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Title: An Association Rule General Analytic System (ARGAS) for hypothesis testing and data mining applications
Authors: Frederick Parente1,2  JohnChristopher Finley1,3  Christopher Magalis, M.A.1,4
Affiliations: 1Department of Psychology, Towson University, 8000 York Rd. Towson MD, 21252, United States of America
Contact email: 2Corresponding author: HYPERLINK "mailto:FParente@towson.edu" FParente@towson.edu
3Second author: HYPERLINK "mailto:jfinle2@students.towson.edu" jfinle2@students.towson.edu
4Third author: HYPERLINK "mailto:cmagalis@towson.edu" cmagalis@towson.edu
Journal: Methodological Innovations
ABSTRACT:
This paper describes an Association Rule General Analytic System (ARGAS) as an alternative to the General Linear Model (GLM) for hypothesis testing. We illustrate how the ARGAS model can be used to analyze a variety of common research designs. Specifically, this study demonstrates how ARGAS can be used to analyze: canonical analyses, multivariate and univariate group comparisons, multiple predictions, bivariate predictions, and two group comparisons. The advantage of the ARGAS approach is that it can test hypotheses with any design that is commonly analyzed with the GLM. It also does not have any distributional assumptions. The ARGAS calculations are welldeveloped and there are a variety of computer software applications available that expedite the computations. The current study provides a framework of how ARGAS can be applied to these designs and how to interpret the results.
Keywords: ARGAS, GLM, Hypothesis, Supervised, Unsupervised
Introduction
The general linear model (GLM) is a commonly used data analytic system that can accommodate a variety of different research models (Christensen, 2011; Nelder & Baker, 1972). One category of research involves predicting one set of variable(s) from another, and another examines mean differences among groups. Although the GLM is the most ubiquitous data analytic system for testing hypotheses, it is not without limitations. Perhaps the biggest problem with the GLM is the fact that it is based on restrictive assumptions, e.g. homogeneity of variance, normality of distributions etc. Unfortunately, these assumptions are not always evaluated before applying one of the variants of the GLM. Although there are transformation procedures available for modifying data to meet these assumptions, they are not commonly used nor do they always achieve their goal (Hutcheson & Sofroniou, 1999).
Another limitation to the GLM is that significance testing is generally based on a test of the Pearson correlation coefficient which is used most often to evaluate linear trends. Many applications of the GLM are designed to detect differences in the arithmetic means of different conditions in an experiment. However, the means are hypothetical values which may not actually exist in the data. Because the GLM is designed for numerical analysis, it is incapable of assessing the nuances of other types of data such as words. Given these issues, it is reasonable to pursue alternative methods of data analyses that can generalize across a variety of research contexts and that are immune to the distributional limitations of the GLM.
Association Rules
We propose that the Association Rule General Analytic System (ARGAS) that provides an alternative to the GLM for analyzing a variety of different research designs. The advantage of the ARGAS approach is that it does not have any distributional assumptions and it does not examine differences between hypothetical mean values. It does evaluate specific patterns of differences or nonlinear relationships among the variables (Han, Pei, & Kamber, 2011). Perhaps, the primary drawback to using the ARGAS as a general data analytic model is the dearth of published research that details its use and interpretation of the results it provides. The goal of the current study is to provide a general context for the ARGAS approach and to illustrate how it can be used as a data analytic tool for hypothesis testing. As a prelude to this discussion, we begin with a basic description of association rule analysis.
The concept of Association Rules is often explained via a market basket analysis example, which is used by many retailers (Agrawal & Srikant, 1994; Webb, 2003). Each persons purchases can be conceptualized as a vector of items which are amalgamated into a larger data base of words that represent each consumers collective purchases. This larger data structure can be analyzed in aggregate to extract association rules that derive from the frequency with which two or more items are purchased together. The analysis allows the retailer to predict what products will be purchased along with others. Though, initially intended for marketing, association rules have been recently applied to other areas of research, such as measuring clustering of words and symbols in memory (Parente & Finley, 2018). However, there have been few other applications of ARGAS outside of marketing parlance.
ARGAS measures the joint probabilities that predict the cooccurrence of two or more items. Joint probabilities can be divided into independent and dependent types. The independent joint probability model assumes that the occurrence of one event is independent of the occurrence of others. Computing independent joint probabilities amounts to multiplying the probability of occurrence of one item times that of another, e.g., p(A) x p(B). Three and four way probabilities are simply the multiplication of the various probabilities in sequence, e.g., p(A) x p(B) x p(C), etc.. ARGAS modeling however, is based on dependent probabilities which assume that the occurrence of one event affects the occurrence of others. Computing dependent joint probabilities is therefore much more complex. For example, recall of item A given item B is P(AB) = P(A&B)/P(B).
Conventional ARGAS analyses provide a variety of measures that associate the antecedent and consequent items. Most of these statistics are hybrids of the dependent conditional probability computations mentioned above (Balcazar & Dogbey, 2013; Parente & Finley, 2018). The various measures are different ways to assess the associative relationships among a set of variables. The measure of Support is an index of the proportion of cases in the data that include both or all of the antecedent and consequent values under consideration. Coverage is a measure of how extensively a given item occurs in the antecedent portion of the rule. Confidence is a conditional probability of recalling item B given that the person also recalls item A. The Lift measure is an index of the predictive value of the rule relative to using no rule at all. Leverage measures the extent to which two items are recalled together versus what would be expected if the items were recalled independently.
Association rules have been welldocumented for use with nonnumerical data, e.g., words that represent purchases in the market basket example. However, the model can also be applied to data with words that represent numerical measures (e.g, abovethe median or belowthemedian). Yet, there is little research using association rules with numerical data that has been transformed in this manner (Agrawal, ImieliDski, & Swami, 1993; Imberman & Domanski, 2001). Moreover, there is literally no published research addressing the issue of how association rule analyses could be used to test research hypotheses once the data has been transformed into words.
The following paragraphs provide a description of different applications of ARGAS outside of the market basket framework. As such, this paper will explain how the ARGAS can be used in lieu of the GLM. Table 1 presents a breakdown of several common research paradigms. The left side of the table identifies common GLM analyses that are appropriate for use in that paradigm. The right side of the table presents the alternative ARGAS alternative analysis that would be appropriate once the numerical data has been transformed into words. In the following simplified examples, the authors simply transformed the data to either abovethemedian or belowthemedian. However, it is reasonable to suggest that a more specific breakdown may be more appropriate with different data sets.
Comparison of ARGAS and the GLM
One of two broad categories of analysis is summarized in Table 1. The first category (see Table 1) includes supervised analyses for which the antecedent (predictor or independent variables) and consequent (outcome or dependent variables) are specified in advance. These analyses are referred to as supervised because the experimental design is constrained to assess relationships between the two sets of variables. A description of unsupervised models in which each variable is considered both an antecedent and consequent is described in detailed in a separate paper.
Table 1. GLM vs. ARGAS Approaches to Hypothesis Testing
GLM ARGAS
Canonical Analysis
Multiple Predictors  Multiple Outcomes Multiple Antecedents Multiple Consequents
e.g., Canonical Correlation Analysis e.g., Supervised learning of associations between antecedents and consequents.
Multiple Prediction
One or more predictors One Outcome Multiple antecedents one consequent
e.g. Multiple Regression e.g., supervised learning of associations
among antecedents and a single consequent.
Multivariate Group Comparison
One or more Independent Variables  Multiple Groups Multiple Consequents
Multiple dependent variables e.g. Supervised learning of association between
e.g., MANOVA antecedents and consequents.
Univariate Group Comparison
One or more Independent Variables Multiple Groups One consequent
One dependent variable e.g., Supervised learning of associations
e.g. ANOVA among group labels and a single consequent.
Bivariate Correlation
One predictor One outcome One antecedent One Consequent
e.g. Bivariate Correlation e.g. Supervised learning of associations
between a single antecedent and a single consequent.
Two Group Comparison
One Independent Grouping Variable One Antecedent Grouping Variable 
One outcome measure One Consequent
e.g., ttest e.g., Supervised learning of associations between group name and a single consequent.
We begin with a discussion of how ARGAS can be used to analyze canonical data structures in which there are multiple antecedent and multiple consequent variables. We also provide an example of the use of ARGAS as an alternative to conventional multiple regression analysis in which there are multiple antecedents and a single consequent variable. We next discuss how these two analyses can be modified to accommodate multivariate and univariate group comparison designs. In each of these examples we provide a comparison of the ARGAS with the corresponding GLM procedure regarding interpretation of the results.
Canonical Analysis
When analyzing numerical data, the canonical analysis assesses the relationship between a set of predictors and a set of outcome variables. The research hypothesis is that one or more of the predictors will predict one or more of the outcomes. The analysis provides an overall measure of association (canonical correlation) as well as indices of association between the individual predictors and outcomes with one another. In essence, the canonical correlation is a Pearson product moment correlation between the linear combination of the predictor variables and a corresponding linear combination of outcome measures. Specifically, the set of predictor scores is collapsed into a weighted combination (called a variate) which is correlated with the outcome variate. The canonical correlation is, therefore, a bivariate correlation between these two variates. A significant canonical correlation indicates that the set of predictors and the set of outcomes are significantly correlated. The analysis also yields weighting coefficients (called loadings) that are indices of the extent to which each individual variable in the predictor and outcome sets contributes to its variate and another set of weights (called cross loadings) that are indices of how well the same variable predicts the opposite variate.
To illustrate this process, we performed a canonical analysis on a public domain data set that assessed the correlative relationship among several student/teacher evaluations. This data set included student evaluations for entire classes of students from 50 universities. The unit of measure was the average rating for several different measures. The purpose of the study was to assess the relationship between the set of predictors that included: 1.) the students perception of the quality of the exams, 2.) the average grade in the course 3.) the enrollment in the course and 4.) the perceived knowledge of the instructor. The set of outcomes included: 1.) the perceived teaching quality and 2.) overall evaluation of the course. Half of the variables met the distributional assumptions of the GLM, whereas the remaining variables did not, even after transforming the variables.
The canonical correlation analysis revealed a significant Rsqr (.782) between the predictor and outcome sets. The standardized coefficients for the individual predictors indicated that two variables (i.e., perceived exam quality and perceived knowledge of the instructor) significantly predicted the outcomes (i.e., overall evaluation and teaching competence). Redundancy analysis indicated that predictors accounted for 55% of the variance in the outcome set and about 30% of the variance in the predictor set.
The same data were then evaluated to derive association rules that related the same predictors: Enrollment, Exam quality, Average Grade in the class, and Perceived Knowledge as antecedents with: Perceived Teaching Ability and Overall Quality of the course as consequents. This was a supervised analysis because the variables were first split into antecedent and consequent sets and the association rules were restricted to the relationships between these two sets. The individual variables were first split at their respective medians and the data points for each variable were recoded into words that identified each value as either above or belowthemedian. For example, the word OVAbove indicated that any particular value for the Overall evaluation variable was abovethemedian for that variable. The word ExmBelow indicated that the numerical value for perceived Exam Quality fell belowthemedian for that variable. It is reasonable to use other labeling schemas and a more specific transformation of the numeric variables (e.g., quartiles instead of a median split); however, dichotomizing at the median was used in this example and those that follow to simplify the explanation of the ARGAS process.
The analysis began with a random segregation of the data into training and verification samples of equal size. The software (Magnum Opus; Webb, 2007) was configured to search for the rules with the highest lift values and to select only those that were significant (p < .05). This analysis identified six rules that replicated in the training and verification samples. A comparison of the results of the two analyses is presented in Table 3, below:
Table 2. Comparison of canonical correlation and ARGAS analysis results.
Canonical Correlation ARGAS
Model Fit R = .782 Six significant rules
Significant
Predictors: Exams, Knowledge Knowledge, Exams, Enrollment
___________________________________________________________________
Canonical Analysis and ARGAS Results
Six binary rules (one antecedent and one consequent) related the antecedents to the Overall variable. High Perceived Knowledge or High Enrollment predicted a high Overall rating. Low Perceived Knowledge or Low Enrollment ratings predicted low Overall ratings. The analysis also indicated a direct relationship between Exam Quality and Teaching Competence. High Exam Quality predicted High Teaching Competence. Low Exam Quality predicted low Teaching Competence.
These results are generally in agreement with the canonical correlation analysis. Both analyses revealed a significant overall relationship between the predictor/outcome variables or, alternatively, the antecedent/consequent variables. The canonical correlation analysis relied on the correlation between the variate sets as a measure of an overall relationship, whereas the overall measure of association in the ARGAS analysis was the number of significant rules that associated the antecedent with the consequent variables. Each analysis also showed significant relationships between the Exam Quality and Perceived Knowledge predictors and the Overall Quality and Teaching Quality outcomes
The results, however, differ in several ways. First, the results of the ARGAS analysis indicate that Enrollment was also a significant predictor of the consequent variables which was not identified in the canonical analysis. Second, several of the numerical variances violated the assumptions of the GLM whereas the ARGAS analysis was not constrained by these assumptions. Third, the canonical analysis assumes a linear relationship between the predictors and the outcomes whereas the ARGAS model does not make this assumption.
Multiple Regression
The analyses described above are designed for canonical data structures in which multiple predictors are related to multiple outcomes. There are, however, several modifications of the canonical model that are appropriate with other types of experimental designs. For example, Multiple Regression Analysis relates multiple predictors to a single outcome. By analogy, the ARGAS procedure with multiple antecedents and a single consequent would also be appropriate for this type of analysis. The number of significant association rules between the antecedents and consequent would reflect the overall relationship among the variables. This statistic is analogous to the multiple Rsquare, which is the percent of variance in the outcome measure that can be accounted for by the predictors. The beta weights for each predictor in the multiple regression analysis are an index of the predictive utility of that variable. The lift values for the antecedent variables in the ARGAS analysis are indices of the value of that variable as a predictor.
Multiple Regression and ARGAS Results
The data set that was used in the canonical analysis above was reanalyzed using multiple regression procedures and the corresponding ARGAS analysis of the same antecedents predicting the Overall variable. Although the multiple Rsquare of .755 for the regression analysis was significant, F(5,44) = 27.184, p < .05, Average Grade, Enrollment, and Exam Quality did not pass tests of normality. Only two of the variables, Teaching Competence and Instructor Knowledge, were significant predictors.
Table 3 compares the results from the two analytic models. The ARGAS analysis of the same variables began by randomly selecting half of the cases to be used as a verification sample. The remaining cases were used as a training sample. The ARGAS analysis was then performed on both data sets to determine which rules replicated. The analysis generally indicated a significant and replicable relationship between the antecedent variables and the Overall consequent variable. Six rules replicated in the training and verification sample.
Table 3. Comparison of multiple regression and ARGAS data analyses results
Multiple Regression ARGAS
Model Fit Rsqr = .755 Six significant rules
Significant predictors Perceived Knowledge (a) Perceived Knowledge (a)
Teaching Competence (b) Teaching Competence (b)
Enrollment (c)
Effect size Beta Weights
(a) = .622, b (b) = .325 Lift values (a) = 1.4
(b) = 1.8, (c) = 1.6
Both analyses revealed that the Overall evaluation was directly related to Teaching Competence, and Perceived Knowledge. The beta weights for the regression model indicated that Perceived Knowledge was a better predictor than Teaching Competence. The ARGAS analysis also identified a significant relationship between the antecedents and the consequent variables however; the Lift values suggest that Teaching Competence was the best individual predictor. In addition, the ARGAS analysis indicated that that Enrollment was also a significant antecedent.
Group Comparison Analyses
The ARGAS analysis can also be used for comparing groups. As an example of this procedure, data were collected from three groups of participants who were asked to memorize a list of 12 words after hearing them presented with different procedures. One group (Rehearsal) heard each word repeated twice as the list was read to them. Another group (Control) heard the words presented once. A third group (Imagery) was asked to form a mental image of the words as they heard them presented once. Each persons recall of the words was tested immediately and again after a half hour delay. The purpose of the experiment was to assess the effect of rehearsal and mental imagery on immediate and delayed memory relative to a control condition that received neither.
With three independent groups and two dependent measures, conventional Multivariate Analysis of Variance procedures computed on these data indicated a significant overall difference among the groups Wilks Lambda = .184, F(4,60) = 184.5, p < .05, h2 = .297, power = 1.0. However, the assumption of equality of variance was violated for both dependent variables. Indeed, all but one of the participants in the imagery group recalled all twelve items immediately and ten of them after the delay interval. All but 2 participants in the Rehearsal group recalled all 12 items as well. Average recall for the Rehearsal and Imagery conditions was higher than occurred in the Control condition but with much less variability. Participants in the Imagery condition produced slightly better recall relative to the Rehearsal condition and much better recall relative to the Control.
The ARGAS analysis of the same data began with splitting the larger sample in half producing a training and verification sample. The immediate and delayed memory variables were then each split at the median in each sample and the ARGAS was applied to the words that identified each data point as either abovethe median or belowthemedian for that consequent variable (immediate and delayed recall). The antecedent variable was a list of words that identified group membership (Rehearsal, Control, and Imagery). The consequent variables were lists of words that identified individual data points as either above or below their respective medians.
The results of an overall ARGAS analysis produced four rules that associated the antecedent (rehearsal, control, imagery) and consequent variables (immediate and delayed recall). Each of these rules was significant in both the training and verification data sets. These results are presented in Table 5, below:
Table 5. ARGAS rules that predict relationships among antecedent and consequent variables
Group Rule
Control Below median for immediate recall
Control Below median for delayed recall
Imagery Above median for immediate recall
Imagery Above median for delayed recall
__________________________________________
The fact that there were four significant rules indicates an overall difference among the groups, i.e., the group membership antecedent predicted the consequent variables (immediate and delayed recall). The individual rules showed that Imagery produced an immediate improvement in recall and the effect persisted with the delayed variable. The rules also indicated a significant relative decrease in recall in the control condition for both the immediate and delayed recall variables. Separate two group analyses that compared the Imagery and Rehearsal conditions did not produce any significant and replicated rules that that differentiated the Imagery versus the Rehearsal conditions. The results of the ARGAS analysis indicated that mental imagery produced significantly better immediate and delayed recall relative to doing nothing at all.
Bivariate Correlation
The Pearson Product Correlation is one appropriate method for assessing the linear relationship between two numerical variables. However, the correlation model requires meeting restrictive assumptions that are often violated which limits the interpretation and generalizability of the results. For example, it is commonly assumed that average grade in a course is directly related to the overall evaluation. We then tested this hypothesis with a conventional Pearson correlation which produced a significant correlation (r (48) = .31, p < .05) supporting the hypothesis. The distributions of average grade were not normally distributed and significantly skewed even after transformation. We therefore applied the ARGAS technique with average grade as an antecedent and overall evaluation as a consequent.
This analysis did not produce a significant rule structure that replicated in both the training and verification samples. To further investigate the relationship described by the Pearson bivariate correlation, the analysis was rerun using the same split samples that were used in the ARGAS analysis. In the training sample, the correlation of .46 was significant (p < .05), however, the same correlation was not significant in the verification sample (r = .17, p > .05). The inconsistencies in these analyses indicate that the relationship between student evaluations and average grade in the course was not replicable. Although the initial analysis of the total data set showed a significant correlation, when the sample was split, even though each split sample had adequate power, the correlation did not replicate across the training and verification samples. The ARGAS analysis also did not identify any significant and replicable rules in the training or verification samples. It is likely that the original correlation of .41 that obtained with the total sample was not reliable.
Two Group Comparison
In conventional linear modeling, a ttest is often used assess the relationship between a continuous dependent measure and a dichotomy which represents a classification, e.g., male versus female, experimental versus control. Again, in this type of analysis, violated assumptions often do not permit a meaningful interpretation of the results. For example, in the MANOVA example presented above, the researchers performed post hoc two group comparisons of the imagery vs control, and the rehearsal vs the control. With two dependent variables (immediate and delayed recall) this analysis is analogous to a multivariate ttest otherwise known as Hotellings Tsquare. As such, when using the same design with each individual outcome, the analysis devolves into a univariate ttest. For example, individual ttests performed on the immediate and delayed recall data for the comparison of the imagery and control conditions revealed significant differences for each dependent variable. For the Immediate Recall variable, the t statistic of 4.39 with 62 df was significant (p < .05). Likewise, for the Delayed Recall variable, the same analysis also produced a significant t value (t (62) = 3.52, p < .05). However, tests of the assumptions of the t statistic using the Levenes test were also significant which indicated that the ttest was invalid.
Two Group Comparison and ARGAS Results
When using the ARGAS method, the analysis can still be performed, thereby producing the following results for the immediate and delayed variables when comparing the Rehearsal and Imagery conditions (Table 5).
Table 4. Comparison of Ttests and ARGAS data analyses results
Antecedent Comparison Significant Consequent
Imagery Above Median Immediate Recall
Imagery Above Median Delayed Recall
___________________________________________________
These rules indicate a significant difference between the Control and Imagery conditions for both Immediate and Delayed Recall. Specifically, Imagery produced significantly more abovethemedian words relative to the Control group for both consequent measures.
Discussion
The purpose of the present study was to illustrate the use of association rule modeling as a general data analytic system for hypothesis testing. Because the GLM is constrained by several statistical assumptions including: linearity, homoscedasticity, normality and random error terms, adequate power; statistical conclusions depend on the extent to which these assumptions are met (Weathington, Cunningham, & Pittenger, 2010). Often, however, these assumptions are either overlooked or ignored. It is therefore reasonable to suggest that the validity of conventional GLM results may be compromised if any of the assumptions are not appropriately evaluated. Our purpose was to demonstrate an alternative to the GLM which can be used to test univariate and multivariate hypotheses and which are not subject to the usual restrictive assumptions listed above. The corpus of this paper illustrated how these analyses would progress and how a researcher would interpret the results. We continue below with a discussion of advantage/disadvantages of the ARGAS and considerations when using the system along with a discussion of other applications of ARGAS.
We have discussed several advantages that make ARGAS a useful alternative the GLM. Perhaps the biggest advantage is that it does not have the same restrictive assumptions that plague the GLM. Another advantage is that the ARGAS, as a general analytic system, can be used to test hypotheses with a variety of different supervised research paradigms in which the system associates a set of specific antecedents with a specific set of consequent variables including: 1.) canonical relationships with multiple antecedents and consequents 2.) multiple regression designs which include multiple antecedents and a single consequent 3.) multivariate and univariate group comparison designs, 4.) simple bivariate relationships with one antecedent and one consequent, 5.) simple two group comparisons with a single outcome variable. Finally, the results of these analyses are not restricted to linear relationships.
Interpretation
Generally, the number of significant rules is an index of the strength of the relationship between the antecedent and consequent variables. The individual rules provide information about specific associations that exist within the antecedent and consequent variable sets. Those variables that are identified by significant association rules should be maintained whereas, those that do not produce a significant association can be eliminated without lessening the overall association between the antecedent and consequent variable sets. These rules can then be broken down further using cross tabulation procedures to assess specific relationships among the variables.
One difference when interpreting ARGAS is the possibility that an association rule may not be symmetrical. This means that a rule may associate abovethemedian scores for an antecedent predicting above or belowthemedian values for a consequent. However, the obverse relationship may not be significant. These asymmetrical relationships may be difficult to interpret. Generally, however, asymmetrical rules suggest that the relationship between the variables exists only in one end of the data distributions. For example, a rule that describes purchase preferences for persons with above and belowthemedian income for a highend BMW versus a Toyota Corolla may indicate that persons with abovethemedian income purchase significantly more BMWs. However, those with belowthemedian income are not necessarily more likely to purchase a Toyota Corolla versus any number of other low cost alternatives. Our experience suggests that asymmetric relationships occur in situations where the original numeric data distributions are highly skewed.
Another interpretation difficulty concerns the meaning of complex rules. For example, it is possible that multiple antecedents can predict a single consequent or multiple consequents. These relationships may be difficult to interpret and will obviously depend on the theories that are being tested. We have also found that more complex rules are less likely to replicate in both the training and verification samples.
We have used the Lift measure as our primary index of association however; there other measures such as Strength, Confidence, Leverage, and Coverage may also be useful for interpreting the results. The Lift value is analogous to a beta weigh in GLM parlance. We chose the lift measure because it is relatively easy to interpret. A value of 1.0 indicates that using the rule is no better than not using the rule. Associations with the largest Lift values are the most replicable.
Interpretation of relationships identified by the association rules can be facilitated with cross tabulation procedures. Conventional measures of association significance such as chi square and other measures of association and cross tabulation procedures for the median split variables can be used to break down significant relationships that are identified by the association rules. These analyses are perhaps most appropriate for simple rules and less useful for interpreting complex rules. Nevertheless, these additional tests of significance computed on the cross tabulations may be especially useful because they are generally familiar to most researchers.
All of the ARGAS analyses reported here included a training and verification sample. Simple logic dictates that split sample verification is desirable in any data analysis. Specifically, if an effect does not replicate with a verification sample that is selected from the same population that spawned the training sample, then the reliability of the effect is questionable. We suggest generating enough data so that the data set can be split into at least a training sample and a verification sample. With the GLM, each sample should include enough data to ensure adequate power. In our experience, an adequate sample has at least twice as many cases as variables and is seldom less than 100. Whenever possible, the original sample should be subdivided into more than two smaller verification samples to identify rules that consistently replicate.
We used the median split procedure to create the word transformations for the various numeric variables. Because there is no published literature that uses association rule modeling to test research hypotheses, it is impossible to speculate about other variable splitting procedures that may provide better interpretation of the effects. For example, splitting the variables into quartiles or even deciles may produce a clearer picture of the results. It is therefore necessary to investigate this issue in future research.
10.1 Other Applications
This paper provided examples of how the ARGAS system can be to test research hypotheses. Clearly, there is a larger argosy of issues that still require research and development; for example, power analysis, simple effects tests, complex experimental designs, covariance analysis etc.. Moreover, the ARGAS model is also more general that has been portrayed here. For example, the system can be used for unsupervised learning applications where each variable in the data base is considered both an antecedent and a consequent and the purpose of the analysis is to derive association rules that exist among the entire variable set. This application of ARGAS is analogous to conventional clustering algorithms in the GLM. The ARGAS analysis produces clusters of variables that share common association similar to those derived from factor analysis, cluster analysis, or multidimensional scaling. For example, we have used this technique as a validation tool with questionnaire data. The analysis provides groups of questions that are associated with one another just as factor analysis groups items into related factors or components. The ARGUS method however is free of correlational assumptions and can detect relationships that are not linear. The technique can also be used with a transposed data matrix that would cluster participants into related types.
Another application involves use of ARGAS for time series analysis in which lagged variables are used to predict the future values of a variable. ARGAS can identify rules that associate specific lagged variables that are significant antecedents of the future values of the time series. For example, we have compared this application of ARGAS with conventional autoregression procedures for predicting temperature and humidity data from the national weather service. Both models identified six significant lags. The autoregression created an equation that was used to predict future values of the series. The researcher may then plug in values for the lagged variables and equation would generate the predicted future value of the series. The ARGAS model produced a set of rules, e.g., abovethemedian on lag 3 > abovethe median on the series variable. The researcher would then know that a value for the current day that is abovethemedian would predict abovethemedian score three days hence. Using all of the rules would produce a more accurate prediction. Although the autoregression forecasting approach is welldeveloped and generally accepted, it is also prone to all of the same distributional assumptions that plague any GLM model. It is especially prone to violation of the assumption of independence of error terms and it is not typically used for multivariate predictions. By contrast, the ARGAS method is distribution free and is capable of handling multivariate time series (e.g., temperature, humidity etc).
Yet another application of ARGAS concerns use of the model for qualitative research. These types of studies require participants to generate unique word descriptors that describe their experience with different treatment conditions. For example, St. Pierre and Parente (2017) investigated mock jurors perception of guilt and innocence for hypothetical crimes committed by defendants who were described having sustained either a severe traumatic brain injury (Severe TBI), a Mild TBI, or NoTbi. Participants were asked to generate their own words that described their perception of the defendant and the ARGAS was computed on these words to determine rules that discriminated defendants with Severe, Mild or No TBI. Significant rules that predicted severe TBI were: cognitively impaired, compulsive, irritable. The words frustrated and depressed predicted Mild TBI, and healthy, impulsive and murderer predicted No TBI. The rules indicated that participants had very different attitudes towards persons with different levels of brain injury. Those with severe TBI were viewed as cognitively impaired whereas those with mild TBI were seen as emotionally impaired although those without TBI were viewed as healthy but more likely to be labeled as murderers.
Finally, the ARGAS model can be used for data mining applications. For example, the canonical research design outlined above can be extended to include literally hundreds of potentially related variables. Given that the data base is large enough, the ARGAS can be used to identify significant rules that associate the antecedents and consequent variables. The goal in this type of study is to generate hypotheses that can be later tested with well controlled studies. The analysis can also serve as a data reduction tool allowing the researcher to extract from a large data base only those variables for which the association is not only significant but also replicable.
Although the ARGAS procedure has a number of advantages relative to the GLM, it is not without its problems. However, most of these issues are the result of lack of research into questions regarding, for example, power, underlying assumptions, overlearning without verification sampling, and methods for using the derived models for prediction. These questions may take quite a while to answer. In the meantime, we assert that the ARGAS model is ready for application to common research designs and may be especially useful in situations where convention GLM analyses may not be appropriate.
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