The Analysis of Categorical Data

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The Analysis of Categorical Data
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Categorical variables

• When both predictor and response variables are
  categorical
• Presence or absence
• Color, etc.
• The data in such a study represents counts or
  frequencies- of observations in each category

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Analysis
Data Analysis
A single categorical predictor variable Organized as two way contingency tables, and tested with chi-square or G-test
Multiple predictor variables (or complex models) Organized as a multi-way contingency tables, and analyzed using either log-linear models or classification trees
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Two way Contingency Tables

• Analysis of contingency tables is done correctly
  only on the raw counts, not on the percentages,
  proportions, or relative frequencies of the data

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Wildebeest carcasses from the Serengeti (Sinclair
and Arcese 1995)
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Sex, cause of death, and bone marrow type

• Sex (males / females)
• Cause of death (predation / other)
• Bone marrow type
• Solid white fatty (healthy animal)
• Opaque gelatinous
• Translucent gelatinous

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Data
Sex Marrow Death by predation
Male SWF Yes
Male OG Yes
Male TG Yes

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Brief format
SEX MARROW DEATH COUNT
FEMALE SWF PRED 26
MALE SWF PRED 14
FEMALE OG PRED 32
MALE OG PRED 43
FEMALE TG PRED 8
MALE TG PRED 10
FEMALE SWF NPRED 6
MALE SWF NPRED 7
FEMALE OG NPRED 26
MALE OG NPRED 12
FEMALE TG NPRED 16
MALE TG NPRED 26
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Contingency table

• Sex Death Crosstabulation

Dead
Sex NPRED PRED Total
FEMALE 48 66 114
MALE 45 67 112
Total 93 133 226
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Contingency table

• Sex Marrow Crosstabulation

Marrow
Sex OG SWF TG Total
FEMALE 58 32 24 114
MALE 55 21 36 112
Total 113 53 60 226
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Contingency table

• Death Marrow Crosstabulation

Marrow
Death OG SWF TG Total
NPRED 38 13 42 93
PRED 75 40 18 133
Total 113 53 60 226
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Are the variables independent?

• We want to know, for example, whether males are
  more likely to die by predation than females
• Specifying the null hypothesis
• The predictor and response variable are not
  associated with each other. The two variables are
  independent of each other and the observed degree
  of association is not stronger than we would
  expect by chance or random sampling

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Calculating the expected values

• The expected value is the total number of
  observations (N) times the probability of a
  population being both males and dead by predation

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The probability of two independent events
Because we have no other information than the
data, we estimate the probabilities of each of
the right hand terms from the equation from the
marginal totals
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Contingency table

• Sex Death expected values

Dead
Sex NPRED PRED P
FEMALE 46.91 67.09 114 0.5044
MALE 46.09 65.91 112 0.4956
93 133
P 0.4115 0.5885 N226
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Testing the hypothesis Pearsons Chi-square test
0.0866, P0.7685
0.0253, P0.8736
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The degrees of freedom
1
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Calculating the P-value

• We find the probability of obtaining a value of
  ?2 as large or larger than 0.0866 relative to a
  ?2 distribution with 1 degree of freedom
• P 0.769

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An alternative

• The likelihood ratio test It compares observed
  values with the distribution of expected values
  based on the multinomial probability distribution

0.0866
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Two way contingency tables

• Sex Death Crosstabulation
• Sex Marrow Crosstabulation
• Marrow Death Crosstabulation

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Which test to chose?
Model Rows/ Columns Sample size Test
I II Not fixed Fixed/not fixed small G-test, with corrections
I II Not fixed Fixed/not fixed large G-test, Chi square test
III Fixed Fisher exact test
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Log-linear modelsMulti-way Contingency Tables
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Multiple two-way tables
Females Marrow
Death OG SWF TG Total
PRED 32 26 8 66
NPRED 26 6 16 48
Total 58 32 24 114
Males Marrow
Death OG SWF TG Total
PRED 43 14 10 67
NPRED 12 7 26 45
Total 55 21 36 112
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Log-linear models

• They treat the cell frequencies as counts
  distributed as a Poisson random variable
• The expected cell frequencies are modeled against
  the variables using the log-link and Poisson
  error term
• They are fit and parameters estimated using
  maximum likelihood techniques

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Log-linear models

• Do not distinguish response and predictor
  variables all the variables are considered
  equally as response variables

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However

• A logit model with categorical variables can be
  analyzed as a log-linear model

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Two way tables

• For a two way table (I by J) we can fit two
  log-linear models
• The first is a saturated (full) model
• Log fij constant ?ix ?ky ?jkxy
• fij is the expected frequency in cell ij
• ?ix is the effect of category i of variable X
• ?ky is the effect of category k of variable Y
• ?jkxy is the effect any interaction between X
  and Y
• This model fit the observed frequencies perfectly

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Note

• The effect does not imply any causality, just the
  influence of a variable or interaction between
  variables on the log of the expected number of
  observations in a cell

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Two way tables

• The second log-linear model represents
  independence of the two variables (X and Y) and
  is a reduced model
• Log fij constant ?ix ?ky
• The interpretation of this model is that the log
  of the expected frequency in any cell is a
  function of the mean of the log of all the
  expected frequencies plus the effect of variable
  x and the effect of variable y. This is an
  additive linear model with no interactions
  between the two variables

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Interpretation

• The parameters of the log-linear models are the
  effects of a particular category of each variable
  on the expected frequencies
• i.e. a larger ? means that the expected
  frequencies will be larger for that variable.
• These variables are also deviations from the mean
  of all expected frequencies

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Null hypothesis of independence

• The Ho is that the sampling or experimental units
  come from a population of units in which the two
  variables (rows and columns) are independent of
  each other in terms of the cell frequencies
• It is also a test that ?jkxy 0
• There is NO interaction between two variables

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Test

• We can test this Ho by comparing the fit of the
  model without this term to the saturated model
  that includes this term
• We determine the fit of each model by calculating
  the expected frequencies under each model,
  comparing the observed and expected frequencies
  and calculating the log-likelihood of each model

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Test

• We then compare the fit of the two models with
  the likelihood ratio test statistic ?
• However the sampling distribution of this ratio
  (? ) is not well known, so instead we calculate
  G2 statistic
• G2 -2log?
• G2 Follows a ?2 distribution for reasonable
  sample sizes and can be generalized to
• - 2(log-likelihood reduced model --
  log-likelihood full model)

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Degrees of freedom

• The calculated G2 is compared to a ?2
  distribution with (I-1)(J-1) df.
• This df (I-1)(J-1) is the difference between the
  df for the full model (IJ-1) and the df for the
  reduced model (I-1)(j-1)

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Akaike information criteria
Hirotugu Akaike
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The full model
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Complete table
Model G2 df P AIC
1 DSM 42.76 7 0.001 28.76
2 DS 42.68 6 0.001 30.68
3 DM 13.24 5 0.021 3.24
4 SM 37.98 5 0.001 27.98
5 DSDM 13.16 4 0.01 5.16
6 DSSM 37.89 4 0.001 29.89
7 DMSM 8.46 3 0.037 2.46
8 DSDMSM 7.19 2 0.027 3.19
9 Saturated full model 0 0
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Two way interactions (marginal independence)
DSM 42.76 reference d.f P
DS 1vs 2 42.6759 42.76-42.680.084 7-6 1 0.769
DM 1vs 3 13.24 42.76-13.2429.520 7-5 2 lt0.001
SM 1 vs 4 37.98 42.76-37.984.778 7-5 2 0.092
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Three way interaction

• DeathSexMarrow
• Models compared 8 vs 9
• G2 7.19
• df 2
• P0.027

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Conditional independence
term Models compared G2 df P
DS 7 vs 8 1.28 1 0.259
DM 6 vs 8 30.71 2 0.001
SM 5 vs 8 5.97 2 0.051
Death and marrow have a partial association
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Conditional independence
Females Marrow
Death OG SWF TG Total
PRED 32 26 8 66
NPRED 26 6 16 48
Total 58 32 24 114
Males Marrow
Death OG SWF TG Total
PRED 43 14 10 67
NPRED 12 7 26 45
Total 55 21 36 112
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Males 95 CI Females
OG vs TG 0.107 0.041-0.283 0.406 0.150-1.097
SWF vs TG 0.192 0.060-0.616 0.115 0.034-0.395
SWF vs OG 0.558 0.184-1.693 3.521 1.261-9.836

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Complete independence

• Models compared 1 vs 8
• G235.57
• df 5
• Plt0.001

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Warning

• Always fit a saturated model first, containing
  all the variables of interest and all the
  interactions involving the (potential) nuisance
  variables. Only delete from the model the
  interactions that involve the variables of
  interest.