Likelihood & K-M estimate
The following is times (months) to relapse of 10 leukemia patients who had bone marrow transplantation. Starred observations are censoring times.
\(5,8,12,24,32,17,16^{*}, 17^{*}, 19^{*}, 30^{*}\)
a) Assume that the time to relapse has an exponential distribution with hazard rate \(\lambda\) Construct the likelihood function for \(\lambda\). Find the maximum likelihood estimate (m.l.e.) of \(\lambda\) and the mlie of the mean relapse time of the patient population.
b) Calculate by hand the K-M estimate of the survival function \(S(t)\) and the variance estimate of \(\mathrm{S}(\mathrm{t})\)
c) Find the nonparametric estimate for the mean relapse time of the patient population.
Homework Answers
Please show me the R code or process of c). Thanks 3.6 The following data consists...
Please show me the R code or process of c). Thanks
3.6 The following data consists of the times to relapse and the times to death following relapse of 10 bone marrow transplant patients. In the sample patients 4 and 6 were alive in relapse at the end of the study and patients 7-10 were alive, free of relapse at the end of the study Suppose the time to relapse had an exponential distribution with hazard rate A and the...
Fifty patients with leukemia received a bone marrow transplant with marrow from a sibling. Some o...
Fifty patients with leukemia received a bone marrow transplant with marrow from a sibling. Some of these patients died or relapsed and some were alive without relapse at the end of the study Interest is in the probabilty of being alive without relapse after a year. In particular, interest is in whether there is significant evidence that this probability is less than the corresponding probability for individuals who had their own bone marrow reinfused. Past data indicates that the latter...
2. Fifty patients with leukemia received a bone marrow transplant with marrow from a sibling. Som...
please do a) b) and c)
2. Fifty patients with leukemia received a bone marrow transplant with marrow from a sibling. Some of these patients died or relapsed and some were alive without relapse at the end of the study Interest is in the probabilty of being alive wit hout relapse after a year. In particul ar, interest is in whether there is significant evidence that this probability is less than the corresponding probability for individuals who had their own...
Weibull Likelihood function
\(\mathrm{T}\) is a failure time following a Weibull distribution. Consider \(\mathrm{Y}=\log \mathrm{T}\) where \(\mathrm{Y}\) has an extreme value distribution with survival function$$ S_{Y}(y)=e^{-\epsilon^{\frac{n \mu}{\sigma}}} $$where \(-\infty<\mu<\infty\) is the location parameter and \(\sigma>0\) is the scale parameter. Expressing with parameters \(\mu\) and \(\varphi=\log \sigma\). Assume that failure times of subjects under study arise from Weibull distribution. Let \(x_{1}, \ldots, x_{n}\) be the observed failure or right censoring times for n subjects. Each subject i (i = \(1, \ldots, \mathrm{n}\) ) has...
The following table contains data from a study on the survival times T (in days) of...
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2. Multiple myeloma, or blood plasma cancer, is characterized by increased blood vessel formulation in the...
2. Multiple myeloma, or blood plasma cancer, is characterized by increased blood vessel formulation in the bone marrow that is a predictive factor in survival. One treatment approach used for multiple myeloma is stem cell transplantation with the patient’s own stem cells. The measurements were taken immediately prior to the stem cell transplant and at the time the complete response was determined. At the 0.05 level of significance, is there evidence that the mean bone marrow microvessel density is higher...
Question 1. We wish to characterise the risk of default (going bankrupt) of individual companies Data...
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Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t,...
In this problem, we will model the likelihood of a particular client of a financial firm defaulting on his or her loans based on previous transactions. There are only two outcomes, "Yes" or &...
In this problem, we will model the likelihood of a particular client of a financial firm defaulting on his or her loans based on previous transactions. There are only two outcomes, "Yes" or "No", depending on whether the client eventually defaults or not. It is believed that the client's current balance is a good predictor for this outcome, so that the more money is spent without paying, the more likely it is for that person to default. For each x,...
3. Suppose that the 5-year survival probability, X, for women with breast cancer who live in...
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Please show me the R code or process of c). Thanks
3.6 The following data consists of the times to relapse and the times to death following relapse of 10 bone marrow transplant patients. In the sample patients 4 and 6 were alive in relapse at the end of the study and patients 7-10 were alive, free of relapse at the end of the study Suppose the time to relapse had an exponential distribution with hazard rate A and the...
Fifty patients with leukemia received a bone marrow transplant with marrow from a sibling. Some of these patients died or relapsed and some were alive without relapse at the end of the study Interest is in the probabilty of being alive without relapse after a year. In particular, interest is in whether there is significant evidence that this probability is less than the corresponding probability for individuals who had their own bone marrow reinfused. Past data indicates that the latter...
please do a) b) and c)
2. Fifty patients with leukemia received a bone marrow transplant with marrow from a sibling. Some of these patients died or relapsed and some were alive without relapse at the end of the study Interest is in the probabilty of being alive wit hout relapse after a year. In particul ar, interest is in whether there is significant evidence that this probability is less than the corresponding probability for individuals who had their own...
Question 1. We wish to characterise the risk of default (going bankrupt) of individual companies Data are available on the survival times in years), that is, time until default. We model the survival times as realisations from an independent and identically distributed (TID) exponen- tial random variable with mean 1/0 and density f(310) Be for 0 > 0,2 > 0. (i) Calculate the maximum likelihood estimate (MLE) for 8 from the data 11,-. In (1) Calculate the expected information for...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t,...
In this problem, we will model the likelihood of a particular client of a financial firm defaulting on his or her loans based on previous transactions. There are only two outcomes, "Yes" or "No", depending on whether the client eventually defaults or not. It is believed that the client's current balance is a good predictor for this outcome, so that the more money is spent without paying, the more likely it is for that person to default. For each x,...
3. Suppose that the 5-year survival probability, X, for women with breast cancer who live in a rural county follows Beta distribution with probability density function (pdf) fx (20) = 0.00-1 where 0 < x < 1 and parameter 6 > 0. Let X1, ..., X, be a random sample of size n from a population of rural counties. Researchers intend to make statistical inference on the parameter 6 using collected data X1, ..., (a) Let Y; = – log(Xi)...
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