# A Comparison of maSAE to forestinventory

2020-03-13, 13:16:56

## What is this about?

In 2016, Andreas Hill published package forestinventory. We thought about merging the packages, but never actually got to it: `maSAE` is S4, `forestinventory` is S3, both of us are busy doing other stuff. So I am trying to at least compare functionality of both packages.

    library("forestinventory")
    library("maSAE")

## Setup

I need some helpers for checking and comparing results from `maSAE` and `forestinventory`:

    clean <- function(x, which = NULL) {
        if (identical(FALSE, which)) {
            res <- as.matrix(unname(x[TRUE, TRUE]))
        } else {
            if (is.null(which)) {
                if (all(c( "small_area", "prediction", "variance") %in% names(x))) {
                    res <- as.matrix(unname(x[TRUE, 1:3]))
                } else {
                    res <- as.matrix(unname(x[["estimation"]][TRUE, c("area", "estimate", "g_variance")]))
                }
            } else {
                res <- as.matrix(unname(x[TRUE, c(1, grep(which, names(x)))]))
            }
        }
        return(res)
    }

    compare <- function(maSAE, forestinventory, message = NULL) {
        if(! isTRUE(all.equal(clean(maSAE), clean(forestinventory),
                              check.attributes = FALSE))) {
            message <- c("Differing results from maSAE and forestinventory: ",
                         message)
            warning(message)
            return(FALSE)
        }
        return(TRUE)
    }

I use the grisons data set from `forestinventory`:

    data(grisons, package = "forestinventory")

I define regression models for simple and cluster sampling:

    formula.s0 <- tvol ~ mean # reduced model:
    formula.s1 <- tvol ~ mean + stddev + max + q75 # full model
    formula.clust.s0 <- basal ~ stade
    formula.clust.s1 <- basal ~ stade + couver + melange

## Two-Phase

Some data handling, true means are taken from forestinventory+s vignette. :

    truemeans.G <- data.frame(Intercept = rep(1, 4),
                              mean = c(12.85, 12.21, 9.33, 10.45),
                              stddev = c(9.31, 9.47, 7.90, 8.36),
                              max = c(34.92, 35.36, 28.81, 30.22),
                              q75 = c(19.77, 19.16, 15.40, 16.91))
    rownames(truemeans.G) <- c("A", "B", "C", "D")

    # data adjustments
    s1 <- grisons[grisons[["phase_id_2p"]] == 1, ]
    s2 <- grisons[grisons[["phase_id_2p"]] == 2, ]
    s12 <- rbind(s1, s2)
    s12$s1 <- s12$phase_id_2p %in% c(1, 2)
    s12$s2 <- s12$phase_id_2p == 2

    tm <- truemeans.G
    tm[["smallarea"]] <- row.names(tm)
    tm[["Intercept"]] <- NULL

    truemeans.G.partially <- truemeans.G[, -which(names(truemeans.G) %in% c("stddev", "mean"))]
    tm.partially <- tm[, -which(names(tm) %in% c("stddev", "mean"))]

### No Exhaustive Auxiliary Information

This is the estimation given on bottom of page 15 of forestinventory+s vignette:

    summary(twophase(formula = formula.s1,
                     data = grisons,
                     phase_id = list(phase.col = "phase_id_2p", terrgrid.id = 2),
                     small_area = list(sa.col = "smallarea",
                                       areas = c("A", "B","C", "D"),
                                       unbiased = TRUE),
                     boundary_weights = "boundary_weights"
                     ))

    ##
    ## Two-phase small area estimation
    ##
    ## Call:
    ## twophase(formula = formula.s1, data = grisons, phase_id = list(phase.col = "phase_id_2p",
    ##     terrgrid.id = 2), small_area = list(sa.col = "smallarea",
    ##     areas = c("A", "B", "C", "D"), unbiased = TRUE), boundary_weights = "boundary_weights")
    ##
    ## Method used:
    ## Extended pseudosynthetic small area estimator
    ##
    ## Regression Model:
    ## tvol ~ mean + stddev + max + q75 + smallarea
    ##
    ## Estimation results:
    ##  area estimate ext_variance g_variance  n1 n2 n1G n2G r.squared
    ##     A 391.9356     995.5602   1017.633 306 67  94  19 0.6526503
    ##     B 419.7231    1214.6053   1019.191 306 67  81  17 0.6428854
    ##     C 328.8600     916.2266   1036.791 306 67  66  15 0.6430018
    ##     D 373.9497    1272.7056   1110.245 306 67  65  16 0.6556178
    ##
    ## 'boundary_weight'- option was used to calculate weighted means of auxiliary variables

Now I+m using both packages to make predictions:

    maSAE <- predict(saObj(data = s12,
                           f = update(formula.s1, ~ . | smallarea),
                           auxiliaryWeights = "boundary_weights",
                           s2 = 's2')
    )

    forestinventory <- twophase(formula = formula.s1,
                                data = grisons,
                                phase_id = list(phase.col = "phase_id_2p", terrgrid.id = 2),
                                small_area = list(sa.col = "smallarea",
                                                  areas = c("A", "B","C", "D"),
                                                  unbiased = TRUE),
                                boundary_weights = "boundary_weights"
                                )

Both packages deliver the same results:

    compare(maSAE, forestinventory, "two-phase, ext. pseudo sae")

    ## [1] TRUE

Now I benchmark them, calculating the small, synthetic and extended synthetic estimators:

    wrap_two <- function(...) {
        dots <- list(...)
        dots$small_area$unbiased <- TRUE
        ex <- do.call(twophase, dots)$estimation
        dots$psmall <- TRUE
        small <- do.call(twophase, dots)$estimation
        dots$psmall <- FALSE
        dots$small_area$unbiased <- FALSE
        synth <- do.call(twophase, dots)$estimation
        cbind(ex[TRUE, c("estimate", "g_variance")],
              synth[TRUE, c("estimate", "g_variance")],
              small[TRUE, c("estimate", "g_variance")])

    }
    mbmb <- microbenchmark::microbenchmark
    mb <- mbmb(
               forestinventory = wrap_two(formula = formula.s1,
                                        data = grisons,
                                        phase_id = list(phase.col = "phase_id_2p", terrgrid.id = 2),
                                        small_area = list(sa.col = "smallarea",
                                                          areas = c("A", "B","C", "D"),
                                                          unbiased = TRUE)
                                        ),
               maSAE = predict(saObj(data = s12,
                                     f = update(formula.s1, ~ . | smallarea),
                                     s2 = 's2'
                                     ))[, -1],
               check = "equivalent"
               )

`maSAE` seems a bit faster:

    print(mb)

    ## Unit: milliseconds
    ##             expr      min        lq      mean    median        uq       max
    ##  forestinventory 99.82881 100.87952 111.31355 110.27632 112.14146 235.67957
    ##            maSAE 46.87428  47.32818  52.04255  48.83249  57.21007  69.40275
    ##  neval
    ##    100
    ##    100

    microbenchmark:::autoplot.microbenchmark(mb)

    ## Coordinate system already present. Adding new coordinate system, which will replace the existing one.

Figure 1. plot of chunk unnamed-chunk-10

### Full Exhaustive Auxiliary Information

The estimation given on page 17 of forestinventory+s vignette is (there are no boundary weights here):

    forestinventory <- twophase(formula = formula.s1,
                                data = grisons,
                                phase_id = list(phase.col = "phase_id_2p",
                                                terrgrid.id = 2),
                                small_area = list(sa.col ="smallarea",
                                                  areas = c("A", "B", "C", "D"),
                                                  unbiased = TRUE),
                                exhaustive = truemeans.G)
    summary(forestinventory)

    ##
    ## Two-phase small area estimation
    ##
    ## Call:
    ## twophase(formula = formula.s1, data = grisons, phase_id = list(phase.col = "phase_id_2p",
    ##     terrgrid.id = 2), small_area = list(sa.col = "smallarea",
    ##     areas = c("A", "B", "C", "D"), unbiased = TRUE), exhaustive = truemeans.G)
    ##
    ## Method used:
    ## Extended synthetic small area estimator
    ##
    ## Regression Model:
    ## tvol ~ mean + stddev + max + q75 + smallarea
    ##
    ## Estimation results:
    ##  area estimate ext_variance g_variance  n1 n2 n1G n2G r.squared
    ##     A 372.6930     744.3658   696.5739 Inf 67 Inf  19 0.6526503
    ##     B 387.5116     693.8576   708.1105 Inf 67 Inf  17 0.6428854
    ##     C 334.8314     838.3953   801.4303 Inf 67 Inf  15 0.6430018
    ##     D 405.9667     940.3149   890.9536 Inf 67 Inf  16 0.6556178

Again, predictions from both packages are identical:

    maSAE <- predict(saObj(data = s12,
                           f = update(formula.s1, ~ . | smallarea),
                           s2 = 's2',
                           smallAreaMeans = tm))
    compare(maSAE, forestinventory, "two-phase, ext. sae")

    ## [1] TRUE

The benchmarks again:

    mb <- mbmb(
               forestinventory = wrap_two(formula = formula.s1,
                                                data = grisons,
                                                phase_id = list(phase.col = "phase_id_2p",
                                                                terrgrid.id = 2),
                                                small_area = list(sa.col ="smallarea",
                                                                  areas = c("A", "B", "C", "D"),
                                                                  unbiased = TRUE),
                                                exhaustive = truemeans.G),
               maSAE = predict(saObj(data = s12,
                                           f = update(formula.s1, ~ . | smallarea),
                                           s2 = 's2',
                                           smallAreaMeans = tm))[, -1],

               check = "equivalent")
    print(mb)

    ## Unit: milliseconds
    ##             expr      min       lq     mean   median       uq       max neval
    ##  forestinventory 77.70466 78.43906 87.69586 87.39151 89.08470 228.46098   100
    ##            maSAE 46.08046 46.53541 52.91265 49.40216 56.94793  77.34464   100

    microbenchmark:::autoplot.microbenchmark(mb)

    ## Coordinate system already present. Adding new coordinate system, which will replace the existing one.

Figure 2. plot of chunk unnamed-chunk-13

## Three-Phase

Some data handling, true means are taken from forestinventory+s vignette. :

    truemeans.G <- data.frame(Intercept = rep(1, 4),
                              mean = c(12.85, 12.21, 9.33, 10.45))
    rownames(truemeans.G) <- c("A", "B", "C", "D")

    ## data adjustments
    s12_3p <- grisons[grisons[["phase_id_3p"]] %in% c(1,2), ]
    s0 <- grisons[grisons[["phase_id_3p"]] ==0 , ]
    s12_3p$s1 <- s12_3p$phase_id_3p %in% c(1, 2)
    s12_3p$s2 <- s12_3p$phase_id_3p == 2
    s0$s1 <- s0$s2 <- FALSE
    predictors_s0 <- all.vars(formula.s0)[-1]
    predictors_s1 <- all.vars(formula.s1)[-1]
    eval(parse(text=(paste0("s0$",
                            setdiff(predictors_s1, predictors_s0),
                            " <- NA"))))
    s012 <- rbind(s0, s12_3p)
    tm <- truemeans.G
    tm[["smallarea"]] <- row.names(tm)
    tm[["Intercept"]] <- NULL

### No Exhaustive Auxiliary Information

The estimation given on page 23 of forestinventory+s vignette is:

    summary(threephase(formula.s0,
                       formula.s1,
                       data = grisons,
                       phase_id = list(phase.col = "phase_id_3p", s1.id = 1, terrgrid.id = 2),
                       small_area=list(sa.col = "smallarea", areas = c("A", "B", "C", "D"),
                                       unbiased = TRUE),
                       boundary_weights = "boundary_weights"
                       ))

    ##
    ## Three-phase small area estimation
    ##
    ## Call:
    ## threephase(formula.s0 = formula.s0, formula.s1 = formula.s1,
    ##     data = grisons, phase_id = list(phase.col = "phase_id_3p",
    ##         s1.id = 1, terrgrid.id = 2), small_area = list(sa.col = "smallarea",
    ##         areas = c("A", "B", "C", "D"), unbiased = TRUE), boundary_weights = "boundary_weights")
    ##
    ## Method used:
    ## Extended pseudosynthetic small area estimator
    ##
    ## Full Regression Model:
    ## tvol ~ mean + stddev + max + q75 + smallarea
    ##
    ## Reduced Regression Model:
    ## tvol ~ mean + smallarea
    ##
    ## Estimation results:
    ##  area estimate ext_variance g_variance  n0  n1 n2 n0G n1G n2G r.squared_reduced
    ##     A 395.1882    1901.2107  1858.2042 306 128 40  94  38  12         0.5454824
    ##     B 389.8329    1846.9952  1816.6552 306 128 40  81  34  11         0.5354637
    ##     C 321.9967     722.7413   763.0731 306 128 40  66  28   8         0.5282291
    ##     D 365.4938    2248.9395  1930.6881 306 128 40  65  28   9         0.5339996
    ##  r.squared_full
    ##       0.7242913
    ##       0.7171512
    ##       0.7172375
    ##       0.7268820
    ##
    ## 'boundary_weight'- option was used to calculate weighted means of auxiliary variables

Wait, the `ext_variance` differs, but that+s a problem with `forestinventory`…

I make predictions omitting the boundary weights:

    forestinventory <- threephase(formula.s0,
                                  formula.s1,
                                  data = grisons,
                                  phase_id = list(phase.col = "phase_id_3p", s1.id = 1, terrgrid.id = 2),
                                  small_area=list(sa.col = "smallarea", areas = c("A", "B", "C", "D"),
                                                  unbiased = TRUE),
                                  boundary_weights = "boundary_weights"
                                  )

    maSAE <- predict(saObj(data = s012,
                           f = update(formula.s1, ~ . | smallarea),
                           s1 = 's1',
                           auxiliaryWeights = "boundary_weights",
                           s2 = 's2'))

    ## n(s0) >> n(s1) should hold, but you've given s1 resulting in n(s1)/n(s0) =  0.418300653594771

    compare(maSAE, forestinventory, "three-phase, ext. pseudo sae")

    ## [1] TRUE

The benchmarks again:

    wrap_three <- function(...) {
        dots <- list(...)
        dots$small_area$unbiased <- TRUE
        ex <- do.call(threephase, dots)$estimation
        dots$psmall <- TRUE
        small <- do.call(threephase, dots)$estimation
        dots$psmall <- FALSE
        dots$small_area$unbiased <- FALSE
        synth <- do.call(threephase, dots)$estimation
        cbind(ex[TRUE, c("estimate", "g_variance")],
              synth[TRUE, c("estimate", "g_variance")],
              small[TRUE, c("estimate", "g_variance")])

    }
    mb <- mbmb(
               forestinventory = wrap_three(formula.s0,
                                  formula.s1,
                                  data = grisons,
                                  phase_id = list(phase.col = "phase_id_3p", s1.id = 1, terrgrid.id = 2),
                                  small_area=list(sa.col = "smallarea", areas = c("A", "B", "C", "D"),
                                                  unbiased = TRUE)
                                  ),
               maSAE = predict(suppressMessages(saObj(data = s012,
                           f = update(formula.s1, ~ . | smallarea),
                           s1 = 's1',
                           s2 = 's2')))[, -1],

               check = "equivalent")
    print(mb)

    ## Unit: milliseconds
    ##             expr       min       lq      mean   median        uq      max neval
    ##  forestinventory 167.78761 171.6586 184.15908 178.2274 186.79755 336.3791   100
    ##            maSAE  77.24488  78.4896  87.95672  87.8197  90.44643 130.4617   100

    microbenchmark:::autoplot.microbenchmark(mb)

    ## Coordinate system already present. Adding new coordinate system, which will replace the existing one.

Figure 3. plot of chunk unnamed-chunk-18

## Partially Exhaustive Auxiliary Information

Funny: `forestinventory` can+t deal with partially exhaustive auxiliary information for two-phase sampling:

    try(twophase(formula = formula.s1,
                 data = grisons,
                 phase_id = list(phase.col = "phase_id_2p",
                                 terrgrid.id = 2),
                 small_area = list(sa.col ="smallarea",
                                   areas = c("A", "B", "C", "D"),
                                   unbiased = TRUE),
                 exhaustive = truemeans.G.partially))

    ## Error in names(Z_bar_1G) <- colnames(design_matrix.s2) :
    ##   'names' attribute [6] must be the same length as the vector [4]

    predict(saObj(data = s12,
                  f = update(formula.s1, ~ . | smallarea),
                  s2 = 's2',
                  smallAreaMeans = tm.partially))

    ##   small_area prediction variance   psynth var_psynth   psmall var_psmall
    ## 1          A   371.0402 712.3895 401.8953   287.3151 373.9803  1048.5674
    ## 2          B   398.9159 944.4190 398.5562   303.4478 399.4579   996.5841
    ## 3          C   330.4869 863.1605 334.5188   261.8389 330.6825  1104.9620
    ## 4          D   384.7398 999.1490 345.1298   227.8379 380.9174  1203.8859

Whereas `maSAE` can+t deal with partially exhaustive auxiliary information for three-phase sampling:

    summary(forestinventory <- threephase(formula.s0,
                                  formula.s1,
                                  data = grisons,
                                  phase_id = list(phase.col = "phase_id_3p", s1.id = 1, terrgrid.id = 2),
                                  small_area = list(sa.col = "smallarea", areas = c("A", "B", "C", "D"),
                                                    unbiased = TRUE),
                                  exhaustive = truemeans.G))

    ##
    ## Three-phase small area estimation
    ##
    ## Call:
    ## threephase(formula.s0 = formula.s0, formula.s1 = formula.s1,
    ##     data = grisons, phase_id = list(phase.col = "phase_id_3p",
    ##         s1.id = 1, terrgrid.id = 2), small_area = list(sa.col = "smallarea",
    ##         areas = c("A", "B", "C", "D"), unbiased = TRUE), exhaustive = truemeans.G)
    ##
    ## Method used:
    ## Extended synthetic small area estimator
    ##
    ## Full Regression Model:
    ## tvol ~ mean + stddev + max + q75 + smallarea
    ##
    ## Reduced Regression Model:
    ## tvol ~ mean + smallarea
    ##
    ## Estimation results:
    ##  area estimate ext_variance g_variance  n0  n1 n2 n0G n1G n2G r.squared_reduced
    ##     A 380.7982    1642.0551  1524.8061 Inf 128 40 Inf  38  12         0.5454824
    ##     B 368.8658    1501.2108  1530.6216 Inf 128 40 Inf  34  11         0.5354637
    ##     C 325.7081     640.2232   541.0235 Inf 128 40 Inf  28   8         0.5282291
    ##     D 389.3585    1961.1322  1753.9986 Inf 128 40 Inf  28   9         0.5339996
    ##  r.squared_full
    ##       0.7242913
    ##       0.7171512
    ##       0.7172375
    ##       0.7268820

    try(predict(saObj(data = s012,
                  f = update(formula.s1, ~ . | smallarea),
                  s2 = 's2',
                  s1 = 's1',
                  smallAreaMeans = tm)))

    ## n(s0) >> n(s1) should hold, but you've given s1 resulting in n(s1)/n(s0) =  0.418300653594771

    ## Error in h(simpleError(msg, call)) :
    ##   error in evaluating the argument 'object' in selecting a method for function 'predict': invalid class "saeObj" object: Got both partial true means and s1. There is no theoretical description for this so far!

I do not see what three-phase partially exhaustive information would be. So: is partially exhaustive auxiliary information two- or three-phase?

### Is Partially Exhaustive Auxiliary Information Two- or Three-Phase?

The (first) estimation given on page 22 of forestinventory+s vignette is:

    extsynth_3p <- threephase(formula.s0, formula.s1, data = grisons,
                              phase_id = list(phase.col = "phase_id_3p",
                                              s1.id = 1, terrgrid.id = 2),
                              small_area = list(sa.col = "smallarea", areas = c("A", "B", "C", "D"),
                                                unbiased = TRUE),
                              exhaustive = truemeans.G,
                              boundary_weights = "boundary_weights"
                              )
    extsynth_3p$estimation

    ##   area estimate ext_variance g_variance  n0  n1 n2 n0G n1G n2G
    ## 1    A 382.6405    1642.0551  1518.7407 Inf 128 40 Inf  38  12
    ## 2    B 368.9013    1501.2108  1530.5759 Inf 128 40 Inf  34  11
    ## 3    C 325.3720     640.2232   543.2681 Inf 128 40 Inf  28   8
    ## 4    D 388.0325    1961.1322  1756.0906 Inf 128 40 Inf  28   9
    ##   r.squared_reduced r.squared_full
    ## 1         0.5454824      0.7242913
    ## 2         0.5354637      0.7171512
    ## 3         0.5282291      0.7172375
    ## 4         0.5339996      0.7268820

    s12_3p$s1 <- NULL
    s12_3p$phase_id_2p <- NULL
    s12_3p$phase_id_3p <- NULL
    maSAE <- predict(saObj(data = s12_3p,
                           f = update(formula.s1, ~ . | smallarea),
                           auxiliaryWeights = "boundary_weights",
                           s2 = 's2', smallAreaMeans = tm)
    )
    compare(maSAE, extsynth_3p, "three-phase, ext. sae")

    ## [1] TRUE

So this is a two-phase setup, `forestinventory` seems to need the three-phase setup (the reduced model) to identify the partially exhaustive part of the auxiliary information. The corresponding publication is

    Daniel Mandallaz, Jochen Breschan, and Andreas Hill. New regression
    estimators in forest inventories with two-phase sampling and partially
    exhaustive information: a design-based Monte Carlo approach with applications
    to small-area estimation.
    In: Canadian Journal of Forest Research 43.11 (2013), pp. 1023-1031.
    doi: 10.1139/cjfr-2013-0181.

The benchmarks again:

    mb <- mbmb(
               forestinventory = wrap_three(formula.s0,
                                  formula.s1,
                                  data = grisons,
                                  phase_id = list(phase.col = "phase_id_3p", s1.id = 1, terrgrid.id = 2),
                                  small_area = list(sa.col = "smallarea", areas = c("A", "B", "C", "D"),
                                                    unbiased = TRUE),
                                  exhaustive = truemeans.G),
               maSAE = predict(suppressMessages(saObj(data = s12_3p,
                           f = update(formula.s1, ~ . | smallarea),
                           s2 = 's2', smallAreaMeans = tm)))[, -1],
               check = "equivalent")
    print(mb)

    ## Unit: milliseconds
    ##             expr       min        lq      mean   median        uq      max
    ##  forestinventory 158.09697 160.52206 165.98454 161.8343 167.84556 207.4526
    ##            maSAE  74.87462  75.96267  86.72364  85.5007  87.85716 269.2849
    ##  neval
    ##    100
    ##    100

    microbenchmark:::autoplot.microbenchmark(mb)

    ## Coordinate system already present. Adding new coordinate system, which will replace the existing one.

Figure 4. plot of chunk unnamed-chunk-23

## Cluster Sampling

I adapt data from `maSAE` to section 6.2 of forestinventory+s vignette:

    suppressWarnings(rm("s1" ,"s2", "s12"))
    data("s1", "s2", package = "maSAE")
    s12 <- bind_data(s1, s2)
    # adapt for forestinventory
    s12[["g"]][is.na(s12[["g"]])] <- "a"
    s12[["phase"]]  <-  s12[["phase1"]] + s12[["phase2"]]
    maSAE <- predict(suppressMessages(saObj(data = s12, f = y ~x1 + x2 + x3 | g,
                  s2 = "phase2", cluster = "clustid")))
    extpsynth.clust <- twophase(y ~x1 + x2 + x3,
             data = s12,
             cluster = "clustid",
             phase_id = list(phase.col = "phase", s1.id = 1, terrgrid.id = 2),
             small_area = list(sa.col = "g", areas = c("a", "b"),
                               unbiased = TRUE))
    compare(maSAE, extpsynth.clust, "three-phase, ext. sae")

    ## [1] TRUE

    mb <- mbmb(
               forestinventory = clean(twophase(y ~x1 + x2 + x3,
             data = s12,
             cluster = "clustid",
             phase_id = list(phase.col = "phase", s1.id = 1, terrgrid.id = 2),
             small_area = list(sa.col = "g", areas = c("a", "b"),
                               unbiased = TRUE))),
               maSAE = clean(predict(suppressMessages(saObj(data = s12, f = y ~x1 + x2 + x3 | g,
                  s2 = "phase2", cluster = "clustid")))),

               check = "equivalent")
    print(mb)

    ## Unit: milliseconds
    ##             expr       min        lq      mean    median        uq       max
    ##  forestinventory 110.35891 120.21952 122.49845 121.84451 123.32627 257.67614
    ##            maSAE  43.12993  43.79442  47.65953  44.19827  52.97338  80.74552
    ##  neval
    ##    100
    ##    100

    microbenchmark:::autoplot.microbenchmark(mb)

    ## Coordinate system already present. Adding new coordinate system, which will replace the existing one.

Figure 5. plot of chunk unnamed-chunk-24

Now I use the example given in section 6.2 of forestinventory+s vignette:

    data("zberg", package = "forestinventory")
    forestinventory <- forestinventory::twophase(
      formula = basal ~ stade + couver + melange, data = zberg,
      phase_id = list(phase.col = "phase_id_2p", terrgrid.id = 2),
      cluster = "cluster",
      small_area = list(
        sa.col = "ismallold", areas = c("1"),
        unbiased = TRUE
      )
    )

    ## Warning: At least one terrestrial cluster not entirely included within the small area 1.
    ## Zero mean residual assumption for small area maybe violated.
    ## Check mean_Rc_x_hat_G and consider alternative estimator 'psmall'

    s1 <- zberg[zberg[["phase_id_2p"]] == 1, ]
    s2 <- zberg[zberg[["phase_id_2p"]] == 2, ]
    s12 <- rbind(s1, s2)
    s12[["s1"]] <- s12[["phase_id_2p"]] %in% c(1, 2)
    s12[["s2"]] <- s12[["phase_id_2p"]] == 2
    object <- maSAE::saObj(
      data = s12,
      f = basal ~ stade + couver + melange | ismallold,
      s2 = "s2",
      cluster = "cluster"
    )

    ## include is NULL, automatically adding it as TRUE to data.

    maSAE <- maSAE::predict(object)
    compare(maSAE[2,], forestinventory, "clustered, ext. sae")

    ## Warning in compare(maSAE[2, ], forestinventory, "clustered, ext. sae"):
    ## Differing results from maSAE and forestinventory: clustered, ext. sae

    ## [1] FALSE

Obviously, something went wrong. The difference to the previous example? `zberg` contains nominally scaled predictors. If I ignore the cluster design, both packages give the same result:

    forestinventory <- forestinventory::twophase(
      formula = basal ~ stade + couver + melange, data = zberg,
      phase_id = list(phase.col = "phase_id_2p", terrgrid.id = 2),
      small_area = list(
        sa.col = "ismallold", areas = c("1"),
        unbiased = TRUE
      )
    )
    object <- maSAE::saObj(
      data = s12,
      f = basal ~ stade + couver + melange | ismallold,
      s2 = "s2",
    )
    maSAE <- maSAE::predict(object)
    compare(maSAE[2,], forestinventory, "clustered, ext. sae")

    ## [1] TRUE

I do not know where that comes from. But since `maSAE` has very structured code compared to `forestinventory` (`maSAE` needs about 500 lines of code for its prediction functions, `forestinventory` more than 2200), I am biased to believe `maSAE`.

## Conclusion

- Both packages give mostly identical results, but different ones for clustered sampling designs with nominally scaled predictors.

- `forestinventory` views partially exhaustive auxiliary information as a three-phase setup, `maSAE` (following Mandallaz' publications) uses a two-phase setup, but both packages give identical results after some data tweaking.

- `maSAE` seems to be a bit faster.

- `forestinventory` has global estimators implemented.