Instructors of introductory and intermediate statistics courses often teach the use of analysis of variance (ANOVA) for the purpose of comparing more than two group means and pairwise comparison procedures (PCPs) to determine which group means differ from one another following a statistically significant ANOVA test. SPSS provides 18 PCPs. The purpose of this study was to determine which PCP has the best power and maintains Type I error control both when assumptions of equal sample size and equal variance are met and when they are violated, so as to provide a single resource for use in introductory and intermediate applied statistics courses. Testing the PCPs with simulated data revealed that only the four tests developed to be used under assumption violations adequately controlled Type I error, so we recommend using one of these procedures. Power results were similar for all four of these tests, but were slightly higher for the Games-Howell test than for the others.
James Madison University has used dedicated Assessment Days for more than 30 years to collect longitudinal data on student learning outcomes. Our model ensures all incoming students are tested twice: once before beginning classes and again after accumulating 45-70 credit hours. Although each student completes only four instruments during a 2-hour testing period, 25 different assessments are administered, thereby allowing for the examination of student growth on a variety of different outcomes. This article describes our model and outlines the logistics involved in planning for Assessment Day, including the physical and human resources needed for its success. We also address changes we have made over the years and the challenges we continue to encounter. Our intention is to share lessons learned and encourage readers to consider how our model might be adapted for the assessment of programs both large and small at their own institutions.
ModelsThere are two primary purposes for the use of a random process model. First of all, a model provides a framework for statistical inference, such as estimation or detection. An area of statistical inference which we consider here is discrimination, where a decision is to be made between two or more possible sources for the observed data. Second, a model may needed for simulation, or, specifically, for the generation of synthetic data to be used in computer simulations. Fidelity of the model to the precise statistical nature of the process is more important for simulation than for discrimination. However, for discrimination, tractability of the model probability distributions is usually required.One of the most universally used models is the transformed Gaussian model, where one assumes that if { X ; } is the observed time series, then the series {Zi} where 2; = $-'(XI) is Gaussian. The lognormal process provides a typical example. This approach provides both simplicity (usually resulting in mathematical tractability) and generality. In such a model, one typically measures and matches univariate marginal distributions and autocorrelation. The density function may be evaluated in an especially efficient manner if the Gaussian process is further assumed to be kth-order autoregressive process. If the process {X,} assumes only positive values, then the transformed Gaussian model may not be the most desirable. The linear dependency structure of the Gaussian process becomes distorted by the transformation. An alternative is to assume that the underlying process {Z;} has a distribution other than a Gaussian distribution; in particular, we may assume a x2 distribution for {Zi}. Since the x2 distribution also admits only positive values, the dependency structure is likely to be less distorted by the transformation. Multivariate x2 random variables which are correlated may be generated simply by Zi = U?i + . . . + U:& where each {Uli,z E N} has a multivariate Gaussian distribution. The transformed x2 model yields a dependency structurewhich differs significantly from the transformed Gaussian model, even when the autocorrelation is the same. Although the density involved in the x 2 model is more difficult to evaluate than the transformed Gaussian density, with certain assumptions, the transformed x2 density may also be efficiently computed. Discriminat ionIf a transformed Gaussian model is assumed, where the Gaussian process is a kth-order stationary autoregressive process, then the LRT for any discrimination problem may be evaluated as a sum of k-step memory nonlinearities. That is, we have n Tn = gO(x1,. . . > Z k ) + g(xj,. . x j -k ) . j = k + lwhere Tn is the log-likelihood ratio Tn = log(f,(")/fi")). This result is obtained by using the factorization Q = LLT of the inverse of the covariance matrix, where L is a lower triangular matrix with L,, = 0 for 12 -j l > k. Thus the likelihood ratio (LR) may be efficiently evaluated. The method works equally efficiently for testing multiple hypotheses, as in pattern recogni...
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