Background

Neonatal, infant, and under-five mortality are critical indicators of a nation’s healthcare system, socioeconomic development, and overall well-being. In India, despite significant advancements in healthcare infrastructure and interventions, high mortality rates among newborns and young children persist, particularly in rural and marginalized communities. Accurate estimation of these mortality rates is essential for policy formulation, resource allocation, and monitoring progress towards Sustainable Development Goal 3 (SDG 3)—ensuring healthy lives and promoting well-being for all at all ages. Policy planners and health administrators rely on such data to design development strategies, address population health needs, and evaluate public health initiatives. But vital registration system often fails to capture all events due to limited administrative reach, illiteracy, and lack of awareness. While direct surveys is valuable for collecting detailed information but are expensive, time-consuming, and logistically challenging to conduct on a large scale. Given these limitations, indirect estimation methods offer a practical and cost-effective alternative by using readily available demographic, health, and socioeconomic indicators to infer mortality patterns and trends.

Demographers have been compelled to develop methods for predicting fundamental demographic parameters from erroneous or partial data due to the lack of reliable population counts and registration of important events. As we know, Brass focuses on methods for estimating mortality rates when data are limited or flawed [1]. His finding suggests techniques to derive reliable estimates of mortality even when data are incomplete or of poor quality. Another technique was used for estimating child mortality from information on proportion of dead children amongst children ever born to women with reference to the length of her marriage [2]. The alternate indirect techniques, such as those proposed for situations where mortality has been declining include the works of [3–5]. Islam and Alam used child mortality index (Ml) to calculate child mortality [6]. Neupert proposed a method to estimate infant mortality rates in small areas with limited data [7]. By using indirect estimation techniques based on fertility, childlessness, and total mortality data, it provides accurate estimates of infant mortality rates for various small geographic areas.

It is common knowledge that the percentage of dead among all children born to women in standard age groups can be used to estimate child mortality. It was first given by (Coal and Demeny, Brass and Coale) widely known as the Brass method [1,8]. On the other hand, the proportion of births of order four or higher (4 + birth order) has been consistently linked to elevated child mortality risks through mechanisms such as “resource dilution” where parental resources and attention are spread across more children, and “maternal depletion” where closely spaced and frequent pregnancies reduce maternal health reserves [9]. Higher birth order, defined as the fourth or subsequent birth within a family, has been associated with an increased risk of child mortality [10]. While some studies have reported a clear association between higher birth order and elevated mortality rates among children [11]. In 2003, Yadava and Tiwari applied an indirect estimation technique to calculate infant and under-five mortality rates using the proportion of dead children alone for India and Bangladesh [12]. Therefore, in this paper, we have evaluated the adequacy of this technique using recent NFHS data and extended it by incorporating the proportion of higher birth-order (4+) along with the proportion of dead children to improve estimation accuracy. The regression line analysis was conducted and tested using data from fifth round of NFHS.

Materials and methods

Methodology

The essentially method is based on the technique of regression line. The regression line concept has been utilized to investigate the relationship between the dependent and independent variable(s).

The straight-line simple regression model with respect to population parameters β₀ and β₁ can be explained by:

Where X and Y are independent and dependent variables respectively.

β₀ indicates the average value of Y when X=0, and

β₁ the slope of the line which indicates the change in the Y for per unit change in the X

For more than one independent variables, multiple regression analysis is used. The mathematical form of multiple regression model as:

Where:

Y_i is the ith value of dependent variable.

β₀ is the intercept term, and

β_1, β_2,…. β_n are regression coefficients representing the change in Y associated with a one-unit change in X_1, X_2,…., X_n respectively

ε_i is the error term

For the present study neonatal mortality rate (NMR)/ infant mortality rate (IMR)/ under five mortality rate (U5MR) was utilized as a dependent variable (Y). And the information on proportion of dead children among all children born to married mothers in the 15−49 age-group and proportion of 4 + birth orders considered as independent variable(s) namely X₁ and X₂ respectively [1,12]. The line of regression is drawn by taking major states of India using NFHS-5 data.

Mathematical form of fitted equations

For Neonatal mortality rates (per thousand live births),

Y = 458.41 X₁ - 3.23, (R² = 0.82) [X₁ is the Proportion of dead children]

Y = 67.31 X₂ + 10.28, (R² = 0.80) [X₂ is the Proportion of 4+ birth order]

Y = 267.66 X₁ + 36.31 X₂ + 0.875, (R² = 0.91) [Both predictors X₁ and X₂ taken together]

For Infant mortality rates (per thousand live births),

Y = 599.23 X₁ - 2.19, (R² = 0.81) [X₁ is the Proportion of dead children]

Y = 89.79 X₂ + 15.13, (R² = 0.83) [X₂ is the Proportion of 4+ birth order]

Y = 325.62 X₁ + 52.08 X₂ + 3.68, (R² = 0.92) [Both predictors X₁ and X₂ taken together]

For Under-five mortality rates (per thousand live births),

Y = 685.03 X₁ + 0.386, (R² = 0.86) [X₁ is the Proportion of dead children]

Y = 100.53 X₂ + 20.59, (R² = 0.84) [X₂ is the Proportion of 4+ birth order]

Y = 400.68 X₁ + 54.13 X₂ + 6.51, (R² = 0.96) [Both predictors X₁ and X₂ taken together]

Thus, the proposed predictors (proportion of dead children along with the proportion of 4 + birth order) improve the value of R² and provide better explanation of considered child mortalities estimates. The R² values indicate the proportion of variation in mortality explained by the regression models. Higher R² values represent better model fit, suggesting that the selected predictors account for a substantial proportion of the variability in mortality estimates.

Model validation of the fitted model

1. 1. Cross Validation predication power

A predictive model’s estimated performance in practice or the degree to which the suggested model is population-stabilized must be determined. In this regard, Herzberg’s cross validity prediction power (CVPP) technique has been applied [14], and it is described as [15]:

where n is the number of instances, c is the correlation coefficient between the estimated and observed values of the dependent variables, and p is the number of explanatory factors in the model.

1. 2. Shrinkage and Stability of R²

The model’s shrinkage is indicated by the standard adjustment applied to the coefficient of determination to account for the arbitrary effects of further sampling:

Where ρ²_υ is Cross Validation predication power (CVPP) and R² is the coefficient of determination of the model. The stability of R² of the model is equal to (1- Shrinkage) means lower shrinkage value provides more stability of the model.

Note: All data management, data analysis, and statistical analyses were performed using Microsoft Excel and Stata Statistical Software, Version 16 (StataCorp LLC, College Station, TX, USA).

Results

Table 1 shows the proportion of dead children among all children born to married mothers in the 15–49 age groups and proportion of 4 + birth orders for India. Haryana, Gujarat, Maharashtra, Rajasthan, Punjab, Tamil Nadu, Karnataka, Orissa, West Bengal, Assam: these states exhibit lower mean numbers of children ever born compared to the national average. The Proportion of dead children is found to be high in four states, i.e., MP, UP, Bihar and Orissa. These states also have higher proportion of 4 + births.

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Table 1. Proportion of dead children to currently married females in the age group 15-49 year and proportion of 4 + birth order for India and its some major states.

https://doi.org/10.1371/journal.pone.0353613.t001

Table 2 indicates the estimates of neonatal mortality rate (NMR) for India and its major states. Results presented in the table shows that the estimated values of NMR for most of the states are closest to observed value of NMR. The neonatal mortality rate for India is estimated 23.0 by using proposed all three regression models, but there are wide variations found among the considered states. Three states located in central and eastern part of country like MP, UP and Bihar has relatively high neonatal mortality rates. India and its 13 major states under study the difference between observed and estimated value of NMR is less than 10% for 5 (India, Gujarat, Maharashtra, Rajasthan and Karnataka), 7 (India, UP, MP, Bihar, Haryana, Gujarat and Assam) and 10 (India, UP, MP, Bihar, Haryana, Gujarat, Maharashtra, Tamil Nadu, Karnataka and Orissa) for different predictors like proportion of dead children, proportion of 4 + birth order and both proportion of dead children and 4 + birth order taken together respectively. To evaluate the suitability of the suggested approach, a cut-off difference of 10% was set between the observed and estimated values.

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Table 2. Estimated and observed Neonatal mortality rates (NMR) per 1000 live birth for India and its some major states.

https://doi.org/10.1371/journal.pone.0353613.t002

Table 3 shows the estimates for infant mortality rate under the proposed predictors for India and its major states. To evaluate the suitability of the suggested approach, a cut-off difference of 10% was set between the observed and estimated values for the infant mortality rates. Out of 14 observed values for India and its states, there are 8 states (India, Bihar, Haryana, Gujarat, Maharashtra, Rajasthan, Punjab and Karnataka), 7 states (India, MP, Haryana, Gujarat, Maharashtra, Karnataka and Assam) and 11 states (India, Uttar Pradesh, Madhya Pradesh, Bihar, Haryana, Gujarat, Maharashtra, Rajasthan, Karnataka, Orissa and Assam) were found to be below 10% for different predictors, i.e., proportion of dead children, proportion of 4 + birth order and both proportion of dead children and 4 + birth order taken together respectively. Utilizing this standard, we noted the estimated values and observed values showed that they were fairly near to one another.

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Table 3. Estimated and observed Infant mortality rates (IMR) per 1000 live birth for India and its some major states.

https://doi.org/10.1371/journal.pone.0353613.t003

Table 4 reveals that on an average the estimated value of U5MR for most of the states and India are closer to observed value. Again, utilizing the cut off 10% difference, there are only 11 states are found to be within the limit (India, Uttar Pradesh, Madhya Pradesh, Bihar, Haryana, Gujarat, Maharashtra, Punjab, Tamil Nadu, Karnataka and Assam), 9 states (India, Uttar Pradesh, Madhya Pradesh, Bihar, Haryana, Gujarat, Tamil Nadu, West Bengal and Assam)and 13 considered states except Punjab for proportion of dead children, proportion of 4 + birth order and both proportion of dead children and 4 + birth order taken together respectively.

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Table 4. Estimated and observed Under-five mortality rates (U5MR) per 1000 live birth for India and its some major states.

https://doi.org/10.1371/journal.pone.0353613.t004

The Table 2–4 also provides the 95% confidence interval for estimates of NMR, IMR and U5MR. For NMR, the observed values lie in the 95% confidence intervals for almost all states as well as India for proposed predictors, whereas the Yadava and Tiwari (2003) model and the higher-order birth alone as a predator miss the observed values in two and three states, respectively. For IMR, the regression model using both predictors demonstrate the strongest performance, with observed values consistently falling within its confidence limits and exhibiting smaller deviations from the true estimates. The proportion of dead-children model also performs reasonably well but shows notable mismatches in states such as Tamil-nadu and Orissa. The proportion of higher-order birth model proves less reliable, failing to capture observed values in approximately three states, including Bihar, Rajasthan, and West Bengal. For U5MR, the combined model yields the most consistent and accurate results, nearly always encompassing the observed values within its 95% confidence intervals and minimizing deviations. The proportion of dead-children model performs moderately well but with larger discrepancies in states such as Punjab and Orissa, while the higher-order birth model fails in around three states, notably Bihar, Rajasthan, and Maharashtra.

Table 5 presents regression models along with the coefficient of determination (R²), shrinkage, and stability of different mathematical models for estimation of neonatal mortality rate, infant mortality rate, and under-five mortality rate per 1,000 live births for India and some of its major states, based on data from the NFHS conducted between 2019−21. Among NMR, the stability values are 0.7652, 0.7442 and 0.8288 for first, second and third model respectively and shrinkage is minimum for third model. Similarly, among IMR, shrinkage is found to be 0.2454, 0.2239 and 0.1711 for 1^st, 2^nd and 3^rd model respectively. And for U5MR, the stability is found to be lowest for combined model.

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Table 5. Regression models, R² (coefficient of determination), Shrinkage and Stability for India and its some major states.

https://doi.org/10.1371/journal.pone.0353613.t005

Discussion

The present study considers the proportion of dead children among all children born to married women aged 15–49 years with the proportion of 4 + birth order children provide a more accurate and robust estimation of neonatal mortality rate (NMR), infant mortality rate (IMR), and under-five mortality rate (U5MR) in India than relying on either predictor alone. This approach advances earlier demographic research that highlighted the importance of multi-variable indirect estimation techniques in data-limited settings.

A notable strength of our approach is that indirect estimates were within ±10% of observed NFHS-5 values for most states, demonstrating practical utility where direct estimates are infeasible. This confirms the relevance of indirect methods with easily available data. Importantly, our analysis also illustrates how indirect methods can fill interim data gaps. The ability to generate reliable subnational estimates using readily available variables is particularly valuable for monitoring progress toward Sustainable Development Goal (SDG)- 3.

From a policy perspective, the proposed approach offers a rapid, low-cost means to estimate child mortality indicators in settings where CRVS systems remain weak. State and district health planners can use such interim estimates to identify high-burden regions, allocate resources for maternal and child health interventions, and monitor SDG 3 progress between NFHS rounds. At the global level, our approach shares conceptual similarities with the estimation framework employed by the UN Inter-agency Group for Child Mortality Estimation (UN IGME), which also relies on indirect indicators and survey-based measures to overcome data limitations [16]. However, unlike the UN IGME, which uses complex statistical modelling and Bayesian smoothing techniques to produce harmonised cross-country estimates, our model is specifically designed for subnational applications within India. It emphasizes methodological simplicity, transparency, and ease of replication at the state level, making it a practical tool for decentralized health planning. In this sense, it complements rather than replaces global estimation efforts. In summary, the study highlights the value of strengthening traditional indirect estimation methods by integrating multiple predictors, thereby enhancing both precision and applicability. The proposed model can serve as a cost-effective tool for policymakers in resource-limited settings, enabling evidence-based decision-making and more targeted child health interventions.

Conclusion

On the basis of the above results, it is reasonable to conclude that the information on proportion of dead children among all children born to married mothers in the 15–49 age-group and proportion of 4 + birth orders provide the better estimates of NMR, IMR and U5MR. The results also reveals that the estimates of these child mortalities are more closed to the observed values compare to estimates calculated using Yadava and Tiwari. While there is some shrinkage in the R² values across the models, the stability remains relatively high for the model that when both predictors taken together, indicating robustness in its predictive ability. The percent difference between observed and estimated values of child mortalities by proposed regression models and positions of observed values in confidence interval indicate the accuracy of technique.

On the basis of above study, we may conclude that the proposed regression has the potential to provide quick and better estimates of neonatal, infant and under-five mortality from a limited and easily available data.