Transcript The Impact of Inequality on Personal Life Chances

Statistical Analysis
ANOVA
Roderick Graham
Fashion Institute of Technology
ANOVA

ANOVA means Analysis Of Variance.

With ANOVA, we have one variable that is ungrouped,
and one categorical, grouped variable.

The question that we try to answer with ANOVA is: “for
any collection of groups, is at least one group different
from the others?”
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This is hypothesis testing. This time your critical statistic
will be an F ratio.
The types of procedures we’ve used…
Regression
ChiSquare
ANOVA
X=
ungrouped/scale
X=
grouped/category
X=
grouped/category
Y=
ungrouped/scale
Y=
grouped/category
Y=
ungrouped/scale
Steps in Conducting ANOVA tests
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Step 1 – State the null hypothesis and critical region
Step 2 – Identify the critical statistic
Step 3 – Compute the test (calculated) statistic
Step 4 – Interpret results
Example: Capital Punishment and
Religious Affiliation
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Suppose we ask 20 people to rank their support for
capital punishment on a scale from 1 to 30.
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1 being absolutely no support under no circumstances
30 being always support under any circumstance
Suppose these 20 people were equally divided into 5
religions (Protestant, Catholic, Jewish, Buddhist, and
other)
Now, we have an ungrouped, scale variable (support for
capital punishment) and a grouped, categorical variable
(religion). This is the perfect situation for ANOVA!
Example: Capital Punishment and
Religious Affiliation
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Now let’s say we got this data:
Protestant
Catholic
Jewish
Buddhist
Other
8
12
12
15
10
12
20
13
16
18
13
25
18
23
12
17
27
21
28
12
What do we do?
Example: Capital Punishment and
Religious Affiliation
Step 1 – State the null hypothesis and critical region
H0: “The population means for each category are the
same.”
H1: “At least one of the population means is different.”
Let’s set our critical region at .05 (Meaning we will accept
95% of all findings, and if we get a calculated statistic that
falls in the .05 region, we will reject the null hypothesis).
Example: Capital Punishment and
Religious Affiliation
Step 2 – Identify the critical statistic.
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To get the critical statistic, (F critical), we must look for two
values.
dfb (degrees of freedom between) – columns for F chart
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dfb = (k – 1), with k = number of categories
dfw (degrees of freedom within) – rows for F chart
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dfw = (N – k), with N = number of respondents, and k = number of
categories
Example: Capital Punishment and
Religious Affiliation
Step 2 – Identify the critical statistic.
For our example we have a k of 5 (five categories) and an N of
20 (20 respondents). Thus, dfb = (5 – 1) = 4
dfw = (20 – 5) = 15, F(critical) = 3.06
Example: Capital Punishment and
Religious Affiliation
What the upcoming symbols mean:
 SST = sum of squares total
 SSW = sum of squares within
 SSB = sum of squares between
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MSB = mean squares between
MSW = mean squares within
Example: Capital Punishment and
Religious Affiliation
Step 3 – Compute the test/calculated statistic requires
using these equations. Important…these formulas should
be used in this order!!!
SST   X  N X
2
2
SSB   N k ( X k  X ) 2
SSW  SST  SSB
SSB
MSB 
k 1
SSW
MSW 
nk
MSB
F (ratio ) 
MSW
Example: Capital Punishment and
Religious Affiliation
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Let’s take a closer look at the beginning formulas
SST   X  N X
2
Total number
of cases in a
sample
2
SSB   N k ( X k  X )
Number of cases
in EACH category
Mean value of
EACH category
2
Mean value for
the entire
sample
Note: SSB will have to be calculated for each category
and then summed. In our example, we have five religious
groups, so we will compute SSB five times, and then sum.
Example: Capital Punishment and
Religious Affiliation
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Understanding the final formula. It is a ratio of the
differences between groups and the differences within
groups
MSB
F (ratio ) 
MSW
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When MSB and MSW are similar, the F is low, and the less
chance we will reject the null.
But when these two values differ, the F increases. We
then begin to believe that at least one of the populations
that these samples represent is different from the other
populations.
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Example: Capital Punishment and
Religious Affiliation
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This is all we need to being building our table for
ANOVA.
Protestant
Catholic
Jewish
Buddhist
Other
8
12
12
15
10
12
20
13
16
18
13
25
18
23
12
17
27
21
28
12
SST   X  N X
2
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2
SSB   N k ( X k  X ) 2
What new information do we need in order to use our
starting formulas?
Example: Capital Punishment and
Religious Affiliation
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Setting up the table for ANOVA…
Protestant
∑
Catholic
Jewish
Buddhist
Other
x
x2
x
x2
x
x2
x
x2
x
x2
8
64
12
144
12
144
15
225
10
100
12
144
20
400
13
169
16
256
18
324
13
169
25
625
18
324
23
529
12
144
17
289
27
729
21
441
28
784
12
144
50
66
84
1898
64
1078
82
1794
52
712
X k 12.5
21.0
16.0
20.5
13.0
X = 16.6
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You also need:
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The N (sample), = 20, and Nk (no. of cases per category) = 4
The mean of the entire sample, X = (50+84+64+82+52)/20 =
16.6
Our F(critical) was….3.06
Our F(test) was…2.57
Thus, we do not reject the null
hypothesis.
Your turn…
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Are sexually active teens better informed about AIDS than teenagers who
are sexually inactive? A 15 item test of knowledge about sex was
administered to teens who were “inactive”, “active with one partner” and
“active with more than one partner”. Here are the results. Test at the .05
level.
Inactive
Active – 1
Partner
Active – More
than 1 Partner
10
11
12
12
11
12
8
6
10
10
5
4
8
15
3
5
10
15
Give H0, H1, F-critical, F-ratio, Concluding Results
END