Psychometrics & Statistical Concepts PSYC 101 Dr. Gregg Fall, 2006

Vocabulary Item:1 question or task Scale:Set of items that measure a single trait or characteristic Test:Usually large set of items that measure one or several traits May consist of several scales or “subtests” (IQ; SAT; ACT)

Likert Scale Item with following response forms:Strongly AgreeAgreeDisagreeDisagree [ ] [ ] [ ] [ ] Strongly Agree [ ][ ] [ ] [ ] [ ] [ ] [ ] Disagree

Psychometrics: Test Design Theory-based strategy:Galton Prediction-based strategy:Binet

Psychometrics: Test Design Theory-based strategy: Create items based on theory “Some people are born with an urge to jump from high places.”

Psychometrics: Test Design Prediction-based strategy: 1. Identify criterion group (with trait) & group without trait. 2. Select items criterion group answers differently than non- criterion group.

Psychometrics: Designing an Accurate Test Reliability:Does test consistently measure what it measures? Validity:Does test measure what it aims to measure?

Reliability Does test consistently measure what it measures? Internal consistency Test-retest reliability

Validity Does test measure what it aims to measure? Convergent Validity:Correlations with other measures of same trait. Divergent Validity:Non- correlation with measures of different traits.

Need to Understand Correlation Regression Factor Analysis  Key concept: variance

Types of Variables

Nominal / Categorical: each value is distinct category [gender, blood type, city] Scale / Interval: linear measure, same interval between each value [age, weight, IQ, GPA, SAT, income] Ordinal: ranking, un-equal intervals between values [Likert scale, preference ranking]

Variables & Statistical Tests Variable TypeExampleCommon Stat Method Nominal by nominal Blood type by gender Chi-square Scale by nominalGPA by gender GPA by major T-test Analysis of Variance Scale by scaleWeight by height GPA by SAT Regression Correlation

Strength of association of scale measures r = -1 to 0 to perfect positive correlation -1 perfect negative correlation 0 no correlation Interpret r in terms of variance

Mean & Variance

Survey of Class n = 42 Height Mother’s height Mother’s education SAT Estimate IQ Well-being (7 pt. Likert) Weight Father’s education Family income G.P.A. Health (7pt Likert)

Frequency Table for:HEIGHT Valid Cum Value Label Value Frequency Percent Percent Percent Total Valid cases 42 Missing cases 0

Frequency Table for:HEIGHT Valid Cum Value Label Value Frequency Percent Percent Percent Total Valid cases 42 Missing cases 0 Descriptive Statistics for:HEIGHT Valid Variable Mean Std Dev Variance Range Minimum Maximum N HEIGHT mean

Variance   x i - Mean ) 2 Variance = s 2 = N Standard Deviation = s =  variance

Frequency Table for:WEIGHT Valid Cum Value Label Value Frequency Percent Percent Percent Total Valid cases 42 Missing cases 0 Descriptive Statistics for:WEIGHT Valid Variable Mean Std Dev Variance Range Minimum Maximum N WEIGHT mean

Relationship of weight & height: Regression Analysis

“Least Squares” Regression Line Dependent = ( B ) (Independent) + constant weight = ( B ) ( height ) + constant

Regression line

Regression:WEIGHTonHEIGHT Multiple R R Square Adjusted R Square Standard Error Analysis of Variance DF Sum of Squares Mean Square Regression Residual F = Signif F = Variables in the Equation Variable B SE B Beta T Sig T HEIGHT (Constant) [ Equation:Weight = 3.3 ( height ) - 73 ]

Regression line W = 3.3 H - 73

Strength of Relationship “Goodness of Fit”: Correlation How well does the regression line “fit” the data?

Frequency Table for:WEIGHT Valid Cum Value Label Value Frequency Percent Percent Percent Total Valid cases 42 Missing cases 0 Descriptive Statistics for:WEIGHT Valid Variable Mean Std Dev Variance Range Minimum Maximum N WEIGHT mean

Regression line mean

Correlation: “Goodness of Fit” Variance (average sum of squared distances from mean) = 454 “Least squares” (average sum of squared distances from regression line) = – 295 = / 454 =.35 Variance is reduced 35% by calculating from regression line

r 2 = % of variance in WEIGHT “explained” by HEIGHT Correlation coefficient = r

Correlation:HEIGHTwith WEIGHT HEIGHT WEIGHT HEIGHT ( 42) ( 42) P=. P=.000 WEIGHT ( 42) ( 42) P=.000 P=.

r =.59 r 2 =.35 HEIGHT “explains” 35% of variance in WEIGHT

Heretibility % of variance in measures of a trait (such as height or IQ) that is “attributable to” genes

Multiple Regression Problem: relationship of weight and calorie consumption Both weight and calorie consumption related to height Need to “control for” height

Regression line mean Multiple Regression

Form of relationship --regression line: Weight = 3.3 ( height ) - 73 Each inch of height “adds” 3.3 pounds of weight Strength of relationship -- correlation: r =.59r 2 =.35% Height “explains” 35% of variance in weight

Statistical Significance

What is the probability that the relationship observed in the sample does not exist in the universe from which the sample was drawn? What are the chances that the sample could be a “quirky” one, which doesn’t reflect the real state of affairs in the larger world?

If the probably of having drawn a “quirky,” non-representative sample is less than 5 in 100, the finding from the sample can be said to be statistically significant. p <.05

Stat Sig of Height–Weight Correlation ( sample n = 42 ) In sample, r =.59 What are chances a sample with r =.59 could come from a population in which there is NO correlation between height and weight?

Statistical Significance Need to know: distribution of possible samples of 42 from population in which height and weight are NOT correlated: Sampling Distribution Is probability of drawing a sample in which r =.59 less than.05? r =.59p <.001

Distinguish Between Relationship -- slope of regression line Strength of the relationship – “goodness of fit” -- % of variance explained Statistical significancep <.05

Regression line

Height and Weight Relationship (regression line) Weight = 3.3 Height - 73 Strength of relationship (correlation) r =.59r 2 =.35 35% variance “explained” Statistical significance( p <.05 ) p <.001

Factor Analysis Charles Spearman

Believed IQ inherited Eugenics advocate Created factor analysis: Showed intercorrelation among Binet’s sub-tests Two-factor theory: g + s-s

Survey of Class n = 42 Height Mother’s height Mother’s education SAT Estimate IQ Well-being (7 pt. Likert) Weight Father’s education Family income G.P.A. Health (7pt Likert) How many pieces of cherry pie could you eat if you had to?

HeightFather Height Mother Height WeightPie Pieces Father Educ Mother Educ G.P.A.S.A.T.I.Q.IncomeHealthHappy Height *.57***.59**.57*** F Height *.37* *-.01 M Height Weight *** Pie * F Educ *** *-.06 M Educ G.P.A ***.51*** S.A.T *** I.Q Income Health * Happy 1.0

WeightPie PiecesG.P.A.S.A.T.I.Q.HealthHappy Height.59**.57*** Weight.54*** Pie Pieces * G.P.A..63***.51*** S.A.T..67*** I.Q Health.36*

WeightPie Pieces G.P.A.S.A.T.I.Q.HealthHappy Height.59**.57*** Weight.54*** Pie Pieces * G.P.A..63***.51*** S.A.T..67*** I.Q Health.36*

Three Factors “Size” “Smarts” “Good Life”

Lewis Thurstone Invented factor rotation technique Found 7 factors – “Primary Mental Abilities”

Thurstone: Primary Mental Abilities Verbal comprehension Word fluency Computational ability Spatial visualization Associative memory Perceptual speed Reasoning

Theories of Intelligence Single Galton Cattell Goddard Terman Spearman Herrnstein & Muray Multiple Binet Thurstone Gardner

David Wechsler Developed W.A.I.S. (Wechsler Adult Intelligence Scale)

W. A. I. S.