Sampling in Research Series 1: Basic Concepts in Estimating Sample Size

1Former HOD, Community Medicine, BHU, Varanasi and Former Dean FMHS and Dean Academic affairs SGT University. 2Assistant Professor, Community Medicine, FMHS, SGT University, Gurgaon. DOI: https://doi.org/10.24321/2394.6539.202003 Sampling is a process/ method of drawing a representative group of units or cases from a particular population. Sampling is based on probability theory.It is very important for each researcher in all fields to understand and select a statistically and epidemiologically valid sample; which has become extinct from the cognitive arena of researcher today.Some people take a thumb rule of 100 samples. If asked why 100, why not 99 or 101; usually there is no answer.If there is no valid response to “why this number of sample”, then the study is to be discarded.Even many journals today publish papers without proper sample size being considered in the study. In fact such journals should also be made to stop publishing. Thus it was imperative to highlight the value of sample size calculation as CME. Sampling based on probability theory determines result of chance.The methodology used to select a sample from a larger population includes special procedure and calculations that needs to be followed.


Introduction
Quality assurance in epidemiological and managerial studies or studies related to Health System Research (HSR)or any kind of study, survey or research methodology, sampling selection of a subset of individuals from within a statistical population to estimate characteristics of the whole population is Vital and unfortunately not followed in most of the theses or researches today. One can observe irrational sampling studies being published also in many journals some also call them as International journal erroneously. Unfortunately, the researchers do not understand the advantages of sampling such as lower cost, faster data collection as well as quality assurance of a study without which it loses the title as a "Study". Therefore, in most of the cases in daily life, business and industry the information are gathered by means of proper Sampling method with adequate size for accepting the study. The results of a properly taken sample enable the investigator to arrive at generalisations that are valid for the entire population. The process of generalising sample results to the population is called Statistical Inference.

Basic Terminology
Before discussing the sampling theory and distribution of a statistic, we shall be discussing basic definitions of the important terms which are very helpful to understand the fundamentals of statistical inference.

Population
In general sense "population means a large group of people who live in a particular geographical area. For example, J. the group of people who live in Uttarakhand, the group of people working in a college, students enrolled in the MBBS programme in any Medical college etc. In statistics, population is a collection or group of individuals/ items/ units/ observations under study. For example, the collection of books in a library, the particles in a salt bag, the rivers in India, the students in a classroom etc. are considered as populations in Statistics. The population is classified into two types as given below.

Finite and Infinite Population
A countable number of units or observations seen in a population, is called a "finite population", say 1.3 crore Indians,30 patients of Gall bladder cancer…and so on. Even non-living thing studied can also be named as "finite population" such as books in the library, number of trees in Tarai or Trains of India etc.
On the other hand, if a population contains uncountable units or observations, it is entitled as"infinite population". The grains of Bengal grams in a bag and the stars on the sky (as compared to finite Moon in the sky) etc. are "infinite population".
Often times people take a sample of 100. When asked why 100 why not 99 or 101. There is usually no answer to this question. Sample size or type cannot be taken from the air, it must be derived from statistics.

Normal Distribution
As Per the Normal Distribution curve, the distribution of observations follows a rule approximately 68% of all observations fall within±1 SD (standard deviation) of the mean. Approximately 95% of all observations fall within ± 2 SD of the mean. Approximately 99.7% of all observations fall within±3 SD of the mean. Figure number one depicts this concept.
part of the population is selected from the population which is called a sample. Thus, the sample can be defined as: "A part/ fraction/ subset of the total population."

Statistical Errors
The inaccuracies or errors in any statistical investigation, i.e., in the collection, processing, analysis and interpretation of the data may be broadly classified as follows: • Sampling errors • Non-sampling errors

Sampling Errors
In a sample survey, since only a small portion of the population is studied, its results are bound to differ from the census results and thus have a certain amount of error. This error would always be there, no matter that the sample is drawn at random and is highly representative. This error is attributed to fluctuations of sampling and is called sampling error. Sampling error due to the fact that only a subset of the population has been used to estimate the population parameters and draw inferences about the population. Thus sampling error is present only in a samples survey and is completely absent in census method.
Sampling errors are primarily due to the following reasons: • Faulty selection of the sample • Substitution • Faulty demarcation of the sample units • Error due to bias in the estimation method • Variability of the population

Non-Sampling Errors
Non-sampling errors are not mistakes attributed to chance and are a consequence of certain factors which are which were otherwise within human control. In other words, they are due to some human errors during the study at some stage or other and are possible to be traced but created due to ignorance or circumvention. This could be undertake at any stages such asplanning, execution, management or evaluation of the study, data collection or analysis or even report writing. Someimportant factors/ situationsof nonsampling errors are given below: • Faulty planning of the study • Faulty definitions of disease or situation. A good example is the definition of diarrhoea in under-five. It's now as per UNICF, " If the mother says diarrhoea in under-five, it is diarrhoea", while most people take diarrhoea as per frequency or constituents of the stool of the child. If the definition is not used properly entire data collected will be lost. Same as the definition of ORS since the homemade sugar-salt solution is not ORS as per the belief of many! • Incomplete sampling frame

Types of Sampling
There are two major types of sampling i.e. probability and non-probability sampling,

Probability Sampling
Probability sampling is a scientific technique of drawing samples from the population following the role of chance for each unit pre-assigned to be included in the sample with a Chance otherwise known as "probability". In such a sample, each sample has an equal probability of being chosen in the sample lot.

Non-Probability Sampling
There are five types of non-probability sampling technique that can be taken as adequate. However one has to consult an Epidemiologist who can advise the type of non-probability sample that can be chosen for the study. They could be convenience sampling, purposive sampling and snowball sampling, etc. Type of snowball sampling in-hospital study and community settings differ in methodology.

Sample Size Calculation
For quantitative analysis we have a given formula to calculate the appropriate sample size:

N=
Where z=confidence interval, p=prevalence rate, q=1-p and D=precision error.
Example 1: A study is to be performed to determine a certain parameter in a community. From a study a standard deviation of 46 was obtained. If a sample error of up to 5 is to be accepted. How many subjects should be included in this study at 95% level of confidence?
Example 2: It was desired to estimate the proportion of anaemic children in a certain preparatory school. In a similar study at another school, a proportion of 28 % was detected. Compute the minimal sample size required at a confidence limit of 95% and accepting a difference of up to 4% of the true population. For the matter of Error, it cannot be more tha 10% either. Then

Solution"
N= or = N= =484.04 ~ 484 or its usually taken as 4pq / L 2 where p=Prevalence; q=!00 or 1 -Prevalence; and L = permissible limit of error. However permissible limit of error should maximum be within 10%, a higher value of which always damages the quality of the study . If Error is taken as 5%, it means 5% of the prevalence ie 0.04. Taking the absolute value of 5 is incorrect and cannot be taken since that introduces Sampling Error. Today many wise investigators make mistake at this point by taking Absolute value of the error. Let us examine in the same example if the absolute value of 4 instead of 0.04 the sample size will become =0.032 as the sample size in place previous sample as 484. So one has to be careful in calculating the Sample Size. Thus n=z 2 .(1-P) /e 2 P)=4610 women Example 5: The prevalence of Alcoholism in a village was seen to be 25%. The villagers were given health education in different modalities and style for one year. How many people should be studied to see the reduction in alcoholism, is it is desired to be 90% sure of detecting a rate of 20% at 5% level of significance? Solution: Earlier study rate= (P 0 )=25%, Anticipated rate=(P 1 )=20%, Level of significance needed (α)=5%, Power of test (1-β)= 90% Thus n={(1-α)[P 0 (1-P 0 ) +Z (1-β).P α(1-Pα )} 2 (P 0 -P α ) 2 = 601

The usefulness of the Pilot Study
This sampling technique helps to gain initial primary data J. Adv. Res. Med. Sci. Tech. 2020; 7(1)   ISSN: 2394-6539  DOI: https://doi.org/10.24321/2394.6539.202003 about a topic. These findings can serve as pointers and help the investigator to decide the further line of action. This is more important for evaluating a probable prevalence of a diseases/condition if it is not known. But pilot study data and the sample size calculated thereof are not valuable for universal use and remains specific for the particular area of study. But a number of pilot studies taken at different situations can be further authenticated by a Meta-analysis of the multiple observations even if taken by pilot studies.

Census Method
In the census method, we resort to 100% inspection and enumeration of each and every unit of the population. The census method seems to provide the more accurate and exact information as compared to sample enumeration as the information. Moreover, it affords a more extensive and detailed study. For instance, the population census conducted by the Government of India every ten years collects information not only about the population but also obtains data relating to age, marital status, occupation, religion, education etc.

When to Use Census Method?
Census method is recommended in the following situations: 1. If the information is required about each and every unit of the population, there is no way but to resort to 100% enumeration. 2. In any manufacturing process in industry, 100% enumeration should be taken recourse to under the following condition: a. The occurrence of a defect may cause loss of life or serious casualty to personnel. b. A defect may cause serious malfunction of the equipment.
Quantitative Research-Sample Size: When conducting probability sampling it is important to use a sample size that is appropriate to the aims and objectives of the research.
A common error is to assume that the sample should be a certain percentage of the population, for example, 10%. In reality, there is no such relationship and it only the size of the sample that is important.
A probability sample size of 100+ is considered a large enough sample to conduct Statistical analysis.

Statistics and Samples
1. When presenting your research you need to be able to demonstrate, how representative of the whole population the sample data you have collected is. 2. There is two statistical tests used to do this: • Standard Error

Standard Error
• Using the standard deviation of the population and the sample size a statistical calculation can measure the degree of error likely to occur between the results of a sample and the results of a census, this is call the standard error? • The larger the sample the lower the standard error.
• When a probability sample of 100+ is undertaken the distribution can usually be assumed to be normal • When the sample has normal distribution, we can use the z score approach to obtain confidence limits for the sample mean.

Standard Error of Mean
The square root of the variance of the sample means: SE of sample mean = SD/ √n SE of sample proportion = Application of Standard Error of Mean: • To determine whether a sample is drawn from the same population or not when its mean is known. • To work out the limits of desired confidence within which the population mean should lie.
Confidence Interval or Fiducial Limits: Confidence limits are two extremes of measurements within which 95% of observations would lie.
• Lower confidence limit=mean -(t 0.05 × SEM) • Upper confidence limit=mean + (t 0.05 × SEM) The important difference between 'p' value and confidence interval is confidence interval represents clinical significance and 'p' value indicates statistical significance.