For women with lymph node-negative, ER-positive breast cancer, obesity was not associated with a material increase in recurrence risk or a change in tamoxifen efficacy. However, because obesity was associated with increased risks of contralateral breast cancer, of other primary cancers, and of overall mortality, it may influence long-term outcomes for breast cancer survivors.

For women with node-negative, ER-breast cancer from clinical trials, obesity did not increase recurrence risk, but was associated with greater risk for second cancers, CBC, and mortality, particularly non-breast cancer deaths. Less favorable prognosis for black women persists in clinical trials, and is in part attributable to non-breast cancer outcomes.

Let M be a randomly chosen n × n permutation matrix. For a fixed arc of the unit circle, let X be the number of eigenvalues of M which lie in the specified arc. We calculate the large n asymptotics for the mean and variance of X, and show that X − E X / Var X 1/2 is asymptotically normally distributed. In addition, we show that for several fixed arcs I 1 I m , the corresponding random variables are jointly normal in the large n limit.

Let U be an n × n random matrix chosen from Haar measure on the unitary group. For a fixed arc of the unit circle, let X be the number of eigenvalues of M which lie in the specified arc. We study this random variable as the dimension n grows, using the connection between Toeplitz matrices and random unitary matrices, and show that (X − E[X])/(Var(X)) 1/2 is asymptotically normally distributed. In addition, we show that for several fixed arcs I 1 , . . . , I m , the corresponding random variables are jointly normal in the large n limit.

For event time data involving multiple mutually exclusive competing causes of failure, classic competing risks results show that marginal survival distributions are not identifiable. In a related instance, one or more failure modes may be observed provided that the failure events occur in a specific order. In such situations, sometimes referred to as semi-competing risks problems, the observations may under realistic assumptions lend information about parameters of interest that would be nonidentifiable in the strict competing risks case. Here, we present an approach that makes use of partially observable multiple modes of failures to obtain an estimate of the marginal distribution of one event type that may occur prior to the occurrence of another event type or be precluded by it. We apply the proposed method to the problem of estimating the distribution of time to tumour recurrence at specific sites among breast cancer patients participating in randomized clinical trials.