1. Multiplicative Interaction Models in R
Heather Turner and David Firth
Department of Statistics
The University of Warwick, UK
8th August 2005

2. Multiplicative Interaction Models
– Interaction terms in generalized linear models can be over-complex
ª use multiplicative interactions instead
– e.g. Goodmans’ row-column association model
log µrc = αr + βc + γr δc
– e.g. UNIDIFF model
log µijk = αik + βjk + γk δij
– e.g. GAMMI model
K
link(µrc ) = αr + βc + σk γkr δkc
k=1

3. Fitting Multiplicative Interaction Models
– Multiplicative interactions are non-linear
ª difficult to fit using standard software
– Llama (Log Linear and Multiplicative Analysis) written by Firth (1998)
ª log link only
ª categorical variables only
ª standard multiplicative interactions
ª XLisp package

4. The gnm R Package
– Provides framework for estimating generalised nonlinear models
ª in-built mechanism for multiplicative terms
ª works with “plug-in” functions to fit other types of nonlinear terms
– Designed to be glm-like
ª common arguments, returned objects, methods, etc
– Estimates model parameters using maximum likelihood
– Uses over-parameterised representations of models

5. Working with Over-Parameterised Models
– gnm does not impose any identifiability constraints on the nonlinear parameters
ª the same model can be represented by an infinite number of
parameterizations, e.g.
log µrc = αr + βc + γr δc
= αr + βc + (2γr )(0.5δc )
′ ′
= αr + βc + γr δc
ª gnm will return one of these parameterisations, at random
– General rule for applying constraints not required
– Fitting algorithm must be able to handle singular matrices

6. Parameter Estimation
– Wish to estimate the predictor
η = η(β)
which is nonlinear, so we have a local design matrix
∂η
X(β) =
∂β
where rank(X) < p, the no. of parameters, due to over-parameterisation
– Use maximum likelihood estimation: want to solve the likehood score equations
U (β) = ∇l(β) = 0

7. Fitting Algorithm
– Use a two stage procedure:
1. one-parameter-at-a-time Newton method to update nonlinear parameters
2. full Newton-Raphson to update all parameters but with the Moore-Penrose
pseudoinverse (X T W X)− since X is not of full rank
– Starting values are obtained in two ways
for the linear parameters use estimates from a glm fit
for the nonlinear parameters generate randomly
ª parameterization determined by starting values for nonlinear parameters

8. Model Specification
– Models in R are usually specified by a symbolic formula, such as
y ∼a+b+a:b
– gnm introduces two functions to specify nonlinear terms
ª Mult for standard multiplicative interactions, e.g.
counts ∼ row + column +
Mult(-1 + row, -1 + column)
ª Nonlin for other terms that require a “plug-in” function, e.g.
counts ∼ row + column +
Nonlin(MultHomog(row, column))

9. Example: Occupational Status Data
– Study of occupational status taken from Goodman (1979)
– Cross-classified by occupational status of father (origin) and son (destination)
> occupationalStatus
destination
origin 1 2 3 4 5 6 7 8
1 50 19 26 8 7 11 6 2
2 16 40 34 18 11 20 8 3
3 12 35 65 66 35 88 23 21
4 11 20 58 110 40 183 64 32
5 2 8 12 23 25 46 28 12
6 12 28 102 162 90 554 230 177
7 0 6 19 40 21 158 143 71
8 0 3 14 32 15 126 91 106

10. Homogeneous Row-Column Association Model
Call:
gnm(formula = Freq ˜ origin + destination + Diag(origin, destination) +
Nonlin(MultHomog(origin, destination)), family = poisson,
data = occupationalStatus)
Coefficients:
(Intercept) origin2
0.01031 0.52684
...
origin8 destination2
1.29563 0.94586
...
destination8 Diag(origin, destination)1
1.87101 1.52667
...
Diag(origin, destination)8 MultHomog(origin, destination).1
0.38848 -1.54112
...
MultHomog(origin, destination).8
1.04786

11. Homogeneous Row-column Association Model (Contd.)
Deviance: 32.56098
Pearson chi-squared: 31.20718
Residual df: 34
– Compare to model with heterogeneous multiplicative effects
> RC <- gnm(Freq ˜ origin + destination + Diag(origin, destination) +
+ Mult(origin, destination), family = poisson,
+ data = occupationalStatus)
> RChomog$dev - RC$dev
[1] 3.411823
> RChomog$df.residual - RC$df.residual
[1] 6

12. Estimate Identifiable Contrasts
– Unconstrained parameters of homogeneous multiplicative factor not identifiable
– Contrast effects with lowest level
> round(getContrasts(RChomog, rev(coefs.of.interest))[[1]], 5)
estimate se
MultHomog(origin, destination).8 2.58898 0.18869
MultHomog(origin, destination).7 2.34541 0.17263
MultHomog(origin, destination).6 1.92927 0.15737
MultHomog(origin, destination).5 1.41751 0.17195
MultHomog(origin, destination).4 1.40034 0.16029
MultHomog(origin, destination).3 0.81646 0.16665
MultHomog(origin, destination).2 0.21829 0.23465
MultHomog(origin, destination).1 0.00000 0.00000

13. Concluding Remarks
– gnm has many more useful features
ª for multiplicative models: functional factors, Exp, multiplicity
ª other: symmetric interactions, diagonal reference models, eliminate, ...
– More details can be found on
http://www.warwick.ac.uk/go/dfirth/software/gnm
– Available on CRAN http://cran.r-project.org