# Theory and Methodology in a research study

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- Theory AND METHODOLOGY
- 3.1 Survey Process
- 3.1.1 Sampling technique – Stratified random trying
- 3.1.2 Sampling size finding
- 3.1.3 Questionnaire Design
- 3.1.4 Types of inquiries used in the questionnaire
- 3.1.5 Post Stratification weights
- 3.2 Trials of Associations
- 3.2.1 Kendall’s Tau rank correlativity trial
- 3.2.2 Spearman rank correlativity trial
- 3.2.3 Mann-Whitney U trial
- 3.3 Advanced Analysis
- 3.3.1 Factor Analysis
- 3.3.2 Factor analysis on the correlativity matrix
- 3.3.3 Method of Extraction – Principal Component method
- 3.3.4 Factor Rotation
- 3.3.5 Determination of figure of factors to maintain
- 3.3.6 Measures of rightness of Factor Analysis
- 3.3.7 Multiple Regression Model

**Chapter 3**

## Theory AND METHODOLOGY

This chapter includes the theories and methodological analysiss that have used throughout this research survey. This subdivision includes the theories that were used under the Survey procedure, Trials of associations and Advanced analysis.

## 3.1 Survey Process

Under this subdivision, the theories that have used in the study procedure are included.

### 3.1.1 Sampling technique – Stratified random trying

Graded random sampling is a probabilistic sampling technique which applies when there is a variableness in the response variable between stratums. This technique involves division of the whole population into smaller homogenous subgroups known as strata.

Any component in the population can be assigned into merely one strata. Each population component must be included in any strata. Then for each strata simple random sampling or system sampling can be applied.

**Proportional Allocation method**

Proportional allotment is a method which is used in graded random trying to find the sample size that is to be drawn from each stratum. This technique allocates sample size into the stratums proportionally.

Sample sizes for each stratum are calculated by utilizing the undermentioned equation.

N_{I}= ( N_{I}/N ) * n

Where, n_{I}is the sample size for I^{Thursday}stratum.

Nitrogen_{I}is the entire figure of units in the I^{Thursday}stratum.

N is the population size.

N is the entire sample size.

### 3.1.2 Sampling size finding

When finding an appropriate sample size three standards should be considered. Those are Level of preciseness ( or trying mistake ) , Confidence degree and Degree of variableness in the properties being measured ( Miaulis, George & A ; Michener, 1976 ) . Following equation was developed by Cochran ( 1963 ) , to find the sample size. Here he has considered the supra indicated standards when finding the sample size.

Ns =.

n = sample size requiredv =

N = Population sized = border of mistake

P = Estimated discrepancy in population

### 3.1.3 Questionnaire Design

When planing a questionnaire, following stairss should be followed.

- Formulate the aims.
- Explicate a program of analysis.
- Make a list of information needed.
- Design the subdivisions of the questionnaire.
- Design appropriate inquiries harmonizing to the aims.
- Decide the order of the inquiries that should be included.
- Complete the questionnaire by adding relevant instructions for the respondents.
- Confirm that all aims are covered by inquiries asked.
- Conduct a pilot survey.
- Harmonizing to the consequences of the pilot survey, polish the questionnaire.

### 3.1.4 Types of inquiries used in the questionnaire

**Open ended inquiries**

Open ended inquiries allow respondents to give their sentiments without any limitations. No preset replies are provided in this type of inquiries.

**Closed ended inquiries**

In this type of inquiries set of replies are provided and respondents are allowed to take replies merely from the given set. Usually these types of inquiries are known as multiple pick inquiries.

**Likert Scale inquiries**

This is the most widely used method for scaling responses. These types of inquiries are used to mensurate either positive or negative impact for a peculiar statement. In this method a numerical value is assigned to each pick. Normally Likert graduated tables have 5 picks ( Very Satisfied, Satisfied, Moderate, Dissatisfied, Very Dissatisfied ) but it can be changed harmonizing to the different state of affairss.

### 3.1.5 Post Stratification weights

In study research, it is really of import to hold a representative sample of the population. Sometimes due to low response rates, some groups can be oversampled and others can be under sampled. Thus the manner a certain feature is distributed in the sample can differ from the manner it is distributed in the population. By making a station stratification this job can be remedied because it gives a weight lower than 1 for those who are oversampled and a weight greater than 1 for those who are under sampled ( Little, 1993 ) .

## 3.2 Trials of Associations

### 3.2.1 Kendall’s Tau rank correlativity trial

Kendall’s rank correlativity trial is a distribution free trial ( Non-parametric trial ) which can be used to mensurate the association between two ordinal variables. In this instance ordinal variables should non needfully be intervals. Same as the other correlativity trials, this correlativity besides varies from -1 to +1. Furthermore Kendall’s correlativity trial can be divided into two trials based on the nature of the variables. That is Kendall’s Tau-b and Tau-c trials. Tau-b trial can be used when the both ordinal variables have same degrees. When the degrees are different Tau-c trial can be used ( Chok, 2010 ) .

Hypothesis tested under this trial are,

Hydrogen_{0}: There is no association between the two variables

Hydrogen_{1}: There is an association between the two variables

This non-parametric trial measures the association based on the figure of accordant and discordant braces in the mated observations.

The two mated observations ( X_{I}, Y_{I}) and ( X_{J}, Y_{J}) are said to be accordant if,

- Ten
_{I}& lt ; X_{J}and Y_{I}& lt ; Y_{J}or - Ten
_{I}& gt ; X_{J}and Y_{I}& gt ; Y_{J}or - ( Ten
_{I}– Ten_{J}) ( Y_{I}– Yttrium_{J}) & gt ; 0

The two mated observations ( X_{I}, Y_{I}) and ( X_{J}, Y_{J}) are said to be discordant if,

- Ten
_{I}& lt ; X_{J}and Y_{I}& gt ; Y_{J}or - Ten
_{I}& gt ; X_{J}and Y_{I}& lt ; Y_{J}or - ( Ten
_{I}– Ten_{J}) ( Y_{I}– Yttrium_{J}) & lt ; 0

Kendall’s rank correlativity ( ? ) =

Where N_{degree Celsiuss}= Number of accordant braces

N_{vitamin D}= Number of discordant braces

### 3.2.2 Spearman rank correlativity trial

Lapp as the Kendall’s rank correlativity trial, this trial is besides a non-parametric trial and can be used to prove the association between two ranked variables. And besides this trial can be used to prove the association between two ordinal and uninterrupted variables. Spearman correlativity coefficient besides varies from -1 to +1 ( Chok, 2010 ) .

R_{s}=

Where*rank ( ten*_{I}*)*and*rank ( Y*_{I}*)*are the ranks of the observations.

### 3.2.3 Mann-Whitney U trial

This trial is the non-parametric version option to the parametric t-test for two independent samples. This trial can be used to compare the difference between two independent samples particularly when dependent variable is either ordinal or uninterrupted. The difference can be identified by comparing the medians of the two samples. There are 4 premises that should be considered before using the Mann-Whitney trial. Those are mentioned below.

**Premises: –**

- Dependent variable should be either ordinal or uninterrupted.
- Independent variable should hold two classs.
- Observations should be independent from each other. ( i.e. There should non be any relationship between observations in each group and between groups themselves )
- Two variables should non be usually distributed.

## 3.3 Advanced Analysis

This subdivision explains the theories that have used under the advanced analysis.

### 3.3.1 Factor Analysis

Harmonizing to Rencher ( 2002 ) , Factor analysis is a multivariate technique and it can be used to cut down a big and complex dataset into smaller meaningful drumhead steps. Or it can be used to place the forms in the dataset ( i.e. this is a dimension decrease technique ) . Factor Analysis trades with non merely the discrepancies, but besides the covariances ( or correlativities ) between variables. Some research workers have said that factor analysis is an extension of chief constituent method but really PCA technique is one method of work outing factor analysis equations.

**The general factor analysis theoretical account**

This theoretical account assumes that ‘p’ figure of variables and ‘m’ implicit in common factors ( m & lt ; P ) , denoted by F_{1}, F_{2}, …… , F_{m}

Ten_{J}– µ_{J}= ?_{j1}F_{1}+ ?_{j2}F_{2}+ _ _ _ _ _ _ + ?_{jm}F_{m}+ ?_{J}for J = 1, 2, …… , P

Ten_{J}= µ_{J}++ ?_{J}for J = 1, 2, …… , P

Whereµ_{J}= Mean of X_{J}

F_{I}= I^{Thursday}unobservable common factor

?_{J}= J^{Thursday}unobservable specific factor

In matrix notation,

Ten_{p?1}= µ_{p?1}+ ?_{p?m}F_{m?1}+ ?_{p?1}

WhereX = ( X_{1}, Ten_{2}, ……. , Ten_{P})^{/}

µ = ( µ_{1}, µ_{2}, ….. , µ_{P})^{/}

F = ( F_{1}, F_{2}, …… , F_{m})^{/}

? = ( ?_{1}, ?_{2}, …… , ?_{P})^{/}

Model premises ( In matrix notation ) : –

- F ~ ( 0, I ) ? iid ( 0,1 )
- ? ~ ( 0, ? ) ? independent merely

Where ? = ?_{1}0 …….. 0

- ?
_{2}…… . 0

. . .

. . .

- 0 …….. ?
_{P}

- F and ? should be independent.
- Datas should non be multivariate normal.

Then X = ?F + ?

**Communality**

The proportion of variableness of the J^{Thursday}standardized variable that is explained by ‘m’ common factors is called the communality of the J^{Thursday}standardized variable and which is given by

### 3.3.2 Factor analysis on the correlativity matrix

Normally factor analysis is applied for the standardised information. The ground for this is if the variables have different measuring units the factor analysis would non give an accurate consequence.

Since,

Sample covariance between I^{Thursday}and k^{Thursday}= Sample correlativity between I^{Thursday}and k^{Thursday}

standardized variablesoriginal variables

We can utilize the correlativity matrix of the original dataset alternatively of utilizing the covariance matrix of standardised dataset.

### 3.3.3 Method of Extraction – Principal Component method

Let S ( Sample covariance matrix ) Eigen value – Eigen vector ( normalized ) braces ( ?_{I}, a_{I}) for I = 1,2, …….. , p. Where ?_{1}? ?_{2}? …….. ? ?_{P}? 0. Let m & lt ; p is the figure of common factors. Then the matrix of estimated factor burdens is given by,

? =Then S = ? ?** ^{/}**+ 0

This is of the signifier S = ? ?** ^{/}**+ ? with ? = 0

Now ?_{jk}is the ( jk )^{Thursday}component of ? = burden of the J^{Thursday}variable on the K^{Thursday}common factor.

Therefore,= S –^{/}

=_{1}0 ……….. 0

0_{2}……… . 0

. . .

. ..

0 0_{P}

Communalities are estimated as,

Communalities =

= 1 –_{J}

### 3.3.4 Factor Rotation

After acquiring the unrotated factor burdens, factor rotary motion is applied to do as many factor burdens as possible near zero and maximise as many of the others as possible. And it besides be noted that response variables are non loaded to a great extent on more than one factor. The most common method for factor rotary motion is Varimax rotary motion. It tries to coerce as many burdens as possible towards zero and coercing others towards one.

### 3.3.5 Determination of figure of factors to maintain

Harmonizing to Ledesma and Mora ( 2007 ) , points of position there are several methods that have been proposed by field experts to find the figure of factors to maintain. Two of often utilizing methods of them are described below.

**( 1 )****Kaieser’s Eigen value greater than one regulation**

Harmonizing to this regulation, it keeps merely the factors which have characteristic root of a square matrixs greater than or equal to one. It should besides be noted that this works merely for standardized informations.

**( 2 ) Cattell’s Scree secret plan**

Figure 3.1: Scree Plot

This secret plan works for both original information every bit good as standardized information. This method determines the factors to maintain by diagrammatically stand foring the characteristic root of a square matrixs and they are presented in the descending order. Then by analyzing the graph, determines the point at which the important interruption takes topographic point.

### 3.3.6 Measures of rightness of Factor Analysis

There are chiefly two trials that could be used to prove the rightness of a factor analysis. Those are Kaiser-Meyer-Olkin ( KMO ) trial and Bartlett ‘s trial.

**( 1 ) Kaiser-Meyer-Olkin ( KMO )**

This measures the trying adequateness of the variables. As stated by Hutcheson and Sofroniou ( 1999 ) , for an equal sample this value should be greater than 0.5. Harmonizing to his position the values between 0.5 and 0.7 are mediocre, 0.7 and 0.8 are good, 0.8 and 0.9 are great and values above 0.9 are superb.

**( 2 )****Bartlett ‘s trial**

This trial identifies the strength of the relationship between the variables.

Hypothesis tested under this trial are,

Hydrogen_{0}: Correlation matrix is an Identity matrix

Hydrogen_{1}: Correlation matrix is non an Identity matrix

Therefore to use the factor analysis, void hypothesis should be rejected.

### 3.3.7 Multiple Regression Model

As stated by Rawlings, Pantula and Dickey ( 1998 ) , multiple arrested development analysis is one of the most widely used statistical technique and there several forecasters are used to pattern a individual response variable. Whenever building arrested development theoretical accounts merely a limited figure of independent variables should be included in the theoretical account for any state of affairs of involvement.

Let see a theoretical account with thousand independent variables X_{1}, Ten_{2}, …. , Ten_{K}

Then the additive arrested development theoretical account could be defined as follows,

*Y** _{I}*=

*?*

_{0}+

*?*

_{1}

*ten*

_{1}

*+*

_{I}*?*

_{2}

*ten*

_{2}

*+ …….. +*

_{I}*?*

_{K}*ten*

*+ ?*

_{qi}*;*

_{I}*I*= 1, 2, …….. , N

Then the theoretical account in matrix signifier,

**Y = X****?****+****?**

*Yttrium*_{1}1*Ten*_{11}*Ten*_{12}…..*Ten*_{k1}*?*_{1}?_{1}

*Yttrium*_{2}1*Ten*_{12}*Ten*_{22}…..*Ten*_{K2}*?*_{2}?_{2}

**Y =**.**Ten =**. .**? =**.**? =**.

. . . . .

*Yttrium*_{N}1*Ten*_{1n}*Ten*_{2n}…..*Ten*_{kn}*?*_{N}?_{N}

**Premises:**

. Tocopherol ( ? ) = 0

?^{2}0

?^{2}

2. V ( ? ) = ?^{2}I_{N}= .

.

0 ?^{2}

3. ? ~ N ( 0, ?^{2}I_{N})

**Adequacy of the fitted theoretical account**

Once the theoretical account being fitted, it should be ensured that the fitted theoretical account is equal or non. Following two methods can be used in order to prove the adequateness of the fitted theoretical account.

**Residual Analysis**

After suiting a theoretical account, it is of import to analyze the rightness of the theoretical account for the information before pulling decisions about the theoretical account being fitted. Residual analysis is one of the largely utilizing statistical technique to prove the adequateness of the theoretical account.

Remainders are the differences between the ascertained and the fitted values. That is*vitamin E*_{I}=*Y*_{I}-For the arrested development theoretical account, mistakes are assumed to be independent normal random variables with average 0 and discrepancy ?^{2}. Therefore if the theoretical account is equal, the ascertained remainders should reflect this belongings assumed for the mistakes*?*_{I}. This belongings can be identified by pulling a residuary secret plan. Residual secret plan can be drawn by taking standardised remainders as the y-axis and fitted values as the x-axis.

Here, Standardized Residuals =

EMS = Error Mean Square

**Normal Probability Plot**

After suiting the theoretical account, it is besides of import to place the normalcy of the mistake footings. The best manner to place this is to build a Normal Probability Plot of the remainders. In this graphical manner, each remainder is plotted against its expected value when the distribution is Normal. If the secret plan is about additive, it suggests understanding with normalcy and whenever it departs well from the one-dimensionality suggests that remainders are non normal.

**Variable choice process**

Several processs have been developed in order to choose the variables which should be included in the concluding theoretical account. These all processs have characteristic that the variables are included and deleted from the arrested development equation one at a clip. There are chiefly three possible choice processs in arrested development analysis. Those are Forward choice, Backward choice and Stepwise choice. In this research survey backward choice process was used and the method has described below in item.

**Backward Selection**

This process starts with holding all the variables of involvement in the theoretical account. Then the least important variable should be dropped from the theoretical account. This process continues until happen the optimum set of variables which should be included in the theoretical account. One drawback of this process is, sometimes variables dropped from the theoretical account would be important when it is added back to the reduced theoretical account.