Elevated plasma concentrations of several one-carbon metabolites are associated with increased CVD risk. Both diet-induced regulation and dietary content of one-carbon metabolites can influence circulating concentrations of these markers. We cross-sectionally analysed 1928 patients with suspected stable angina pectoris (geometric mean age 61), representing elevated CVD risk, to assess associations between dietary macronutrient composition (FFQ) and plasma one-carbon metabolites and related B-vitamin status markers (GC–MS/MS, LC–MS/MS or microbiological assay). Diet-metabolite associations were modelled on the continuous scale, adjusted for age, sex, BMI, smoking, alcohol and total energy intake. Average (geometric mean (95 % prediction interval)) intake was forty-nine (38, 63) energy percent (E%) from carbohydrate, thirty-one (22, 45) E% from fat and seventeen (12, 22) E% from protein. The strongest associations were seen for higher protein intake, i.e. with higher plasma pyridoxal 5’-phosphate (PLP) (% change (95 % CI) 3·1 (2·1, 4·1)), cobalamin (2·9 (2·1, 3·7)), riboflavin (2·4 (1·1, 3·7)) and folate (2·1 (1·2, 3·1)) and lower total homocysteine (tHcy) (–1·4 (–1·9, −0·9)) and methylmalonic acid (MMA) (–1·4 (–2·0, −0·8)). Substitution analyses replacing MUFA or PUFA with SFA demonstrated higher plasma concentrations of riboflavin (5·0 (0·9, 9·3) and 3·3 (1·1, 5·6)), tHcy (2·3 (0·7, 3·8) and 1·3 (0·5, 2·2)) and MMA (2·0 (0·2, 3·9) and 1·7 (0·7, 2·7)) and lower PLP (–2·5 (–5·3, 0·3) and −2·7 (–4·2, −1·2)). In conclusion, a higher protein intake and replacing saturated with MUFA and PUFA were associated with a more favourable metabolic phenotype regarding metabolites associated with CVD risk.

EPA and DHA are essential for maternal and fetal health, but epidemiological data are sparse in China. We examined the trends of EPA alone and a combination of EPA plus DHA in pregnant and lactating women in three distinct geographic regions in China and explored their potential influencing factors. A total of 1015 healthy women during mid-pregnancy, late pregnancy or lactation were recruited from Weihai (coastland), Yueyang (lakeland) and Baotou (inland) cities of China between May and July of 2014. Maternal EPA and DHA concentrations (percentage of total fatty acids) in plasma and erythrocytes were measured by capillary GC. Adjusted EPA plus DHA concentrations in both plasma and erythrocytes significantly declined from mid-pregnancy (2·92 %, 6·95 %) to late pregnancy (2·20 %, 6·42 %) and lactation (2·40 %, 6·29 %) (P trend < 0·001); and both concentrations were highest in coastland, followed by lakeland, and lowest in inland (P < 0·001). Regarding EPA alone, the concentrations were higher in women during lactation or late pregnancy and in women in coastland and inland areas. Moreover, concentrations of EPA or EPA plus DHA were higher in women with older age, higher education, higher annual family income per capita and higher dietary intake of marine aquatic product and mutton. In lactating women, erythrocyte EPA concentration was higher in those having breast-feeding partially v. exclusively. In conclusion, maternal plasma and erythrocyte concentrations of EPA plus DHA or EPA alone differed with geographic regions, physiological periods and maternal characteristics, indicating a need of population-specific health strategies to improve fatty acids status in pregnant and lactating women.

Inflammation confounds the interpretation of several micronutrient biomarkers resulting in estimates that may not reflect the true burden of deficiency. We aimed to assess and compare the micronutrient status of a cohort of Indonesian infants (n 230) at aged 6, 9 and 12 months by ignoring inflammation (unadjusted) and adjusting four micronutrient biomarkers for inflammation with C-reactive protein (CRP) and α-1-glycoprotein (AGP) using the following methods: (1) arithmetic correction factors with the use of a four-stage inflammation model; and (2) regression modelling. Prevalence of infants with any inflammation (CRP>5 mg/l and/or AGP>1 g/l) was about 25% at each age. Compared with unadjusted values, regression adjustment at 6, 9 and 12 months generated the lowest (P<0·001) geometric mean (GM) for serum ferritin (26·5, 14·7, 10·8 μg/l) and the highest GM for serum retinol-binding protein (0·95, 1·00, 1·01 μmol/l) and Zn (11·8, 11·0, 11·5 μmol/l). As a consequence, at 6, 9 and 12 months regression adjustment yielded the highest prevalence of Fe deficiency (20·3, 37·8, 59·5 %) and the lowest prevalence of vitamin A (26·4,16·6, 17·3 %) and Zn (16·9, 20·6, 11·0 %) deficiency, respectively. For serum Se, irrespective of adjustment, GM were low (regression: 0·73, 0·78, 0·81 μmol/l) with prevalence of deficiency >50 % across all ages. In conclusion, without inflammation adjustment, Fe deficiency was grossly under-estimated and vitamin A and Zn deficiency over-estimated, highlighting the importance of correcting for the influence of such, before implementing programmes to alleviate micronutrient malnutrition. However, further work is needed to validate the proposed approaches with a particular focus on assessing the influence of varying degrees of inflammation (i.e. recurrent acute infections and low-grade chronic inflammation) on each affected nutrient biomarker.

Exact upper and lower bounds on the difference between the arithmetic and geometric means are obtained. The inequalities providing these bounds may be viewed, respectively, as a reverse Jensen inequality and an improvement of the direct Jensen inequality, in the case when the convex function is the exponential.

Let A, B be non-negative bounded self-adjoint operators, and let a be a real number such that 0 < a < 1. The Loewner–Heinz inequality means that A ≤ B implies that Aa ≦ Ba. We show that A ≤ B if and only if (A + λ)a ≦ (B + λ)a for every λ > 0. We then apply this to the geometric mean and spectral order.

Stratified and simple random sampling (or testing) are two common methods used to investigate the number or proportion of items in a population with a particular attribute. Although it is known that cost factors and information about the strata in the population are often crucial in deciding whether to use stratified or simple random sampling in a given situation, the stochastic precedence ordering for random variables can also provide the basis for an interesting criteria under which these methods may be compared. It may be particularly relevant when we are trying to find as many special items as possible in a population (for example individuals with a disease in a country). Properties of this total stochastic order on the class of random variables are discussed, and necessary and sufficient conditions are established which allow the comparison of the number of items of interest found in stratified random sampling with the number found in simple random sampling in the stochastic precedence order. These conditions are compared with other results established on stratified and simple random sampling (testing) using different stochastic-order-type criteria, and applications are given for the comparison of sums of Bernoulli random variables and binomial distributions.

Testing in order to produce software of high reliability is an area of major concern in software engineering. In an effort to find efficient methods of testing, the comparison of partition and random sampling testing methods has received considerable attention in the literature. A standard criterion for comparisons between random and partition testing, based on their expected efficacy in program debugging, is the probability of detecting at least one failure causing input in the program's domain. However, the goal in software testing is usually to find as many faults as possible in a reasonable period of time, and therefore stochastic comparisons of the number of faults obtained in partition and random testing may provide more valuable information on which testing procedures to use. We establish various conditions which guarantee that the number of faults discovered in partition testing is stochastically greater than the number discovered in random testing (using a fixed total sample size) for many of the well-established stochastic orders (including the usual stochastic order, the hazard rate order, the likelihood ratio order, and the variability order). The results established also allow us to obtain both upper and lower bounds with these stochastic orders for the sum of n independent Bernoulli random trials (with varying probability of success) in terms of the binomial distribution with parameters n and p.