Answering Descriptive Questions in Multivariate Research

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Answering Descriptive Questions in Multivariate
Research

• When we are studying more than one variable, we
  are typically asking one (or more) of the
  following two questions
• How does a persons score on the first variable
  compare to his or her score on a second variable?
• How do scores on one variable vary as a function
  of scores on a second variable?

2
Making Sense of Scores

• Lets work with this first issue for a moment.
• Lets assume we have Marcs scores on his first
  two Psych 242 exams.
• Marc has a score of 50 on his first exam and a
  score of 50 on his second exam.
• On which exam did Marc do best?

3
Example 1

• In one case, Marcs exam score is 10 points above
  the mean
• In the other case, Marcs exam score is 10 points
  below the mean
• In an important sense, we must interpret Marcs
  grade relative to the average performance of the
  class

Exam1
Exam2
Mean Exam2 60
Mean Exam1 40
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Example 2

• Both distributions have the same mean (40), but
  different standard deviations (10 vs. 20)
• In one case, Marc is performing better than
  almost 95 of the class. In the other, he is
  performing better than approximately 68 of the
  class.
• Thus, how we evaluate Marcs performance depends
  on how much spread or variability there is in the
  exam scores

Exam1
Exam2
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Standard Scores

• In short, what we would like to do is express
  Marcs score for any one exam with respect to (a)
  how far he is from the average score in the class
  and (b) the variability of the exam scores
• how far a person is from the mean
• (X M)
• variability in scores
• SD

6
Standard Scores

• Standardized scores, or z-scores, provide a way
  to express how far a person is from the mean,
  relative to the variation of the scores.
• (1) Subtract the persons score from the mean.
  (2) Divide that difference by the standard
  deviation.
• This tells us how far a person is from the
  mean, in the metric of standard deviation units

Z (X M)/SD
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Example 1
Marcs z-score on Exam1 z (50 - 40)/10
1 (one SD above the mean) Marcs z-score on
Exam2 z (50 - 60)/10 -1 (one SD below the
mean)
Exam1
Exam2
Mean Exam2 60 SD 10
Mean Exam1 40 SD 10
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Example 2
An example where the means are identical, but the
two sets of scores have different spreads Marcs
Exam1 Z-score (50-40)/5 2 Marcs Exam2
Z-score (50-40)/20 .5
Exam1 SD 5
Exam2 SD 20
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Some Useful Properties of Standard Scores

• (1) The mean of a set of z-scores is always zero
• Why? If we subtract a constant, C, from each
  score, the mean of the scores will be off by that
  amount (M C). If we subtract the mean from
  each score, then mean will be off by an amount
  equal to the mean (M M 0).

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(2) The SD of a set of standardized scores is
always 1 Why? SD/SD 1
if x 60,
M 50 SD 10
50
60
70
80
40
30
20
x
0
1
2
3
-1
-2
-3
z
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A Normal Distribution
(3) The distribution of a set of standardized
scores has the same shape as the unstandardized
(raw) scores beware of the normalization
misinterpretation
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The shape is the same
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Some Useful Properties of Standard Scores

• (4) Standard scores can be used to compute
  centile scores the proportion of people with
  scores less than or equal to a particular score.
  

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The area under a normal curve
50
34
34
14
14
2
2
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Some Useful Properties of Standard Scores

• (5) Z-scores provide a way to standardize very
  different metrics (i.e., metrics that differ in
  variation or meaning). Different variables
  expressed as z-scores can be interpreted on the
  same metric (the z-score metric). (Each score
  comes from a distribution with the same mean
  zero and the same standard deviation 1.)

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Multiple linear indicators Caution
(Recall this slide from a previous lecture?)

• Variables with a large range will influence the
  latent score more than variable with a small
  range
• Person Heart rate Time spent talking
  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 total
  score more (i.e., the score on the latent
  variable)

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