A Quantile-Score Test for Experimental Design jvb

A Quantile-Score Test for Experimental Design

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A THESIS Presented to the Graduate Faculty of Mathematics and ICT Department College of Arts and Sciences Cebu Normal University Cebu City, Philippines

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In Partial Fulfillment of the Requirements for the Course RESEARCH IN MATHEMATICS EDUCATION

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CAPUYAN, RUBENA DANO, NOREEN JUMUAD, RICHARD MARK SECOP, KENT DATE

TABLE OF CONTENTS

TITLE PAGE

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TABLE OF CONTENTS

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LIST OF NOTATIONS 1 INTRODUCTION 1.1 Background of the Study . . . . 1.2 Statement of the Problem . . . 1.3 Objectives of the Study . . . . . 1.4 Significance of the Study . . . . 1.5 Scope and Limitations . . . . . 1.6 Basic Concepts and Preliminary

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1 1 1 2 2 2 2

2 REVIEW OF RELATED LITERATURE

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BIBLIOGRAPHY

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LIST OF NOTATIONS

Notation

Description

Yijk

score obtained by k-th student corresponding in rowi and columnj

nr

number of observations per row

nc

number of observations per column

Yr••

sum of all observations in row 1. The two dots replacing c and n means tha

Y r••

mean for row r =

Y•c•

sum of all observations for column c

Y•••

grand sum of all the scores in all the cells from the first to the last row; fro

Y •b••

grand mean by dividing Y••• by total numver of observations

Yrc•

sum of observations in a given cell

Y rc•

mean for a given cell

Y2•• n2••

CHAPTER 1 INTRODUCTION

1.1

Background of the Study

Nonparametric rank tests for two-way analysis of variance have been studied and used as important tools for experimental designs, examples of which are rank transformation and the normal- score rank test. However, there were some studies and controversies for some empirical power and robustness properties of the rank tests, specifically the RT (Rank Transform) technique, in testing interaction in a two- way layout. (Blair, et.al (1987)) Hence, Van der Waerden (1952/1953) proposed the use of the quantities of a standard normal distribution instead of using the ranks of the observations. And this notion was supported by the several rank tests of Mansouri and Chang (1995) which showed good results by using the quantile- score test for an interaction term in two- way experimental designs. 1.2

Statement of the Problem

This aims to test interactions in two- way analysis of variance.

2 1.3

Objectives of the Study

To test an interaction in a two- way analysis of variance using quantilescore test. 1.4

Significance of the Study

The use of quantile- score test gives a clearer understanding in an experimental design and provides a transparent analysis of interactions. Hence, the significance of this study is to be able to analyze critically an interactive phenomena. 1.5

Scope and Limitations

This study focuses on a nonparametric quantile- score test using quantile estimation for testing interaction in two- way analysis of variance. 1.6

Basic Concepts and Preliminary Notions

Quantile score One of the class of values of a variate that divides the total frequency of a sample or population into a given number of equal proportions. Quantiles are values taken at regular intervals from the inverse function of the cumulative distribution function (CDF) of a random variable. Dividing ordered data into

3 q essentially equal-sized data subsets is the motivation for q-quantiles; the quantiles are the data values marking the boundaries between consecutive subsets.

The quantiles can be used as cutoff values for grouped data in

approximately equal size groups. Quantiles can also be applied to continuous data, providing a way to generalize rank statistics to continuous variables.

A kth q-quantile for a random variable is a value x such that the probability that the random variable will be less than x is at most k/q and the probability that the random variable will be greater than x is at most (qk)/q = 1(k/q). There are q1 of the q−quantiles, one for each integer k satisfying 0 < k < q. In some cases the value of a quantile may not be uniquely determined, as can be the case for the median of a uniform probability distribution on a set of even size.

CHAPTER 2 REVIEW OF RELATED LITERATURE

BIBLIOGRAPHY

[1] R.B. Allan and R. Laskar, On domination and independent domination numbers of a graph, Discrete Mathematics, Vol.23, No.2, 73-76, 1978. [2] Ameenal Bibi K, Selvakumar R, The Inverse Domination in Semitotal Block Graphs, IJCA Vol.8 No.8, (2010), 4-7.