Multivariate Descriptive Research

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Multivariate Descriptive Research

• In the previous lecture, we discussed ways to
  quantify the relationship between two variables
  when those variables are continuous.
• What do we do when one or more of the variables
  is categorical?

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Categorical Variables

• Fortunately, this situation is much easier to
  deal with because we can use the same techniques
  that weve discussed already.
• Lets consider a situation in which we are
  interested in how one continuous variable varies
  as a function of a categorical variable.
• Example How does mood vary as a function of sex
  (male vs. female)?

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• In this case, we want to know how the average
  womans score compares to that of the average
  mans score.
• level of a categorical variable

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Participants Mood score
Males
A 4
B 3
C 4
D 3
M 3.5, SD .5
Females
A 5
B 4
C 5
D 4
M 4.5 , SD .5
First, find the average score for each level of
the categorical variable separately. (Also find
the SD.) Second, find the difference between the
means of each group. This is called a mean
difference. (4.5 3.5 1.0) Third, express this
mean difference relative to the SD. This is
called a standardized mean difference. 1/.5
2 In this example, women score 2 SD higher than
the men.
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Participants Mood score
Males
A 4
B 3
C 4
D 3
M 3.5, SD .5
Females
A 5
B 4
C 5
D 3
M 4.25 , SD .83
Note If the SDs for the two groups are
different, you can simply average the two
SDs. Here, the two SDs are .5 and .83.
Averaged, these are (.5 .83)/2 .66. The
standardized mean difference is (4.25 3.5)/.66
.75/.66 1.13 Thus, on average, women score
1.13 SDs higher than men on this mood variable.
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Cohens d

• If we divide the mean difference by the average
  SD of the two groups, we obtain a standardized
  mean difference or Cohens d.

Pooled standard deviation
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Bargraph
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Bargraph More than two categorical variables
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Both variables are categorical

• When two variables are categorical, it is
  sometimes most useful to express the data as
  percentages.
• Example Lets assume that depression is a
  categorical variable, such that some people are
  depressed and others are not.
• What is the relationship between biological sex
  and depression?

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Depression status Depression status
Sex Not Depressed Depressed row total
Male 600 60 660
Female 40 300 340
column total 640 360 1000
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Depression status Depression status
Sex Not Depressed Depressed row total
Male .60 .06 .66
Female .04 .30 .34
column total .64 .36 1.00
In this table, weve expressed each cell as a
proportion of the total.
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Depression status Depression status
Sex Not Depressed Depressed row total
Male .60 .06 .66
Female .04 .30 .34
column total .64 .36 1.00
.60/.64 .94 .06/.36 .16
Here, weve expressed the association with
respect to sex. For example, we can see here
that 16 of people who are depressed are male.
Moreover, 94 of people who are not depressed are
male.
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Depression status Depression status
Sex Not Depressed Depressed row total
Male .60 .06 .66 .06/.66 .09
Female .04 .30 .34 .30/.34 .88
column total .64 .36 1.00
Here, weve expressed the association with
respect to depression status. For example, we
can see here that 9 of men are depressed and 88
of women are depressed.
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Phi

• It is possible to quantify the association among
  these variables using a correlation coefficient
  when the two variables are binary.
• This statistic is sometimes referred to as phi.
• (Phi is .78 in this example)

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Variable 1 Variable 1
Variable 2 0 1 row total
0 a b n3
1 c d n4
Col total n1 n2
Phi (ad) (bc) / sqrt(n1n2n3n4)
Online calculator at http//www.quantitativeskill
s.com/sisa/statistics/twoby2.htm