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STATISTICAL ANALYSIS.
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Paper Abstract:
The scientific method as the underlying basis. Elements of the scientific method: concepts, definition, hypotheses theory. Statistical analysis as the process through which data becomes knowledge. Alternative models for statistical analysis. Procedures in classical statistics analysis. Samples, validity, reliability. Uses of statistics.
Paper Introduction:
STATISTICAL ANALYSIS: THEORIES, CONCEPTS, AND PROCEDURES — A REVIEW
Introduction
The underlying basis of statistical analysis is the scientific method. The foundations of the scientific method are (1) concepts, (2) definition, (3) hypotheses, and (4) theory. Concepts are basic to all thought and communication. The application of concepts provides the basis for the development of hypothesis statements. The success of scientific research depends upon (1) how clearly the investigator conceptualizes the problem and (2) how well others understand the concepts used by the researcher. A construct is a concept designed specifically for a particular research purpose or theory-building purpose (Pfaffenberger & Patterson, 1999).
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(1986). New York: W. This approach to the conduct of scholarly research isparticularly relevant to the objectives of contemporary statisticalanalysis.Causality Model versus Plausibility Model Over the years the dominant philosophy of scholarly research has beenthe causality model. M. (199 ). Hypothesis testing falls in the realm of inferential statistics.The basic theoretical principle underlying inferential statistics isprobability. S., McCabe, G. The advocated shift infocus requires a production of evidence and argument that makes a favoredcause a plausible explanation for the available data. Cardon, L. Informal scholarly research, the research design is carefully planned andfollowed to the maximum possible extent. Thus,to be valid, data provided by an instrument designed to measure a tendencytoward abnormal behavior would have to be consistent with those of otherinstruments of proven reliability in the measurement of tendencies towardabnormal behavior (Lockhart, 1998). M., Fares, M., Hodge, W. Non-probability samples include (a) convenience samples, wherein the researcherselects who or what ever is at hand, (b) judgment samples, wherein theresearcher selects population elements because they are thought to servethe research purpose, and (c) quota samples, wherein the researcher selectspopulation elements from specified population sub-groups (Ott & Mendenhall,1994). S., & Guba, E. The foundations of the scientific method are (1) concepts, (2) definition,(3) hypotheses, and (4) theory. (199 ). In qualitative research, the issue of replicability is important, andthe issue is addressed through the confidence one has in the investigator'sinterpretation of reality. Quantitative analytical procedures provide a positivist result - aspecific quantitative relationship between two or more variables isestablished. Probability, random variables, and random signal principles. The practice of statistics. In most instances ofthe application of inferential statistics in hypothesis testing, populationdistributions are unknown. The logic of scientific discovery. (1999). A measurement procedure is,thus, reliable to the extent that it supplies consistent results. Random or equivalent procedures are employed to eliminate bias to theextent possible from the research sample in formal scholarly research.Random procedures are mathematically-based actions that assure that everymember of a population to be sampled has a chance equal to that of everyother member of the population to be selected as a part of the sample. In the context of discovery, there is free reignfor speculation, creative thought, and subjective interpretation. Rockville, Maryland: Agency For Health Care Policy Research, pp. Posterior analysis is the process by which thevalue of sample data is established before the sample data are collected.The basic procedure involves considering all of the possible sampleoutcomes, determining the estimated value in the decision process of eachof the possible sample outcomes, and, finally, determining the expectedvalues of the sample data, by weighting each of these different values bythe probability that the associated sample outcome will actually occur(Peebles, 1993). Therefore, the central limit theorem assumes ahigh-level of importance in hypothesis testing. The use of qualitative research methods permits the investigation ofphenomena of expert practice in context and from the eyes of the actors.This approach to research is "in concert with investigations in ...disciplines in which activities are focused on analyzing expertise andidentifying aspects of practical knowledge, that is, knowledge that emergesfrom observing and investigating practice" (Elstein, Shulman, & Sprafka,199 , p. In principle, thus,positive inquiry is independent of any particular ethical position ornormative value. H. > The probability distribution of the statistic upon which the analysis is based is not dependent upon specific information or assumptions about the population(s) from which the sample(s) are drawn. The elements of social scientific thinking. R. (1994). The basis forassessing the validity of this measuring instrument calibrated in feet andinches would its ability to accurately measure feet and inches inaccordance with an external standard, such as a master measurementinstrument maintained by the National Bureau of Standards. Homewood, Illinois: Irwin.Popper, K. Procedures in Classical Statistical Analysis Descriptive statistics involve the determination of basic datacharacteristics, such as mean, median, range, and standard deviation. Although statistical inference techniques are directly concerned withestimating values or with the testing of hypotheses concerning populationparameters, the results of these procedures relate to alternative coursesof action or to alternative decisions. C., & Patterson, J. Within the realm of contemporary research, system norms tend to favorquantification, universalistic criteria, rationality, and objective andimpersonal treatment over qualitative factors, particularistic oridiosyncratic criteria, emotionality, and personalized treatment (Ashforth,1992). Alternatively, qualitativeinvestigators are an integral part of developing the research design,collecting the data, and analyzing the data. With a philosophical context, however, scholarly inquiry may beapproached from a position of either positivism or normativism (Barzun &Graff, 1977). (3rd ed.). (3rd ed.). Thesignificance of Bayes' theorem is that it is applied in the context ofsequential events, and further, that the computational version of theformula provides the basis for determining the conditional probability ofan event having occurred in the first sequential position given that aparticular event has been observed in the second sequential position(Peebles, 1993). Qualitative evaluative criteria, however, may also be appliedto quantitative data (Hoover, 1995). Freeman Company.Myers, J. (1977). (1989). 385). Freeman and Company.Ott, R. The central limit theorem holds that the distribution of thetotals (and therefore the means) of random samples will be normal no matterwhat the distribution in the population is like, provided only that thesamples are large enough (Ott & Mendenhall, 1994). Thus,hypothesis testing applied to sample data in inferential statisticalapplications can provide valid and reliable information about the entirepopulation when the research design assures that the sample isrepresentative of the population (Oehlert, 2 ). Randomization is an approach typically used for evaluation of two ormore groups or systems. Probability sampling is characterized byrandom sampling procedures that assure that all elements of a population tobe investigated have an equal chance of being selected for the sample. (2 ). The mean, median,and mode are all at the same point for a normally distributed variable. S., & McCabe, G. Belmont, California: Wadsworth Publishing.Guba, E. Newbury Park, California: Sage Publications.Guba, E. Fourth generation evaluation. An objective ofruling in a cause may require an expansion of the search for data or achange in the type of data sought in order to acquire relevant evidence(Hoover, 1995). (1995). > The inference does not concern a parameter in the population distribution (e.g., a hypothesis that a time-ordered set of observations exhibits a random pattern). Z., Jr. 477). In thecase of the measurement of tendencies toward abnormal behavior inindividuals, an instrument, to be reliable, would have to provideconsistent scores for the same individuals within a short span of time whenthose individuals were both tested and retested with the instrument (Myers& Well, 1995). A case might be made in such an instance that, todecrease the spousal battery rate more people must be put back to work.Such an argument might be extended to contend that increased governmentalspending to stimulate the local economy was the best way to attack theincreasing spousal battery rate. H. Qualitative data may also be analyzed through the application ofeither quantitative or qualitative analytical procedures. Perhaps the most important use of statistics and statisticalprocedures is the testing hypotheses - one of the procedures that permitthe making of inferences. On the otherhand, if an instrument does measure the height of the same individual asbeing the same in measurement-after-measurement, the instrument is areliable instrument. S. J. New York: McGraw-Hill, Inc.Peter, L. New York: Basic Books.Shao, S. Control of extraneous variablesis an important feature of the research design. The point at which two linear functionscross indicates the point of indifference in respect to making a choicebetween the two alternative courses of action, because the conditionalvalues of the acts are equal at that point. Rather, there are two-probability and non-probability samplingmethods. Experimental research is not, however, a feasible option forsuch an inquiry. Friedrich,Kierniesky, and Cardon (1989), however, investigated what they referred toas the "naturalistic fallacy" (p. The reliability of a data collection procedure is a contributor to thevalidity of that procedure, rather than the other way around. ReferencesAshforth, B. percent chance that the actualvalue will be higher. The posterior probability distribution is the probability distributionthat is applicable subsequent to the observation of sample data, andsubsequent to the use of these data to revise the prior probabilitydistribution through application of the Bayes' theorem. Contriving to rule in theplausible, as opposed to ruling out the false, however, strikes at theheart of scholarly inquiry as the approach has long been understood. Unfortunately, Charles Grosvenor utteredsomething quite similar at an earlier date, to wit: "Figures won't lie, butliars will figure" (Peter, 1977, p. D. Qualitative analysis can offer some plausible explanations forwhat has happened or is happening. To illustratethese points, consider, again, the case of the instrument calibrated infeet and inches used to measure the height of a group of individuals. . Hillsdale, New Jersey: Erlbaum Publishers.Oehlert, G. percent chance thatthe actual value will be lower and a 5 . Thus, empirical psychologicalresearch attempts to establish cause-and-effect relationships betweendependent and independent variables. Obviously, researchers must useboth exploratory and confirmatory research. The instrument calibrated in feet andinches would be reliable, if it yielded consistent results in themeasurement time-and-time-again of the same group of individuals. Evaluation and the Health Professions, 13(1), 5-36.Embrey, D. 3 ). Postacute phase. Some qualitative researchers appear to be fixated on the exploratorystage of the process of scholarly inquiry. Typically, the conditional probabilities are determined by the use ofsome standard probability distribution according to the character of thesampling situation-Bayesian formulas for the determination of posteriordistribution probabilities differ according to the conditions of the priordistribution probability. Rather, probability depends only on general assumptions, such as a continuous and/or symmetric population distribution. S. Theory provides a basisfor the narrowing of the body of facts required to investigate a problem(Hoover, 1995). With respect to sampling methods, there are not four primary samplingmethods. Although classicalstatistical techniques are directly concerned with estimating values orwith the testing of hypotheses concerning population parameters, theresults of these procedures are related to alternative courses of action orto alternative decisions (Peebles, 1993). compuserve.com/homepages/Rainer_Wuerlaender/ statquot.htmYates, D., Moore, D. 3). Einhorn & Hogarth (1986) found acorrespondence between everyday judgments of probable cause and bothphilosophical and empirical literature on the topic. 414). At the polar extreme, such norms mayundermine perceived equity by precluding "soft" but nonetheless relevantfactors (Ashforth, 1992, p. Aconstruct is a concept designed specifically for a particular researchpurpose or theory-building purpose (Pfaffenberger & Patterson, 1999). Statistical analysis, therefore, is the process through which databecomes knowledge. R., White, O. (199 ). W., & Oatis, C. Thus, both quantitative and qualitative analysis is useful for theresearcher. They concluded thatcausality becomes a positive rather than a residual inference. When formulated for empirical testing, a propositionbecomes a hypothesis (Hoover, 1995). The success of scientific researchdepends upon (1) how clearly the investigator conceptualizes the problemand (2) how well others understand the concepts used by the researcher. 477). One of the principal applications of Bayesian statistical analysis isthe derivation of alternative probabilities to those that may be observedin sample data. G. A first course in design and analysis of experiments. Probability is a number expressing the likelihood ofoccurrence of a specific event (Ott & Mendenhall, 1994). A quantitative variableis one than can be measured numerically (annual income, as an example),while a qualitative variable is non-numeric, such as gender or ethnic group(Pfaffenberger & Patterson, 1999). , which is an indication of nocorrelation whatever, to 1. Thequantitative investigator would present the positivist result without avalue judgment - the spousal battery rate and unemployment each increased.A qualitative evaluation of the result, however, would depend upon thenormative evaluative criteria applied because a qualitative evaluation is asubjective evaluation. Additional quantitative research, then,can test hypotheses based on the findings of the qualitative analysis. Internal validity isassociated with causal relationships. , which is an indication of a perfectcorrelation. (6th ed.). One result ofthis difference in emphasis is that, in regression analysis, therequirement is that the measurement of both the independent and thedependent variables occur through the use an interval scale, whereas, inanalysis of variance procedures, only the measurement of the dependentvariable must be interval in character (Oehlert, 2 ). Hypotheses may beboth descriptive and relational in form (Ott & Mendenhall, 1994). (1989). As is true for regression analysis, correlation analysis requires adependent variable and at least one independent variable for the structureof the calculations, although the variable labeled as the dependentvariable may not actually be dependent upon the actions of the variablesincluded in the calculations as independent variables. Analysis of variance is a statistical technique that assesses theeffects of one or more categorical independent variables (factors),measured at any level upon a continuous dependent variable (typically theassumption is that it is the product of an interval level measurement (Ott& Mendenhall, 1994). In the context of such a decision situation,prior probability distribution is that probability distribution which isapplicable before any sample data are collected. Popper (1959) distinguished between the context of discovery and thecontext of justification. The bestdecision (act) may be identified by substituting the value of the priormean in each function, and choosing the decision with the highest expectedvalue. Twain, however, also added to the body of humor surroundingstatistical analysis, when he said: "Figures don't lie, but liars figure"(Wuerlaender, 2 1, p. Judging probable cause. Bayes' theorem is used for the purpose of alteringprobabilities associated with an entire set of probable events, or states,in a decision situation. Reliability refers to the accuracy and precision of a ofdata collection procedure. Use of the randomization model isappropriate when at least one of the following criteria is applicable to aninvestigation (Pfaffenberger & Patterson, 1999): > The data entering the analysis are enumerative (data represent the number of observations in each category or cross-category). Correlation analysis is quite similar to regression analysis in majorportions of the calculations. Statistical analysis is not the data that one analyzes. At one level, such norms safeguard perceived equity by encouragingequal and impartial treatment. A change of focus has been advocated to shift the research emphasisfrom one of ruling out rival causes or plausible rival hypotheses to arelatively stronger emphasis on ruling in causes. As such, a theory is an attempt toexplain the interrelated actions within a system. Thus, examiner bias is aconcern in the conduct of qualitative research (Embrey, Guthrie, White, &Dietz, 1996).Positivist Model versus Normative Model The role of values in scholarly inquiry is a source of dispute.Specifically, the issue is whether evaluation is, or ever could be, valuefree, and if it is not, whether evaluation should not operate openly in theservice of particular values. 4). The paradigm dialog. Suppose that it was found that an increase in the spousal battery ratein a given community from 268 occurrences per 1 , persons to 364occurrences per 1 , persons occurred simultaneously with an increase inthe rate of unemployment from 6.2 percent to 1 .1 percent. Choice of perspectives, criteria, measures,methods, and so forth, they contend, must necessarily reflect values. Bayesian statistical analysis is concerned with the derivation of aconditional probability - an alteration of the probability found throughthe application of classical statistical analysis. The existence of a linear function is an indication that the expectedvalue associated with a specific act is a linear function with respect tothe uncertain level of the state. S., Shulman, L., & Sprafka, S. (2 ). If aresearcher were able to manipulate the unemployment variable, such a causalrelationship theoretically could be established through quantitativeanalysis. statistical analysis: theories, concepts, and procedures - a review Introduction The underlying basis of statistical analysis is the scientific method. In Bayesian decisionanalysis, this type of probability distribution is often subjective, inthat the data upon which it is based is itself based on the judgments ofindividuals. Therational model is consistent with the discipline of statistical analysis.Quantitative Model versus Qualitative Model The evaluation of data may rely on the application of eitherquantitative or qualitative analytical procedures (Guba, 199 ).Quantitative approaches are more easily defined than are qualitativeprocedures because qualitative research may refer to either the way dataare measured or the way such data are evaluated. One may enhance the probability ofsample representativeness through the application of random procedures insample selection. The major threatto the external validity of an experiment or study typically is therepresentativeness of the sample. and less than 1. Statistics for the behavioral sciences. > The data are measured and/or analyzed using an ordinal scale of measurement. Mathematics and quantitative methods. The practice of business statistics. To accomplish the objective of validity in measurement, it isnecessary to have some standard that is external to the measurementprocedure, in order to evaluate the validity of the procedure. Another definition of statistical analysis is that itis a science to assist one in making decisions under conditions ofuncertainty. Newbury Park, California: Sage Publications.Hoover, K. Qualitative research is characterized by an inherent limitation. Since the estimate of the event is based on an informed judgment,there is no mathematical theorem that would justify the use of the normaldistribution in respect to such a judgment in any specific situation(Peebles, 1993) For judgment situations in which an informed decision maker is awareof a number of uncertain factors which could influence the value of theeventual outcome in one direction or the other, however, the use of thenormal distribution has been found to be a satisfactory approximation ofthe uncertainty inherent in the estimate. The quantitative analysis illustrateddoes not establish any causal relationship between the two variables. External validity is concerned withthe generalizability of the study findings; that is the ability to applythe study findings beyond the research sample. New York: Harcourt Brace Jovanovich, Inc., 1977.Cordray, D. (1959). Normativism is value oriented. These alternative logic bases are reviewed in thediscussions that follow.Rational Model versus Naturalistic Model Lincoln and Guba (1989) contended that the rational model of scholarlyinquiry is now dead. (2 2). In Bayesian statistical analysis, however, the priorprobability distribution may also be based on historical data (Peebles,1993). 17). (1993). The acceptabilityof a specific correlation coefficient as a descriptor of a relationshipdepends, to a great extent, on the past experience and practice within aspecific field of investigation; however, in most social scienceinvestigations, a correlation minimum coefficient of .7 is often requiredbefore the descriptor is considered to be acceptable (Moore, McCabe,Duckworth, & Sclove, 2 2). Three procedures that are widely used are (1)analysis of variance (ANOVA), (2) linear regression analysis, and (3)correlation analysis (Ott & Mendenhall, 1994). Freeman Company. The basis of the randomization approach is the re-sampling of observations without replacement when changing groupmemberships (Hoover, 1995).Classical Model The classical model of statistical analysis is concerned with theanalysis of sampled data. G., & Lincoln, Y. (5th ed.). F. Further, aninstrument cannot be valid if it is not also reliable; however, justbecause it is reliable, it is not, of necessity, valid. Clinical decision making by experienced and inexperienced pediatric physical therapists for children with diplegic cerebral palsy. Ifthe instrument does not measure the height of the same individual as beingthe same in measurement-after-measurement, it is not a reliable measurementinstrument. By contrast, however, qualitative analysis can explore indepth the potential relationship between the two variables, and, possibly,can develop a plausible explanation for a cause and effect relationshipbetween unemployment (as a cause) and spousal battery (as an effect). G., Guthrie, M. New York: W. (Ed.). E., Riley, P. Introduction to statistics and data analysis for the behavioral sciences. Advocates of seeking to rule in plausible causes contend that, asscholarly research in the long-run seeks to generalize about causalrelationships, such generalizations will be supported by positiveaffirmation of the causal relationships. L., & Mendenhall, W. The application ofsampling procedures is one means of controlling bias and extraneousvariables (Pfaffenberger & Patterson, 1999).Bayesian Model Some statisticians contend that one means of improving probabilityestimates is the application of Bayes' Theorem. Thus,for conducting research into contemporary issues, the causality model isthe preferred paradigm for statistical analysis (Hoover, 1995).Exploratory Model versus Confirmatory Model The distinction made between exploratory and confirmatory researchrepresents the divergent properties of two complementary and sequentialstages of the scientific process, as opposed to two alternative procedures. A normal distribution is definedby identifying the mean and the standard deviation of the distribution.The mean of the prior distribution may be obtained by asking the decisionmaker to identify the most likely value of the random event or by askingthe decision maker for that value which reflects a 5 . Research methodology: Strengthening causal interpretations of nonexperimental data. Classical statisticalanalysis is concerned with the analysis of sampled data. There are several statistical procedures used inthe testing of hypotheses. Essentially, the choice between probability levels of Type I (Alpha)and Type II (Beta) errors is the basis for assessing the relativeimportance of two alternative types of mistakes in hypothesis testing instatistical inference analysis (Ott & Mendenhall, 1994). Themajor application of statistics in decision-making, however, involvesinference (Pfaffenberger & Patterson, 1999). Personality and Social Psychology Bulletin, 15, 414-425.Givens-Heiss, D. With respect to internal validity, the experimental researcher isconcerned with matters such as the stability of the sample over time andsubjects' maturation during the course of the experiment. Physical Therapy, 76(1), 2 -33.Friedrich, J., Kierniesky, N., &. If, however, this identical time-after-timemeasurement is not in agreement with a previously validated externalstandard, the reliable instrument is not a valid instrument. Thus, the reliability and validity of datacollected for use in statistical analysis is of paramount importance.Validity refers to the extent to which data or a data collection instrumentmeasures what it is actually desired to measure. New York: St. Bayesian statistical analysis also is concerned with the selection ofthe best decision. Statistical quotes. (2 1). In such analysis,decision procedures rest entirely on the analysis of data collected througha random sampling of the total population (Yates, Moore, & McCabe, 1999). Thus, ifthe objective were to measure the height of a group of elementary schoolstudents in terms of feet and inches, it would be necessary to have ameasuring instrument calibrated in feet and inches. The interval level requirement for measurement of thedependent variable means that an equality of interval exists between thepoints on the scale with which the variable is measured (Moore, McCabe,Duckworth, & Sclove, 2 2). The principal techniquesof statistical inference are interval estimation and hypothesis testing(Moore, McCabe, Duckworth, & Sclove, 2 2). One consideration of vital importance in defining statistical analysisis the necessity to avoid confusing data with statistics. (1999). Other approaches to sampling are subsidiary classifications ofthe two primary sampling methods. New York: William Morrow and Company, Inc.Pfaffenberger, R. Inference is among the most important of the uses made of quantitativemethods. The situation, according to Cordray (1986) is made worseby a researcher inclination to treat quasi-experiments as only impoverishedversions of true experiments rather than exploiting quasi-experiments fullyfor the value of their information. Onone hand, qualitative investigators typically become immersed in the datato carefully examine the research topic. If the instrument does not measure the height of the sameindividual as being the same in measurement-after-measurement, theinstrument, obviously, does not always yield results which are in agreementwith the external standard, such as a master instrument maintained by theNational Bureau of Standards. 151-172.Einhorn, H. The quality of any statistical analysis can be only as good as is thequality of the data analyzed. Qualitative dataoften may be analyzed in conjunction with quantitative data through theapplication of non-parametric quantitative procedures, such as chi-square.In such analyses, the quantitative distributions of qualitative variablesform either the columns or the rows in a chi square matrix. In almost all cases, the correlation coefficient will be morethan . Validity, thus, is the extent to whichdifferences found through a particular data collection procedure reflecttrue differences among the measured variables (Gravetter & Wallnau, 2 ). In order to applythe Bayes' theorem, the prior probability of an uncertain event and theconditional probability of the sample result must be known (Peebles, 1993). Drawing moral inferences from descriptive science: The impact of attitudes on naturalistic fallacy errors. (1992, October). J., & Hogarth, R. Administration & Society, 24(3), 375-4 8.Barzun, J., & Graff, H. Retrieved from the Internet 2 3- 1-22 at: http://ourworld. Hypotheses are declarative statementsthat are both tentative and conjectural in character. The analysis of sampled data permits theresearcher to make inferences concerning total populations, with theexclusion of any personal judgment or opinions. Cordray(1986) suggested that the usual criteria for positing causal relationshipsare impoverished. (6th ed.). A study must be internallyvalid for its findings to be generalizable (Gravetter & Wallnau, 2 ). (1994). (1998). Bivariate correlation analysis, however,goes further than does regression analysis. In scientific inquiry, a proposition is a statement concerning aconcept that one may evaluate as either true or false when it refers toobservable phenomena. (5th ed.). (5th ed.). A. The specific value of therandom variable may be established quantitatively (algebraically) bysetting the two equations (for alternative decision choices) equal to oneanother and solving for the random variable (Peebles, 1993). The emphasis in the causal model is on falsification,or the quest for residual truth. Freeman Company.Moore, D. Thus, it is important that the investigatorpresent a coherent, illuminating description of reality that is consistentwith a detailed study of the situation (Givens-Heiss, Krebs, Strickland,Fares, Hodge, Mann, & Oatis, 1992). E. Thus, in researchinto contemporary issues, both the exploratory paradigm and theconfirmatory paradigm should be observed as appropriate to the specificstage of research being performed through the process of statisticalanalysis (Hoover, 1995).Randomization Model The randomization model requires the application of non-parametricprocedures for statistical analysis. J., & Wallnau, L. In inferential statistical applications, probability is largely a functionof sampling, as well as of relationships, because it is usually notfeasible or even possible to measure an entire population. Neither isstatistical analysis a particular form of the discipline, such asdescriptive statistical analysis. Understanding statistics. Onceagain, consider the desire to measure the height of a group of elementaryschool students in feet and inches. Theory, in scientific inquiry, is, in effect, a descriptiveexplanation of how something may work. S. The central limit theorem and central tendency areamong the assumptions underlying the validity of the results of statisticalanalysis. L., Krebs, D. Nevertheless, correlation analysis does notestablish causal relationships (Gravetter & Wallnau, 2 ) Correlation coefficients range from . O., Strickland, E. The use of Bayesian decision analysis in cases where normaldistributions apply requires that only two decision acts be evaluated atone time and that the functions associated with these two alternatives arelinear. The modern researcher. Strengthening causal interpretations of nonexperimental data: The role of meta-analysis. Correlation analysis providesa single number that summarizes the relationship between two variables.This single number expresses the strength of the relationship between thevariables (Yates, Moore, & McCabe, 1999). The first approach is, essentially, a request forthe mode of the probability distribution, while the second approach is arequest for the median of the distribution (Shao, 1994). Whereas classical statistical analysis decisionprocedures are based entirely on the analysis of data collected through arandom sampling of a total population, the decision procedures associatedwith Bayesian statistical analysis may include the analysis of sample data,but are not dependent upon the availability of such data (Peebles, 1993). R., & Dietz, J. Martin's Press.Lincoln, Y. Twain repeated the statement in hisautobiography. (1996 January). (1977). P., Duckworth, W., & Sclove, S. H. The rational model, based on objective analysis, appears to be thebetter paradigm for the conduct of research on most contemporary issues.There exists a role for qualitative approaches, but not to the extent thatresearcher-induced suggestion can influence subject responses. Research design and statistical analysis. It is not the event thatfollows the probability distribution; rather, it is the estimate of theevent. New York: W. Therefore, in this example, because themeasuring instrument is not reliable, it cannot be valid. Positivism, by contrast with normativism, is, in theory, value neutral(Barzun & Graff, 1977). Simplified to the extreme, positivism may be said to dealwith life as it is, while normativism may be said to deal with life as itshould be. Reliability, as opposed to validity, is concerned with the accuracyand precision of a measurement procedure. W. New York: W. Statistical methods for business and economics. Having made this statement, however, one mustrecognize that: > Some people will continue to confuse data and statistical analysis (considering, as an example the runs scored by the Toronto Blue Jays as statistics, as opposed to data); while > Others will continue to confuse the parts of statistical analysis with the whole (narrowing the definition of statistical analysis and, in the process, narrowing the scope of the discipline). Rather, it is best to define statisticalanalysis in a holistic context wherein the definition includes the entirescope of the process. Data are nothingmore than crude information. Therefore,reliability in an instrument is required before that instrument can bevalid, but that reliability in an instrument alone cannot guarantee thatsuch instrument will be valid. The prior probability distribution is descriptive of theuncertainty that is associated with the decision maker's estimate of theprobability of occurrence of a random event. Naturalistic inquiry. Alternative Models for Statistical Analysis There are several alternative models upon which to base the process ofstatistical analysis. In Sechrest, L., Perris, E., & Bunker, J. Rigorous hypothesis testing isa waste of effort in the absence of theory development; however, theorydevelopment is pointless in the absence of verification. Bayes' theorem is avariation on the general formula for conditional probability. B. The perceived inequity of systems. Use of the expected value criterion is adistinct procedure (involving the use of expected values) from that of theBayesian theorem (involving the revision of prior probability values). Concepts are basic to all thought andcommunication. The application of concepts provides the basis for thedevelopment of hypothesis statements. (1985). The goal of positive inquiry is to describe andpredict what has happened or will happen in the presence of specificconditions, regardless of the values involved. Indeed, in someinstances, the purpose of a correlation analysis is not to determine thestrength of the association between a dependent and an independentvariable, but, rather, to determine the strength of the relationshipbetween two variables for which dependency characteristics are unknown.More often than not, however, the researcher will have strong support forlabeling one or other of the variables as the dependent variable in thecorrelation analysis. Peter's quotations: Ideas for our time. P. Bayesian statistics have been expanded to incorporate the expectedvalue criterion (Peebles, 1993). As has been stated, the correlationcoefficient represents the strength of a relationship between variables.An r of .25 (a simple coefficient of correlation is represented by thesymbol r, while a multiple coefficient of correlation is represented by thesymbol R) would represent a relatively weak relationship, while an r of .85 would represent a relatively strong relationship. Guba & Lincoln (1989) insist that evaluationcan never be value free. Cincinnati: South-Western Publishing Company.Wuerlaender, R. Defining Statistical Analysis Even the humor surrounding statistical analysis is controversial.Americans frequently attribute to Mark Twain the following saying byBenjamin Disraeli: "There are three kinds of lies: lies, damned lies, andstatistics" (Peter, 1977, p. The analysis ofsampled data permits the researcher to make inferences concerning totalpopulations, with the exclusion of any personal judgment or opinions. In otherinstances, the quantitative distributions of qualitative variables may formboth the columns and the rows of a chi square matrix (Ott & Mendenhall,1994). It is with respect to samples and their characteristics, and their usein hypothesis testing, that the central limit theorem and central tendencymeasures are important. A., Mann, R. Belmont, California: Duxbury Press.Peebles, P. The process of statistical analysistransforms data to "information, from information to facts, and finallyfrom facts to knowledge" (Hoover, 2 1, p. For use ininferential statistics, this probability must be statistically independent. Two of the other statistical procedures that arewidely used are t tests and chi square analysis (Ott & Mendenhall, 1994). Ordinal, interval, and ratio scalesreflect quantitative data, while nominal scales reflect qualitative data(Ott & Mendenhall, 1994). Evaluation ofthe validity of an instrument designed to measure some psychologicalcharacteristic or factor would rely on its ability to yield resultsconsistent with those of another instrument of established validity. (Eds.). In addition to the use of ANOVA, correlation analysis, and regressionanalysis in the testing of hypotheses, several other quantitativeprocedures are also used. Medical problem solving: A ten-year retrospective. H. A major difference between the ANOVA procedure and regression analysisis that, in analysis of variance, the emphasis is on analysis of thevariations in the independent variable, as opposed to the joint interactionof the variations in dependent and independent variables. P. Beverly Hills, California: Sage Publications.Lockhart, R. Psychological Bulletin, 9, 3-19.Elstein, A. As an example, the inability toreject a null hypothesis in statistical inference analysis that indicatedthat patient outcomes associated with a proposed therapy likely would beinferior to those associated with existing therapies typically would beassociated with a decision not to implement the proposed therapy.Rejection of the null hypothesis, however, would provide a basis foracceptance of the alternative hypothesis that adoption of the proposedtherapy would lead to improved patient outcomes (Yates, Moore, & McCabe,1999). Physical Therapy, 72(1 ), 7 -71 .Gravetter, F. Theprimary concern of Bayesian decision analysis is the choice of a decisionact (Peebles, 1993). The naturalistic model is qualitative in approach,and tends to infer moral guides from empirical findings. Yet another definition of statistical analysis is that it is"a set of methods that are used to collect, analyze, present, and interpretdata" (Hoover, 2 1, p. > The data are measured and/or analyzed using a nominal scale of measurement. Quantitative analysis can identify what has happened or ishappening. H. (1995). (1992, November). L., & Well, A. Tothe contrary, Bayesian statistical analysis purposefully incorporatesinformed judgments into the analyses of data (Peebles, 1993).
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