# This function helps the "two phase" cluster sampling estimator function when the first phase sample size is finite.
# This corresponds to the situation where wall2wall info is known for post-stratification
# The external (large sample version for linear models) and g-weight variances are calulated
# INPUTS:
# formula = model formula
# data = data.frame (plot level)
# exhaustive = the vector of auxiliary variable means known exhaustively (should ideally be weighted mean adjusted for proportion of pixels in interpretation are in the forest)
#
# OUTPUTS:
# estimate = point estimate
# ext_variance = external variance
# g_variance = g-weight variance
# n1 = Inf
# n2 = sample size
# r.squared = R squared of model
global_nonexhaustive2p_cluster <- function(formula, data, phase_id, cluster, boundary_weights){
n2<- sum(data[[phase_id[["phase.col"]]]] == phase_id[["terrgrid.id"]])
n1<- nrow(data)
design_matrix.s1_plot_level <- design_matrix.s1_return(formula=formula, data=data)
data_ext <- data.frame(design_matrix.s1_plot_level)
data_ext[,all.vars(formula)[1]] <- data[,all.vars(formula)[1]]
data_ext[,cluster] <- data[,cluster]
if(!is.na(boundary_weights)){
data_ext[,boundary_weights]<- data[,boundary_weights]
}
cluster_weights <- aggregate(data_ext[,all.vars(formula)[1]], list(cluster = data_ext[,cluster]), length) # the M(x)
cluster_means <- aggregate(data_ext[,-which(names(data_ext)==cluster)], list(cluster = data_ext[,cluster]), mean)
# for weighted cluster-means:
if(!is.na(boundary_weights)){
# weight the auxiliary information, but not the response:
clustmeans.temp<- ddply(data_ext[,-c(which(names(data_ext) == all.vars(formula)[1]))],
.(cluster), function(x) colwise(weighted.mean, w = x[[boundary_weights]]) (x))
clustmeans.temp<- clustmeans.temp[, -which(names(clustmeans.temp) == boundary_weights)] # remove column with boundary-weights, we don't need them anymore
# merge unweighted cluster-means of response:
cluster_means<- merge(clustmeans.temp, cluster_means[,c(cluster, all.vars(formula)[1])], by=cluster)
}
data_clust_s1 <- merge(cluster_means, cluster_weights, by=cluster) #right hand column is M(x)
data_clust_s1_groundindicator <- merge(data_clust_s1, aggregate(data[,c(cluster, phase_id[["phase.col"]])], list(data[,cluster]), unique)[-1], by=cluster, all.x=TRUE)
# we need the s2 design matrix from the cluster-level dataset "data_clust"
data_clust_s2 <- data_clust_s1_groundindicator[data_clust_s1_groundindicator[[phase_id[["phase.col"]]]]==phase_id[["terrgrid.id"]],] #THIS SHOULD HAVE PHASE_ID AT SOME POINT
Yc_x <- data_clust_s2[, all.vars(formula)[1]]
design_matrix.s1 <- data_clust_s1[, -1*c(which(names(data_clust_s1) %in% c(cluster,all.vars(formula)[1])), ncol(data_clust_s1))]
design_matrix.s2 <- as.matrix(data_clust_s2[, -1*c(which(names(data_clust_s2) %in% c(cluster,all.vars(formula)[1],phase_id[["phase.col"]])), ncol(data_clust_s2)-1)])
M_x.s1 <- data_clust_s1[, ncol(data_clust_s1)]
M_x.s2 <- data_clust_s2[, ncol(data_clust_s2)-1] # the number of columns minus one location of M_x due to the merge order
n1_clusters <- length(M_x.s1)
n2_clusters <- length(Yc_x)
As2inv <- solve( (t(design_matrix.s2) %*% (M_x.s2*design_matrix.s2)) / n2_clusters ) # (1/n2)SUMx[Mc(x)Zc(x)t(Zc(x))]
beta_s2 <- t(As2inv %*% t(t(M_x.s2*Yc_x) %*% design_matrix.s2 / n2_clusters))
Yc_x_hat_s2 <- design_matrix.s2 %*% t(beta_s2)
Yc_x_hat_s1 <- as.matrix(design_matrix.s1) %*% t(beta_s2)
Rc_x_hat <- Yc_x - Yc_x_hat_s2
MR_square <- as.vector((M_x.s2^2)*(Rc_x_hat^2)) # eq [64] p. 24 Mandallaz Small Area technical report
middle_term <- ((t(as.matrix(design_matrix.s2)) %*% ( MR_square*(as.matrix(design_matrix.s2)))) / n2_clusters^2) # eq [64] p. 24 Mandallaz Small Area technical report
cov_beta_s2 <- As2inv %*% middle_term %*% As2inv # eq [64] p. 24 Mandallaz Small Area technical report
Z_bar_1 <- apply(design_matrix.s1, 2, weighted.mean, w=M_x.s1) # eq [65] p. 25 Mandallaz Small Area technical report
#prepare matrix calculation for cov_Z_bar_1 by centering design matrix and scaling it with M_x ...eq [67] p. 25 Mandallaz Small Area technical report
design_matrix.s1_centered <- t(apply(design_matrix.s1, 1, function(x){x-Z_bar_1}))
design_matrix.s1_centered_weighted <- apply(design_matrix.s1_centered, 2, function(col){as.vector(M_x.s1 / mean(M_x.s1))*col})
cov_Z_bar_1 <- t(design_matrix.s1_centered_weighted) %*% design_matrix.s1_centered_weighted / (n1_clusters*(n1_clusters-1))
w_s2 <- (M_x.s2 / mean(M_x.s2))
w_s1 <- (M_x.s1 / mean(M_x.s1))
weighted_mean_Yc_x_hat <- sum(M_x.s1*Yc_x_hat_s1)/sum(M_x.s1)
weighted_mean_Rc_x_hat <- sum(M_x.s2*Rc_x_hat)/sum(M_x.s2)
estimate <- Z_bar_1 %*% t(beta_s2)
ext_variance <- ((1/n1_clusters)*(1/(n1_clusters-1))*sum((w_s1^2)*(Yc_x_hat_s1 - weighted_mean_Yc_x_hat)^2)) + ((1/n2_clusters)*(1/(n2_clusters-1))*sum((w_s2^2)*(Rc_x_hat - weighted_mean_Rc_x_hat)^2)) #approximation in cluster case
g_variance <- (t(Z_bar_1) %*% cov_beta_s2 %*% Z_bar_1) + (beta_s2 %*% cov_Z_bar_1 %*% t(beta_s2)) #eq [67] p. 25 Mandallaz Small Area technical report
r.squared <- summary(lm(formula, data, y=TRUE))$r.squared # plot level R square, normally worse than R-square on cluster level
## ------- create outputs ------------------------------------------------- ##
# summarize sample size info:
samplesizes<- data.frame(cbind (n1_clusters, n2_clusters, n1, n2))
colnames(samplesizes)<- c("n1_clust", "n2_clust", "n1", "n2")
rownames(samplesizes)<- "plots"
estimation<- data.frame(estimate=estimate, ext_variance=ext_variance, g_variance=g_variance,
n1=samplesizes$n1_clust, n2=samplesizes$n2_clust,
r.squared=r.squared)
# ... to store inputs used:
inputs<- list()
inputs[["data"]]<- data
inputs[["formula"]]<- formula
inputs[["boundary_weights"]]<- boundary_weights
inputs[["method"]]<- "non-exhaustive"
inputs[["cluster"]]<- TRUE
inputs[["exhaustive"]]<- FALSE
# save warning-messages:
warn.messages<- NA
result<- list(input=inputs,
estimation=estimation,
samplesizes=samplesizes,
coefficients=beta_s2,
cov_coef=cov_beta_s2,
Z_bar_1=Z_bar_1,
cov_Z_bar_1G=cov_Z_bar_1,
Rc_x_hat=Rc_x_hat,
mean_Rc_x_hat=weighted_mean_Rc_x_hat,
warn.messages=warn.messages)
class(result)<- c("global", "twophase")
return(result)
}
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