Comprehensive Notes on Tests and Measurement

Why Measurement? An Introduction

• Historical Context of Testing
  • Evidence of testing dates back to 2200 BCE in China, where citizens took civil service exams in writing, arithmetic, horsemanship, and archery.
  • Modern civil service systems in Britain (1830) and America (1889) were modeled after these early competitive systems.
  • Charles Darwin's Origin of Species (1859) catalyzed interest in individual differences, which is the core of most psychological tests.
  • Francis Galton, Darwin's cousin, established the first anthropometric lab to measure physical variables like strength and hand steadiness.
  • James Cattell coined the term "mental test" and founded the Psychological Corporation in the 1920s.
  • Alfred Binet and Theodore Simon created the first intelligence test in 1905 to identify Parisian schoolchildren in need of specialized instruction.
• Definition of a Test
  • A test is a tool, procedure, device, examination, investigation, assessment, or measure of an outcome (usually behavior).
  • Formats range from 50-question multiple-choice exams to 30-minute qualitative interviews.
• Purposes of Testing
  • Selection: Identifying individuals for specific roles (e.g., jet pilots).
  • Placement: Assigning individuals to appropriate levels (e.g., college math classes).
  • Diagnosis: Identifying mental disorders or specific strengths/weaknesses.
  • Hypothesis Testing: Validating "if…then" scientific statements.
  • Classification: Helping individuals choose career paths based on aptitude.
• Modern Economic Scale
  • The Brookings Institution estimates that $1.7 billion is spent annually on assessment for the $K–12$ population in the United States alone.

The Psychology of Psychometrics

Levels of Measurement

• Nominal Level
  • Variables are categorical or discrete.
  • Names or labels are assigned (e.g., "Republican" vs. "Democrat," "Nurse" vs. "Non-nurse").
  • Categories must be mutually exclusive; a score cannot belong to more than one group.
• Ordinal Level
  • Variables are ordered along a continuum.
  • Indicates "more than" or "less than" relationships (e.g., ranking childhood fears).
  • Does not provide information on the distance between ranks.
• Interval Level
  • Values are based on an underlying continuum with equal intervals.
  • Allows for the calculation of differences between scores (e.g., IQ scores where the difference between 100 and 102 is the same as 102 and 104).
  • Does not have an absolute zero.
• Ratio Level
  • Characterized by all properties of nominal, ordinal, and interval scales, plus an absolute zero (absence of the trait).
  • Examples: Height, weight, number of finger taps.
  • Hard to apply to psychological constructs (e.g., a 0 on a spelling test does not mean zero spelling ability).

Reliability: Consistency of Measurement

• The Reliability Formula
  • Observed Score = True Score + Error Score.
  • True Score: The theoretical, 100% accurate reflection of underlying knowledge.
  • Error Score: Difference between observed and true scores, consisting of Trait Error (individual factors like fatigue) and Method Error (situational factors like unclear instructions).
  • Conceptual Reliability Formula: $\text{Reliability} = \frac{\text{True Score}}{\text{True Score} + \text{Error Score}}$.
• Types of Reliability
  • Test-Retest: Consistency over time; measured by correlating scores from two different time points.
  • Parallel Forms: Similarity between two different versions of the same test.
  • Internal Consistency: Whether items in a single test measure one dimension. Tools include:
    • Split-Half: Correlating odd vs. even items.
    • Cronbach’s Alpha ($\alpha$): Correlating each item with the total score for non-binary items (e.g., Likert scales).
    • Kuder-Richardson (KR20): Internal consistency for binary (Right/Wrong) items.
  • Interrater Reliability: Level of agreement between two or more observers. Formula: $\frac{\text{Number of Agreements}}{\text{Number of Possible Agreements}}$.
• The Spearman-Brown Formula
  • Used to correct split-half reliability because shortening a test reduces its reliability. Formula: $r<em>t = \frac{2r</em>h}{1 + r_h}$.

Validity: Accuracy of Measurement

• General Definition
  • The extent to which inferences made from a test are appropriate, meaningful, and useful.
  • A test must be reliable before it can be valid, but a reliable test is not necessarily valid.
• Types of Validity
  • Content Validity: Whether items sample the entire universe of possible items for the domain (essential for achievement tests).
  • Criterion Validity: How well a test correlates with an external criterion.
    • Concurrent: Criterion is measured at the same time.
    • Predictive: Test predicts future performance (e.g., GRE scores predicting grad school GPA).
  • Construct Validity: Most complex; whether a test measures an underlying theoretical construct (e.g., shyness).
    • Multitrait-Multimethod Matrix: Uses multiple traits and methods to establish Convergent (high similarity) and Discriminant (low similarity) validity.

Norms, Percentiles, and Standard Scores

• Percentiles ($P_r$)
  • Indicates the point below which a certain percentage of scores fall. Formula: $P_r = \frac{B}{N} \times 100$, where $B$ is the number of lower values and $N$ is total observations.
• Stanines
  • Divides a distribution into nine equal segments. Mean = 5, SD = 2.
• Standard Scores
  • z Score: Represents the number of standard deviations a score is from the mean. Formula: $z = \frac{X - \bar{X}}{s}$.
  • T Score: Transformed score to eliminate negatives and decimals. Formula: $T = 50 + 10z$.
• Standard Error of Measurement (SEM)
  • Measure of variability in an individual's score upon repeated testing. Formula: $SEM = s\sqrt{1 - r}$.

Item Response Theory (IRT)

• Core Concept
  • Focuses on the characteristics of individual items rather than total scores. Also called "Latent Trait Theory."
• The Item Characteristic Curve (ICC)
  • A graph with $Theta (\theta)$ on the x-axis representing underlying ability and $P(\theta)$ on the y-axis representing the probability of a correct response.
  • Difficulty ($b$): The point on the x-axis where the probability of success is 0.50.
  • Discrimination ($a$): Represented by the steepness of the curve.
  • Guessing ($c$): The probability of low-ability test takers getting the item right by chance.

The Tao and How of Testing: Item Construction

• Short-Answer and Completion Items
  • Best for lower-level thinking (memorization, facts).
  • Advantage: Minimizes guessing (no options provided).
• Essay Items
  • Best for higher-order thinking (synthesis, analysis).
  • Open-ended vs. Closed-ended (restricted) formats.
  • Scoring requires batched grading, model answers, and anonymity to reduce bias.
• Multiple-Choice Items
  • Anatomy: Stem (premise), Key (correct alternative), and Distracters (plausible incorrect options).
  • Difficulty Index ($D$): Percentage who got the item right. $D = \frac{N<em>h + N</em>l}{T}$.
  • Discrimination Index ($d$): How well the item separates high from low performers. $d = \frac{N<em>h - N</em>l}{0.5T}$.
• Matching and True-False
  • Matching: Uses premises (Column A) and options (Column B).
  • True-False: Dichotomous format. Correction for guessing: $CS = R - W$.
• Portfolios
  • Systematic collections of work showing progress over time. Both formative (ongoing) and summative (final) evaluation.

Areas of Assessment

• Intelligence Tests
  • Historically based on Spearman’s g factor (general factor) vs. Thurstone’s Primary Mental Abilities.
  • Robert Sternberg’s Triarchic Theory: Componential, Experiential, and Contextual.
  • Howard Gardner’s Multiple Intelligences: Musical, Bodily-Kinesthetic, Logical-Mathematical, Linguistic, Spatial, Interpersonal, Intrapersonal, and Naturalist.
  • Emotional Intelligence (Goleman): Focuses on self-awareness and empathy.
• Neuropsychological Testing
  • Assessment of cognitive skills relating to brain function.
  • Areas covered: Memory, Language, Visuospatial ability, and Executive Function (e.g., Stroop Test).
• Personality Testing
  • Objective: Clear stimuli (e.g., MMPI-2, NEO-4).
  • Projective: Ambiguous stimuli (e.g., Rorschach Inkblot, Thematic Apperception Test or TAT).
• Career Choices
  • John Holland’s Hexagon (RIASEC model): Realistic, Investigative, Artistic, Social, Enterprising, Conventional.

Legal and Ethical Issues

• Major Legislation
  • NCLB (2002): No Child Left Behind; focused on closing achievement gaps through high-stakes testing.
  • IDEA (1997/PL 94-142): Individuals with Disabilities Education Act; guarantees free appropriate public education in the "Least Restrictive Environment" (LRE).
  • FERPA (1974): Protects the privacy of student education records.
  • Truth in Testing: New York law requiring disclosure of items and scoring processes for admissions tests.
• Ethics
  • Key principles: No physical or psychological harm, informed consent, confidentiality, anonymity, and appropriate use of incentives.