Comparison of generalized estimating equations and Gaussian copula regression results using data from the randomized control trial

doi:10.21203/rs.3.rs-1970527/v1

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Comparison of generalized estimating equations and Gaussian copula regression results using data from the randomized control trial

https://doi.org/10.21203/rs.3.rs-1970527/v1

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Background:

In repeated measures data the observations tend to be correlated within each subject and such data are often analysed using Generalized Estimating Equations which is robust to assumptions that many methods hold. Correlation within-subject can be accounted for in working correlation structures. The main limitation of GEE is that its method of estimation is quasi-likelihood. The latest framework of the copula is very popular for handling correlated data. The likelihood-based analysis for correlated data can be obtained using Gaussian copula regression. The main advantage of copula regression model is that there are no boundaries on the probability distributions that can be used. The purpose of this study is to compare the findings of GEE and Gaussian copula regression using randomized controlled trial data for a continuous outcome along with different correlation structures.

Methods:

The prospective, double-blinded, randomized controlled trial data for this study was obtained from the Department of Anaesthesia, Christian Medical College, Vellore. ASA I and II patients were randomized into three groups. Hemodynamic parameters were obtained for 88 patients at thirteen time points. The outcome of interest was mean arterial pressure. Both GEE and Gaussian copula regression were compared assuming four different correlation structures. The optimal correlation structures were selected with the Akaike Information Criterion (AIC) and Correlation Information Criterion (CIC) goodness of fit criteria according to the method of estimation of Gaussian copula regression and GEE respectively.

Results:

The correlation structures unstructured and autoregressive were found to be optimal using simulation studies for Gaussian copula regression and GEE based on AIC and CIC criteria values respectively. Comparison between the estimated values of the selected models showed no major differences, except that Gaussian copula regression identifies interaction term, intrathecal morphine over time having significant association with MAP, this significance is considered to be important as the study uses a randomized controlled trial data.

Conclusions:

Both methods have almost similar results, but Gaussian copula regression provides better results by identifying significant variables associated with the outcome variable using maximum likelihood estimation that GEE fails to identify using quasi-likelihood estimation.

Correlated data

Correlation structures

Gaussian copula regression

GEE

Repeated measures data

Repeated data arise frequently in biomedical and many other research fields. In the longitudinal studies, the correlation usually occurs when data are collected sequentially from the same individual over time. Statistical procedures that fail to account for the correlation between repeated data points are likely to yield invalid conclusions due to parameter estimates may be inconsistent with wrong standard error estimates [1]. Over the years, there are many statistical methods available for handling and analysing repeated data such like Repeated Measures ANOVA (RM-ANOVA), Mixed Effects Model (MEM), and Generalised Estimating Equations (GEE). These statistical procedures will be used under some specific conditions and properties [2–4].

Repeated measure studies are always important and offer the chance to observe individual patterns of change. The purpose of longitudinal data analysis is to model the relationship of the repeated measurements of each individual to the associated covariates or factors. RM-ANOVA has a lot of assumptions that can be difficult to meet in all situations. One of the problem with repeated measures ANOVA is missing data on the response variable. There are two types of methods, generalized estimating equations and mixed-effect models, which are widely used in practice for analysing correlated data [2, 5, 6]. The advanced method of Generalized Estimating Equations [7] and Mixed Effects Models have started to replace the traditional methods such as Repeated Measures ANalysis Of Variance [8] as the older method is not flexible to accommodate all of features of repeated measure designs. But in GEE, the model selection is difficult due to lack of an absolute goodness of fit test to support in selecting the best model among several models [9–11].

The recent growing framework of the copula approach is very popular and used to handle repeated measure data [12]. The word copula means ‘a bond, link or tie‘, and was first employed in a statistical or mathematical sense by Abe Sklar in 1959 [12, 13]. The copula method is a relatively different and applied in various fields such as spatial [14], medical [15–18], financial [19], and air pollution [20]. A Copula is a function that permits us to combine univariate distributions to get a joint distribution with a particular dependence structure [21–23]. The main advantage of copula methodology in modelling correlated data are pointed out here as (i) allowance to model both linear and non-linear dependence, (ii) capability of modelling extreme endpoints, and (iii) arbitrary choice of a marginal distribution [21, 22, 24]. There is little amount of knowledge concerning how to handle and analyses the correlated data using Gaussian copula regression. As an alternative to GEE, a Gaussian copula regression under the copula approach is proposed for modelling longitudinal data [22]. Gaussian copula regression model is very flexible and used to analyse correlated data with missing values even though the sample size is low. The Gaussian copula estimation procedure for the estimation of regression parameters in continuous correlated response data using various working correlation structure compared with the generalized estimating equations. The aim of this study is to compare the results obtained from GEE and Gaussian copula regression using data from a Randomized Control Trial (RCT) study. In this illustration, we model the relationship between a continuous response (mean arterial pressure) and covariates such as hypertension and group.

Data

The data used for this study is from a prospective, double-blinded, randomized control trial that was conducted over two years between November 2016 and November 2018 in ASA I and II patients aged between 18–60 years with a normal renal function who underwent substitutional urethroplasty with buccal mucosal graft. Patients were randomized into three groups, group-A received systemic morphine (0.1mg/kg), group-B received epidural morphine (3mg) and group-C received intrathecal morphine (150µg). Repeated measurements of hemodynamic parameters systolic blood pressure, diastolic blood pressure, and mean arterial pressure were collected for each patients at thirteen various time points. The mean arterial pressure was considered as an outcome for analysis. This study was approved by the Institutional Review Board (IRB Number-10285; dated 21-09-2016) and ethics committee of Christian Medical College (CMC) and Hospital, Vellore.

Study variables

Baseline demographic variables, associated co-morbidities and other study characteristics of interest included were age (years), BMI (kg/m²), previous surgery (yes/no), allergies (yes/no), alcohol consumption(yes/no), smoker(yes/no), diabetes Mellitus(yes/no), hypertension(yes/no), and mean arterial pressure (MAP). Mean arterial pressure was considered as the outcome.

Statistical analysis

The descriptive statistics were reported as Mean ± SD for continuous and number (percentage) for categorical data. The Pearson Chi-square test and Fisher’s exact test (less cell frequency) were used to find the association between categorical variables. One-way analysis of variance was performed to find the difference between groups on the continuous data. Further, the only significant predictor variable hypertension was included in the model. Line plot was used to visualize a trend in MAP between three groups over time. Two analytical methods of GEE and Gaussian copula regression under the copula framework were compared. Unadjusted and adjusted estimates with 95% confidence intervals and p-values were reported for both GEE and Gaussian copula regression. All statistical tests were two-sided at α = 0.05 level of significance. All analyses were performed using R software version 4.1.0.

Generalized Estimating Equations

Many simple approaches are existing for handling the repeated data and analysis but the limit is the incapability of including covariates. GEE is a method to fit a marginal model for longitudinal data analysis, and it has been widely used in biomedical and clinical trial research [25, 26]. The GEE is a population-level approach based on a quasi-likelihood function and provides the population-averaged estimates of the parameter [27]. In Generalized Estimating Equations [28], the within-subject correlation structure is treated as a ‘nuisance’ variable (i.e. as a covariate). The simplest equation is extended as

$${Y}_{it = } {\beta }_{0}+\sum _{j=1}^{J}{\beta }_{1j} {X}_{ijt}+{\beta }_{2}t +\dots + {CORR}_{it}+ {ϵ}_{it}$$

Where,

${Y}_{it}$ are observations for i ^th subject at time‘t’, ${\beta }_{0}$ is the intercept, ${X}_{ijt}$ is the independent variable j for i ^th subject at time ‘t’. ${\beta }_{1j}$ is the coefficient for independent variable j and J is the number of independent variables. ${\beta }_{2}$ is the coefficient for time ,${CORR}_{it}$ is working correlation structure, and ${ϵ}_{it}$ is the error for i ^th subject at time ‘t’.

GEE has many features [29–31]. 1) The variance-covariance matrix of responses is treated as nuisance parameters in GEE and therefore this model fitting turns out to be easier than another approach of mixed-effect models [32]. 2) Specifically, if the treatment effect is of primary interest of the research study, the GEE approach is preferred [33]. 3) GEE relaxes the distribution assumption and only needs the correct specification of marginal mean and variance as well as the link function which connects the covariates of interest and marginal means [31].

However, generalized estimating equations are still in controversy in many aspects like [7] some issues on inconsistent estimation of within-subject correlation coefficient under a misspecified “working” correlation structure [34]. The estimation of the correlation coefficients using the moment-based method is not efficient. Such limitations lead investigators or researcher to actively work on this area to develop novel methods. Many alternative methods for estimating the correlation coefficients have been offered; among those, one method was based on the “Gaussian” estimation approach [20, 21].

Gaussian Copula

There are different types of copulas used in modelling. Majorly copulas are considered to be two families: (i) Archimedean copula family, and (ii) elliptical copula family. Copula function has some effect on the shape of the joint distribution so an appropriate and reasonable copula function should be selected because properties exhibited by different copula functions will vary [19]. Gaussian copula under the elliptical copula family has been successfully employed in several complex applications arising, for example, in correlated data analysis. For longitudinal data applications, elliptical copulas are more useful than archimedean copulas [35]. For longitudinal data, dependence among observations within a subject is relative to the time. Elliptical copulas have a correlation structure described with a correlation matrix that can handle the time-series behaviour of longitudinal data.

The Gaussian copula function is defined as

${ C}_{R}\left({u}_{1} , {u}_{2}, \dots ,{u}_{n}\right)= {ɸ}_{G }[{ɸ}^{-1}$ (${u}_{1}$),${ɸ}^{-1}$(${u}_{2}$),…,${ɸ}^{-1}$(${u}_{n}$)]

Where ${ɸ}_{G }$is the multivariate normal cumulative distribution function with zero means and unit variances. ${ɸ}^{-1}$(${u}_{1}$) is the inverse of the standard normal cumulative distribution function. ${ɸ}_{G }$is the Gaussian copula with the correlation matrix R.

Gaussian Copula Regression

Longitudinal data can be analysed using Gaussian copula regression. In Gaussian copula regression, the modelling with the response and independent variables is done with Gaussian copula. The Gaussian copula is first proposed for modelling selectivity in the context of continuous but non-normal distributions. Gaussian copula regression provides a general framework for modelling dependent variables which may belong to any distribution families. It is flexible enough to allow for both positive and negative dependencies. Dependence observed can be expressed with a convenient working correlation structure like autoregressive and exchangeable [21, 22, 24]. Inference for continuous response is carried out through the likelihood approach, for non-continuous response variable numerical approximations are used. In Gaussian copula regression, the dependence between the variables is modelled with a Gaussian copula so that the joint data cumulative distribution function is given by

P (Y₁ ≤ y₁, Y₂ ≤ y₂… Y_n ≤ y_n) = ɸ_n (${\in }_{1},{\in }_{2, } ,\dots ,{\in }_{n};P)$

Where

${\in }_{i}$ = $ɸ$⁻¹( F(y_i | x_i) ), with $ɸ(.)$ indicating the univariate standard normal cumulative distribution function.

${ɸ}_{n}$ (. ; P) the n dimensional multivariate standard normal cumulative distribution function with correlation matrix P.

Consider a regression model that links each variable Y_i to a vector of covariates x_i through the relationship

Y_i = h (x_i ,${ϵ}_{i})$

Where ${ϵ}_{i}$ is a stochastic error term. In many possible specifications of the function h (.) and error ${ϵ}_{i}$, the Gaussian copula regression model assumes that

h (${x}_{i}$, ${ϵ}_{i})$ = ${F}^{-1}$ {ɸ (${ϵ}_{i}$) | ${x}_{i}$}_,

the vector of error terms $ϵ$ = (${ϵ}_{1},{ϵ}_{2},\dots {ϵ}_{n})$ ^T has a multivariate standard normal distribution with correlation matrix P.

Demographic characteristics

A total of 93 male patients were randomised for the study into three groups, with 31 patients in Group 1 and 31 patients in Group 2 and 31 patients in Group 3. Of these, 2 patients were excluded in Group I and II, and 1 in Group 3, respectively, after randomization. For analysis, 88 male patients were included aged between 18 and 67 years. The mean age of the patient is 39 years (SD 12.2). Table 1 describes the summary statistics of baseline demographic and other characteristics of study patients who were allocated into three groups receiving systemic morphine, epidural morphine, and intrathecal morphine. We found that only the hypertension was significantly associated with group (P = 0.049).

Table 1

Baseline demographic and other characteristics of study patients (n = 88)
Variable	Systemic morphine (0.1mg/kg) (n = 29)	Epidural morphine (3mg) (n = 29)	Intrathecal morphine (150µg) (n = 30)	p-value
Age(years)^*	38.17 ± 11.90	40.69 ± 12.93	39.47 ± 12.12	0.740
BMI (kg/m²)^*	24.82 ± 3.31	24.72 ± 3.83	25.63 ± 4.28	0.604
Previous surgery Yes No	11(37.9) 18(62.1)	20(69.0) 9(31.0)	17(56.7) 13(43.3)	0.620
Allergies Yes No	1(3.4) 28(96.6)	3(10.3) 26(89.7)	0(0.0) 30(100.0)	0.123
Alcohol consumption Yes No	4(13.8) 25(86.2)	7(24.1) 22(75.9)	7(23.3) 23(76.7)	0.553
Smoker Yes No	4(13.8) 25(86.2)	8(27.6) 21(72.4)	8(26.7) 22(73.3)	0.373
Diabetes Mellitus Yes No	4(13.8) 25(86.2)	6(20.7) 23(79.3)	1(3.3) 29(96.7)	0.121
Hypertension Yes No	7(24.1) 22(75.9)	5(17.2) 24(82.8)	1(3.3) 29(96.7)	0.049
Values are presented as number (percentage) and the p-value is obtained from the Chi-square test and Fisher’s exact test (less cell count).
*Values are presented as Mean ± SD and the p-value is obtained from One-way ANOVA. SD: Standard Deviation.

Table 2 describes the mean arterial pressure that was compared between groups at each time point. We found that statistically significant difference between groups on MAP at 1.5 hours. Figure 1 shows the correlation between sets of time points on the MAP. The correlogram allows that to see which pairs have the highest correlation. Figure 2 explains graphically how the changes are varying over time of thirteen-time points between three groups. From that, it was noted that till 30 minutes there were no major changes between groups. After 30 minutes, there is a drastic drop in the mean arterial pressure for the patients in an intrathecal morphine group compared to systemic morphine and epidural morphine.

Table 2

Summary statistics for hemodynamic parameter mean arterial pressure
Time points	Systemic morphine (0.1mg/kg) (n = 29)	Epidural morphine (3mg) (n = 29)	Intrathecal morphine (150µg) (n = 30)	p-value
	Mean ± SD	Mean ± SD	Mean ± SD
Baseline	96.69 ± 10.70	94.10 ± 6.72	96.40 ± 6.35	0.419
at 5 mins	89.34 ± 11.06	87.24 ± 11.42	87.67 ± 12.61	0.771
at 30 mins	90.07 ± 10.42	86.72 ± 10.50	84.70 ± 10.32	0.143
at 1 hour	89.21 ± 9.22	87.07 ± 10.65	84.53 ± 7.96	0.163
at 1.5 hours	90.24 ± 7.94	86.83 ± 9.07	84.27 ± 7.17	0.021
at 2 hours	88.52 ± 6.51	88.69 ± 7.69	86.93 ± 6.60	0.565
at discharge	88.76 ± 7.74	88.21 ± 8.04	86.80 ± 6.81	0.590
on arrival	89.97 ± 8.06	87.55 ± 7.82	86.50 ± 6.39	0.195
at 2 hours after arrival	90.00 ± 8.11	87.69 ± 8.48	85.13 ± 7.54	0.073
at 4 hours	88.28 ± 6.35	87.86 ± 8.33	86.13 ± 7.12	0.492
at 6 hours	88.90 ± 8.26	87.59 ± 7.65	85.67 ± 7.13	0.273
at 12 hours	88.17 ± 8.08	88.83 ± 7.76	85.33 ± 6.56	0.168
at 24 hours	89.66 ± 7.22	89.21 ± 7.82	87.70 ± 6.92	0.563
SD: Standard Deviation.

Simulation study was performed to evaluate and compare the correlation structures of the proposed Gaussian copula regression with generalized estimating equation. Data was simulated based on the continuous outcome for longitudinal data with working correlation matrix structure for GEE and Gaussian copula regression. We generated data for n = 300, 600, 900, 1500 and n = 3000 units with three group mean values of MAP observed at time (t) = 13 time points and covariance.

Data simulated using the “mvrnorm ( )” R function. Each group data set was generated with respective group mean values at 13 time points and covariance. The simulated data is similar to the real data in terms of number of groups, number of time points and MAP outcome. We included one binary variable similar to real data analysis setting with each group proportion of hypertension as 0.24, 0.17 and 0.03. We evaluated and compared CIC and QIC based on a working correlation matrix with GEE and Gaussian approach. Simulation study results are given in the Table 3 and Table 4. Table 3 shows QIC and CIC of the estimate of the association between outcome and factors under consideration of different working correlation structures for unadjusted and adjusted models. Overall for unadjusted and adjusted model of GEE, the autoregressive correlation structure is working better compared to other correlation structure. Similarly, Table 4 shows AIC of the estimate of the association between outcome and factors under consideration of various working correlation structures for unadjusted and adjusted models. Unstructured correlation structure is working better in Gaussian copula approach compared to other correlation structures.

Table 3

Simulation results for longitudinal data with correlation structure for GEE
Criterion	Correlation structure - Unadjusted				Correlation structure - Adjusted
Criterion	Independence	AR (1)	Exchangeable	Unstructured	Independence	AR(1)	Exchangeable	Unstructured
n = 300 QIC CIC	291699.2 23.6	310200 3.86	291672 10	303464.4 6.8	291501.8 29.1	310000 4.477	291465.3 10.9	303000 7.72
n = 600 QIC CIC	587446.8 23.5	631000 4.08	587419.2 9.7	626000 6.54	587397.6 29.7	630000 4.89	587359.4 10.6	625561.3 7.6
n = 900 QIC CIC	863573 23	923213.4 3.9	864000 9.26	919000 6.59	863587.9 30.4	923283.5 4.8	863547.9 10.4	919000 7.86
n = 1500 QIC CIC	1424506.3 23.8	1505759.6 4.2	1424478.8 10.1	1455602.9 17.4	1424206.4 30.7	1510000 5.09	1424167.1 11.1	1450492 872
n = 3000 QIC CIC	2870000 23.4	3050000 4.15	2870000 9.51	2960000 7.78	2870000 30.4	3050190 5	2870000 10.6	3030000 8.04

Table 4

Simulation results for longitudinal data with correlation structure for Gaussian copula regression
Criterion	Correlation structure- Unadjusted				Correlation structure- Adjusted
Criterion	Independence	AR (1)	Exchangeable	Unstructured	Independence	AR(1)	Exchangeable	Unstructured
n = 300 AIC	27909	26118	26211	24280.27	27908	26119	26213	24285.18
n = 600 AIC	55858	52468	52409	48732.3	55859	52470	52411	48668.13
n = 900 AIC	83544	78479	78511	72917.07	83546	78480	78513	72893.02
n = 1500 AIC	139030	130301	130256	120895.7	139027	130302	130258	120886.1
n = 3000 AIC	278299	261431	261031	243682.7	278299	261433	261033	243791.6

Generalized Estimating Equation Analysis

GEE analysis was done by assuming common four correlation structures, namely independence, autoregressive, exchangeable, and unstructured. The time at which repeated measurements have been taken for each patient and treatment groups of the patients were considered as a factor, time, and group respectively. As identified in the simulation, the Autoregressive (AR1) correlation structure was used for the data.

In Table 5, unadjusted GEE analysis was carried out for AR (1) correlation structure considering the factors time, group, and an interaction term of groups over time. There is a significant change over time on mean arterial pressure, and the group intrathecal morphine was shown to be having a reduction in mean arterial pressure when compared to systemic morphine especially assuming the AR (1) correlation structures. Since, the hypertension presence among the patients showed a significant association between groups from the baseline analysis, hypertension was considered in the adjusted GEE model along with time, groups, and interaction between groups and time which is given in Table 6. Similar to the results of the GEE unadjusted model, in the GEE adjusted model also there was a significant change in the mean arterial pressure over time and hypertension.

Table 5

Unadjusted parameter estimates using a GEE and Gaussian copula for mean arterial pressure
Variable	GEE		Gaussian copula
	Autoregressive		Unstructured
	β (95% CI)	p - value	β (95% CI)	p - value
Time	-0.51 (-0.66,-0.36)	< 0.001	-0.21 (-0.36,-0.05)	0.016
Group Epidural morphine Intrathecal morphine Systemic morphine	-1.54 (-4.61,1.54) -2.16 (-5.03,0.70) Ref.	0.328 0.140	-2.46 (-5.83,0.89) -2.63 (-6.01,0.75) Ref.	0.151 0.117
Group * Time Epidural morphine * Time Intrathecal morphine * Time Systemic morphine * Time	0.18 (-0.19,0.56) -0.11 (-0.47,0.25) Ref.	0.340 0.542	-0.07 (-0.36,0.22) -0.25 (-0.55,0.04) Ref.	0.564 0.044
95% CI: 95% Confidence Interval

Table 6

Adjusted parameter estimates using a GEE and Gaussian copula for mean arterial pressure
Variable	GEE		Gaussian copula
	Autoregressive		Unstructured
	β (95% CI)	p - value	β (95% CI)	p - value
Hypertension Yes No	5.82 (2.70,8.94) Ref.	0.0002	4.21 (-0.80,9.22) Ref.	0.050
Time	-0.53 (-0.82,-0.23)	0.0005	-0.10 (-0.35,0.14)	0.374
Group Epidural morphine Intrathecal morphine Systemic morphine	-2.44 (-6.28,1.41) -0.23 (-3.91,3.44) Ref.	0.213 0.900	-1.85 (-6.42,2.70) 0.39 (-4.25,5.04) Ref.	0.368 0.837
Group * Time Epidural morphine * Time Intrathecal morphine * Time Systemic morphine * Time	0.18 (-0.19,0.56) -0.11 (-0.47,0.25) Ref.	0.333 0.549	-0.14 (-0.45,0.17) -0.37 (-0.70,-0.05) Ref.	0.312 0.009
95% CI: 95% Confidence Interval

Gaussian Copula Regression Analysis

The Gaussian copula regression was performed to handle the repeated measures data. In Table 5, unadjusted Gaussian copula regression was performed with time, group, and interaction between groups over time same as in GEE analysis. There was a significant reduction in mean arterial pressure over time. In addition, in the unstructured correlation structure, over time the intrathecal morphine has a reduction in the mean arterial pressure by 25% compared to the systemic morphine and which is statistically significant. The Gaussian copula was an efficient statistical method to capture this finding but GEE failed to identify it. The adjusted Gaussian copula regression model with hypertension is given in Table 6. Under an unstructured correlation structure, over time the intrathecal morphine has a reduction in the mean arterial pressure by 37% than systemic morphine, which is statistically significant. The Gaussian copula was able to capture this finding but the GEE model was failed to capture it.

The goodness of fit comparison was given in Table 7 for the GEE and Gaussian copula regressions. The AIC and CIC criteria measures were used to select the optimal correlation structure under Gaussian copula regression, and GEE respectively. The goodness of fit criteria may not select the same correlation structure as the method of assessing differs between each criterion. An unstructured correlation structure from Gaussian copula regression and autoregressive correlation structure from GEE was found to be optimal models as they have smaller values. According to the AIC and CIC criteria, estimated values of the unstructured correlation structure in Gaussian copula regression with autoregressive correlation structure in GEE were compared. Overall, there was no major difference in the performance of the two methods.

Table 7

Goodness of fit criteria
Correlation structure	Gaussian copula regression AIC (smaller is better)	Generalized Estimating Equation CIC (smaller is better)
Independence	8114	29.3
Autoregressive [AR(1)]	7647	4.67
Exchangeable	7647	10.5
Unstructured	7277	7.76

In GEE, with an autoregressive correlation structure in Table 6, significance was seen with variables hypertension and time. In the unstructured correlation structure through Gaussian copula regression, significance was observed in variables hypertension and as well as interaction intrathecal morphine over time, which was considered as the important significant change as the study was a randomized controlled trial. But the method GEE failed to identify any of the groups to be having a significant change in the mean arterial pressure.

This present study mainly focused to explore the performance of different correlation structures under both GEE and Gaussian copula regression using simulation data. The performance of these two methods assessed with application to correlated data from a RCT study with an outcome of mean arterial pressure. GEE had been compared to an approximation of Gaussian copula regression via vector generalized linear model (VGLM) which was found to be more efficient than GEE [36]. GEE estimates using autoregressive correlation structure was compared to verify with estimates obtained from Gaussian copula regression with autoregressive correlation structure [24]. Misleading estimates can be avoided when using maximum likelihood estimation by Gaussian copula regression which is a major strength [37].

Many statistical procedures with maximum likelihood estimation can be used to analyse the correlated data but they may not be as robust as Gaussian copula regression. For instance, linear mixed-effects models are shown to have the impossibility of joint modelling [23]. In this study, we compared all correlation structures between two statistical methods. However, choosing the same optimum correlation structure for two methods differs due to goodness of fit criteria and which was found to be a limitation [38].

In this study, the Gaussian copula regression identified certain significantly associated variables using maximum likelihood estimation that even widely used GEE failed to capture using quasi-likelihood estimation. Although comparison was done between two different methods of estimation, model showed similar performance comparatively. From this study, we can find that over time intrathecal morphine plays a significant role in the reduction of mean arterial pressure post-surgery than systemic morphine.

This study focused to analyse and compare the findings of GEE that uses a quasi-likelihood method of estimation and Gaussian copula regression that uses the maximum likelihood method of estimation when applied to correlated data from a randomized controlled trial study. For continuous outcomes in correlated data, GEE and Gaussian copula regression are highly comparable and lead to valid results. In conclusion, both methods showed similar performance, an alternative to GEE that uses quasi-likelihood estimation, Gaussian copula regression identifies a few more significant variables using the maximum likelihood method of estimation but GEE fails.

AIC: Akaike Information Criterion; CMC: Christian Medical College; CIC: Correlation Information Criterion; CI: Confidence Interval; GEE: Generalized Estimating Equations; MEM: Mixed Effects Model; IRB: Institutional Review Board; MAP: Mean Arterial Pressure; RCT: Randomized Control Trial; RM-ANOVA: Repeated Measures ANOVA; SD: Standard Deviation; VGLM: Vector Generalized Linear Model.

Acknowledgements

Not applicable

Authors’ contributions

Conceptualization: RK, Methodology, Formal analysis: RK, GS and JR, Clinical data collection and clinical assessment performed: JPJJ and RM, Supervision: RK, PSP. Writing – original draft: RK and GS. Review & Editing: RK, and PSP.

The author(s) read and approved the final manuscript.

Funding

Institutional research board of Christian Medical College funded the study.

Availability of data and materials

The dataset used and/or analysed during the current study are not publicly available due to institutional policy. Data are however available from the authors upon reasonable request and with permission of institution.

Ethics approval and consent to participate

The study design and procedures were approved by the Institutional Review Board of the Christian Medical College and Hospital, Vellore, India (IRB Number-10285; dated 21-09-2016). The study was conducted in adherence to the Declaration of Helsinki. The informed consent form was obtained from patients.

Consent for publication

Not applicable

Competing interests

The authors declare that they have no competing interests.

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No competing interests reported.

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Comparison of generalized estimating equations and Gaussian copula regression results using data from the randomized control trial

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