Thesis (M.B.A.)--North-West University, Potchefstroom Campus, 2010.

Thesis (M.B.A.)--North-West University, Potchefstroom Campus, 2010.

Covariate quality has been primarily theory driven in propensity score matching with a general adversity to the interpretation of group prediction. However, effect sizes are well supported in the literature and may help to inform the method. Specifically, I index can be used as a measure of effect size in logistic regression to evaluate group prediction. As such, simulation was used to create 35 conditions of I, initial bias and sample size to examine statistical differences in (a) post-matching bias reduction and (b) treatment effect sensitivity. The results of this study suggest these conditions do not explain statistical differences in percent bias reduction of treatment likelihood after matching. However, I and sample size do explain statistical differences in treatment effect sensitivity. Treatment effect sensitivity was lower when sample sizes and I increased. However, this relationship was mitigated within smaller sample sizes as I increased above I = .50.

The goal of this study was to establish empirical benchmarks for Cohen's d in child counseling research. After initial review of over 1,200 child intervention research studies published from 1990 to 2016, 41 randomized clinical trials were identified in which intervention and control groups were compared with children 3-12 years old (N = 3,586). Upon identification or calculation of a Cohen's d for each study, I calculated a weighted mean d by multiplying the effect size of each study by the number of participants in that study then dividing by total number of effect sizes. The weighted mean accounted for study sample size and served as the suggested medium effect size benchmark. Results indicated effect size is impacted in large part by type of reporter, with parents apparently most sensitive to improvement and yielding higher effect sizes overall; teachers relatively less sensitive, perhaps due to difficulty observing change in a classroom setting; and children self-reporting lowest levels of improvement, perhaps reflecting a lack of sufficient measures of child development. Suggested medium benchmarks for Cohen's d in child counseling literature are .70. for parent report, .50 for teacher report, and .36 for child self-report. Small and large benchmarks are suggested based on the use of standard deviations of the mean Cohen's d for each reporter.

This study examined the statistical consequences of employing various methods of computing and cumulating effect sizes in meta-analysis. Six methods of computing effect size, and three techniques for combining study outcomes, were compared. Effect size metrics were calculated with one-group and pooled standardizing denominators, corrected for bias and for unreliability of measurement, and weighted by sample size and by sample variance. Cumulating techniques employed as units of analysis the effect size, the study, and an average study effect. In order to determine whether outcomes might vary with the size of the meta-analysis, mean effect sizes were also compared for two smaller subsets of studies. An existing meta-analysis of 60 studies examining the effectiveness of computer-based instruction was used as a data base for this investigation. Recomputation of the original study data under the six different effect size formulas showed no significant difference among the metrics. Maintaining the independence of the data by using only one effect size per study, whether a single or averaged effect, produced a higher mean effect size than averaging all effect sizes together, although the difference did not reach statistical significance. The sampling distribution of effect size means approached that of the population of 60 studies for subsets consisting of 40 studies, but not for subsets of 20 studies. Results of this study indicated that the researcher may choose any of the methods for effect size calculation or cumulation without fear of biasing the outcome of the metaanalysis. If weighted effect sizes are to be used, care must be taken to avoid giving undue influence to studies which may have large sample sizes, but not necessarily be the most meaningful, theoretically representative, or elegantly designed. It is important for the researcher to locate all relevant studies on the topic under investigation, since selective or even random sampling may bias the results of small meta-analyses.

The present journal article formatted dissertation assessed the characteristics of effect sizes of commonly used statistical techniques. In the first study, the author examined the American Educational Research Journal (AERJ) and select American Psychological Association (APA) and American Counseling Association (ACA) journals to provide an historical account and synthesis of which statistical techniques were most prevalent in the fields of education and psychology. These reviews represented a total of 17,698 techniques recorded from 12,012 articles. Findings point to a general decrease in the use of the tvtest and ANOVA/ANCOVA and a general increase in the use of regression and factor/cluster analysis. In the second study, the author compared the efficacy of one Pearson r2 and seven multiple R2 correction formulas for the Pearson r2. The author computed adjustment bias and precision under 108 conditions (6 population p2 values, 3 shape conditions and 6 sample size conditions). The Pratt and the Olkin-Pratt Extended formulas more consistently provided unbiased estimates across sample sizes, p2 values and the shape conditions investigated. In the third study, the author evaluated the robustness of estimates of practical significance (n2, e2 and w2) in one-way between subjects univariate ANOVA. There were 360 simulation conditions (5 population Cohen's d values, 4 group proportion ratios, 3 shape conditions, 3 variance conditions, and 2 total sample size conditions) for each of three group configurations (2, 3 and 4 groups). Three indices of practical significance (n2, e2, w2) and two indices of statistical significance (Type I error and power) were computed for each of the 5,400, 000 (5,000 replications x 360 simulation conditions x 3 group configurations). Simulation findings for n2 under heterogeneous variance conditions indicated that for the k=2 and k=3 condition Cohen's d values up to 0.2 (up to 0.5 for k=4) tend to produce overestimated population n2 values. Under heterogeneous variance conditions for e2 and w2 at Cohen's d = 0.0 and 0.2, the negative variance pairing overestimated and the positive variance pairing underestimated the parameter n2 but at Cohen's d greater than or equal to 0.5, both the positive and negative variance conditions resulted in underestimated parameter n2 values.

The distributional properties of improvement-over-chance, I, effect sizes derived from linear and quadratic predictive discriminant analysis (PDA) and from logistic regression analysis (LRA) for the two-group univariate classification were examined. Data were generated under varying levels of four data conditions: population separation, variance pattern, sample size, and prior probabilities. None of the indices provided acceptable estimates of effect for all the conditions examined. There were only a small number of conditions under which both accuracy and precision were acceptable. The results indicate that the decision of which method to choose is primarily determined by variance pattern and prior probabilities. Under variance homogeneity, any of the methods may be recommended. However, LRA is recommended when priors are equal or extreme and linear PDA is recommended when priors are moderate. Under variance heterogeneity, selecting a recommended method is more complex. In many cases, more than one method could be used appropriately.

This thesis is submitted to the University of Manchester for the degree of Doctor of Philosophy (PhD). This study developed an experimental small group teaching method in the Realistic Mathematics Education tradition for teaching fractions using models and contexts to year 7 children in Brunei (N=89) whose effectiveness was evaluated using a treatment-control design: the E1 group was given the experimental lessons, the E2 group who was given “normal” lessons taught by the experimenter, and a whole class (E3) group which acted as the control group. The experimental teaching was video recorded and subject to semiotic analysis, aiming to describe the objectifications that realized ‘learning of fractions’ by the groups.The research addresses two research questions:1. How effective was the experimental teaching in helping learners make sense of fractions, with respect to equivalence of fractions and flexibility of unitizing?2. What were the semiotic learning and teaching processes in the experimental group of the RME-like lessons? This study used a mixed method approach with a quasi-experimental design (QED) for the quantitative side, and a semiotic analysis for the qualitative side. Quantitatively, the experimental teachings proved to be relatively effective with an effect size of 0.6 from the pre- to the delayed post-teaching test, compared to the E2 and the control groups.The basic findings pertaining to the semiotic analyses were:a. The mediation of the production of fractions in terms of length, from the production of fractions in terms of the number of parts which led to equivalence of fractions;b. The use of language and gesture help to objectify the equivalence of fractions and the flexibility of unitizing–in some case it involved gesturing to the self;c. The role of the Hour-Foot clock (HFC) as a model in a realistic context; andd. The complexity of the required chains of objectifications reflects the difficulties of the topic.

This dissertation: (a) investigated the degree to which the squared canonical correlation coefficient is biased in multivariate nonnormal distributions and (b) identified formulae that adjust the squared canonical correlation coefficient (Rc2) such that it most closely approximates the true population effect under normal and nonnormal data conditions. Five conditions were manipulated in a fully-crossed design to determine the degree of bias associated with Rc2: distribution shape, variable sets, sample size to variable ratios, and within- and between-set correlations. Very few of the condition combinations produced acceptable amounts of bias in Rc2, but those that did were all found with first function results. The sample size to variable ratio (n:v)was determined to have the greatest impact on the bias associated with the Rc2 for the first, second, and third functions. The variable set condition also affected the accuracy of Rc2, but for the second and third functions only. The kurtosis levels of the marginal distributions (b2), and the between- and within-set correlations demonstrated little or no impact on the bias associated with Rc2. Therefore, it is recommended that researchers use n:v ratios of at least 10:1 in canonical analyses, although greater n:v ratios have the potential to produce even less bias. Furthermore,because it was determined that b2 did not impact the accuracy of Rc2, one can be somewhat confident that, with marginal distributions possessing homogenous kurtosis levels ranging anywhere from -1 to 8, Rc2 will likely be as accurate as that resulting from a normal distribution. Because the majority of Rc2 estimates were extremely biased, it is recommended that all Rc2 effects, regardless of which function from which they result, be adjusted using an appropriate adjustment formula. If no rationale exists for the use of another formula, the Rozeboom-2 would likely be a safe choice given that it produced the greatest number of unbiased Rc2 estimates for the greatest number of condition combinations in this study.

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