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NATIONAL UNIVERSITY OF RWANDA

FACULTY OF ECONOMICS AND MANAGEMENT

DEPARTMENT OF APPLIED STATISTICS

ACADEMIC YEAR: 2011-2012, BACC: IV

ASSIGNMENT OF MULTIVARIATE ANALYSIS

Presented by:

NDIBANJE Gilbert UG10105473

Lecturer: Mr. NIRAGIRE Francois

Done at Huye, The14th , May2012

BASIC CONCEPTS OF EXPERIMENTAL DESIGN AND

ONE WAY ANALYSIS OF VARIANCE



PART I: BASIC CONCEPTS OF EXPERIMENTAL DESIGN

1. DEFINITION

Experimental design is the process of planning a study to meet

specified objectives (SAS, 2005). It includes both Strategies for

organizing data collection and data analysis procedures matched

to those data collection strategies .



2. DESCRIPTION OF EXPERIMENTAL DESIGN

2.1. Designing an Experiment

When design the experiment the following steps will be

perfomed:

Define the problem and the questions to be addressed.

Define the population of interest.

Determine the need for sampling.

Define the experimental design.



2.1.1. Define the problem and the questions to be addressed

This first step of experimental design is consisting in identifying

clearly the specific questions that the researcher is lanning to

examine .The researcher should identify the sources of valiability in

experimental conditions. The objectives of design experiment:

To partition the effects of the sources of variability into distinct

components in order to evaluate the specific questions on which the

researcher is interested.

To improve the precision of the results in order to examine or to test

the research hypothesis.



2.1.2. Define the population of interest

A population is a collective or a set of all people, animals,

plants, or other items that researchers collect data from.

The designed experiment should designate the population

for which the problem will be examined. The whole

population on which the researcher will based in making

inferences will be the focus of the experimental design.



2.1.3. Determine the need for sampling

A sample is finite part of a statistical population whose

properties are studied to gain information about the whole

population.The results from a sample are then used to draw

valid inferences or conclusions about the population.



2.1.4. Define the experimental design

A clear definition of the details of the experiment makes the

desired statistical analyses possible, and almost always

improves the usefulness of the results. It is consisting of the

following steps:

Identify the experimental unit.

Identify the types of variables.

Define the treatment structure.

Define the design structure.



3. ASSUMPTIONS OF EXPERIMENTAL DESIGN

Data from experiments are analyzed using linear regression and

analysis of variance . The standard assumptions for data analysis

that apply to linear regression and the analysis of variance are now

summarized as follows:

1. No model specification error

• The response Y is the dependent variable.

• The independent variables, x1,..,xp, influence Y.

• The form of the relationship between Y and (x1,..,xp) is linear (not

nonlinear) in the parameters.

2. No measurement error• The dependent variable(s) are interval or ratio data (not ordinal or

nominal).

• The independent variables are measured accurately.



Assumptions’ .3.No collinearity (a high correlation between any two independent

variables is not allowed).

4.The error term,residuals,are well-behaved when the following

conditions hold:

• A zero mean and Homoscedasticity

• No autocorrelation (usually of most concern with time series or

spatial data)

• No large correlations between any of the independent variables and

Normal distribution



4. ILLUSTRATION The example is for a study that can be perfomed in evaluating the

contribution of NGOs in economic growth of their beneficiaries.

Grobal fund in Nyamagabe and Huye Districts is taken as case study.

4.1. Define the problem and the question to be addressed

The problem was to know how Grobal fund as NGOs is

contributing on the economic growth,as the number of NGOs in

Rwanda is increasing and the poverty remains the serious

problem. The following questions:

Is the income generated by the beneficairies depending on the

kind of support they are given?



4.2. Defining the population of interest

Within this study the whole population on which the research will

be concerned is the set of 60000 beneficiaries of Grobal fund

from Nyamagabe and Huye Districts.

4.3. Determine the need for sampling

Using techniques such as random selection after stratification or

blocking is often preferred. The sample of 100 beneficiaries

was randomly selected from all two Districts.

Sample size was been calculated as follows :

83.996001

60000

)1.0(600001

60000

)(122

e N

N n



4.4. Defining the experimental design

The experimental units in this study are the beneficiaries of

Grobal fund. The variable that can be identified at this level are so many;

Primary variables (income, kind of suport given), background

variables ( age,sex, number of family members,),etc Treatment structure: the study will treat the income of

beneficiaries according to the kind of suport they are given.

The design structure that will be used is CompletelyRandomized Design (CRD) where Subjects are assigned to

treatments completely at random.



Completely Randomized Design (CRD)

Suppose the study is going to be conducted on the evaluation of the

contribution of Grobal Fund on the economic growth of their

beneficiaies in 2 Districts (Nyamagabe and Huye). Their

Beneficiaries from 2 Districts of sourthern provinces forCompletely

Randomized Design (CRD) will be randomly assigned to one of

two treatment groups (income and kind of support provided) .The

total number of beneficiaries in two districts, on which to conduct aresearch is 100. Randomly assign 1/2 of them, or 50 beneficiaries, to

each of the 2types of evaluations.



PART II: ONE-WAY ANALYSIS OF VARIANCE

2.1. Definition

Analysis of Variance is a linear model that relates nominal

predictor variables to a continuous outcome variable.

According to Silicon Genetics (2003), One-way analysis of

variance (ANOVA) tests is allowing determining if one

given factor, such as drug treatment, has a significant effect

on gene expression behavior across any of the groups under

study.



Description

The analysis of variance model examines the association between

nominal predictor variables or factors and a continuous outcome

variable or dependent. The mathematical model that describes

the relationship between the response and treatment for theone-way ANOVA is given by :

Where Y ij represents the j-th observation ( j = 1, 2, ...ni) on the i-th

treatment (i = 1, 2, ..., k levels. τij is the common effect for the whole experiment, i represents the i-th treatment effect and ε ij

represents the random error present in the j-th observation on

the i-th treatment.

ijiij y



Zero mean value of disturbances: E(u)=0

No serial correlation, or cov(ui,u j)=0

Homoscedasticity, or var(ui)=σ2

Error term are normaly distributed with (0, σ2 )

2.3. Assumptions

2 4 EXAMPLE



2.4. EXAMPLE In the example of completely randomized design of evaluating the

contribution of Grobla Fund in economic growth, we evaluate the

dependance between the income of 100 beneficiaries of Grobla Fund

and their kind of suport (Health insurance, Livestock, Construction of

house and schooll fees)the data cfr: word doc.

H0: μ1 = μ2 = μ3 = μ4, i.e. the income of beneficiaries of gaining

different suports is the same or the income of beneficiaries is

equally affected by the kind of suport they are provided.

H1: The means are not all equal.



Summary of data

TA1=315102

TA2=266278

TA3=213978TA4=216837

01024538718100

)1012196()( 22

N

yij

311003960302 ij y

9102591229421

216837

22

213978

25

266278

32

31510222222

Aj

Aj

N

T

758573123

)(2

2

N

y

ySSTotalij

ij

13735769)(22

N

y

N

T SSBetween

Aj

Aj

Aj

744837354

2

2

AJ

Aj

ij N

T ySSWithin



The ANOVA Table:

Source SS DF Mean squre F

A(orTreatment,orexplained

SS Between

=13735769

J-1

4-1=3

SS Between/(J-1)

Error(Residual) SS Within=744837354

N-J100-4=96

SS Within/(N-J)

Total SS Total=758573123

N-1100-1=99

SS Total/(N-1)

MSwithin

MSBetween

576.45785893

13735769

438.775872296

744837354

590122.0438.7758722

576.4578589



Using SPSSUsing SPSS, the following is the output provided:

As conclusion, we fail to reject the null hypothesis

because the p-value of 0.623 is greater than the level of signicance of 5%.



References http://stat.stanford.edu/~jtaylo/courses/stats203 http://stattrek.com/experiments/ http://support.sas.com/

SAS (2005), Concepts of Experimental Design, USA www.sas.com Zar, J. (1999) Biostatistical Analysis (4th ed.) Upper Saddle River,

NJ, Prentice Hall. http://www.themeasurementgroup.com

http://www.smartersolutions.com Israel, Glenn D. 1992. Sampling the Evidence Of ExtensionProgram Impact. Program Evaluation and OrganizationalDevelopment, IFAS, University of Florida. PEOD-5. October.

http://www.sas.com/

http://www.smartersolutions.com/

http://www.smartersolutions.com/

http://www.sas.com/



THANK YOU FOR YOUR KIND ATTENTION!!!!

Questions, ideas andcomments are welcome!!!!

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