Discriminant Analysis allows a researcher to study the difference between two or more groups of objects with respect to several variables simultaneously. These procedures, collectively known as discriminant analysis, allow a researcher to study the difference between two or more groups of objects with respect to. functions, classification functions and procedures. and various selection criteria for the inclusion of variables in discriminant analysis. Professor. Klecka derives.
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The use of stepwise methodologies has been sharply criticized by several researchers, yet their popularity, especially in educational and psychological research, continues unabated. Stepwise methods have been considered particularly well suited for use in regression and discriminant analyses, however their use in discriminant analysis predictive discriminant analysis and descriptive discriminant analysis has not been the direct focus of as much written commentary.
Viscriminant, predictive discriminant analysis and descriptive discriminant analysis are discussed in general, and then their relevance with respect to stepwise techniques is examined.
There are several problems associated with the use of stepwise methods. Stepwise methods hold out the promise of assisting researchers with such important tasks as variable selection and variable ordering. However, the promise is almost always unfulfilled and researchers are cautioned against using stepwise methodologies. Some alternatives to the present use of stepwise methods are discussed.
Huberty stated that discriminant analysis DA includes a set of response variables and a set of one or more grouping or nominally scaled variables. As Huberty ‘ s analusis and Klecka ‘ s prerequisites in the analhsis paragraph imply, discriminant analysis has two analysia of techniques based on the purpose of the analysis, i.
In fact, anyone who is familiar with the basic goals and techniques of multiple regression can easily understand the association between multiple regression and discriminant analysis. Unfortunately, as Kachiganp. This practice of squandering variance information, by dichotomization or polychotomization disriminant continuous variables, has been strongly criticized in the viscriminant Kerlinger, ; Thompson, Despite the close association between DA and MR, it is klekca to note that some researchers have recognized that all parametric procedures can be derived from the same linear model which involves the use of least squares weights Cohen, ; Knapp, As Knapp noted, ” virtually all of the commonly encountered parametric tests of significance can be treated as special cases of canonical correlation analysis, which is the general procedure for investing differences between two sets of variables ” p.
Thompson pointed out that every parametric procedure involves the creation of a synthetic score s for each individual on some latent construct. In discriminant analysis the synthetic scores are annalysis discriminant scores created with the discriminant function coefficients Pedhazur, A researcher must make choices about the variables that will be involved in an analysis.
Oftentimes, the researcher may want a to select a subset of variables from the original set or b to determine the relative importance of the set of analyiss even if no variables are to eliminated. Some researchers erroneously believe that stepwise methods can be used to accomplish either of these tasks Huberty, Several researchers Huberty, ; Huberty, ; and Thompson, have noted the common use of stepwise analyses.
According to Thompsonp” stepwise analytic methods may be among the most popular research practices employed in both substantive and validity research. Snyder, have advanced strong arguments against the use of stepwise methodologies.
A discussion ,lecka the problems associated with stepwise methodologies in discriminant analysis is best understood with a basic understanding of discriminant analysis itself.
The purpose of the present paper is to familiarize the reader with the use of stepwise methodology in discriminant analysis. Therefore, a brief history of DA and a description of discriminant analysis is offered first. Third, stepwise methodologies, as applied to DA, and the inherent problems in their use are discussed.
Last, a number of alternative suggestions to the use of stepwise procedures are offered. The ideas associated with discriminant analysis can be traced back to the s and work completed discrominant the English statistician Karl Pearson, and others, on intergroup distances, e.
STEPWISE METHODOLOGY USE IN DISCRIMINANT ANALYSIS
In the s R. Fisher translated multivariate intergroup distance into a linear combination of variables to aid in discrimihant discrimination. Methodologists from Harvard University contributed much to the interest in application of discriminant analysis in education and psychology in the s and s Huberty, Klecka provided several historical references klec,a deal mostly with early applications of DA.
The two types of discriminant analysis, i. According to Huberty” discriminant analysis for the first three or four decades focused on the prediction of group membership, ” PDA, whereas DDA usage did not appear until the s and ” its use has been very limited in applied research settings over the past two decades. The following sections on descriptive discriminant analysis and predictive discriminant analysis are deliberately limited as regards technical and mathematical descriptions.
The reader is encouraged to consult the numerous texts on DA referred to by Hubertypp. In addition, many texts on multivariate data analysis have sections or chapters on discriminant analysis; however, some of these texts, especially earlier ones, do not make clear distinctions between PDA and DDA.
Predictive discriminant analysis is similar to multiple regression analysis except that PDA is discdiminant when the criterion variable is categorical and nominally scaled. As in multiple regression, in PDA a set of rules is formulated which consists of as many linear combinations of predictors as there are categories, or groups Huberty, For example, a school district might be interested in predicting which pre-kindergarten students are likely to have difficulty learning to read by second grade.
A prediction rule would be generated using such predictors as scores on a kindergarten readiness test, ratings on age at which developmental milestones were reached, family socio-economic status, and gender. Predictor weights for two linear combinations, one associated with each group, are determined Huberty, Two probabilities of group membership can be calculated for subsequent students based on the two linear combinations; the student is assigned to the group with the larger linear combination score.
In predictive discriminant analysis each object will have a single score on the discriminant function in place of its scores on the various predictor variables.
At the same time a cutoff score will be determined such that when the criterion groups are compared with respect to the discriminant scores the errors of classification are minimized Kachigan,p. Table 3 provides an example of a classification table used to report results from an application of a prediction rule. This heuristic provides information about the accuracy of the prediction rule, i. The overall percentage of correct classifications is The percentage of correct classifications must be judged against chance probabilities.
Are our results better than chance? DDA includes a collection of techniques involving two or more criterion variables and a set of one or more grouping variables, each with two or more levels, whose effects are assessed through MANOVA. That is, the roles of the two types of variables involved in a multivariate, multigroup setting in DDA are reversed from the roles in PDA ” Huberty,p. In DDA the total ” between-groups ” association in MANOVA is broken down into additive pieces through the use of uncorrelated linear combinations of the original variables discriminant functions Stevens,p.
As can be seen from the eiscriminant example in Kledka 1, lambda at a given step equals 1 – R 2 and, conversely, Klecks 2 equals 1 – lambda. Once the analysis changes to a DDA with more than 2-groups, the calculations become more complex and are no longer directly analogous to regression results.
Again, Table 2 shows the relationship between DDA structure coefficients and regression structure coefficients for the above mentioned case.
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Although values are not identical and are arbitrarily scaled in the opposite direction, their relative magnitudes within each column are the same. Stevens pointed out that DA makes descriptions parsimonious because 5 groups can be compared on 10 variables, for example, where the groups differ mainly on only two major dimensions discriminant functions.
In DDA linear combinations are used to distinguish groups. If k is the number of groups and p is the number of dependent variables, then the number of possible discriminant functions is the minimum of p and k – 1 Stevens,p.
The linear composites linear discriminant functions, LDFs can be used to identify outcome variable ” constructs or latent variables that underlie the group differences, that is, that underlie the grouping variable effect ” Huberty,p. Huberty stated that ” the predominant method of identifying latent constructs in multivariate analyses–this includes factor analysis and canonical correlation–is to examine correlations between linear composite scores and scores on the individual variables in the composite.
These LDF-variable correlations are often called structurer’s ” p. Aside from the differences in purpose, variable roles, and two aspects of DA, the sampling designs may be also be different Huberty,p. One of the most important differences for the researcher is that of purpose. However, Klecka presented a case that examined Senatorial factions and utilized both procedures.
But, as Huberty and Bartonp. But, generally, research questions are of the descriptive type or of the predictive type; only seldom would both types of questions be addressed in a given research situation. PDA is appropriate when the researcher is interested in assigning units individuals to groups based on composite scores on several predictor variables, i.
The accuracy of such prediction can be assessed by examining ” hit rates ” as against chance, for example. The most basic question answered by PDA is ” given the individuals scores on several predictor variables, which group represents their true membership group? As Huberty and Barton noted with respect to PDA, ” One is basically interested in determining a classification rule and assessing its accuracy “. Some researchers incorrectly use a series of post hoc ANOVAs to investigate statistically significant MANOVA effects, but this is inappropriate since univariate methods can not be used to explore multivariate effects.
According to Huberty” stepwise analysis is believed to have been first advanced by Efroymsonand is fully described by Draper and Smithchap. However, the default settings usually result in a forward selection analysis Huberty,p. Stepwise methodologies have enjoyed popular usage, especially in educational and psychological research settings. Huberty noted the widespread use of stepwise methods in empirically based journal articles.
Thompson suggested that ” stepwise analytic methods may be among the most popular research practices employed in both substantive and validity research ” p. Researchers erroneously use stepwise methods to evaluate the relative importance of variables in a particular study or to choose variables to retain for future analyses.
However, a number of researchers have cautioned against using stepwise methodologies because they fail to achieve the aforementioned two purposes, namely, to evaluate variable importance or to select variables. In addition, there are problems associated with stepwise methodologies in a variety of statistical contexts.
The problems with stepwise methods described below are just as relevant within a univariate context, such as regression, as they are in any multivariate case Moore, SAS, and SPSS, include programs to conduct a ” stepwise multiple regression analysis ” and a ” stepwise discriminant analysis.
Several researchers Huberty,; Snyder, ; Thompson,have highlighted three basic problems inherent in the use of stepwise methodologies, i.
Although these problems are germane to stepwise methodologies in certain univariate cases, e. First, as noted above, incorrect degrees of freedom are used in the calculation of statistical tests for discriminant function analysis by most computer packages that employ stepwise methods.
Although some researchers have challenged traditional interpretations of statistical significance testing Carver, ; Chronbach, ; Cohen,; Meehl, ; Shaver, ; Thompson,they are still part of many analyses. When incorrect degrees of freedom are used the results of statistical tests of significance are systematically biased in favor of spuriously high statistical significance Thompson, Students and researchers should be cautioned against interpreting potentially fallible results commonly generated by computer packages.
Thompson remarked that ” degrees of freedom in statistical analyses reflect the number of unique pieces of information present for a given research situation. These degrees of freedom constrain the number of inquiries we may direct at our data and are the currency we spend in analysis ” p.
In any computerized stepwise procedure the pre-set degrees of freedom are ” one ” for each variable included in the analysis. Thompson drew the analogy of the pre-set degrees of freedom as coins that we can spend to explore out data, or rather, we are charged one degree of freedom for every predictor variable used.
However, at each step all predictors from the original variable set are considered for inclusion. So, at each step the correct number of degrees of freedom should be the same as the total number of variables from the predictor set. If the original number of predictor variables was ten than the correct ” charge ” is ten. In the statistical test of significance, there are three calculations for degrees of freedom, i. The computer packages calculate the correct ” total ” degrees of freedom n In regression the ” explained ” degrees of freedom are erroneously entered as the number of predictor variables i.