# Standard Scores _Z-scores_

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```					• Reliability, the Properties of Random Errors,
and Composite Scores
Reliability

• Reliability: the extent to which measurements
are free of random errors.
• Random error: nonsystematic mistakes in
measurement
– observer looks away when coding behavior
– response scale not quite fitting
Reliability

• What are the implications of random
measurement errors for the quality of our
measurements?
Reliability

•O = T + E + S
O = a measured score (e.g., performance on an exam)
T = true score (e.g., the value we want)
E = random error
S = systematic error
•O = T + E
Reliability

• O=T+E
• The error becomes a part of what we’re measuring
• This is a problem if we’re operationally defining our
variables using equivalence definitions because part
of our measurement is based on the true value that
we want and part is based on error.
• Once we’ve taken a measurement, we have an
equation with two unknowns. We can’t separate the
relative contribution of T and E.
10 = T + E
Reliability: Do random errors
accumulate?
• Question: If we aggregate or average
multiple observations, will random errors
accumulate?
Reliability: Do random errors
accumulate?
• Answer: No. If E is truly random, we are just
as likely to overestimate T as we are to
underestimate T.
• Height example
5’   5’   5’   5’   5’   5’   5’   5’   5’   5’   6    6’   6’   6’   6’   6’   6’   6’   6’   6’
2    3    4    5    6    7    8    9    10   11        1    2    3    4    5    6    7    8    9
62   63   64   65   66   67   68   69   70   71   72   73   74   75   76   77   78   79   80   81
Reliability: Do random errors
accumulate?
O=       T     +    E
Obs. 1   10       10          0
Obs. 2    9       10         -1
Obs. 3   10       10          0
Obs. 4   11       10         +1
Obs. 5    8       10         -2
Obs. 6   10       10          0
Obs. 7   12       10         +2
Average   10       10          0

Note: The average of the seven O’s is equal to T
Composite scores

• These demonstrations suggest that one important
way to help eliminate the influence of random errors
of measurement is to aggregate multiple
measurements of the same construct. Composite
scores.

– use multiple questionnaire items in surveys of an attitude,
behavior, or trait
– use more than one observer when quantifying behaviors
– use observer- and self-reports when possible
•   Example: Self-esteem survey items

•   1. I feel that I'm a person of worth, at least on an equal plane with others.
Strongly Disagree        1             2             3            4             5   Strongly Agree

2. I feel that I have a number of good qualities.
Strongly Disagree        1           2              3             4             5   Strongly Agree

4. I am able to do things as well as most other people.
Strongly Disagree      1             2            3               4             5   Strongly Agree
•   Example: Self-esteem survey items

•   1. I feel that I'm a person of worth, at least on an equal plane with others.
Strongly Disagree        1             2             3            4             5   Strongly Agree

2. I feel that I have a number of good qualities.
Strongly Disagree        1           2              3             4             5   Strongly Agree

4. I am able to do things as well as most other people.
Strongly Disagree      1             2            3               4             5   Strongly Agree

Composite self-esteem score = (4 + 5 + 3)/3 = 4
Two things to note about aggregation

• Some measurements are keyed in the
direction opposite of the construct of interest.
High values represent low values on the trait
of interest.
•   Example: Self-esteem survey items

•   1. I feel that I'm a person of worth, at least on an equal plane with others.
Strongly Disagree        1             2             3            4             5   Strongly Agree

2. I feel that I have a number of good qualities.
Strongly Disagree        1           2              3             4             5   Strongly Agree

3. All in all, I am inclined to feel that I am a failure.
Strongly Disagree        1             2             3            4             5   Strongly Agree

4. I am able to do things as well as most other people.
Strongly Disagree      1             2            3               4             5   Strongly Agree

5. I feel I do not have much to be proud of.
Strongly Disagree      1           2                3             4             5   Strongly Agree

Inappropriate composite self-esteem score =
(5 + 5+ 1 + 4 + 1)/5 = 3.2
Reverse keying: Transform the measures such
that high scores become low scores and vice
versa.
•   Example: Self-esteem survey items

•   1. I feel that I'm a person of worth, at least on an equal plane with others.
Strongly Disagree        1             2             3            4             5   Strongly Agree

2. I feel that I have a number of good qualities.
Strongly Disagree        1           2              3             4             5   Strongly Agree

3. All in all, I am inclined to feel that I am a failure.
Strongly Disagree        1             2             3            4             5   Strongly Agree

4. I am able to do things as well as most other people.
Strongly Disagree      1             2            3               4             5   Strongly Agree

5. I feel I do not have much to be proud of.
Strongly Disagree      1           2                3             4             5   Strongly Agree

Appropriate composite self-esteem score =
(5 + 5+ 5 + 4 + 5)/5 = 4.8
• A simple algorithm for reverse keying in
SPSS or Excel

New X = Max + Min - X

• Max represents the highest possible value (5
on the self-esteem scale). Min represents the
lowest possible value (1 on the self-esteem
scale).
Two things to note about aggregation

• Be careful when averaging measurements
that are not on the same scale or metric.
• Example: stress

Person   Heart rate   Complaints    Average
A        80           2              41
B        80           3              42
C        120          2              61
D        120          3              62

Beats per minute

Number of
complaints
Two things to note about aggregation

• Two problems
• First, the resulting metric for the psychological
variable doesn’t make much sense.

Person A: 2 complaints + 80 beats per minute
= 41 complaints/beats per minute???
Two things to note about aggregation

• Second, the variables may have different
ranges.
• If this is true, then some indicators will “count”
more than others.
• Variables with a large range will influence the composite score
more than variable with a small range

Person   Heart rate       Complaints                Average
A        80               2                          41
B        80               3                          42
C        120              2                          61
D        120              3                          62

* Moving between lowest to highest scores matters more for one variable
than the other
* Heart rate has a greater range than time spent talking and, therefore,
influences the composite score more
Two things to note about aggregation

• One common solution to this problem is to
standardize the variables before aggregating
them.

• Constant mean and variance
• Variables with a large range will influence the composite score
more than variable with a small range

Person   Heart rate(z)    Complaints(z)        Average
A        -.87            -.87                   -.87
B        -.87             .87                   0
C        .87              -.87                  0
D        .87              .87                   .87
Reliability: Estimating reliability

• Question: How can we quantify the reliability
of our measurements?
(a) test-retest reliability
(b) internal consistency reliability
Reliability: Estimating reliability

• Test-retest reliability: Reliability assessed by
measuring something at least twice at different time
points. Test-retest correlation.
• The logic is as follows: If the errors of measurement
are truly random, then the same errors are unlikely to
be made more than once. Thus, to the degree that
two measurements of the same thing agree, it is
unlikely that those measurements contain random
error.
Less error         More error
(off by 1 point)   (off by 2 points)
Time     Time      Time      Time
1        2         1         2
Person A     1        2         1         3
Person B     7        6         7         5
Person C     2        3         2         4
Person D     6        5         6         4
Person E     3        4         3         5
Person F     5        4         5         3
Person G     4        5         4         6

r = .92              r = .27
Reliability: Estimating reliability

• Internal consistency: Reliability assessed by
measuring something at least twice within the

Split-half: based on an arbitrary split (e.g, comparing odd and
even, first half and second half). Split-half correlation.
Cronbach’s alpha (): based on the average of all possible
split-half correlations.
Ave r = .50
The reliability of the
Ave r = .25
composite (a) increases
as the number of items
Ave r = .10
(k) increases.
In fact, the reliability of
the composite can get
relatively high even if
the items themselves do
not correlate strongly.
Ave r = .10

Ave r = .10
Reliability: Final notes

• An important implication: As you increase the number
of measures, the amount of random error in the
averaged measurement decreases.
• An important assumption: The entity being measured
is not changing.
• An important note: Common indices of reliability
range from 0 to 1—in the metric of correlation
coefficients; higher numbers indicate better reliability
(i.e., less random error).

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 views: 0 posted: 3/28/2013 language: English pages: 30