This vignette explains the usage of the `ipf()`

function, which has been used for calibrating the labour force survey of Austria for several years. It is based on the Iterative Proportional Fitting algorithm and gives some flexibility about the details of the implementation. See (Meraner, Gumprecht, and Kowarik 2016) or `vignette("methodology")`

for more details.

We will assume the output of `demo.eusilc()`

is our population. From this population, a sample without replacement is drawn. The sample covers 10 percent of the population. We assign a weight of one for all observations of the population and a weight of ten for all observations of the sample.

We will start with an example where we want to adapt the weights of `pop_sample`

such that the weighted number of males and females matches the ones of `population`

. We can see that this is currently not the case.

```
(gender_distribution <- xtabs(pWeight ~ gender, population))
#> gender
#> male female
#> 7267 7560
xtabs(pWeight ~ gender, pop_sample)
#> gender
#> male female
#> 7280 7540
```

Due to random sampling (rather than stratified sampling), there are differences between the gender distributions. We can pass `gender_distribution`

as a parameter to `ipf()`

to obtain modified weights.

The resulting dataset, `pop_sample_c`

is similar to `pop_sample`

but has an additional column with the adjusted weights.

```
dim(pop_sample)
#> [1] 1482 30
dim(pop_sample_c)
#> [1] 1482 31
setdiff(names(pop_sample_c), names(pop_sample))
#> [1] "calibWeight"
```

We can now calculate the weighted number of males and females according to this new weight. This will result in a match for the constraints.

```
xtabs(calibWeight ~ gender, pop_sample_c)
#> gender
#> male female
#> 7267 7560
xtabs(pWeight ~ gender, population)
#> gender
#> male female
#> 7267 7560
```

In this simple case, `ipf`

just performs a post stratification step. This means, that all males and all females have the same weight.

```
xtabs(~ calibWeight + gender, pop_sample_c)
#> gender
#> calibWeight male female
#> 9.98214285714286 728 0
#> 10.026525198939 0 754
```

All males have been weighted down (`calibWeight < 10`

) to compensate for the overrepresentation in the sample.

Let’s now assume that we want to put constraints on the number of males and females for each age group. The numbers from the original population can be obtained with `xtabs()`

.

```
(con_ga <- xtabs(pWeight ~ gender + age, population))
#> age
#> gender (-Inf,16] (16,25] (25,45] (45,65] (65, Inf]
#> male 1528 855 2165 1822 897
#> female 1375 848 2255 1845 1237
xtabs(pWeight ~ gender + age, pop_sample)
#> age
#> gender (-Inf,16] (16,25] (25,45] (45,65] (65, Inf]
#> male 1300 850 2280 1840 1010
#> female 1210 970 2350 1830 1180
```

Again, we can see that those constraints are not met. Supplying the contingency table `con_ga`

to `ipf()`

will again resolve this.

```
pop_sample_c2 <- ipf(pop_sample, conP = list(con_ga), w = "pWeight")
xtabs(pWeight ~ gender + age, population)
#> age
#> gender (-Inf,16] (16,25] (25,45] (45,65] (65, Inf]
#> male 1528 855 2165 1822 897
#> female 1375 848 2255 1845 1237
xtabs(calibWeight ~ gender + age, pop_sample_c2)
#> age
#> gender (-Inf,16] (16,25] (25,45] (45,65] (65, Inf]
#> male 1528 855 2165 1822 897
#> female 1375 848 2255 1845 1237
```

Now we assume that we know the number of persons living in each nuts2 region from registry data.

However, those registry data does not provide any information about age or `gender`

. Therefore, the two contingency tables (`con_ga`

and `registry_table`

) have to be specified independently. This can be done by supplying a list to `conP`

```
pop_sample_c2 <- ipf(pop_sample, conP = list(con_ga, registry_table), w = "pWeight")
xtabs(pWeight ~ gender + age, population)
#> age
#> gender (-Inf,16] (16,25] (25,45] (45,65] (65, Inf]
#> male 1528 855 2165 1822 897
#> female 1375 848 2255 1845 1237
xtabs(calibWeight ~ gender + age, pop_sample_c2)
#> age
#> gender (-Inf,16] (16,25] (25,45] (45,65] (65, Inf]
#> male 1528 855 2165 1822 897
#> female 1375 848 2255 1845 1237
xtabs(pWeight ~ region, population)
#> region
#> Burgenland Carinthia Lower Austria Salzburg Styria
#> 549 1078 2804 924 2295
#> Tyrol Upper Austria Vienna Vorarlberg
#> 1317 2805 2322 733
xtabs(calibWeight ~ region, pop_sample_c2)
#> region
#> Burgenland Carinthia Lower Austria Salzburg Styria
#> 549.0000 1077.9999 2804.0000 924.0002 2294.9999
#> Tyrol Upper Austria Vienna Vorarlberg
#> 1316.9996 2805.0003 2322.0001 733.0001
```

this time, the constraints are not matched perfectly. That is, because we provided more than one constraint. therefore, the `ipf()`

algorithm had to work iteratively.

If the dataset has a household structure, household constraints can be passed via the parameter `conH`

. If this parameter is used, it is also necessary to supply `hid`

, which defines the column names that contains household ids.

```
(conH1 <- xtabs(pWeight ~ hsize + region, data = population[!duplicated(hid)]))
#> region
#> hsize Burgenland Carinthia Lower Austria Salzburg Styria Tyrol
#> (0,1] 58 117 325 103 264 118
#> (1,2] 82 126 345 102 260 149
#> (2,3] 37 80 189 55 187 79
#> (3,4] 33 63 169 71 122 102
#> (4,5] 14 22 82 18 49 37
#> (5,Inf] 2 17 21 12 34 11
#> region
#> hsize Upper Austria Vienna Vorarlberg
#> (0,1] 262 431 67
#> (1,2] 321 355 72
#> (2,3] 203 175 44
#> (3,4] 168 96 53
#> (4,5] 79 35 27
#> (5,Inf] 35 15 7
pop_sample_hh <- ipf(pop_sample, hid = "hid", conH = list(conH1), w = "pWeight",
bound = 10)
xtabs(calibWeight ~ hsize + region, data = pop_sample_hh[!duplicated(hid)])
#> region
#> hsize Burgenland Carinthia Lower Austria Salzburg Styria Tyrol
#> (0,1] 0 0 0 0 0 0
#> (1,2] 0 0 0 0 0 0
#> (2,3] 0 0 0 0 0 0
#> (3,4] 0 0 0 0 0 0
#> (4,5] 0 0 0 0 0 0
#> (5,Inf] 0 0 0 0 0 0
#> region
#> hsize Upper Austria Vienna Vorarlberg
#> (0,1] 0 0 0
#> (1,2] 0 0 0
#> (2,3] 0 0 0
#> (3,4] 0 0 0
#> (4,5] 0 0 0
#> (5,Inf] 0 0 0
```

If `conP`

or `conH`

contain several contingency tables or if `conP`

and `conH`

are used at the same time, the ipf algorithm will operate iteratively. This means that the calibrated dataset will satisfy the constraints only approximately. The default tolerances of the approximation can be overwritten using the parameters `conP`

and `conH`

.

Lowering the tolerances will improve the match between the constraints and the contingency tables according to the calibrated weights. However, lower tolerances will also make it so more iterations are necessary until a convergence is met. If the constraints are too small, ipf will return with a warning that indicates that a convergence could not be reached.

```
ipf(pop_sample, conP = list(con_ga, registry_table), w = "pWeight",
verbose = TRUE, epsP = 0.01)
#> Iteration stopped after 2 steps
#> Convergence reached
ipf(pop_sample, conP = list(con_ga, registry_table), w = "pWeight",
verbose = TRUE, epsP = 0.0001)
#> Iteration stopped after 4 steps
#> Convergence reached
```

We see that changing the tolerances from `0.01`

(one percent) to `0.0001`

increases the number of required iterations.

Meraner, Angelika, Daniela Gumprecht, and Alexander Kowarik. 2016. “Weighting Procedure of the Austrian Microcensus Using Administrative Data.” *Austrian Journal of Statistics* 45 (June): 3. https://doi.org/10.17713/ajs.v45i3.120.