Optimal Data Analysis, LLC

Lysozyme levels in gastric juice of peptic ulcer patients were compared against normal controls by *t*-test, finding *p*<0.05. Because standard deviations differed by a factor of two between groups, and were proportional to the means, analysis of natural logarithms was instead deemed appropriate: the resulting *t*-test wasn’t statistically significant. Analyzed by ODA no statistically significant between-group difference emerged, and results obtained for raw data and for natural logarithms were identical because ODA results (i.e., *p* and ESS) are invariant over all monotonic transformations of the data.

Optimal Data Analysis, LLC

Four examiners independently recorded the DMFS (decayed, missing, filled surfaces) scores of ten patients. Inter-examiner correspondence of DMFS scores was evaluated using Pearson correlation and novometric analysis. Whereas essentially perfect correlation models were unable to accurately predict DMFS scores in training analysis, novometric models were consistently perfect in both training and reproducibility analysis.

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Foundational to the ODA algorithm when used with an ordered attribute is the identification of the optimal threshold—the specific cutpoint that yields the most accurate (weighted) classification solution for a sample of observations. ODA models involving a single optimal threshold will henceforth be called “fixed-threshold” models. This note proposes a new “relative-threshold” ODA model for an inter-examiner reliability study in which four examiners independently rate teeth condition for a sample of ten patients: “An important inferential question is whether the rater effects differ significantly from one another” (p. 19). In the original study, analysis of variance showed rater C assigned the greatest mean rating across patients: “The inference is therefore drawn that differential measurement bias exists (i.e., the *k* examiners differ systematically from one another in their mean levels of measurement)” (pp. 20-21). ODA was used to compare the entire response distribution (not only means) between raters. A fixed-threshold model identified no effects. A relative-threshold model tested the hypothesis that, for each observation in the sample considered separately, the rating by rater X will be less than (or equal to) the rating by rater Y. Analysis showed that the distribution of ratings made by rater C was nearly perfectly greater than corresponding (non-discriminable) ratings made by raters A, B, and D. This finding hints of possible development of optimal analogues of multidimensional scaling and facet theory methodologies.

Linden Consulting Group, LLC, Loyola University Chicago & Optimal Data Analysis, LLC

Recent research compared the ability of various classification algorithms [logistic regression (LR), random forests (RF), support vector machines (SVM), boosted regression (BR), multi-layer perceptron neural net model (MLP), and classification tree analysis (CTA)] to correctly fail to identify a relationship between a binary class (dependent) variable and ten randomly generated attributes (covariates): only CTA failed to find a model. We use the same ten-variable N=1,000 dataset to assess training classification accuracy of models developed by logistic discriminant analysis (LDA), generalized structural equation modelling (GSEM), and robust diagonally-weighted least-squares (DWLS) SEM for binary outcomes. Except for CTA, all machine-learning algorithms assessed thus far have identified training effects in random data.

]]>Linden Consulting Group, LLC & Optimal Data Analysis, LLC

Prior research contrasted the ability of different classification algorithms [logistic regression (LR), random forests (RF), boosted regression (BR), support vector machines (SVM), classification tree analysis (CTA)] to correctly fail to identify a relationship between a binary class (dependent) variable and ten randomly generated attributes (covariates): only CTA found no relationship. In this paper, using the same ten-variable N=1,000 dataset, a Weka multi-layer perceptron (MLP) neural net model using its default tuning parameters yielded (area under the curve) AUC=0.724 in training analysis, and AUC=0.507 in ten-fold cross-validation. With the exception of CTA, all machine-learning algorithms assessed thus far have identified training effects in completely random data.

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After any algorithm which controls the growth of a classification tree model has completed, the resulting model must be pruned in order to explicitly maximize predictive accuracy normed against chance. This article illustrates manually-conducted maximum-accuracy pruning of a classification and regression tree (CART) model that was developed to predict the functional capacity of lower limb prosthesis users.

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This study compares linear regression vs. novometric models of the association of education and income for a sample of 32 observations. Regression analysis identified a relatively strong effect (R-squared=56.4), but only 25% of point predictions fell within a 20% band of actual income. Novometric analysis identified a strong effect (ESS=81.7%) which was stable in jackknife validity analysis: the model correctly classified 91.7% of observations earning income less than $12,405, and 90.0% of those earning greater income. For people with an income which is less than the optimal threshold, and for those earning greater income, factors other than the number of years of education influenced earned income.

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In a recent paper, we assessed the ability of several classification algorithms (logistic regression, random forests, boosted regression, support vector machines, and classification tree analysis [CTA]) to correctly not identify a relationship between the dependent variable and ten covariates generated completely at random. Only classification tree analysis correctly observed that no relationship existed. In this study, we examine whether various randomly derived subsets of the original N=1000 dataset change the ability of these models to correctly observe that no relationship exists. The randomly drawn samples were 250 and 500 observations. We further test the hold-out validity of these models by applying the generated model’s logic onto the remaining sample and computing the area under the receiver operator’s characteristics curve (AUC). Our results indicate that limiting the sample size has no effect on whether classification algorithms correctly determine that a relationship does not exist between variables in randomly generated data. Only CTA consistently identified that the data were random.

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This note illustrates the disorder and confusion attributable to analytic ethos whereby a smorgasbord of different statistical tests are used to test identical or parallel statistical hypotheses. Herein four classic methods are used for an application with a binary class (dependent) variable and an ordered attribute (independent variable) measured using a five-point scale. Legacy methods reach different conclusions—which is correct? In absolute contrast, for a given sample and hypothesis novometric analysis identifies every statistically viable model (models vary as functions of precision and complexity) which reproducibly maximizes the predictive accuracy for the sample.

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The adaptability of novometric analysis is illustrated for an example involving three class categories and two ordered attributes.

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