1. HoelandWalburg(1972)reportresults ofanexperimenttostudytheeffects ofradiationonlifelengths ofmice.Miceweregivenadoseof300radsofradiationat5-6weeks of age and followed to death. At death each mouse was necropsied to determine if the cause ofdeathwasthymiclymphoma,reticulumcellsarcoma, oranothercause. Theagesofthemiceatdeathareshown below:

CauseofDeath andAgeatDeath(Days)

Thymiclymphoma(X_1,n_1=27,k=1):

158,192,193,194,195,202,212,215,229,230,237,240,244,

247,259,300,301,337,415,444,485,496,529,537,624,707,800

Reticulumcellsarcoma(X_2,n_2=15,k=2):

430,590,606,638,655,679,691,693,696,747,752,760,778,

821,986

Othercauses(X_3,n_3=37,k=3):

136,246,255,376,421,565,616,617,652,655,658,660,662,

675,681,734,736,737,757,769,777,801,807,825,855,857,

864,868,870,873,882,895,910,934,942,1015,1019

1. (10 points) For each of the three competing risks estimate the cumulative incidence functionCI_k(t), k= 1,2, 3, at 200, 300, . . ., 1,000 days by considering the two other risks as a single competingrisk.
2. (5points)Shownumericallythatthesumofthethreecumulativeincidencefunctionsfoundin part a is equal to 1 minus the Kaplan-Meier estimate of the overall survival function for this set of data, i.e., using the formula (4.7.1, p.128) for CI(t) to show CI₁(t) + CI₂(t) +CI₃(t)=1ST(t),whereT =min(X₁,X₂,X₃).
3. (12 points) Repeat part (a) using the complement of the marginal Kaplan-Meier estimatesto estimate marginal distributionsF_Xk(t), k=1,2, 3.What are the quantities estimatingandhowdifferentfromtheresultsfoundinpart(a)aretheseestimates?
4. (8 points) Compute the conditional probability function for thymic lymphoma (k =1) at500and800days.Whatarethequantitiesestimating?