1 Class 4 Basic Psychometric Characteristics: Variability, Reliability, Interpretability October 15,...

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Class 4

Basic Psychometric Characteristics:Variability, Reliability, Interpretability

October 15, 2009

Anita L. StewartInstitute for Health & Aging

University of California, San Francisco

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Overview of Class 4

Concepts of error, sources of error and bias in measures.

Indicators of variability and reasons for poor variability

Indicators of reliability Interpretability of scores

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Components of an Individual’s Observed Item Score

(Simplistic view)

Observed true item score score

= + error

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Components of an Individual’s Observed Item Score


= + error

“score that would be obtained over repeated testings”

Nunnally, 1994, p211

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Random versus Systematic Error


= + error random

systematic

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Random versus Systematic Error


= + error random

systematic

Relevant to reliability

Relevant to validity

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Components of Variability in Item Scores of a Group of Individuals

Observed true score score variance variance

Total variance (sum of all observed item scores)

= + errorvariance

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Components of Variability in Item Scores of a Group of Individuals


Total variance (sum of all observed item scores)

= +(Random)

errorvariance

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Combining Items into Multi-Item Scales

When items are combined into a summated scale, random error to some extent “cancels out”– Error variance reduced as # items increases

– Reducing random error increases amount of “true score” variance

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Sources of Error

Subjects Observers or interviewers Measure or instrument

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Example: Measuring Weight of Children

Observed score is a linear combination of many sources of variation for an individual

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Measuring Weight in Pounds (Without Shoes) of One Child

Scale ismiscalibrated

True weight

80 lbs

Amount of water

past 30 min

Weightof clothes

Observed weight

Person weighing children

is not very precise

= + +

+ +

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Measuring Weight in Pounds (Without Shoes) of One Child

Scale ismiscalibrated

+.1 lb

True weight80 lbs

Amount of water

past 30 min+.25 lb

Weightof clothes

+.70 lb

Observed weight82.1 lbs

Person weighing children

is not very precise+1 lb

= + +

+ +

82.1 = 80 +.25 +.70 +.1 +1

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Sources of Error in Measuring Weight of Children

Weight of clothes– Subject source of random error

Scale is miscalibrated– Instrument source of systematic error

Person weighing child is not precise– Observer source of random error

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Measuring Depressive Symptoms (past 4 weeks) in an Asian or Latino Man

Unwillingnessto tell

interviewer

“True” depression

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Hard to choose number on the 1-6

response choice scale

Observed depression

score

Measuremisses 2

culturally-bound symptoms

= +

+ +Poor

memoryof feelings

+

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Measuring Depressive Symptoms (past 4 weeks) in an Asian or Latino Man

Unwillingto tell

interviewer-2

“True” depression

16

Hard to choose number on the 1-6

response choice scale+1

Observed depression

score12

Measuremisses 2

culturally-bound symptoms

-2

= +

+ +

12 = 16 +1 -2 -1 -2

Poor memory

of feelings-1

+

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Sources of Error in Measuring Depression

Hard to choose one number on 1-6 response scale– Subject source of random error

Unwilling to tell interviewer, poor memory of feelings– Subject sources of systematic error (underreport true

depression) Measure misses culturally-bound symptoms

– Instrument source of systematic error (underestimate true depression)

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Four Types of Memory Errors: From Cognitive Psychology

Encoding– Information inadequately stored in memory

Storage– Memory eroded over time

Retrieval– Some events/feelings harder to recall

Reconstruction – Errors filling in missing pieces

R Torangeau, Chap 3, in AA Stone et al. (eds)The Science of Self-Report, London: Lawrence Erlbaum, 2000

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Memory and Time Autobiographical memory – memory of

things in time and space Events not encoded with their calendar dates

– Thus time is a poor retrieval method Numerous errors remembering “when” and

“how often” something occurred within a particular time frame

N Bradburn, Chap 4, The Science of Self-Report

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Memory and Emotion

Tend to remember– positive more than negative experiences– more emotionally intense than neutral

experiences– non-threatening events more than

threatening, sensitive events

Kihlstrom et al, Chap 6, The Science of Self-Report

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Overview

Concepts of error Basic psychometric characteristics

– Variability

– Reliability

– Interpretability

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Variability

Good variability– All (or nearly all) scale levels are represented– Distribution approximates bell-shaped normal

Variability is a function of the sample– Need to understand variability of a measure in

sample similar to one you are studying Review criteria

– Adequate variability on the latent variable that is relevant to your study

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Indicators of Variability

Range of scores Mean, median, mode Standard deviation (or standard error) Interquartile range Skewness statistic % at floor (lowest possible score) % at ceiling (highest possible score)

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Range of Scores: Possible and Observed

Especially important for multi-item measures Example:

– CES-D possible range is 0-30– Wong et al. study of mothers of young

children: observed range was 0-23» missing entire high end of the distribution (none

had high levels of depression)

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Mean, Median, Mode

Mean - average Median - midpoint Mode - most frequent score In normally distributed measures, these are

all the same In non-normal distributions, they will vary

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Mean and Standard Deviation Most information on variability is from

mean and standard deviation– Can envision how measure is distributed

on the possible range

– Mean + 1 SD = 64% of the scores

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Interquartile Range (IR)

Difference between the 3rd and 1st quartiles

IR = Quartile 3 - Quartile 1 This range contains the middle 50% of the

distribution– 25% of the sample is above and 25% is

below this range

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Quartiles

Divide distribution into 4 parts with 25% of the sample in each part (quartiles)

Quartile 1 - the scale score at the boundary of the lowest 25% of the distribution

Quartile 2 - the score that divides the distribution in half (same as the median)

Quartile 3 - the score at the boundary of the highest 25% (25% of the sample scores above this point)

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Set of Scores on 12 people

1 2 3 4 5 6 7 8 9 10 11 12 2 3 8 1 7 4 4 3 2 7 5 3

4 9 1 8 2 12 7 6 11 10 5 3 1 2 2 3 3 3 4 4 5 7 7 8

12 people (red), 12 scores (black)

Re-arrange scores in numeric order

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Example of Quartiles: Set of Scores on 12 people

1 2 2 3 3 3 4 4 5 7 7 8

Q1=lowest 25% (lowest 3 people)Q2= median (50% below, 50% above)Q3=highest 25% (highest 3 people)

2.5Q1

6Q3

3.5Q2

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Example of Quartiles: Set of Scores on 12 people

1 2 2 3 3 3 4 4 5 7 7 8

Interquartile range - quartile 3 - quartile 1 = 6 - 2.5 = 3.5

2.5Q1

6Q3

3.5Q2

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Skewness

Positive skew - scores bunched at low end, long tail to the right

Negative skew - opposite pattern Skewness coefficient ranges from - infinity to +

infinity– the closer to zero, the more normal

Scores +2.0 are cause for concern

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Ceiling and Floor Effects: Similar to Skewness Information

Ceiling effects: substantial number of people get highest possible score

Floor effects: opposite More helpful for single-item measures or

coarse scales with only a few levels

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… to what extent did health problems limit you in everyday physical activities (such as walking and climbing stairs)?

0

10

20

30

40

50

Not at all Slightly Moderately Quite a bit Extremely

%

49% not limited at all (can’t improve)

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SF-36 Variability Information in Patients with Chronic Conditions (N=3,445)

Physicalfunction10 items

Role-physical

4 items

Mental health5 items

Vitality (energy)

5 items

Mean (SD) 80 (27) 75 (41) 71 (21) 54 (22)

Skewness - .99 - .26 - .83 - .24

% floor < 1 24 <1 <1

% ceiling 19 37 4 <1

McHorney C et al. Med Care. 1994;32:40-66.

All on 0-100 scales, higher is better

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Evidence of Floor and Ceiling Effects in One SF-36 Scale

Physicalfunction10 items

Role-physical

4 items

Mental health5 items

Vitality (energy)

5 items

Mean (SD) 80 (27) 75 (41) 71 (21) 54 (22)

Skewness - .99 - .26 - .83 - .24

% floor < 1 <1 <1

% ceiling 19 4 <1

McHorney C et al. Med Care. 1994;32:40-66.

All on 0-100 scales, higher is better

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Reasons for Poor Variability

Low variability in construct being measured in that “sample” (true low variation)

Items not adequately tapping construct– If only one item, especially hard

Items not detecting variation at one end What to do:

– If developing measures, add items– If selecting measures – find another one

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Advantages of Multi-item Scales Revisited

Using multi-item scales minimizes likelihood of ceiling/floor effects

Even if items are skewed, multi-item scale “normalizes” the skew

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Percent with “Best” Score on 5 Items in the MOS MHI-5

6-level response scale - all of the time to none of the time:

Stewart A. et al., Measuring Functioning and Well-Being, 1992

%

Very nervous person (none of the time) 34

Felt calm and peaceful (all of the time) 4

Felt downhearted and blue (none of the time) 33

Happy person (all of the time) 10

So down in the dumps nothing could cheer you up (none of the time) 63

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%






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%






5-itemscale:

only 5%had

highestscore

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Overview


– Variability

– Reliability


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Reliability

Extent to which an observed score is free of random error– Produces the same score each time it is administered (all

else being equal) Population-specific - reliability affected by:

– sample size– variability in scores (dispersion)– a person’s level on the scale

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Back to Components of Variability in Item Scores of a Group of Individuals


Total variance (Variation is the sum of all observed item scores)

= + errorvariance

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Reliability Depends on True Score Variance

Reliability is a group-level statistic Reliability:

– Reliability = 1 – (error variance)– OR

Proportion of variance due to true score Total variance

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Reliability Depends on True Score Variance

Reliability of .70 means 30% of variancein observed scores is due to error

Reliability = total variance – error variance.70 = 1.0 – .30

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Reliability Coefficient

Typically ranges from .00 - 1.00 Higher scores indicate better reliability

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Importance of Reliability

Necessary for validity (but not sufficient)– Low reliability (or high measurement error)

attenuates correlations with other variables – May conclude that two variables are not

related when they are Greater reliability = greater power

– The more reliable your scales, the smaller sample size you need to detect an association

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Reliable Scale?

NO! There is no such thing as a “reliable” scale We accumulate “evidence” of reliability in a

variety of populations in which it has been tested

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How Do You Know if a Scale or Measure Has Adequate Reliability?

Adequacy of reliability judged according to standard criteria

– Criteria depend on type of coefficient

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Types of Reliability Tests

Internal-consistency Test-retest Inter-rater Intra-rater

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Internal Consistency Reliability: Cronbach’s Alpha

Requires multiple items supposedly measuring same construct to calculate

Extent to which all items measure the same construct (same latent variable)

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Internal-Consistency Reliability

For multi-item scales Cronbach’s alpha

– for scales using ordinal items (e.g., 1-5) Kuder Richardson 20 (KR-20)

– for scales using dichotomous items

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Minimum Standardsfor Internal Consistency Reliability

For group comparisons (e.g., regression, correlational analyses)– .70 or above is minimum (Nunnally, 1978)– .80 is optimal– above .90 is unnecessary

For individual assessment (e.g., treatment decisions)– .90 or above (.95) is preferred (Nunnally, 1978)

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Internal-Consistency Reliability Can be Spurious

Based on only those who answered all questions in the measure– If a lot of people are having trouble with the

items and skip some, they are not included in test of reliability

Important to compare sample size in reliability calculation to total sample

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Internal-Consistency Reliability is a Function of Number of Items in Scale

Increases with the number of items Very large scales (20 or more items) can

have high reliability without other good psychometric properties

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Example: 20 item Beck Depression Inventory (BDI)

BDI 1978 version (asks about past week)– Internal consistency reliability = .86

Beck AT et al. J Clin Psychol. 1984;40:1365-1367

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Example: 20 item Beck Depression Inventory (BDI)

BDI 1978 version (asks about past week)– Internal consistency reliability = .86

– BUT: 3 items correlated < .30 with other items in the scale

Beck AT et al. J Clin Psychol. 1984;40:1365-1367

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Reliability Varies by Level on Measure

Reliability can be poorer for those scoring at one end of the scale

Example: Number of visits to doctor in past 12 months– More reliable for those with fewer visits

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Test-Retest Reliability

Repeat assessment on individuals not expected to change

Time between assessments should be:– Short enough so no change occurs– Long enough so subjects don’t recall first response

Only reliability test for single item measures Coefficient: correlation between 2

measurements

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Appropriate Test-Retest Coefficients by Type of Scale

Continuous scales (ratio or interval scales, multi-item Likert scales):– Pearson

Ordinal or non-normally distributed scales:– Spearman or Kendall’s tau

Dichotomous (categorical) measures:– Phi or Kappa

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Minimum Standards for Test-Retest Reliability

Magnitude of a test-retest correlation is important, not significance

Criterion: similar to that for internal consistency

– >.70 is desirable

– >.80 is optimal

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Observer or Rater Reliability

Inter-rater reliability (across two or more raters)– Consistency (correlation) between two or more

observers of the same subjects (one point in time)

Intra-rater reliability (within one rater)– Consistency within one observer– Correlation among repeated values obtained by the

same observer (over time)

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Observer or Rater Reliability

Sometimes Pearson correlations are used – scores on a group of individuals obtained by one observer correlated with scores obtained by another observer– Assesses association only

.65 to .95 are typical correlations >.85 is considered acceptable

McDowell I et al. Measuring Health, 2006, p. 45.

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Association vs. Agreement When Correlating Scores from Two Times or Ratings

Association: degree to which scores of one rater linearly predict scores of 2nd rater

Agreement: extent to which same score obtained on 2nd measurement (retest, 2nd rater)

Can have high correlation and poor agreement– If second score is consistently higher for all

subjects, can obtain high correlation– Need second test of mean differences

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Hypothetical Scores on 4 Subjects by 2 Observers

1

2

3

4

5

6

7

S1 S2 S3 S4

Subjects

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Example of Association and Agreement

Scores by observer 1 are exactly 2 points above scores by observer 2– Correlation (association) would be perfect

(r=1.0)

– Agreement is poor (no agreement on score in all cases - a difference of 2 between scores on each subject

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Intraclass Correlation Coefficient (Kappa) for Testing Inter-rater Reliability

Coefficient indicates level of agreement of two or more judges, exceeding that which would be expected by chance

Appropriate for dichotomous (categorical) scales and ordinal scales

Several forms of kappa:– e.g., Cohen’s kappa: 2 judges, dichotomous scale

Sensitive to number of observations, distribution of data

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Interpreting Magnitude of Kappa: Level of Reliability

<0.00

.00 - .20

.21 - .40

.41 - .60

.61 - .80

.81 - 1.00

Poor

Slight

Fair

Moderate

Substantial

Almost perfect

.60 or higher is acceptable (Landis, 1977)

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Reliability Often Poorer in Lower SES or Low Literacy Groups

More random error due to Reading problems Difficulty understanding complex

questions Unfamiliarity with questionnaires and

surveys

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Advantages of Multi-item Scales Revisited

Using multi-item scales improves reliability

Random error is “canceled out” across multiple items

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What Makes a Measure Reliable?

Preventing measurement error easier than assessing its effects

Measure– Clear items, appropriate response choices, etc.

Format– Make instrument easily understood

Method of administration– Train raters to do their job– Adhere to standard administration procedures

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Overview


– Variability

– Reliability


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Interpretability: What does a Score Mean?

What are the endpoints? What does a high score mean?

(direction of scoring) Compared to norms - is score low or high?

Single items, more easily interpretable

Multi-item scales, no inherent meaning to scores

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Endpoints

What is minimum and maximum possible?– Enable interpretation of mean score

When scores are added, endpoints depend on number of items & number of response choices– 5 items, 4 response choices = 5 to 20– 3 items, 5 response choices = 3 to 15

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Compare Results to Norms

Comparing your means to published norms helps interpret the mean of your sample

SF-36 has numerous norms, e.g.– General population

» By age group, gender, and chronic disease

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SF-36 in MOS Patients versus Population Norms

Physicalfunction

Role-physical

Mental health

Vitality (energy)

MOS patients

Mean (SD) 80 (27) 75 (41) 71 (21) 54 (22)

NORMS

Gen pop 84 (23) 81 (34) 75 (18) 61 (21)

Age 75+ 53 (30) 45 (42) 74 (20) 50 (24)

JE Ware et al, SF-36 Health Survey Manual andInterpretation Guide, The Health Institute, 1993.

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Direction of Scoring

What does a high score mean? Where in the range does the mean score

lie?– Toward top, bottom?

– In the middle?

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Descriptive Statistics for ~3,000 Women

M (SD) Min Max

Age 46.2 (2.7) 42.0 52.9

Activity 7.7 (1.8) 3.0 14.0

Stress 8.6 (2.9) 4.0 19.0

Med Care, 2003;41:1262-1276

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Descriptive Statistics for ~3,000 Women

M (SD) Min Max

Age 46.2 (2.7) 42.0 52.9

Activity 7.7 (1.8) 3.0 14.0

Stress 8.6 (2.9) 4.0 19.0

Med Care, 2003;41:1262

Activity: no measure mentionedStress: Perceived stress scale (Cohen, 1983)

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Perceived Stress Scale (Cohen 1983): Hard to Find

Available in JSTOR– Can print one page at a time

Searched article “on line” – Could not find scoring information other than

reverse 7 of the 14 items and sum them» Possible score range of 0-56

– Could not find response choices

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Another Example: Mean Scores in a Sample of Older Adults

Physical functioning 45.0Sleep problems 28.1Disability 35.7

Mean

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Making it Easier to Interpret

Physical functioning 45.0Sleep problems 28.1Disability 35.7

* All scores 0-100

Mean*

84

Making it Easier to Interpret

Physical functioning (+) 45.0Sleep problems (-) 28.1Disability (-) 35.7

* All scores 0-100 (+) indicates higher score is better health(-) indicates lower score is better health

Mean*

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Confusion Introduced by Labels:

SF-36 Bodily Pain scale– Higher score is no pain or limitations due to pain– Rationale: so 8 subscales scored in same direction

Social Adjustment Scale (Weissman) Functional Status Index (Jette)

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Mean Has to be Interpreted Within Possible Range

M SD

Parents’ harsh discipline practices* Interviewers’ ratings of mother 2.55 .74 Husbands’ reports of wife 5.32 3.30

*Note: high score indicates more harsh practices

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Mean Has to be Interpreted Within Possible Range (Add Range)

M SD

Parents’ harsh discipline practices* Interviewers’ ratings of mother (1-5) 2.55 .74 Husbands’ reports of wife (1-7) 5.32 3.30


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Mean Has to be Interpreted Within Possible Range

M SD

Parents’ harsh discipline practices* Interviewers’ ratings of mother (1-5) 2.55 .74 Husbands’ reports of wife (1-7) 5.32 3.30

Interviewer: 1 2 3 4 5

Husband: 1 2 3 4 5 6 7


2.55

5.32

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Transforming a Summated Scale to a 0-100 Scale

Works with any ordinal or summated scale Transforms it so 0 is the lowest possible and

100 is the highest possible Eases interpretation across numerous scales

100 x (observed score - minimum possible score)

(maximum possible score - minimum possible score)

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Homework

Complete rows 13-19 on matrix for both measures– Interpretability, nature of samples on which

it has been tested, variability and central tendency, reliability

1 Class 4 Basic Psychometric Characteristics: Variability, Reliability, Interpretability October 15,...

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