Dref: Specify a Diagonal Reference Term in a gnm Model Formula

Description

Dref is a function of class "nonlin" to specify a diagonal reference term in the formula argument to gnm.

Usage

Dref(..., delta = ~ 1)

Value

A list with the anticipated components of a "nonlin" function:

predictors: the factors passed to Dref and the formulae for the weights.
common: an index to specify that common effects are to be estimated across the factors.
term: a function to create a deparsed mathematical expression of the term, given labels for the predictors.
start: a function to generate starting values for the parameters.
call: the call to use as a prefix for parameter labels.

Arguments

...: a comma-separated list of two or more factors.
delta: a formula with no left-hand-side specifying the model for each factor weight.

Author

Heather Turner

Details

Dref specifies 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 delta argument (constant weights by default). The returned parameters are those in the model for $\delta_f$, rather than the implied weights $w_f$. The DrefWeights function will take a fitted gnm model and return the weights $w_f$, along with their standard errors.

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)

## 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 ~ -1 + Dref(origin, destination), 
                     family = binomial, data = voting)
DrefWeights(classMobility)

## 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 ~ -1 + Dref(origin, destination,
                                       delta = ~ 1 + downward + upward),
                      family = binomial, data = voting)
DrefWeights(socialMobility)

## fit separate weights for downwardly mobile groups only
downwardMobility <- gnm(yvar ~ -1 + Dref(origin, destination,
                                         delta = ~ 1 + downward),
                        family = binomial, data = voting)
DrefWeights(downwardMobility)

if (FALSE) {	       
### 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 + 
         Dref(MOPLM, FOPLF), family = gaussian, data = conformity,
         verbose = FALSE)
A
## Call:
## gnm(formula = MCFM ~ -1 + AGEM + MRMM + FRMF + MWORK + MFCM + 
##     Dref(MOPLM, FOPLF), family = gaussian, data = conformity, 
##     verbose = FALSE)
## 
## Coefficients:
##                     AGEM                      MRMM  
##                  0.06363                  -0.32425  
##                     FRMF                     MWORK  
##                 -0.25324                  -0.06430  
##                     MFCM  Dref(MOPLM, FOPLF)delta1  
##                 -0.06043                  -0.33731  
## Dref(MOPLM, FOPLF)delta2   Dref(., .).MOPLM|FOPLF1  
##                 -0.02505                   4.95121  
##  Dref(., .).MOPLM|FOPLF2   Dref(., .).MOPLM|FOPLF3  
##                  4.86329                   4.86458  
##  Dref(., .).MOPLM|FOPLF4   Dref(., .).MOPLM|FOPLF5  
##                  4.72343                   4.43516  
##  Dref(., .).MOPLM|FOPLF6   Dref(., .).MOPLM|FOPLF7  
##                  4.18873                   4.43378  
## 
## Deviance:            425.3389 
## Pearson chi-squared: 425.3389 
## Residual df:         576    
 
## Weights as in Table 4
DrefWeights(A)
## Refitting with parameters of first Dref weight constrained to zero
## $MOPLM
##    weight        se
## 0.4225636 0.1439829
## 
## $FOPLF
##    weight        se
## 0.5774364 0.1439829 

F <- gnm(MCFM ~ -1 + AGEM + MRMM + FRMF + MWORK + MFCM + 
         Dref(MOPLM, FOPLF, delta = ~1 + MFCM), family = gaussian,
         data = conformity, verbose = FALSE)
F	 
## Call:
## gnm(formula = MCFM ~ -1 + AGEM + MRMM + FRMF + MWORK + MFCM + 
##     Dref(MOPLM, FOPLF, delta = ~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, delta = ~ . + MFCM).delta1(Intercept)  
##                                                  1.03515  
##           Dref(MOPLM, FOPLF, delta = ~ 1 + .).delta1MFCM  
##                                                 -1.77756  
## Dref(MOPLM, FOPLF, delta = ~ . + MFCM).delta2(Intercept)  
##                                                 -0.03515  
##           Dref(MOPLM, FOPLF, delta = ~ 1 + .).delta2MFCM  
##                                                  2.77756  
##              Dref(., ., delta = ~ 1 + MFCM).MOPLM|FOPLF1  
##                                                  4.82476  
##              Dref(., ., delta = ~ 1 + MFCM).MOPLM|FOPLF2  
##                                                  4.88066  
##              Dref(., ., delta = ~ 1 + MFCM).MOPLM|FOPLF3  
##                                                  4.83969  
##              Dref(., ., delta = ~ 1 + MFCM).MOPLM|FOPLF4  
##                                                  4.74850  
##              Dref(., ., delta = ~ 1 + MFCM).MOPLM|FOPLF5  
##                                                  4.42020  
##              Dref(., ., delta = ~ 1 + MFCM).MOPLM|FOPLF6  
##                                                  4.17957  
##              Dref(., ., delta = ~ 1 + MFCM).MOPLM|FOPLF7  
##                                                  4.40819  
## 
## Deviance:            420.9022 
## Pearson chi-squared: 420.9022 
## Residual df:         575 
##
##

## Standard error for MFCM == 1 lower than reported by Van der Slik et al
DrefWeights(F)
## Refitting with parameters of first Dref weight constrained to zero
## $MOPLM
##   MFCM     weight        se
## 1    1 0.02974675 0.2277711
## 2    0 0.74465224 0.2006916
## 
## $FOPLF
##   MFCM    weight        se
## 1    1 0.9702532 0.2277711
## 2    0 0.2553478 0.2006916
}

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