Twoway classification

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Two-way classification
Example Tiger beetles
Two color morphs BR Bright red NBR Not
bright red
Four seasons ESpring Early spring LSpring
Late spring ESum Early summer LSum Late summer
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Observed distribution of color morphs
H0 The distribution of color morphs is
independent of season
H1 The distribution of color morphs is not
independent of season
a 0.05
Test ?2-test, G-test or logistic regression
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G-test
Predicted distribution of color morphs assuming
independence between factors
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The log-linear model
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The saturated model
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The saturated model
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• The saturated model has as many parameters as
  there are cells in the table.
• The saturated model perfectly fits the observed
  data.
• Consequently, its log likelihood function is 0

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The reduced model
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G -2(ln Lsaturated ln Lreduced) 2 ln
Lreduced 28.596
with 3 df
P lt 0.0001
Conclusion color morphs do not occur
independently of season
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How to do it with SAS
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/ Example from Sokal and Rohlf 1995 p
738/ /Log linear model/ Title
'Cicindila' data LOGLIN infile
'H\Forsøgsplanlægning\2005\Log linear
models\two-way table.prn' firstobs 2 input
Season Color Obs proc catmod weight
Obs model SeasonColor_response_ /
noresponse predfreq prob / the noresponse
option is used because log linear models do not
distinguish between response and non-response
variables / loglin Season Color /
SeasonColor is omitted from the saturated model
/ run
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Maximum Likelihood Analysis of
Variance
Source DF Chi-Square Pr gt
ChiSq
________________________________________________
_____ Season
3 499.18 lt.0001
Color 1
8.80 0.0030

Likelihood
Ratio 3 28.60 lt.0001



Analysis of Maximum Likelihood Estimates

Standard Chi-
Effect
Parameter Estimate Error Square
Pr gt ChiSq
__________________________________________________
__________ Season
1 -0.8972 0.1276 49.42
lt.0001
2 -0.9225 0.1289 51.24
lt.0001
3 1.5538 0.0698 496.19
lt.0001 Color 4
0.1153 0.0389 8.80 0.0030

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Maximum Likelihood Predicted Values for
Frequencies


-------Observed------
------Predicted------

Standard Standard
Season Color Frequency
Error Frequency Error Residual
____________________________________
_________________________ ESPRING
BR 29 5.267509 22.29508
3.50348 6.704918 ESPRING NBR
11 3.289327 17.70492
2.820956 -6.70492 ESUM
BR 8 2.811516 21.7377
3.459946 -13.7377 ESUM
NBR 31 5.437629 17.2623
2.784941 13.7377 LSPRING BR
273 12.72511 258.623
11.11879 14.37705 LSPRING NBR
191 11.68896 205.377
10.35382 -14.377 LSUM
BR 64 7.608921 71.34426
6.180843 -7.34426 LSUM
NBR 64 7.608921 56.65574
5.129978 7.344262
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Since the response variable (color morphs) has
two levels, an alternative to log-linear analysis
is logistic regression
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SAS program
Title 'Cicindela' data Logistic infile
'H\Forsøgsplanlægning\2005\Log linear
models\two-way table logistic).prn' firstobs
2 input Season BR NBR Total BRNBR /
Total is sum of Bright red and non-bright red
individuals / proc genmod class season model
BR/Total Season / link logit dist binomial
type3 obstats run
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Criteria For Assessing Goodness Of Fit
Criterion DF
Value Value/DF
Deviance 0
0.0000 . Scaled
Deviance 0 0.0000
. Pearson Chi-Square 0
0.0000 .
Scaled Pearson X2 0 0.0000
. Log Likelihood
-446.3758 Algorithm
converged.
Analysis Of Parameter Estimates
Standard Wald 95
Confidence Chi- Parameter
DF Estimate Error Limits
Square Pr gt ChiSq Intercept
1 0.0000 0.1768 -0.3465
0.3465 0.00 1.0000 Season
ESPRING 1 0.9694 0.3958 0.1937
1.7451 6.00 0.0143 Season ESUM
1 -1.3545 0.4342 -2.2055
-0.5036 9.73 0.0018 Season
LSPRING 1 0.3572 0.2004 -0.0355
0.7499 3.18 0.0746 Season LSUM
0 0.0000 0.0000 0.0000
0.0000 . . Scale
0 1.0000 0.0000 1.0000
1.0000 NOTE The scale parameter was held
fixed. LR Statistics For Type 3 Analysis
Chi-
Source DF Square Pr
gt ChiSq Season 3
28.60 lt.0001

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Three-way classification
Example Carcasses of wildebeests (gnus)
Two mortality causes P Predation NP
Non-predation
Two sexes F Female M Male
Three marrow types SWF Solid White Fat OG
Opaque Gelatinous TG Translucent Gelatineous
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Observed distribution of carcasses
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The saturated log-linear model
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/ Example from Quinn and Keough p 395/ /Data
from Sinclair and Arcese 1995/ /Log linear
model/ Title 'Data from Sinclair and Arcese
1995' data LOGLIN infile 'H\Forsøgsplanlægning\
2005\Log linear models\Sinclair data.prn'
firstobs 2 input Death Sex Marrow
Obs / Death mortality factor (P predation,
NP non-predation) / / Sex (M male, F
female) / Marrow Marrow type (SWF solid
white fatty, OG opaque gelatinous or TG
translucent gelatinous / proc catmod weight
Obs model DeathSexMarrow_response_ /
noresponse predfreq prob loglin Death Sex
Marrow DeathSex DeathMarrow SexMarrow
title2 'Model with no three-way
interactions' run
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No three-way interaction terms
Maximum Likelihood Analysis of Variance Source
DF Chi-Square Pr gt
ChiSq ____________________________________________
______ Death 1 4.22
0.0400 Sex 1
0.11 0.7439 Marrow 2
28.38 lt.0001 DeathSex
1 1.26 0.2610 DeathMarrow 2
27.44 lt.0001 SexMarrow 2
5.79 0.0554 Likelihood Ratio
2 7.19 0.0275
H0 The model fits the data (i.e. deviation is
due to random noise)
A low P-value indicates a poor fit
Goodness-of-fit test
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Maximum Likelihood Predicted
Values for Frequencies
-------Observed------ ------Predicted------

Standard Standard Death Sex
Marrow Frequency Error Frequency Error
residual ________________________________________
_______________ NP F OG
26 4.796754 21.59476 4.08837
4.40524 NP F SWF 6
2.416756 8.614391 2.546923 -2.61439 NP
F TG 16 3.855808
17.79085 3.771567 -1.79085 NP M
OG 12 3.370880 16.40524
3.52073 -4.40524 NP M SWF 7
2.604455 4.385611 1.581861
2.61439 NP M TG 26
4.796754 24.20915 4.410191 1.79085 P
F OG 32 5.241090
36.40524 5.265174 -4.40524 P F
SWF 26 4.796754 23.38561
4.378406 2.61439 P F TG
8 2.777915 6.209150 1.967663
1.79085 P M OG 43
5.900727 38.59476 5.402358 4.40524 P
M SWF 14 3.623913 16.61439
3.687038 -2.61439 P M TG
10 3.091524 11.79085 3.001563
-1.79085
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The reduced model
is found to be the best one
Conclusion Marrow type affects predation
risk, but differently for males and females
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What marrow type has the highest predation risk
given an individual is a female?
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Odds for a female with SWF marrow being predated
is
Odds for a female with OG marrow being predated is
Odds for a female with TG marrow being predated is
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To compare the risk of predation of SWF females
with OG females we use the odds ratio
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Odds ratio for females with SWF versus OG marrow
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Odds ratio for males with SWF versus OG marrow
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Asymptotic confidence limits for an odds ratio
For instance, if a 0.05, then za 1.96
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Confidence limits for the odds ratio for
predation of females with SWF marrow versus OG
marrow
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Confidence limits for females
SWF vs OG
Conclusion Females with SWF marrow have a
significantly higher risk of predation than
females with OG or TG marrow, whereas there is no
difference between females with OG and TG marrow
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Confidence limits for males
SWF vs OG
Conclusion Males with TG marrow have a
significantly lower risk of predation than males
with SWF and OG marrow, whereas there is no
difference between males with SWF and OG marrow