eChapters

Q.1.    What is the most fundamental assumption to be followed by random variable for using the standard method of parametric statistics or parametric inference ?

    Answer
    
    The most fundamental assumption required to be followed by random varibale for using the parametric statistics to have statistical inference drawn from certain set of data is that the random variables have normal distribution.

Cluster Sampling (Meaning and Concept)

Cluster sampling is a special type of sampling which is used when we have to make a large survey such as a survey for the whole country and whole state etc. to know the privilege. Under this method, the total population is divided into some recognizable subdivisions known as clusters. Out of all the clusters, a given number of clusters are chosen at random. All the items covered by the selected clusters are included in the sample.

A sampling technique in which we divide the whole population into groups or subgroups in such a way that there is heterogeneity within each group and homogeneity among the groups and then a simple random sample of n groups is selected from a total of N groups of population and finally all the units of selected clusters from our sample.

Non-Probability sampling methods are also known as non-random sampling methods or deliberate sampling. These types of methods are reliant on the researcher’s ability to select members at random. These sampling methods are not fixed or pre-defined selection processes making it difficult for all elements of the population to have equal opportunities to be included in the sample.

Types of non-probability sampling techniques

1. Judgmental or Purposive sampling

This technique involves the purposive selection of units of samples based on the judgement of the researcher. The researcher selects the sample on the basis of his a priori knowledge and experience.

Yet now we have discussed probability sampling techniques in which sampling units are assigned with equal probabilities. But sometimes in practice if want to draw a sample with unequal probabilities i.e., every sampling unit is assigned with varying probabilities and we draw a sample from these units with the help of equal probability methods, then the results are not as much efficient as compared to unequal probability sampling.

Simple random sampling is a procedure wherein the chance of selection of every unit in the required sample is equal. However, the use of simple random sampling is not advisable when the units differ substantially in size as the significance of unit size is ignored.

The basic idea of this type of probability technique is that chance of each unit of a population being selected in the sampling is proportional to its size. The units vary significantly in size and the variable under study is largely related to the size of the unit. Therefore, the probabilities fixed for each unit are proportional to their sizes.

In our previous chapters, we have discussed various sampling designs or techniques such as simple random sampling, stratified sampling, systematic sampling and cluster sampling recently we have discussed unequal probability sampling i.e., probability proportional to size sampling. In all these sampling designs the variable of interest (generally there is only one variable of interest) which we have considered and perform our estimation based on that single variable. There are some other variables which are related to our variable of interest and if we utilize those variables, we may gain some efficiency in our estimator i.e., if we take some correlated variables with our variable of interest or study variable by this, we can enhance our estimation. The variable which is correlated with the study variable is usually known auxiliary variable or supporting variable or supplementary variable or benchmark variable. This auxiliary variable is utilized in two stages either in the design stage or estimation stage.

Graunt (1662) was the first who used auxiliary information to estimate the population size of England. After this Laplace used auxiliary information for the estimation of the population in France.

Non-Response

Non response occurs when respondents are not available or do not occur in the survey and refuse to respond by any means. This non response is mainly of two types either item non response or unit non response.

If we are conducting an online survey and mailing a questionnaire to various persons. But sometimes we get a reply for some questions and sometimes not i.e., in case of questions in which income is concerned. In some cases, the mail is ignored due to lack of time. So, if the whole questionnaire is ignored by any means, then it is known as non-response and if some of the questions are ignored due to their fussy nature such type of response is known as a unit non response. Such types of non-response problems are very common in surveys. Whenever we carry out a survey non response always exists because it is almost impossible to get every response from each individual.

The individual chosen for the sample is not ready to participate in the survey is known as non-response. This type of selection bias is unit/item non response.

R software is a programming language used for statistical analysis, graphical representation and reporting, and is currently developed by the R Development Core Team. R software is available freely under the GNU General Public License, and the binary versions are provided for the Windows, Linux and Mac operating systems. To develop R, effort from all over the world is made in a collaborative manner.

What is GNU?

GNU stands for GNU’s Not Unix. It is a Unix like computer operating system, but different from Unix. GNU consists of GNU packages which are program specific and released by GNU project. Free software means that the users of the software have freedom to run and use the software and also can copy, change, improve and distribute the software freely.

Development of R

R software was initially written by Robert Gentleman and Ross Ihaka also known as “R & R” of the statistics department of the University of Auckland, New Zealand. R appeared first in 1993. Since then various contribution to R has been made by a large group of talented individuals by sending code and bug reports. A core group, known as the “R Core Team” modify the R source code archive since 1997. Thus R is a collaborative project with many contributors.

Preparation of base map being in mind, the forest, agriculture, water, road and boundary are almost same for all but theme may be different as per requirement. Base information is to survey and need terrain information, the preparation includes land forms by separation of water, kart area, coastal area. Meanwhile preparation of focus might be on DEM (Digital Elevation Model) and water movement helps in demarcation of boundaries on map.

Aerial photography is objective of getting panoramic view of earth surface to analyze the features of earth and their interrelation at particular time period which may be changed in further events. In this concerned, the observation may play a vital role in achieving the goals of photography. Undoubtedly it is widely used in social function and so on, but still awaited for scientific or research use. Drone can measure the geometry of ground surface at lower altitude with more accuracy in the form of visual feature in natural colour. flight height of drone mainly limits the large area coverage due to lower flying and drone can work perfectly at low altitude planning with good quality feature/photography.

Whenever we design a scientific study it is designed in a specific contextual background to solve one or another scientific problem. Subsequently it also involves a set of variables/phenomenon operating in a particular context, set of relationships among these variables and the conditions under which they operate. Certainly, to address an unsolved scientific problem we need to carry out research by employing well defined research methodology and have an idea about possible outcomes.  For example when a extension scientist is asked to study the impact of a novel programme, he needs to understand the background and the context under which the programme was introduced, what are the developmental inputs given under the programme, to whom, when (variables), with what purpose (context), what kind of impact he/she is supposed to investigate (presumed relationship), what methods he is going to use to investigate the impact of developmental programme (research methodology) and what are the possible outcomes of the investigation and how programme interventions are related to the outcome (relationship).

10.1. Introduction

Decision-making is a fundamental aspect of human life, influencing choices in both personal and professional settings. Every individual, whether consciously or unconsciously, engages in decision-making daily, ranging from simple tasks like selecting a meal to complex strategic planning in organizations. However, decision-making becomes increasingly challenging when multiple conflicting criteria are involved. The presence of numerous factors, each with varying degrees of importance, often creates ambiguity and uncertainty, making it difficult to arrive at the most suitable option. To address these challenges, structured decision-making methodologies have been developed to provide a systematic and logical framework for evaluating alternatives.

One such widely recognized method is the Analytical Hierarchy Process (AHP), introduced by Thomas L. Saaty in the 1970s. AHP was designed to assist decision-makers in breaking down complex problems into a structured and systematic hierarchy. This approach allows for a step-by-step evaluation of different factors, making it easier to analyze their relative importance. By structuring a problem into different levels - such as goal, criteria, sub-criteria, and alternatives - AHP provides clarity in decision-making and enhances consistency in judgments. The method is particularly beneficial when dealing with decisions that involve both tangible and intangible factors, as it allows subjective preferences to be quantified systematically (Saaty, 2008).

Variable is any character or attribute of the given object/set of objects which varies from object to object or from one point of time to another point of time. For example height of individuals is a variable; it varies from person to person.  Kerlinger (1973) defined variable as a symbol to which numerals or values are assigned.

Importance of variables: In previous sections, we discussed about formulating the research problem and development of research hypothesis. Variables are the building blocks of any scientific research problem and research hypothesis. When we consider a hypothesis to be relational hypothesis, we mean that it make claims about relationship between two or more variables. When we test hypothesis, as such we do not test the hypothesis, rather we test the significance of hypothesised relationship between two or more variables. Since variables play central role in observation, measurement and testing of relationship, it is of prime importance to understand the meaning of variables, different types and the way we define these variables.

15.1. Introduction

Meta-analysis is a statistical technique used to combine the results of multiple studies to produce a single, more precise estimate of the effect size of a particular intervention or association. It is a powerful tool in research, particularly in f ields such as medicine, psychology, education, and social sciences, where it is often necessary to synthesize evidence from numerous studies to draw meaningful conclusions. This chapter provides a comprehensive overview of meta-analysis, including its history, methodology, advantages, limitations, and applications.

15.2. Historical Background

The concept of meta-analysis dates back to the early 20th century, but it was not until the 1970s that the term was coined by Gene V. Glass. Glass defined meta-analysis as “the statistical analysis of a large collection of analysis results from individual studies for the purpose of integrating the findings” (Glass, 1976). Since then, meta-analysis has evolved significantly, with advancements in statistical methods and software making it more accessible and robust.

15.3. The Need for Meta-Analysis

15.3.1. Accumulation of Evidence

In many fields, particularly in medicine and psychology, a large number of studies are conducted on the same or similar research questions. However, individual studies may have small sample sizes, leading to low statistical power and inconclusive results. Meta-analysis allows researchers to pool data from multiple studies, increasing the overall sample size and statistical power, thereby providing a more reliable estimate of the effect size.

In previous sections we discussed about the validity and estimation of validity. Now the question is whether the reliable and valid measuring instrument is sufficient condition to ensure the reliable and valid results in actual experiments? What if the measuring instruments are perfectly reliable and valid but the experiment is conducted in less controlled conditions? Can we expect accurate predictions under such conditions? The answer is No.

In previous section we discussed about the ability of the research design to precisely detect the relationship between cause and effect under controlled conditions. But in social research it becomes difficult to conduct the research under highly controlled conditions and moreover many of the researches in social science aim at prediction rather than understanding.

As we discussed earlier, aim of science is development theory, understanding, prediction and explanation of natural phenomenon. In many of the cases we are more interested in predicting the occurrence of social phenomenon rather than explaining it or understanding its relationship with other phenomenon. Under such conditions internal validity is of less use because a relationship that is established in artificially controlled conditions (with high internal validity) may fail to express itself in natural setup due to effect of extraneous variables. When the relationship between cause and effect is weak enough to be expressed in natural setup then it is of less use in predicting the phenomenon. Under such conditions we require experiments that are conducted in nearly natural setup, so that we can generalize the findings of the research to the wider domain.

Sampling is the process of drawing a part of population supposed to be the representative of that population.

Types: Sampling process can be either random or non-random. Most frequently we use random sampling method in our studies. But sometimes the purpose of the study demands for non-random sampling in one or other phase of sampling (Example: When we want to study the impact of life skills development programme on the disabled female beneficiaries)

Hypothesis testing is the statistical procedure used in order to accept or reject statistical null hypotheses. One of the major purpose for which we formulate the hypothesis is, to test the statistical significance of the relation, difference or variation etc. among given sample/s at a given point of time or at different time points.

True/False

Q.1.    Statistics in singular is used to denote a subject (branch) of science.
Q.2.    Statistics in plural is used to denote “data or methods”
Q.3.    Statistics is defined as a science of collection, compilation, analysis and interpretation of numerical data.
Q.4.    Statistics can be used for two major purposes: descriptive and infe-rential.

True/False

Q.1.    A set containing all possible outcome of a trial in a study is known as “Sample space”.
Q.2.    If we are interested in observing the blood type of a child born in a hospital then sample space may be a set S is of the form S = {A, B, AB, O}.
Q.3.    If outcomes are continuous in nature, the sample space is specified as an interval that contains all possible outcome values.
Q.4.    When the occurrence of one event excludes the occurrence of other events, the events are said to be “Mutually exclusive events”.
Q.5.    In case of testing the type of blood of a newly born baby in a hospital, the blood types A, B, AB, O or gender (male, female) are exclusive events.

Chi-square test is a non-parametric test as the null hypothesis does not contain any assumption regarding the population parameter. Chi-square is also written as c₂. In this chapter two uses of chi-square test will be dealt with. They are the following :