Generalized Iterative Proportional Fitting for Relational Models

Description

The package provides an iterative scaling procedure that computes the maximum likelihood estimates of the cell frequencies and of the model parameters under a relational model, with or without the overall effect.

Details

The iterative proportional fitting procedure is called by the function g.ipf.

Note

Tamas Rudas was supported in part by Grant K-106154 from the Hungarian National Scientific Research Fund (OTKA). The authors wish to thank Juraj Medzihorsky for his help with building this package.

Author(s)

Anna Klimova, Tamas Rudas

Maintainer: Anna Klimova <aklimova25@gmail.com>

References

A.Klimova, T.Rudas, A.Dobra, Relational models for contingency tables. J. Multivariate Anal., 2012, 104, 159–173.

A.Klimova, T.Rudas, Iterative proportional scaling for curved exponential families. Scand. J. Statist., 2015, 42, 832–847.

A. Klimova, Coordinate-Free Exponential Families on Contingency Tables. PhD thesis. Advisers: Tamas Rudas and Thomas Richardson.

A.Agresti, Categorical Data Analysis. Wiley, New York, 1990.

J.Aitchison, S.D.Silvey, Maximum-likelihood estimation procedures and associated tests of significance. J. Roy. Statist. Soc. Ser.B, 1960, 22, 154–171.

G.Kawamura, T.Matsuoka, T.Tajiri, M.Nishida, M.Hayashi, Effectiveness of a sugarcane-fish combination as bait in trapping swimming crabs. Fisheries Research, 1995, 22, 155–160.

Examples

### Multiplicative model from Aitchison and Silvey (1960)

A = matrix(c(1, 0, 0, 1, 0, 1, 1, 
             0, 1, 0, 1, 1, 0, 1,
             0, 0, 1, 0, 1, 1, 1), byrow=TRUE, nrow=3) ## the model matrix 

y = c(46,24,7,15,3,4,1) ## the observed data

g.ipf(A, y, 1e-4, "probabilities", "grid")
g.ipf(A, y, 1e-6, "probabilities", "bisection")

Search for the adjustment factor corresponding to the MLE using the bisection method

Description

bisection.update computes the segment that is known to contain the adjustment factor corresponding to the MLE and finds this adjustment factor using the bisection method. It is needed only for relational models for probabilities.

Usage

bisection.update(ModelMx, ObsTbl, tolerance)

Arguments

Value

Author(s)

Anna Klimova, Tamas Rudas

References

A. Klimova, Coordinate-Free Exponential Families on Contingency Tables. PhD thesis. Advisers: Tamas Rudas and Thomas Richardson.

D. Bertsekas, Non-Linear Programming. Athena Scientific, 1999.

Examples

### Multiplicative model from Aitchison and Silvey (1960)

A = matrix(c(1, 0, 0, 1, 0, 1, 1, 
             0, 1, 0, 1, 1, 0, 1,
             0, 0, 1, 0, 1, 1, 1), byrow=TRUE, nrow=3) ## the model matrix 

y = c(46,24,7,15,3,4,1) ## the observed data

bisection.update(A, y, 1e-4)



## The model of independence for a 2 by 2 contingency table

A = matrix(c( 1,1,0,0,
              0,0,1,1,
              1,0,1,0,
              0,1,0,1), byrow=TRUE, nrow=4) ## the model matrix

y = c(1,2,3,4)  ## the observed data


bisection.update(A, y, 1e-5)

Generalized Iterative Proportional Fitting for Relational Models

Description

g.ipf computes the maximum likelihood estimates of the cell frequencies and of the model parameters under a relational model specified by a model matrix.

Usage

g.ipf(ModelMatrix, ObsTable, tol, estimand, adjustment)

Arguments

Value

Author(s)

Anna Klimova, Tamas Rudas

References

A.Klimova, T.Rudas, A.Dobra, Relational models for contingency tables. J. Multivariate Anal., 2012, 104, 159–173.

A.Klimova, T.Rudas, Iterative proportional scaling for curved exponential families. Scand. J. Statist., 2015, 42, 832–847.

A. Klimova, Coordinate-Free Exponential Families on Contingency Tables. PhD thesis. Advisers: Tamas Rudas and Thomas Richardson.

A.Agresti, Categorical Data Analysis. Wiley, New York, 1990.

J.Aitchison, S.D.Silvey, Maximum-likelihood estimation procedures and associated tests of significance. J. Roy. Statist. Soc. Ser.B, 1960, 22, 154–171.

G.Kawamura, T.Matsuoka, T.Tajiri, M.Nishida, M.Hayashi, Effectiveness of a sugarcane-fish combination as bait in trapping swimming crabs. Fisheries Research, 1995, 22, 155–160.

Examples

### Multiplicative model from Aitchison and Silvey (1960)

A = matrix(c(1, 0, 0, 1, 0, 1, 1, 
             0, 1, 0, 1, 1, 0, 1,
             0, 0, 1, 0, 1, 1, 1), byrow=TRUE, nrow=3) ## the model matrix 

y = c(46,24,7,15,3,4,1) ## the observed data

g.ipf(A, y, 1e-6, "probabilities", "bisection")
g.ipf(A, y, 1e-4, "probabilities", "grid")



### Bait study in swimming crabs, see Kawamura at al. (1995) :  

A = matrix(c(1,1,0,1,0,1), 2,3,byrow=TRUE) ## the model matrix

y1 = c(36,2,11)  ## the observed data for Charybdis japonica
y2 = c(71,3,44)  ## the observed data for Portunuspelagicus 

## If the model is for intensities, the adjustemnt factor is equal to one;
## the adjustment parameter is redundant.

g.ipf(A, y1, 1e-6, "intensities", "none")
g.ipf(A, y2, 1e-6, "intensities", "none")

Search for the adjustment factor corresponding to the MLE using a grid

Description

grid.update computes the segment that is known to contain the adjustment factor corresponding to the MLE and, using a grid on this segment, finds this adjsutment factor. It is needed only for relational models for probabilities.

Usage

grid.update(ModelMx, ObsTbl, tolerance)

Arguments

Value

Author(s)

Anna Klimova, Tamas Rudas

References

A. Klimova, Coordinate-Free Exponential Families on Contingency Tables. PhD thesis. Advisers: Tamas Rudas and Thomas Richardson.

Examples

### Multiplicative model from Aitchison and Silvey (1960)

A = matrix(c(1, 0, 0, 1, 0, 1, 1, 
             0, 1, 0, 1, 1, 0, 1,
             0, 0, 1, 0, 1, 1, 1), byrow=TRUE, nrow=3) ## the model matrix 

y = c(46,24,7,15,3,4,1) ## the observed data

grid.update(A, y, 1e-4)


## The model of independence for a 2 by 2 contingency table

A = matrix(c( 1,1,0,0,
              0,0,1,1,
              1,0,1,0,
              0,1,0,1), byrow=TRUE, nrow=4) ## the model matrix

y = c(1,2,3,4)  ## the observed data


grid.update(A, y, 1e-5)

Iterative Proportional Fitting in Relational Models, with a Given Adjustment Factor

Description

For a given model matrix and a given vector of observed cell frequencies, ipf.gamma computes the vector of frequencies whose subset sums are equal to the observed subset sums times the adjustment factor and whose relative frequencies satisfy the multiplicative structure prescribed by the model.

Usage

ipf.gamma(ModelMatrix, ObsTable, gamma, tol, estimand)

Arguments

Value

Author(s)

Anna Klimova, Tamas Rudas

References

A.Klimova, T.Rudas, A.Dobra, Relational models for contingency tables. J. Multivariate Anal., 2012, 104, 159–173.

A.Klimova, T.Rudas, Iterative proportional scaling for curved exponential families. Scand. J. Statist., 2015, 42, 832–847.

A. Klimova, Coordinate-Free Exponential Families on Contingency Tables. PhD thesis. Advisers: Tamas Rudas and Thomas Richardson.

Examples

## The model of independence for a 2 by 2 contingency table

A = matrix(c( 1,1,0,0,
              0,0,1,1,
              1,0,1,0,
              0,1,0,1), byrow=TRUE, nrow=4) ## the model matrix

y = c(1,2,3,4)  ## the observed data

ipf.gamma(A, y, 1, 1e-6, "intensities")   ## Adjustment factor is set to 1 

ipf.gamma(A, y, 0.5, 1e-6, "intensities")   ## Adjustment factor is set to 0.5 

Single Cells under a Relational Model

Description

The function finds all single cells under a relational model. Such cells appear as the only positive entries in their row and column in the model matrix.

Usage

single.cells(ModelMatrix)

Arguments

Value

the row and column indices of the single cells.

Author(s)

Anna Klimova

Examples

G = matrix(c( 2,0,0,0,0,
              0,1,1,0,0,
              0,0,1,1,1), byrow=TRUE, nrow=3) ## a given matrix

single.cells(G)  

Sufficient Statistics under a Relational Model

Description

For an I by J model matrix of a relational model and a vector of frequencies of the length I, the function computes sufficient statistics under the model (subset sums).

Usage

suff.stat(ModelMatrix, Table)

Arguments

Value

a vector of subset sums.

Author(s)

Anna Klimova

References

A.Klimova, T.Rudas, A.Dobra, Relational models for contingency tables. J. Multivariate Anal., 104, 159–173.

Examples

### Multiplicative model from Aitchison and Silvey (1960)

A = matrix(c(1, 0, 0, 1, 0, 1, 1, 
             0, 1, 0, 1, 1, 0, 1,
             0, 0, 1, 0, 1, 1, 1), byrow=TRUE, nrow=3) ## the model matrix 

y = c(46,24,7,15,3,4,1) ## the observed data

suff.stat(A,y) 

### pneumonia infection in calves:

A=matrix(c(2,1,0,0,1,1), 2,3,byrow=TRUE) ## the model matrix

y=c(30,63,63)  ## the observed data

suff.stat(A,y)