de Silva and Condrey: Patterns in patchy data discerned from Brevoortia patronus bycatch 
195 
Table 1 
Hypothetical example used to explain odds ratios and conditional probabili- 
ties. 7i jiu is the conditional probability of observing bycatch given a particular 
area. n j2 , { is the conditional probability of observing no bycatch given a par- 
ticular area. 
Conditional 
Bycatch probabilities 
Presence (jf Absence (jf Total sets n }1 , ( n j2 , ; 
Area A (qj 4 6 10 0.40 0.60 
Area B (i 2 ) 2 4 6 0.33 0.67 
Total 6 10 16 
£ th cell, and let m k = E(n^) represent 
the expected value where k = 1 N. 
The probabilities (jt I for that mul- 
tinomial distribution form the joint dis- 
tribution of two categorical responses. 
These two responses are statistically 
independent when n }J - jz 1+ k +j for all i 
and j. 
If there is a dependence between 
the two variables, then all expected 
values of each cell (m- ) are > 0. The 
yj 
loglinear model for this two-way 
table can be written as 
log m ii =ii + ^+xy J + ^ t 
where p = Z -Z- log m tj / IJ; 
\ x = Z ; log m ij /'J- p; 
= Z- log m i . 7 1 - p; and 
\ xy = log m ij - X* - x/ + p. 
This model perfectly describes any set of positive 
expected frequencies and is referred to as the satu- 
rated model. The right-hand side of this equation 
resembles the formula for the cell-means ANOVA. 
The parameters [X x \ and {Ay v } are deviations about a 
mean and I, X- xy = Z, X, xy = Z, X x = Z, Xf = 0. This model 
can also be described in the notation form as [XY]. 
A saturated loglinear model always expresses a 
given table of categorical data perfectly. This model 
has the maximum achievable log likelihood because 
it is the most general model, with as many para- 
meters as observations. However, it is possible that 
a simpler model may provide a fit as statistically good 
as that of the saturated model. How well this model 
fits is represented by the scaled deviance, a function 
of twice the difference in the log likelihoods of the 
saturated model and the simpler model. In addition 
to testing the fit of a model, one can use the deviance 
to diagnose lack of fit through residual analysis. 
For example, consider a three-dimensional satu- 
rated model with variables X, Y, and Z. For this model 
[ XYZ ], log = p + Xf + Xf + X/f + X tJ xy + X,ff" + 
Xjk yz + X ijk xyz . When X l} }^ yz - 0, there is no three-factor 
interaction, and the association between two vari- 
ables is identical at each level of the third variable 
and reduces to the loglinear model [XY XZ YZ ]. Fur- 
ther, if Xijk xyz - 0 and X jk yz = 0, then for any given 
level of X, Y and Z are conditionally independent [AY 
XZ\. Similarly if X lJ ^ xyz = 0 , Xj^ yz = 0 and X^ 2 - 0, then 
Z is jointly independent of X and Y [ XY Z]. Finally if 
hjk xyz = 0, X jk yz = 0, X ik xz = 0, and X tJ xy = 0, then X, Y 
and Z are mutually independent [XYZ]. With these 
criteria and beginning with a saturated model, we 
used a stepwise model selection procedure with de- 
viance in the form of the G 2 test statistic to find a 
simpler model that fits as well as the saturated 
model. This simpler model would enable one to ex- 
plore multidimensional tables to find simpler repre- 
sentations of the information contained therein. 
Another advantage of loglinear models is that when 
one of the variables can be modeled as a response, 
and the others as explanatory variables, certain 
loglinear models are equivalent to logit models with 
categorical explanatory variables. Such logit models 
enable us to study the problem of interest in a man- 
ner analogous to ANOVA. 
Many categorical response variables have only two 
categories. The response can be classified either as a 
success or a failure. The Bernoulli distribution, which 
belongs to the natural exponential family, forms the 
basis of modeling the logit model. For such a dichoto- 
mous variable, the probability of observing response 
0 can be defined as P(Y-0) - k, and the probability of 
observing response 1 , as P( Y= 1 ) = 1 - 7i. The link func- 
tion for this model, log k i / ( l-7t-), known as the logit, 
is equivalent to the log odds. 
Consider the following example, where we exam- 
ine the presence or absence of bycatch in two areas. 
In this example (Table 1), the 2x2 table has rows 
i x (area 1), and i 2 ( area 2) and columns j 1 ( presence of 
bycatch) and j 2 (absence of by catch). The counts in 
the cells of the table are the number of units of effort 
(individual sets) observed in each category. 
In this case, the odds, Q. iv of observing^ ( presence 
of by catch) given you are in category i 1 (area 1) is 
computed as the ratio of the conditional probabili- 
ties of observing a set with bycatch to that of observ- 
ing a set with no bycatch in area 1: 
{^m/^2ia} is 0.4/0.6 = 0.67. 
Similarly, the odds, O j2 , of observing j ^presence of 
by catch) given you are in category i 2 ( area 2) is com- 
puted as the ratio of the conditional probabilities of 
