Conditional maximization routine for the indirect estimation of origin-destination-migrant type migration flow tables with known origin and destination margins.

The cm3 function finds the maximum likelihood estimates for parameters in the log-linear model: $$ \log y_{ijk} = \log \alpha_{i} + \log \beta_{j} + \log m_{ijk} $$ as introduced by Abel (2005). The $\alpha_{i}$ and $\beta_{j}$ represent background information related to the characteristics of the origin and destinations respectively. The $m_{ijk}$ factor represents auxiliary information on origin-destination migration flows by a migrant characteristic (such as age, sex, disability, household type, economic status, etc.). This method is useful for combining data from detailed data collection processes (such as a Census) with more up-to-date information on migration inflows and outflows (where details on movements by migrant characteristics are not known).

Usage

cm3(
  row_tot = NULL,
  col_tot = NULL,
  m = NULL,
  tol = 1e-06,
  maxit = 500,
  verbose = TRUE
)

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: Array of auxiliary data. By default set to 1 for all origin-destination-migrant typology 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.

Value

Parameter estimates were obtained using the conditional maximization of the likelihood, as discussed by Abel (2005) and Raymer et. al. (2007).

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

Returns a list object with

N: Origin-Destination matrix of indirect estimates
theta: Collection of parameter estimates

References

Abel, G. J. (2005) The Indirect Estimation of Elderly Migrant Flows in England and Wales (MS.c. Thesis). University of Southampton

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.

Author

Guy J. Abel

Examples

## over two tables
r <- LETTERS[1:2]
y <- cm3(row_tot = c(18, 20) * 2, col_tot = c(16, 22) * 2,
         m = array(c(5, 1, 2, 7, 4, 2, 5, 9), dim = c(2, 2, 2),
                   dimnames = list(orig = r, dest = r, type = c("ILL", "HEALTHY"))))
#> iteration: 0 
#> alpha parameters: 
#> beta parameters: 1 1 
#> 
#> iteration: 1 
#> alpha parameters: 2.25 2.105263 
#> beta parameters: 1.204557 0.8900719 
#> max difference: 1.25 
#> 
#> iteration: 2 
#> alpha parameters: 2.108776 2.240291 
#> beta parameters: 1.245143 0.8694605 
#> max difference: 0.1412239 
#> 
#> iteration: 3 
#> alpha parameters: 2.081826 2.2667 
#> beta parameters: 1.253107 0.8654605 
#> max difference: 0.02694999 
#> 
#> iteration: 4 
#> alpha parameters: 2.076582 2.271864 
#> beta parameters: 1.254664 0.8646796 
#> max difference: 0.005244326 
#> 
#> iteration: 5 
#> alpha parameters: 2.075558 2.272874 
#> beta parameters: 1.254969 0.864527 
#> max difference: 0.001024239 
#> 
#> iteration: 6 
#> alpha parameters: 2.075357 2.273071 
#> beta parameters: 1.255028 0.8644972 
#> max difference: 0.0002001792 
#> 
#> iteration: 7 
#> alpha parameters: 2.075318 2.273109 
#> beta parameters: 1.25504 0.8644914 
#> max difference: 3.912878e-05 
#> 
#> iteration: 8 
#> alpha parameters: 2.075311 2.273117 
#> beta parameters: 1.255042 0.8644902 
#> max difference: 7.648661e-06 
#> 
#> iteration: 9 
#> alpha parameters: 2.075309 2.273118 
#> beta parameters: 1.255043 0.86449 
#> max difference: 1.495122e-06 
#> 
#> iteration: 10 
#> alpha parameters: 2.075309 2.273119 
#> beta parameters: 1.255043 0.86449 
#> max difference: 2.922594e-07 
#> 
# display with row, col and table totals
y
#> $n
#> , , type = ILL
#> 
#>     dest
#> orig         A         B
#>    A 13.023008  3.588167
#>    B  2.852862 13.755619
#> 
#> , , type = HEALTHY
#> 
#>     dest
#> orig         A         B
#>    A 10.418407  8.970418
#>    B  5.705723 17.685795
#> 
#> 
#> $theta
#>   alpha1   alpha2    beta1    beta2 
#> 2.075309 2.273119 1.255043 0.864490 
#> 

## over three tables
y <- cm3(row_tot = c(170, 120, 410), col_tot = c(500, 140, 60),
         m = array(c(5, 1, 2, 7,  4, 2, 5, 9,  5, 4, 3, 1), dim = c(2, 2, 3),
                   dimnames = list(orig = r, dest = r, type = c("0--15", "15-60", ">60"))),
                   verbose = FALSE)
# display with row, col and table totals
y
#> $n
#> , , type = 0--15
#> 
#>     dest
#> orig        A         B
#>    A 116.2709  9.922764
#>    B  24.9202 37.217838
#> 
#> , , type = 15-60
#> 
#>     dest
#> orig        A        B
#>    A 93.01674 24.80691
#>    B 49.84040 47.85151
#> 
#> , , type = >60
#> 
#>     dest
#> orig        A         B
#>    A 116.2709 14.884147
#>    B  99.6808  5.316834
#> 
#> 
#> $theta
#>        alpha1        alpha2         beta1         beta2 
#> 3.798846e-171 4.071009e-171 6.121381e+171 1.306024e+171 
#>