eChapters

People all around the world have been conducting genetic association studies for a long time. Still the literature to practically understand the statistical analysis of such data using software packages is scarce. Statistical analysis of molecular data is a vast topic. So we will be undertaking one of the important basic and preliminary topics “Statistical analysis of genetic association studies among unrelated individuals” in this chapter. Overall, we will be discussing population association studies in which qualitative and quantitative data on unrelated individuals for various economic traits is collected and these individuals are also typed at a number of Single Nucleotide Polymorphism (SNP) markers. This excludes family-based association studies, admixture mapping or linkage studies. Various software packages used in this chapter to analyze molecular data of a population are given below in Table 1.

Statistical Inference

It is the process of drawing valid inference or conclusion about the population on the basis of a sample drawn from the population. In other words, Statistical inference is the process of using a sample to infer the properties of a population. It requires studies pertaining to three aspects:

• Point Estimation • Interval Estimation

• Testing of Hypothesis • Point Estimation: It is to find a single value, which is used as an estimator of unknown parameter. For example, sample mean may be used as an estimate of the population mean µ.

• Interval Estimation: It is to find confidence limits based on sample values, within which the unknown parameter lies with confidence coefficient described in probability term. In other words, we estimate an unknown parameter using an interval of values that is likely to contain the true value of that parameter and state how confident we are that this interval indeed captures the true value of the parameter.

• Testing of Hypothesis: The tests used to ascertain whether the differences between estimator and the parameter or between two estimators are real or due to chance are called tests of hypothesis or tests of significance. In other words, the procedures, which enable us to decide whether to accept or reject hypothesis, are called tests of hypothesis or tests of significance. Two types of problems mainly arise in tests of significance:

• Testing the significance of the difference between estimator and the pa rameter.

Introduction to Statistics

The word statistics has been derived from Latin word “status”, the German word “statistik”, the Italian word “statistica” or the French word “statistique”. During early periods, it was used by emperors and kings only for collecting the information about state and other information which was needed about their people, their number, land, status of people, revenue of the state etc. Gradually, it meaning and uses extended and there onwards its nature changed.

Some scientist/ researchers were given definition of statistics as per time and place.

• An ancient date, A.L. Bowley gives some definitions according their use and place. He seems to be inadequate because these do not include all aspects of statistics. As per time, place or requirement, he gave three definitions of statistics:

• Statistics may be called the science of counting

• Statistics may rightly be called the science of average

• Statistics as the science of counting and average

Stem cell research is one of the most important and promising areas of research in biological science in recent times. The stem cell technology has revolutionized modern biology. It is playing a major role for improving human as well as animal health by controlling number of incurable degenerative diseases and restoration of damage to organs via transplantation therapy. Stem cells are a specific group of cells that remain in an undifferentiated state and can be found in embryos, fetuses and adult individuals. Stem cells are primitive, undifferentiated cells present in almost every tissue that have the ability to differentiate into different types of tissue like bone, tendon, ligament, cartilage etc. Bone marrow stem cell is the ideal stem cell that is unspecialized and can undergo lineage specific differentiation into bone or blood cells under different signals or environment with new special function. Several verities of stem cells have been isolated and identified in vivo and in vitro. The two broad types of mammalian stem cells are embryonic stem cells that are found in blastocyst, and adult stem cells that are found in adult tissue, bone marrow.

Introduction
Stem cells are the undifferentiated cells of the body which have inherent capacity to differentiate and proliferate into any cell type under programmed command. These cells perpetuate themselves through self-renewal by programmed pathways to give rise to new adult cells (Remya et al., 2014). According to plasticity, the stem cells are categorised into totipotent, pluripotent, multipotent and unipotent. Broadly they can also be classified as embryonic stem cells and adult somatic stem cells. Totipotent stem cells are those which can give rise to all types of cells along with placental membranes whereas pluripotent stem cells give rise to all cell types except placental membranes. The multipotent set of stem cells can give rise to more than one cell type and unipotent ones can differentiate into only one type of cells. Mesenchymal stem cells (MSCs), which fall under the multipotent stem cells category, have the multilineage potential to differentiate into many mesenchymal lineages like osteocytes, chondrocytes, adipocytes etc. (Kim et. al., 2013). They can be easily isolated from the adipose tissues, bone marrows, muscles, liver, umbilical cords, lungs etc. and can be easily proliferated in vitro. MSCs were first identified from the bone marrow in a pioneering study by Fredenstein et al (1976) who isolated the bone forming progenitor cells from rat bone marrow. Although the MSCs represent very small fraction of 0.001-0.01% of total nucleated marrow cells, they can be isolated and differentiated into multiple lineages under defined culture conditions. They are generally isolated from iliac crest, tibial and femoral marrow compartments along with thoracic and lumbar spine in large animals. Stem cells have been used for cellular therapy and tissue engineering for quite a period of time owing to their regeneration potential. They are unspecialized cells with the ability to renew themselves for long periods without significant changes in their general properties. These cells under the influence of appropriate signals can differentiate into specialized cells with a phenotype distinct from that of the precursor (Kim et al., 2013). It may be that stem

Identify main reasons and purpose of curriculum development and key areas needed by stakeholders for curriculum change. This can be done by adopting participatory methods like Focus group discussion/ field level workshops. Carry out stakeholder analysis and identify specific potential areas suggested by stakeholders to be included in the curriculum for better performance at the f ield level and their likely role in the process. In order to encourage greater stakeholder involvement invite stakeholders such as staff and students from various farm universities, field extension workers and farmers.

CHI- SQUARE TEST (χ₂ TEST)

The chi square test is used to determine if the two attributes are independent of each other. Chi square is a measure to evaluate the difference between observed frequencies and expected frequencies to examine whether the difference so obtained is due to a chance factor or due to sampling error.

In a contingency table if each attribute is divided into two classes it is known as 2×2 contingency table. When one attribute is divided into two classes and another into r or c resultant contingency table is known as r×2 or 2×c contingency table. Here, r denotes the number of rows and c the number of columns. For such data, the statistical hypothesis under test is that the two attribute are independent of one another. To test this hypothesis we use the test statistic

FOR GROUPED DATA

Objective: Data regarding lives of two new models of refrigerators collected in a recent survey are

COEFFICIENT OF VARIATION:

The most important of all the relative measure of dispersion is the coefficient of variation. This word is variation not variance. There is no such thing as coefficient of variance. The coefficient of variation( C.V.) is defined

CORRELATION

In case of bivariate or multivariate normal distributions, we may be interested in discovering and measuring the magnitude and direction of the relationship between two or more variables. For this purpose we use correlation. The sys- tematic interrelationship between the variables is termed as correlation. When only two variables are involved the correlation is known as simple correlation. If more than two variables are involved the correlation is known as multiple correlation. An increase in one variable may cause an increase in the other variable, or a decrease in one variable may cause a decrease in the other vari- able, then they are said to be positively correlated. If a decrease in one variable causes an increase in the other variable or vise versa, the variables are said to be negatively correlated.

F TEST (VARIANCE RATIO TEST)

F distribution is applied in several tests of significance relating to the equality of two sampling variances drawn on the basis of independent samples from a normal population.

KURTOSIS

Kurtosis refers to the extent to which unimodal frequency curve is peaked. Kurtosis is a measure that refers to the peakedness of the top of the curve. Kurtosis gives the degree of flatness in the region about the mode of a frequency distribution.

Types of Kurtosis

There are three categories of kurtosis that can be displayed by a set of data. All measures of kurtosis are compared against a standard normal distribution, or bell curve.

The first category of kurtosis is a mesokurtic distribution. This type of kurtosis is the most similar to a standard normal distribution in that it also resembles a bell curve. However, a graph that is mesokurtic has fatter tails than a standard normal distribution and has a slightly lower peak. This type of kurtosis is considered normally distributed but is not a standard normal distribution.

The second category is a leptokurtic distribution. Any distribution that is leptokurtic displays greater kurtosis than a mesokurtic distribution. Characteristic of this type of distribution is one with extremely thick tails and a very thin and tall peak. The prefix of “lepto-” means “skinny,” making the shape of a leptokurtic distribution easier to remember. T-distributions are leptokurtic.

The final type of distribution is a platykurtic distribution. These types of distributions have slender tails and a peak that’s smaller than a mesokurtic distribution. The prefix of “platy-” means “broad,” and it is meant to describe a short and broad-looking peak. Uniform distributions are platykurtic.

COEFFICIENT OF THE MEAN DEVIATION

A relative measure of dispersion based on the mean deviation is called the coefficient of the mean deviation or the coefficient of dispersion. It is defined as the ratio of the mean deviation to the average used in the calculation of the mean deviation.

MEAN DEVIATION

The mean deviation is the first measure of dispersion that we will use that ac- tually uses each data value in its computation. It is the mean of the distances between each value and the mean. It gives us an idea of how spread out from the center the set of values is.

The mean deviation is based on all the observations, a property which is not possessed by the range and the quartile deviation. The formula of the mean de- viation gives a mathematical impression that is a better way of measuring the variation in the data. Any suitable average among the mean, median or mode can be used in its calculation but the value of the mean deviation is minimum if the deviations are taken from the median.

Objective: The distribution of age of males at the time of marriage was as follows

Measures of Central Tendency

The term ‘Mean” or ‘average’ is something, we have been familiar with from a very early age when we start analyzing our marks on report card. We add together all of our test results and then divide it by the sum of the total number of marks there are. We often call it the average. However, statistically it’s the Mean!

The Median is the ‘middle value’. When the totals of the list are odd, the me- dian is the middle entry in the list after sorting the list into increasing order. When the totals of the list are even, the median is equal to the sum of the two middle (after sorting the list into increasing order) numbers divided by two. Thus, remember to line up values, the middle number is the median!

The Mode in a list of numbers refers to the list of numbers that occur most frequently. A trick to remember this one is to remember that mode starts with the same first two letters that most does. Most frequently - Mode.