# Using ipfr" In ipfr: List Balancing for Reweighting and Population Synthesis

```knitr::opts_chunk\$set(
collapse = TRUE,
comment = "#>"
)
```

# Introduction

Iterative proportional fitting (IPF) is used in many disciplines to adjust an initial set of weights to match various marginal distributions. This package implements the iterative proportional updating algorithm based on the paper from Arizona State University (IPU). In survey raking or population synthesis, the IPU algorithm has the added advantage of being able to match household- and person-level marginal targets.

The primary functions of the `ipfr` package are discussed below.

• `ipu_matrix()`
• `ipu()`
• `synthesize()`
```suppressPackageStartupMessages({
library(ipfr)
library(dplyr)
})
```

## Example: 2D / Matrix

The first example shows how to use `ipfr` to solve one of the most common IPF problems: Given a matrix of starting weights and a set of row and column targets, balance the matrix in order to satisfy both row and column targets.

```set.seed(42)
mtx <- matrix(data = runif(9), nrow = 3, ncol = 3)
row_targets <- c(3, 4, 5)
column_targets <- c(5, 4, 3)
mtx
```

The resulting matrix satisfies both row and column targets while retaining a similar relative distribution of weights in each cell.

```result <- ipu_matrix(mtx, row_targets, column_targets)
result
rowSums(result)
colSums(result)
```

## Example: Household survey (simple)

The matrix example is common and conceptually simple, but makes thinking about higher-dimension problems difficult. A much better way to think about the problem (whether in two or ten dimensions) is as a table.

This example creates a household survey that will act as the seed (like the matrix from the last example). The number of persons and autos for each household is listed in the `size` and `autos` columns. A starting weight of 1 is also included.

```survey <- tibble(
size = c(1, 2, 1, 1),
autos = c(0, 2, 2, 1),
weight = 1
)

survey
```

Assume that we know the total number of households by size and households by auto-ownership from the Census. These target marginal distributions are just like our row and column totals from the previous example. Targets are passed into the `ipu` function as shown below. Take the size targets, for example:

• The list name `size` corresponds to the `size` column in the survey.
• The column names `1` and `2` correspond to the unique values in the survey's `size` column.
```targets <- list()
targets\$size <- tibble(
`1` = 75,
`2` = 25
)
targets\$autos <- tibble(
`0` = 25,
`1` = 50,
`2` = 25
)

targets
```

A call to `ipu()` returns a named list of results.

```result <- ipu(survey, targets)
names(result)
```

The first element is the resulting weight table. The primary seed has three columns appended:

• weight
• The new/expanded weight of the record.
• avg_weight
• The average weight, which is the total target / number of seed records.
• weight_factor
• Each record's weight divided by the average weight. This is a useful diagnostic to measure how extreme a weight is.
```result\$weight_tbl
```

The second element is a histogram of the `weight_factor`. This provides a quick overview of the distribution of weights. Many large or small weights can indicate underlying problems even if the ipu converged and matches targets. For instance, a record with a weight factor of 50 means that it is greatly over-represented in the re-weighted survey.

```result\$weight_dist
```

The next element is a comparison back to the targets provided. With complex seed and target tables, this makes investigating results quick and easy.

```result\$primary_comp
```

If secondary targets are provided to `ipu()`, a fourth item in the list will contain a `secondary_comp` table.

In household survey expansion, it is common to want to control for certain features that describe households (like size) while controlling for other attributes that describe people (like age). This is possible with the `ipu()` function.

This example is taken directly from the Arizona paper on page 20: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.537.723&rep=rep1&type=pdf

In this example, household type could represent size (e.g. 1-person and 2-person households). Person type could represent age groups (e.g. under 18, between 18 and 50, and over 50).

`setup_arizona()` creates the seed and target tables used in the example.

```result <- setup_arizona()
hh_seed <- result\$hh_seed
hh_targets <- result\$hh_targets
per_seed <- result\$per_seed
per_targets <- result\$per_targets
```
• The household seed table is the `primary_seed`
• The household target list is the `primary_target`
• The person seed table is the `secondary_seed`
• The person target list is the `secondary_target`

In the interest of keeping vignette build time short, the `ipu()` algorithm is only run for 30 iterations. After running for 400 or more iterations, the results match closely to those shown in the paper.

```result <- ipu(hh_seed, hh_targets, per_seed, per_targets, max_iterations = 30)
```

The comparison table for the secondary (person) marginals is now included in `result`. Feel free to run the code chunk above for 400 or more iterations and then look again.

```result\$weight_tbl

result\$primary_comp

result\$secondary_comp
```

## Example 3: Using multiple geographies

`ipu()` allows different geographies to be specified for different marginal tables. There are a few rules that make this possible, but in short, a geography field in each target table tells the algorithm which scale to constrain to. If no geography field is present in a target table, it applies to the entire seed table (i.e. region).

All of the following rules are checked by the function and a warning message will show which (if any) is violated.

• The primary seed table must contain all geography fields used by any target tables.
• Do not duplicate geography fields on the secondary seed table.
• This prevents potential errors/inconsistencies between seed tables.
• geo_tract
• geo_region
• geo_state

To demonstrate, the Arizona example is modified to add two different tracts for household controls but to still control the person targets at the regional level.

```# Repeat the hh_seed to create tract 1 and 2 households
new_hh_seed <- hh_seed %>%
mutate(geo_tract = 1)
new_hh_seed <- bind_rows(
new_hh_seed,
new_hh_seed %>%
mutate(geo_tract = 2, id = id + 8)
)
new_hh_seed\$geo_region <- 1

new_hh_seed

# Repeat the household targets for two tracts
new_hh_targets <- hh_targets
new_hh_targets\$hhtype <- bind_rows(hh_targets\$hhtype, hh_targets\$hhtype)
new_hh_targets\$hhtype <- new_hh_targets\$hhtype %>%
mutate(geo_tract = c(1, 2))

new_hh_targets\$hhtype

# Repeat the per_seed to create tract 1 and 2 persons
new_per_seed <- bind_rows(
per_seed,
per_seed %>%
mutate(id = id + 8)
)

new_per_seed

# Double the regional person targets
new_per_targets <- per_targets
new_per_targets\$pertype <- per_targets\$pertype %>%
mutate_all(list(~. * 2)) %>%
mutate(geo_region = 1)

new_per_targets\$pertype
```

Run the IPU algorithm. Again, for vignette build time, only 30 iterations are performed. Run the code yourself with `max_iterations` set to 600 to see the converged result.

```result <- ipu(
new_hh_seed, new_hh_targets,
new_per_seed, new_per_targets,
max_iterations = 30
)
```

The tables below show the results compared back to targets.

```result\$primary_comp

result\$secondary_comp
```

`ipu` offers some advanced options for controlling the behavior. Weights can be capped with `min_ratio` and `max_ratio` options. Secondary target importance can be adjusted using the `secondary_importance` option. For info on using these options appropriately, see the `common_ipf_problems` vignette.

## Synthetic Populations

The `weight_tbl` from `ipu()` can be fed into the `synthesize()` function to generate a synthetic population. The code block below takes the output from the previous example to demonstrate. The `group_by` option tells the function to group by the `geo_tract` field before random sampling takes places. This will ensure that the number of sampled records in each tract equals the total weight in each tract.

A `new_id` is created for each synthetic member (usually a household), but the old ID is preserved to facilitate joins or other operations where it could be useful.

```set.seed(42)
synthesize(result\$weight_tbl, group_by = "geo_tract") %>%
```

A tidyverse-style call also works, as shown below.

```set.seed(42)
result\$weight_tbl %>%
group_by(geo_tract) %>%
synthesize() %>%