# ipf3: Iterative proportional fitting routine for the indirect... In migest: Methods for the Indirect Estimation of Bilateral Migration

## Description

The `ipf3` function finds the maximum likelihood estimates for fitted values in the log-linear model:

\log y_{ijk} = \log α_{i} + \log β_{j} + \log λ_{k} + \log γ_{ik} + \log κ_{jk} + \log m_{ijk}

where m_{ijk} is a set of prior estimates for y_{ijk} and is no more complex than the matrices being fitted.

## Usage

 ```1 2 3 4 5 6 7 8``` ```ipf3( row_tot = NULL, col_tot = NULL, m = NULL, tol = 1e-05, 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 typologies 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

Iterative Proportional Fitting routine set up in a similar manner to Agresti (2002, p.343). The arguments `row_tot` and `col_tot` take the row-table and column-table specific known margins.

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

Returns a `list` object with

 `mu ` Array of indirect estimates of origin-destination matrices by migrant characteristic `it ` Iteration count `tol ` Tolerance level at final iteration

Guy J. Abel

## References

Abel and Cohen (2019) Bilateral international migration flow estimates for 200 countries Scientific Data 6 (1), 1-13

Azose & Raftery (2019) Estimation of emigration, return migration, and transit migration between all pairs of countries Proceedings of the National Academy of Sciences 116 (1) 116-122

Abel, G. J. (2013). Estimating Global Migration Flow Tables Using Place of Birth. Demographic Research 28, (18) 505-546

Agresti, A. (2002). Categorical Data Analysis 2nd edition. Wiley.

`ipf3_qi`, `ipf2`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32``` ```## create row-table and column-table specific known margins. dn <- LETTERS[1:4] P1 <- matrix(c(1000, 100, 10, 0, 55, 555, 50, 5, 80, 40, 800 , 40, 20, 25, 20, 200), nrow = 4, ncol = 4, byrow = TRUE, dimnames = list(pob = dn, por = dn)) P2 <- matrix(c(950, 100, 60, 0, 80, 505, 75, 5, 90, 30, 800, 40, 40, 45, 0, 180), nrow = 4, ncol = 4, byrow = TRUE, dimnames = list(pob = dn, por = dn)) # display with row and col totals addmargins(P1) addmargins(P2) # run ipf y <- ipf3(row_tot = t(P1), col_tot = P2) # display with row, col and table totals round(addmargins(y\$mu), 1) # origin-destination flow table round(sum_od(y\$mu), 1) ## with alternative offset term dis <- array(c(1, 2, 3, 4, 2, 1, 5, 6, 3, 4, 1, 7, 4, 6, 7, 1), c(4, 4, 4)) y <- ipf3(row_tot = t(P1), col_tot = P2, m = dis) # display with row, col and table totals round(addmargins(y\$mu), 1) # origin-destination flow table round(sum_od(y\$mu), 1) ```