Psychology 203

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Psychology 203

• Semester 1, 2007
• Week 12
• Lecture 24

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Beyond the parameters 2

• Battlestar Nonparametrica A New Hope

Gravetter Wallnau, Chapters 19 20
Siegel Castellan (1988) Nonparametric
Statistics for the Behavioral Sciences
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Who gets the most action?
High masculine
Low masculine
4
Chi-square test for Independence

• Used to test whether there is a relationship
  between two variables

50
Classified as either masculine looking or not
60
23
48
28
11
n 110
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Chi-square test for Independence

• The Null hypothesis
• If two variables are independent then there is a)
  no predictable relationship between them and b)
  the distributions do not differ
• Calculating Chi-square
• Observed frequencies
• Expected frequencies
• H0 states there is no difference in the
  distributions for high low masculine men

6
Calculating e

• Work out overall proportion in each category
• 0 23/110 0.21 (21)
• 1 48/110 0.44 (44)
• 4-10 28/110 0.25 (25)
• 10 11/110 0.10 (10)
• Calculate fe for each category
• High Masculine
• 0 0.21 x 50 10.5
• 1 0.44 x 50 21.8
• 4-10 0.25 x 50 12.7
• 10 0.10 x 50 5.0

Row totals tell us how many in each group
10.5
21.8
12.7
5.0
50
12.5
26.2
15.3
6.0
60
23
48
28
11
n 110
Column totals describe the overall distribution
of proportions in each category
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Degrees of freedom

• df number of cells in table can choose freely

df (R - 1)(C - 1)
df (2 - 1)(4 - 1)
?
?
?
df 3
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The formula
exactly the same as for Goodness of Fit
Chi-square!
?2 crit for df3, ?.05 is 7.81
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Drawing your conclusions

• ?2 obtained ?2 critical
• There is a significant relationship between
  masculinity and the number of sexual partners
• More masculine men have significantly more sexual
  partners than do less masculine men

10
Effect size for ?2 test of independence

• phi-coefficient F
• measure of association (correlation) for two
  dichotomous variables (see last weeks lectures)
• measures strength of association, not significance

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SPSS output - Crosstabs
12
Assumptions of the Chi-square tests

• Size of the expected frequencies
• Dont do chi-square when the expected frequency
  (e) of any cell is less than 5
• because ?2 is too sensitive when have very small
  e i.e. get large values of ?2 for small
  differences between e o
• Observations must be independent
• each observed frequency is produced by a
  different participant
• so not appropriate for repeated measures designs

13
To para or not to para?

• Q. When should I use a nonparametric test?
• A. When you have no other choice!
• Because your data are nominal or ordinal
• Your data horribly violate the assumptions for a
  parametric test and cannot be fixed (by removal
  of outliers, transforming data etc.)

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Previously on Beyond the Parameters
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Serious Research Hypothesis
You are significantly less likely to survive a
mission to an alien planet if you are not a main
character!
Chi-square test for independence!
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Calculate Chi-square
All our expected frequencies are less than
5!!! Violates the assumptions of Chi-square
Ingenious solution! Proceed Mr Spock, at warp
speed.
Logic determines we should use the Fishers exact
test, Captain
ALERT
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Fishers Exact Probability Test

• Can use when
• Have cells with expected values less than 5
• and have a design with 2 rows and 2 columns (2 x
  2)
• Extra added bonus!
• can be used as a directional test, unlike
  Chi-square
• Problem
• Calculations laborious, tedious time-consuming
  ?
• Solution
• Use the Starship Enterprises computer! (or your
  laptop) ?

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The Logic of the Fisher Exact Test

• Null hypothesis states there is no relationship
  between the variables
• e.g. character status (main vs unknown) has no
  effect on your likelihood of surviving a mission
• How likely is it that you would get our data if
  their were no relationship?
• Work out all the possible combinations that would
  give you our row and column totals

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Fisher Exact Test
Positive association
Negative association
No association
Extreme unlikely outcome
Extreme unlikely outcome
Likely outcomes
Number of possibilities as extreme or more
extreme than our outcome that we would get by
chance
Probability of our outcome

Total number of possible outcomes
http//faculty.vassar.edu/lowry/ch8a.html
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Fisher Exact Test - SPSS
Keep an eye out for this!
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Assumption Buster II

• The other assumption for the chi-square test of
  independence was independent observations i.e. no
  repeated measures
• But repeated measures rock!
• Is there a nonparametric equivalent of the
  repeated measures (related samples) t-test?
• The McNemar Change Test

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Tom Cruise Box Office Gold
This guy OPENS movies! It can be a stinking pile
of festering horse _at_! But Joe Public will
flock to it in droves cos its a Tom Cruise
movie
Ari Gold Agent to the Stars
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Then Tom went on Oprah
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Tom Cruise Box Office Poison
This guy will kill your movie! I dont care if
its Titanic 2! Joe Public will stay at home
watching televised knitting rather than go see
this guy!
Ari Gold Agent to the Stars
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Did people really go off Tom?
People were asked their opinion twice, once
before Oprah and once afterwards
Like related samples t, we are interested in
differences, i.e. the people who change
AT Number change away from Tom 13
TT Number change toward Tom 4
Null hypothesis states Among those who change
their preference the number who change away from
Tom will not differ from those who change toward
him
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The Formula
AT Number change away from Tom 13
TT Number change toward Tom 4
?2 crit, df1, ?.05 is 3.84
Significant effect
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McNemar in SPSS
http//home.clara.net/sisa/pairwhlp.htmMcNemar
http//www.fon.hum.uva.nl/Service/Statistics/McNem
ars_test.html
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Its black and white! The Binomial Test

• We live in a binary world!
• Yes/no
• Pass/fail
• Good/evil
• Binomial data measurement scale has only 2
  categories

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The Binomial Test

• Two categories A B
• evaluates hypotheses about values of p q for
  the population
• Null hypotheses
• Chance e.g. coin toss, p1/2, q1/2
• No difference e.g. psame as known population,
  qsame as known population

Probability of A p
Probability of B p
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The Binomial Distribution

• When pn qn 10
• the binomial distribution is approximately normal
• ? pn
• Which means we can calculate z-scores and do a
  z-test!

Formula for binomial test
Proportion in Category A
Expected Proportion in Category A if H0 true
Standard error
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Do children with autism have a theory of mind?

• Being able to impute beliefs to others and
  predict their behaviour
• e.g. the Sally Ann task
• Pass or fail task
• Pass ToM
• Fail no ToM

?
?
Failure to take Sallys perspective
Baron-Cohen et al, (1985) Cognition, 21, 37-46.
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Binomial Test Theory of Mind

• Compare the pass/fail rates of children with
  autism to the known rates of typical children
  (mental age matched)
• H0
• Pass 86 (Fail 14)
• Data for our autism sample, n80
• Pass 16 (Fail 64)
• pn68.8, qn11.2, so both 10

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Calculating z for the binomial test

• Critical z for ?.05, is /- 1.96

• obtained z critical z

• Reject null hypothesis

Children with autism fail significantly more
often on the Sally Ann task than we would predict
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Ordinal Data
BJ and Tyler You are Team Number 1!
Fran and Barry I am afraid you are the last team
to arrive and you have been eliminated from the
race
Uchenna Joyce You are the first team to arrive!
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Do same gender couples do better on The Amazing
Race?
Amazing Race 9

• Rank/ordinal data violate assumptions for
  parametric tests
• Provide info about direction of one score
  relative to another, but not the distance
• But are simple easy to obtain
• Variety of non-parametric tests can use with
  ordinal data
• These are different tests than those for
  categorical data
• Mann-Whitney U like independent samples t-test

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Mann-Whitney U-Test
Effect of Group
No Effect of Group
Same gender clustered at one end
Evenly mixed
Different gender clustered at other end
H0 Ranks of one group not systematically higher
or lower than other
H1 Ranks of one group systematically higher or
lower than other
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Calculating U

• Assign points to each group
• Mann-Whitney U is whichever value is smallest

Check that
nSame6
nDifferent5
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Interpreting U

• If there was no overlap between the samples U0
• Because one group would get no points and so the
  smallest U would be zero
• The more intermixed the groups are, the bigger
  the value of U
• So, the smaller the U the more likely we have a
  significant difference!
• Critical values of U is 1 (for ?.05 nSame 6,
  ndifferent 5)
• Obtained U (10) Critical U (1)
• Fail to reject H0

Opposite direction to usual