An Introduction to Vedic Mathematics

Vedic Mathematics is an ancient system of mathematics, rediscovered last century by Sri Bharati Krsna Tirthaji (henceforth referred to as Bharati Krsna).

The Sanskrit word "veda" means "knowledge". The Vedas are ancient writings whose date is disputed but which date from at least several centuries BC. According to Indian tradition the content of the Vedas was known long before writing was invented and was freely available to everyone. It was passed on by word of mouth. The writings called the Vedas consist of a huge number of documents (there are said to be millions of such documents in India, many of which have not yet been translated) and these have recently been shown to be highly structured, both within themselves and in relation to each other (see Reference 2). Subjects covered in the Vedas include Grammar, Astronomy, Architecture, Psychology, Philosophy, Archery etc., etc.

A hundred years ago Sanskrit scholars were translating the Vedic documents and were surprised at the depth and breadth of knowledge contained in them. But some documents headed "Ganita Sutras", which means mathematics, could not be interpreted by them in terms of mathematics. One verse, for example, said "in the reign of King Kamse famine, pestilence and unsanitary conditions prevailed". This is not mathematics they said, but nonsense.

Bharati Krsna was born in 1884 and died in 1960. He was a brilliant student, obtaining the highest honours in all the subjects he studied, including Sanskrit, Philosophy, English, Mathematics, History and Science. When he heard what the European scholars were saying about the parts of the Vedas which were supposed to contain mathematics he resolved to study the documents and find their meaning. Between 1911 and 1918 he was able to reconstruct the ancient system of mathematics which we now call Vedic Mathematics.

He wrote sixteen books expounding this system, but unfortunately these have been lost and when the loss was confirmed in 1958 Bharati Krsna wrote a single introductory book entitled "Vedic Mathematics". This is currently available and is a best-seller.

You need to seeVedic Mathematics in action to appreciate it fully the many special aspects and features. The main points are:

1) The system rediscovered by Bharati Krsna is based on sixteen formulae (or Sutras) and some sub-formulae (sub-Sutras). These Sutras are given in word form: for example Vertically and Crosswise and By One More than the One Before. These Sutras can be related to natural mental functions such as completing a whole, noticing analogies, generalisation and so on.

2) Not only does the system give many striking general and special methods, previously unknown to modern mathematics, but it is far more coherent and integrated as a system.

3) Vedic Mathematics is a system of mental mathematics (though it can also be written down).

Many of the Vedic methods are new, simple and striking. They are also beautifully interrelated so that division, for example, can be seen as an easy reversal of the simple multiplication method (similarly with squaring and square roots).

This is in complete contrast to the modern system. Because the Vedic methods are so different to the conventional methods, and also to gain familiarity with the Vedic system, it is best to practice the techniques as you go along.

Statistics Reporting of Subgroup Analyses in Clinical Trials

Medical research relies on clinical trials to assess therapeutic benefits. Because of the effort and cost involved in these studies, investigators frequently use analyses of subgroups of study participants to extract as much information as possible. Such analyses, which assess the heterogeneity of treatment effects in subgroups of patients, may provide useful information for the care of patients and for future research. However, subgroup analyses also introduce analytic challenges and can lead to overstated and misleading results. This report outlines the challenges associated with conducting and reporting subgroup analyses, and it sets forth guidelines for their use in the Journal. Although this report focuses on the reporting of clinical trials, many of the issues discussed also apply to observational studies.

Subgroup Analyses and Related Concepts

Subgroup Analysis

By "subgroup analysis," we mean any evaluation of treatment effects for a specific end point in subgroups of patients defined by baseline characteristics. The end point may be a measure of treatment efficacy or safety. For a given end point, the treatment effect — a comparison between the treatment groups — is typically measured by a relative risk, odds ratio, or arithmetic difference. The research question usually posed is this: Do the treatment effects vary among the levels of a baseline factor?

A subgroup analysis is sometimes undertaken to assess treatment effects for a specific patient characteristic; this assessment is often listed as a primary or secondary study objective. For example, Sacks et al. conducted a placebo-controlled trial in which the reduction in the incidence of coronary events with the use of pravastatin was examined in a diverse population of persons who had survived a myocardial infarction. In subgroup analyses, the investigators further examined whether the efficacy of pravastatin relative to placebo in preventing coronary events varied according to the patients' baseline low-density lipoprotein (LDL) levels.

Subgroup analyses are also undertaken to investigate the consistency of the trial conclusions among different subpopulations defined by each of multiple baseline characteristics of the patients. For example, Jackson et al. reported the outcomes of a study in which 36,282 postmenopausal women 50 to 79 years of age were randomly assigned to receive 1000 mg of elemental calcium with 400 IU of vitamin D₃ daily or placebo. Fractures, the primary outcome, were ascertained over an average follow-up period of 7.0 years; bone density was a secondary outcome. Overall, no treatment effect was found for the primary outcome; that is, the active treatment was not shown to prevent fractures. The effect of calcium plus vitamin D supplementation relative to placebo on the risk of each of four fracture outcomes was further analyzed for consistency in subgroups defined by 15 characteristics of the participants.

Heterogeneity and Statistical Interactions

The heterogeneity of treatment effects across the levels of a baseline variable refers to the circumstance in which the treatment effects vary across the levels of the baseline characteristic. Heterogeneity is sometimes further classified as being either quantitative or qualitative. In the first case, one treatment is always better than the other, but by various degrees, whereas in the second case, one treatment is better than the other for one subgroup of patients and worse than the other for another subgroup of patients. Such variation, also called "effect modification," is typically expressed in a statistical model as an interaction term or terms between the treatment group and the baseline variable. The presence or absence of interaction is specific to the measure of the treatment effect.

The appropriate statistical method for assessing the heterogeneity of treatment effects among the levels of a baseline variable begins with a statistical test for interaction. For example, Sacks et al. showed the heterogeneity in pravastatin efficacy by reporting a statistically significant (P=0.03) result of testing for the interaction between the treatment and baseline LDL level when the measure of the treatment effect was the relative risk. Many trials lack the power to detect heterogeneity in treatment effect; thus, the inability to find significant interactions does not show that the treatment effect seen overall necessarily applies to all subjects. A common mistake is to claim heterogeneity on the basis of separate tests of treatment effects within each of the levels of the baseline variable. For example, testing the hypothesis that there is no treatment effect in women and then testing it separately in men does not address the question of whether treatment differences vary according to sex. Another common error is to claim heterogeneity on the basis of the observed treatment-effect sizes within each subgroup, ignoring the uncertainty of these estimates.

Multiplicity

It is common practice to conduct a subgroup analysis for each of several — and often many — baseline characteristics, for each of several end points, or for both. For example, the analysis by Jackson and colleagues of the effect of calcium plus vitamin D supplementation relative to placebo on the risk of each of four fracture outcomes for 15 participant characteristics resulted in a total of 60 subgroup analyses.

When multiple subgroup analyses are performed, the probability of a false positive finding can be substantial. For example, if the null hypothesis is true for each of 10 independent tests for interaction at the 0.05 significance level, the chance of at least one false positive result exceeds 40%. Thus, one must be cautious in the interpretation of such results. There are several methods for addressing multiplicity that are based on the use of more stringent criteria for statistical significance than the customary P<0.05.A less formal approach for addressing multiplicity is to note the number of nominally significant interaction tests that would be expected to occur by chance alone. For example, after noting that 60 subgroup analyses were planned, Jackson et al. pointed out that "Up to three statistically significant interaction tests (P<0.05)> on the basis of chance alone," and then they incorporated this consideration in their interpretation of the results.

Prespecified Analysis versus Post Hoc Analysis

A prespecified subgroup analysis is one that is planned and documented before any examination of the data, preferably in the study protocol. This analysis includes specification of the end point, the baseline characteristic, and the statistical method used to test for an interaction. For example, the Heart Outcomes Prevention Evaluation 2 investigators conducted a study involving 5522 patients with vascular disease or diabetes to assess the effect of homocysteine lowering with folic acid and B vitamins on the risk of a major cardiovascular event. The primary outcome was a composite of death from cardiovascular causes, myocardial infarction, and stroke. In the Methods section of their article, the authors noted that "Prespecified subgroup analyses involving Cox models were used to evaluate outcomes in patients from regions with folate fortification of food and regions without folate fortification, according to the baseline plasma homocysteine level and the baseline serum creatinine level." Post hoc analyses refer to those in which the hypotheses being tested are not specified before any examination of the data. Such analyses are of particular concern because it is often unclear how many were undertaken and whether some were motivated by inspection of the data. However, both prespecified and post hoc subgroup analyses are subject to inflated false positive rates arising from multiple testing. Investigators should avoid the tendency to prespecify many subgroup analyses in the mistaken belief that these analyses are free of the multiplicity problem.

Subgroup Analyses in the Journal — Assessment of Reporting Practices

As part of internal quality-control activities at the Journal, we assessed the completeness and quality of subgroup analyses reported in the Journal during the period from July 1, 2005, through June 30, 2006. A detailed description of the study methods can be found in the available with the full text of this article at www.nejm.org. In this report, we describe the clarity and completeness of subgroup-analysis reporting, evaluate the authors' interpretation and justification of the results of subgroup analyses, and recommend guidelines for reporting subgroup analyses.

Among the original articles published in the Journal during the period from July 1, 2005, through June 30, 2006, a total of 95 articles reported primary outcome results from randomized clinical trials. Among these 95 articles, 93 reported results from one clinical trial; the remaining 2 articles reported results from two trials. Thus, results from 97 trials were reported, from which subgroup analyses were reported for 59 trials (61%). summarizes the characteristics of the trials. We found that larger trials and multicenter trials were significantly more likely to report subgroup analyses than smaller trials and single-center trials, respectively. With the use of multivariate logistic-regression models, when ranked according to the number of participants enrolled in a trial and compared with trials with the fewest participants, the odds ratio for reporting subgroup analyses for the second quartile was 1.38 (95% confidence interval [CI], 0.45 to 4.20), for the third quartile was 1.98 (95% CI, 0.62 to 6.24), and for the fourth quartile was 8.90 (95% CI, 2.10 to 37.78) (P=0.02, trend test). The odds ratio for reporting subgroup analyses in multicenter trials as compared with single-center trials was 4.33 (95% CI, 1.56 to 12.16).