To comprehend the total factor productivity growth of health systems in African LDCs, the theory of constraints by Goldratt and Cox (1984) is adopted. According to Ochiel (2019), the premise of this theory is that the transformation of inputs into outputs is through the production process is faced with constraints. Thus the elimination of these constraints is the ultimate goal of each health systems to witness progress in their total factor productivity growth ( Aguilar-Escobar & Garrido-Vega, 2016; Ochiel, 2019). This is why Ochiel (2019) suggests that several steps like investing in technology can be adopted to counteract the negative effects of the constraints in health systems and witness progress in the total factor productivity growth of these health systems.

Several studies have assessed the total factor productivity growth of health systems comparing several countries from different regions of the world like Organization for Economic Co-operation and Development (OECD) countries ( Adang & Borm, 2007; Kim et al., 2016), European and Central Asian countries ( Hsu, 2014), Visegrád group countries ( Grausová et al., 2014), Continental African Countries ( Kirigia et al., 2007), World Health Organization countries from the Eastern Mediterranean region ( Masri & Asbu, 2018), Upper Middle Income Countries with focus on Iran ( Mohamadi et al., 2020), countries from the Association of South East Asian Nations ( Singh et al., 2021) and developed countries ( Almessabi, 2020). They used several inputs like number of medical personnel ( Adang & Borm, 2007; Almessabi, 2020), health expenditure ( Almessabi, 2020; Kim et al., 2016; Masri & Asbu, 2018), number of hospital beds ( Almessabi, 2020; Grausová et al., 2014), education ( Kim et al., 2016; Kirigia et al., 2007). Several outputs like infant mortality rate ( Almessabi, 2020; Kim et al., 2016; Masri & Asbu, 2018), under five mortality rate ( Almessabi, 2020), life expectancy ( Adang & Borm, 2007; Grausová et al., 2014; Hsu, 2014) and maternal mortality ratio ( Ibrahim et al., 2019) have been used as well.

All these studies employed the Data Envelopment Analysis (DEA) based Malmquist index and established variations in the total factor productivity growth of their health systems, with health systems of some countries experiencing a regress in productivity ( Almessabi, 2020; Hsu, 2014; Masri & Asbu, 2018) while others demonstrating progress in productivity ( Adang & Borm, 2007; Kim et al., 2016; Kirigia et al., 2007). The regress in productivity was attributed increase or decrease in efficiency ( Almessabi, 2020; Hsu, 2014; Masri & Asbu, 2018) and an increase or decrease in technology ( Almessabi, 2020; Hsu, 2014; Masri & Asbu, 2018). Similarly, the progress in productivity was attributed to an increase or decrease in efficiency ( Adang & Borm, 2007; Kim et al., 2016; Kirigia et al., 2007) and an increase or decrease in technology ( Adang & Borm, 2007; Kim et al., 2016; Kirigia et al., 2007).

The contribution and originality of this paper is based on Sajadi et al. (2020) suggestion of selecting the best input and output combinations which is so crucial in the estimation of the total factor productivity growth of health systems. According to Wagner and Shimshak (2007), most of the studies assessing the total factor productivity growth of health systems consider the input and output combinations as simply “givens” based on literature and do not focus on the choice of the best input and output combinations. Yet if the choice of input and output combinations is not given the attention it deserves, results of total factor productivity growth are biased and inconsistent. Adang and Borm (2007) further note that much of the critique of the 2000 World Health Report from the World Health Organization (2000) had to do with completeness of the production function and the choice of the inputs and output combinations. This study addresses this gap by using correlational analysis as used by Rooijakkers (2018); Kizza (2012); Hisali and Yawe (2011) and Yawe (2006) to select the best input and output combinations for the estimation of the total factor productivity growth of health systems.

Following Kizza (2012) and Yawe (2006) each African LDCs is considered to be a Decision Making Unit (DMU) or unit of analysis. Twenty-nine African LDCs are considered for this study based on the availability of data (see Table 1). According to Table 1, of the twenty-nine African LDCs, twelve are found in west Africa, eight are found in east Africa, six are found in south Africa and three in central Africa. According to Wale-Oshinowo et al. (2022), the geographic configurations resulting from the colonial and post-colonial delineation of these regions of Africa are to blame for the high proportion of African LDCs in West and East Africa.

Based on studies like: Hadad et al. (2013); Çelik et al. (2017): Ibrahim et al. (2019); Masri and Asbu (2018); Behr and Theune (2017); Retzlaff-Roberts et al. (2004) and Mohamadi et al. (2020), four inputs and outputs are considered for this study. Since the production of health at a macro level is complicated, health outcomes are used as health outputs ( Çelik et al., 2017; Ng, 2008; Peacock et al., 2001). The input, output data and their definitions based on the World Bank (2021) and World Health Organization (2019) are shown in Table 2.

The estimation of the total factor productivity growth of health systems require the use of output variables that capture good health outcomes. To conform to isotonicity and devise output variables that capture good health outcomes in infants, mothers and children under five years ( Ibrahim et al., 2019; Zhou et al., 2020). The Infant Mortality Rate (IMR); Maternal Mortality Ratio (MMR) and under-five mortality rate (U5MR) values are converted to infant survival rate (ISR) ( ISR = 1,000 − IMR / IMR ), maternal survival ratio (MSR) ( MSR = 100,000 − MMR / MMR ) and under five survival rate (U5SR) ( U 5 SR = 1,000 − U 5 SR / U 5 SR ).

Correlation analysis recommended by Cetin and Bahce (2016); Yawe (2006) and Kizza (2012) is used to select the best input and output combinations. According to Cetin and Bahce (2016), input and output combinations that are highly correlated and significant are redundant and dropped from the further analysis of the total factor productivity growth of health systems. Furthermore, following Kizza (2012) and Yawe (2006) input and output combinations that provide the highest average total factor productivity growth are chosen for the DEA based Malmquist model.

According to Santín and Sicilia (2017) and Cordero et al. (2013) endogeneity occurs when the technical efficiency scores are strongly correlated with any one input. Correlation analysis suggested by Dhaoui (2019) is used to test for potential endogeneity in the assessment of the total factor productivity growth of health systems in African LDCs.

The theoretical framework for estimating the total factor productivity growth of health systems in African LDCs is based on Solow (1957) model which is summarized as: Q t = A t F X t Y t (1)

Where Q t is the output and A t is the total factor productivity, which measures the shift in the production function at given the X , Y inputs and technology set. This total factor productivity ( A t ) is measured using a non-parametric index number ( Hulten, 1986). Following Ojwang’Oyieke and Karamagi (2023), since this approach does not impose a specific form on the production function, equation (1) is converted to a (logarithmic) differential of the production function as: Q ∗ Q t = ∂ Q ∂ X X t Q t X t ∗ Q t + ∂ Q ∂ + Y t ∗ Y t + A ∗ A t (2)

According to Hulten (1986), the growth rate of real output can be factored out into the growth rate of X and Y inputs weighted by their output elasticities and the growth rate of the hicksian efficiency index.

By total differentiation of equation (2), Solow (1957) showed that the hicksian efficiency index is a residual growth rate of output that is not accounted for by the growth in inputs which is given as: R t = Q ∗ Q t − S t k = S t ∗ X t − S t l + Y t ∗ Y t = A t ∗ A t (3)

Where S t = x t y t : x t can produce y t Is the feasible technology set which contains a combination of x t inputs and x t outputs. Thus the solow residual R t = the hicksian index. Solow concluded that, theoretically, this growth rate was equal to the growth rate of the hicksian efficiency parameter A t ∗ A t ( Ojwang’Oyieke & Karamagi, 2023). Abramovitz (1956) called this residual as a measure of the degree of our ‘ignorance.’ “This ignorance could be wanted (technical, scale and technological innovation) or unwanted like (measurement errors, omitted variables, aggregation bias, and model misspecification)” ( Ojwang’Oyieke & Karamagi, 2023). For the case of African LDCs, assuming that the unwanted ignorance is minimal, and hence attribute the solow residual to technical, scale and technological innovation in the decomposition of total factor productivity growth ( Zofio, 2007).

Following Masri and Asbu (2018), the output-oriented Variable Returns to Scale Data Envelopment Analysis (VRS-DEA) based Malmquist total factor productivity index is adopted for estimating the total factor productivity growth of health systems in African LDCs. The output-oriented VRS malmquist total factor productivity index is chosen over the input-oriented Constant Returns to Scale (CRS) approach because it is better suited for least developed countries while the input oriented is better suited for developed countries that have better health outputs ( Dhaoui, 2019; Dingake, 2017).

If countries produce multiple outputs y t using multiple inputs x t , productivity change is measured using total factor productivity index also called multifactor productivity index. The output distance function each country over a given period of time is given as D 0 t x t y t = inf θ : x t y t / θ ∈ T t (4) where T t is the production set. Following Chowdhury et al. (2011), defining the multifactor productivity index requires the definition of the distance function with respect to two different time periods, such as: D 0 t x t + 1 y t + 1 = inf θ : x t + 1 y t + 1 / θ ∈ T t (5) and D 0 t + 1 x t y t = inf θ : x t y t / θ ∈ T t + 1 (6) where T t and T t + 1 are the production sets in periods 1 and 2 respectively.

The first distance function, in equation (5), measures the maximum proportional change in outputs required to make x t + 1 y t + 1 feasible in relation to the technology at the previous period t . Similarly, the second mixed-period distance function, equation (6), measures the maximum proportional change in output required to make x t y t feasible in relation to the technology at t + 1 which we call D 0 t + 1 x t y t .

The Malmquist total factor productivity index (MTFP) measures total factor productivity (TFP) change between two time points in terms of ratios of distance functions. The MTFP between two time periods ( t and t + 1 ) using period t and period t + 1 technologies respectively is given as for period t M 0 t = D 0 t x t + 1 y t + 1 D 0 t x t y t (7)

For period t + 1 M 0 t + 1 = D 0 t + 1 x t + 1 y t + 1 D 0 t + 1 x t y t (8)

Where;

M 0 t and M 0 t + 1 denote the MTFP in period t and t + 1 respectively;

D 0 t x t + 1 y t + 1 refers to the output distance function which evaluates period t + 1 data relative to the technology in period t ;

D 0 t x t y t is output distance function evaluating period t data relative to technology in period t ;

D 0 t + 1 x t + 1 y t + 1 is the output distance function evaluating period t + 1 data relative to technology in period t + 1 ;

D 0 t + 1 x t y t is the output distance function evaluating period t data relative to technology in period t + 1 ;

Using period t and t + 1 technologies, the MTFP is defined as the geometric mean if the equations as follows; M 0 t = x t + 1 y t + 1 x t y t = D 0 t x t + 1 y t + 1 D 0 t x t y t D 0 t + 1 x t + 1 y t + 1 D 0 t + 1 x t y t 1 2 (9)

The MTFP in equation (9) is further decomposed into efficiency change EFFCH and technical/technological TECHCH change as follows M 0 t = x t + 1 y t + 1 x t y t = D 0 t + 1 x t + 1 y t + 1 D 0 t x t y t D 0 t x t + 1 y t + 1 D 0 t + 1 x t + 1 y t + 1 D 0 t x t y t D 0 t + 1 x t y t 1 2 (10)

That is: M 0 t = TEFFCH × TECHCH . The MTFP index value greater than 1 indicates growth in productivity, whereas a value less than 1 indicates a decline in productivity between periods t and t + 1 . A value of 1 denotes stagnation in productivity ( Kirigia et al., 2007). Likewise for efficiency change and technical change, if TEFFCH > < and TECHCH > < then there is an increase (decrease) in efficiency and technical progress (regress).

The DEA based Malmquist model is estimated using DEAP version 2.1 a free DEA Program developed by Coelli (1996). STATA version 15 by Stata Corp (2015) is used for the pre estimation techniques (choice of the best input/output combinations and checking for endogeneity issues regarding the total factor productivity growth of health systems). R, a free software environment for statistical computing and graphics, can be used for this analysis as well. Please see Underlying data ( Musoke et al., 2023) for access to the specific datasets used in the study.

There is variation among the chosen inputs and outputs for various Africa LDCs (see Table 3).

The minimum and maximum amounts for domestic general government health spending were 0.927 and 89.097 million US dollars, respectively, while the minimum and maximum amounts for external health spending were 1.121 and 74.705 million US dollars. The difference between domestic private health spending and out-of-pocket medical expenses is even greater, with minimum and maximum values of 2.182 and 1.825 million US dollars and 139.601 and 147.569 million US dollars, respectively. For the health outputs, the average life expectancy at birth is 59.056 years, with a range of 43.384 to 68.7 years. With minimum values of 0.005, -0.405, and 7.734 and maximum values of 0.046, 8.259, and 35.63, respectively, the average under-five survival rate, maternal survival ratio, and infant survival rate are 0.013, 1.438, and 17.453, respectively.

To determine the interrelationships between various input and output variables, the Pearson’s correlation matrix for the input and output variables in Table 4 is calculated.

Several input/output combinations for three (3) DEA model specifications based on the output orientation and Variable Returns to Scale (VRS) assumption are presented in Table 5. The results in Table 5 are in light of the results of the Pearson’s correlation matrix in Table 4. Only two outputs and all inputs are included in the DEA Model 1. Under five survival rate and maternal survival rate are dropped from DEA Model 1 as outputs because they have a strong significant positive correlation r = 0.837 > 0.5 p < 0.001 .

DEA malmquist model 2 has two outputs and all inputs. DEA malmquist model 2’s outputs life expectancy at birth and infant survival rate are dropped due to their strong significant positive correlation r = 0.775 > 0.5 p < 0.001 . DEA malmquist model 3 only has two inputs and four outputs. Due to their significant positive correlation r = 0.980 > 0.5 p < 0.001 , domestic private health expenditure and out-of-pocket health expenditure inputs were dropped for DEA malmquist model 3.

Results of the total factor productivity growth of the three estimated DEA Malmquist models based on the several input and output combinations for DEA malmquist model specifications in Table 5 are presented in Table 6. Over the 2008-2018 period, the average total factor productivity changes in DEA malmquist model 1 is 0.990 indicating a 1% regress in productivity. Similarly, the average total factor productivity changes in DEA malmquist model 3 is 0.983 indicating a 1.7% regress in productivity. The average total factor productivity change in DEA malmquist model 2 is 1.003 which indicates a productivity progress of 0.3%.

A comparison of all the three (3) DEA Malmquist models in Table 6 indicated that model 2 is the most preferred model with an average total factor productivity growth of 1.003 and 16 of 29 African LDCs on the frontier.

Test for endogeneity for the most preferred DEA malmquist model 2

Results of the pearsons correlation between inputs and technical efficiency scores based on VRS for the most popular DEA Malmquist model 2, are presented in Table 7.

Since endogeneity typically denotes a strong correlation between inputs and the technical efficiency scores based on the VRS ( Orme & Smith, 1996). According to Table 7, there isn’t much of a correlation between the input variables and technical efficiency scores. As a result, the DEA Malmquist Model 2 does not have an endogeneity issue and can be adopted for analysis.

Results of the Malmquist index summary of annual means for DEA Malmquist Model 2 of the African LDCs from 2008 to 2018 are presented in Table 8. According to Kizza (2012) and Yawe (2006), the annual means of the Malmquist Index are geometric in nature and represent the efficiency change, technical change and total factor productivity change.

The findings in Table 8 demonstrate that over time, the average technical efficiency change of health systems in African LDCs improved by 1.2%, the average pure efficiency change of African LDCs’ health systems improved by 0.9%. The average scale efficiency change and total factor productivity change improved by 0.3%. The mean pure efficiency change and mean scale efficiency change were responsible for the 1.2% increase in the average technical efficiency change of health of health systems. The highest progress in technical efficiency change of 11.9% was registered in the year 2013 while the highest regress of 12.1% was registered in the year 2017.

The findings in Table 8 also indicate a 0.9% regress in the technological change of health systems in African LDCs. The highest progress and regress of 18.4% and 11.4% of technological change were registered in the years 2017 and 2018 respectively. Furthermore, the average total factor productivity change for health systems in African LDCs in Table 8 was 1.003, representing a 0.3% increase in total factor productivity. The highest progress in total factor productivity change of 13.4% was in the year 2016 while the lowest of 0.1 was during the 2008-2010 period. The highest regress in total factor productivity change of 5.3% was in the year 2012 while the lowest regress in total factor productivity change was in the year 2014. The growth in total factor productivity over the years was largely from the technical efficiency change than the technical change.

According to Kizza (2012), group averages like the malmquist index summary of annual means for the best DEA Malmquist Model 2 in Table 8 hide individual results. As a result, it is crucial to run estimates for summary means for each African LDC for the 2008-2018 period (see Table 9). Results in Table 9 indicate a 1.2% progress in the technical efficiency of health systems for African LDCs over the 2008-2018 period. since technical efficiency = pure efficiency change × scale efficiency change, the mean technical efficiency change of 1.012 was as a result of 1.009 and 1.003 progress in pure efficiency change and scale efficiency change respectively.

Seventeen (Angola: 2.8%, Benin: 2%, Burundi: 3.8%, Central African Republic: 2.2%, Eritrea: 4.5%, Gambia: 5.6%, Guinea: 6.4%, Guinea Bissau: 7.3%, Liberia: 0.4%, Madagascar: 2.4%, Malawi: 2.7%, Mauritania: 0.1%, Mali: 5%, Niger: 3.4%, Uganda: 8.4%, Tanzania: 3.9%, Zambia: 5.5%) African LDCs had progress in technical efficiency change meaning that they moved towards the frontier. Five (Burkina Faso, Democratic Republic of Congo, Mozambique, Rwanda and Sierra Leone) neither registered regress or progress in the technical efficiency change. Seven (Chad: 0.8%, Djibouti: 4.7%, Ethiopia: 0.6%, Lesotho: 3.8%, Senegal: 4.5%, Sudan: 11.4% and Togo: 4%) African LDCs registered regress in the technical efficiency change. These results are consistent with those of studies like Kim et al. (2016) and Hsu (2014) who also reported a progress in the technical efficiency change. However, they are in disagreement with those of Singh et al. (2021) and Kirigia et al. (2007) who reported a decline in the technical efficiency change. A possible explanation for this is the efficient use of resources in countries that demonstrated a progress and inefficient use of resources in countries that demonstrated regress.

All African LDCs experienced a 0.9% mean reduction or regress in technology indicating that technical change for the African LDCs was less than one (<1). This meant that the technology (production) frontier shifted downwards. Sixteen African LDCs (Central African Republic, Democratic Republic of Congo, Eritrea, Ethiopia, Gambia, Guinea, Guinea Bissau, Liberia, Malawi, Mali, Mozambique, Rwanda, Sierra Leone, Tanzania, Togo and Uganda) had a regress in technical change while twelve African LDCs (Angola, Benin, Burkina Faso, Burundi, Chad, Djibouti, Lesotho, Madagascar, Niger, Senegal, Sudan and Zambia) had progress or improvement in technical change over the 2008-2018 period. Mauritania is the only African LDC that had stagnation in technical Change. The regress in technical change experienced by the African LDCs during the 2008-2018 period is attributed to low adoption of technologies and to the use of outdated technologies. Similar results are reported by ( Hsu, 2014; Masri & Asbu, 2018; Singh et al., 2021).

Over period 2008 to 2018 period, there was a 0.3% progress in the total factor productivity change of health systems in African LDCs. This progress was due to 1.2% progress in technical efficiency change and 0.9% regress in technical change. Sixteen African LDCs (Angola = 6.2%, Benin = 2.2%, Burkina Faso = 0.5%, Burundi = 7.7%, Central African Republic = 0.7%, Chad = 0.3%, Eritrea = 1.8%, Gambia = 4.9%, Madagascar = 8.4%, Malawi = 1.3%, Mali = 2.4%, Mauritania = 0.1%, Niger = 6.2%, Uganda = 7.5%, Tanzania = 3.7% and Zambia = 7.1%) registered progress in the total factor productivity change of health systems in African LDCs. Thirteen African LDCs (Democratic Republic of Congo, Djibouti, Ethiopia, Guinea, Guinea Bissau, Lesotho, Liberia, Mozambique, Rwanda, Senegal, Sierra Leone, Sudan and Togo) registered regress in the total factor productivity change of health systems in African LDCs. These findings are in agreement with those of Ibrahim et al. (2019), Kim et al. (2016) and Hsu (2014) and in disagreement with those of Kirigia et al. (2007). A possible explanation for this according to Kim et al. (2016) and Cashin and Dossou (2021) are the several health policy reforms such as (easy access to primary care, better treatment procedures implementation of information technology and payment systems) amongst the African LDCs.