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gnm: a Package for Generalized Nonlinear
                          Models

                                 Heather Turner and David Firth

                                        Department of Statistics
                                       University of Warwick, UK




Heather Turner and David Firth                                     University of Warwick
gnm: a Package for Generalized Nonlinear Models
Generalized Nonlinear Models

         Same form as generalized linear models:

                                          g(E(Y )) = g(µ) = η(x; β)
                                           Var(Y ) = f (µ)

         except η(x; β) can be nonlinear in β.
         E.g. row-column association model (Goodman, 1979)

                                            log µrc = αr + βc + γr δc



Heather Turner and David Firth                                          University of Warwick
gnm: a Package for Generalized Nonlinear Models
Further Examples
         Diagonal reference (Sobel, 1981)

                                            e.g. µrc = w1 γr + w2 γc

         Stereotype (Anderson, 1984)

                                  e.g. log µic = β0c + γc (β1 x1i + β2 x2i )

         UNIDIFF (Erikson & Goldthorpe, 1992; Xie, 1992)

                                     e.g. log µijk = αik + βjk + γk δij

         GAMMI (van Eeuwijk, 1995)

                                 e.g. µij = αi + βj + σ1 γ1i δ1j + σ2 γ2i δ2j
Heather Turner and David Firth                                                  University of Warwick
gnm: a Package for Generalized Nonlinear Models
The gnm Package


         Provides framework for fitting generalized nonlinear models
         Model-fitting function gnm
                 in-built mechanism to fit multiplicative terms
                 works with “plug-in” functions to fit other nonlinear terms
         Designed to be glm-like
                 common arguments, gnm objects inherit from glm objects, etc
         Uses over-parameterized representations of models




Heather Turner and David Firth                                      University of Warwick
gnm: a Package for Generalized Nonlinear Models
Model Specification

         gnm introduces two functions to specify nonlinear terms
                 Mult for standard multiplicative interactions, e.g.
                 counts ∼ row + column +
                     Mult(-1 + row, -1 + column)
                 Nonlin for other terms that require a “plug-in” function, e.g.
                 counts ∼ row + column +
                     Nonlin(MultHomog(row, column))
         Also functions to create factors for structured linear interactions
                 Diag (diagonal), Symm (symmetric), Topo (topological)



Heather Turner and David Firth                                       University of Warwick
gnm: a Package for Generalized Nonlinear Models
Working with Over-Parameterized Models

         gnm imposes minimal identifiability constraints
                 the same model can be represented by an infinite number of
                 parameterisations, e.g.

                                          log µrc = αr + βc + γr δc
                                                  = αr + βc + (2γr )(0.5δc )
                                                  = αr + βc + γr δc

                 gnm will return one of these parameterisations, at random
         Rules for constraining nonlinear parameters not required


Heather Turner and David Firth                                                 University of Warwick
gnm: a Package for Generalized Nonlinear Models
Estimating Identifiable Parameter
                              Combinations


         Through gnm
                 using arguments constrain and constrainTo
         From gnm object
                 estimate simple contrasts using getContrasts
                 estimate linear combinations of parameters using se
         both getContrasts and se check estimability first




Heather Turner and David Firth                                     University of Warwick
gnm: a Package for Generalized Nonlinear Models
Example: Occupational Status Data
         Study of occupational status taken from Goodman (1979)
         Cross-classified by occupational status of father: orig[in] and
         son: dest[ination]
                         > status
                               dest
                         orig     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

Heather Turner and David Firth                                                    University of Warwick
gnm: a Package for Generalized Nonlinear Models
Row-Column Association Model

         Model diagonal effects separately and use standard multiplicative
         interaction

                                 log µrc = αr + βc + θr (r = c) + γr δc
         > RC <- gnm(Freq ~ orig + dest + Diag(orig, dest) +
         +           Mult(-1 + orig, -1 + dest),
         +           family = poisson, data = status)

         Print summary of model
         > summary(RC)



Heather Turner and David Firth                                            University of Warwick
gnm: a Package for Generalized Nonlinear Models
Row-Column Association Model Summary
        Call: ...

        Deviance Residuals: ...

        Coefficients:
                            Estimate Std. Error z value Pr(>|z|)
        (Intercept)          0.19649         NA      NA       NA
        orig2                0.46730         NA      NA       NA
        ...
        orig8                1.15890         NA      NA       NA
                                                                     orig main effects
        dest2                0.98771         NA      NA       NA
        ...
        dest8                1.67662         NA      NA       NA
                                                                     dest main effects
        Diag(orig, dest)1    1.47923    0.45401   3.258 0.00112 **
        ...                                                          diagonal effects
        Diag(orig, dest)8    0.40731    0.21930   1.857 0.06327 .
        Mult1.Factor1.orig1 1.80430          NA      NA       NA
        ...                                                          orig multiplier
        Mult1.Factor1.orig8 -1.48489         NA      NA       NA
        Mult1.Factor2.dest1 1.23943          NA      NA       NA
        ...                                                          dest multiplier
        Mult1.Factor2.dest8 -0.82438         NA      NA       NA

        ...
        Residual deviance: 29.149 on 28 degrees of freedom
        AIC: 423.49

        Number of iterations: 9
Heather Turner and David Firth                                                     University of Warwick
gnm: a Package for Generalized Nonlinear Models
Homogeneous Row-Column Association Model
         Compare to model with homogeneous multiplicative interaction

                                       log µij = αi + βj + θij + δi δj
         > RChomog <- update(RC, . ~ . - Mult(-1 + orig, -1 + dest)
         +                   + Nonlin(MultHomog(orig, dest)),
         +                   ofInterest = "Mult")
         >
         > anova(RChomog, RC, test = "Chisq")
         Analysis of Deviance Table

         Model 1:         Freq ~ orig + ... + Nonlin(MultHomog(orig, dest))
         Model 2:         Freq ~ orig + ... + Mult(orig, dest)
           Resid.         Df Resid. Dev Df Deviance P(>|Chi|)
         1                34     32.561
         2                28     29.149 6     3.412     0.756
Heather Turner and David Firth                                           University of Warwick
gnm: a Package for Generalized Nonlinear Models
Homogeneous Model Summary
        > summary(RChomog)
        Call:

        gnm(formula = Freq ~ orig + dest + Diag(orig, dest) + Nonlin(MultHomog(orig,
            dest)), ofInterest = "Mult", family = poisson, data = status)


        Deviance Residuals:
            Min       1Q   Median          3Q        Max
        -1.6588 -0.4297     0.0000     0.3862     1.7208

        Coefficients of interest:
                                Estimate Std. Error z value Pr(>|z|)
        MultHomog(orig, dest).1 -1.50270         NA      NA       NA
        MultHomog(orig, dest).2 -1.28440         NA      NA       NA
        MultHomog(orig, dest).3 -0.68624         NA      NA       NA
        MultHomog(orig, dest).4 -0.10236         NA      NA       NA
        MultHomog(orig, dest).5 -0.08519         NA      NA       NA        coefficients of interest
        MultHomog(orig, dest).6 0.42657          NA      NA       NA
        MultHomog(orig, dest).7 0.84271          NA      NA       NA
        MultHomog(orig, dest).8 1.08629          NA      NA       NA

        Std. Error is NA where coefficient has been constrained or is unidentified

        Residual deviance: 32.561 on 34 degrees of freedom
        AIC: 414.9

        Number of iterations: 7

Heather Turner and David Firth                                                         University of Warwick
gnm: a Package for Generalized Nonlinear Models
Using getContrasts
         > contr <- getContrasts(RChomog,         ofInterest(RChomog))
         > round(contr[[1]]$qvframe, 3)
                                 estimate            SE quasiSE quasiVar
         MultHomog(orig, dest).1    0.000         0.000   0.157    0.025
         MultHomog(orig, dest).2    0.218         0.235   0.119    0.014
         MultHomog(orig, dest).3    0.816         0.167   0.061    0.004
         MultHomog(orig, dest).4    1.400         0.160   0.052    0.003
         MultHomog(orig, dest).5    1.418         0.172   0.080    0.006
         MultHomog(orig, dest).6    1.929         0.157   0.036    0.001
         MultHomog(orig, dest).7    2.345         0.173   0.080    0.006
         MultHomog(orig, dest).8    2.589         0.189   0.110    0.012


         Quasi standard errors are independent of the parameterisation -
         for more detail see Firth and de Menezes (Biometrika, 2004)
Heather Turner and David Firth                                    University of Warwick
gnm: a Package for Generalized Nonlinear Models
More on gnm ...


         Package available on CRAN: http://cran.r-project.org
         demo(gnm) covers example from this talk
         vignette("gnmOverview", package = "gnm") covers more
         examples and details of further features
                 for Mult terms: Exp, multiplicity
                 other: Dref, eliminate, ...
         Vignette and slides from short course available from
         http://www.warwick.ac.uk/go/heatherturner/gnm



Heather Turner and David Firth                             University of Warwick
gnm: a Package for Generalized Nonlinear Models
Further Work

         Integrate models which currently need work-arounds
                 > 1 instance of homogeneous multiplicative interaction, e.g.

                                       log µij = αi + βj + θij + δi δj + θi θj

                 “sum-of-exponentials” models, e.g.

                                 y = α + exp(β1 + γ1 x) + exp(β2 + γ2 x) + e

         Handle exponentials and reciprocals as standard
                 add Exp and Inv terms
                 allow nested nonlinear terms?


Heather Turner and David Firth                                                   University of Warwick
gnm: a Package for Generalized Nonlinear Models

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gnm: a Package for Generalized Nonlinear Models

  • 1. gnm: a Package for Generalized Nonlinear Models Heather Turner and David Firth Department of Statistics University of Warwick, UK Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 2. Generalized Nonlinear Models Same form as generalized linear models: g(E(Y )) = g(µ) = η(x; β) Var(Y ) = f (µ) except η(x; β) can be nonlinear in β. E.g. row-column association model (Goodman, 1979) log µrc = αr + βc + γr δc Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 3. Further Examples Diagonal reference (Sobel, 1981) e.g. µrc = w1 γr + w2 γc Stereotype (Anderson, 1984) e.g. log µic = β0c + γc (β1 x1i + β2 x2i ) UNIDIFF (Erikson & Goldthorpe, 1992; Xie, 1992) e.g. log µijk = αik + βjk + γk δij GAMMI (van Eeuwijk, 1995) e.g. µij = αi + βj + σ1 γ1i δ1j + σ2 γ2i δ2j Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 4. The gnm Package Provides framework for fitting generalized nonlinear models Model-fitting function gnm in-built mechanism to fit multiplicative terms works with “plug-in” functions to fit other nonlinear terms Designed to be glm-like common arguments, gnm objects inherit from glm objects, etc Uses over-parameterized representations of models Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 5. Model Specification gnm introduces two functions to specify nonlinear terms Mult for standard multiplicative interactions, e.g. counts ∼ row + column + Mult(-1 + row, -1 + column) Nonlin for other terms that require a “plug-in” function, e.g. counts ∼ row + column + Nonlin(MultHomog(row, column)) Also functions to create factors for structured linear interactions Diag (diagonal), Symm (symmetric), Topo (topological) Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 6. Working with Over-Parameterized Models gnm imposes minimal identifiability constraints the same model can be represented by an infinite number of parameterisations, e.g. log µrc = αr + βc + γr δc = αr + βc + (2γr )(0.5δc ) = αr + βc + γr δc gnm will return one of these parameterisations, at random Rules for constraining nonlinear parameters not required Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 7. Estimating Identifiable Parameter Combinations Through gnm using arguments constrain and constrainTo From gnm object estimate simple contrasts using getContrasts estimate linear combinations of parameters using se both getContrasts and se check estimability first Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 8. Example: Occupational Status Data Study of occupational status taken from Goodman (1979) Cross-classified by occupational status of father: orig[in] and son: dest[ination] > status dest orig 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 Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 9. Row-Column Association Model Model diagonal effects separately and use standard multiplicative interaction log µrc = αr + βc + θr (r = c) + γr δc > RC <- gnm(Freq ~ orig + dest + Diag(orig, dest) + + Mult(-1 + orig, -1 + dest), + family = poisson, data = status) Print summary of model > summary(RC) Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 10. Row-Column Association Model Summary Call: ... Deviance Residuals: ... Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 0.19649 NA NA NA orig2 0.46730 NA NA NA ... orig8 1.15890 NA NA NA orig main effects dest2 0.98771 NA NA NA ... dest8 1.67662 NA NA NA dest main effects Diag(orig, dest)1 1.47923 0.45401 3.258 0.00112 ** ... diagonal effects Diag(orig, dest)8 0.40731 0.21930 1.857 0.06327 . Mult1.Factor1.orig1 1.80430 NA NA NA ... orig multiplier Mult1.Factor1.orig8 -1.48489 NA NA NA Mult1.Factor2.dest1 1.23943 NA NA NA ... dest multiplier Mult1.Factor2.dest8 -0.82438 NA NA NA ... Residual deviance: 29.149 on 28 degrees of freedom AIC: 423.49 Number of iterations: 9 Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 11. Homogeneous Row-Column Association Model Compare to model with homogeneous multiplicative interaction log µij = αi + βj + θij + δi δj > RChomog <- update(RC, . ~ . - Mult(-1 + orig, -1 + dest) + + Nonlin(MultHomog(orig, dest)), + ofInterest = "Mult") > > anova(RChomog, RC, test = "Chisq") Analysis of Deviance Table Model 1: Freq ~ orig + ... + Nonlin(MultHomog(orig, dest)) Model 2: Freq ~ orig + ... + Mult(orig, dest) Resid. Df Resid. Dev Df Deviance P(>|Chi|) 1 34 32.561 2 28 29.149 6 3.412 0.756 Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 12. Homogeneous Model Summary > summary(RChomog) Call: gnm(formula = Freq ~ orig + dest + Diag(orig, dest) + Nonlin(MultHomog(orig, dest)), ofInterest = "Mult", family = poisson, data = status) Deviance Residuals: Min 1Q Median 3Q Max -1.6588 -0.4297 0.0000 0.3862 1.7208 Coefficients of interest: Estimate Std. Error z value Pr(>|z|) MultHomog(orig, dest).1 -1.50270 NA NA NA MultHomog(orig, dest).2 -1.28440 NA NA NA MultHomog(orig, dest).3 -0.68624 NA NA NA MultHomog(orig, dest).4 -0.10236 NA NA NA MultHomog(orig, dest).5 -0.08519 NA NA NA coefficients of interest MultHomog(orig, dest).6 0.42657 NA NA NA MultHomog(orig, dest).7 0.84271 NA NA NA MultHomog(orig, dest).8 1.08629 NA NA NA Std. Error is NA where coefficient has been constrained or is unidentified Residual deviance: 32.561 on 34 degrees of freedom AIC: 414.9 Number of iterations: 7 Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 13. Using getContrasts > contr <- getContrasts(RChomog, ofInterest(RChomog)) > round(contr[[1]]$qvframe, 3) estimate SE quasiSE quasiVar MultHomog(orig, dest).1 0.000 0.000 0.157 0.025 MultHomog(orig, dest).2 0.218 0.235 0.119 0.014 MultHomog(orig, dest).3 0.816 0.167 0.061 0.004 MultHomog(orig, dest).4 1.400 0.160 0.052 0.003 MultHomog(orig, dest).5 1.418 0.172 0.080 0.006 MultHomog(orig, dest).6 1.929 0.157 0.036 0.001 MultHomog(orig, dest).7 2.345 0.173 0.080 0.006 MultHomog(orig, dest).8 2.589 0.189 0.110 0.012 Quasi standard errors are independent of the parameterisation - for more detail see Firth and de Menezes (Biometrika, 2004) Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 14. More on gnm ... Package available on CRAN: http://cran.r-project.org demo(gnm) covers example from this talk vignette("gnmOverview", package = "gnm") covers more examples and details of further features for Mult terms: Exp, multiplicity other: Dref, eliminate, ... Vignette and slides from short course available from http://www.warwick.ac.uk/go/heatherturner/gnm Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models
  • 15. Further Work Integrate models which currently need work-arounds > 1 instance of homogeneous multiplicative interaction, e.g. log µij = αi + βj + θij + δi δj + θi θj “sum-of-exponentials” models, e.g. y = α + exp(β1 + γ1 x) + exp(β2 + γ2 x) + e Handle exponentials and reciprocals as standard add Exp and Inv terms allow nested nonlinear terms? Heather Turner and David Firth University of Warwick gnm: a Package for Generalized Nonlinear Models