Sidney C. Port

Systolic blood pressure and mortality

Summary Background The current systolic blood-pressure threshold for hypertension treatment is 140 mm Hg for all adults. WHO and the International Society of Hypertension have proposed that normal pressure be lower than 130 mm Hg, with an optimum pressure of less than 120 mm Hg. These recommendations are based largely on the assumption that cardiovascular and overall mortality depend in a strictly increasing manner on systolic blood pressure. The Framingham study was instrumental in establishing this viewpoint. We reassessed data from that study to find out whether the relation is strictly increasing or whether there is a threshold in this relation. Methods We used logistic splines to model the relation of risk of cardiovascular and all-cause death with systolic blood pressure, using age-specific and sex-specific rates. We tested for the independence of the slope parameters from age and sex, and the reduced model with common slopes was used to produce a model different from the conventional linear logistic model. Findings Against the predictions of the linear logistic model, neither all-cause nor cardiovascular deaths depended on systolic blood pressure in a strictly increasing manner. The linear logistic model was rejected by the Framingham data. Instead, risk was independent of systolic blood pressure for all pressures lower than a threshold at the 70th percentile for a person of a given age and sex. Risk sharply increased with pressure higher than the 80th percentile. Since systolic blood pressure steadily increases with age, the threshold increases with age, but more rapidly in women than in men. Interpretation The Framingham data contradict the concept that lower pressures imply lower risk and the idea that 140 mm Hg is a useful cut-off value for hypertension for all adults. There is an age-dependent and sex-dependent threshold for hypertension. A substantial proportion of the population who would currently be thought to be at increased risk are, therefore, at no increased risk.

Theoretical probability for applications

Fundamentals of Probability Theory Discrete Models Non-Discrete Models Multivariate Normal Model Limit Concepts Appendix.

Relative impact of insulin resistance and obesity on cardiovascular risk factors in polycystic ovary syndrome

Polycystic ovary syndrome (PCOS) affects 5% to 7% of women of reproductive age. Insulin resistance and obesity are components of this important syndrome that may contribute to excess cardiovascular risk. We analyzed data from 69 patients with PCOS who had undergone quantitative assessment of insulin sensitivity, blood pressure, lipid profiles, and androgen levels to determine the impact of insulin resistance and obesity on parameters of cardiovascular risk. Homeostasis model assessment (HOMA) was used to stratify patients in terms of insulin resistance. To obtain a reference population, we used data from the National Health and Nutrition Examination Study (NHANES III, 1988 to 1994). The most insulin-resistant tertile of patients exhibited higher body mass index (BMI), androgen levels, systolic and diastolic blood pressure (DBP), triglyceride (TG) levels, and decreased high-density lipoprotein cholesterol (HDL-C) levels. Insulin resistance, not BMI, was the main determinant of HDL-C and TG levels and systolic blood pressure (SBP) in PCOS. Among normal women, both BMI and insulin resistance influenced cardiovascular risk factors. Insulin resistance was a more significant predictor of TGs in women with PCOS than in normal women (P =.008). In contrast to normal women, insulin resistance in PCOS appears to be the prime determinant of abnormal lipids, blood pressure, and androgens. Thus, early detection of insulin resistance, as well as weight reduction, should be emphasized for all patients with PCOS.

An elementary probability approach to fluctuation theory

Abstract : Problems connected with the fluctuation of the sums of independent and identically distributed random variables are discussed. Basically these problems consist in finding the distribution of various functions which are definable in terms of the sums and which give a measure (in some sense) of the amount of oscillation which the sums undergo.

Stable processes with drift on the line

The stable processes on the line having a drift are investigated. Except for the symmetric Cauchy processes with drift these are all transient and points are nonpolar sets. Explicit information about the potential kernel is obtained and this is used to obtain specific results about hitting times and places for various sets. 1. Statement of results Let be a process with stationary independent increments having log characteristic function -*|0|°(l-iAsgn(0)), where 0 0 . There is an enormous literature on the behavior of drift free stable processes. Surprisingly, little has been written on stable processes with drift. In view of the extensive results known for the potential theory [6] and the path behavior [3, 4] for infinitely divisible processes in general, the interest today in special processes such as stable processes with drift lies in the fact that for such processes rather explicit results may be obtained. The incorporation of a drift term changes the behavior of the process as compared to a drift free process. However the drift acts substantially different for processes with index a 1 . In all cases except for a 1 and s = 0 (i.e. the symmetric Cauchy process) the processes with drift are transient and hit points with positive probability. This fact follows from the general theory [3 and 6]. This is in contrast to the drift free processes. For drift free processes if a < 1 the processes are transient but do not hit points, Received by the editors July 31, 1986 and, in revised form, February 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 60J30; Secondary 60J45.

Optimal procedures for the installation of a unit subject to stochastic failures

Suppose we have a unit which we observe at discrete periods of time, and which at any given moment may be in one of three possible states, 0, 1, or 2, where 0 is the “failed” state, 1 is the “turned-off” state, and 2 is the “good” state. If a unit is in state 0 then we may decide to either leave it alone or to replace it with a new unit; and in the latter case the replacement will take a time X, where, in general, X will be a random variable. A unit which is in state 2 may be “turned off” (which we assume to be instantaneous), and this transforms a unit from state 2 to state 1. On the other hand, a unit in state 2 may be allowed to “run,” and in this case it has probability /I”-l( 1 ,LI) of surviving K time periods. Finally, if a unit is in state 1 we may either leave it in state 1 or we may “restart” the unit which we assume takes one period of time. At the end of this restart time we shall find the unit in state 2 with probability 01 or in state 0 with probability 1 cy. Successive “restarts” are to be independent Bernoulli trials with probability OL for success. Successive replacements X1 , Xs , . . . and successive failures are to be independent random variables, and the Xi are to have the common distribution

Representations of bounded harmonic functions

SummaryAn open subsetD ofRd,d≧2, is called Poissonian iff every bounded harmonic function on the set is a Poisson integral of a bounded function on its boundary. We show that the intersection of two Poissonian open sets is itself Poissonian and give a sufficient condition for the union of two Poissonian open sets to be Poissonian. Some necessary and sufficient conditions for an open set to be Poissonian are also given. In particular, we give a necessary and sufficient condition for a GreenianD to be Poissonian in terms of its Martin boundary.

The first hitting distribution of a sphere for symmetric stable processes

We will always work with the version of the process X(t) which is a standard Markov process. (See Chapter 1 of [1] for a complete description of a standard process.) For any r>0 let S,={y E RN IyI =r} denote the sphere of center 0 and radius r. Set T, = inf {t > O : I X(t)I = r}, and, as usual, set T, = oo if I X(t)l I #r for all t >0. The hitting measure and Greens function of S, are respectively the quantities, Hr(x, de) = Px(X(Tr) e de, Tr < oo) and gr(x, y), where gr(x, y) is the density of the measure

Equilibrium systems of recurrent Markov processes

Let &, , n 3 0, be a Markov process with states in a measurable space (x, F), where F is a u-algebra containing all one-point sets, and let P”(x, B) be the nth step transition function of the process. In this paper we will always assume that P has at least one u-finite invariant measure h (henceforth taken as fixed), and that the process is recurrent in the sense that for any B of positive h measure,

Limit theorems for transient Markov chains

Abstract An investigation of the asymptotic behavior, for large n, of the quantities P x ( n V B r ∞ ; X V B r = y ) where VBr is the time of the r-th visit to a finite, non-empty set B of states in a countable state space transient Markov chain Xn. These results are shown to yield information on the asymptotic behavior for solutions to the exterior Dirichlet problem for B, and on the error in approximating Px(Nn(B) N ( B ) = lim ⁡ n N n ( B ) Applications are given to several specific classes of Markov chains.