Analysis of Frequencies

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Analysis of Frequencies So far --gt response
variable (Y) - continuous Now look at response
variable that is discrete/categorical eg
presence-absence or yes-no --gt ask questions
about frequencies in each category
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Chi -Square Test - compare OBSERVED frequency
of a trait to EXPECTED frequency of a trait
- by counting
- based on randomness or theory
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Testing for randomness Lets say were
interested in the ratio of females to males in
this class. Is the composition a RANDOM sample
of the UWO population?
Ho the class is a random sample of UWO
students HA the class is not a random sample of
UWO students
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I will have two observed frequencies, one for
each of females and males, And two expected
frequencies, one for each of females and males.
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Where do the frequencies come from? Observed Co
unt all females and males
Observed frequency of females
Count Females 132 Males 76 Total 208
Observed frequency of males
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Where do the frequencies come from? Expected Let
s say femalesmales at UWO is 5050, half are
females and half are males. So, in a RANDOM
sample, expect 50 female and 50 male.
Observed Expected Females 132 Males
76 Total 208
2080.5104 2080.5104
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Observed Expected Females 132 Males
76 Total 208
104 104 208
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Therefore reject Ho, not a random sample of UWO
students
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Expected Lets say femalesmales at UWO is
6040. So, in a RANDOM sample, expect 60 female
and 40 male.
Observed Expected Females 132 Males
76 Total 208
2080.6124.8 2080.483.2
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Observed Expected Females 132 Males
76 Total 208
124.8 83.2 208
Therefore, do not reject Ho, the class is a
random sample of UWO students
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Testing to see if your data fit a theoretical
distribution
Hardy-Weinberg expectations for Mendels Peas
Dihybrid cross testing for independent assortment
of traits
smooth-yellow 9 smooth-green 3 wrinkled-yellow
3 wrinkled-green 1
Ho The peas occur in a ratio of 9331 HA The
peas do not occur in a ratio of 9331
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Observed
Count smooth-yellow 152 smooth-green
53 wrinkled-yellow 39 wrinkled-green 6 TOTAL 25
0
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Expected
Count Expected smooth-yellow 152 0.5625250
140.625 smooth-green 53 0.187525046.875 wrink
led-yellow 39 0.187525046.875 wrinkled-green 6
0.062525015.625 TOTAL 250
9331 ---gt 9/16 3/16 3/16 1/16 0.5625
0.1875 0.1875 0.0625
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Observed Expected O-E smooth-yellow 152 140.6
25 11.375 smooth-green 53 46.875 6.125 wrinkle
d-yellow 39 46.875 7.875 wrinkled-green 6 15.62
5 9.625 TOTAL 250 250
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Therefore, reject Ho, the sample is not in H-W
Eqm
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Often, collect data on more than one variable
simultaneously --gt contingency table
analysis --gt test for independence among the
variables
2 X 2 Contingency Tables 2 categorical
variables, each with two levels
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R1 sum of observed in Row 1 R2 sum of
observed in Row 2 C1 sum of observed in Column
1 C2 sum of observed in Column 2 C3 sum of
observed in Column 3 Total sum of all observed
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Expected calculation
/
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For example, A rare tree species can be
rooted in serpentine or non-serpentine
soil have pubescent or smooth leaves
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Is leaf morphology independent of soil type?
Ho Leaf type is independent of soil type HA
Leaf type is not independent of soil type
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Again, want to compare OBSERVED to EXPECTED
Observed
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Expected
3428/100
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Expected
3472/100
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Expected
6628/100
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Expected
6672/100
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Therefore, do not reject Ho leaf type is
independent of soil
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For example, varieties of tiger beetles found
during four times of the year
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Ho The occurrence of tiger beetle colour types
is not dependent upon time of year HA The
occurrence of tiger beetle colour types is
dependent upon time of year
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Therefore, reject Ho, occurrence of tiger beetle
colour type is dependent upon time of year
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Logistic Regression --gt used to describe the
relationship between Discrete Y Continuous X
Y --gt usually dichotomous or binary
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Examples of dichotomous Y variable Presence -
Absence Alive - Dead Male - Female Red -
Green
Scored as 0s and 1s
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Y -0.070.01(X)
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Cigs
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