Makeham's Law: Negative B Values Explored

The Gompertz-Makeham Law of Mortality, also known as the Gompertz Law of Mortality, describes the age dynamics of human mortality. The law states that the human death rate is the sum of an age-dependent component (the Gompertz function) and an age-independent component (the Makeham term). The Gompertz-Makeham Law accurately describes mortality rates for individuals between the ages of 30 and 80. The law's applicability at more advanced ages is a subject of debate among studies. This raises the question: can the age-independent component, or 'b' in Makeham's Law, ever be negative?

The Gompertz-Makeham law of mortality

The Gompertz-Makeham law can be expressed mathematically as:

\[ h(x) = \alpha e^{\beta x} + \lambda\]

Where $$\backslash (\backslash alpha\backslash )$$ and $$\backslash (\backslash beta\backslash )$$ are parameters estimated from data, and $$\backslash (\backslash lambda\backslash )$$ represents the age-independent component. The $$\backslash (\backslash beta\backslash )$$-parameter indicates the rate of increase in the mortality hazard with age. For example, a $$\backslash (\backslash beta\backslash )$$-parameter value of 0.085 implies that mortality hazard doubles every 8 years.

While the Gompertz-Makeham law accurately describes human mortality patterns for ages between 30 and 80, its applicability at more advanced ages is debated. Some studies suggest a late-life mortality deceleration, where death rates increase more slowly at very old ages. However, more recent studies challenge this idea, indicating that mortality rates may continue to rise exponentially even at older ages.

The Gompertz-Makeham law has been applied to various species to understand their lifespan and mortality risks. For example, Ruby et al. used this model to analyze naked mole rat data, but surprisingly found that their mortality hazard did not increase with age, contrary to what would be expected for mammals. This discovery highlights the exceptional longevity of naked mole rats, which do not exhibit the typical signs of aging seen in other species.

Age-dependent and age-independent components of death rates

The Gompertz-Makeham law of mortality states that the human death rate is the sum of two components: an age-dependent component (the Gompertz function, named after Benjamin Gompertz) and an age-independent component (the Makeham term, named after William Makeham).

The age-dependent component, or the Gompertz function, suggests that death rates increase exponentially with age. Benjamin Gompertz proposed this component in 1825, noting that death rates tend to increase exponentially with age. This component accurately describes the age dynamics of human mortality from around 30 to 80 years of age. However, at more advanced ages, some studies have observed a phenomenon known as late-life mortality deceleration, where death rates increase more slowly.

The age-independent component, or the Makeham term, represents external causes of death that are not directly related to age. In certain controlled environments, such as laboratory conditions or low-mortality countries, this component becomes negligible, and the overall law simplifies to the Gompertz law of mortality.

The Gompertz-Makeham law can be expressed mathematically as:

> {\displaystyle h(x)=\alpha e^{\beta x}+\lambda}

Here, the beta-parameter implies a doubling of mortality approximately every eight years.

It is important to note that the population growth rate is influenced by both birth and death rates, which, in turn, are dependent on the variance of age-at-death. This relationship allows for the separation of the impacts of age patterns of fertility and mortality on population growth.

To compare the health of different groups or the same group at different times, health statistics are often adjusted for factors like age and sex. Age-adjustment is the most common type of adjustment, as older individuals are generally more susceptible to illness, while younger people are more prone to injuries. These adjustments are crucial for accurate comparisons between groups.

The law's accuracy in describing age dynamics of human mortality

The Gompertz-Makeham law of mortality describes the age dynamics of human mortality with some degree of accuracy in the age window from about 30 to 80 years of age. The law states that the human death rate is the sum of an age-dependent component (the Gompertz function) and an age-independent component (the Makeham term). The Gompertz function, named after Benjamin Gompertz, suggests an exponential increase in death rates with age. This function was proposed by Gompertz in 1825 and holds true in environments where external causes of death are rare, such as in laboratory conditions or low-mortality countries. In these cases, the formula simplifies to the Gompertz law of mortality, which indicates a doubling of mortality every 8 years.

However, the accuracy of the Gompertz-Makeham law has been questioned, particularly when applied to human populations. The "aging rate" estimated by the Gompertz formula is not stable throughout the entire lifespan and changes with certain patterns. For instance, some studies have found that at more advanced ages, death rates increase more slowly, a phenomenon known as late-life mortality deceleration.

Additionally, the law's applicability may be limited by factors such as infant and child mortality, public health measures, medical care, and socioeconomic conditions, which can vary significantly across different populations and historical periods. For example, English life expectancy at birth averaged about 36 years in the 17th and 18th centuries, while in the 20th century, the average lifespan in the United States increased by more than 30 years, largely due to improvements in public health.

Furthermore, the Gompertz-Makeham law may not fully capture the complexity of age-specific mortality rates. More robust approaches, such as the Lee-Carter model, have been developed to forecast time series of age-specific mortality rates and derive measures such as survival curves or life expectancies at different ages.

In summary, while the Gompertz-Makeham law provides a reasonable description of age dynamics in human mortality within a certain age range, it may not capture the full complexity of human lifespan and mortality. The law's accuracy is influenced by various factors, including environmental conditions, age-specific mortality patterns, and improvements in public health and medical care.

Late-life mortality deceleration

Several causes have been proposed for late-life mortality deceleration. Population heterogeneity, first proposed in Beard (1959), is the most common explanation. Even if individual hazard rates follow the Gompertz law, if the population is heterogeneous, the population hazard rate may exhibit late-life deceleration. Redundancy exhaustion is another proposed cause. In the reliability theory of aging, an organism's redundancy (reserves) are exhausted at extremely old ages, so every random hit results in death. However, this explanation may only hold for relatively simple organisms, as deceleration is less present or absent in more complex organisms with various subsystems. Other causes include less risky behaviour and more sheltered environments for older people, as well as various evolutionary explanations.

Recent studies have suggested that late-life mortality deceleration and plateaus may be primarily caused by low-frequency errors in demographic data. These errors can include age estimation and cohort blending errors, which can lead to spurious mortality deceleration and plateaus. Correction for these sources of error using a mixed linear model can eliminate late-life mortality deceleration and plateaus without the need for biological or evolutionary models. For example, studies using simulation and direct age validation found that longevity records for ages 105 years and older are often incorrect and may lead to spurious mortality deceleration. Therefore, it is suggested that all mortality estimates for ages above 105 years should be treated with caution until new, more strict methodologies of data quality control are developed and tested.

Hazard function for the Gompertz-Makeham distribution

The Gompertz-Makeham law of mortality states that the human death rate is the sum of two components: an age-dependent component (the Gompertz function) and an age-independent component (the Makeham term). The age-dependent component increases exponentially with age, as proposed by Benjamin Gompertz in 1825. On the other hand, the age-independent component remains relatively constant, especially in protected environments with low external causes of death.

The hazard function for the Gompertz-Makeham distribution is often represented as:

H(x) = αe^βx + λ

In this equation, h(x) represents the hazard function, α is a constant, β denotes the rate of increase in the hazard with age, and λ represents the age-independent component. The beta-parameter is empirically estimated to be approximately 0.085, suggesting that mortality doubles every 8 years.

The Gompertz-Makeham law accurately describes mortality patterns for individuals aged between 30 and 80 years. However, at advanced ages beyond this range, there is some debate. While certain studies indicate a slower increase in death rates, known as late-life mortality deceleration, more recent research challenges this notion.

In summary, the hazard function for the Gompertz-Makeham distribution provides a mathematical framework to understand how human death rates change with age, incorporating both age-dependent and age-independent factors.

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