Calculate the likelihood ratio and log-likelihood difference

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Assignment - Principles Of Statistical Inference (PSI) Exercises

Exercise 1: In a particular study we observe n₁= 78 mothers having full-term births and n₂ = 51 mothers having premature births. The mothers having full-term births had a total of Σx_i = 207 pregnancy attempts, and the mothers having premature births had a total of Σy_i = 212 pregnancy attempts.

a. Substitute the above sample values into the likelihood function and hence plot the likelihood function for p.

b. Calculate the following two possible estimates of p

(n₁+n₂)/(Σx_i+Σy_i) and ½(n₁/Σx_i+ n₂/Σy_i).

What do each of these estimates represent?

c. Calculate the likelihood ratio and log-likelihood difference for comparing the support for each of these values of p. Which estimate is better supported by the data?

d. Mark both estimates on your plot of the likelihood function from part a. Does there seem to be a better supported value than either of these?

Exercise 2: Suppose now that we are not prepared to assume that the fertility parameters are the same in the two groups, and that we want to compare the fertility using the relative fertility rate γ.

a. Using the sample described in Exercise 1, write down the likelihood and log-likelihood functions for p₁ and p₂, i.e., L(p₁, p₂) and l(p₁, p₂).

b. Using the relationship p₁ = p₂γ, rewrite the likelihood function and log-likelihood functions in terms of p₂ and γ, i.e., L(γ, p₂) and l(γ, p₂).

c. A possible estimate of γ is y^-/x^- = 1.57. Suppose we want to compare whether this value is better supported by the data than the value 1 (the value 1 would indicate no difference in the fertility of mothers having full-term births and mothers having premature births). Calculate the likelihood ratio comparing these two values, first assuming p₂ = 0.25 and secondly assuming of p₂ = 0.50? Does the result depend on the value of p₂ that we assume?

d. Suppose you are told that for any given value of γ, the best estimate of p₂ is

p^{^}₂(γ) = 1/x^-γ = 0.377/γ.

Substitute p^{^}₂(γ) into the log-likelihood in place of p₂ to end up with a function of γ only. Plot this function, which is called the profile log-likelihood function.

e. Based on the plot in part d, what is the most supported value for γ, based on the profile log-likelihood? What is the interpretation of this value in terms of the relative fertility of the two groups of mothers?

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Reference no: EM132250141

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