Statistical Method Quotes

There is one and only one statistical method for one objective in research.

The statistical method shows the facts in the light of the ideal average but does not give us a picture of their empirical reality. While reflecting an indisputable aspect of reality, it can falsify the actual truth in a most misleading way. This is particularly true of theories which are based on statistics. The distinctive thing about real facts, however, is their individuality. Not to put too fine a point on it, once could say that the real picture consists of nothing but exceptions to the rule, and that, in consequence, absolute reality has predominantly the character of irregularity.

There is no rule that is true under all circumstances, for this is the real and not a statistical world. Because the statistical method shows only the average aspects, it creates an artificial and predominantly conceptual picture of reality.

IQ is a statistical method for quantifying specific kinds of problem-solving ability, mathematically convenient but not necessarily corresponding to a real attribute of the human brain, and not necessarily representing whatever it is that we mean by ‘intelligence’.

A certain elementary training in statistical method is becoming as necessary for everyone living in this world of today as reading and writing.

It is not easy to become an educated person.

The statistics all point towards the same conclusion: we have a global outbreak of fuckarounditis.

One of the methods used by statists to destroy capitalism consists in establishing controls that tie a given industry hand and foot, making it unable to solve its problems, then declaring that freedom has failed and stronger controls are necessary.”=Ayn Rand

The applications of knowledge, especially mathematics, reveal the unity of all knowledge. In a new situation almost anything and everything you ever learned might be applicable, and the artificial divisions seem to vanish.

A good way to do econometrics is to look for good natural experiments and use statistical methods that can tidy up the confounding factors that Nature has not controlled for us.

What happened to me personally is only anecdotal evidence," Harry explained. "It doesn't carry the same weight as a replicated, peer-reviewed journal article about a controlled study with random assignment, many subjects, large effect sizes and strong statistical significance.

In the course catalogue of the psychology department at my own university, the first required course in the curriculum is ‘Introduction to Statistics and Methodology in Psychological Research’. Second-year psychology students must take ‘Statistical Methods in Psychological Research’. Confucius, Buddha, Jesus and Muhammad would have been bewildered if you’d told them that in order to understand the human mind and cure its illnesses you must first study statistics.

The statistical methods underlying productivity measurements tend to factor out gains by essentially concluding that we still get only one dollar of products and services for a dollar, despite the fact that we get much more for that dollar.

You end up with a machine which knows that by its mildest estimate it must have terrible enemies all around and within it, but it can't find them. It therefore deduces that they are well-concealed and expert, likely professional agitators and terrorists. Thus, more stringent and probing methods are called for. Those who transgress in the slightest, or of whom even small suspicions are harboured, must be treated as terrible foes. A lot of rather ordinary people will get repeatedly investigated with increasing severity until the Government Machine either finds enemies or someone very high up indeed personally turns the tide... And these people under the microscope are in fact just taking up space in the machine's numerical model. In short, innocent people are treated as hellish fiends of ingenuity and bile because there's a gap in the numbers.

Every criminal was humiliated, neglected, or abused in childhood, but few of them can admit to it. Many genuinely do not know that they were. Thus denial gets in the way of statistical surveys based on the question-and-answer method, none of which will have any practical prophylactic effect as long as our eyes and ears remain closed to the issues posed by childhood.

Makridakis and Hibon reached the sad conclusion that "statistically sophisticated or complex methods do not necessarily provide more accurate forecasts than simpler ones.

The combination of Bayes and Markov Chain Monte Carlo has been called "arguably the most powerful mechanism ever created for processing data and knowledge." Almost instantaneously MCMC and Gibbs sampling changed statisticians' entire method of attacking problems. In the words of Thomas Kuhn, it was a paradigm shift. MCMC solved real problems, used computer algorithms instead of theorems, and led statisticians and scientists into a worked where "exact" meant "simulated" and repetitive computer operations replaced mathematical equations. It was a quantum leap in statistics.

Including the differential mortgage loan approval rates between Asian Americans and whites shows that the same methods to conclude that that blacks are discriminated against in mortgage lending would also lead to the conclusion that whites are discriminated against in favor of Asian Americans, reducing this whole procedure to absurdity, since no one believes that banks are discriminating against whites..."[W]hen loan approval rates are not cited, but loan denial rates are, that creates a larger statistical disparity, since most loans are approved. Even if 98 percent of blacks had their mortgage loan applications approved, if 99 percent of whites were approved than by quoting denial rates alone it could be said that blacks were rejected twice as often as whites.

Perhaps the most flagrant testimony to the intellectual shallowness of statism is that the typical statist believes that the fantastically hypothetical threat of a corporation monopolizing the supply of water is a devastating objection to libertarianism, but the painfully real threat of a state methodically exterminating tens of millions of individuals is not a devastating objection to statism.

What was all this? He knew only too well. The statistical method! Probability theory! There was a greater probability of finding enemies among people of a non-proletarian background. And it was on these same grounds – probability theory – that the German Fascists had destroyed whole peoples and nations. The principle was inhuman, blind and inhuman. There was only one acceptable way of relating to people – a human way.

The brain cannot reach its inner conclusions by any logic of certainty. In place of this, the brain must do two things. It must be content to accept less than certain knowledge. And it must have statistical methods which are different in kind from ours, by which it reaches its acceptable level of uncertainty. By these means, the brain constructs a picture of the world which is less than certain yet highly interlocked in its parts.

Thinking Statistically by Uri Bram   How to Lie with Statistics by Darrell Huff   Turning Numbers into Knowledge by Jonathan G. Koomey, PhD For an examination of more advanced methods of analysis, Principles of Statistics by M. G. Bulmer is a useful reference.

This will be a frequent dilemma for historians trying to apply the comparative method to problems of human history: apparently too many potentially independent variables, and far too few separate outcomes to establish those variables’ importance statistically.

But psychology is passing into a less simple phase. Within a few years what one may call a microscopic psychology has arisen in Germany, carried on by experimental methods, asking of course every moment for introspective data, but eliminating their uncertainty by operating on a large scale and taking statistical means. This method taxes patience to the utmost, and could hardly have arisen in a country whose natives could be bored. Such Germans as Weber, Fechner, Vierordt, and Wundt obviously cannot ; and their success has brought into the field an array of younger experimental psychologists, bent on studying the elements of the mental life, dissecting them out from the gross results in which they are embedded, and as far as possible reducing them to quantitative scales. The simple and open method of attack having done what it can, the method of patience, starving out, and harassing to death is tried ; the Mind must submit to a regular siege, in which minute advantages gained night and day by the forces that hem her in must sum themselves up at last into her overthrow. There is little of the grand style about these new prism, pendulum, and chronograph-philosophers. They mean business, not chivalry. What generous divination, and that superiority in virtue which was thought by Cicero to give a man the best insight into nature, have failed to do, their spying and scraping, their deadly tenacity and almost diabolic cunning, will doubtless some day bring about. No general description of the methods of experimental psychology would be instructive to one unfamiliar with the instances of their application, so we will waste no words upon the attempt.

This was the strategy that concerned Podhorzer, who worried that the approach reflected “an overconfidence in the liberal mind that only a fool would vote for Bush again, so therefore all we had to do was base turnout.” He returned to the AFL dispirited by his exchange with the Kerry campaign and his sense that Democrats had chosen to ignore winnable votes in part because the statistical methods used to identify them were not properly understood by campaign decision makers. “At the time,” says Podhorzer, “all of this just sounded like alien talk to most people.

In 1900, Planck came up with an equation, partly using what he called “a fortuitous guess,” that described the curve of radiation wavelengths at each temperature. In doing so he accepted that Boltzmann’s statistical methods, which he had resisted, were correct after all. But the equation had an odd feature: it required the use of a constant, which was an unexplained tiny quantity (approximately 6.62607 × 10–34 joule-seconds), that needed to be included for it to come out right. It was soon dubbed Planck’s constant, h, and is now known as one of the fundamental constants of nature.

Using more traditional methods of tallying assaults, the statistics showed that Border Patrol agents did not experience the highest assault rate among law enforcement officers. They experienced the lowest. The death rate among Border Patrol agents was about one-third that of the nation’s law enforcement officers who policed residents.

Price mostly meanders around recent price until a big shift in opinion occurs, causing price to jump up or down. This is crudely modeled by quants using something called a jump-diffusion process model. Again, what does this have to do with an asset’s true intrinsic value? Not much. Fortunately, the value-focused investor doesn’t have to worry about these statistical methods and jargon. Stochastic calculus, information theory, GARCH variants, statistics, or time-series analysis is interesting if you’re into it, but for the value investor, it is mostly noise and not worth pursuing. The value investor needs to accept that often price can be wrong for long periods and occasionally offers interesting discounts to value.

Before I repeat any statistical claim, I first try to take note of how it makes me feel. It’s not a foolproof method against tricking myself, but it’s a habit that does little harm and is sometimes a great deal of help. Our emotions are powerful. We can’t make them vanish, nor should we want to. But we can, and should, try to notice when they are clouding our judgment.

the most common approach to curriculum design is to address the needs of the so-called “average student.” Of course this average student is a myth, a statistical artifact not corresponding to any actual individual. But because so much of the curriculum and teaching methods employed in most schools are based on the needs of this mythical average student, they are also laden with inadvertent and unnecessary barriers to learning.

In trying to comprehend and judge moral dilemmas of this scale, people often resort to one of four methods. The first is to downsize the issue. To understand the Syrian civil war as though it were occurring between two foragers, for example, one imagines the Assad regime as a lone person and the rebels as another person; one of them is bad and one of them is good. The historical complexity of the conflict is replaced by a simple, clear plot.4 The second method is to focus on a touching human story that ostensibly stands for the whole conflict. When you try to explain to people the true complexity of the conflict by means of statistics and precise data, you lose them, but a personal story about the fate of one child activates the tear ducts, makes the blood boil, and generates false moral certainty.5 This is something that many charities have understood for a long time. In one noteworthy experiment, people were asked to donate money to help a poor seven-year-old girl from Mali named Rokia. Many were moved by her story and opened their hearts and purses. However, when in addition to Rokia’s personal story the researchers also presented people with statistics about the broader problem of poverty in Africa, respondents suddenly became less willing to help. In another study, scholars solicited donations to help either one sick child or eight sick children. People gave more money to the single child than to the group of eight.6

If, for instance, I determine the weight of each stone in a bed of pebbles and get an average weight of 145 grams, this tells me very little about the real nature of the pebbles. Anyone who thought, on the basis of these findings, that he could pick up a pebbles of 145 grams at the first try would be in for a serious disappointment. Indeed, it might well happen that however long he searched he would not find a single pebble weighing exactly 145 grams. The statistical method shows the facts in the light of the ideal average but does not give us a picture of their empirical reality. While reflecting an indisputable aspect of reality, it can falsify the actual truth in a most misleading way. This is particularly true of theories which are based on statistics. The distinctive thing about real facts, however, is their individuality...one could say that the real picture consists of nothing but exceptions to the rule, and that, in consequence, absolute reality has predominantly the characteristic of *irregularity* (The Undiscovered Self)

I found out that telling researchers "This is where your methods work very well" is vastly better than telling them "This is what you guys don't know." So when I presented to what was until then the most hostile crowd in the world, members of the American Statistical Association, a map of the four quadrants, and told them: your knowledge works beautifully in these three quadrants, but beware the fourth one, as this is where the Black Swans breed, I received instant approval, support, offers of permanent friendship, refreshments (Diet Coke), invitations to come present at their sessions, even hugs.

But we don’t correct for the difference in science, medicine, and mathematics, for the same reasons we didn’t pay attention to iatrogenics. We are suckers for the sophisticated. In institutional research, one can selectively report facts that confirm one’s story, without revealing facts that disprove it or don’t apply to it—so the public perception of science is biased into believing in the necessity of the highly conceptualized, crisp, and purified Harvardized methods. And statistical research tends to be marred with this one-sidedness. Another reason one should trust the disconfirmatory more than the confirmatory.

While a 10x improvement is gargantuan, Teller has very specific reasons for aiming exactly that high. “You assume that going 10x bigger is going to be ten times harder,” he continues, “but often it’s literally easier to go bigger. Why should that be? It doesn’t feel intuitively right. But if you choose to make something 10 percent better, you are almost by definition signing up for the status quo—and trying to make it a little bit better. That means you start from the status quo, with all its existing assumptions, locked into the tools, technologies, and processes that you’re going to try to slightly improve. It means you’re putting yourself and your people into a smartness contest with everyone else in the world. Statistically, no matter the resources available, you’re not going to win. But if you sign up for moonshot thinking, if you sign up to make something 10x better, there is no chance of doing that with existing assumptions. You’re going to have to throw out the rule book. You’re going to have to perspective-shift and supplant all that smartness and resources with bravery and creativity.” This perspective shift is key. It encourages risk taking and enhances creativity while simultaneously guarding against the inevitable decline. Teller explains: “Even if you think you’re going to go ten times bigger, reality will eat into your 10x. It always does. There will be things that will be more expensive, some that are slower; others that you didn’t think were competitive will become competitive. If you shoot for 10x, you might only be at 2x by the time you’re done. But 2x is still amazing. On the other hand, if you only shoot for 2x [i.e., 200 percent], you’re only going to get 5 percent and it’s going to cost you the perspective shift that comes from aiming bigger.” Most critically here, this 10x strategy doesn’t hold true just for large corporations. “A start-up is simply a skunk works without the big company around it,” says Teller. “The upside is there’s no Borg to get sucked back into; the downside is you have no money. But that’s not a reason not to go after moonshots. I think the opposite is true. If you publicly state your big goal, if you vocally commit yourself to making more progress than is actually possible using normal methods, there’s no way back. In one fell swoop you’ve severed all ties between yourself and all the expert assumptions.” Thus entrepreneurs, by striving for truly huge goals, are tapping into the same creativity accelerant that Google uses to achieve such goals. That said, by itself, a willingness to take bigger risks

[…] a student in our class asked disdainfully why quantitative methodologists do not openly criticize qualitative methods. He scoffed, 'They don't even mention it. But in courses in qualitative methods, quantitative methods always come up.' […] I pointed out that the lack of critical remarks and the absence of any mention of qualitative research in 'methods' courses indicate the hegemony of the quantitative approach. Were not his statistics professors making a strong statement about the place of qualitative methods by omitting them entirely? Qualitative researchers, then, have to legitimate their perspective to students in order to break the methodological silence coming from the other side.

In Bohr’s model of the atom, electrons could change their orbits (or, more precisely, their stable standing wave patterns) only by certain quantum leaps. De Broglie’s thesis helped explain this by conceiving of electrons not just as particles but also as waves. Those waves are strung out over the circular path around the nucleus. This works only if the circle accommodates a whole number—such as 2 or 3 or 4—of the particle’s wavelengths; it won’t neatly fit in the prescribed circle if there’s a fraction of a wavelength left over. De Broglie made three typed copies of his thesis and sent one to his adviser, Paul Langevin, who was Einstein’s friend (and Madame Curie’s). Langevin, somewhat baffled, asked for another copy to send along to Einstein, who praised the work effusively. It had, Einstein said, “lifted a corner of the great veil.” As de Broglie proudly noted, “This made Langevin accept my work.”47 Einstein made his own contribution when he received in June of that year a paper in English from a young physicist from India named Satyendra Nath Bose. It derived Planck’s blackbody radiation law by treating radiation as if it were a cloud of gas and then applying a statistical method of analyzing it. But there was a twist: Bose said that any two photons that had the same energy state were absolutely indistinguishable, in theory as well as fact, and should not be treated separately in the statistical calculations.

It will be noticed that the fundamental theorem proved above bears some remarkable resemblances to the second law of thermodynamics. Both are properties of populations, or aggregates, true irrespective of the nature of the units which compose them; both are statistical laws; each requires the constant increase of a measurable quantity, in the one case the entropy of a physical system and in the other the fitness, measured by m, of a biological population. As in the physical world we can conceive the theoretical systems in which dissipative forces are wholly absent, and in which the entropy consequently remains constant, so we can conceive, though we need not expect to find, biological populations in which the genetic variance is absolutely zero, and in which fitness does not increase. Professor Eddington has recently remarked that 'The law that entropy always increases—the second law of thermodynamics—holds, I think, the supreme position among the laws of nature'. It is not a little instructive that so similar a law should hold the supreme position among the biological sciences. While it is possible that both may ultimately be absorbed by some more general principle, for the present we should note that the laws as they stand present profound differences—-(1) The systems considered in thermodynamics are permanent; species on the contrary are liable to extinction, although biological improvement must be expected to occur up to the end of their existence. (2) Fitness, although measured by a uniform method, is qualitatively different for every different organism, whereas entropy, like temperature, is taken to have the same meaning for all physical systems. (3) Fitness may be increased or decreased by changes in the environment, without reacting quantitatively upon that environment. (4) Entropy changes are exceptional in the physical world in being irreversible, while irreversible evolutionary changes form no exception among biological phenomena. Finally, (5) entropy changes lead to a progressive disorganization of the physical world, at least from the human standpoint of the utilization of energy, while evolutionary changes are generally recognized as producing progressively higher organization in the organic world.

Good science is more than the mechanics of research and experimentation. Good science requires that scientists look inward--to contemplate the origin of their thoughts. The failures of science do not begin with flawed evidence or fumbled statistics; they begin with personal self-deception and an unjustified sense of knowing. Once you adopt the position that personal experience is the "proof of the pudding," reasoned discussion simply isn't possible. Good science requires distinguishing between "felt knowledge" and knowledge arising out of testable observations. "I am sure" is a mental sensation, not a testable conclusion. Put hunches, gut feelings, and intuitions into the suggestion box. Let empiric methods shake out the good from the bad suggestions.

The leftist is always a statist. He has all sorts of grievances and animosities against personal initiative and private enterprise. The notion of the state doing everything (until, finally, it replaces all private existence) is the Great Leftist Dream. Thus it is a leftist tendency to have city or state schools—or to have a ministry of education controlling all aspects of education. For example, there is the famous story of the French Minister of Education who pulls out his watch and, glancing at its face, says to his visitor, “At this moment in 5,431 public elementary schools they are writing an essay on the joys of winter.” Church schools, parochial schools, private schools, or personal tutors are not at all in keeping with leftist sentiments. The reasons for this attitude are manifold. Here not only is the delight in statism involved, but the idea of uniformity and equality is also decisive; i.e., the notion that social differences in education should be eliminated and all pupils should be given a chance to acquire the same knowledge, the same type of information in the same fashion and to the same degree. This should help them to think in identical or at least in similar ways. It is only natural that this should be especially true of countries where “democratism” as an ism is being pushed. There efforts will be made to ignore the differences in IQs and in personal efforts. Sometimes marks and report cards will be eliminated and promotion from one grade to the next be made automatic. It is obvious that from a scholastic viewpoint this has disastrous results, but to a true ideologist this hardly matters. When informed that the facts did not tally with his ideas, Hegel once severely replied, “Um so schlimmer für die Tatsachen”—all the worse for the facts. Leftism does not like religion for a variety of causes. Its ideologies, its omnipotent, all-permeating state wants undivided allegiance. With religion at least one other allegiance (to God), if not also allegiance to a Church, is interposed. In dealing with organized religion, leftism knows of two widely divergent procedures. One is a form of separation of Church and State which eliminates religion from the marketplace and tries to atrophy it by not permitting it to exist anywhere outside the sacred precincts. The other is the transformation of the Church into a fully state-controlled establishment. Under these circumstances the Church is asphyxiated, not starved to death. The Nazis and the Soviets used the former method; Czechoslovakia still employs the latter.

This irrelevance of molecular arrangements for macroscopic results has given rise to the tendency to confine physics and chemistry to the study of homogeneous systems as well as homogeneous classes. In statistical mechanics a great deal of labor is in fact spent on showing that homogeneous systems and homogeneous classes are closely related and to a considerable extent interchangeable concepts of theoretical analysis (Gibbs theory). Naturally, this is not an accident. The methods of physics and chemistry are ideally suited for dealing with homogeneous classes with their interchangeable components. But experience shows that the objects of biology are radically inhomogeneous both as systems (structurally) and as classes (generically). Therefore, the method of biology and, consequently, its results will differ widely from the method and results of physical science.

To understand my doctor’s error, let’s employ Bayes’s method. The first step is to define the sample space. We could include everyone who has ever taken an HIV test, but we’ll get a more accurate result if we employ a bit of additional relevant information about me and consider only heterosexual non-IV-drug-abusing white male Americans who have taken the test. (We’ll see later what kind of difference this makes.) Now that we know whom to include in the sample space, let’s classify the members of the space. Instead of boy and girl, here the relevant classes are those who tested positive and are HIV-positive (true positives), those who tested positive but are not positive (false positives), those who tested negative and are HIV-negative (true negatives), and those who tested negative but are HIV-positive (false negatives). Finally, we ask, how many people are there in each of these classes? Suppose we consider an initial population of 10,000. We can estimate, employing statistics from the Centers for Disease Control and Prevention, that in 1989 about 1 in those 10,000 heterosexual non-IV-drug-abusing white male Americans who got tested were infected with HIV.6 Assuming that the false-negative rate is near 0, that means that about 1 person out of every 10,000 will test positive due to the presence of the infection. In addition, since the rate of false positives is, as my doctor had quoted, 1 in 1,000, there will be about 10 others who are not infected with HIV but will test positive anyway. The other 9,989 of the 10,000 men in the sample space will test negative. Now let’s prune the sample space to include only those who tested positive. We end up with 10 people who are false positives and 1 true positive. In other words, only 1 in 11 people who test positive are really infected with HIV.

To counter all these biases, both in my readers, and in myself, I try to move my estimates in the following directions. I try to be less confident, to expect typical outcomes to be more ordinary, but also to expect more deviations from typical outcomes. I try to rely more on ordinary methods, sources, and assumptions, and also more on statistics or related systems and events. I expect bigger deviations from traditional images of the future, but also rely less on strange, exotic, unlikely-seeming, and hypothetical possibilities. Looking backward, future folk should see their world as changing less from their past than we might see looking forward. Seen up close and honestly, I expect the future usually to look like most places: mundane, uninspiring, and morally ambiguous, with grand hopes and justifications often masking lives of quiet desperation. Of course, lives of quiet desperation can still be worth living.

I would advise those who think that self-help is the answer to familiarize themselves with the long history of such efforts in the Negro community, and to consider why so many foundered on the shoals of ghetto life. It goes without saying that any effort to combat demoralization and apathy is desirable, but we must understand that demoralization in the Negro community is largely a common-sense response to an objective reality. Negro youths have no need of statistics to perceive, fairly accurately, what their odds are in American society. Indeed, from the point of view of motivation, some of the healthiest Negro youngsters I know are juvenile delinquents. Vigorously pursuing the American dream of material acquisition and status, yet finding the conventional means of attaining it blocked off, they do not yield to defeatism but resort to illegal (and often ingenious) methods.... If Negroes are to be persuaded that the conventional path (school, work, etc.) is superior, we had better provide evidence which is now sorely lacking.

Due to the various pragmatic obstacles, it is rare for a mission-critical analysis to be done in the “fully Bayesian” manner, i.e., without the use of tried-and-true frequentist tools at the various stages. Philosophy and beauty aside, the reliability and efficiency of the underlying computations required by the Bayesian framework are the main practical issues. A central technical issue at the heart of this is that it is much easier to do optimization (reliably and efficiently) in high dimensions than it is to do integration in high dimensions. Thus the workhorse machine learning methods, while there are ongoing efforts to adapt them to Bayesian framework, are almost all rooted in frequentist methods. A work-around is to perform MAP inference, which is optimization based. Most users of Bayesian estimation methods, in practice, are likely to use a mix of Bayesian and frequentist tools. The reverse is also true—frequentist data analysts, even if they stay formally within the frequentist framework, are often influenced by “Bayesian thinking,” referring to “priors” and “posteriors.” The most advisable position is probably to know both paradigms well, in order to make informed judgments about which tools to apply in which situations.

When he applied this approach to a gas of quantum particles, Einstein discovered an amazing property: unlike a gas of classical particles, which will remain a gas unless the particles attract one another, a gas of quantum particles can condense into some kind of liquid even without a force of attraction between them. This phenomenon, now called Bose-Einstein condensation,* was a brilliant and important discovery in quantum mechanics, and Einstein deserves most of the credit for it. Bose had not quite realized that the statistical mathematics he used represented a fundamentally new approach. As with the case of Planck’s constant, Einstein recognized the physical reality, and the significance, of a contrivance that someone else had devised.49 Einstein’s method had the effect of treating particles as if they had wavelike traits, as both he and de Broglie had suggested. Einstein even predicted that if you did Thomas Young’s old double-slit experiment (showing that light behaved like a wave by shining a beam through two slits and noting the interference pattern) by using a beam of gas molecules, they would interfere with one another as if they were waves. “A beam of gas molecules which passes through an aperture,” he wrote, “must undergo a diffraction analogous to that of a light ray.

To come now to the last point: can we call something with which the concepts of position and motion cannot be associated in the usual way, a thing, or a particle? And if not, what is the reality which our theory has been invented to describe? The answer to this is no longer physics, but philosophy, and to deal with it thoroughly would mean going far beyond the bounds of this lecture. I have given my views on it elsewhere. Here I will only say that I am emphatically in favour of the retention of the particle idea. Naturally, it is necessary to redefine what is meant. For this, well-developed concepts are available which appear in mathematics under the name of invariants in transformations. Every object that we perceive appears in innumerable aspects. The concept of the object is the invariant of all these aspects. From this point of view, the present universally used system of concepts in which particles and waves appear simultaneously, can be completely justified. The latest research on nuclei and elementary particles has led us, however, to limits beyond which this system of concepts itself does not appear to suffice. The lesson to be learned from what I have told of the origin of quantum mechanics is that probable refinements of mathematical methods will not suffice to produce a satisfactory theory, but that somewhere in our doctrine is hidden a concept, unjustified by experience, which we must eliminate to open up the road.

Modern statistics is built on the idea of models — probability models in particular. [...] The standard approach to any new problem is to identify the sources of variation, to describe those sources by probability distributions and then to use the model thus created to estimate, predict or test hypotheses about the undetermined parts of that model. […] A statistical model involves the identification of those elements of our problem which are subject to uncontrolled variation and a specification of that variation in terms of probability distributions. Therein lies the strength of the statistical approach and the source of many misunderstandings. Paradoxically, misunderstandings arise both from the lack of an adequate model and from over reliance on a model. […] At one level is the failure to recognise that there are many aspects of a model which cannot be tested empirically. At a higher level is the failure is to recognise that any model is, necessarily, an assumption in itself. The model is not the real world itself but a representation of that world as perceived by ourselves. This point is emphasised when, as may easily happen, two or more models make exactly the same predictions about the data. Even worse, two models may make predictions which are so close that no data we are ever likely to have can ever distinguish between them. […] All model-dependant inference is necessarily conditional on the model. This stricture needs, especially, to be borne in mind when using Bayesian methods. Such methods are totally model-dependent and thus all are vulnerable to this criticism. The problem can apparently be circumvented, of course, by embedding the model in a larger model in which any uncertainties are, themselves, expressed in probability distributions. However, in doing this we are embarking on a potentially infinite regress which quickly gets lost in a fog of uncertainty.

Bose’s creative use of statistical analysis was reminiscent of Einstein’s youthful enthusiasm for that approach. He not only got Bose’s paper published, he also extended it with three papers of his own. In them, he applied Bose’s counting method, later called “Bose-Einstein statistics,” to actual gas molecules, thus becoming the primary inventor of quantum-statistical mechanics. Bose’s paper dealt with photons, which have no mass. Einstein extended the idea by treating quantum particles with mass as being indistinguishable from one another for statistical purposes in certain cases. “The quanta or molecules are not treated as structures statistically independent of one another,” he wrote.48 The key insight, which Einstein extracted from Bose’s initial paper, has to do with how you calculate the probabilities for each possible state of multiple quantum particles. To use an analogy suggested by the Yale physicist Douglas Stone, imagine how this calculation is done for dice. In calculating the odds that the roll of two dice (A and B) will produce a lucky 7, we treat the possibility that A comes up 4 and B comes up 3 as one outcome, and we treat the possibility that A comes up 3 and B comes up 4 as a different outcome—thus counting each of these combinations as different ways to produce a 7. Einstein realized that the new way of calculating the odds of quantum states involved treating these not as two different possibilities, but only as one. A 4-3 combination was indistinguishable from a 3-4 combination; likewise, a 5-2 combination was indistinguishable from a 2-5. That cuts in half the number of ways two dice can roll a 7. But it does not affect the number of ways they could turn up a 2 or a 12 (using either counting method, there is only one way to roll each of these totals), and it only reduces from five to three the number of ways the two dice could total 6. A few minutes of jotting down possible outcomes shows how this system changes the overall odds of rolling any particular number. The changes wrought by this new calculating method are even greater if we are applying it to dozens of dice. And if we are dealing with billions of particles, the change in probabilities becomes huge.

Was this luck, or was it more than that? Proving skill is difficult in venture investing because, as we have seen, it hinges on subjective judgment calls rather than objective or quantifiable metrics. If a distressed-debt hedge fund hires analysts and lawyers to scrutinize a bankrupt firm, it can learn precisely which bond is backed by which piece of collateral, and it can foresee how the bankruptcy judge is likely to rule; its profits are not lucky. Likewise, if an algorithmic hedge fund hires astrophysicists to look for patterns in markets, it may discover statistical signals that are reliably profitable. But when Perkins backed Tandem and Genentech, or when Valentine backed Atari, they could not muster the same certainty. They were investing in human founders with human combinations of brilliance and weakness. They were dealing with products and manufacturing processes that were untested and complex; they faced competitors whose behaviors could not be forecast; they were investing over long horizons. In consequence, quantifiable risks were multiplied by unquantifiable uncertainties; there were known unknowns and unknown unknowns; the bracing unpredictability of life could not be masked by neat financial models. Of course, in this environment, luck played its part. Kleiner Perkins lost money on six of the fourteen investments in its first fund. Its methods were not as fail-safe as Tandem’s computers. But Perkins and Valentine were not merely lucky. Just as Arthur Rock embraced methods and attitudes that put him ahead of ARD and the Small Business Investment Companies in the 1960s, so the leading figures of the 1970s had an edge over their competitors. Perkins and Valentine had been managers at leading Valley companies; they knew how to be hands-on; and their contributions to the success of their portfolio companies were obvious. It was Perkins who brought in the early consultants to eliminate the white-hot risks at Tandem, and Perkins who pressed Swanson to contract Genentech’s research out to existing laboratories. Similarly, it was Valentine who drove Atari to focus on Home Pong and to ally itself with Sears, and Valentine who arranged for Warner Communications to buy the company. Early risk elimination plus stage-by-stage financing worked wonders for all three companies. Skeptical observers have sometimes asked whether venture capitalists create innovation or whether they merely show up for it. In the case of Don Valentine and Tom Perkins, there was not much passive showing up. By force of character and intellect, they stamped their will on their portfolio companies.

regression as dummy variables Explain the importance of the error term plot Identify assumptions of regression, and know how to test and correct assumption violations Multiple regression is one of the most widely used multivariate statistical techniques for analyzing three or more variables. This chapter uses multiple regression to examine such relationships, and thereby extends the discussion in Chapter 14. The popularity of multiple regression is due largely to the ease with which it takes control variables (or rival hypotheses) into account. In Chapter 10, we discussed briefly how contingency tables can be used for this purpose, but doing so is often a cumbersome and sometimes inconclusive effort. By contrast, multiple regression easily incorporates multiple independent variables. Another reason for its popularity is that it also takes into account nominal independent variables. However, multiple regression is no substitute for bivariate analysis. Indeed, managers or analysts with an interest in a specific bivariate relationship will conduct a bivariate analysis first, before examining whether the relationship is robust in the presence of numerous control variables. And before conducting bivariate analysis, analysts need to conduct univariate analysis to better understand their variables. Thus, multiple regression is usually one of the last steps of analysis. Indeed, multiple regression is often used to test the robustness of bivariate relationships when control variables are taken into account. The flexibility with which multiple regression takes control variables into account comes at a price, though. Regression, like the t-test, is based on numerous assumptions. Regression results cannot be assumed to be robust in the face of assumption violations. Testing of assumptions is always part of multiple regression analysis. Multiple regression is carried out in the following sequence: (1) model specification (that is, identification of dependent and independent variables), (2) testing of regression assumptions, (3) correction of assumption violations, if any, and (4) reporting of the results of the final regression model. This chapter examines these four steps and discusses essential concepts related to simple and multiple regression. Chapters 16 and 17 extend this discussion by examining the use of logistic regression and time series analysis. MODEL SPECIFICATION Multiple regression is an extension of simple regression, but an important difference exists between the two methods: multiple regression aims for full model specification. This means that analysts seek to account for all of the variables that affect the dependent variable; by contrast, simple regression examines the effect of only one independent variable. Philosophically, the phrase identifying the key difference—“all of the variables that affect the dependent variable”—is divided into two parts. The first part involves identifying the variables that are of most (theoretical and practical) relevance in explaining the dependent

eigenvalue of a factor is the sum of correlations (r) of each variable with that factor. This correlation is also called loading in factor analysis. Analysts can define (or “extract”) how many factors they wish to use, or they can define a statistical criterion (typically requiring each factor to have an eigenvalue of at least 1.0). The method of identifying factors is called principal component analysis (PCA). The results of PCA often make it difficult to interpret the factors, in which case the analyst will use rotation (a statistical technique that distributes the explained variance across factors). Rotation causes variables to load higher on one factor, and less on others, bringing the pattern of groups better into focus for interpretation. Several different methods of rotation are commonly used (for example, Varimax, Promax), but the purpose of this procedure is always to understand which variables belong together. Typically, for purposes of interpretation, factor loadings are considered only if their values are at least .50, and only these values might be shown in tables. Table 18.4 shows the result of a factor analysis. The table shows various items related to managerial professionalism, and the factor analysis identifies three distinct groups for these items. Such tables are commonly seen in research articles. The labels for each group (for example, “A. Commitment to performance”) are provided by the authors; note that the three groupings are conceptually distinct. The table also shows that, combined, these three factors account for 61.97 percent of the total variance. The table shows only loadings greater than .50; those below this value are not shown.6 Based on these results, the authors then create index variables for the three groups. Each group has high internal reliability (see Chapter 3); the Cronbach alpha scores are, respectively, 0.87, 0.83, and 0.88. This table shows a fairly typical use of factor analysis, providing statistical support for a grouping scheme. Beyond Factor Analysis A variety of exploratory techniques exist. Some seek purely to classify, whereas

SUMMARY A vast array of additional statistical methods exists. In this concluding chapter, we summarized some of these methods (path analysis, survival analysis, and factor analysis) and briefly mentioned other related techniques. This chapter can help managers and analysts become familiar with these additional techniques and increase their access to research literature in which these techniques are used. Managers and analysts who would like more information about these techniques will likely consult other texts or on-line sources. In many instances, managers will need only simple approaches to calculate the means of their variables, produce a few good graphs that tell the story, make simple forecasts, and test for significant differences among a few groups. Why, then, bother with these more advanced techniques? They are part of the analytical world in which managers operate. Through research and consulting, managers cannot help but come in contact with them. It is hoped that this chapter whets the appetite and provides a useful reference for managers and students alike. KEY TERMS   Endogenous variables Exogenous variables Factor analysis Indirect effects Loading Path analysis Recursive models Survival analysis Notes 1. Two types of feedback loops are illustrated as follows: 2. When feedback loops are present, error terms for the different models will be correlated with exogenous variables, violating an error term assumption for such models. Then, alternative estimation methodologies are necessary, such as two-stage least squares and others discussed later in this chapter. 3. Some models may show double-headed arrows among error terms. These show the correlation between error terms, which is of no importance in estimating the beta coefficients. 4. In SPSS, survival analysis is available through the add-on module in SPSS Advanced Models. 5. The functions used to estimate probabilities are rather complex. They are so-called Weibull distributions, which are defined as h(t) = αλ(λt)a–1, where a and 1 are chosen to best fit the data. 6. Hence, the SSL is greater than the squared loadings reported. For example, because the loadings of variables in groups B and C are not shown for factor 1, the SSL of shown loadings is 3.27 rather than the reported 4.084. If one assumes the other loadings are each .25, then the SSL of the not reported loadings is [12*.252 =] .75, bringing the SSL of factor 1 to [3.27 + .75 =] 4.02, which is very close to the 4.084 value reported in the table. 7. Readers who are interested in multinomial logistic regression can consult on-line sources or the SPSS manual, Regression Models 10.0 or higher. The statistics of discriminant analysis are very dissimilar from those of logistic regression, and readers are advised to consult a separate text on that topic. Discriminant analysis is not often used in public

The test statistics of a t-test can be positive or negative, although this depends merely on which group has the larger mean; the sign of the test statistic has no substantive interpretation. Critical values (see Chapter 10) of the t-test are shown in Appendix C as (Student’s) t-distribution.4 For this test, the degrees of freedom are defined as n – 1, where n is the total number of observations for both groups. The table is easy to use. As mentioned below, most tests are two-tailed tests, and analysts find critical values in the columns for the .05 (5 percent) and .01 (1 percent) levels of significance. For example, the critical value at the 1 percent level of significance for a test based on 25 observations (df = 25 – 1 = 24) is 2.797 (and 1.11 at the 5 percent level of significance). Though the table also shows critical values at other levels of significance, these are seldom if ever used. The table shows that the critical value decreases as the number of observations increases, making it easier to reject the null hypothesis. The t-distribution shows one- and two-tailed tests. Two-tailed t-tests should be used when analysts do not have prior knowledge about which group has a larger mean; one-tailed t-tests are used when analysts do have such prior knowledge. This choice is dictated by the research situation, not by any statistical criterion. In practice, two-tailed tests are used most often, unless compelling a priori knowledge exists or it is known that one group cannot have a larger mean than the other. Two-tailed testing is more conservative than one-tailed testing because the critical values of two-tailed tests are larger, thus requiring larger t-test test statistics in order to reject the null hypothesis.5 Many statistical software packages provide only two-tailed testing. The above null hypothesis (men and women do not have different mean incomes in the population) requires a two-tailed test because we do not know, a priori, which gender has the larger income.6 Finally, note that the t-test distribution approximates the normal distribution for large samples: the critical values of 1.96 (5 percent significance) and 2.58 (1 percent significance), for large degrees of freedom (∞), are identical to those of the normal distribution. Getting Started Find examples of t-tests in the research literature. T-Test Assumptions Like other tests, the t-test has test assumptions that must be met to ensure test validity. Statistical testing always begins by determining whether test assumptions are met before examining the main research hypotheses. Although t-test assumptions are a bit involved, the popularity of the t-test rests partly on the robustness of t-test conclusions in the face of modest violations. This section provides an in-depth treatment of t-test assumptions, methods for testing the assumptions, and ways to address assumption violations. Of course, t-test statistics are calculated by the computer; thus, we focus on interpreting concepts (rather than their calculation). Key Point The t-test is fairly robust against assumption violations. Four t-test test assumptions must be met to ensure test validity: One variable is continuous, and the other variable is dichotomous. The two distributions have equal variances. The observations are independent. The two distributions are normally distributed. The first assumption, that one variable is continuous and the other dichotomous,

usually does not present much of a problem. Some analysts use t-tests with ordinal rather than continuous data for the testing variable. This approach is theoretically controversial because the distances among ordinal categories are undefined. This situation is avoided easily by using nonparametric alternatives (discussed later in this chapter). Also, when the grouping variable is not dichotomous, analysts need to make it so in order to perform a t-test. Many statistical software packages allow dichotomous variables to be created from other types of variables, such as by grouping or recoding ordinal or continuous variables. The second assumption is that the variances of the two distributions are equal. This is called homogeneity of variances. The use of pooled variances in the earlier formula is justified only when the variances of the two groups are equal. When variances are unequal (called heterogeneity of variances), revised formulas are used to calculate t-test test statistics and degrees of freedom.7 The difference between homogeneity and heterogeneity is shown graphically in Figure 12.2. Although we needn’t be concerned with the precise differences in these calculation methods, all t-tests first test whether variances are equal in order to know which t-test test statistic is to be used for subsequent hypothesis testing. Thus, every t-test involves a (somewhat tricky) two-step procedure. A common test for the equality of variances is the Levene’s test. The null hypothesis of this test is that variances are equal. Many statistical software programs provide the Levene’s test along with the t-test, so that users know which t-test to use—the t-test for equal variances or that for unequal variances. The Levene’s test is performed first, so that the correct t-test can be chosen. Figure 12.2 Equal and Unequal Variances The term robust is used, generally, to describe the extent to which test conclusions are unaffected by departures from test assumptions. T-tests are relatively robust for (hence, unaffected by) departures from assumptions of homogeneity and normality (see below) when groups are of approximately equal size. When groups are of about equal size, test conclusions about any difference between their means will be unaffected by heterogeneity. The third assumption is that observations are independent. (Quasi-) experimental research designs violate this assumption, as discussed in Chapter 11. The formula for the t-test test statistic, then, is modified to test whether the difference between before and after measurements is zero. This is called a paired t-test, which is discussed later in this chapter. The fourth assumption is that the distributions are normally distributed. Although normality is an important test assumption, a key reason for the popularity of the t-test is that t-test conclusions often are robust against considerable violations of normality assumptions that are not caused by highly skewed distributions. We provide some detail about tests for normality and how to address departures thereof. Remember, when nonnormality cannot be resolved adequately, analysts consider nonparametric alternatives to the t-test, discussed at the end of this chapter. Box 12.1 provides a bit more discussion about the reason for this assumption. A combination of visual inspection and statistical tests is always used to determine the normality of variables. Two tests of normality are the Kolmogorov-Smirnov test (also known as the K-S test) for samples with more than 50 observations and the Shapiro-Wilk test for samples with up to 50 observations. The null hypothesis of

12.2. The transformed variable has equal variances across the two groups (Levene’s test, p = .119), and the t-test statistic is –1.308 (df = 85, p = .194). Thus, the differences in pollution between watersheds in the East and Midwest are not significant. (The negative sign of the t-test statistic, –1.308, merely reflects the order of the groups for calculating the difference: the testing variable has a larger value in the Midwest than in the East. Reversing the order of the groups results in a positive sign.) Table 12.2 Independent-Samples T-Test: Output For comparison, results for the untransformed variable are shown as well. The untransformed variable has unequal variances across the two groups (Levene’s test, p = .036), and the t-test statistic is –1.801 (df = 80.6, p =.075). Although this result also shows that differences are insignificant, the level of significance is higher; there are instances in which using nonnormal variables could lead to rejecting the null hypothesis. While our finding of insignificant differences is indeed robust, analysts cannot know this in advance. Thus, analysts will need to deal with nonnormality. Variable transformation is one approach to the problem of nonnormality, but transforming variables can be a time-intensive and somewhat artful activity. The search for alternatives has led many analysts to consider nonparametric methods. TWO T-TEST VARIATIONS Paired-Samples T-Test Analysts often use the paired t-test when applying before and after tests to assess student or client progress. Paired t-tests are used when analysts have a dependent rather than an independent sample (see the third t-test assumption, described earlier in this chapter). The paired-samples t-test tests the null hypothesis that the mean difference between the before and after test scores is zero. Consider the following data from Table 12.3. Table 12.3 Paired-Samples Data The mean “before” score is 3.39, and the mean “after” score is 3.87; the mean difference is 0.54. The paired t-test tests the null hypothesis by testing whether the mean of the difference variable (“difference”) is zero. The paired t-test test statistic is calculated as where D is the difference between before and after measurements, and sD is the standard deviation of these differences. Regarding t-test assumptions, the variables are continuous, and the issue of heterogeneity (unequal variances) is moot because this test involves only one variable, D; no Levene’s test statistics are produced. We do test the normality of D and find that it is normally distributed (Shapiro-Wilk = .925, p = .402). Thus, the assumptions are satisfied. We proceed with testing whether the difference between before and after scores is statistically significant. We find that the paired t-test yields a t-test statistic of 2.43, which is significant at the 5 percent level (df = 9, p = .038 < .05).17 Hence, we conclude that the increase between the before and after scores is significant at the 5 percent level.18 One-Sample T-Test Finally, the one-sample t-test tests whether the mean of a single variable is different from a prespecified value (norm). For example, suppose we want to know whether the mean of the before group in Table 12.3 is different from the value of, say, 3.5? Testing against a norm is akin to the purpose of the chi-square goodness-of-fit test described in Chapter 11, but here we are dealing with a continuous variable rather than a categorical one, and we are testing the mean rather than its distribution. The one-sample t-test assumes that the single variable is continuous and normally distributed. As with the paired t-test, the issue of heterogeneity is moot because there is only one variable. The Shapiro-Wilk test shows that the variable “before” is normal (.917, p = .336). The one-sample t-test statistic for testing against the test value of 3.5 is –0.515 (df = 9, p = .619 > .05). Hence, the mean of 3.39 is not significantly

categorical and the dependent variable is continuous. The logic of this approach is shown graphically in Figure 13.1. The overall group mean is (the mean of means). The boxplots represent the scores of observations within each group. (As before, the horizontal lines indicate means, rather than medians.) Recall that variance is a measure of dispersion. In both parts of the figure, w is the within-group variance, and b is the between-group variance. Each graph has three within-group variances and three between-group variances, although only one of each is shown. Note in part A that the between-group variances are larger than the within-group variances, which results in a large F-test statistic using the above formula, making it easier to reject the null hypothesis. Conversely, in part B the within-group variances are larger than the between-group variances, causing a smaller F-test statistic and making it more difficult to reject the null hypothesis. The hypotheses are written as follows: H0: No differences between any of the group means exist in the population. HA: At least one difference between group means exists in the population. Note how the alternate hypothesis is phrased, because the logical opposite of “no differences between any of the group means” is that at least one pair of means differs. H0 is also called the global F-test because it tests for differences among any means. The formulas for calculating the between-group variances and within-group variances are quite cumbersome for all but the simplest of designs.1 In any event, statistical software calculates the F-test statistic and reports the level at which it is significant.2 When the preceding null hypothesis is rejected, analysts will also want to know which differences are significant. For example, analysts will want to know which pairs of differences in watershed pollution are significant across regions. Although one approach might be to use the t-test to sequentially test each pair of differences, this should not be done. It would not only be a most tedious undertaking but would also inadvertently and adversely affect the level of significance: the chance of finding a significant pair by chance alone increases as more pairs are examined. Specifically, the probability of rejecting the null hypothesis in one of two tests is [1 – 0.952 =] .098, the probability of rejecting it in one of three tests is [1 – 0.953 =] .143, and so forth. Thus, sequential testing of differences does not reflect the true level of significance for such tests and should not be used. Post-hoc tests test all possible group differences and yet maintain the true level of significance. Post-hoc tests vary in their methods of calculating test statistics and holding experiment-wide error rates constant. Three popular post-hoc tests are the Tukey, Bonferroni, and Scheffe tests.

A NONPARAMETRIC ALTERNATIVE A nonparametric alternative to one-way ANOVA is Kruskal-Wallis’ H test of one-way ANOVA. Instead of using the actual values of the variables, Kruskal-Wallis’ H test assigns ranks to the variables, as shown in Chapter 11. As a nonparametric method, Kruskal-Wallis’ H test does not assume normal populations, but the test does assume similarly shaped distributions for each group. This test is applied readily to our one-way ANOVA example, and the results are shown in Table 13.5. Table 13.5 Kruskal-Wallis’ H-Test of One-Way ANOVA Kruskal-Wallis’ H one-way ANOVA test shows that population is significantly associated with watershed loss (p = .013). This is one instance in which the general rule that nonparametric tests have higher levels of significance is not seen. Although Kruskal-Wallis’ H test does not report mean values of the dependent variable, the pattern of mean ranks is consistent with Figure 13.2. A limitation of this nonparametric test is that it does not provide post-hoc tests or analysis of homogeneous groups, nor are there nonparametric n-way ANOVA tests such as for the two-way ANOVA test described earlier. SUMMARY One-way ANOVA extends the t-test by allowing analysts to test whether two or more groups have different means of a continuous variable. The t-test is limited to only two groups. One-way ANOVA can be used, for example, when analysts want to know if the mean of a variable varies across regions, racial or ethnic groups, population or employee categories, or another grouping with multiple categories. ANOVA is family of statistical techniques, and one-way ANOVA is the most basic of these methods. ANOVA is a parametric test that makes the following assumptions: The dependent variable is continuous. The independent variable is ordinal or nominal. The groups have equal variances. The variable is normally distributed in each of the groups. Relative to the t-test, ANOVA requires more attention to the assumptions of normality and homogeneity. ANOVA is not robust for the presence of outliers, and it appears to be less robust than the t-test for deviations from normality. Variable transformations and the removal of outliers are to be expected when using ANOVA. ANOVA also includes three other types of tests of interest: post-hoc tests of mean differences among categories, tests of homogeneous subsets, and tests for the linearity of mean differences across categories. Two-way ANOVA addresses the effect of two independent variables on a continuous dependent variable. When using two-way ANOVA, the analyst is able to distinguish main effects from interaction effects. Kruskal-Wallis’ H test is a nonparametric alternative to one-way ANOVA. KEY TERMS   Analysis of variance (ANOVA) ANOVA assumptions Covariates Factors Global F-test Homogeneous subsets Interaction effect Kruskal-Wallis’ H test of one-way ANOVA Main effect One-way ANOVA Post-hoc test Two-way ANOVA Notes   1. The between-group variance is

In Depth Types of Effect Size Indicators Researchers use several different statistics to indicate effect size depending on the nature of their data. Roughly speaking, these effect size statistics fall into three broad categories. Some effect size indices, sometimes called dbased effect sizes, are based on the size of the difference between the means of two groups, such as the difference between the average scores of men and women on some measure or the differences in the average scores that participants obtained in two experimental conditions. The larger the difference between the means, relative to the total variability of the data, the stronger the effect and the larger the effect size statistic. The r-based effect size indices are based on the size of the correlation between two variables. The larger the correlation, the more strongly two variables are related and the more of the total variance in one variable is systematic variance related to the other variable. A third category of effect sizes index involves the odds-ratio, which tells us the ratio of the odds of an event occurring in one group to the odds of the event occurring in another group. If the event is equally likely in both groups, the odds ratio is 1.0. An odds ratio greater than 1.0 shows that the odds of the event is greater in one group than in another, and the larger the odds ratio, the stronger the effect. The odds ratio is used when the variable being measured has only two levels. For example, imagine doing research in which first-year students in college are either assigned to attend a special course on how to study or not assigned to attend the study skills course, and we wish to know whether the course reduces the likelihood that students will drop out of college. We could use the odds ratio to see how much of an effect the course had on the odds of students dropping out. You do not need to understand the statistical differences among these effect size indices, but you will find it useful in reading journal articles to know what some of the most commonly used effect sizes are called. These are all ways of expressing how strongly variables are related to one another—that is, the effect size. Symbol Name d Cohen’s d g Hedge’s g h 2 eta squared v 2 omega squared r or r 2 correlation effect size OR odds ratio The strength of the relationships between variables varies a great deal across studies. In some studies, as little as 1% of the total variance may be systematic variance, whereas in other contexts, the proportion of the total variance that is systematic variance may be quite large,

But most other fields do not use protocol registration, and researchers have the freedom to use whatever methods they feel appropriate. For example, in a survey of academic psychologists, more than half admitted to deciding whether to collect more data after checking whether their results were significant, usually concealing this practice in publications.

More generally, a data scientist is someone who knows how to extract meaning from and interpret data, which requires both tools and methods from statistics and machine learning, as well as being human. She spends a lot of time in the process of collecting, cleaning, and munging data, because data is never clean. This process requires persistence, statistics, and software engineering skills — skills that are also necessary for understanding biases in the data, and for debugging logging output from code. Once she gets the data into shape, a crucial part is exploratory data analysis, which combines visualization and data sense. She’ll find patterns, build models, and algorithms — some with the intention of understanding product usage and the overall health of the product, and others to serve as prototypes that ultimately get baked back into the product. She may design experiments, and she is a critical part of data-driven decision making. She’ll communicate with team members, engineers, and leadership in clear language and with data visualizations so that even if her colleagues are not immersed in the data themselves, they will understand the implications.

the use of statistical process control tools to evaluate variation, correlate root cause, forecast capacity, and anticipate throughput barriers. By measuring incidence of preventable venous

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In the US, not only did the rise of postwar Neoclassicism represent the consolidation of a particular ‘triumvirate’ in economics—mathematics, formalism, and physics envy (Bateman 1998), but it also replaced an important interwar pluralism[26] that allowed for more than one single approach in economics to co-exist with certain prestige and influence.[27] Amongst the reasons for such a shift, historians of economic thought expose a complex story that involves changes in the epistemology, methodology, and sociology of the economics discipline (Morgan & Rutherford 1998). For the US, it involved inter alia a change in how mathematics began to dominate economics scholarship, and particularly how mathematical formalism began to be closely associated with the concept of scientific neutrality, objectivity, and universal applicability (Furner 1975). This led to an expansion of method-oriented analyses, reinforced by the adoption of econometrics and statistical analysis in economics.

Ifin.Ordered.Set.SΦRT.Time.Returned is a computer paradigm in a methodology that acts like an operating system in a statistical method or function in a controller that returns a set of Hashed Objects in a standard normal distribution.

The suggestion that the fundamental logic underlying these methods is broken should be terrifying.

If we take a science such as experimental physics, where studies tend to have high statistical power, methods are well defined and de facto preregistered, then the failure to reproduce a previous result is considered a major cause for concern. But in a weaker science where lax statistical standards and questionable research practices are the norm, attempts to reproduce prior work will often fail, and it should therefore come as no surprise that retraction is rare.

Basically, action is, and always will be, faster than reaction. Thus, the attacker is the one that dictates the fight. They are forcing the encounter with technique after technique that are designed to overcome any defensive techniques initiated by the defender. Much of this exchange, and determining which of the adversaries is victorious, is all a matter of split seconds. That is the gap between action and reaction. That attacker acts; the defender reacts. Military history is saturated with an uneven amount of victorious attackers compared to victorious defenders. It is common to observe the same phenomenon in popular sports, fighting competitions, in the corporate world of big business. The list goes on and on. So, how do we effectively defend ourselves when we can easily arrive at the conclusion that the defender statistically loses? It is by developing the mentality that once attacked that you immediately counter-attack. That counter-attack has to be ferocious and unrelenting. If someone throws a punch, or otherwise initiates battle with you, putting you, for a split second, on the wrong side of the action versus reaction gap. Your best chance of victory is to deflect, smoother, parry, or otherwise negate their attack and then immediately launch into a vicious counter-attack. Done properly, this forces your adversary into a reactive state, rather than an action one. You turn the table on them and become the aggressor. That is how to effectively conceptualizes being in a defensive situation. Utilizing this method will place you in a greater position to be victorious. Dempsey, Sun Tzu and General Patton would agree. Humans are very violent animals. As a species, we are capable of high levels of extreme violence. In fact, approaching the subject of unarmed combatives, or any form of combatives, involves the immersion into a field that is inherently violent to the extreme of those extremes. It is one thing to find yourself facing an opponent across a field, or ring, during a sporting match. Those contests still pit skill verses skill, but lack the survival aspects of an unarmed combative encounter. The average person rarely, if ever, ponders any of this and many consider various sporting contests as the apex of human competition. It is not. Finding yourself in a life-or-death struggle against an opponent that is completely intent on ending your life is the greatest of all human competitions. Understanding that and acknowledging that takes some degree of courage in today’s society.

The first, or predictive, approach could also be called the qualitative approach, since it emphasizes prospects, management, and other nonmeasurable, albeit highly important, factors that go under the heading of quality. The second, or protective, approach may be called the quantitative or statistical approach, since it emphasizes the measurable relationships between selling price and earnings, assets, dividends, and so forth. Incidentally, the quantitative method is really an extension—into the field of common stocks—of the viewpoint that security analysis has found to be sound in the selection of bonds and preferred stocks for investment. In our own attitude and professional work we were always committed to the quantitative approach. From the first we wanted to make sure that we were getting ample value for our money in concrete, demonstrable terms. We were not willing to accept the prospects and promises of the future as compensation

Dan Corrieri trained in engineering graphics, statistical process control, accounting manufacturing methods, finance, management, engineering economy, marketing, design processes in technology and engineering mechanics. Daniel Corrieri was a veteran of the United States Marine Corps Reserves with an honorable discharge. In addition, Daniel Corrieri has gained plenty of volunteer experience with Toys for Tots through the Marine Corps, community service for the American Cancer Society, church service and other excellent opportunities.

His greatest academic achievement was the discovery of the theory of statistical decision functions. Today, Wald is known as the founder of sequential analysis. He also published some of the first papers on game theory.2 The methods of sequential analysis he developed were widely applied to the US WWII effort.3 Wald was known

Collecting data using improper methods can spoil any statistical analysis

First, purpose—what you hope to accomplish. In the case of Beane, he sought a better method for predicting baseball success. Second, scope—what to include or exclude in arriving at the decision. Beane decided to include past performance statistics and exclude aesthetic qualities. He reduced the scope of his search to data on performance. Third, perspective—your point of view in approaching this decision and how others might approach it.

Data Science is a multidisciplinary field that combines various techniques and methods to extract knowledge and insights from data. It involves the application of statistical analysis, machine learning algorithms, and computational tools to analyze and interpret complex data sets.

Statistics to the layman can appear rather complex, but the concept behind what is used today is so simple that my French mathematician friends call it deprecatorily "cuisine". It is all based on one simple notion; the more information you have the more you are confident about the outcome. Now the problem: by how much? Common statistical method is based on the steady augmentation of the confidence level, in nonlinear proportion to the number of observations. That is, for an n time increase in the sample size, we increase our knowledge by the square root of n. Suppose i'm drawing from an urn containing red and black balls. My confidence level about the relative proportion of red and black balls after 20 drawings in not twice the one I have after 10 drawings; it's merely multiplied by the square root of 2.

He moved instead to an investment strategy that required no great macroeconomic insight. Instead, he explained, “As time goes on, I get more and more convinced that the right method in investment is to put fairly large sums into enterprises which one thinks one knows something about and in the management of which one thoroughly believes.

Even the statistics suggesting that more violent crime—especially on innocent victims—was occurring in urban Black neighborhoods were based on a racist statistical method rather than reality. Drunk drivers, who routinely kill more people than violent urban Blacks, were not regarded as violent criminals in such studies, and 78 percent of arrested drunk drivers were White males in 1990.

The bureaucratic method can be defined as one that (a) administers human beings as if they were things and (b) administers things in quantitative rather than qualitative terms, in order to make quantification and control easier and cheaper. The bureaucratic method is governed by statistical data: the bureaucrats base their decisions on fixed rules arrived at from statistical data, rather than on response to the living beings who stand before them; they decide issues according to what is statistically most likely to be the case, at the risk of hurting the 5 or 10 percent of those who do not fit into that pattern. Bureaucrats fear personal responsibility and seek refuge behind their rules; their security and pride lie in their loyalty to rules, not in their loyalty to the laws of the human heart.

As time goes on, I get more and more convinced that the right method in investment is to put fairly large sums into enterprises which one thinks one knows something about and in the management of which one thoroughly believes.” Forget what the economy is doing; just find well-managed companies, buy some shares, and don’t try to be too clever. And if that approach sounds familiar, it’s most famously associated with Warren Buffett, the world’s richest investor—and a man who loves to quote John Maynard Keynes.

His favoured objects of contemplation were economic facts, usually in statistical form. He used to say that his best ideas came to him from ‘messing about with figures and seeing what they must mean’. Yet he was famously sceptical about econometrics – the use of statistical methods for forecasting purposes. He championed the cause of better statistics, not to provide material for the regression coefficient, but for the intuition of the economist to play on.

It was the income-determination model, based on the multiplier, together with the consequent development of national income statistics, which made Keynesian economics acceptable to policy-makers, since it offered them a seemingly secure method of forecasting and controlling the movement of such ‘real’ variables as investment, consumption, and employment.

the well of inspiration has run dry, it’s because you need a deeper well full of examples, illustrations, stories, statistics, diagrams, analogies, metaphors, photos, mindmaps, conversation notes, quotes—anything that will help you argue for your perspective or fight for a cause you believe in.

Therefore, in stead of candle patterns any other set of signals or indicators could also be used for the advise function and compared with the PATTERN method, or with other machine learning functions like decision trees, perceptrons, or neural networks from the R statistics package.

It appears that the reporter has passed along some words without inquiring what they mean, and you are expected to read them just as uncritically for the happy illusion they give you of having learned something. It is all too reminiscent of an old definition of the lecture method of classroom instruction: a process by which the contents of the textbook of the instructor are transferred to the notebook of the student without passing through the heads of either party.

[On Gauss studies on the methods of least squares] ... and he was a wizard in using it in every thinkable way.

[About Gauss studies on the methods of least squares] ... and he was a wizard in using it in every thinkable way.

[About Gauss' studies on the methods of least squares] ... and he was a wizard in using it in every thinkable way.

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proof of the social genealogy of AI: the first artificial neural network – the perceptron – was born not as the automation of logical reasoning but of a statistical method originally used to measure intelligence in cognitive tasks and to organise social hierarchies accordingly.

best market research companies in Myanmar As organizations keep on growing their arrive at across the globe, it's essential to have a profound comprehension of the business sectors they are focusing on. One vital method for acquiring this understanding is best market research companies in Myanmar , there are various statistical surveying organizations to browse, yet which ones are awesome? Assuming you're searching for the best market research companies in Myanmar , look no farther than AMT Statistical surveying. With long stretches of involvement and a profound comprehension of the South East Asian market, AMT Statistical surveying is the believed accomplice you want to open the bits of knowledge you want to succeed. Why Pick AMT Statistical surveying? AMT Statistical surveying has a demonstrated history of outcome in Myanmar and all through South East Asia. They have worked with many clients, from little new companies to enormous global partnerships, assisting them with acquiring a profound comprehension of the business sectors they are focusing on. However, what separates AMT Statistical surveying from other statistical surveying organizations in Myanmar? As far as one might be concerned, they have a group of experienced specialists who are specialists in their field. They can plan and execute research concentrates on that give the bits of knowledge you want to go with informed choices. Also, AMT Statistical surveying comprehends the remarkable difficulties of carrying on with work in Myanmar and all through South East Asia. They have major areas of strength for an of neighborhood contacts and a profound comprehension of the social subtleties that can influence business achievement. Administrations Advertised AMT Statistical surveying offers many administrations to assist organizations with acquiring a profound comprehension of the business sectors they are focusing on. These administrations include: 1. Statistical surveying: AMT Statistical surveying conducts an extensive variety of statistical surveying studies, including shopper research, contender investigation, industry examination, from there, the sky is the limit. They utilize an assortment of exploration strategies, including overviews, center gatherings, and top to bottom meetings, to acquire a profound comprehension of the market. 2. Information Assortment: AMT Statistical surveying has a group of experienced information gatherers who can accumulate information rapidly and productively. They utilize different information assortment techniques, including on the web overviews, phone meetings, and eye to eye interviews. 3. Information Examination: When the information has been gathered, AMT Statistical surveying utilizes progressed information investigation methods to uncover experiences and patterns. They utilize measurable investigation, relapse examination, and different strategies to give significant bits of knowledge to their clients. 4. Counseling: AMT Statistical surveying additionally gives counseling administrations to assist organizations with pursuing informed choices in view of the experiences acquired through statistical surveying. They work intimately with their clients to foster systems and strategies that will assist them with prevailing in their objective business sectors. End Assuming you're searching for the best market research companies in Myanmar , look no farther than AMT Statistical surveying. With long periods of involvement and a profound comprehension of the South East Asian market, AMT Statistical surveying is the believed accomplice you really want to open the experiences you really want to succeed. Reach them today to become familiar with their administrations and how they can assist your business with succeeding.

prepare data for analysis, including cleaning, transformation, and normalization. Data cleaning involves identifying and correcting errors in the data, such as incorrect values, missing data, or duplicate records. It is crucial to ensure that data is accurate, complete, and consistent before proceeding with analysis. Data transformation involves converting data into a more suitable format for analysis. This may include converting data types, scaling data, and handling outliers. For example, data may need to be normalized to ensure that all values are on the same scale. Data normalization involves scaling data so that it falls within a specific range. This is important because many statistical methods assume that data is normally distributed, and normalization helps to achieve this. Common methods of normalization include z-score normalization and min-max normalization

processed, and transformed into a format that is suitable for analysis. This often involves removing duplicate data, correcting errors, and dealing with missing values. After data is prepared, exploratory data analysis is performed to better understand the data and identify patterns, trends, and outliers. Descriptive statistics, data visualization, and data clustering techniques are often used to explore data. Once the data is understood, statistical methods such as hypothesis testing and regression analysis can be applied to identify relationships and make predictions.

Fewer are the ways of helping people than of harming them; it is the nature of things, not a consequence of the statistical method. Our world does not stand halfway between heaven and hell; it seems much closer to hell.

T20 World Cup Betting: A Quick Guide The T20 World Cup is one of the most exciting events in the cricketing calendar, bringing together top teams from around the world for a fast-paced, action-packed tournament. For many fans, placing a bet can add even more excitement to the games. Here’s a brief guide to get you started with T20 World Cup betting. What is T20 World Cup Betting? T20 World Cup betting involves wagering on various outcomes related to the tournament. This can range from predicting the overall winner of the World Cup to specific match outcomes or player performances. Types of Bets Match Bets: Wager on the outcome of individual matches. Outright Bets: Bet on which team will win the entire tournament. Prop Bets: Bet on specific events, like the top run-scorer in a match. How to Bet Choose a Platform: Select a reputable betting site like Bet365 or Betway. Create an Account: Sign up and verify your details. Deposit Funds: Add money to your account using a secure payment method. Place Bets: Choose your bets based on your research and predictions. Tips for Successful Betting Research: Study team form, player statistics, and match conditions. Set a Budget: Only bet what you can afford to lose. Stay Informed: Keep up with the latest news and updates. Responsible Betting Betting should always be fun and done responsibly. Set limits for yourself and seek help if betting becomes a problem. Betting on the T20 World Cup can enhance your enjoyment of the game, but always remember to bet wisely and responsibly.

Measuring replication rates across different experiments requires that research be reviewed in some fashion. Research reviews can be classified into four types. A type 1 review simply identifies and discusses recent developments in a field, usually focusing on a few exemplar experiments. Such reviews are often found in popular-science magazines such as Scientific American. They are also commonly used in skeptical reviews of psi research because one or two carefully selected exemplars can provide easy targets to pick apart. The type 2 review uses a few research results to highlight or illustrate a new theory or to propose a new theoretical framework for understanding a phenomenon. Again, the review is not designed to be comprehensive but only to illustrate a general theme. Type 3 reviews organize and synthesize knowledge from various areas of research. Such narrative reviews are not comprehensive, because the entire pool of combined studies from many disciplines is typically too large to consider individually. So again, a few exemplars of the “best” studies are used to illustrate the point of the synthesis. Type 4 is the integrative review, or meta-analysis, which is a structured technique for exhaustively analyzing a complete body of experiments. It draws generalizations from a set of observations about each experiment.1 Integration Meta-analysis has been described as “a method of statistical analysis wherein the units of analysis are the results of independent studies, rather than the responses of individual subjects.”2 In a single experiment, the raw data points are typically the participants’ individual responses. In meta-analysis, the raw data points are the results of separate experiments.

The ‘quantitative revolution’ in geography required the discipline to adopt an explicitly scientific approach, including numerical and statistical methods, and mathematical modelling, so ‘numeracy’ became another necessary skill. Its immediate impact was greatest on human geography as physical geographers were already using these methods. A new lexicon encompassing the language of statistics and its array of techniques entered geography as a whole. Terms such as random sampling, correlation, regression, tests of statistical significance, probability, multivariate analysis, and simulation became part both of research and undergraduate teaching. Correlation and regression are procedures to measure the strength and form, respectively, of the relationships between two or more sets of variables. Significance tests measure the confidence that can be placed in those relationships. Multivariate methods enable the analysis of many variables or factors simultaneously – an appropriate approach for many complex geographical data sets. Simulation is often linked to probability and is a set of techniques capable of extrapolating or projecting future trends.

When you feel stuck in your creative pursuits, it doesn’t mean that there’s something wrong with you. You haven’t lost your touch or run out of creative juice. It just means you don’t yet have enough raw material to work with. If it feels like the well of inspiration has run dry, it’s because you need a deeper well full of examples, illustrations, stories, statistics, diagrams, analogies, metaphors, photos, mindmaps, conversation notes, quotes—anything that will help you argue for your perspective or fight for a cause you believe in.

Biologist Carl T. Bergstrom and information scientist Jevin West teach a class called “Calling Bullshit” at the University of Washington, and published a book with the same name. “Bullshit involves language, statistical figures, data graphics, and other forms of presentation intended to persuade by impressing and overwhelming a reader or listener, with a blatant disregard for truth and logical coherence,” in their definition. They offer a simple, three-step method for bullshit detection, which involves these questions: Who is telling me this? How do they know it? What are they trying to sell me?

As Cochran recalls in a blog post he wrote about his experiment for the Harvard Business Review, these simple statistics got him thinking about the rest of his company. Just how much time were employees of Atlantic Media spending moving around information instead of focusing on the specialized tasks they were hired to perform? Determined to answer this question, Cochran gathered company-wide statistics on e-mails sent per day and the average number of words per e-mail. He then combined these numbers with the employees’ average typing speed, reading speed, and salary. The result: He discovered that Atlantic Media was spending well over a million dollars a year to pay people to process e-mails, with every message sent or received tapping the company for around ninety-five cents of labor costs. “A ‘free and frictionless’ method of communication,” Cochran summarized, “had soft costs equivalent to procuring a small company Learjet.” Tom Cochran’s experiment yielded an interesting result about the literal cost of a seemingly harmless behavior. But the real importance of this story is the experiment itself, and in particular, its complexity.

In my view, you can increase your career options if you start by studying a “hard” subject instead of a “soft” one. A hard subject is one that is more rigorous, where there is generally a right answer and a wrong answer, such as physics. You should learn hard subjects first because they teach you the fundamental skills that are needed to evaluate softer subjects. For example, if you first obtain a rigorous education in statistical methods, you will have some of the tools necessary to analyze the impact of many public policies. Furthermore, softer subjects are easier to learn on your own. You might be able to pick up the main ideas of a certain field of sociology by reading some papers, but you probably can’t learn the fundamentals of neurophysiology without formal instruction.