However, the spread is more than that of the standard normal distribution. The larger the degrees of freedom, the closer the t-density is to the normal density. Why Is Every Thing Priced One Penny Off the Dollar. Yet another motivation Psychologically 9. 99 might look better than 10. 00, but there is a more basic reason too. The assistant has to give you change from your ten dollars, and has to ring the sale up through his her cash register to get at the one cent.
This forces the transaction to go through the books, you get a receipt, and the assistant can t just pocket the 10 him herself. There s sales tax for that. Mind you, there s nothing to stop a particularly untrustworthy employee going into work with a pocketful of cents. For either price at least in the USyou ll have to pay sales tax too. So that solves the problem of opening the cash register.
That, plus the security cameras. There has been some research in marketing theory on the consumer s behavior at particular price points. Essentially, these are tied up with buyer expectations based on prior experience. A critical case study in UK on price pointing of pantyhose tights shown that there were distinct demand peaks at buyer anticipated price points of 59p, 79p, 99p, Ј1.
In the UK, for example, prices of wine are usually set at key price points. Demand at intermediate price points was dramatically below these anticipated points for similar quality goods. The wine retailers also confirm that sales at different prices even a penny or so different does result in dramatically different sales volumes. Other studies showed the opposite where reduced price showed reduced sales volumes, consumers ascribing quality in line with price.
Other similar research turns on the behavior of consumers to variations in price. The key issue here is that there is a Just Noticeable Difference JND below which consumers will not act on a price increase. However, it is not fully tested to determine if sales volume continued to increase with price. This has practical application when increasing charge rates and the like. As an empirical experiment, try overcharging clients by 1, 2. The JND is typically 5 and this provides the opportunity for consultants etc to increase prices above prior rates by less than the JND without customer complaint.
5, 6 and watch the reaction. Conversely, there is no point in offering a fee reduction of less than 5 as clients will not recognize the concession you have made. Up to 5 there appears to be no negative impact. A Short History of Probability and Statistics. Equally, in periods of price inflation, price rises should be staged so that the individual price rise is kept under 5perhaps by raising prices by 4 twice per year rather than a one off 8 rise.
The birth of statistics occurred in mid-17 th century. A commoner, named John Graunt, who was a native of London, begin reviewing a weekly church publication issued by the local parish clerk that listed the number of births, christenings, and deaths in each parish. These so called Bills of Mortality also listed the causes of death. Graunt who was a shopkeeper organized this data in the forms we call descriptive statistics, which was published as Natural and Political Observation Made upon the Bills of Mortality.
Thus, statistics has to borrow some concepts from sociology, such as the concept of Population. Shortly thereafter, he was elected as a member of Royal Society. It has been argued that since statistics usually involves the study of human behavior, it cannot claim the precision of the physical sciences.
Probability has much longer history. Probability is derived from the verb to probe meaning to find out what is not too easily accessible or understandable. The word proof has the same origin that provides necessary details to understand what is claimed to be true. Probability originated from the study of games of chance and gambling during the sixteenth century. Probability theory was a branch of mathematics studied by Blaise Pascal and Pierre de Fermat in the seventeenth century.
Currently; in 21 st century, probabilistic modeling are used to control the flow of traffic through a highway system, a telephone interchange, or a computer processor; find the genetic makeup of individuals or populations; quality control; insurance; investment; and other sectors of business and industry. Professor Bradley Efron expressed this fact nicely During the 20 th Century statistical thinking and methodology have become the scientific framework for literally dozens of fields including education, agriculture, economics, biology, and medicine, and with increasing influence recently on the hard sciences such as astronomy, geology, and physics.
In other words, we have grown from a small obscure field into a big obscure field. New and ever growing diverse fields of human activities are using statistics; however, it seems that this field itself remains obscure to the public. Further Readings Daston L.Classical Probability in the EnlightenmentPrinceton University Press, 1988. The book points out that early Enlightenment thinkers could not face uncertainty. A mechanistic, deterministic machine, was the Enlightenment view of the world.Philosophical Theories of ProbabilityRoutledge, 2000.
Covers the classical, logical, subjective, frequency, and propensity views.The Emergence of ProbabilityCambridge University Press, London, 1975. A philosophical study of early ideas about probability, induction and statistical inference.Counting for Something Statistical Principles and PersonalitiesSpringer, New York, 1987. It teaches the principles of applied economic and social statistics in a historical context.
Featured topics include public opinion polls, industrial quality control, factor analysis, Bayesian methods, program evaluation, non-parametric and robust methods, and exploratory data analysis.The Rise of Statistical Thinking1820-1900, Princeton University Press, 1986. The author states that statistics has become known in the twentieth century as the mathematical tool for analyzing experimental and observational data. Enshrined by public policy as the only reliable basis for judgments as the efficacy of medical procedures or the safety of chemicals, and adopted by business for such uses as industrial quality control, it is evidently among the products of science whose influence on public and private life has been most pervasive.
This new field of mathematics found so extensive a domain of applications. Statistical analysis has also come to be seen in many scientific disciplines as indispensable for drawing reliable conclusions from empirical results.The History of Statistics The Measurement of Uncertainty Before 1900U. It covers the people, ideas, and events underlying the birth and development of early statistics.
of Chicago Press, 1990.The Statistical PioneersSchenkman Books, New York, 1984. This work provides the detailed lives and times of theorists whose work continues to shape much of the modern statistics. Different Schools of Thought in Statistics. The Birth Process of a New School of Thought. The process of devising a new school of thought in any field has always taken a natural path. Birth of new schools of thought in statistics is not an exception.
Given an already established school, one must work within the defined framework. A crisis appears, i. The birth process is outlined below.some inconsistencies in the framework result from its own laws. Reluctance to consider the crisis. Try to accommodate and explain the crisis within the existing framework. Conversion of some well-known scientists attracts followers in the new school.
The perception of a crisis in statistical community calls forth demands for foundation-strengthens. After the crisis is over, things may look different and historians of statistics may cast the event as one in a series of steps in building upon a foundation. So we can read histories of statistics, as the story of a pyramid built up layer by layer on a firm base over time.
Other schools of thought are emerging to extend and soften the existing theory of probability and statistics. Some softening approaches utilize the concepts and techniques developed in the fuzzy set theory, the theory of possibility, and Dempster-Shafer theory. The arrows in this figure represent some of the main criticisms among Objective, Frequentist, and Subjective schools of thought.
To which school do you belong. The following Figure illustrates the three major schools of thought; namely, the Classical attributed to LaplaceRelative Frequency attributed to Fisherand Bayesian attributed to Savage. Read the conclusion in this figure. What Type of Statistician Are You. Click on the image to enlarge it. Further Readings Plato, Jan von, Creating Modern ProbabilityCambridge University Press, 1994.
This book provides a historical point of view on subjectivist and objectivist probability school of thoughts. Tanur, The Subjectivity of Scientists and the Bayesian ApproachWiley, 2001. Comparing and contrasting the reality of subjectivity in the work of history s great scientists and the modern Bayesian approach to statistical analysis.
Weatherson B.Begging the question and Bayesians, Studies in History and Philosophy of Science30 4687-697, 1999. Bruno de Finetti, in the introduction to his two-volume treatise on Bayesian ideas, clearly states that Probabilities Do not Exist. Bayesian, Frequentist, and Classical Methods. By this he means that probabilities are not located in coins or dice; they are not characteristics of things like mass, density, etc.
Some Bayesian approaches consider probability theory as an extension of deductive logic including dialogue logic, interrogative logic, informal logic, and artificial intelligence to handle uncertainty. It purports to deduce from first principles the uniquely correct way of representing your beliefs about the state of things, and updating them in the light of the evidence. The laws of probability have the same status as the laws of logic. These Bayesian approaches are explicitly subjective in the sense that they deal with the plausibility which a rational agent ought to attach to the propositions he she considers, given his her current state of knowledge and experience.
However, the Bayesian is better able to quantify the true uncertainty in his analysis, particularly when substantial prior information is available. Bayesians are willing to assign probability distribution function s to the population s parameter s while frequentists are not. From a scientist s perspective, there are good grounds to reject Bayesian reasoning. The problem is that Bayesian reasoning deals not with objective, but subjective probabilities.
The result is that any reasoning using a Bayesian approach cannot be publicly checked -- something that makes it, in effect, worthless to science, like non replicative experiments. Bayesian perspectives often shed a helpful light on classical procedures. It is necessary to go into a Bayesian framework to give confidence intervals the probabilistic interpretation which practitioners often want to place on them. This insight is helpful in drawing attention to the point that another prior distribution would lead to a different interval.
A Bayesian may cheat by basing the prior distribution on the data; a Frequentist can base the hypothesis to be tested on the data. For example, the role of a protocol in clinical trials is to prevent this from happening by requiring the hypothesis to be specified before the data are collected. In the same way, a Bayesian could be obliged to specify the prior in a public protocol before beginning a study. In a collective scientific study, this would be somewhat more complex than for Frequentist hypotheses because priors must be personal for coherence to hold.
A suitable quantity that has been proposed to measure inferential uncertainty; i.to handle the a priori unexpected, is the likelihood function itself. If you perform a series of identical random experiments e.coin tossesthe underlying probability distribution that maximizes the probability of the outcome you observed is the probability distribution proportional to the results of the experiment. This has the direct interpretation of telling how relatively well each possible explanation modelwhether obtained from the data or not, predicts the observed data.
If the data happen to be extreme atypical in some way, so that the likelihood points to a poor set of models, this will soon be picked up in the next rounds of scientific investigation by the scientific community. No long run frequency guarantee nor personal opinions are required. There is a sense in which the Bayesian approach is oriented toward making decisions and the frequentist hypothesis testing approach is oriented toward science.
For example, there may not be enough evidence to show scientifically that agent X is harmful to human beings, but one may be justified in deciding to avoid it in one s diet. In almost all cases, a point estimate is a continuous random variable. Therefore, the probability that the probability is any specific point estimate is really zero. This means that in a vacuum of information, we can make no guess about the probability.
Even if we have information, we can really only guess at a range for the probability. Therefore, in estimating a parameter of a given population, it is necessary that a point estimate accompanied by some measure of possible error of the estimate. The widely acceptable approach is that a point estimate must be accompanied by some interval about the estimate with some measure of assurance that this interval contains the true value of the population parameter.
For example, the reliability assurance processes in manufacturing industries are based on data driven information for making product-design decisions. Objective Bayesian There is a clear connection between probability and logic both appear to tell us how we should reason. But how, exactly, are the two concepts related. Objective Bayesians offers one answer to this question. According to objective Bayesians, probability generalizes deductive logic deductive logic tells us which conclusions are certain, given a set of premises, while probability tells us the extent to which one should believe a conclusion, given the premises certain conclusions being awarded full degree of belief.
According to objective Bayesians, the premises objectively i. uniquely determine the degree to which one should believe a conclusion. Further Readings Bernardo J. Smith, Bayesian Theory, Wiley, 2000.Bayesian Statistical Modelling, Wiley, 2001. Williamson, Foundations of BayesianismKluwer Academic Publishers, 2001. Contains Logic, Mathematics, Decision Theory, and Criticisms of Bayesianism.Operational Subjective Statistical MethodsWiley, 1996.
Presents a systematic treatment of subjectivist methods along with a good discussion of the historical and philosophical backgrounds of the major approaches to probability and statistics.Subjective and Objective Bayesian Statistics Principles, Models, and ApplicationsWiley, 2002. Zimmerman H.Fuzzy Set TheoryKluwer Academic Iqoption blog, 1991. Fuzzy logic approaches to probability based on L. Zadeh and his followers present a difference between possibility theory and probability theory.
Rumor, Belief, Opinion, and Fact. As a necessity the human rational strategic thinking has evolved to cope with his her environment. The rational strategic thinking which we call reasoning is another means to make the world calculable, predictable, and more manageable for the utilitarian purposes. In constructing a model of reality, factual information is therefore needed to initiate any rational strategic thinking in the form of reasoning.
However, we should not confuse facts with beliefs, opinions, or rumors. The following table helps to clarify the distinctions. Rumor, Belief, Opinion, and Fact Rumor Belief Opinion Fact One says to oneself I need to use it anyway This is the truth. I m right This iqoption blog my view This is a fact One says to others It could be true. You re wrong That is yours I can explain it to you. Beliefs are defined as someone s own understanding. In belief, I am always right and you are wrong.
There is nothing that can be done to convince the person that what they believe is wrong. With respect to belief, Henri Poincarй said, Doubt everything or believe everything these are two equally convenient strategies. With either, we dispense with the need to think. Believing means not wanting to know what is fact. Human beings are most apt to believe what they least understand.
Therefore, you may rather have a mind opened by wonder than one closed by belief. The greatest derangement of the mind is to believe in something because one wishes it to be so. The history of mankind is filled with unsettling normative perspectives reflected in, for example, inquisitions, witch hunts, denunciations, and brainwashing techniques. The sacred beliefs are not only within religion, but also within ideologies, and could even include science.
In much the same way many scientists trying to save the theory. For example, the Freudian treatment is a kind of brainwashing by the therapist where the patient is in a suggestive mood completely and religiously believing in whatever the therapist is making of him her and blaming himself herself in all cases. There is this huge lumbering momentum from the Cold War where thinking is still not appreciated. Nothing is so firmly believed as that which is least known. The history of humanity is also littered with discarded belief-models.
However, this does not mean that someone who didn t understand what was going on invented the model nor had no utility or practical value. The main idea was the cultural values of any wrong model. The falseness of a belief is not necessarily an objection to a belief. The question is, to what extent is it life-promoting, and life enhancing for the believer. Opinions or feelings are slightly less extreme than beliefs however, they are dogmatic.
An opinion means that a person has certain views that they think are right. Also, they know that others are entitled to their own opinions. People respect others opinions and in turn expect the same. In forming one s opinion, the empirical observations are obviously strongly affected by attitude and perception. However, opinions that are well rooted should grow and change like a healthy tree.
Fact is the only instructional material that can be presented in an entirely non-dogmatic way. Everyone has a right to his her own opinion, but no one has a right to be wrong in his her facts. Public opinion is often a sort of religion, with the majority as its prophet. Moreover, the profit has a short memory and does not provide consistent opinions over time.
Rumors and gossip are even weaker than opinion. Now the question is who will believe these. For example, rumors and gossip about a person are those when you hear something you like, about someone you do not. Here is an example you might be familiar with Why is there no Nobel Prize for mathematics. It is the opinion of many that Alfred Nobel caught his wife in an amorous situation with Mittag-Leffler, the foremost Swedish mathematician at the time. Therefore, Nobel was afraid that if he were to establish a mathematics prize, the first to get it would be M-L.
The story persists, no matter how often one repeats the plain fact that Nobel was not married. To understand the difference between feeling and strategic thinkingconsider carefully the following true statement He that thinks himself the happiest man really is so; but he that thinks himself the wisest is generally the greatest fool. Most people do not ask for facts in making up their decisions. They would rather have one good, soul-satisfying emotion than a dozen facts.
This does not mean that you should not feel anything. Notice your feelings. But do not think with them. Facts are different than beliefs, rumors, and opinions. By contrast, at least some non-Bayesian approaches consider probabilities as objective attributes of things or situations which are really out there availability of data. Facts are the basis of decisions. A fact is something that is right and one can prove to be true based on evidence and logical arguments.
Facts are always subject to change. A fact can be used to convince yourself, your friends, and your enemies. Data becomes information when it becomes relevant to your decision problem. Information becomes fact when the data can support it. Fact becomes knowledge when it is used in the successful completion of a structured decision process.
However, a fact becomes an opinion if it allows for different interpretations, i.different perspectives. Note that what happened in the past is fact, not truth. Truth is what we think about, what happened i. Business Statistics is built up with facts, as a house is with stones. But a collection of facts is no more a useful and instrumental science for the manager than a heap of stones is a house.
Science and religion are profoundly different. Religion asks us to believe without question, even or especially in the absence of hard evidence. Indeed, this is essential for having a faith. Science asks us to take nothing on faith, to be wary of our penchant for self-deception, to reject anecdotal evidence. Science considers deep but healthy skepticism a prime feature. One of the reasons for its success is that science has built-in, error-correcting machinery at its very heart.
Learn how to approach information critically and discriminate in a principled way between beliefs, opinions, and facts. Critical thinking is needed to produce well-reasoned representation of reality in your modeling process. Analytical thinking demands clarity, consistency, evidence, and above all, a consecutive, focused-thinking. Further Readings Boudon R.The Origin of Values Sociology and Philosophy of BeliefTransaction Publishers, London, 2001.
Castaneda C.The Active Side of InfinityHarperperennial Library, 2000. Wright, Decision Analysis for Management JudgmentWiley, 1998.The Hoax of Freudism A Study of Brainwashing the American Professionals and LaymenPhiladelphia, Dorrance, 1974.Religions in Four Dimensions Existential and Aesthetic, Historical and ComparativeReader s Digest Press, 1976. What is Statistical Data Analysis.
Data are not Information. Vast amounts of statistical information are available in today s global and economic environment because of continual improvements in computer technology. To compete successfully globally, managers and decision makers must be able to understand the information and use it effectively. A Bayesian and a classical statistician analyzing the same data will generally reach the same conclusion. Jurjevich R. Statistical data analysis provides hands on experience to promote the use of statistical thinking and techniques to apply in order to make educated decisions in the business world.
The statistical software package, SPSS, which is used in this course, offers extensive data-handling capabilities and numerous statistical analysis routines that can analyze small to very large data statistics. Computers play a very important role in statistical data analysis. The computer will assist in the summarization of data, but statistical data analysis focuses on the interpretation of the output to make inferences and predictions.
Collecting the data 3. Defining the problem 2. Analyzing the data 4. Defining the Problem. Reporting the results. An exact definition of the problem is imperative in order to obtain accurate data about it. It is extremely difficult to gather data without a clear definition of the problem. Collecting the Data. We live and work at a time when data collection and statistical computations have become easy almost to the point of triviality. Studying a problem through the use of statistical data analysis usually involves four basic steps.
Paradoxically, the design of data collection, never sufficiently emphasized in the statistical data analysis textbook, have been weakened by an apparent belief that extensive computation can make up for any deficiencies in the design of data collection. One must start with an emphasis on the importance of defining the population about which we are seeking to iqoption blog inferences, all the requirements of sampling and experimental design must be met.
Two important aspects of a statistical study are Population - a set of all the elements of interest in a study Sample - a subset of the population Statistical inference is refer to extending your knowledge obtain from a random sample from a population to the whole population. Designing ways to collect data is an important job in statistical data analysis. This is known in mathematics as an Inductive Reasoning. That is, knowledge of whole from a particular.
Its main application is in hypotheses testing about a given population. The purpose of statistical inference is to obtain information about a population form information contained in a sample. It is just not feasible to test the entire population, so a sample is the only realistic way to obtain data because of the time and cost constraints. Data can be either quantitative or qualitative.
Qualitative data are labels or names used to identify an attribute of each element. For the purpose of statistical data analysis, distinguishing between cross-sectional and time series data is important. Cross-sectional data re data collected at the same or approximately the same point in time. Time series data are data collected over several time periods. Data can be collected from existing sources or obtained through observation and experimental studies designed to obtain new data.
In an experimental study, the variable of interest is identified. Then one or more factors in the study are controlled so that data can be obtained about how the factors influence the variables. Quantitative data are always numeric and indicate either how much or how many. A survey is perhaps the most common type of observational study. Analyzing the Data. Statistical data analysis divides the methods for analyzing data into two categories exploratory methods and confirmatory methods.
Exploratory methods are used to discover what the data seems to be saying by using simple arithmetic and easy-to-draw pictures to summarize data. In observational studies, no attempt is made to control or influence the variables of interest. Confirmatory methods use ideas from probability theory in the attempt to answer specific questions. Probability is important in decision making because it provides a mechanism for measuring, expressing, and analyzing the uncertainties associated with future events.
The majority of the topics addressed in this course fall under this heading. Through inferences, an estimate or test claims about the characteristics of a population can be obtained from a sample. The results may be reported in the form of a table, a graph or a set of percentages. Because only a small collection sample has been examined and not an entire population, the reported results must reflect the uncertainty through the use of probability statements and intervals of values.
To conclude, a critical aspect of managing any organization is planning for the future. Good judgment, intuition, and an awareness of the state of the economy may give a manager a rough idea or feeling of what is likely to happen in the future. Statistical data analysis helps managers forecast and predict future aspects of a business operation. However, converting that feeling into a number that can be used effectively is difficult.
The most successful managers and decision makers are the ones who can understand the information and use it effectively. Data Processing Coding, Typing, and Editing. Coding the data are transferred, if necessary to coded sheets. Typing the data are typed and stored by at least two independent data entry persons. For example, when the Current Population Survey and other monthly surveys were taken using paper questionnaires, the U.
Census Bureau used double key data entry. Editing the data are checked by comparing the two independent typed data. The standard practice for key-entering data from paper questionnaires is to key in all the data twice. Ideally, the second time should be done by a different key entry operator whose job specifically includes verifying mismatches between the original and second entries.
It is believed that this double-key verification method produces a 99. 8 accuracy rate for total keystrokes. Types of error Recording error, typing error, transcription error incorrect copyingInversion e. 45 is typed as 123. 54Repetition when a number is repeatedDeliberate error. Type of Data and Levels of Measurement. Qualitative data, such as eye color of a group of individuals, is not computable by arithmetic relations. They are labels that advise in which category or class an individual, object, or process fall.
They are called categorical variables. Quantitative data sets consist of measures that take numerical values for which descriptions such as means and standard deviations are meaningful. They can be put into an order and further divided into two groups discrete data or continuous data. Discrete data are countable data, for example, the number of defective items produced during a day s production.
For example, measuring the height of a person. The first activity in statistics is to measure or count. Continuous data, when the parameters variables are measurable, are expressed on a continuous scale. Measurement counting theory is concerned with the connection between data and reality. A set of data is a representation i.a model of the reality based on a numerical and mensurable scales.
Data are called primary type data if the analyst has been involved in collecting the data relevant to his her investigation. Otherwise, it is called secondary type data. Data come in the forms of Nominal, Ordinal, Interval and Ratio remember the French word NOIR for color black. Data can be either continuous or discrete.
While the unit of measurement is arbitrary in Ratio scale, its zero point is a natural attribute. Both zero and unit of measurements are arbitrary in the Interval scale. The categorical variable is measured on an ordinal or nominal scale. Measurement theory is concerned with the connection between data and reality. Both statistical theory and measurement theory are necessary to make inferences about reality.
Since statisticians live for precision, they prefer Interval Ratio levels of measurement. Problems with Stepwise Variable Selection. It yields R-squared values that are badly biased high. The F and chi-squared tests quoted next to each variable on the printout do not have the claimed distribution. The method yields confidence intervals for effects and predicted values that are falsely narrow.
It yields P-values that do not have the proper meaning and the proper correction for them is a very difficult problem It gives biased regression coefficients that need shrinkage, i.the coefficients for remaining variables are too large. It has severe problems in the presence of collinearity. It is based on methods e. F-tests for nested models that were intended to be used to test pre-specified hypotheses. Increasing the sample size does not help very much. Note also that the all-possible-subsets approach does not remove any of the above problems.
Further Reading Derksen, S. Keselman, Backward, forward and stepwise automated subset selection algorithms, British Journal of Mathematical and Statistical Psychology45, 265-282, 1992. An Alternative Approach for Estimating a Regression Line. Further Readings Cornish-Bowden A.Analysis of Enzyme Kinetic DataOxford Univ Press, 1995.A History of Mathematical Statistics From 1750 to 1930Wiley, New York, 1998. Among others, the author points out that in the beginning of 18-th Century researches had four different methods to solve fitting problems The Mayer-Laplace method of averages, The Boscovich-Laplace method of least absolute deviations, Laplace method of minimizing the largest absolute residual and the Legendre method of minimizing the sum of squared residuals.
The only single way of choosing between these methods was to compare results of estimates and residuals. Exploring the fuzzy data picture sometimes requires a wide-angle lens to view its totality. At other times it requires a closeup lens to focus on fine detail. Multivariate Data Analysis. The graphically based tools that we use provide this flexibility.
Most chemical systems are complex because they involve many variables and there are many interactions among the variables. Therefore, chemometric techniques rely upon multivariate statistical and mathematical tools to uncover interactions and reduce the dimensionality of the data. Multivariate analysis is a branch of statistics involving the consideration of objects on each of which are observed the values of a number of variables.
Multivariate techniques are used across the whole range of fields of statistical application in medicine, physical and biological sciences, economics and social science, and of course in many industrial and commercial applications. Principal component analysis used for exploring data to reduce the dimension. Generally, PCA seeks to represent n correlated random variables by a reduced set of uncorrelated variables, which are obtained by transformation of the original set onto an appropriate subspace.
The uncorrelated variables are chosen to be good linear combination of the original variables, in terms of explaining maximal variance, orthogonal directions in the data. Two closely related techniques, principal component analysis and factor analysis, are used to reduce the dimensionality of multivariate data.
In these techniques correlations and interactions among the variables are summarized in terms of a small number of underlying factors. The methods rapidly identify key variables or groups of variables that control the system under study. The resulting dimension reduction also permits graphical representation of the data so that significant relationships among observations or samples can be identified.
Other techniques include Multidimensional Scaling, Cluster Analysis, and Correspondence Analysis.Statistical Strategies for small Sample Research, Thousand Oaks, CA, Sage, 1999. Further Readings Chatfield C. Collins, Introduction to Multivariate AnalysisChapman and Hall, 1980. Krzanowski W.Principles of Multivariate Analysis A User s PerspectiveClarendon Press, 1988.
Bibby, Multivariate AnalysisAcademic Press, 1979. The Meaning and Interpretation of P-values what the data say. P-value Interpretation P 0. 01 very strong evidence against H0 0. 05 moderate evidence against H0 0. 10 suggestive evidence against H0 0. 10 P little or no real evidence against H0. This interpretation is widely accepted, and many scientific journals routinely publish papers using this interpretation for the result of test of hypothesis.
For the fixed-sample size, when the number of realizations is decided in advance, the distribution of p is uniform assuming the null hypothesis. We would express this as P p x x. That means the criterion of p 0. 05 achieves a of 0. When a p-value is associated with a set of data, it is a measure of the probability that the data could have arisen as a random sample from some population described by the statistical testing model.
The smaller the p-value, the more evidence you have. A p-value is a measure of how much evidence you have against the null hypothesis. One may combine the p-value with the significance level to make decision on a given test of hypothesis. In such a case, if the p-value is less than some threshold usually. 05, sometimes a bit larger like 0. 1 or a bit smaller like. 01 then you reject the null hypothesis.
Understand that the distribution of p-values under null hypothesis H0 is uniform, and thus does not depend on a particular form of the statistical test. In a statistical hypothesis test, the P value is the probability of observing a test statistic at least as extreme as the value actually observed, assuming that the null hypothesis is true. The value of p is defined with respect to a distribution.
Therefore, we could call it model-distributional hypothesis rather than the null hypothesis. In short, it simply means that if the null had been true, the p value is the probability against the null in that case. The p-value is determined by the observed value, however, this makes it difficult to even state the inverse of p. Further Readings Arsham H.Kuiper s P-value as a Measuring Tool and Decision Procedure for the Goodness-of-fit Test, Journal of Applied StatisticsVol.
3, 131-135, 1988. Accuracy, Precision, Robustness, and Quality. The robustness of a procedure is the extent to which its properties do not depend on those assumptions which you do not wish to make. This is a modification of Box s original version, and this includes Bayesian considerations, loss as well as prior. The central limit theorem CLT and the Gauss-Markov Theorem qualify as robustness theorems, but the Huber-Hempel definition does not qualify as a robustness theorem.
We must always distinguish between bias robustness and efficiency robustness. One needs to be more specific about what the procedure must be protected against. If the sample mean is sometimes seen as a robust estimator, it is because the CLT guarantees a 0 bias for large samples regardless of the underlying distribution. This estimator is bias robust, but it is clearly not efficiency robust as its variance can increase endlessly.
That variance can even be infinite if the underlying distribution is Cauchy or Pareto with a large scale parameter. This is the reason for which the sample mean lacks robustness according to Huber-Hampel definition. The problem is that the M-estimator advocated by Huber, Hampel and a couple of other folks is bias robust only if the underlying distribution is symmetric. In the context of survey sampling, two types of statistical inferences are available the model-based inference and the design-based inference which exploits only the randomization entailed by the sampling process no assumption needed about the model.
Unbiased design-based estimators are usually referred to as robust estimators because the unbiasedness is true for all possible distributions. It seems clear however, that these estimators can still be of poor quality as the variance that can be unduly large. However, others people will use the word in other imprecise ways. 2, Advanced Theory of Statistics, also cites Box, 1953; and he makes a less useful statement about assumptions. In addition, Kendall states in one place that robustness means merely that the test size, aremains constant under different conditions.
Kendall s Vol. This is what people are using, apparently, when they claim that two-tailed t-tests are robust even when variances and sample sizes are unequal. I find it easier to use the phrase, There is a robust differencewhich means that the same finding comes up no matter how you perform the test, what justifiable transformation you use, where you split the scores to test on dichotomies, etc.or what outside influences you hold constant as covariates.
Influence Function and Its Applications. It is main potential application of the influence function is in comparison of methods of estimation for ranking the robustness. A commonsense form of influence function is the robust procedures when the extreme values are dropped, i.data trimming. There are a few fundamental statistical tests such as test for randomness, test for homogeneity of population, test for detecting outliner sand then test for normality.
For all these necessary tests there are powerful procedures in statistical data analysis literatures. Moreover since the authors are limiting their presentation to the test of mean, they can invoke the CLT for, say any sample of size over 30. The concept of influence is the study of the impact on the conclusions and inferences on various fields of studies including statistical data analysis. This is possible by a perturbation analysis. For example, the influence function of an estimate is the change in the estimate when an infinitesimal change in a single observation divided by the amount of the change.
It acts as the sensitivity analysis of the estimate. The influence function has been extended to the what-if analysis, robustness, and scenarios analysis, such as adding or deleting an observation, outliners s impact, and so on. For example, for a given distribution both normal or otherwise, for which population parameters have been estimated from samples, the confidence interval for estimates of the median or mean is smaller than for those values that tend towards the extremities such as the 90 or 10 data.
While in estimating the mean on can invoke the central limit theorem for any sample of size over, say 30. However, we cannot be sure that the iqoption blog variance is the true variance of the population and therefore greater uncertainty creeps in and one need to sue the influence function as a measuring tool an decision procedure. Further Readings Melnikov Y.Influence Functions and MatricesDekker, 1999. What is Imprecise Probability. What is a Meta-Analysis. a Especially when Effect-sizes are rather small, the hope is that one can gain good power by essentially pretending to have the larger N as a valid, combined sample.
b When effect sizes are rather large, then the extra POWER is not needed for main effects of design Instead, it theoretically could be possible to look at contrasts between the slight variations in the studies themselves. For example, to compare two effect sizes r obtained by two separate studies, you may use. It seems obvious to me that no statistical procedure can be robust in all senses. where z 1 and z 2 are Fisher transformations of r, and the two n i s in the denominator represent the sample size for each study.
If you really trust that all things being equal will hold up. The typical meta study does not do the tests for homogeneity that should be required. there is a body of research data literature that you would like to summarize. one gathers together all the admissible examples of this literature note some might be discarded for various reasons. most important would be the effect that has or has not been found, i.how much larger in sd units is the treatment group s performance compared to one or more controls.
call the values in each of the investigations in 3. mini effect sizes. across all admissible data sets, you attempt to summarize the overall effect size by forming a set of individual effects. and using an overall sd as the divisor. I, personally, do not like to call the tests robust when the two versions of the t-test, which are approximately equally robust, may have 90 different results when you compare which samples fall into the rejection interval or region.
certain details of each investigation are deciphered. thus yielding essentially an average effect size. in the meta analysis literature. sometimes these effect sizes are further labeled as small, medium, or large. across different factors and variables. but, in a nutshell, this is what is done. I recall a case in physics, in which, after a phenomenon had been observed in air, emulsion data were examined. The theory would have about a 9 effect in emulsion, and behold, the published data gave 15.
As it happens, there was no significant difference practical, not statistical in the theory, and also no error in the data. It was just that the results of experiments in which nothing statistically significant was found were not reported. You can look at effect sizes in many different ways. This non-reporting of such experiments, and often of the specific results which were not statistically significant, which introduces major biases. This is also combined with the totally erroneous attitude of researchers that statistically significant results are the important ones, and than if there is no significance, the effect was not important.
We really need to differentiate between the term statistically significantand the usual word significant. Meta-analysis is a controversial type of literature review in which the results of individual randomized controlled studies are pooled together to try to get an estimate of the effect of the intervention being studied. It s not easy to do well and there are many inherent problems. Further Readings Lipsey M. It increases statistical power and is used to resolve the problem of reports which disagree with each other.
Wilson, Practical Meta-AnalysisSage Publications, 2000. What Is the Effect Size. Therefore, the ES is the mean difference between the control group and the treatment group. Effect size permits the comparative effect of different treatments to be compared, even when based on different samples and different measuring instruments. Howevere, by Glass s method, ES is mean1 - mean2 SD of control group while by Hunter-Schmit s method, ES is mean1 - mean2 pooled SD and then adjusted by instrument reliability coefficient.
ES is commonly used in meta-analysis and power analysis. Further Readings Cooper H. Hedges, The Handbook of Research SynthesisNY, Russell Sage, 1994. What is the Benford s Law. What About the Zipf s Law. This can be observed, for instance, by examining tables of Logarithms and noting that the first pages are much more worn and smudged than later pages. Bias Reduction Techniques.
This implies that a number in a table of physical constants is more likely to begin with a smaller digit than a larger digit. According to legend, Baron Munchausen saved himself from drowning in quicksand by pulling himself up using only his bootstraps. The statistical bootstrap, which uses resampling from a given set of data to mimic the variability that produced the data in the first place, has a rather more dependable theoretical basis and can be a highly effective procedure for estimation of error quantities in statistical problems.
Bootstrap is to create a virtual population by duplicating the same sample over and over, and then re-samples from the virtual population to form a reference set. Very often, a certain structure is assumed so that a residual is computed for each case. The purpose is often to estimate a P-level. Then you compare your original sample with the reference set to get the exact p-value.
What is then re-sampled is from the set of residuals, which are then added to those assumed structures, before some statistic is evaluated. Jackknife is to re-compute the data by leaving on observation out each time. Leave-one-out replication gives you the same Case-estimates, I think, as the proper jack-knife estimation. Jackknifing does a bit of logical folding whence, jackknife -- look it up to provide estimators of coefficients and error that you hope will have reduced bias.
Bias reduction techniques have wide applications in anthropology, chemistry, climatology, clinical trials, cybernetics, and ecology. Further Readings Efron B.The Jackknife, The Bootstrap and Other Resampling PlansSIAM, Philadelphia, 1982. Tu, The Jackknife and BootstrapSpringer Verlag, 1995. Tibshirani, An Introduction to the BootstrapChapman Hall now the CRC Press1994.
Number of Class Interval in Histogram. k the smallest integer greater than or equal to 1 Log n Log 2 1 3. Area Under Standard Normal Curve. To have an optimum you need some measure of quality - presumably in this case, the best way to display whatever information is available in the data. The sample size contributes to this, so the usual guidelines are to use between 5 and 15 classes, one need more classes if you one has a very large sample. You take into account a preference for tidy class widths, preferably a multiple of 5 or 10, because this makes it easier to appreciate the scale.
This assumes you have a computer and can generate alternative histograms fairly readily. There are often management issues that come into it as well. For example, if your data is to be compared to similar data - such as prior studies, or from other countries - you are restricted to the intervals used therein. Beyond this it becomes a matter of judgement - try out a range of class widths and choose the one that works best.
Use narrow classes where the class frequencies are high, wide classes where they are low. The following approaches are common. If the histogram is very skewed, then unequal classes should be considered. Let n be the sample size, then number of class interval could be. Thus for 200 observations you would use 14 intervals but for 2000 you would use 33. Alternatively. Find the range highest value - lowest value. Divide the range by a reasonable interval size 2, 3, 5, 10 or a multiple of 10.
Aim for no fewer than 5 intervals and no more than 15. Structural Equation Modeling. A structural equation model may apply to one group of cases or to multiple groups of cases. When multiple groups are analyzed parameters may be constrained to be equal across two or more groups. When two or more groups are analyzed, means on observed and latent variables may also be included in the model.
As an application, how do you test the equality of regression slopes coming from the same sample using 3 different measuring methods. You could use a structural modeling approach. 1 - Standardize all three data sets prior to the analysis because b weights are also a function of the variance of the predictor variable and with standardization, you remove this source. 2 - Model the dependent variable as the effect from all three measures and obtain the path coefficient b weight for each one.
3 - Then fit a model in which the three path coefficients are constrained to be equal. If a significant decrement in fit occurs, the paths are not equal. Further Readings Schumacker R.
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