Piaget's stages of Intellectual development

Piaget’s Stages of Cognitive Development

Sensory Motor Period (0 – 24 months)
Stage-Age	Characteristic Behavior
Reflexive Stage (0-2 months)	Simple reflex activity such as grasping, sucking.
Primary Circular Reactions(2-4 months)	Reflexive behaviors occur in stereotyped repetition such as opening and closing fingers repetitively.
Secondary Circular Reactions (4-8 months)	Repetition of change actions to reproduce interesting consequences such as kicking one’s feet to more a mobile suspended over the crib.
Coordination of Secondary Reactions (8-12 months)	Responses become coordinated into more complex sequences. Actions take on an “intentional” character such as the infant reaches behind a screen to obtain a hidden object.
Tertiary Circular Reactions (12-18 months)	Discovery of new ways to produce the same consequence or obtain the same goal such as the infant may pull a pillow toward him in an attempt to get a toy resting on it.
Invention of New Means Through Mental Combination (18-24 months)	Evidence of an internal representational system. Symbolizing the problem-solving sequence before actually responding. Deferred imitation.
The Preoperational Period (2-7 years)
Stage-Age	Characteristic Behavior
Preoperational Phase (2-4 years)	Increased use of verbal representation but speech is egocentric. The beginnings of symbolic rather than simple motor play. Transductive reasoning. Can think about something without the object being present by use of language.
Intuitive Phase (4-7 years)	Speech becomes more social, less egocentric. The child has an intuitive grasp of logical concepts in some areas. However, there is still a tendency to focus attention on one aspect of an object while ignoring others. Concepts formed are crude and irreversible. Easy to believe in magical increase, decrease, disappearance. Reality not firm. Perceptions dominate judgment.In moral-ethical realm, the child is not able to show principles underlying best behavior. Rules of a game not develop, only uses simple do’s and don’ts imposed by authority.
Period of Concrete Operations (7-12 years) Characteristic Behavior: Evidence for organized, logical thought. There is the ability to perform multiple classification tasks, order objects in a logical sequence, and comprehend the principle of conservation. thinking becomes less transductive and less egocentric. The child is capable of concrete problem-solving. Some reversibility now possible (quantities moved can be restored such as in arithmetic: 3+4 = 7 and 7-4 = 3, etc.) Class logic-finding bases to sort unlike objects into logical groups where previously it was on superficial perceived attribute such as color. Categorical labels such as “number” or animal” now available.
Period of Formal Operations (12 years and onwards) Characteristic Behavior: Thought becomes more abstract, incorporating the principles of formal logic. The ability to generate abstract propositions, multiple hypotheses and their possible outcomes is evident. Thinking becomes less tied to concrete reality. Formal logical systems can be acquired. Can handle proportions, algebraic manipulation, other purely abstract processes. If a + b = x then a = x – b. If ma/ca = IQ = 1.00 then Ma = CA. Prepositional logic, as-if and if-then steps. Can use aids such as axioms to transcend human

Basic Statistics

DESCRIPTIVE STATISTICS

 Statistics used to describe the pattern of data. It includes central tendency (mean, median mode), variability (Range, SD, Variance, Skewness, Kurtosis), frequency and percentage distribution.

Correlation: It is the linear relation between the variables. r is the index of correlation. It indicates strength and direction of relation. Direction is indicated by the sign advanced of r. There are few assumptions of r.

a)  The correlation coefficient r assumes that the two variables measured
form a bivariate normal distribution population.

b) Correlation does not measure nonlinear association, only linear association. The correlation coefficient is appropriate only for quantitative variables, not ordinal or categorical variables, even if their values are numerical.

c)Correlation is a measure of association, not causation

d) Scatterplot or scatter diagram: The correlation coefficient r is close to 1 if the data cluster tightly around a straight line that slopes up from left to right. The correlation coefficient is close to -1 if the data cluster tightly around a straight line that slopes down from left to right. If the data do not cluster around a straight line, the correlation coefficient r is close to zero, even if the variables have a strong nonlinear association. 

e) Some scatterplots show curved patterns. Such scatterplots are said to show nonlinear association between the two variables. The correlation coefficient does not reflect nonlinear relationships between variables, only linear ones. For example, even if the association is quite strong, if it is nonlinear, the correlation coefficient r can be small or zero.

f) The correlation coefficient r measures only linear associations: how nearly the data
falls on a straight line. It is not a good summary of the association if the scatterplot has a nonlinear
(curved) pattern.

g) Scatterplots in which the scatter in Y is about the same in different vertical slices are called homoscedastic (equal scatter). Data are homoscedastic if the SD in vertical slices through the scatterplot is about the same, regardless of where you take the slice. Homoscedastic means "same scatter." In contrast, if the vertical SD varies a great deal depending on where you take the slice through the scatterplot, the data are heteroscedastic. The SD is a measure of the scatter in the list. So far, all the plots in this section have been homoscedastic. The next scatterplot shows heteroscedasticity: the scatter in vertical slices depends on where you take the slice.

h) The correlation coefficient r is not a good summary of association if the data have outliers.

REGRESSION
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC374386/


History: http://www.amstat.org/publications/jse/v9n3/stanton.html


INFERENTIAL
Statistics used for testing hypotheses

Correlation







regression
rank order correlation
STATISTICAL INFERENCE
ANOVA
ANCOVA

SPSS
REPORT WRITING

BASIC CONCEPTS OF SURVEY RESEARCH

BASIC CONCEPTS OF SURVEY RESEARCH
Debdulal Dutta Roy
Indian Statistical Institute, Kolkata
Venue: Andhra University
Date: 16.2. 2014

What is Survey ?

•       It is an investigation about the characteristics of a given population by means of collecting data from a sample of that population and estimate their characteristics through the systematic use of statistical methodology or technique.

What is sample survey ?

•       A sample survey is sampling method in which a portion only and not the whole proportion is surveyed.

 Sample

•       It is a subset of a frame where elements are selected based on a randomized process with a known probability of selection.

 •       Representative sample: A representative sample is one that has all the important characteristics of the population from which it is drawn.

 What is sampling frame ?

It is a list of all members of a population used as a basis for sampling.

What is sampling ?

It is the research strategy of collecting data from a part of population with a view to drawing inferences about the whole.

Sample size

The number of sampling units which are to be included in the sample.

Sampling unit

It is one of the units into which an aggregate is divided for the purpose of sampling, each unit being regarded as individual or individuals.

Sampling fraction

The ratio of the sample size to the population size.

Probability sampling

•       In probability sampling, each population element has a known and nonzero probability of being selected. Selection probabilities arise from the use of a randomized procedure, such as random number tables.

•       It requires the existence of a sampling frame from which the sample can be drawn. Major advantage is that statistical theory can be employed to derive the properties of the sample estimators. Bias in sample selection is avoided.

Non-probability sampling

•       It is any form of sampling that fails to meet the conditions for probability sampling.

Non-probability sampling techniques

•         Haphazard, convenience or accidental sampling: The sampled elements are chosen for convenience or haphazardly, with the purpose of making inference about some general population. Examples include a sample of volunteers, street corner interviews, pull-out questionnaires in a magazine.

•         Judgment or purposive sampling or expert choice: Sampled units are selected carefully to provide a ‘ representative sample’. This is possible when expert has a good deal of information about the population element. For example, rather than relying on random choice, the individual is selected purposefully.

•         Quota sampling:  Researcher has quotas of respondents of different types to interview. For example, an interviewer may require 7 men under 55 years and 5 men 55 years or older.

Probability sampling

•        Before I can explain the various probability methods we have to define some basic terms. These are:

•        N = the number of cases in the sampling frame

•        n = the number of cases in the sample

•        NCn = the number of combinations (subsets) of n from N

•        f = n/N = the sampling fraction

•        That's it. With those terms defined we can begin to define the different probability sampling methods.

Simple Random Sampling

•        This is the simplest form of probability sampling. To select a simple random sample you need to make a numbered list of all the units in the population from which you want to draw a sample or use an already existing one (sampling frame).

•        Objective: To select n units out of N such that each NCn has an equal chance of being selected.

NCn = the number of combinations (subsets) of n from N

•        Procedure: Use a table of random numbers, a computer random number generator, or a mechanical device to select the sample.

Systematic sampling

•       Selection : In systematic sampling, sampled units are chosen at regular intervals from the sampling frame. For this method we randomly select a number to tell us where to start selecting individuals from the list.

•       Example:  a systematic sample is to be selected from 1,200 students at a school. The sample size selected is 100. The sampling fraction is 1200/100. The sampling interval is therefore 12. The number of the first student to be included in the sample is chosen randomly, for example, by blindly picking one out of 12 pieces of paper, numbered 1 to 12. If number 6 is picked, then every twelfth student will be included in the sample, starting with student number 6, until 100 students are selected. The numbers selected would be 6, 18, 30, 42, etc.

•       Advantage:  Systematic sampling is usually less time-consuming and easier to perform than simple random sampling.

•       Disadvantage: However, there is a risk of bias, as the sampling interval may coincide with a systematic variation in the sampling frame. For instance, if we want to select a random sample of days on which to count clinic attendance, systematic sampling with a sampling interval of 7 days would be inappropriate, as all study days would fall on the same day of the week, which might, for example, be a market day.

Stratified sampling

•          Assumption:  This technique requires classification of the total population into strata.  Here, sampling frame is divided into strata with assumption that strata will have significant effect on change in dependent variable.

•          Requirement:    It requires large sample. Stratified sampling is only possible when we know what proportion of the study population belongs to each group we are interested in. An advantage of stratified sampling is that it is possible to take a relatively large sample from a small group in the study population. This makes it possible to get a sample that is big enough to enable researchers to draw valid conclusions about a relatively small group without having to collect an unnecessarily large (and hence expensive) sample of the other, larger groups. However, in doing so, unequal sampling fractions are used and it is important to correct for this when generalizing our findings to the whole study population.

•          Example:  A survey is conducted on self-medication practices in a district comprising 20,000 households, of which 20% are urban and 80% rural. It is suspected that in urban areas self-medication is less common due to the vicinity of health centres. A decision is made to include 100 urban households (out of 4,000, which gives a 1 in 40 sample) and 200 rural households (out of 16,000, which gives a 1 in 80 sample). This allows for a good comparison between urban and rural self-medication practices. Because we know the sampling fraction for both strata, the rates for self-medication for all the district households can be calculated. 

Proportional Stratified Random Sampling

•         Proportional or quota random sampling, involves dividing your population into homogeneous subgroups and then taking a simple random sample in each subgroup. In more formal terms:

•         Objective: Divide the population into non-overlapping groups (i.e., strata) N1, N2, N3, ... Ni, such that N1 + N2 + N3 + ... + Ni = N. Then do a simple random sample of f = n/N in each strata.

•         Advantage:  It represent not only the overall population, but also key subgroups of the population, especially small minority groups.

•         Strata properties: Each strata must be homogenous otherwise lot of statistical precisions are required.

Cluster (Area) Random Sampling

•         Use:   when we have to sample a population that's disbursed across a wide geographic region is that you will have to cover a lot of ground geographically in order to get to each of the units you sampled.

•         Advantage:  It reduces travelling  time, money and energy of researcher.

•         Steps:

–      divide population into clusters (usually along geographic boundaries)

–      randomly sample clusters

–      measure all units within sampled clusters

 Multi-Stage Sampling

•         It is the combination of all sampling methods.

•         Example: consider the idea of sampling New York State residents for face-to-face interviews. Clearly we would want to do some type of cluster sampling as the first stage of the process. We might sample townships or census tracts throughout the state. But in cluster sampling we would then go on to measure everyone in the clusters we select. Even if we are sampling census tracts we may not be able to measure everyone who is in the census tract. So, we might set up a stratified sampling process within the clusters. In this case, we would have a two-stage sampling process with stratified samples within cluster samples. Or, consider the problem of sampling students in grade schools. We might begin with a national sample of school districts stratified by economics and educational level. Within selected districts, we might do a simple random sample of schools. Within schools, we might do a simple random sample of classes or grades. And, within classes, we might even do a simple random sample of students. In this case, we have three or four stages in the sampling process and we use both stratified and simple random sampling. By combining different sampling methods we are able to achieve a rich variety of probabilistic sampling methods that can be used in a wide range of social research contexts.

Summary

•       There are two types of sampling techniques – probabilistic and non-probabilistic.

•       Probabilistic sampling requires basic assumption of normal probability curve where as non probability sampling  does not need.