Handling Anticipated Variance

When optimizing the stratification of a sampling frame, values of the target variables Y’s are supposed to be available for the generality of the units in the frame, or at least for a sample of them by means of which it is possible to estimate means and standard deviation of Y’s in atomic strata. Of course, this assumption is seldom expected to hold. The situation in which some proxy variables are available in the frame is much more likely to happen. In these situations, instead of directly indicating the real target variables, proxy ones are named as Y’s. By so doing, there is no guarantee that the final stratification and allocation can ensure the compliance to the set of precision constraints.
In order to take into account this problem, and to limit the risk of overestimating the expected precision levels of the optimized solution, it is possible to carry out the optimization by considering, instead of the expected coefficients of variation related to proxy variables, the anticipated coefficients of variation (ACV) that depend on the model that is possile to fit on couples of real target variables and proxy ones.

In the current implementation, the following restrictions hold:

• only models linking continuous variables can be considered;
• only linear or loglinear models;
• only one explanatory variable for each target variable.

The definition and the use of these models is the same that has been implemented in the package stratification (see Baillargeon and Rivest 2014).

In particular, the reference here is to two different models, the linear model with heteroscedasticity:

\[Z=\beta Y + \epsilon\]

where

\[\epsilon \sim N(0,\sigma^2 Y^\gamma)\]

(in case \(\gamma = 0\), then the model is homoscedastic)

and the loglinear model:

\[Z= \exp (\beta log(Y) + \epsilon)\]

where

\[\epsilon \sim N(0,\sigma^{2})\]

In SamplingStrata, the use of models to calculate the anticipated variance implies the execution of the following steps:

• for each couple target variable Z and available variable in the frame Y fit a model (linear or loglinear) using available data;
• in case of linear model, evaluate the heteroscedasticity of residuals (with the function computeGamma);
• with the above, define the model dataframe that is given as argument to the optimStrata function.

The model dataframe must contain a number of rows equal to the number of the Y variables.The coupling is positional: the first row refers to the Y1, the second row to Y2 and so on.

This is the structure of the model dataframe:

Handling heteroscedasticity in linear models

When dealing with linear models, an evaluation of the quantity of heteroscedasticity in the residuals is of the utmost importance in order to correctly evaluate the anticipated variance (see Henry and Valliant (2006) and Knaub (2019)). To this aim, a dedicated function has been developed: computeGamma. This function accepts as arguments the following:

• vector e of residuals;
• vector x of the values of the explanatory variable;
• number nbins of groups of residuals/values with which to fit a model.

The function produces a vector of 3 values:

• the heteroscedasticity index;
• the model standard deviation be used together with the heteroscedasticity index;
• the \(R^{2}\) of the fitted model used to estimate the two above values.

A suitable number to be given to the nbins parameter can be found by trying different values and choosing the one with which the value of the \(R^{2}\) is the highest. Experience shows that a choice of nbins ranging between 10 to 16 works quite well.

Example

Consider the following example, based on the dataset swissmunicipalities available in the package.

library(SamplingStrata)
data("swissmunicipalities")
swissmunicipalities$id <- c(1:nrow(swissmunicipalities))
swissmunicipalities$dom <- 1
head(swissmunicipalities[,c(3,4,5,6,9,22)])

##    COM        Nom HApoly Surfacesbois Airbat POPTOT
## 1  261     Zurich   8781         2326   2884 363273
## 2 6621     Geneve   1593           67    773 177964
## 3 2701      Basel   2391           97   1023 166558
## 4  351       Bern   5162         1726   1070 128634
## 5 5586   Lausanne   4136         1635    856 124914
## 6  230 Winterthur   6787         2807    972  90483

Let us assume that in the sampling frame only variables total population (POPTOT) and total area (HApoly) is available for all municipalities, while buildings area (Airbat) and woods area (Surfacesbois) are available only on a sample of 500 municipalities.

set.seed(1234)
swiss_sample <- swissmunicipalities[sample(c(1:nrow(swissmunicipalities)),500),]

In this subset we can fit models between POPTOT and the two variables that we assume are the target of our survey.

One model for buildings area and total population:

mod_Airbat_POPTOT <- lm(swiss_sample$Airbat ~ swiss_sample$POPTOT)
summary(mod_Airbat_POPTOT)

## 
## Call:
## lm(formula = swiss_sample$Airbat ~ swiss_sample$POPTOT)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -143.380  -12.223   -4.391    7.336  208.728 
## 
## Coefficients:
##                       Estimate Std. Error t value            Pr(>|t|)    
## (Intercept)         19.6788127  1.3203745   14.90 <0.0000000000000002 ***
## swiss_sample$POPTOT  0.0119216  0.0002854   41.78 <0.0000000000000002 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 25.49 on 498 degrees of freedom
## Multiple R-squared:  0.778,  Adjusted R-squared:  0.7776 
## F-statistic:  1745 on 1 and 498 DF,  p-value: < 0.00000000000000022

and one model for woods area and total area:

mod_Surfacesbois_HApoly <- lm(swiss_sample$Surfacesbois ~ swiss_sample$HApoly)
summary(mod_Surfacesbois_HApoly)

## 
## Call:
## lm(formula = swiss_sample$Surfacesbois ~ swiss_sample$HApoly)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3979.1  -152.5   -87.7    98.0  2107.3 
## 
## Coefficients:
##                       Estimate Std. Error t value             Pr(>|t|)    
## (Intercept)         154.503747  19.210398   8.043  0.00000000000000646 ***
## swiss_sample$HApoly   0.199703   0.007415  26.932 < 0.0000000000000002 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 371.1 on 498 degrees of freedom
## Multiple R-squared:  0.5929, Adjusted R-squared:  0.5921 
## F-statistic: 725.4 on 1 and 498 DF,  p-value: < 0.00000000000000022

We calculate the heteroscedasticity index and associated prediction standard error for both models:

Airbat <- computeGamma(mod_Airbat_POPTOT$residuals,
             swiss_sample$POPTOT,
             nbins = 10)

Airbat

##     gamma     sigma  r.square 
## 0.6419641 0.1220129 0.9535845

Surfacesbois <- computeGamma(mod_Surfacesbois_HApoly$residuals,
             swiss_sample$HApoly,
             nbins = 10)

Surfacesbois

##     gamma     sigma  r.square 
## 0.8911053 0.3582789 0.9931305

We now proceed in building the model dataframe using the above models:

model <- NULL
model$beta[1] <- mod_Airbat_POPTOT$coefficients[2]
model$sig2[1] <- Airbat[2]^2
model$type[1] <- "linear"
model$gamma[1] <- Airbat[1]
model$beta[2] <- mod_Surfacesbois_HApoly$coefficients[2]
model$sig2[2] <- Surfacesbois[2]^2
model$type[2] <- "linear"
model$gamma[2] <- Surfacesbois[1] 
model <- as.data.frame(model)
model

##         beta       sig2   type     gamma
## 1 0.01192162 0.01488715 linear 0.6419641
## 2 0.19970325 0.12836378 linear 0.8911053

We define the sampling frame in this way:

frame <- buildFrameDF(swissmunicipalities,
                      id = "COM",
                      domainvalue = "dom",
                      X = c("POPTOT", "HApoly"),
                      Y = c("POPTOT", "HApoly"))
frame$Airbat <- swissmunicipalities$Airbat
frame$Surfacesbois <- swissmunicipalities$Surfacesbois

Note that the explanatory variables in the models have been set as both target variables Y and stratification variables X.

(Since in this exercise the true values of the variables of interest are known for each unit of the population, they have been included in the frame in order to allow a performance evaluation in the next step. It is quite clear that this knowledge is not available in real cases.)

We set 5% precision constraint on both variables:

cv <- as.data.frame(list(DOM=rep("DOM1",1),
                         CV1=rep(0.05,1),
                         CV2=rep(0.05,1),
                         domainvalue=c(1:1)
                    ))
cv

##    DOM  CV1  CV2 domainvalue
## 1 DOM1 0.05 0.05           1

We can now proceed with the optimization step:

set.seed(1234)
solution <- optimStrata(
    method = "continuous",
    errors = cv , 
    framesamp = frame, 
    model = model,
    iter = 25, 
    pops = 20, 
    parallel = FALSE,
    nStrata = 5)

# *** Domain :  1   1
#  Number of strata :  2896
# GA Settings
#   Population size       = 20
#   Number of Generations = 25
#   Elitism               = 4
#   Mutation Chance       = 0.111111111111111
# 
# 
# 
#  *** Sample cost:  295.1979
#  *** Number of strata:  5
#  *** Sample size :  296
#  *** Number of strata :  5
# ---------------------------

What about the expected CVs? We attribute the real values of Airbat and Surfacesbois to the Ys of obtained framenew and run the simulation:

outstrata <- solution$aggr_strata
framenew <- solution$framenew
framenew$Y3 <- framenew$AIRBAT
framenew$Y4 <- framenew$SURFACESBOIS
results <- evalSolution(framenew, outstrata, 500, progress = FALSE)
results$coeff_var
#      CV1    CV2    CV3    CV4  dom
# 1 0.0369 0.0241 0.0352 0.0366 DOM1

The first two CVs pertain to the proxy available variables, namely total population and total area, while the last two regard respectively buildings area and woods area: they are more than compliant with the precision constraint of 5%.

What if we did not use the information contained in models? We run the same optimization step without indication of model parameter:

set.seed(1234)
solution <- optimStrata(
    method = "continuous",
    errors = cv , 
    framesamp = frame, 
    model = NULL,
    iter = 25, 
    pops = 20, 
    parallel = FALSE,
    nStrata = 5)

#  *** Domain :  1   1
#  Number of strata :  2896
# GA Settings
#   Population size       = 20
#   Number of Generations = 25
#   Elitism               = 4
#   Mutation Chance       = 0.111111111111111
# 
# 
# 
#  *** Sample cost:  151.9848
#  *** Number of strata:  5
#  *** Sample size :  152
#  *** Number of strata :  5

We obtain a solution that requires a much lower sample size to satisfy the precision constraints on the Ys. But as we did not consider the anticipated variance on the real target variables, we pay a price in terms of expected CVs on them:

outstrata <- solution$aggr_strata
framenew <- solution$framenew
framenew$Y3 <- framenew$AIRBAT
framenew$Y4 <- framenew$SURFACESBOIS
results <- evalSolution(framenew, outstrata, 500, progress = FALSE)
results$coeff_var
#      CV1    CV2    CV3    CV4  dom
# 1 0.0501 0.0491 0.0502 0.0686 DOM1

Norwithstanding the non inclusion of the models in the optimization step, the CV related to buildings area is still inside the limit of the 5%: this is most likely due to the high correlation between buildings area and total population. While the lower correlation between the total area and woods area determines the non compliance of the expected CV of woods area that was instead guaranted using the related model.