# Simulation Tools for Small Area Estimation

#### 2019-03-28

Tools for the simulation of data in the context of small area estimation. Combine all steps of your simulation - from data generation over drawing samples to model fitting - in one object. This enables easy modification and combination of different scenarios. You can store your results in a folder or start the simulation in parallel.

Two external resources may be of interest in addition to this vignette:

## An initial Example

Consider a linear mixed model. It contains two components. A fixed effects part, and an error component. The error component can be split into a random effects part and a model error.

library(saeSim)
setup <- sim_base() %>%
sim_gen_x() %>%
sim_gen_e() %>%
sim_gen_v() %>%
sim_resp_eq(y = 100 + 2 * x + v + e) %>%
sim_simName("Doku")
setup
## # Description: data.frame [10,000 × 6]
##     idD   idU      x     e     v     y
## * <int> <int>  <dbl> <dbl> <dbl> <dbl>
## 1     1     1 -2.51  -3.22 0.235  92.0
## 2     1     2  0.735 -4.23 0.235  97.5
## 3     1     3 -3.34  -4.14 0.235  89.4
## 4     1     4  6.38  -4.74 0.235 108.
## 5     1     5  1.32  -2.00 0.235 101.
## 6     1     6 -3.28  -2.10 0.235  91.6
## # … with 9,994 more rows

sim_base() is responsible to supply a data.frame to which variables can be added. The default is to create a data.frame with indicator variables idD and idU (2-level-model), which uniquely identify observations. ‘D’ stands for the domain, i.e. the grouping variable. ‘U’ stands for unit and is the identifier of single observations within domains. sim_resp will add a variable y as response.

The setup itself does not contain the simulated data but the functions to process the data. To start a simulation use the function sim. It will return a list containing data.frames as elements:

dataList <- sim(setup, R = 10)

You can coerce a simulation setup to a data.frame with as.data.frame.

simData <- sim_base() %>%
sim_gen_x() %>%
sim_gen_e() %>%
as.data.frame
simData

## Naming and structure

Components in a simulation setup should be added using the pipe operator %>% from magrittr. A component defines ‘when’ a specific function will be applied in a chain of functions. To add a component you can use one of: sim_gen, sim_resp, sim_comp_pop, sim_sample, sim_comp_sample, sim_agg and sim_comp_agg. They all expect a simulation setup as first argument and a function as second and will take care of the order in which the functions are called. The reason for this is the following:

setup <- sim_base() %>%
sim_gen_x() %>%
sim_gen_e() %>%
sim_resp_eq(y = 100 + 2 * x + e)

setup1 <- setup %>% sim_sample(sample_fraction(0.05))
setup2 <- setup %>% sim_sample(sample_number(5))

You can define a rudimentary scenario and only have to explain how scenarios differ. You do not have to redefine them. setup1 and setup2 only differ in the way samples are drawn. sim_sample will take care, that the sampling will take place at the appropriate place in the chain of functions no matter how setup was composed.

If you can’t remember all function names, use auto-complete. All functions to add components start with sim_. And all functions meant to be used in a given phase will start with the corresponding prefix, i.e. if you set the sampling scheme you use sim_sample – all functions to control sampling have the prefix sample.

## Exploring the setup

You will want to check your results regularly when working with sim_setup objects. Some methods are supplied to do that:

• show - this is the print method for S4-Classes. You don’t have to call show explicitly. You may have noticed that only a few lines of data are printed to the console if you evaluate a simulation setup. show will print the head of the resulting data of one simulation run.
• plot - for visualizing the data
• autoplot - Will imitate smoothScatter with ggplot2
setup <- sim_base_lmm()
plot(setup)
autoplot(setup)
autoplot(setup, "e")
autoplot(setup %>% sim_gen_vc())

## sim_gen

### Semi-custom data

saeSim has an interface to standard random number generators in R. If things get more complex you can always define new generator functions.

base_id(2, 3) %>%
sim_gen(gen_generic(rnorm, mean = 5, sd = 10, name = "x", groupVars = "idD"))
## # Description: data.frame [6 × 3]
##     idD   idU      x
##   <int> <int>  <dbl>
## 1     1     1  -3.48
## 2     1     2  -3.48
## 3     1     3  -3.48
## 4     2     1  13.2
## 5     2     2  13.2
## 6     2     3  13.2

You can supply any random number generator to gen_generic and since we are in small area estimation you have an optional group variable to generate ‘area-level’ variables. Some short cuts for data generation are sim_gen_x, sim_gen_v and sim_gen_e which add normally distributed variables named ‘x’, ‘e’ and ‘v’ respectively. Also there are some function with the prefix ‘gen’ which will be extended in the future.

library(saeSim)
setup <- sim_base() %>%
sim_gen_x() %>% # Variable 'x'
sim_gen_e() %>% # Variable 'e'
sim_gen_v() %>% # Variable 'v' as a random-effect
sim_gen(gen_v_sar(name = "vSp")) %>% # random-effect following a SAR(1)
sim_resp_eq(y = 100 + x + v + vSp + e) # Computing 'y'
setup
## # Description: data.frame [10,000 × 7]
##     idD   idU     x     e     v     vSp     y
## * <int> <int> <dbl> <dbl> <dbl>   <dbl> <dbl>
## 1     1     1 -3.26 -5.80 -1.29 0.00300  89.7
## 2     1     2  4.26  4.77 -1.29 0.00300 108.
## 3     1     3 -3.07  1.02 -1.29 0.00300  96.7
## 4     1     4  5.21  2.32 -1.29 0.00300 106.
## 5     1     5  1.91  8.53 -1.29 0.00300 109.
## 6     1     6 -2.66  3.76 -1.29 0.00300  99.8
## # … with 9,994 more rows

### Contaminated data

For contaminated data you can use the same generator functions, however, instead of using sim_gen you use sim_gen_cont which will have some extra arguments to control the contamination. To extend the above setup with a contaminated spatially correlated error component you can add the following:

contSetup <- setup %>%
sim_gen_cont(
gen_v_sar(sd = 40, name = "vSp"), # defining the model
nCont = 0.05, # 5 per cent outliers
type = "area", # whole areas are outliers, i.e. all obs within
areaVar = "idD", # var name to identify domain
fixed = TRUE # if in each iteration the same area is an outlier
)

Note that the generator is the same but with a higher standard deviation. The argument nCont controls how much observations are contaminated. Values < 1 are interpreted as probability. A single number as the number of contaminated units (can be areas or observations in each area or observations). When given with length(nCont) > 1 it will be interpreted as number of contaminated observations in each area. Use the following example to see how these things play together:

base_id(3, 4) %>%
sim_gen_x() %>%
sim_gen_e() %>%
sim_gen_ec(mean = 0, sd = 150, name = "eCont", nCont = c(1, 2, 3)) %>%
as.data.frame
##    idD idU          x           e      eCont   idC
## 1    1   1  2.1238928  2.85552209    0.00000 FALSE
## 2    1   2  1.8224797 -3.70320125    0.00000 FALSE
## 3    1   3 -0.7466757 -4.63509843    0.00000 FALSE
## 4    1   4  0.2864107 -0.51238608  174.16195  TRUE
## 5    2   1 -4.1711873 -1.64389880    0.00000 FALSE
## 6    2   2 -0.9948926  2.68706944    0.00000 FALSE
## 7    2   3 -4.4030395  0.08884468  131.30190  TRUE
## 8    2   4  1.4103819  0.77040266 -111.81580  TRUE
## 9    3   1 -0.3921937  0.88877112    0.00000 FALSE
## 10   3   2 -5.7884702 -1.54674283   47.70671  TRUE
## 11   3   3 -4.8635870  1.35621664 -306.40766  TRUE
## 12   3   4 -5.7893455  4.26584633  305.57991  TRUE

## sim_comp

Here follow some examples how to add components for computation to a sim_setup. Three points can be accessed with

• sim_comp_pop - add a computation before sampling
• sim_comp_sample - add a computation after sampling
• sim_comp_agg - add a computation after aggregation
base_id(2, 3) %>%
sim_gen_x() %>%
sim_gen_e() %>%
sim_gen_ec() %>%
sim_resp_eq(y = 100 + x + e) %>%
# the mean in each domain:
sim_comp_pop(comp_var(popMean = mean(y)), by = "idD")
## # Description: data.frame [6 × 7]
##     idD   idU       x     e idC       y popMean
## * <int> <int>   <dbl> <dbl> <lgl> <dbl>   <dbl>
## 1     1     1   0.575 6.62  FALSE  107.    108.
## 2     1     2   3.77  0.226 FALSE  104.    108.
## 3     1     3  12.6   1.07  FALSE  114.    108.
## 4     2     1   6.64  5.01  FALSE  112.    106.
## 5     2     2  -0.842 3.41  FALSE  103.    106.
## 6     2     3   4.50  0.359 FALSE  105.    106.

The function comp_var is a wrapper around dplyr::mutate so you can add simple data manipulations. The argument by is a little extension and lets you define operations in the scope of groups identified by a variable in the data. In this case the mean of the variable ‘y’ is computed for every group identified with the variable ‘idD’. This is done before sampling, hence the prefix ‘pop’ for population.

By adding computation functions you can extend the functionality to wrap your whole simulation. This will separate the utility of this package from only generating data.

comp_linearPredictor <- function(dat) {
dat\$linearPredictor <- lm(y ~ x, dat) %>% predict
dat
}

sim_base_lm() %>%
sim_comp_pop(comp_linearPredictor)
## # Description: data.frame [10,000 × 6]
##     idD   idU      x      e     y linearPredictor
##   <int> <int>  <dbl>  <dbl> <dbl>           <dbl>
## 1     1     1 -8.50  -0.637  90.9            91.6
## 2     1     2 -0.732  3.09  102.             99.3
## 3     1     3  1.11   6.99  108.            101.
## 4     1     4  3.82   4.38  108.            104.
## 5     1     5  0.615 -0.191 100.            101.
## 6     1     6  0.382 -3.60   96.8           100.
## # … with 9,994 more rows

Or, should this be desirable, directly produce a list of lm objects or add them as attribute to the data. The intended way of writing functions is that they will return the modified data of class ‘data.frame’.

sim_base_lm() %>%
sim_comp_pop(function(dat) lm(y ~ x, dat)) %>%
sim(R = 1)
## [[1]]
##
## Call:
## lm(formula = y ~ x, data = dat)
##
## Coefficients:
## (Intercept)            x
##    100.0421       0.9872
comp_linearModelAsAttr <- function(dat) {
attr(dat, "linearModel") <- lm(y ~ x, dat)
dat
}

dat <- sim_base_lm() %>%
sim_comp_pop(comp_linearModelAsAttr) %>%
as.data.frame

attr(dat, "linearModel")
##
## Call:
## lm(formula = y ~ x, data = dat)
##
## Coefficients:
## (Intercept)            x
##     100.017        1.032

If you use any kind of sampling, the ‘linearPredictor’ can be added after sampling. This is where small area models are supposed to be applied.

sim_base_lm() %>%
sim_sample() %>%
sim_comp_sample(comp_linearPredictor)
## # Description: data.frame [500 × 6]
##     idD   idU      x     e     y linearPredictor
## * <int> <int>  <dbl> <dbl> <dbl>           <dbl>
## 1     1    98  5.47  -1.31 104.            106.
## 2     1    27 -0.462  8.40 108.             99.7
## 3     1    47  0.358 -2.91  97.4           101.
## 4     1     5 -6.90   3.15  96.2            93.1
## 5     1    35  2.40  -4.40  98.0           103.
## 6     2    50 -1.19   2.28 101.             98.9
## # … with 494 more rows

Should you want to apply area level models, use sim_comp_agg instead.

sim_base_lm() %>%
sim_sample() %>%
sim_agg() %>%
sim_comp_agg(comp_linearPredictor)
## # Description: data.frame [100 × 5]
##     idD      x      e     y linearPredictor
## * <dbl>  <dbl>  <dbl> <dbl>           <dbl>
## 1     1  0.202  2.20  102.            100.
## 2     2  0.103 -2.58   97.5           100.
## 3     3 -2.03   2.49  100.             97.9
## 4     4  1.52  -1.41  100.            101.
## 5     5 -0.649  0.413  99.8            99.3
## 6     6 -1.14   1.45  100.             98.8
## # … with 94 more rows

## sim_sample

After the data generation you may want to draw a sample from the population. Use the function sim_sample() to add a sampling component to your sim_setup.

• sample_number - wrapper around dplyr::sample_n
• sample_fraction - wrapper around dplyr::sample_frac
base_id(3, 4) %>%
sim_gen_x() %>%
sim_sample(sample_number(1L))
## # Description: data.frame [1 × 3]
##     idD   idU     x
##   <int> <int> <dbl>
## 1     1     1 -7.73
base_id(3, 4) %>%
sim_gen_x() %>%
sim_sample(sample_number(1L, groupVars = "idD"))
## # Description: data.frame [3 × 3]
##     idD   idU      x
## * <int> <int>  <dbl>
## 1     1     2  0.299
## 2     2     4  1.63
## 3     3     4 -1.87
# simple random sampling:
sim_base_lm() %>% sim_sample(sample_number(size = 10L))
## # Description: data.frame [10 × 5]
##     idD   idU      x     e     y
##   <int> <int>  <dbl> <dbl> <dbl>
## 1    37    63 -0.976 -2.66  96.4
## 2    59    79 -1.14   4.38 103.
## 3     7    19  4.97  -2.24 103.
## 4    28    36  2.00   4.49 106.
## 5    94    47  5.84  -2.69 103.
## 6    46    59  6.11  -3.75 102.
## # … with 4 more rows
sim_base_lm() %>% sim_sample(sample_fraction(size = 0.05))
## # Description: data.frame [500 × 5]
##     idD   idU      x      e     y
##   <int> <int>  <dbl>  <dbl> <dbl>
## 1    65    24 -0.538  2.33  102.
## 2    21    10  2.87  -4.87   98.0
## 3    95    71 -1.01   1.80  101.
## 4     7    49  4.77  -0.567 104.
## 5    93    68  7.60   1.86  109.
## 6    94    56  7.05   2.70  110.
## # … with 494 more rows
# srs in each domain/cluster
sim_base_lm() %>% sim_sample(sample_number(size = 10L, groupVars = "idD"))
## # Description: data.frame [1,000 × 5]
##     idD   idU     x     e     y
## * <int> <int> <dbl> <dbl> <dbl>
## 1     1    26 -1.90 -1.80  96.3
## 2     1    49  9.35  3.79 113.
## 3     1    95 -6.71  5.98  99.3
## 4     1    41  1.57 -5.40  96.2
## 5     1    42 -8.54 -1.21  90.2
## 6     1     3  1.49  1.48 103.
## # … with 994 more rows
sim_base_lm() %>% sim_sample(sample_fraction(size = 0.05, groupVars = "idD"))
## # Description: data.frame [500 × 5]
##     idD   idU       x     e     y
## * <int> <int>   <dbl> <dbl> <dbl>
## 1     1    49  5.45    2.94 108.
## 2     1    12  0.0983 -2.76  97.3
## 3     1    19 -0.247   1.67 101.
## 4     1     5 -5.93    1.10  95.2
## 5     1    45  4.52   -8.22  96.3
## 6     2    32  0.0987  6.23 106.
## # … with 494 more rows