# Research & Statistical Methods (Free Access)

Research process is divided in the following manner:

- Research Problem
- Meaning and Characteristics of a Research Problem
- Usually in a question form and should be
- clear
- specific
- answerable
- interconnected
- substantively relevant

- Two or more variables
- Must show some relationship

- Can be tested by empirical methods

- Usually in a question form and should be

- Meaning and Characteristics of a Research Problem

- Hypothesis
- Any testable proposition to a research problem
- Based on literature
- conceptual clarity
- must be testable
- should be economical and parsimonious
- should be related to existing theory or fact
- logical unity
- general inscape
- available scientific tools and techniques
- must be similar to other hypothesis

- 4 major types
- Universal – when the relationship holds true for all variable for any time and place
- Existential – relationship holds true for at least one case
- Based on goal and causation:
- Causal – causal influence in relationship
- Descriptive – shows some characteristic or goal for observation

- Other types are
- Simple – one or two variables
- Complex – more than two variables
- Research – derived from a theory, also known as working hypothesis
- (H
_{0}) Null – Denial of a relationship, also known as no effect, negative difference - (H
_{1}) Statistical (alternate) – existence of a relationship, makes numerical expressions of null and research hypotheses, operational statement

- Any testable proposition to a research problem

- Variables
- Variables are the characteristics or conditions that are manipulated, controlled or observed by the experimenter
- Classification:
- Dependent – is the variable about which the experimenter makes a prediction
- Independent also known as stimulus variable – is the variable the experimenter manipulates, selects and measures for the purpose of producing observable change in dependent variable
- 2 types:
- Type E – directly manipulated
- Type S – manipulated through selection

- Qualitative
- categories that cannot be ordered in magnitude
- precise measurement in numerical terms is not possible

- Quantitative
- categories that can be ordered in magnitude
- Precise measurements in numerical form can be made
- 2 categories
- Continuous variable – categories that can be measured in any arbitrary degree of fineness or exactness
- E.g. marks in an exam, height

- Discrete also known as categorical variable – cannot be measured by fineness because clear gap exists
- E.g. gender, sex, educational level

- Continuous variable – categories that can be measured in any arbitrary degree of fineness or exactness

- Moderator/Mediating variable
- Mediating – it is affected by the IV, it affects the DV, indirectly via the relationship between IV and DV.
- Moderating- it affects the DV but not through any influence of the IV

- Active/Attribute
- Active- the variable manipulated by the experimenter
- Attribute- not manipulated by experimenter

- Reliability [Charles Spearman]
- consistency of scores obtained by the same person when they are examined on a test multiple times
- error of measurement
- It is measured by understanding error scores

- measures of reliability make it possible to estimate what proportion of total variance is error variance (irrelevant conditions)
- Correlational coefficient
- degree of consistency or agreement between two sets of scores

- Types of Reliability
- Test-Retest Reliability
- same test, given twice (correlated)
- interval between the two is kept short
- usually a fortnight

- practice will cause error variance
- due to recall

- temporal stability coefficient
- good for speed and power tests
- Sources of error variance (SOEV)
- Time sampling: errors will occur due to time differences

- Alternate form Reliability
- Also known as parallel form/equivalent form/comparative form
- two different, yet similar tests are administered
- interval is important
- more time and different test reduces coaching or teaching effects

- coefficient of equivalence
- SOEV- immediate – content sampling- dependent on the closeness of the two tests
- SOEV long term – time sampling

- Split Half Reliability
- Also known as coefficients of internal consistency
- one test, divided into parts, usually odd and even, which are then correlated
- coefficient of only half the test
- the longer the test, the higher the reliability
- source of error variance – content sampling
- Spearman-Brown formula can test the coefficient

- Kuder Richardson and Coefficient Alpha
- single administration
- Performance is rising in each item
- When homogenous items present in the test KR<Split half
- For yes-no items, 120 items is considered optimum
- for multiple choice use
- coefficient alpha (Cronbach alpha)

- difference of KR and SH may show in heterogeneity of tests
- KR underestimates coefficient source
- SOEV-content sampling

- Interscorers Reliability
- two or more scorers review the same test
- coefficients of the two are calculated
- useful for subjective tests

- Test-Retest Reliability
- Validity
- what the test is meant to measure and how well it does so
- Types:
- Content validity
- the measurement of whether the test content covers a behaviour that is to be measured
- objectives of the test need to be broad and well covered
- not good for aptitude and personality tasks
- intrinsic, relevance, circular and representativeness
- requires item and sampling validity

- Face validity
- the superficial appearance of what the test measures

- Criterion related validity
- effectiveness of the test in predicting an individual’s performance
- Concurrent – criterion data for measuring performance are already available
- Predictive – criterion may not be presently available but will be available in the near future to make a comparison
- Empirical or Statistical- predicts future behaviour
- Predictive validity is lower than construct

- Construct Validity
- Also known as factorial or test validity
- It encompasses the entire test
- It is the extent to which a test can measure theoretical construct or trait
- 2 types
- Convergent
- should correlate with other related tests

- Divergent
- should not correlate with other unrelated tests

- Mixing the two creates a multi-trait-multimethod matrix

- Convergent

- Content validity

- Types of research designs
- Quantitative
- Hypothesis derived based on an existing theory,
- tested through data analysis
- Identify the cause-effect relationship

- Experimental: is lab based
- measures effects or results on the dependent variable by manipulating the independent variable

- pre-decided steps
- causality between independent variable and dependent variable
- 3 principles
- Replication
- helps revalidation
- identical procedures, place, irrespective of time
- avoids experimental error (based on faulty experimental design)

- Randomization
- it ensures independence of observation
- improves validity

- Local control- done in three ways:
- Grouping- refers to placing similar (homogenous) subjects into a group
- Blocking – creating different blocks for attainment of grouping
- Balancing- grouping and blocking should create designs that are balanced

- Replication
- Non-experimental: No causation or effect but building relationships between many factors
- Methods of Collecting Data
- Survey: Questionnaires are sent to many people to gain information usually in a short space of time
- Diary Method: same questionnaires are sent out to same number of participants at different times

- Quantitative Data Analysis
- Advanced statistical techniques and softwares IBM SPSS/AMOS
- Causal Modeling: relations between given set of variables – helps to tests specific hypothesis
- Mediation: An outcome can be explained by the effect of a third factor known as mediator.
- Sobel’s test: compares difference of the outcome with and without mediators

- Moderation: shows the strength or direction of the relationship of the third variable with the dependent variable

- Hypothesis derived based on an existing theory,
- Qualitative
- Generate and analyse data which are not reducible to numbers
- Focus on meaning and interpretation
- Inductive – theory generating
- Sensitive to the context
- Recognize researcher’s perspective and subjectivity
- Data collection
- Interviews – interactions based on Q&A, structured or unstructured
- Focus Group – group discussion to get opinions
- Naturally occurring data – based on observing people in their natural day-today environment
- Observation – natural, to understand in an interrelated events
- Structured: Created by the researcher to fit a context
- Structured methods of data collection – open ended questionnaire

- Qualitative Data Analysis
- Narrative Analysis – understanding data from stories
- Discourse Analysis – is the understanding that different situations create different meaning
- Archival Research – Using past information such as written stories, past census, personal diaries etc.

- Ethological Research
- scientific and objective study of animal behaviour focused on behaviour under natural conditions and as an evolutionary adaptive traits

- Mixed Method
**–**combining both qualitative and quantitative methods - Triangulation – multiple methods of data collection and analysis to arrive at conclusive results

- Speed and Power tests – no perfect score
- Speed test
- Result is dependent on time
- low item difficulty
- no single trait reliability tasks (odd even, KR)

- Power test
- time limited
- steeply graded difficulty (from easy to hard, usually)
- may include items that are too difficult

- Speed test
**(Study Tip:**Speed = how quickly, infinite time, Power= how many, finite time**)**- Classical test theory
- any observed score is equal to the score (T) plus error score
- errors of measurement are random
- errors of measurement cannot be correlated with other scores

- Item analysis
- A net of procedures, that is applied to know the indices of truthfulness (validity) of the items
- Item analysis shows
- which items are difficult, easy or moderate (index of difficulty)
- ability of the item to discriminate between inferior and superior
- Indicates how well multiple choices create distractions

- This can be done via:
- Structural Equation Modeling
- Hypothesized casual relations

- Item difficulty
- the method to differentiate the correct answer from the incorrect answer
- Value is discerned from the percentage of persons who answer correctly
- Maximum number of discrimination is 50X50=2500, occurs when ID is at 50%
- Must have normal distribution of difficulty
- Variance should be 0.25, Standard Deviation 0.5

- Power test item difficulty
- Power test
- do not have a set time limit
- arranged from easy to hard questions

- Item difficulty
- the difficulty value is discerned by the percentage of individuals who answer the item correctly
- maximum number of discrimination is 50 x 50 = 2500
- this occurs when independent variable is 50%
- must have a normal distribution of difficulty

- Power test
- Empirical method
- P = R/N
- P = index of difficulty
- R = number of correct responses
- N = total number
- For speed test

- P=R/N, where R is number of people with similar attempts

- P = R/N
- Method of judgement
- judgement by experts

- Index of Discrimination also known as Item validity index
- ability of the item to divide between superiors and inferiors
- positively discriminating (correct answers higher in upper group)
- negatively discriminating (correct answer is lower in upper group)
- non-discriminating (equal in both groups)
- these items are usually dropped

- Structural Equation Modeling
- 2 methods of calculating Index of discrimination
- A test of significance of difference between two percentages/propositions
- top 27% and bottom 27%, N = 370 (normal curve)
- using critical ratio
- Guilford suggests using chi square when there are an equal number of people in each group
- Marshall and Hales
- Net D Index of Discrimination

- Correlational Techniques
- each item is validated against internal criteria of total score called item total correlation
- closer relationship suggests better discrimination
- product moment, biserial, point biserial, tetrachoric, and phi coefficients are employed

- Item Characteristics Curve
- the graphic representation of the probability of giving the correct answer to an item as a function of the level of attribute assessed by the test
- used to illustrate discriminator power and item difficulty
- slope = discrimination
- position = difficulty

- used to illustrate discriminator power and item difficulty

- the graphic representation of the probability of giving the correct answer to an item as a function of the level of attribute assessed by the test
- Item Response Theory
- Latent trait theory
- Item characteristic curve
- Each item on a test has an independent item characteristic curve that describes the probability of getting each item right or wrong, given the certain level of the examinee
- IRT > CTT
- It can help in making predictions

- Latent trait theory

- Classical test theory
**Other Statistical Techniques**- Significance of difference between two means – T-Test
- depends on the size of the sample
- related or independent variables
- Process:
- set up null hypothesis → level of significance → SE of difference → compute Z scores or E ratio → retain or reject type

- Chi Square (Helmert; Karl Pearson)
- Often used for goodness of fit
- Is actually a test of significance
- (x
^{2}) – used when data is in frequency or percentage is discrete, in categories, data is non parametric, or to test the goodness of fit - For one variable – (
*x*) distribution can be used to determine how well the experimentally obtained results fit the results expected theoretically^{2} - Degrees of freedom = (
*r – 1)(c – 1)**r*= number of rows in contingency table*c*= number of columns in contingency table

- Same procedure for two independent variables
- First null hypothesis testing by formula then df by critical value

- Contingency Coefficient- measure of correlation between two variables with each going into two or more variables
- 2 x 2 tables with 1 df = Yates correction

- Methods of Correlation
- (Point)Biserial correlation – used when finding results between a continuous variable and an artificially created dichotomy variable
- Dichotomous means a variable that is separated into two categories
- If dichotomy is natural, then point biserial should be used
- Natural dichotomous variables can be divided into two categories only and not more, therefore biserial correlation can be used.
- Point biserial (rbis) is better than biserial (rpbis
*)*because- rpbismakes no assumption of normality
- can be used for regression
- easy and convenient
- standard error can be determined

- Tetrachoric correlation
- when both variables are dichotomous and cannot be expressed in scores
- Artificial dichotomy

- Phi correlation
- when both variables are naturally dichotomous
- useful for item analysis for item-item correlation

- (Point)Biserial correlation – used when finding results between a continuous variable and an artificially created dichotomy variable
- Partial and Multiple correlation
- Partial correlation
- helps estimating independent reliable relationship between any two variables by eliminating and ruling out any undesirable influence of a third additional variable by controlling them
- 2nd order or 3rd order partial correlation include controlling main extra variables

- Partial correlation
- Multiple correlation
- to assess the relationship between the dependent variable and many independent variables

- T Scores [William A. McCall]
- Refers to the normalized standard scores which are converted into a distribution with mean = 50 & SD = 10
- Hence, 0 is 5SD below mean
- 100 is 5SD above mean

- T score = 102 + 50 = 10 [(x – m)/50] + 50

- Refers to the normalized standard scores which are converted into a distribution with mean = 50 & SD = 10
- Analysis of variance (ANOVA)
- One Way ANOVA
- to test the significance of difference between the means of 3 groups
- gives a composite score
- 2 types of variance
- Within group – the average variance of members of each group around their respective group means
- Between group – the variance of group means around the total or composite mean of all groups

- F ratio
- is the critical ratio for determining the significance of the difference between group mean at a given level of significance
- doesn’t tell which group is better, only that they are different
- Procedure:
- computation of total sum of squares
- computation of between group sum of squares
- computation of within group sum of squares
- computation of F ratio group sum of squares
- use of t-test
- if required, when F is significant

- df= N – K
- (N = number of p in sample; k is 1 per group)

- Significance of difference between two means – T-Test

- Two-way ANOVA
- used when there are two experimental variables
- For example Total variance of effectiveness of teaching methods and school would be calculated in the following manner
- variance due to methods alone (1
^{st}IV) - variance due to school alone (2
^{nd}IV) - residual variance called interaction variance (MS)
- Chance
- uncontrollable variance
- merits of the methods
- If null hypothesis is true, variance due to methods is not very different from interaction variance
- same for school (2nd IV)
- this is analysed by F ratio

- variance due to methods alone (1

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