Tests & Tests & MeasurementsMeasurements

Furr & BacharachFurr & BacharachChapter 2Chapter 2

ScalingScaling

Measurement (Stevens, Measurement (Stevens, 1946)1946)

►““Measurement is the assignment of Measurement is the assignment of numerals to objects or events numerals to objects or events according to rules.”according to rules.”

►F&B -F&B - “Definition is a beehive “Definition is a beehive swarming with conceptual problems.”swarming with conceptual problems.”

Stevens (1946)Stevens (1946)

►“Measurement is never better than the empirical operations by which it is carried out, and operations range from bad to good. Any particular scale, sensory or physical, may be objected to on the grounds of bias, low precision, restricted generality, and other factors, but the objector should remember that these are relative and practical matters and that no scale used by mortals is perfectly free of their taint.”

ScalingScaling

►The way in which numerical values are The way in which numerical values are assigned to psychological attributesassigned to psychological attributes

Scale – How much of the Scale – How much of the attribute?attribute?

►Personality traitsPersonality traits►AttitudesAttitudes► Intelligence Intelligence

►Unobservable psychological Unobservable psychological characteristics:characteristics:

Yes (1)Yes (1) No (2)No (2)

11 2 2 33 44 55

Properties of the numbers used Properties of the numbers used in the construction of scalesin the construction of scales

► IdentityIdentity►OrderOrder►QuantityQuantity

Meaning of zero in scales?Meaning of zero in scales?

►Complex numberComplex number► Implications for interpretation of test Implications for interpretation of test

scoresscores

Property of IdentityProperty of Identity

►Most fundamental form of measurementMost fundamental form of measurement Ability to reflect Ability to reflect samenesssameness vs vs differentnessdifferentness

►CategoriesCategories Within category – Within category – samenesssameness

►Share a psychological featureShare a psychological feature

Between categories – Between categories – differentnessdifferentness ►Those not in category do not share features with Those not in category do not share features with

those in the category…those in the category…

Oppositional ChildrenOppositional Children

► Children in the category - Children in the category - Oppositional Oppositional Frequently say “no” in response to most requests Frequently say “no” in response to most requests

from authority figures and do not complete those from authority figures and do not complete those requestsrequests

Child often tantrums when authority figures Child often tantrums when authority figures attempt to enforce complianceattempt to enforce compliance

► Children NOT in the categoryChildren NOT in the category Unlikely to do the above two thingsUnlikely to do the above two things Generally cooperative with requests and typically Generally cooperative with requests and typically

emotionally stable & cooperative even when emotionally stable & cooperative even when asked to do non-preferred taskasked to do non-preferred task

Two CategoriesTwo Categories

► Sort children into two categoriesSort children into two categories

1.1. Oppositional ChildrenOppositional Children

2.2. Non-oppositional ChildrenNon-oppositional Children

Characteristics of CategoriesCharacteristics of Categories

Characteristics of CategoriesCharacteristics of Categories

►Mutually exclusiveMutually exclusive►Exhaustive – Oppositional, non-Exhaustive – Oppositional, non-

oppositional, & other(???)oppositional, & other(???)►Behaviors within the category must Behaviors within the category must

meet the property of identity:meet the property of identity: All people within the category must be All people within the category must be

“identical” with respect to the feature “identical” with respect to the feature reflected by the category.reflected by the category.

Characteristics of CategoriesCharacteristics of Categories

Three Categories?Three Categories?

1.1. OppositionalOppositional

2.2. CooperativeCooperative

3.3. Not actively defiant, but slow to Not actively defiant, but slow to cooperatecooperate

► Remains categorical and all three rules Remains categorical and all three rules must be met – 1)mutually exclusive, must be met – 1)mutually exclusive, 2)exhaustive, & 3)identical with respect 2)exhaustive, & 3)identical with respect to the feature reflected by the category…to the feature reflected by the category…

Numerals with respect to Numerals with respect to identity?identity?

► Reflect identity of individuals within a Reflect identity of individuals within a categorycategory

► Numerals do not have true mathematical Numerals do not have true mathematical value – value – no quantitative significanceno quantitative significance……

1.1. OppositionalOppositional

2.2. CooperativeCooperative

3.3. Not actively defiant, but slow to Not actively defiant, but slow to cooperatecooperate

Property of OrderProperty of Order

►Conveys information about the relative Conveys information about the relative amount of an attribute amount of an attribute

►Numerals indicate the Numerals indicate the RANK ORDERRANK ORDER of a person relative to other peopleof a person relative to other people

►Relative amount of the attribute (in Relative amount of the attribute (in comparison to others)comparison to others)

NumeralsNumerals used to rank order used to rank order

►Numerals serve as labelsNumerals serve as labels►NO INFORMATION about the exact NO INFORMATION about the exact

amount of the attribute is conveyed…amount of the attribute is conveyed…

11 22 33 44 55 66 77 88 99

Property of QuantityProperty of Quantity

►MagnitudeMagnitude of difference in people is of difference in people is reflected in numeralsreflected in numerals

►Numerals reflect Numerals reflect real numbersreal numbers►““1” = size of basic unit on a scale1” = size of basic unit on a scale►Unit = unit of measurement; Unit = unit of measurement;

standardized quantitystandardized quantity

Standardized QuantitiesStandardized Quantities

►PoundsPounds► InchesInches►SecondsSeconds

Real NumbersReal Numbers

►Continuous – can be divided into Continuous – can be divided into infinitely small parts (or multiplied)infinitely small parts (or multiplied)

►Different than category labels or Different than category labels or numbers used to “rank order”numbers used to “rank order”

►Can be used to measure the Can be used to measure the quantityquantity of the attributeof the attribute

Real Numbers & Psychological Real Numbers & Psychological AttributesAttributes

►Assumption: Test scores have the Assumption: Test scores have the property of quantityproperty of quantity Seldom a good assumption…Seldom a good assumption…

The Number ZeroThe Number Zero

The Number ZeroThe Number Zero

► MeaningsMeanings1.1. Attribute has no existenceAttribute has no existence

2.2. Arbitrary zero: 0Arbitrary zero: 0oo F, Z score mean = 0 F, Z score mean = 0

► Serious problem:Serious problem: Is ZERO relative or absolute?Is ZERO relative or absolute?

Is ZERO relative or absolute?Is ZERO relative or absolute?

►88thth grader grader American history testAmerican history test Score of 0Score of 0 NOT THE CASE that the 8NOT THE CASE that the 8thth grader knows grader knows

nothing of American historynothing of American history►Test has an Test has an absolute zeroabsolute zero = no = no

questions answered correctly; questions answered correctly; howeverhowever,,

►History ability of 8History ability of 8thth grader is likely not grader is likely not equal to zeroequal to zero

Interpretation of Test ScoresInterpretation of Test Scores

► Influenced by type of zero associated Influenced by type of zero associated with the testwith the test

► If absolute zero, can complete If absolute zero, can complete arithmetical operations of arithmetical operations of multiplication and divisionmultiplication and division

► If relative zero, can If relative zero, can onlyonly complete complete arithmetical operations of addition and arithmetical operations of addition and subtractionsubtraction

The Units of Measurement in the The Units of Measurement in the Measurement of AttributesMeasurement of Attributes

Units of Measurement - Units of Measurement - DefinedDefined

►Must be clearly definedMust be clearly defined In physical measurement this is readily In physical measurement this is readily

apparent apparent ►Length of board Length of board units of measurement = units of measurement =

inches, centimetersinches, centimeters

In psychological measurement, units of In psychological measurement, units of measurement are often less obvious measurement are often less obvious ►Measure shyness, impulsiveness, attention, Measure shyness, impulsiveness, attention,

intelligence intelligence units of measurement? units of measurement?

Example – new unit of Example – new unit of measurementmeasurement

►Rod:Rod: Xrod =Xrod =

Xrods:Xrods:

XrodXrod

►Unit was arbitraryUnit was arbitrary►However, once selected, the unit was However, once selected, the unit was

used to measure length in “units of used to measure length in “units of xrods”xrods”

Arbitrary Units of Arbitrary Units of MeasurementMeasurement

►Three ways in which a unit may be Three ways in which a unit may be arbitraryarbitrary Unit size: Unit size:

►xrod length completely arbitraryxrod length completely arbitrary

Not tied to any one type of objectNot tied to any one type of object►Xrod can be used to measure spatial extent of any Xrod can be used to measure spatial extent of any

objectobject

Used to measure different features of the Used to measure different features of the objectobject►Used xrod to measure length, but could also use it Used xrod to measure length, but could also use it

to measure weight…to measure weight…

Standard MeasuresStandard Measures

► Examples (physical attributes)Examples (physical attributes) Pounds, liters, inches, centimeters, millisecondsPounds, liters, inches, centimeters, milliseconds These are arbitrary units These are arbitrary units

► flexibility & generalityflexibility & generality

► Examples (psychological attributes)Examples (psychological attributes) IQ pointsIQ points Arbitrary unit of measurement Arbitrary unit of measurement only in terms of only in terms of

unit sizeunit size, but IS NOT tied to a specific object or , but IS NOT tied to a specific object or dimensiondimension

That is, no referent beyond the test itself (no That is, no referent beyond the test itself (no “intelligence” device or object in the brain)“intelligence” device or object in the brain)

Additivity & CountingAdditivity & Counting

►AdditivityAdditivity Unit size must remain constant – a 1 unit Unit size must remain constant – a 1 unit

increase at one point must equal a 1 unit increase at one point must equal a 1 unit increase at any other point in the increase at any other point in the measurement processmeasurement process

Size of the unit cannot vary and must Size of the unit cannot vary and must remain constant – measure 1 dimension remain constant – measure 1 dimension onlyonly►Whether the piece of wood is wet or dry, hot Whether the piece of wood is wet or dry, hot

or cold, its length in xrods does not varyor cold, its length in xrods does not vary

Conjoint MeasurementConjoint Measurement

►two attributes could be simultaneously two attributes could be simultaneously quantified quantified if and only ifif and only if certain ordinal certain ordinal relations held upon a third attribute relations held upon a third attribute

Conjoint Measurement, Conjoint Measurement, exampleexample

Measurement ParadoxMeasurement Paradox

Measurement ParadoxMeasurement Paradox

►Differing amounts of a psychological Differing amounts of a psychological attribute can be translated into numbers attribute can be translated into numbers that “measure” the attributethat “measure” the attribute We do not directly observe the attribute – not We do not directly observe the attribute – not

measuring OBJECTSmeasuring OBJECTS Numbers (IQ points) have the appearance of Numbers (IQ points) have the appearance of

arbitrary unitsarbitrary units►Are arbitrary only in the sense of sizeAre arbitrary only in the sense of size

Numbers are not useful except in relation to the Numbers are not useful except in relation to the test being used (no flexibility or generality); e.g. test being used (no flexibility or generality); e.g. IQ points – can’t take IQ points over to the IQ points – can’t take IQ points over to the mechanical aptitude test and measure that…mechanical aptitude test and measure that…

Counts: Counts: Qualification as MeasurementQualification as Measurement

► We count things (paper clips, lever presses), not We count things (paper clips, lever presses), not attributesattributes Counting in and of itself may not qualify as measurement Counting in and of itself may not qualify as measurement

(25 paper clips or 25 lever presses)(25 paper clips or 25 lever presses) Counts as measurement when count reflects a feature or Counts as measurement when count reflects a feature or

attribute of object (# of paper clips in the supply closet, # attribute of object (# of paper clips in the supply closet, # of lever presses by a single rat during a training session)of lever presses by a single rat during a training session)

► Count correct answers on a math test (amount of Count correct answers on a math test (amount of math knowledge). The psychological attribute is math knowledge). The psychological attribute is also amount of math knowledge. also amount of math knowledge. Not counting attribute (math knowledge) directly (the Not counting attribute (math knowledge) directly (the

attribute of math knowledge is unobservable and not attribute of math knowledge is unobservable and not directly measurable); RATHER, counting correct answers…directly measurable); RATHER, counting correct answers…

Scales of MeasurementScales of Measurement►ScalingScaling

Linking numbers or symbols to behavioral Linking numbers or symbols to behavioral observations to create a measureobservations to create a measure

►Scaling = measurementScaling = measurement

Stevens (1946)Stevens (1946)

TEXTTEXT

Principle

Level of Measurement

Nominal Ordinal Interval Ratio

Identity X X X X

Order X X X

Quantity X X

Absolute zero

X

Example Sex Class rank Temperature Distance

Nominal ScalesNominal Scales

Principle

Level of Measurement

Nominal

Identity X

Order

Quantity

Absolute zero

Example Sex

Ordinal ScalesOrdinal Scales

Principle

Level of Measurement

Ordinal

Identity X

Order X

Quantity

Absolute zero

Example Class rank

Interval ScalesInterval Scales

Principle

Level of Measurement

Interval

Identity X

Order X

Quantity X

Absolute zero

Example Temperature

Ratio ScalesRatio Scales

Principle

Level of Measurement

Ratio

Identity X

Order X

Quantity X

Absolute zero

X

Example Distance

Nominal ScalesNominal Scales

► Identify groups (or categories) of peopleIdentify groups (or categories) of people► Unique participant number string Unique participant number string

(########) has one member of the (########) has one member of the string assigned based on gender string assigned based on gender (#######(#######11), (#######), (#######22), ), (#######(#######33) )

1.1. MaleMale

2.2. FemaleFemale

3.3. Not reportedNot reported1. Young (<2) 2. Old (>70)

Ordinal ScalesOrdinal Scales

►Qualitative difference in amountsQualitative difference in amounts Labels indicate relative POSITION or Labels indicate relative POSITION or

RANK, RANK, notnot how muchhow much of the quantity is of the quantity is possessedpossessed

►Property of orderProperty of order►Trumpet players – playing ability or Trumpet players – playing ability or

musicianshipmusicianship

Interval ScalesInterval Scales

►Constant distance between unitsConstant distance between units►Arbitrary zeroArbitrary zero►Few psychological tests can be Few psychological tests can be

thought to yield interval-level scoresthought to yield interval-level scores

Ratio ScalesRatio Scales

►Absolute zero pointAbsolute zero point►Distance between two object = 0Distance between two object = 0

No distance between objectsNo distance between objects

►Time between two events = 0Time between two events = 0 No time between eventsNo time between events

►Allows additivity & multiplicative Allows additivity & multiplicative interpretations (20 miles x 2 = twice as interpretations (20 miles x 2 = twice as far)far)

►No psychological tests are on ratio scalesNo psychological tests are on ratio scales

ZeroZero

►Distinguish between zero associated Distinguish between zero associated with test and the zero associated with with test and the zero associated with the characteristic we think we are the characteristic we think we are measuringmeasuring

StevensStevens

►““Stevens’s rules for assigning symbols, Stevens’s rules for assigning symbols, including numbers to behavior including numbers to behavior observations used as tests, should be observations used as tests, should be taken as heuristic devices [taken as heuristic devices [informalinformal] ] rather than algorithms [rather than algorithms [formalformal] for ] for scaling.”scaling.”

► Inexact – fudge factorInexact – fudge factor

Psychological ScalesPsychological Scales

►Represent differences among peopleRepresent differences among people

Category 5… type of scale?Category 5… type of scale?

Saffir-Simpson Hurricane Scale

Category

Wind speed Storm surge

mph ft

5 ≥156 >18

4 131–155 13–18

3 111–130 9–12

2 96–110 6–8

1 74–95 4–5

Additional classifications

Tropical storm 39–73 0–3

Tropical depression 0–38 0