Formulas{bivpois} |
R Documentation |
In the formula1 argument of functions lm.bp
and lm.dibp the response should
always be defined as y1y2. The syntax of formula1 is similar as all formula
objects in R. The operators <+, :, *> are used as in normal formula
objects to define additional main effects, interaction terms, or both. Three
additional arguments/variables are defined internally in lm.bp and lm.dibp in order to help us build the
model: noncommon, l1 and l2.
Ø Variable noncommon
is used to specify which terms will be different for the linear
predictors of ë1 and ë2.
Ø Variable l1
is used to specify which terms will have effect only on ë1 using a syntax of type ‘+I(X*l1)' or ‘+I(l1*X)'.
Ø Variable l2
is used to specify which terms will have effect only on ë2 using a syntax of type ‘+I(X*l2)' or ‘+I(l2*X)'.
Hence the following
combinations can be used
1. `+X:noncommon' or `+X*noncommon': The variable X has different effect on ë1 and ë2.
2. `+X': When additional terms
of the form `+X:noncommon' or `+X*noncommon' are not included in formula then
the variable X has common effect on
both ë1 and ë2.
3. `+I(X*l1)' or `+I(l1*X)': The effect of variable X is
estimated only for the linear predictor of ë1 (while for the linear
predictor of ë2 is set equal to zero).
4. `+I(X*l2)' or `+I(l2*X)': The effect of variable X is
estimated only for the linear predictor of
ë2 (while for the linear predictor of ë1 is set equal to zero).
For the argument formula2 the response should always be
defined as y3 while its syntax is the
same as all formula objects in R.
Some usual models are
the followin
1. y1y2~1 :
Common constant for ë1 and ë2 that is log(ë1i )= â0 and log(ë2i )= â0
.
2. y1y2~noncommon :
Constant but not equal ë1 and ë2 that is log(ë1i )= â1,0
and log(ë2i )= â2,0 with â1,0¹â2,0.
3. y1y2~. : Full model with different parameters for ë1 and ë2.
Finally, in both lm.bp and lm.dibp we can construct models for which
different variables have the same effect on ë1
and ë2. This can be achieved using terms of type c(z1,z2). Such as a term results to a
common parameter for both ë1 and ë2 for the variable z1 and z2
respectively.
Ôhe formula
y1y2 $\sim$ noncommon +
c(z1,z2)+z4:noncommon + z5
will result to the following model
log(ë1) = â1,0 + â12 z1 + â1,4 z4 + â5 z5
log(ë2) = â2,0 + â12 z2 + â2,4 z4
+ â5 z5
1.
Karlis, D. and Ntzoufras, I. (2004). Bivariate Poisson and Diagonal
Inflated Bivariate Poisson Regression Models in S. (submitted). Technical
Report, Athens University of Economics and Business, Athens, Greece.
2.
Karlis, D. and Ntzoufras, I. (2003). Analysis of Sports Data Using
Bivariate Poisson Models. Journal of the Royal Statistical Society, D,
(Statistician), 52, 381 – 393.
1.
Dimitris Karlis, Department of Statistics, Athens University of
Economics and Business, Athens, Greece, e-mail: karlis@aueb.gr
.
2.
Ioannis Ntzoufras, Department of Statistics, Athens University of
Economics and Business, Athens, Greece, e-mail: ntzoufras@aueb.gr .
Ôhe formula
y1y2 $\sim$ noncommon +
c(z1,z2)+z4:noncommon + z5
will result to the following model
log(ë1) = â1,0 + â12 z1 + â1,4 z4 + â5 z5
log(ë2) = â2,0 + â12 z2 + â2,4 z4
+ â5 z5
see also examples 1 – 4: ex1.sim, ex2.sim, ex3.health, ex4.ita91 .