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BY 4.0 license Open Access Published by De Gruyter Open Access April 5, 2023

Investigating the impact of CO2 emissions on the COVID-19 pandemic by generalized linear mixed model approach with inverse Gaussian and gamma distributions

  • Neslihan İyit EMAIL logo , Ferhat Sevim and Ümran Münire Kahraman
From the journal Open Chemistry

Abstract

Carbon dioxide (CO2) rate within the atmosphere has been rising for decades due to human activities especially due to usage of fuel types such as coal, cement, flaring, gas, oil, etc. Especially in 2020, COVID-19 pandemic caused major economic, production, and energy crises all around the world. As a result of this situation, there was a sharp decrease in the global CO2 emissions depending on the fuel types used during this pandemic. The aim of this study was to explore the effects of “CO2 emissions due to the fuel types” on “percentage of deaths in total cases” attributed to the COVID-19 pandemic using generalized linear model and generalized linear mixed model (GLMM) approaches with inverse Gaussian and gamma distributions, and also to obtain global statistical inferences about 169 World Health Organization member countries that will disclose the impact of the CO2 emissions due to the fuel types during this pandemic. The response variable is taken as “percentage of deaths in total cases attributed to the COVID-19 pandemic” calculated as “(total deaths/total confirmed cases attributed to the COVID-19 pandemic until December 31, 2020)*100.” The explanatory variables are taken as “production-based emissions of CO2 from different fuel types,” measured in tonnes per person, which are “coal, cement, flaring, gas, and oil.” As a result of this study, according to the goodness-of-fit test statistics, “GLMM approach with gamma distribution” called “gamma mixed regression model” is determined as the most appropriate statistical model for investigating the impact of CO2 emissions on the COVID-19 pandemic. As the main findings of this study, 1 t CO2 emissions belonging to the fuel types “cement, coal, flaring, gas, and oil” per person cause increase in deaths in total cases attributed to the COVID-19 pandemic by 2.8919, 2.6151, 2.5116, 2.5774, and 2.5640%, respectively.

1 Introduction

COVID-19 pandemic, caused by the SARS-CoV-2 virus, declared as an epidemic by the World Health Organization (WHO) on March 11, 2020 leads to life-threatening diseases especially as severe acute respiratory diseases, pneumonia, cardiovascular diseases, etc. [1,2,3,4,5,6,7,8,9].

In the most basic sense, “CO2 emission” is expressed as the release of carbon to the atmosphere and is also defined as “greenhouse gas emission” or “carbon footprint.” Therefore, although CO2 emission is naturally found in the atmosphere, human and production-based activities have led to a significant increase in the amount of CO2 emission in the atmosphere. In particular, human-induced CO2 emission in the atmosphere has started to gain great importance with the increase in industrialization in the world since the 1850s [10,11,12,13,14].

CO2 emission is seen as the primary actor in the “global climate change.” It is also one of the most important factors considered as a threat to the human life and development [15,16,17,18]. However, the increase in the amount of CO2 emission has a negative impact on the sustainable economic development of the countries in the world [19,20]. On the other hand, CO2 emission, which has negative effects on the climate and economy, arises due to the high rate of energy obtained from fossil fuels and cement production [21,22].

There are many studies in the literature on examining the effects of various CO2 emission types for different countries in the world. Gyamfi et al. [23] dealt with the amount of energy consumption and economic variables that affect the CO2 emission generated by the use of fossil fuels and cement production. Khatun et al. [24] proposed a regression model to predict the effect of solid and liquid fuel consumption on CO2 emissions in Bangladesh. Karakurt and Aydin [25] obtained regression models for BRICS and MINT countries to predict the effect of fossil fuels on CO2 emissions.

It is known that the COVID-19 pandemic has a temporary reducing effect on the CO2 emissions worldwide [26]. Many studies in the literature dealt with the decrease in the CO2 emissions after the declaration of the COVID-19 pandemic due to the major changes in the human activities, including restriction of travel, and lockdown of many commercial and industrial activities [27,28,29,30]. So the COVID-19 pandemic has been a period that has negative effects on the environmental quality as well as the human health [31,32]. The amount of CO2 emissions, which decreased due to the decrease in the human activities, increased rapidly with the removal of the restrictions [33].

Adebayo et al. [34] applied time series regression, cointegration, and causality analysis to assess the decreasing impact of COVID-19 cases on the CO2 emissions. Andreoni [35], in order to reveal the impact of the COVID-19 restrictions on the CO2 emissions, examined the CO2 emission change obtained by multiplying the country’s GDP and the CO2 emissions from production for European countries and found a decreasing trend in the CO2 emissions. Charumathi and Mangaiyarkarasi [36] examined the impact of the COVID-19 pandemic restrictions on CO2 emissions in India. They found that the restrictions had a reducing effect on non-production CO2 emissions. Smith et al. [37] stated that the CO2 emissions in the developed countries decreased with the COVID-19 pandemic, but this effect was limited in the developing countries. Mzoughi et al. [38] developed a VAR model that revealed the existence of a negative correlation between the CO2 emission data compiled by the International Energy Agency and the number of COVID-19 cases.

When the studies in the literature given above are examined, it is generally seen that the main interest in the studies is to reveal the reducing effect of the COVID-19 pandemic on the CO2 emissions. However, it has been observed that the inverse relationship between the CO2 emissions and the COVID-19 pandemic has not been adequately investigated in the literature. In order to eliminate this deficiency in the literature, the first main motivation of this study is to investigate the impact of the CO2 emissions on the COVID-19 pandemic.

For this purpose, based on the lack of studies in the literature examining the impacts of production-based emissions of CO2 on the COVID-19 pandemic, “production-based emissions of CO2 from cement, coal, flaring, gas, and oil” are taken as explanatory variables in order to provide the WHO’s 13th Sustainable Development Goal (SDG) called “Climate Action” by 2030. So as the starting point of this study, it was inspired by the thought that the increase in the amount of production-based emissions of CO2 from coal, cement, flaring, gas, and oil in the atmosphere accelerates the spread of the SARS-CoV-2 virus, which causes COVID-19 pandemic all over the world and can increase the deaths accordingly.

In this study, it is aimed to model the statistical relationships between “percentages of deaths in total cases attributed to the COVID-19 pandemic” and “production-based emissions of CO2 from different fuel types” by “generalized linear model” (GLM) and “generalized linear mixed model” (GLMM) approaches with inverse Gaussian and gamma distributions. The GLM approach with gamma distribution called “gamma regression model” is commonly used for modeling continuous and positively skewed data [39,40]. The GLM approach with inverse Gaussian distribution called “inverse Gaussian regression model” is preferred over gamma distribution when the response variable is extremely positively skewed [41,42].

In the literature, there are many studies in which the GLM and GLMM approaches were applied on the COVID-19 pandemic data as a method. Cobre et al. [43] applied the GLM approach with normal, gamma, and Tweedie distributions to examine the effect of nutrition on the COVID-19 pandemic recovery rates. Jang et al. [44] investigated the relationship between medical expenses and duration of stay of the admitted Korean COVID-19 patients using the GLM approach with negative binomial and gamma distributions. Wagatsuma et al. [45] investigated the human respiratory activity of COVID-19 patients in Japan using the GLM approach with gamma distribution and time series analysis. Kontodimopoulos et al. [46] studied the COVID-19 pandemic in the aspect of the physical and mental health through the GLM approach with gamma distribution. Wang et al. [47] used the GLM approach with gamma distribution to reveal whether the case fatality rates differ between the COVID-19 variants among the countries. Hashimoto et al. [48] applied the GLM approach with inverse Gaussian distribution to reveal the model of length of stay in the health center during the COVID-19 pandemic in Brazil. Ulya and Nugraha [49] investigated the age and sex for cases and deaths from the COVID-19 pandemic in Indonesia by the GLM approach with inverse Gaussian distribution.

On the other hand, it is seen that there are not many studies in the literature, especially on the GLMM approach with inverse Gaussian and gamma distributions, called “inverse Gaussian mixed regression model” and “gamma mixed regression model” applied to the COVID-19 pandemic data. In order to fill this gap in terms of the statistical methodology in the literature, the second main motivation of this study is to investigate the impact of the “production-based emissions of CO2 due to different fuel types” on the COVID-19 pandemic by using “inverse Gaussian regression model” and “gamma regression model” as special cases of GLMs, and also “inverse Gaussian mixed regression model” and “gamma mixed regression model” as special cases of GLMMs where “country” is taken as fixed and random effects, respectively. For this aim, to make global statistical inferences about 169 countries in 6 continents in the world, the response variable is taken as “(total deaths/total confirmed cases attributed to the COVID-19 pandemic until December 31, 2020)*100.” The explanatory variables are taken as production-based emissions of CO2 due to different fuel types, measured in tonnes per person, that are “coal, cement, flaring, gas, and oil” in 2020.

2 Materials

In the material part of this study, the response variable is taken as “percentage of deaths in total cases attributed to the COVID-19 pandemic” until December 31, 2020 [50]. The explanatory variables are taken as CO2 emissions from cement, coal, flaring, gas, and oil measured in tonnes per person in 2020, respectively [51]. The response and explanatory variables used in this study to investigate the impact of CO2 emissions due to the fuel types on the COVID-19 pandemic are given in Table 1.

Table 1

Response and explanatory variables used in this study to model the COVID-19 pandemic in terms of CO2 emissions due to the fuel types

Variables Description
Percentage of deaths in total cases attributed to the COVID-19 pandemic (Total deaths/total confirmed cases attributed to the COVID-19 pandemic until 31 December 2020)*100 [50]
CO2 emissions from cement Annual CO2 emissions from production of cement, in tonnes per person in 2020 [52,53]
CO2 emissions from coal Annual CO2 emissions from production of coal, in tonnes per person in 2020 [52,53]
CO2 emissions from flaring Annual CO2 emissions from production of flaring, in tonnes per person in 2020 [52,53]
CO2 emissions from gas Annual CO2 emissions from production of gas, in tonnes per person in 2020 [52,53]
CO2 emissions from oil Annual CO2 emissions from production of oil, in tonnes per person in 2020 [52,53]

169 countries are taken as subjects in the study; 53 countries from Africa, 40 countries from Asia, 41 countries from Europe, 19 countries from North America, 4 countries from Oceania, and 12 countries from South America Continents as given in Table 2 [54].

Table 2

Countries taken in the study according to the continents in the world [54]

Continents Countries
Africa Algeria, Angola, Benin, Botswana, Burkina Faso, Burundi, Cameroon, Cape Verde, Central African Republic, Chad, Comoros, Congo, Democratic Republic of Congo, Djibouti, Egypt, Equatorial Guinea, Eritrea, Eswatini, Ethiopia, Gabon, Gambia, Ghana, Guinea, Guinea-Bissau, Cote d’Ivoire, Kenya, Lesotho, Liberia, Libya, Madagascar, Malawi, Mali, Mauritania, Mauritius, Morocco, Mozambique, Namibia, Niger, Nigeria, Rwanda, Sao Tome and Principe, Senegal, Sierra Leone, Somalia, South Africa, South Sudan, Sudan, Tanzania, Togo, Tunisia, Uganda, Zambia, and Zimbabwe
Asia Afghanistan, Armenia, Azerbaijan, Bahrain, Bangladesh, Brunei, China, Georgia, India, Indonesia, Iran, Iraq, Israel, Japan, Jordan, Kazakhstan, Kuwait, Kyrgyzstan, Lebanon, Malaysia, Maldives, Mongolia, Myanmar, Nepal, Oman, Pakistan, Philippines, Qatar, Saudi Arabia, Singapore, South Korea, Sri Lanka, Syria, Tajikistan, Thailand, Turkey, United Arab Emirates, Uzbekistan, Vietnam, and Yemen
Europe Albania, Andorra, Austria, Belarus, Belgium, Bosnia and Herzegovina, Bulgaria, Croatia, Cyprus, Czechia, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Moldova, Montenegro, Netherlands, North Macedonia, Norway, Poland, Portugal, Romania, Russia, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Ukraine, and United Kingdom
North America Antigua and Barbuda, Bahamas, Barbados, Belize, Canada, Costa Rica, Cuba, Dominican Republic, El Salvador, Guatemala, Haiti, Honduras, Jamaica, Mexico, Nicaragua, Panama, Saint Lucia, Trinidad and Tobago, and United States
Oceania Australia, Fiji, New Zealand, and Papua New Guinea
South America Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, Guyana, Paraguay, Peru, Suriname, Uruguay, and Venezuela

For easier understanding of the data used in this study, descriptive statistics of the “percentage of deaths in total cases attributed to the COVID-19 pandemic” until December 31, 2020 and the “CO2 emissions due to different fuel types” according to 169 countries in 2020 are given in Table 3.

Table 3

Descriptive statistics of the response and explanatory variables attributed to the CO2 emissions and the COVID-19 pandemic data used in this study

Variables Min. Median Mean value ± SD Max.
Percentages of deaths in total cases attributed to the COVID-19 pandemic 0.0495 1.7707 2.2267 ± 2.5341 29.0615
CO2 emissions from cement 0 0.1009 0.1347 ± 0.1574 0.8829
CO2 emissions from coal 0 0.0623 0.8647 ± 2.3781 25.5041
CO2 emissions from flaring 0 0 0.1042 ± 0.3825 4.1469
CO2 emissions from gas 0 0.1857 1.4968 ± 3.8030 31.2954
CO2 emissions from oil 0.0197 1.1937 1.7046 ± 1.8060 10.3367

Top ten countries with the highest percentages of deaths in total cases attributed to the COVID-19 pandemic are Yemen, Peru, Mexico, Ecuador, Syria, Sudan, Bolivia, Egypt, China, and Chad with 29.0615, 9.1682, 8.8218, 6.6039, 6.2183, 5.7569, 5.7237, 5.5272, 4.9467, and 4.9219, respectively.

Descriptive statistics of the response variable as “percentage of deaths in total cases attributed to the COVID-19 pandemic” until December 31, 2020 according to the six continents in the world are given in Table 4.

Table 4

Descriptive statistics of the percentages of deaths in total cases attributed to the COVID-19 pandemic according to the six continents in the world

Continents Number of countries Min. Median Mean value ± SD Max.
Africa 53 0.2273 1.7049 2.0418 ± 1.3006 5.7569
Asia 40 0.0495 1.3341 2.2550 ± 4.5537 29.0615
Europe 41 0.5040 1.7993 1.9327 ± 0.9230 3.8171
North America 19 1.2307 2.3014 2.5510 ± 1.6710 8.8218
Oceania 4 1.1539 2.2002 2.4090 ± 1.4664 4.0816
South America 12 0.9053 2.6103 3.3797 ± 2.4849 9.1682

Histogram of the “percentages of deaths in total cases attributed to the COVID-19 pandemic” for 169 countries is given in Figure 1. As seen from Figure 1, 11 countries have less than 0.5% of deaths in total cases attributed to the COVID-19 pandemic. 55, 55, and 28 countries have the percentages of deaths in total cases attributed to the COVID-19 pandemic between 0.5 and 1.5, 1.5 and 2.5, 2.5 and 3.5%, respectively. 20 countries have greater than 3.5% of deaths in total cases attributed to the COVID-19 pandemic until 31 December 2020.

Figure 1 
               Histogram of the percentages of deaths in total cases attributed to the COVID-19 pandemic in 169 countries.
Figure 1

Histogram of the percentages of deaths in total cases attributed to the COVID-19 pandemic in 169 countries.

Histogram of the percentages of deaths in total cases attributed to the COVID-19 pandemic in the six continents of the world is given in Figure 2. Country in each continent with the highest percentage of deaths in total cases attributed to the COVID-19 pandemic until 31 December 2020 is as follows: Sudan from Africa, Yemen from Asia, United Kingdom from Europe, Mexico from North America, Fiji from Oceania, and Peru from South America with 5.7569, 29.0615, 3.8171, 8.8218, 4.0816, and 9.1682%, respectively. Country in each continent with the lowest percentage of deaths in total cases attributed to the COVID-19 pandemic until 31 December 2020 is as follows: Eritrea from Africa, Singapore from Asia, Iceland from Europe, Cuba from North America, Papua New Guinea from Oceania, and Venezuela from South America with 0.2273, 0.0495, 0.5040, 1.2307, 1.1538, and 0.9053%, respectively.

Figure 2 
               Histograms of the percentages of deaths in total cases attributed to the COVID-19 pandemic belonging to the six continents in the world.
Figure 2

Histograms of the percentages of deaths in total cases attributed to the COVID-19 pandemic belonging to the six continents in the world.

As can be seen from Figure 1 and Figure 2, the structures of the response variable for both 169 countries, and 6 continents in the world are right-skewed distributed. This means that most data falls to the right or positive side of the histograms.

Bar graph of the production-based emissions of CO2 from different fuel types as coal, cement, flaring, gas, and oil from all over the world in 2020 is given in Figure 3. According to this graph, production-based emissions of CO2 from these fuel types from highest to lowest are determined as 13737.04, 9931.413, 7238.935, 1610.577, and 435.031 million tonnes for coal, oil, gas, cement, and flaring, respectively.

Figure 3 
               Bar graph of the production-based emissions of CO2 from different fuel types measured in million tonnes in 2020.
Figure 3

Bar graph of the production-based emissions of CO2 from different fuel types measured in million tonnes in 2020.

Histograms of the explanatory variables as the production-based emissions of CO2 from different fuel types are given in Figure 4.

Figure 4 
               Histograms of the production-based emissions of CO2 from different fuel types.
Figure 4

Histograms of the production-based emissions of CO2 from different fuel types.

Top ten countries with the highest production-based emissions of CO2 from cement, measured in tonnes per person, are United Arab Emirates, Qatar, Saudi Arabia, Cyprus, Luxembourg, China, Vietnam, South Korea, Turkey, and Oman with 0.8829, 0.8001, 0.7351, 0.6535, 0.6310, 0.5963, 0.5449, 0.4461, 0.4177, and 0.4104, respectively.

Top ten countries with the highest production-based emissions of CO2 from coal, measured in tonnes per person, are Mongolia, Kazakhstan, South Africa, Australia, South Korea, China, Bosnia and Herzegovina, Estonia, Czechia, and Poland with 25.5041, 9.1406, 6.6440, 6.2674, 5.5176, 5.1560, 4.9381, 4.6180, 4.3743, and 4.4296, respectively.

Top ten countries with the highest production-based emissions of CO2 from flaring, measured in tonnes per person, are Equatorial Guinea, Brunei, Libya, Venezuela, Iraq, Australia, Kazakhstan, Gabon, Iran, and Oman with 4.1469, 1.5155, 1.0947, 1.0203, 0.8034, 0.7977, 0.7787, 0.7310, 0.7170, and 0.5990, respectively.

Top ten countries with the highest production-based emissions of CO2 from gas, measured in tonnes per person, are Qatar, Trinidad and Tobago, Bahrain, Brunei, United Arab Emirates, Oman, Kuwait, Saudi Arabia, Canada, and Iran with 31.2954, 22.2481, 18.1184, 17.2098, 11.6300, 10.4106, 9.7055, 7.3884, 5.8060, and 5.1698, respectively.

Top ten countries with the highest production-based emissions of CO2 from oil, measured in tonnes per person, are Kuwait, Saudi Arabia, Luxembourg, Canada, United States, Iceland, Andorra, Bahamas, Australia, and Cyprus with 10.3466, 9.8409, 9.5084, 6.5273, 6.1043, 6.0982, 6.0312, 5.9225, 4.8865, and 4.6589, respectively.

3 Method

Nelder and Wedderburn [55] introduced the GLM as an extension of the traditional linear model where the response variable is assumed to come from normal distribution. In this way, GLM allows the response variable to come from all distributions belonging to the exponential family [56,57,58,59].

GLM constitutes three components, which are the systematic component, random component, and the link function. In the random component, the response variable comes from the exponential family given as follows [56,60,61,62]:

(1) f ( y ; θ , ϕ ) = exp y θ b ( θ ) a ( ϕ ) + c ( y , ϕ ) ,

where θ is the location parameter, ϕ is the dispersion or scale parameter, a ( ) , b ( ) , and c ( ) are the known functions belonging to the exponential family of distributions.

The systematic component as a linear function of explanatory variables is given as follows [57,60,63]:

(2) η = β 0 + β 1 x 1 + + β p x p .

The link function serves as a linear or nonlinear function between the systematic component, and the conditional mean of the response variable as follows [58,64,65]:

(3) g ( μ ) = η or μ = g 1 ( η ) .

These relationships between the systematic component, random component, and the link function are demonstrated in Figure 5.

Figure 5 
               Relationships between the components of the GLM.
Figure 5

Relationships between the components of the GLM.

In this study, the main interest is on the inverse Gaussian distribution and gamma distribution from continuous distributions belonging to the exponential family. For this aim, a ( ) , b ( ) , and c ( ) functions with the canonical link function ( η = g ( μ ) ) , and the inverse link function ( μ = g 1 ( η ) ) for the inverse Gaussian, and gamma distributions are given in Table 5 [58,64,66,67].

Table 5

Some characteristics belonging to the inverse Gaussian and gamma distributions in the GLM

Functions Distributions
Inverse Gaussian Gamma
a ( ϕ ) 1 / γ 1 / v
b ( θ ) 2 θ log ( θ )
c ( y , ϕ ) 1 / 2 [ log ( 2 π y 3 ϕ ) + 1 / ( ϕ y ) ] ϕ 1 log ( y / θ ) log ( y Γ ( ϕ 1 ) )
η = g ( μ ) 1 / μ 1 / η
μ = g 1 ( η ) 1 / μ 2 1 / η 1 / 2

On the other hand, GLMM includes “random effects” in addition to the “fixed effects” in the GLM [61,68,69]. In GLMM, both the fixed effects and random effects are included in the “linear predictor” as the systematic component. The relationships between the fixed effects, random effects, and the systematic component as the linear predictor in the GLMM are given in Figure 6.

Figure 6 
               Relationships between the components of the GLMM.
Figure 6

Relationships between the components of the GLMM.

In addition to the GLM, the link function and the inverse link function of the GLMM that constitutes both the fixed effects and random effects are given as follows [61,68,70]:

(4) g ( μ ̲ ) = η ̲ = X β ̲ + Z u ̲ ,

(5) μ ̲ = g 1 ( η ̲ ) = g 1 ( X β ̲ + Z u ̲ ) .

In this study, the canonical link function and the inverse link function for GLMM with the inverse Gaussian and gamma distributions are given in Table 5.

In the parameter estimation part of this study, iteratively reweighted least squares (IRLS) method and maximum likelihood (ML) method with Laplace approximation are used for GLM and GLMM with inverse Gaussian and gamma distributions, respectively. More details on the parameter estimation methods for both GLM and GLMM can be obtained from the following references [57,61,68,70].

The performances of the GLM and GLMM with the inverse Gaussian and gamma distributions are compared using information criteria (IC) as goodness-of-fit test statistics given in Table 6.

Table 6

IC used as goodness-of-fit-test statistics to compare the performances of the GLM and GLMM with the inverse Gaussian and gamma distributions [71]

IC Reference Formula
Akaike Information Criteria (AIC) [72] 2 l + 2 p
Finite Sample Corrected Akaike Information Criteria (AICc) [73] 2 l + 2 p N N p 1
Bayesian Information Criteria (BIC) [74] 2 l + p ln ( N )
Consistent Akaike Information Criteria (CAIC) [75] 2 l + p ( ln ( N ) + 1 )

In Table 6, p is the number of parameters, l is the maximum value of the log-likelihood function, and N is the number of observations in the model. The smallest values of IC determine the best performed GLM and GLMM with the inverse Gaussian and gamma distributions.

4 Results and discussion

In this study, the statistical relationships between the CO2 emissions from different fuel types measured in tonnes per person in 2020 and percentage of deaths in total cases attributed to the COVID-19 pandemic until December 31, 2020 are modeled by GLM and GLMM approaches with the inverse Gaussian and gamma distributions under the “inverse square” and “inverse” link functions to make statistical inferences about 169 countries in the 6 continents of the world.

In this study, all statistical calculations and the data analysis are done by using RStudio [76].

“Inverse Gaussian regression model” and “gamma regression model” as special cases of GLMs, “inverse Gaussian mixed regression model” and “gamma mixed regression model” as special cases of GLMMs under the “inverse square” and “inverse” link functions are fitted for the response variable as the “percentages of deaths in total cases attributed to the COVID-19 pandemic” in the aspect of the explanatory variables given in Table 1.

The results for the inverse Gaussian regression model using the IRLS parameter estimation method with the Fisher-Scoring (FS) iterative algorithm under the inverse square link function are given in Table 7.

Table 7

Results of the inverse Gaussian regression model for the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types under the inverse square link function

Explanatory variables β ˆ s .e ( β ˆ ) t value P ( > t ) 95% Confidence interval for β
Lower bound Upper bound
Intercept 0.11989 0.04009 2.990117 0.00322** 0.04071 0.19906
CO2 emissions from cement 1.26467 1.03113 1.226492 0.22179 −0.77152 3.30087
CO2 emissions from coal 1.26132 0.95061 1.326861 0.18642 −0.61585 3.13850
CO2 emissions from flaring 1.22826 0.93667 1.311309 0.19161 −0.62139 3.07792
CO2 emissions from gas 1.24821 0.93790 1.330853 0.18511 −0.60388 3.10031
CO2 emissions from oil 1.25455 0.94382 1.329231 0.18564 −0.60922 3.11832

By using the IRLS parameter estimates of the inverse Gaussian regression model under the inverse square link function given in Table 7 belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types, the expected value (mean) of the response variable in the inverse Gaussian regression model is given as follows:

(6) 1 ( μ ˆ ) 2 = 0 .11989 + 1 .26467 ( cement CO 2 Emis . ) + 1 .26132 ( coal CO 2 Emis . ) + 1 .22826 ( flaring CO 2 Emis . ) + 1 .24821 ( gas CO 2 Emis . ) + 1 .25455 ( oil CO 2 Emis . ) ,

or

(7) μ = 1 / 0 .11989 + 1 .26467 ( cement CO 2 Emis ) + 1 .26132 ( coal CO 2 Emis ) + 1 .22826 ( flaring CO 2 Emis ) + 1 .24821 ( gas CO 2 Emis ) + 1 .25455 ( oil CO 2 Emis ) .

The residual graphs for the inverse Gaussian regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types are given in Figure 7.

Figure 7 
               The residual graphs of the inverse Gaussian regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types.
Figure 7

The residual graphs of the inverse Gaussian regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types.

In Figure 7, the scatter graph of the Pearson residuals of the inverse Gaussian regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types shows that the Pearson residuals are intensely scattered around the line zero and are obtained in the range (−1,6). Histogram of the Pearson residuals of the model shows that the Pearson residuals are highly concentrated around 0, and particularly in the range (−0.5,0.5). Q–Q plot of the Pearson residuals of the model shows that the Pearson residuals are closely fitted according to the theoretical quantiles over the q–q line shown in red color. Box plot of the Pearson residuals of the model shows that the Pearson residuals are especially concentrated around 0, and there are so many outliers in the box plot of the Pearson residuals of the model.

The results for the gamma regression model using the IRLS parameter estimation method with the FS iterative algorithm under the inverse square link function are given in Table 8.

Table 8

Results of the gamma regression model for the COVID-19 pandemic data in aspect of the CO2 emissions due to different fuel types under the inverse link function

Explanatory variables β ˆ s .e ( β ˆ ) t - value P ( > t ) 95% Confidence interval for β
Lower bound Upper bound
Intercept 0.35740 0.04993 7.15732 2.71 × 10−11*** 0.25879 0.45601
CO2 emissions from cement 1.13682 1.18305 0.96092 0.33802 −1.19937 3.47301
CO2 emissions from coal 1.08968 1.09071 0.99906 0.31926 −1.06416 3.24352
CO2 emissions from flaring 1.07194 1.07903 0.99343 0.32198 −1.05883 3.20272
CO2 emissions from gas 1.07436 1.07750 0.99709 0.32021 −1.05340 3.20213
CO2 emissions from oil 1.09106 1.08611 1.00456 0.31661 −1.05370 3.23583

By using the IRLS parameter estimates of the gamma regression model under the inverse link function given in Table 8 belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types, the expected value (mean) of the response variable in the gamma regression model is given as follows:

(8) 1 μ = 0 .35740 + 1 .13682 ( cement CO 2 Emis . ) + 1 .08968 ( coal CO 2 Emis . ) + 1 .07194 ( flaring CO 2 Emis . ) + 1 .07436 ( gas CO 2 Emis . ) + 1 .09106 ( oil CO 2 Emis . ) ,

or

(9) μ = 1 0 .35740 + 1 .13682 ( cement CO 2 Emis . ) + 1 .08968 ( coal CO 2 Emis . ) + 1 .07194 ( flaring CO 2 Emis . ) + 1 .07436 ( gas CO 2 Emis . ) + 1 .09106 ( oil CO 2 Emis . ) .

The residual graphs for the gamma regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types are given in Figure 8.

Figure 8 
               The residual graphs of the gamma regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types.
Figure 8

The residual graphs of the gamma regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types.

In Figure 8, the scatter graph of the Pearson residuals of the gamma regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types shows that the Pearson residuals are intensely scattered around the line and are obtained in the range (−1,10). Histogram of the Pearson residuals of the model shows that the Pearson residuals are highly concentrated around 0, and particularly in the range (−1,0). Q–Q Plot of the residuals of the model shows that the Pearson residuals are closely fitted according to the theoretical quantiles over the q–q line shown in red color. Box plot of the Pearson residuals of the model shows that the Pearson residuals are especially concentrated around 0, and there are so many outliers in the box plot of the Pearson residuals of the model.

The results of the inverse Gaussian mixed regression model using the Laplace integral approximation method under the inverse square link function are given in Table 9.

Table 9

Results of the inverse Gaussian mixed regression model for the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types the under the inverse square link function

Explanatory variables β ˆ s .e ( β ˆ ) t value P ( > t ) 95% Confidence interval for β
Lower bound Upper bound
Intercept 0.16916 0.05237 3.23032 0.00124 0.06574 0.27257
CO2 emissions from cement 1.17462 1.12177 1.04712 0.29505 −1.04076 3.39000
CO2 emissions from coal 1.21512 1.04110 1.16715 0.24315 −0.84094 3.27119
CO2 emissions from flaring 1.17602 1.02514 1.14718 0.25131 −0.84853 3.20058
CO2 emissions from gas 1.20483 1.02674 1.17345 0.24062 −0.82288 3.23253
CO2 emissions from oil 1.20517 1.03353 1.16608 0.24358 −0.83594 3.24628

By using the parameter estimates of the inverse Gaussian mixed regression model under the inverse square link function given in Table 9 belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types, the expected value (mean) of the response variable in the inverse Gaussian mixed regression model is given as follows:

(10) 1 ( μ ) 2 = 0 .16916 + 1 .17462 ( cement CO 2 Emis . ) + 1 .21512 ( coal CO 2 Emis . ) + 1 .17602 ( flaring CO 2 Emis . ) + 1 .20483 ( gas CO 2 Emis . ) + 1 .20517 ( oil CO 2 Emis . ) ,

or

(11) μ = 1 / 0 .16916 + 1 .17462 ( cement CO 2 Emis . ) + 1 .21512 ( coal CO 2 Emis . ) + 1 .17602 ( flaring CO 2 Emis . ) + 1 .20483 ( gas CO 2 Emis . ) + 1 .20517 ( oil CO 2 Emis . ) .

The residual graphs for the inverse Gaussian mixed regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types are given in Figure 9.

Figure 9 
               The residual graphs of the inverse Gaussian mixed regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types.
Figure 9

The residual graphs of the inverse Gaussian mixed regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types.

In Figure 9, the scatter graph of the Pearson residuals of the inverse Gaussian mixed regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types shows that the Pearson residuals are randomly scattered around the line zero and are obtained in the range (−1,2). Histogram of the Pearson residuals of the model shows that the Pearson residuals are concentrated in the range (−0.5,0.5). Q–Q Plot of the residuals of the model shows that the Pearson residuals are far fitted according to theoretical quantiles over the q–q line shown in red color. Box plot of the model shows that the Pearson residuals are especially concentrated around 0, and there are so many outliers in the box plot of the Pearson residuals of the model.

The results of the gamma mixed regression model using the Laplace integral approximation method under the inverse link function are given in Table 10.

Table 10

Results of the gamma mixed regression model for the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types under the inverse link function

Explanatory variables β ˆ s .e ( β ˆ ) t value P ( > t ) 95% Confidence interval for β
Lower bound Upper bound
Intercept 0.66506 0.07022 9.47067 <2 × 10−16*** 0.52638 0.80375
CO2 emissions from cement 2.89187 1.13491 2.54810 0.01083* 0.65053 5.13321
CO2 emissions from coal 2.61714 1.05259 2.48637 0.01291* 0.53837 4.69591
CO2 emissions from flaring 2.51163 1.03666 2.42281 0.01540* 0.46433 4.55892
CO2 emissions from gas 2.57743 1.03772 2.48374 0.01300* 0.52803 4.62682
CO2 emissions from oil 2.56400 1.04257 2.45931 0.01392* 0.50503 4.62298

By using the parameter estimates of the gamma mixed regression model under the inverse link function given in Table 10 belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types, the expected value (mean) of the response variable in the gamma mixed regression model is given as follows:

(12) 1 μ = 0 .66506 + 2 .89187 ( cement CO 2 Emis . ) + 2 .61714 ( coal CO 2 Emis . ) + 2 .51163 ( flaring CO 2 Emis . ) + 2 .57743 ( gas CO 2 Emis . ) + 2 .56400 ( oil CO 2 Emis . ) ,

or

(13) μ = 1 0 .66506 + 2 .89187 ( cement CO 2 Emis . ) + 2 .61714 ( coal CO 2 Emis . ) + 2 .51163 ( flaring CO 2 Emis . ) + 2 .57743 ( gas CO 2 Emis . ) + 2 .56400 ( oil CO 2 Emis . ) .

The residual graphs for the gamma mixed regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types are given in Figure 10.

Figure 10 
               The residual graphs of the gamma mixed regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types.
Figure 10

The residual graphs of the gamma mixed regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types.

In Figure 10, the scatter graph of the Pearson residuals of the gamma mixed regression model belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types shows that the Pearson residuals are randomly scattered around the line zero and are obtained in the range (−1,2). Histogram of the Pearson residuals of the model shows that the Pearson residuals are homogeneously distributed in the histogram bars in the range (−0.8,1.4). Q–Q Plot of the residuals of the model shows that the Pearson residuals are closely fitted according to theoretical quantiles over the q–q line shown in red color. Box plot of the model shows that the Pearson residuals are especially concentrated approximately in the range (−0.5,0.5), and there are no outliers in the box plot of the Pearson residuals of the model.

The performances of the “inverse Gaussian regression model” and “gamma regression model” as special cases of GLMs, “inverse Gaussian mixed regression model” and “gamma mixed regression model” as special cases of GLMMs for the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types are compared using goodness-of-fit test statistics belonging to these models given in Table 11.

Table 11

Goodness-of-fit test statistics for the GLMs and GLMMs belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types

Goodness-of-fit test statistics Inverse Gaussian distribution under inverse square link function (inverse Gaussian regression) Gamma distribution under inverse link function (gamma regression) Inverse Gaussian distribution under inverse square link function (inverse Gaussian mixed regression) Gamma distribution under inverse link function (gamma mixed regression)
Log-likelihood −309.6949 −281.9446 −304.2182 −242.3547*
AIC 625.3899 579.8892 626.4364 502.7094*
AICc 625.5353 580.7892 627.5684 503.8415*
BIC 634.7796 604.9284 654.6054 530.8785*
CAIC 637.7796 612.9284 663.6054 539.8785*

As can be seen from Table 11, among the GLMs and GLMMs, the best fitted model is determined as the “gamma mixed regression model” as a special case of GLMMs belonging to the COVID-19 pandemic data in the aspect of CO2 emissions due to different fuel types according to the log-likelihood, AIC, AICc, BIC, and CAIC values with −242.3547, 502.7094, 503.8415, 530.8785, and 539.8785, respectively.

5 Conclusion

In this study, “inverse Gaussian regression model” and “gamma regression model” as special cases of GLMs, “inverse Gaussian mixed regression model” and “gamma mixed regression model” as special cases of GLMMs under the “inverse square” and “inverse” link functions are investigated for determining the statistical relationships between percentages of deaths in total cases attributed to the COVID-19 pandemic and the production-based emissions of CO2 from cement, coal, flaring, gas, and oil, respectively. And then by using the goodness-of-fit test statistics given in Table 11, the suitability of the “gamma mixed regression model,” which is rarely used in the literature, is emphasized for statistical modeling of the COVID-19 pandemic data in the aspect of the given production-based emissions of CO2.

When the studies investigating the global relationships between the COVID-19 pandemic and the CO2 emissions in the literature are examined, the superiority of the “gamma mixed regression model” as a special case of GLMM in empirical modeling of the COVID-19 pandemic in the aspect of CO2 emissions is almost rarely mentioned. In this context, it is thought that this study will make great contributions to fill this gap in the literature.

As the main findings from this study, according to the “gamma mixed regression model” given in Eq. (12) using the “Laplace integral approximation” method under the “inverse” link function.

The percentage of deaths in total cases attributed to the COVID-19 pandemic belonging to the 169 countries in the 6 continents as a whole increases by 2.8919, 2.6151, 2.5116, 2.5774, and 2.5640% by 1 tonne change per person due to the production-based emissions of CO2 from cement, coal, flaring, gas, and oil, respectively.

As a main conclusion of this study, globally, production-based emissions of CO2 from cement, coal, flaring, gas, and oil play a significant role on the rise in percentages of the COVID-19 pandemic deaths for the 169 countries in 6 continents of the world. And specifically, production-based emission of CO2 from cement is the most important production-based emission type of CO2 compared to coal, flaring, gas, and oil investigated in this study. In this aspect, contrary to the studies in the literature investigating the impacts of the COVID-19 pandemic on the CO2 emissions, another superiority of this study is that it is one of the very few studies in the literature that examined the impact of the production-based emission types of CO2 on the COVID-19 pandemic globally as a whole.

Acknowledgments

Neslihan İyit is the main author of this study. Ferhat Sevim continues his M.Sc. degree under the supervision of the first author. The authors are grateful to the editors and the anonymous reviewers for their valuable comments and contributions for the improvement of this article.

  1. Funding information: This study is supported by Selcuk University Scientific Research Projects (BAP) Coordinators with Research Project Number 22401002.

  2. Author contributions: All authors have read and approved the manuscript. Neslihan İyit: supervising, conceptualizing, writing, reviewing and editing the original draft preparation; Ferhat Sevim: data collection and statistical data analysis; and Ümran Münire Kahraman; literature review and visualizing the original draft.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Ethical approval: The conducted research is not related to either human or animal use.

  5. Data availability statement: All the data used in this study are public available in the Github and Our World in Data Repositories [https://github.com/owid/co2-data/blob/master/owid-co2-data.csv, https://ourworldindata.org/coronavirus].

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Received: 2022-12-27
Revised: 2023-02-23
Accepted: 2023-02-24
Published Online: 2023-04-05

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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