ChiSquare

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Chi-Square
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Uses of chi-square

• a mathematical distribution
• a test for categorical data
• (more generally) a test for the fit of data to a
  theory

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The Chi-Square Distribution
where
or
for z an integer
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(matched in beauty only by Students t)
where
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the chi-square family
k1
k2
k4
k8
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Continuous vs. Discrete distributions
k8
these have probability density
these have probability
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from probabilities to densities
k8
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(familiar?) question
Given a large set of data converted to z-scores,
the mean and variance will obviously be 0 and 1.
Now if we square all our scores and compute the
mean, what will it be?
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chi-square and z
mean 0.989 sd 1.390
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chi-square and z (cont.)
so z2 is distributed as
and, further,
is distributed as
(and this is the really useful bit)
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why is this useful?
is distributed as
Because, basically it tells us the sampling
distribution of the difference between what we
expected and what we observed. As is always the
case, if the form of a sampling distribution is
known, a lot of work is saved.
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example how well do these data fit my theory?
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how well do these data fit my theory?

• let m stand for expected value
• calculate s2 in the regular way
• calculate

and compare it to the known sampling distribution
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What does that tell us?
How well do these data fit my theory?turned
intoGiven my theory, what are the odds of
observing data this bad or worse?or, more
technicallyAssuming that my theory is correct,
what is the probability of observing a departure
from the theory at least as large as that
represented by the data?
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Interpretation

• a large chi2 is bad news for your theory
• a small chi2 is good

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Have we been here before?

• yes
• substitute null hypothesis for my theory
• flip interpretation of large small values of
  the statistic

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The chi-square goodness-of-fit test
(and here we go again)
Does the sign on our rear door have any effect?
rear
front
Observed
Expected
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The one-way calculation
rear
front
Observed
Expected
3.52/13.5 3.52/13.5 1.88
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chi-square calc. (cont.)
p 0.17
1.88
k1
so if people ordinarily use both doors with
equal frequency, then we would see behavior at
least this aberrant almost 2 times out of 5.
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Contingency Table Analysis
Eij (RiCj)/N
in parentheses are
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Expected frequencies
p(low) R/N 177/358
Eij (RiCj)/N
(?)
p(guilty) C/N 258/358
p(low, guilty) (R/N)(C/N) (RC) / N2
(low, guilty) p(low, guilty)N (RC)/N
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Contingency table chi-square
and is distributed on (R-1)(C-1) df
35.93
.99
.01
k1
hm
6.6
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other stuff

• continuity correction (it exists)
• small expected values (be careful)
• Assumptions
• independence
• normality (sm. expected frequences)
• including non-occurrences (make sure your
  analyzed data totals to n)

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Likelihood Ratio Tests
(not yet)
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Measures of Association

• Cramers Phi - chi-sq (un)weighted by N
• Odds ratios - good for reporting low relative
  frequencies

odds ratio (a/b)/(c/d) 1.8
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That variance thing
Sample variance is unbiased, but is distributed
as
where
s1.0 s1.06
s1.0 s1.02