Dref: gnm Plug-in Function to Fit Diagonal Reference Terms

Description

A function to provide the objects and functions required to fit a diagonal reference term in a generalized nonlinear model using gnm.

Usage

Dref(..., formula = ~ 1)

Arguments

...

a comma-separated list of two or more factors.

formula

a symbolic description of the model for each factor weight.

Value

A list with the anticipated components of a gnm plug-in function:
starta vector of starting values for the parameters in the diagonal reference term.
labelsa character vector of labels for the parameters in the diagonal reference term.
predictora function that takes estimates of the parameters in the diagonal reference term and returns a matrix whose columns are the fitted additive components of this term.
localDesignFunctiona function that takes the result of the predictor function and returns the local design matrix.

Details

Designed to work as a plug-in function for gnm, Dref produces the objects required to fit diagonal reference terms as introduced by Sobel (1981, 1985). Such terms comprise an additive component for each factor of the form $$w_f\gamma_l$$ where $w_f$ is the weight for factor $f$, $\gamma_l$ is the diagonal effect for level $l$ and $l$ is the level of factor $f$ for the given data point.

The weights are constrained to be nonnegative and to sum to one as follows $$w_f = \frac{e^{\delta_f}}{\sum_i e^{\delta_i}}$$ and the $\delta_f$ are modelled as specified by the formula argument (constant weights by default). The returned parameters are those in the model for $\delta_f$, rather than the implied weights $w_f$. To specify a diagonal reference term in the formula argument to gnm, the symbolic wrapper Nonlin must be used, with a call to Dref as the first argument.

If the factors passed to Dref do not have exactly the same levels, the set of levels in the diagonal reference term is taken to be the union of the factor levels, sorted into increasing order.

References

Sobel, M. E. (1981), Diagonal mobility models: A substantively motivated class of designs for the analysis of mobility effects. American Sociological Review 46, 893--906. Sobel, M. E. (1985), Social mobility and fertility revisited: Some new models for the analysis of the mobility effects hypothesis. American Sociological Review 50, 699--712.

Clifford, P. and Heath, A. F. (1993) The Political Consequences of Social Mobility. J. Roy. Stat. Soc. A, 156(1), 51-61.

Van der Slik, F. W. P., De Graaf, N. D and Gerris, J. R. M. (2002) Conformity to Parental Rules: Asymmetric Influences of Father's and Mother's Levels of Education. European Sociological Review 18(4), 489 -- 502.

Examples

Run this code

### Examples from Clifford and Heath paper
### (Results differ slightly - possible transcription error in
### published data?)
set.seed(1)
data(voting)
## reconstruct counts voting Labour/non-Labour
count <- with(voting, percentage/100 * total)
yvar <- cbind(count, voting$total - count)

## fit diagonal reference model with constant weights
classMobility <- gnm(yvar ~ Nonlin(Dref(origin, destination)), 
                     family = binomial, data = voting)
prop.table(exp(coef(classMobility)[2:3]))

## create factors indicating movement in and out of salariat (class 1)
upward <- with(voting, origin != 1 & destination == 1)
downward <- with(voting, origin == 1 & destination != 1)

## fit separate weights for the "socially mobile" groups
socialMobility <- gnm(yvar ~ Nonlin(Dref(origin, destination,
                                         formula = ~ 1 + downward + upward)),
                      family = binomial, data = voting)
prop.table(exp(coef(socialMobility)[c(4, 7)] + coef(socialMobility)[c(2, 5)]))
prop.table(exp(coef(socialMobility)[c(3, 6)] + coef(socialMobility)[c(2, 5)]))
prop.table(exp(coef(socialMobility)[c(2, 5)]))

## fit separate weights for downwardly mobile groups only
downwardMobility <- gnm(yvar ~ Nonlin(Dref(origin, destination,
                                           formula = ~ 1 + downward)),
                        family = binomial, data = voting)
downwardMobility
prop.table(exp(coef(downwardMobility)[c(3, 5)] + 
               coef(downwardMobility)[c(2, 4)]))
	       prop.table(exp(coef(downwardMobility)[c(2, 4)]))

### Examples from Van der Slik paper
### For illustration only - data not publically available
### Using data in data.frame named 'conformity', with variables
### MCFM - mother's conformity score
### FCFF - father's conformity score
### MOPLM - a factor describing the mother's education with 7 levels
### FOPLF - a factor describing the father's education with 7 levels
### AGEM - mother's birth cohort
### MRMM - mother's traditional role model
### FRMF - father's traditional role model
### MWORK - mother's employment
### MFCM - mother's family conflict score
### FFCF - father's family conflict score

set.seed(1)

## Models for mothers' conformity score as specified in Figure 1
A <- gnm(MCFM ~ -1 + AGEM + MRMM + FRMF + MWORK + MFCM + 
         Nonlin(Dref(MOPLM, FOPLF)), family = gaussian, data = conformity,
         verbose = FALSE)
A
## Call:
## gnm(formula = MCFM ~ -1 + AGEM + MRMM + FRMF + MWORK + MFCM + 
##     Nonlin(Dref(MOPLM, FOPLF)), family = gaussian, data = conformity,
##     verbose = FALSE)
##
##
## Coefficients:
##                    AGEM                      MRMM                      FRMF  
##                 0.06364                  -0.32425                  -0.25324  
##                   MWORK                      MFCM  Dref(MOPLM, FOPLF).MOPLM  
##                -0.06430                  -0.06043                   0.34389  
## Dref(MOPLM, FOPLF).FOPLF      Dref(MOPLM, FOPLF).1      Dref(MOPLM, FOPLF).2  
##                 0.65611                   4.95123                   4.86328  
##     Dref(MOPLM, FOPLF).3      Dref(MOPLM, FOPLF).4      Dref(MOPLM, FOPLF).5  
##                 4.86458                   4.72342                   4.43516  
##     Dref(MOPLM, FOPLF).6      Dref(MOPLM, FOPLF).7  
##                 4.18873                   4.43379  
##
## Deviance:            425.3389 
## Pearson chi-squared: 425.3389 
## Residual df:         576

prop.table(exp(coef(A)[6:7])) ## weights as in Table 4
## Dref(MOPLM, FOPLF).MOPLM Dref(MOPLM, FOPLF).FOPLF 
##               0.4225734                0.5774266

F <- gnm(MCFM ~ -1 + AGEM + MRMM + FRMF + MWORK + MFCM + 
         Nonlin(Dref(MOPLM, FOPLF, formula = ~1 + MFCM)), family = gaussian,
         data = conformity, verbose = FALSE)
F	 
## Call:
## gnm(formula = MCFM ~ -1 + AGEM + MRMM + FRMF + MWORK + MFCM + 
##     Nonlin(Dref(MOPLM, FOPLF, formula = ~1 + MFCM)), family = gaussian, 
##     data = conformity, verbose = FALSE)
##
##
## Coefficients:
##                                                      AGEM  
##                                                   0.05818  
##                                                      MRMM  
##                                                  -0.32701  
##                                                      FRMF  
##                                                  -0.25772  
##                                                     MWORK  
##                                                  -0.07847  
##                                                      MFCM  
##                                                  -0.01694  
## Dref(MOPLM, FOPLF, formula = ~1 + MFCM).MOPLM.(Intercept)  
##                                                   1.03516  
##        Dref(MOPLM, FOPLF, formula = ~1 + MFCM).MOPLM.MFCM  
##                                                  -1.77703  
## Dref(MOPLM, FOPLF, formula = ~1 + MFCM).FOPLF.(Intercept)  
##                                                  -0.03516  
##        Dref(MOPLM, FOPLF, formula = ~1 + MFCM).FOPLF.MFCM  
##                                                   2.77703  
##                 Dref(MOPLM, FOPLF, formula = ~1 + MFCM).1  
##                                                   4.82477  
##                 Dref(MOPLM, FOPLF, formula = ~1 + MFCM).2  
##                                                   4.88066  
##                 Dref(MOPLM, FOPLF, formula = ~1 + MFCM).3  
##                                                   4.83969  
##                 Dref(MOPLM, FOPLF, formula = ~1 + MFCM).4  
##                                                   4.74849  
##                 Dref(MOPLM, FOPLF, formula = ~1 + MFCM).5  
##                                                   4.42019  
##                 Dref(MOPLM, FOPLF, formula = ~1 + MFCM).6  
##                                                   4.17956  
##                 Dref(MOPLM, FOPLF, formula = ~1 + MFCM).7  
##                                                   4.40819  
## 
## Deviance:            420.9022 
## Pearson chi-squared: 420.9022 
## Residual df:         575

prop.table(exp(coef(F))[c(6,8)]) ## weights for MFCM = 0
## Dref(MOPLM, FOPLF, formula = ~1 + MFCM).MOPLM.(Intercept) 
##                                                 0.7446585 
## Dref(MOPLM, FOPLF, formula = ~1 + MFCM).FOPLF.(Intercept) 
##                                                 0.2553415 
prop.table(exp(coef(F)[c(7,9)] + coef(F)[c(6,8)])) ## weights for MFCM = 1
## Dref(MOPLM, FOPLF, formula = ~1 + MFCM).MOPLM.MFCM 
##                                         0.02977851 
## Dref(MOPLM, FOPLF, formula = ~1 + MFCM).FOPLF.MFCM 
##                                         0.97022149

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