In FanWangEcon/PrjOptiAlloc: The Optimal Allocation of Resources Among Heterogeneous Individuals

In Logit Employment Binary Allocation Lalonde Training Example, analyzed optimal allocation when all observables attributes of individuals are used for targeting. Subgroup allocation targeting here, based on one attribute.

Set Up

# dev.off(dev.list()["RStudioGD"])
rm(list = ls(all.names = TRUE))
options(knitr.duplicate.label = 'allow')

library(dplyr)
library(tidyr)
library(tibble)
library(forcats)
library(stringr)
library(broom)
library(ggplot2)
library(REconTools)

library(PrjOptiAlloc)

library(knitr)
library(kableExtra)

bl_save_rda = FALSE
bl_save_img = FALSE

Get Data

spt_img_save <- '../_img/'
spt_img_save_draft <- 'C:/Users/fan/Documents/Dropbox (UH-ECON)/repos/HgtOptiAlloDraft/_img/'

The regression is the same as prior. Subgroup allocation is based on the idea of targeting only a subset of individuals, when we know the marginal effects and needs of all individuals.

# Dataset
data(df_opt_lalonde_training)

# Add a binary variable for if there are wage in year 1975
dft <- df_opt_lalonde_training %>% 
  mutate(re75_zero = case_when(re75 == 0 ~ 1, re75 != 0 ~ 0))

# dft stands for dataframe training
dft <- dft %>% mutate(id = X) %>%
           select(-X) %>%
           select(id, everything()) %>%
           mutate(emp78 =
                    case_when(re78 <= 0 ~ 0,
                              TRUE ~ 1)) %>%
           mutate(emp75 =
                    case_when(re75 <= 0 ~ 0,
                              TRUE ~ 1))

# Generate combine black + hispanic status
# 0 = white, 1 = black, 2 = hispanics
dft <- dft %>%
    mutate(race =
             case_when(black == 1 ~ 1,
                       hisp == 1 ~ 2,
                       TRUE ~ 0))

dft <- dft %>%
    mutate(age_m2 =
             case_when(age <= 23 ~ 1,
                       age >  23~ 2)) %>%
    mutate(age_m3 =
             case_when(age <= 20 ~ 1,
                       age > 20 & age <= 26 ~ 2,
                       age > 26 ~ 3))

dft$trt <- factor(dft$trt, levels = c(0,1), labels = c("ntran", "train"))

summary(dft)

# X-variables to use on RHS
ls_st_xs <- c('age', 'educ',
              'black','hisp','marr', 'nodeg')
svr_binary <- 'trt'
svr_binary_lb0 <- 'ntran'
svr_binary_lb1 <- 'train'
svr_outcome <- 'emp78'
sdt_name <- 'NSW Lalonde Training'

Logit Regression

Prediction with Observed Binary Input

Logit regression with a continuous variable and a binary variable. Predict outcome with observed continuous variable as well as observed binary input variable.

# Regress No bivariate
rs_logit <- glm(as.formula(paste(svr_outcome,
                                 "~", paste(ls_st_xs, collapse="+")))
                ,data = dft, family = "binomial")
summary(rs_logit)
dft$p_mpg <- predict(rs_logit, newdata = dft, type = "response")

# Regress with bivariate
# rs_logit_bi <- glm(as.formula(paste(svr_outcome,
#                                     "~ factor(", svr_binary,") + ",
#                                     paste(ls_st_xs, collapse="+")))
#                    , data = dft, family = "binomial")
rs_logit_bi <- glm(emp78 ~
                     age + I(age^2) + factor(age_m2)
                   # + educ + I(educ^2) +
                   # + educ + black + hisp + marr + nodeg
                   + factor(trt)
                   + factor(age_m2)*factor(trt)
                   , data = dft, family = "binomial")
summary(rs_logit_bi)

# Predcit Using Regresion Data
dft$p_mpg_hp <- predict(rs_logit_bi, newdata = dft, type = "response")

# Predicted Probabilities am on mgp with or without hp binary
scatter <- ggplot(dft, aes(x=p_mpg_hp, y=p_mpg)) +
      geom_point(size=1) +
      # geom_smooth(method=lm) + # Trend line
      geom_abline(intercept = 0, slope = 1) + # 45 degree line
      labs(title = paste0('Predicted Probabilities ', svr_outcome, ' on ', ls_st_xs, ' with or without hp binary'),
           x = paste0('prediction with ', ls_st_xs, ' and binary ', svr_binary, ' indicator, 1 is high'),
           y = paste0('prediction with only ', ls_st_xs),
           caption = paste0(sdt_name, ' simulated prediction')) +
      theme_bw()
print(scatter)

Prediction with Binary set to 0 and 1

Now generate two predictions. One set where binary input is equal to 0, and another where the binary inputs are equal to 1. Ignore whether in data binary input is equal to 0 or 1. Use the same regression results as what was just derived.

Note that given the example here, the probability changes a lot when we

# Previous regression results
summary(rs_logit_bi)

# Two different dataframes, mutate the binary regressor
dft_bi0 <- dft %>% mutate(!!sym(svr_binary) := svr_binary_lb0)
dft_bi1 <- dft %>% mutate(!!sym(svr_binary) := svr_binary_lb1)

# Predcit Using Regresion Data
dft$p_mpg_hp_bi0 <- predict(rs_logit_bi, newdata = dft_bi0, type = "response")
dft$p_mpg_hp_bi1 <- predict(rs_logit_bi, newdata = dft_bi1, type = "response")

# Predicted Probabilities and Binary Input
scatter <- ggplot(dft, aes(x=p_mpg_hp_bi0)) +
      geom_point(aes(y=p_mpg_hp), size=4, shape=4, color="red") +
      geom_point(aes(y=p_mpg_hp_bi1), size=2, shape=8) +
      # geom_smooth(method=lm) + # Trend line
      geom_abline(intercept = 0, slope = 1) + # 45 degree line
      labs(title = paste0('Predicted Probabilities and Binary Input',
                          '\ncross(shape=4)/red is predict actual binary data',
                          '\nstar(shape=8)/black is predict set binary = 1 for all'),
           x = paste0('prediction with ', ls_st_xs, ' and binary ', svr_binary, ' = 0 for all'),
           y = paste0('prediction with ', ls_st_xs, ' and binary ', svr_binary, ' = 1'),
           caption = paste0(sdt_name, ' simulated prediction')) +
      theme_bw()
print(scatter)

Generate and Analyze Individual A and alpha

Prediction with Binary set to 0 and 1 Difference

What is the difference in probability between binary = 0 vs binary = 1. How does that relate to the probability of outcome of interest when binary = 0 for all.

In the binary logit case, the relationship will be hump--shaped by construction between $A_i$ and $\alpha_i$. In the exponential wage cases, the relationship is convex upwards.

# Generate Gap Variable
dft <- dft %>% mutate(alpha_i = p_mpg_hp_bi1 - p_mpg_hp_bi0) %>%
                mutate(A_i = p_mpg_hp_bi0)

dft_graph <- dft
dft_graph$age_m2 <- factor(dft_graph$age_m2, labels = c('Age <= 23', 'Age > 23'))

# Titling
title_line1 <- sprintf("Each circle (cross) represents an individual <= age 23 (> age 23)")
title_line2 <- sprintf("Heterogeneous expected outcome (employment probability) with and without training")
title_line3 <- sprintf("Heterogeneity from logistic regression nonlinearity and heterogeneous age group effects")
title <- expression('The joint distribution of'~A[i]~'and'~alpha[i]~','~'Logistic Regression, Lalonde (AER, 1986)')
caption <- paste0('Logistic regression predictions of the employment effects of a training RCT. Data from Lalonde (AER, 1986).')
# Labels
st_x_label <- expression(A[i]~', '~Probability~of~Employment~without~Training~','~'P(train=0)')
st_y_label <- expression(alpha[i]~','~Marginal~Effects~of~Training~','~'P(train=1) - P(train=0)')

# Binary Marginal Effects and Prediction without Binary
plt_A_alpha <- dft_graph %>% ggplot(aes(x=A_i)) +
      geom_point(aes(y=alpha_i,
                     color=factor(age_m2),
                     shape=factor(age_m2)), size=4) +
      geom_abline(intercept = 0, slope = 1) + # 45 degree line
      scale_colour_manual(values=c("#69b3a2", "#404080")) +
      labs(subtitle = paste0(title_line1,'\n', title_line2, '\n', title_line3),
           x = st_x_label,
           y = st_y_label,
           caption = caption) +
      theme_bw(base_size=8) +
      scale_shape_manual(values=c(1, 4)) +
      guides(color=FALSE)

# Labeling
plt_A_alpha$labels$shape <- "Age Subgroups"

print(plt_A_alpha)

if (bl_save_img) {
  snm_cnts <- 'Lalonde_employ_A_alpha_age.png'
  png(paste0(spt_img_save, snm_cnts),
        width = 135, height = 86, units='mm', res = 300, pointsize=7)
  print(plt_A_alpha)
  dev.off()
  png(paste0(spt_img_save_draft, snm_cnts),
        width = 135, height = 86, units='mm', res = 300,
      pointsize=5)
  print(plt_A_alpha)
  dev.off()
}

Optimal Binary Allocation

Preference Vector

beta_i <- rep(1/dim(dft)[1], times=dim(dft)[1])
ar_rho = c(-100, -0.001,  0.95)
ar_rho <- 1 - (10^(c(seq(-2,2, length.out=30))))
ar_rho <- unique(ar_rho)

Solve for Optimal Allocaions Across Preference Parameters

Invoke the binary optimal allocation function ffp_opt_anlyz_rhgin_bin that loops over rhos.

svr_inpalc <- 'rank'
dft <- cbind(dft, beta_i)
svr_rho_val <- 'rho_val'
ls_bin_solu_all_rhos <-
  ffp_opt_anlyz_rhgin_bin(dft, svr_id_i = 'id',
                          svr_A_i = 'A_i', svr_alpha_i = 'alpha_i', svr_beta_i = 'beta_i',
                          ar_rho = ar_rho,
                          svr_rho = 'rho', svr_rho_val = svr_rho_val,
                          svr_inpalc = svr_inpalc,
                          svr_expout = 'opti_exp_outcome',
                          verbose = TRUE)

df_all_rho <- ls_bin_solu_all_rhos$df_all_rho
df_all_rho_long <- ls_bin_solu_all_rhos$df_all_rho_long

# How many people have different ranks across rhos
it_how_many_vary_rank <- sum(df_all_rho$rank_max - df_all_rho$rank_min)
it_how_many_vary_rank

Change in Rank along rho

# get rank when wage rho = 1
df_all_rho_rho_c1 <- df_all_rho %>% select(id, rho_c1_rk)
# Merge
df_all_rho_long <- df_all_rho_long %>% mutate(rho = as.numeric(rho)) %>%
                      left_join(df_all_rho_rho_c1, by='id')
# Select subset to graph
df_rank_graph <- df_all_rho_long %>%
                    mutate(id = factor(id)) %>%
                    filter((id == 1) |  # utilitarian rank = 1
                           (id == 11) |  # utilitarian rank = 101
                           (id == 5)  |  # utilitarian rank = 201
                           (id == 205) |  # utilitarian rank = 301
                           (id == 42)|  # utilitarian rank = 401
                           (id == 8) |  # utilitarian rank = 501
                           (id == 31) |  # utilitarian rank = 601
                           (id == 134)   # utilitarian rank = 701
                           ) %>%
                    mutate(one_minus_rho = 1 - !!sym(svr_rho_val)) %>%
                    mutate(rho_c1_rk = factor(rho_c1_rk))

df_rank_graph$rho_c1_rk

df_rank_graph$id <- factor(df_rank_graph$rho_c1_rk,
                           labels = c('Rank= 1  ', #200
                                      'Rank= 124', #110
                                      'Rank= 245', #95
                                      'Rank= 320', #217
                                      'Rank= 411', #274
                                      'Rank= 503', #101
                                      'Rank= 595',
                                      'Rank= 700'
                                       ))

# x-labels
x.labels <- c('λ=0.99', 'λ=0.90', 'λ=0', 'λ=-10', 'λ=-100')
x.breaks <- c(0.01, 0.10, 1, 10, 100)

# title line 2
title_line1 <- sprintf("Optimal allocation queue vary λ, allocate using only AGE")
title_line2 <- sprintf("Colored lines = different individuals from the NSW training dataset")
title_line3 <- sprintf("Track ranking changes for eight individuals ranked 1, 101, ..., 701 at λ=0.99")

# Graph Results--Draw
line.plot <- df_rank_graph %>%
  ggplot(aes(x=one_minus_rho, y=!!sym(svr_inpalc),
             group=fct_rev(id),
             colour=fct_rev(id), size=2)) +
  # geom_line(aes(linetype =fct_rev(id)), size=0.75) +
  geom_line(size=0.5) +
  geom_vline(xintercept=c(1), linetype="dotted") +
  labs(subtitle = paste0(title_line1, '\n', title_line2, '\n', title_line3),
       x = 'log10 Rescale of λ, Log10(λ)\nλ=1 Utilitarian (Maximize Average), λ=-infty Rawlsian (Maximize Mininum)',
       y = 'Optimal Allocation Queue Rank (1=highest)',
       caption = 'Based on logistic regression of the employment effects of a training RCT. Data from Lalonde (AER, 1986).') +
  scale_x_continuous(trans='log10', labels = x.labels, breaks = x.breaks) +
  theme_bw(base_size=8)

# Labeling
line.plot$labels$colour <- "At λ=0.99, i's"

# Print
print(line.plot)

if (bl_save_img) {
  snm_cnts <- 'Lalonde_employ_rank_age.png'
  png(paste0(spt_img_save, snm_cnts),
        width = 135, height = 86, units='mm', res = 300, pointsize=7)
  print(line.plot)
  dev.off()
  png(paste0(spt_img_save_draft, snm_cnts),
        width = 135, height = 86, units='mm', res = 300,
      pointsize=5)
  print(line.plot)
  dev.off()
}

Save Results

# Change Variable names so that this can becombined with logit file later
df_all_rho <- df_all_rho %>% rename_at(vars(starts_with("rho_")), funs(str_replace(., "rk", "rk_empage")))
df_all_rho <- df_all_rho %>%
        rename(A_i_empage = A_i, alpha_i_empage = alpha_i, beta_i_empage = beta_i,
               rank_min_empage = rank_min, rank_max_empage = rank_max, avg_rank_empage = avg_rank)

# Save File
if (bl_save_rda) {
  df_opt_lalonde_training_empage <- df_all_rho
  usethis::use_data(df_opt_lalonde_training_empage, df_opt_lalonde_training_empage, overwrite = TRUE)
}

Binary Marginal Effects and Prediction without Binary

What is the relationship between ranking,

# ggplot.A.alpha.x <- function(svr_x, df,
#                              svr_alpha = 'alpha_i', svr_A = "A_i"){
#
#   scatter <- ggplot(df, aes(x=!!sym(svr_x))) +
#         geom_point(aes(y=alpha_i), size=4, shape=4, color="red") +
#         geom_point(aes(y=A_i), size=2, shape=8, color="blue") +
#         geom_abline(intercept = 0, slope = 1) + # 45 degree line
#         labs(title = paste0('A (blue) and alpha (red) vs x variables=', svr_x),
#              x = svr_x,
#              y = 'Probabilities',
#              caption = paste0(sdt_name, ' simulated prediction')) +
#         theme_bw()
#
# return(scatter)
# }

FanWangEcon/PrjOptiAlloc documentation built on Jan. 25, 2022, 6:55 a.m.