Dissertations / Theses on the topic 'Two sample testing'

Doctor of Philosophy
Department of Statistics
Haiyan Wang
Various tests have been created to compare the means of two populations in many scenarios and applications. The two-sample t-test, Wilcoxon Rank-Sum Test and bootstrap-t test are commonly used methods. However, methods for skewed two-sample data set are not well studied. In this dissertation, several existing two sample tests were evaluated and four new tests were proposed to improve the test accuracy under moderate sample size and high population skewness. The proposed work starts with derivation of a first order Edgeworth expansion for the test statistic of the two sample t-test. Using this result, new two-sample tests based on Cornish Fisher expansion (TCF tests) were created for both cases of common variance and unequal variances. These tests can account for population skewness and give more accurate test results. We also developed three new tests based on three transformations (T[subscript i] test, i = 1; 2; 3) for the pooled case, which can be used to eliminate the skewness of the studentized statistic. In this dissertation, some theoretical properties of the newly proposed tests are presented. In particular, we derived the order of type I error rate accuracy of the pooled two-sample t-test based on normal approximation (TN test), the TCF and T[subscript i] tests. We proved that these tests give the same theoretical type I error rate under skewness. In addition, we derived the power function of the TCF and TN tests as a function of the population parameters. We also provided the detailed conditions under which the theoretical power of the two-sample TCF test is higher than the two-sample TN test. Results from extensive simulation studies and real data analysis were also presented in this dissertation. The empirical results further confirm our theoretical results. Comparing with commonly used two-sample parametric and nonparametric tests, our new tests (TCF and Ti) provide the same empirical type I error rate but higher power.

Magister Artium - MA
Recent advancements in child well-being research have shown an increased interest in the importance of subjective well-being. The development of instruments and scales to measure subjective well-being among children and adolescents is in its infancy. Furthermore, there are few existing cross-cultural studies with child and adolescent populations. Validation of existing measures and cross-cultural comparisons has been identified by a number of researchers as critical in contributing to the international dialogue. In the current study, two measures of subjective well-being (Student Life Satisfaction Scale and Personal Well-Being Index-School Children) are tested among a sample of children in the Western Cape region of South Africa. Noting the diversity of experience between children from different socio-economic status groups in South Africa, the study further aims to determine the extent to which the measures are comparable across socio-economic status groups. Data from the Children’s World Survey were used; and includes a sample of 1004 children randomly selected from 15 schools within the Cape Town Metropole. Confirmatory factor analysis was used to test the overall fit structure and multi-group factor analysis, with Scalar and Metric invariance constraints. The results show appropriate fit structure for the overall model, with Scalar and Metric factor invariance tenable across socio-economic status groups. The overall findings suggest that the two measures are appropriate for use with children from low and medium socio-economic status groups in the Western Cape province of South Africa.

In many hypothesis testing problems such as one-sample and two-sample test problems, the test statistics are degenerate U-statistics. One of the challenges in practice is the computation of U-statistics for a large sample size. Besides, for degenerate U-statistics, the limiting distribution is a mixture of weighted chi-squares, involving the eigenvalues of the kernel of the U-statistics. As a result, it’s not straightforward to construct the rejection region based on this asymptotic distribution. In this research, we aim to reduce the computation complexity of degenerate U-statistics and propose an easy-to-calibrate test statistic by using the divide-and-conquer method. Specifically, we randomly partition the full n data points into kn even disjoint groups, and compute U-statistics on each group and combine them by averaging to get a statistic Tn. We proved that the statistic Tn has the standard normal distribution as the limiting distribution. In this way, the running time is reduced from O(n^m) to O( n^m/km_n), where m is the order of the one sample U-statistics. Besides, for a given significance level , it’s easy to construct the rejection region. We apply our method to the goodness of fit test and two-sample test. The simulation and real data analysis show that the proposed test can achieve high power and fast running time for both one and two-sample tests.

The present paper demonstrates how Structural Equation Modelling (SEM) can be used to formulate a test of the difference in means between groups on a number of dependent variables. A Monte Carlo study compared the Type I error rates of the Likelihood Ratio (LR) Chi-square ($\chi\sp2$) statistic (SEM test criterion) and Hotelling's two-sample T$\sp2$ statistic (MANOVA test criterion) in detecting differences in means between two independent samples. Seventy-two conditions pertaining to average sample size ((n$\sb1$ + n$\sb2$)/2), extent of inequality of sample sizes (n$\sb1$:n$\sb2$), number of variables (p), and degree of inequality of variance-covariance matrices ($\Sigma\sb1$:$\Sigma\sb2$) were modelled. Empirical sampling distributions of the LR $\chi\sp2$ statistic and Hotelling's T$\sp2$ statistic consisted fo 2000 samples drawn from multivariate normal parent populations. The actual proportion of values that exceeded the nominal levels are presented. The results indicated that, in terms of maintaining Type I error rates that were close to the nominal levels, the LR $\chi\sp2$ statistic and Hotelling's T$\sp2$ statistic were comparable when $\Sigma\sb1$ = $\Sigma\sb2$ and (n$\sb1$ + n$\sb2$)/2:p was relatively large (i.e., 30:1). However, when $\Sigma\sb1$ = $\Sigma\sb2$ and (n$\sb1$ + n$\sb2$)/2:p was small (i.e., 10:1) Hotelling's T$\sp2$ statistic was preferred. When $\Sigma\sb{1} \not=\Sigma\sb2$ the LR $\chi\sp2$ statistic provided more appropriate Type I error rates under all of the simulated conditions. The results are related to earlier findings, and implications for the appropriate use of the SEM method of testing for group mean differences are noted.

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Esta dissertação concentra-se nos processos estocásticos espaciais definidos em um reticulado, os chamados modelos do tipo Cliff & Ord. Minha contribuição nesta tese consiste em utilizar aproximações de Edgeworth e saddlepoint para investigar as propriedades em amostras finitas do teste para detectar a presença de dependência espacial em modelos SAR (autoregressivo espacial), e propor uma nova classe de modelos econométricos espaciais na qual os parâmetros que afetam a estrutura da média são distintos dos parâmetros presentes na estrutura da variância do processo. Isto permite uma interpretação mais clara dos parâmetros do modelo, além de generalizar uma proposta de taxonomia feita por Anselin (2003). Eu proponho um estimador para os parâmetros do modelo e derivo a distribuição assintótica do estimador. O modelo sugerido na dissertação fornece uma interpretação interessante ao modelo SARAR, bastante comum na literatura. A investigação das propriedades em amostras finitas dos testes expande com relação a literatura permitindo que a matriz de vizinhança do processo espacial seja uma função não-linear do parâmetro de dependência espacial. A utilização de aproximações ao invés de simulações (mais comum na literatura), permite uma maneira fácil de comparar as propriedades dos testes com diferentes matrizes de vizinhança e corrigir o tamanho ao comparar a potência dos testes. Eu obtenho teste invariante ótimo que é também localmente uniformemente mais potente (LUMPI). Construo o envelope de potência para o teste LUMPI e mostro que ele é virtualmente UMP, pois a potência do teste está muito próxima ao envelope (considerando as estruturas espaciais definidas na dissertação). Eu sugiro um procedimento prático para construir um teste que tem boa potência em uma gama de situações onde talvez o teste LUMPI não tenha boas propriedades. Eu concluo que a potência do teste aumenta com o tamanho da amostra e com o parâmetro de dependência espacial (o que está de acordo com a literatura). Entretanto, disputo a visão consensual que a potência do teste diminui a medida que a matriz de vizinhança fica mais densa. Isto reflete um erro de medida comum na literatura, pois a distância estatística entre a hipótese nula e a alternativa varia muito com a estrutura da matriz. Fazendo a correção, concluo que a potência do teste aumenta com a distância da alternativa à nula, como esperado.
This dissertation focus on spatial stochastic process on a lattice (Cliff & Ord--type of models). My contribution consists of using Edgeworth and saddlepoint series to investigate small sample size and power properties of tests for detecting spatial dependence in spatial autoregressive (SAR) stochastic processes, and proposing a new class of spatial econometric models where the spatial dependence parameters that enter the mean structure are different from the ones in the covariance structure. This allows a clearer interpretation of models' parameters and generalizes the set of local and global models suggested by Anselin (2003) as an alternative to the traditional Cliff & Ord models. I propose an estimation procedure for the model's parameters and derive the asymptotic distribution of the parameters' estimators. The suggested model provides some insights on the structure of the commonly used mixed regressive, spatial autoregressive model with spatial autoregressive disturbances (SARAR). The study of the small sample properties of tests to detect spatial dependence expands on the existing literature by allowing the neighborhood structure to be a nonlinear function of the spatial dependence parameter. The use of series approximations instead of the often used Monte Carlo simulation allows a simple way to compare test properties across different neighborhood structures and to correct for size when comparing power. I obtain the power envelope for testing the presence of spatial dependence in the SAR process using the optimal invariant test statistic, which is also locally uniformly most powerful invariant (LUMPI). I have found that the LUMPI test is virtually UMP since its power is very close to the power envelope. I suggest a practical procedure to build a test that, while not UMP, retain good power properties in a wider range for the spatial parameter when compared to the LUMPI test. I find that power increases with sample size and with the spatial dependence parameter -- which agrees with the literature. However, I call into question the consensus view that power decreases as the spatial weight matrix becomes more densely connected. This finding in the literature reflects an error of measure because the hypothesis being compared are at very different statistical distance from the null. After adjusting for this, the power is larger for alternative hypothesis further away from the null -- as one would expect.

The study examined the theoretical and statistical factor structure of the Dean-Woodcock Structured Emotional-Status Exam (D-WESE) and evaluated the prediction of representations of clinical two-point scores representing MMPI codetypes. Analyses provided greater insight into the structure of the measure and its effectiveness in classifying psychopathology. The data of 200 patients referred for a neuropsychological evaluation at a large Midwestern neurology practice was used in this study. Age of patients ranged between 13 and 96 years. An 11-factor exploratory analysis with an oblique rotation and principle axis factoring was compared to the original theoretical model of DSM diagnoses used to develop the 50-items of the measure in order to evaluate the effectiveness of the measure. Regression analyses was incorporated to assess the ability of the relatively brief, 50-item D-WESE to predict representations of 12 codetypes from the 566-item MMPI.
Department of Educational Psychology

碩士
淡江大學
統計學系碩士班
97
In recent years, it''s an important issue for manufacturers to assess, monitor and improve product quality and performance. Process capability indices (PCIs) are used to evaluate whether product quality meets the required level in many industries. Since the lifetime of products may not be normal distribution and it may follow an exponential, gamma or Weibull distribution, etc. On the other hand, the experimenters may not be able to obtain the lifetime of all products which are put on test due to the limitation of cost, time or artificial difficulties in data collection. Therefore, we assess the lifetime performance index of products for a two-parameter exponential distribution base on the multiply type II censored samples. We use 14 estimators including 12 weighted moments estimators (WMEs), approximate maximum likelihood estimator (AMLE) and best linear unbiased estimator (BLUE) to estimate the scale parameter and then we can obtain 14 estimators of the process capability index . The bias and mean square error(MSE) of 14 estimators are simulated and the optimal estimator is identified in the sense of minimum MSE. We also propose a hypothesis testing algorithmic procedure for practitioners to determine whether the lifetime of products meet the required level. Finally, two real-life examples are given to demonstrate the testing algorithmic procedure. 表單編號：ATRX-Q03-001-FM031-01

博士
國立政治大學
統計研究所
98
The Poisson distribution is a well-known suitable model for modeling a rare events in variety fields such as biology, commerce, quality control, and so on. Many applications involve comparisons of two treatment groups and focus on showing the superiority of the new treatment to the conventional one, or the non-inferiority of the experimental implement to the standard implement upon the cost consideration. We aim to develop statistical tests for testing the superiority and non-inferiority by two independent random samples from Poisson distributions. In developing these tests, both computational and theoretical difficulties arise from presence of nuisance parameters. In this study, we first consider the problems with the null hypothesis of equality for simplicity. The problems are extended to have a regular null hypothesis of non-superiority next. Subsequently, the proposed methods are further investigated in establishing the non-inferiority. Two types of Wald test statistics are of our main research interest. The correspondent asymptotic testing procedures are developed by using the normal limiting distribution. In our study, the asymptotic distribution of the test statistics are derived. The asymptotic power functions and the sample size formula are further obtained. Given the power functions, we justify the validity and unbiasedness of the tests. The adequate continuity correction term for these tests is also found to reduce inflation of the type I error rate. On the other hand, the exact testing procedures based on two exact $p$-values, the confidence-interval $p$-value (Berger and Boos (1994)), and the estimated $p$-value (Krishnamoorthy and Thomson (2004)), are also applied in our study. It is known that an exact testing procedure tends to involve complex computations. In this thesis, several strategies are proposed to lessen the computational burden. For the confidence-interval $p$-value, a truncated confidence set is used to narrow the area for finding the $p$-value. Further, the test statistic is verify whether they fulfill the property of convexity. It is shown that under the convexity the exact $p$-value occurs somewhere of the boundary of the null parameter space. On the other hand, for the estimated $p$-value, a simpler point estimate is applied instead of the use of the restricted maximum likelihood estimators, which are less straightforward in this problem. The estimated $p$-value is shown to provide a conservative conclusion. The calculations of the sample sizes required by using the two exact tests are discussed. Intensive numerical studies show that the performances of the asymptotic tests depend on the fraction of the two sample sizes and the continuity correction can be useful in some cases to reduce the inflation of the type I error rate. However, with small samples, the two exact tests are more adequate in the sense of having a well-controlled type I error rate. A data set of breast cancer patients is analyzed by the proposed methods for illustration.