migest (version 2.0.4)

cm2: Conditional maximization routine for the indirect estimation of origin-destination migration flow table with known margins

Description

The cm2 function finds the maximum likelihood estimates for parameters in the log-linear model: $$ \log y_{ij} = \log \alpha_i + \log \beta_j + \log m_{ij} $$ as introduced by Willekens (1999). The \(\alpha_i\) and \(\beta_j\) represent background information related to the characteristics of the origin and destinations respectively. The \(m_{ij}\) factor represents auxiliary information on migration flows, which imposes its interaction structure onto the estimated flow matrix.

Usage

cm2(
  row_tot = NULL,
  col_tot = NULL,
  m = matrix(data = 1, nrow = length(row_tot), ncol = length(col_tot)),
  tol = 1e-06,
  maxit = 500,
  verbose = TRUE,
  rtot = row_tot,
  ctot = col_tot
)

Value

Parameter estimates are obtained using the EM algorithm outlined in Willekens (1999). This is equivalent to a conditional maximization of the likelihood, as discussed by Raymer et. al. (2007). It also provides identical indirect estimates to those obtained from the ipf2 routine.

The user must ensure that the row and column totals are equal in sum. Care must also be taken to allow the dimension of the auxiliary matrix (m) to equal those provided in the row (row_tot) and column (col_tot) arguments.

Returns a list object with

N

Origin-Destination matrix of indirect estimates

theta

Collection of parameter estimates

Arguments

row_tot

Vector of origin totals to constrain the sum of the imputed cell rows.

col_tot

Vector of destination totals to constrain the sum of the imputed cell columns.

m

Matrix of auxiliary data. By default set to 1 for all origin-destination combinations.

tol

Numeric value for the tolerance level used in the parameter estimation.

maxit

Numeric value for the maximum number of iterations used in the parameter estimation.

verbose

Logical value to indicate the print the parameter estimates at each iteration. By default FALSE.

rtot

Depreciated. Use row_tot

ctot

Depreciated. Use col_tot

Author

Guy J. Abel

References

Raymer, J., G. J. Abel, and P. W. F. Smith (2007). Combining census and registration data to estimate detailed elderly migration flows in England and Wales. Journal of the Royal Statistical Society: Series A (Statistics in Society) 170 (4), 891--908.

Willekens, F. (1999). Modelling Approaches to the Indirect Estimation of Migration Flows: From Entropy to EM. Mathematical Population Studies 7 (3), 239--78.

See Also

ipf2

Examples

## with Willekens (1999) data
r <- LETTERS[1:2]
y <- cm2(row_tot = c(18, 20), col_tot = c(16, 22), 
         m = matrix(c(5, 1, 2, 7), ncol = 2, dimnames = list(orig = r, dest = r)))
y

## with all elements of offset equal (independence fit)
y <- cm2(row_tot = c(18, 20), col_tot = c(16, 22))
y

## with bigger matrix
r <- LETTERS[1:4]
y <- cm2(row_tot = c(250, 100, 140, 110), col_tot = c(150, 150, 180, 120),
         m = matrix(data = c(0, 100, 30, 70, 50, 0, 45, 5, 60, 35, 0, 40, 20, 25, 20, 0),
                    nrow = 4, ncol = 4, dimnames = list(orig = r, dest = r), byrow = TRUE))
                    
# display with row and col totals
round(addmargins(y$n))