Survival Analysis & Reliability Theory

Section- A

 

1-Calculate the moment generating function of exponential distribution.

 

The moment generating function (MGF) of a probability distribution is a useful tool for characterizing the distribution and calculating its moments. The MGF of a random variable X is defined as the expected value of e^(tX), where t is a real number called the argument of the MGF.

For the exponential distribution with parameter λ, the probability density function (pdf) is given by:



f(x) = λe^(-λx) for x ≥ 0

The moment generating function (MGF) of the exponential distribution is defined as:

M_X(t) = E(e^(tX)) = ∫ x=0^∞ e^(tx)λe^(-λx) dx

By making the substitution u = λx and du = λ dx, we get:

M_X(t) = ∫ u=0^∞ e^((t/λ)u)e^(-u) du

Integrating with respect to u from 0 to infinity, we get:

M_X(t) = ∫ u=0^∞ e^((t/λ)u-u) du

= e^(-u)[e^((t/λ)u)]u=0^∞

= e^(-1)[e^(t/λ)]^∞

 

2-Write a short note on Desh Pande test.

 

The Desh Pande test is a statistical test used to determine whether a population's age-specific fertility rates have changed over time. The test is named after its creator, Indian demographer Desh Pande. It is a method to determine whether a change in fertility is real or simply due to random variation.

 

The test compares the age-specific fertility rates (ASFR) of two or more time periods and calculates a chi-square statistic to determine whether there is a significant difference between the ASFRs. If the chi-square statistic is greater than the critical value, it suggests that there is a significant difference between the fertility rates of the two time periods.

 

The Desh Pande test is particularly useful for populations with low fertility rates, where small changes in fertility can have a large impact on population growth. It is also useful for populations where data on fertility is limited.

In summary, the Desh Pande test is a statistical method used to determine whether a population's age-specific fertility rates have changed over time, by comparing the fertility rates of two or more time periods. It is a useful method for populations with low fertility and limited data.

 

3-Discuss about the life tables. Also construct the life table.

 

A life table is a statistical tool that is used to estimate mortality rates and life expectancy for a population. It is constructed by using data on the number of deaths and number of individuals in specific age groups over a certain period of time.

The steps to construct a life table are as follows:

1.       Collect data on the number of deaths and number of individuals in specific age groups in a population over a certain period of time.

2.       Calculate the crude death rate (CDR) for each age group by dividing the number of deaths in that age group by the number of individuals in that age group.

3.       Calculate the age-specific death rate (ASDR) for each age group by dividing the number of deaths in that age group by the number of person-years lived by individuals in that age group.

4.       Use the ASDR to calculate the probability of dying (qx) in a given age group.

5.       Use the qx values to calculate the number of survivors (lx) in each age group.

6.       Use the lx values to calculate the life expectancy at birth (e0), which is the number of years a newborn can expect to live, based on current mortality rates.

7.       The final life table will contain columns for age, number of deaths, number of survivors, death rates, and life expectancy.

Life tables are important demographic tools that can be used to estimate mortality rates and life expectancy for a population. They are also useful for studying population aging patterns and for measuring the overall health and well-being of a population. It's important to note that the data used to construct a life table should be as recent and accurate as possible to ensure the most accurate results.

 

Section - B

 

 1-Write short notes on Mentel Haenzel test & Log rank test.

 

The Mantel-Haenszel test and the log-rank test are both statistical tests used to compare the survival rates of two or more groups.

The Mantel-Haenszel test is a chi-square test that is used to determine whether there is a significant difference in survival rates between two groups. It is particularly useful when there are confounding factors that can affect survival rates, such as age or gender. The test adjusts for these confounding factors by stratifying the data and comparing the survival rates within each stratum.

The log-rank test is a non-parametric test that compares the survival times of two or more groups. It is used to determine whether there is a significant difference in survival times between the groups. The test calculates a chi-square statistic based on the number of deaths in each group, and compares it to a critical value to determine whether there is a significant difference in survival times.

Both tests are widely used in medical research to compare the effectiveness of different treatments or interventions on survival rates. They are also used in other fields, such as epidemiology and ecology, to compare the survival rates of different populations.

In summary, both the Mantel-Haenszel test and the log-rank test are statistical methods used to compare the survival rates or survival times of two or more groups. Mantel-Haenszel test is useful when there are confounding factors, while log-rank test is a non-parametric test used to compare the survival times of two or more groups. Both tests are widely used in medical research and other fields to compare the effectiveness of different treatments or interventions on survival rates.

 

2-Describe Weibull distribution with its first four moments.

The Weibull distribution is a widely used probability distribution that is commonly used in reliability and survival analysis. It is named after the Swedish engineer Waloddi Weibull, who first described the distribution in 1951 while studying the strength of materials.

The probability density function (pdf) of the Weibull distribution with shape parameter k and scale parameter λ is given by:

f(x) = k/λ * (x/λ)^(k-1) * e^(-(x/λ)^k) for x ≥ 0

The cumulative distribution function (cdf) is given by:

F(x) = 1 - e^(-(x/λ)^k)

The first four moments of Weibull distribution are:

1.      Mean (expected value): E(X) = λ * Γ(1+1/k) where Γ(x) is the gamma function

2.      Variance: Var(X) = λ^2 * (Γ(1+2/k) - Γ^2(1+1/k))

3.      Skewness: Γ(1+3/k) * λ^3/[(Γ(1+2/k) - Γ^2(1+1/k))^(3/2)]

4.      Kurtosis: λ^4 * [Γ(1+4/k) - 4Γ(1+3/k) Γ(1+1/k) + 6Γ^2(1+2/k) - 3Γ^4(1+1/k)] / [(Γ(1+2/k) - Γ^2(1+1/k))^2]

Where Γ(x) is the gamma function.

One of the main features of Weibull distribution is that its shape parameter k, determines the shape of the distribution and it can take on any positive value. When k < 1, the distribution has a high probability of small values and a low probability of large values, which is called a "fragile" distribution. When k = 1, the distribution is exponential. When k > 1, the distribution has a low probability of small values and a high probability of large values, which is called a "robust" distribution.

In summary, Weibull distribution is a widely used probability distribution that is commonly used in reliability and survival analysis, it is named after the Swedish engineer Waloddi Weibull. The probability density function (pdf) and cumulative distribution function (cdf) of Weibull distribution are given by a specific formula. The first four moments of Weibull distribution are Mean, Variance, Skewness, and Kurtosis and can be calculated by specific formulas. The shape parameter k of Weibull distribution determines the shape of the distribution, when k < 1 the distribution is fragile when k = 1 the distribution is exponential and when k > 1 the distribution is robust.

 

3-What is Ageing Classes. Write its properties.

Ageing classes refer to groups of individuals that share similar age-related characteristics or experiences. These classes are often used in demographic research to study population aging patterns and to understand the social and economic impact of an aging population.

Properties of Ageing classes are:

1.      Ageing classes are usually defined based on chronological age, but they can also be based on other factors such as health status, income, or education level.

2.      Ageing classes are generally divided into three categories: young-old, old-old, and oldest-old. The young-old are typically considered to be between the ages of 65 and 74, the old-old are between 75 and 84, and the oldest-old are 85 and older.

3.      Ageing classes are often used to understand the social and economic impact of an aging population. For example, the old-old and the oldest-old are more likely to require long-term care and to have higher healthcare costs than the young-old.

4.      Ageing classes can also be used to study the different experiences and needs of different age groups. For example, the young-old may be more likely to be active and engaged in the workforce, while the old-old and the oldest-old may be more likely to experience health problems and to require assistance with daily living activities.

5.      Ageing classes can also be used in mortality analysis, to study the mortality risk of different age groups.

Overall, the concept of ageing classes is a useful way to group individuals based on similar age-related characteristics and experiences, and to study population aging patterns, social and economic impact of aging population, and the mortality risk of different age groups.

 

4-Define survival function. Establish its relationship with hazard function.

The survival function, also known as the reliability function or the survivor function, is a probability distribution that describes the probability that an individual or a system will survive beyond a certain time. It is defined as the probability that the time of failure, X, is greater than a specific value, x. Mathematically, it is represented as S(x) = P(X > x).

The survival function is often used in reliability analysis, where it is used to describe the probability that a system or component will continue to function without failure over a specific period of time. It is also used in medical research to describe the probability that a patient will survive a certain period of time after a diagnosis or treatment.

The hazard function, also known as the failure rate or the risk function, is a measure of the instantaneous probability of failure given that the system or component has survived up to a certain time. It is defined as the ratio of the probability of failure in an infinitesimal interval of time to the probability of survival up to that time. Mathematically, it is represented as h(x) = f(x)/S(x), where f(x) is the probability density function of the failure time.

The survival function and the hazard function are related through the following relationship:

S(x) = e^(-∫ h(x)dx)

This relationship is known as the "survival function-hazard function relationship" and it states that the survival function can be calculated by integrating the hazard function over time and taking the negative exponential of the result. This relationship allows us to use the hazard function to estimate the survival function and vice versa.

In summary, the survival function is a probability distribution that describes the probability that an individual or a system will survive beyond a certain time. The hazard function is a measure of the instantaneous probability of failure given that the system or component has survived up to a certain time. The survival function and the hazard function are related through the survival function-hazard function relationship, which states that the survival function can be calculated by integrating the hazard function over time and taking the negative exponential of the result.


---------------------------------------------------------------------------------------------------------

Please reads the answers carefully if any error please show in the comment. This answers are not responsible for any objection. All the answers of Assignment are above of the paragraph. If you like the answer, please comment and follow for more also If any suggestion please comment or E-mail me. 

 Thank You!