gene selections into the construction of a predictor. In the data analysis, we constructed the PI based on three significant genes (ignoring two non-significant genes). We conducted an additional analysis to confirm that the inclusion of non-significant genes (which results in a five-gene PI) do not give additional predictive power over the three-gene PI (Supplement).

Finally, all the results of this article are easily reproduced by the dataset and computer code (R code) available in the Supplementary Materials. The computation time to produce all the numerical results of the article is a few hours. If users wish to apply the copula models for dependent censoring, they can first run the R code, and then modify the dataset and code according to their needs.