###########################################################################
#       Advanced Econometrics                                                 #
#   Spring semester                                                       #
#   dr Marcin Chlebus, dr Rafał Woźniak                                   #
#   University of Warsaw, Faculty of Economic Sciences                    #
#                                                                         #
#                                                                         #
#                 Labs 05: Unordered choice models                        #
#                                                                         #
###########################################################################


Sys.setenv(LANG = "en")

##########################################################

# Binary choice Models

#########################################################
# getwd() 
# setwd("C:\\Users\\asus\\Dysk Google\\WNE Przedmioty\\ADVANCED ECONOMETRICS\\2020\\Lab\\AE_Lab_05")
library("sandwich")
library("zoo")
library("lmtest")
library("MASS")
library("aod")

# install.packages("nnet")
library("nnet")

library("Formula")
library("miscTools")
library("maxLik")
# install.packages("mlogit")
library("mlogit")
library("car")
library("sandwich")
library("survival")
# install.packages("AER")
library("AER")
library("nnet")
library("stargazer")
library("dplyr")

1 Fishing Analysis

# ------------------------------------------
# Lecture slides
# ------------------------------------------

options(scipen = 999)

# ********************************************
# The other way of estimating multinomial 
# logit models
# This method does not allow us to obtain
# marginal effects

fish = read.csv(file="Fishing_mode.csv", sep=",", header=TRUE)
fish %>% as_tibble()

## # A tibble: 1,182 × 16
##    mode    price  crate dbeach dpier dprivate dcharter pbeach ppier pprivate
##    <chr>   <dbl>  <dbl>  <int> <int>    <int>    <int>  <dbl> <dbl>    <dbl>
##  1 charter 183.  0.539       0     0        0        1  158.  158.     158. 
##  2 charter  34.5 0.467       0     0        0        1   15.1  15.1     10.5
##  3 private  24.3 0.241       0     0        1        0  162.  162.      24.3
##  4 pier     15.1 0.0789      0     1        0        0   15.1  15.1     55.9
##  5 private  41.5 0.108       0     0        1        0  107.  107.      41.5
##  6 charter  63.9 0.398       0     0        0        1  192.  192.      28.9
##  7 beach    51.9 0.0678      1     0        0        0   51.9  51.9    192. 
##  8 charter  56.7 0.0209      0     0        0        1   15.1  15.1     21.7
##  9 private  34.9 0.0233      0     0        1        0   34.9  34.9     34.9
## 10 private  28.3 0.0233      0     0        1        0   28.3  28.3     28.3
## # ℹ 1,172 more rows
## # ℹ 6 more variables: pcharter <dbl>, qbeach <dbl>, qpier <dbl>,
## #   qprivate <dbl>, qcharter <dbl>, income <dbl>

fish$mode = as.factor(fish$mode)
fish$income2 = fish$income^2
str(fish)

## 'data.frame':    1182 obs. of  17 variables:
##  $ mode    : Factor w/ 4 levels "beach","charter",..: 2 2 4 3 4 2 1 2 4 4 ...
##  $ price   : num  182.9 34.5 24.3 15.1 41.5 ...
##  $ crate   : num  0.5391 0.4671 0.2413 0.0789 0.1082 ...
##  $ dbeach  : int  0 0 0 0 0 0 1 0 0 0 ...
##  $ dpier   : int  0 0 0 1 0 0 0 0 0 0 ...
##  $ dprivate: int  0 0 1 0 1 0 0 0 1 1 ...
##  $ dcharter: int  1 1 0 0 0 1 0 1 0 0 ...
##  $ pbeach  : num  157.9 15.1 161.9 15.1 106.9 ...
##  $ ppier   : num  157.9 15.1 161.9 15.1 106.9 ...
##  $ pprivate: num  157.9 10.5 24.3 55.9 41.5 ...
##  $ pcharter: num  182.9 34.5 59.3 84.9 71 ...
##  $ qbeach  : num  0.0678 0.1049 0.5333 0.0678 0.0678 ...
##  $ qpier   : num  0.0503 0.0451 0.4522 0.0789 0.0503 ...
##  $ qprivate: num  0.26 0.157 0.241 0.164 0.108 ...
##  $ qcharter: num  0.539 0.467 1.027 0.539 0.324 ...
##  $ income  : num  7.08 1.25 3.75 2.08 4.58 ...
##  $ income2 : num  50.17 1.56 14.06 4.34 21.01 ...

# descriptive statistics
summary(fish)

##       mode         price            crate            dbeach      
##  beach  :134   Min.   :  1.29   Min.   :0.0002   Min.   :0.0000  
##  charter:452   1st Qu.: 15.87   1st Qu.:0.0361   1st Qu.:0.0000  
##  pier   :178   Median : 37.90   Median :0.1643   Median :0.0000  
##  private:418   Mean   : 52.08   Mean   :0.3894   Mean   :0.1134  
##                3rd Qu.: 67.51   3rd Qu.:0.5333   3rd Qu.:0.0000  
##                Max.   :666.11   Max.   :2.3101   Max.   :1.0000  
##      dpier           dprivate         dcharter          pbeach      
##  Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :  1.29  
##  1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.: 26.66  
##  Median :0.0000   Median :0.0000   Median :0.0000   Median : 74.63  
##  Mean   :0.1506   Mean   :0.3536   Mean   :0.3824   Mean   :103.42  
##  3rd Qu.:0.0000   3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:144.14  
##  Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   Max.   :843.19  
##      ppier           pprivate         pcharter          qbeach      
##  Min.   :  1.29   Min.   :  2.29   Min.   : 27.29   Min.   :0.0678  
##  1st Qu.: 26.66   1st Qu.: 13.12   1st Qu.: 42.90   1st Qu.:0.0678  
##  Median : 74.63   Median : 33.53   Median : 61.61   Median :0.2537  
##  Mean   :103.42   Mean   : 55.26   Mean   : 84.38   Mean   :0.2410  
##  3rd Qu.:144.14   3rd Qu.: 72.40   3rd Qu.:102.77   3rd Qu.:0.5333  
##  Max.   :843.19   Max.   :666.11   Max.   :691.11   Max.   :0.5333  
##      qpier           qprivate         qcharter          income       
##  Min.   :0.0014   Min.   :0.0002   Min.   :0.0021   Min.   : 0.4167  
##  1st Qu.:0.0503   1st Qu.:0.0233   1st Qu.:0.0219   1st Qu.: 2.0833  
##  Median :0.0789   Median :0.0897   Median :0.4216   Median : 3.7500  
##  Mean   :0.1622   Mean   :0.1712   Mean   :0.6294   Mean   : 4.0993  
##  3rd Qu.:0.1498   3rd Qu.:0.2413   3rd Qu.:1.0266   3rd Qu.: 5.4167  
##  Max.   :0.4522   Max.   :0.7369   Max.   :2.3101   Max.   :12.5000  
##     income2        
##  Min.   :  0.1736  
##  1st Qu.:  4.3403  
##  Median : 14.0625  
##  Mean   : 22.8607  
##  3rd Qu.: 29.3403  
##  Max.   :156.2500

1.1 数据结构

variable	name variable label	Trans
mode	Fishing mode	钓鱼模式
price	price for chosen alternative	选择的替代方案价格
crate	catch rate for chosen alternative	选择的替代方案捕获率
pbeach	price for beach model	海滩模型价格
ppier	price for pier mode	码头模式价格
pprivate	price for private boat mode	私人船模式价格
pcharter	price for charter boat mode	包船模式价格
qbeach	catch rate for beach mode	海滩模式捕获率
qpier	catch rate for pier mode	码头模式捕获率
qprivate	catch rate for private boat mode	私人船模式捕获率
qcharter	catch rate for charter boat mode	包船模式捕获率
income	monthly income in thousands $	每月收入（千美元）

1.2 multinomial model with multinom

# multinomial model
mlogit = multinom(mode~income+income2, data=fish)

## # weights:  16 (9 variable)
## initial  value 1638.599935 
## iter  10 value 1478.182912
## final  value 1469.643815 
## converged

summary(mlogit)

## Call:
## multinom(formula = mode ~ income + income2, data = fish)
## 
## Coefficients:
##         (Intercept)     income      income2
## charter   0.8619802  0.2071885 -0.022473520
## pier      1.1011571 -0.3174451  0.018404477
## private   0.6227868  0.1497982 -0.005260256
## 
## Std. Errors:
##         (Intercept)    income    income2
## charter   0.3180882 0.1308082 0.01134261
## pier      0.3411857 0.1437094 0.01216713
## private   0.3196997 0.1262830 0.01026384
## 
## Residual Deviance: 2939.288 
## AIC: 2957.288

这段代码展示了使用 R 语言中的 nnet 包（通常是该包提供的 multinom 函数）拟合多项 Logit 模型 (Multinomial Logit Model, MNL) 的过程。该模型用于处理因变量是无序多分类变量的情况。

以下是对代码、参数及输出结果的详细解析：

1.2.1 代码解释

multinom(mode ~ income + income2, data = fish):
- multinom: 这是构建多项 Logit 模型的函数。
- mode ~ income + income2: 这是公式（Formula）。mode 是因变量（即钓鱼方式的选择，如 charter, pier, private, beach 等）；income 和 income2 是自变量。
- data = fish: 指定使用名为 fish 的数据集。
summary(mlogit):
- 用于提取并显示模型的详细汇总信息，包括系数（Coefficients）、标准误（Std. Errors）以及模型整体的拟合统计量。

1.2.2 参数解释 (Coefficients)

多项 Logit 模型通过估计对数几率 (Log-Odds) 来工作。在输出中，每一行代表一个类别相对于参照组（Reference Group）的比较。

参照组 (Reference Group): 在 multinom 的输出中，有一个类别会被自动设为基准（本例中未显示的类别通常是 beach）。所有的系数都是相对于这个基准类别的对数几率变化。
Intercept (截距): 当所有自变量为 0 时，该类别相对于参照组的对数几率。
income: 当收入增加一个单位时，选择该类别相对于参照组的对数几率的变化量。
- 例如，charter 的 income 系数为 $0.207$，表示收入越高，用户越倾向于选择 charter 而非 beach。
- pier 的 income 系数为 $-0.317$，表示收入越高，用户选择 pier 的概率相对于 beach 在下降。
income2: 这通常是收入的平方项，用于捕捉非线性关系（如 U 型或倒 U 型关系）。

1.2.3 输出结果解析

1.2.3.1 系数矩阵 (Coefficients)

该矩阵展示了每个备选项相对于基准组的估计系数 $\beta$。

charter: 收入对选择包船有正向影响，但随着收入进一步增加，income2 的负系数（$-0.022$）暗示这种增长趋势可能会放缓。
pier: 收入对其有显著的负向影响，说明高收入者更不容易选择在码头钓鱼。
private: 收入对其有正向影响，但系数略低于 charter。

1.2.3.2 标准误矩阵 (Std. Errors)

标准误用于评估参数估计的精确度。

你可以通过 $\frac{Coefficients}{Std.Errors}$ 计算 z 统计量。如果 z 值的绝对值大于 $1.96$，则通常认为该系数在 $5\%$ 的显著性水平下是显著的。

1.2.3.3 模型拟合统计量

Residual Deviance (残差离差): $2939.288$。衡量模型未解释的变异量，数值越小表示模型对样本数据的拟合越好。
AIC (赤池信息准则): $2957.288$。用于模型选择。AIC 考虑了模型复杂度和拟合度的平衡，在比较不同模型时，AIC 越小的模型越优。

1.2.4 关于平均效应 (AE) 的补充注释

虽然 summary() 输出的是原始系数（Log-odds），但在经济学和社会科学研究中，我们通常更关心平均边际效应 (Average Marginal Effects, AME/AE)。

原始系数的局限: 多项 Logit 的系数不直接等于概率的变化，只能告诉我们影响的方向和相对程度。
AE 的意义: AE 将这些系数转化为具体的概率变化。例如：“收入每增加一个单位，选择 charter 的概率平均增加几个百分点”。
计算方式: 在 R 中，通常在运行完 multinom 后，使用 margins 包或 effects 包来计算 AE。

注意: 在解释 income 和 income2 时，必须将两者结合看待。总的边际效应应该是 $\frac{\partial P}{\partial income}$，这涉及到对复合函数求导，结果会随 income 取值的不同而变化。

1.2.5 Interpretation of Specific Parameters

1.2.5.1 Intercepts

The intercepts represent the log-odds of choosing a specific mode (charter, pier, or private) over the reference mode (beach) when the predictors income and income2 are equal to zero.

1.2.5.2 Linear Income Effect

The coefficient for income indicates the direction and magnitude of the relationship between income levels and the choice of fishing mode:

Charter (0.207): A one-unit increase in income is associated with an increase in the log-odds of choosing a charter boat over beach fishing.
Pier (-0.317): A one-unit increase in income is associated with a decrease in the log-odds of choosing pier fishing over beach fishing. This suggests that as individuals become wealthier, they are less likely to choose the pier.
Private (0.150): Similar to charter, higher income increases the log-odds of choosing a private boat over beach fishing.

1.2.5.3 Quadratic Income Effect (income2)

The income2 term captures the non-linear relationship between income and choice.

For charter (-0.022) and private (-0.005), the negative quadratic terms suggest a “concave” relationship. While initial increases in income raise the probability of choosing these modes, the effect diminishes at higher income levels.
For pier (0.018), the positive quadratic term combined with the negative linear term suggests that the decrease in the probability of choosing the pier happens at a decreasing rate as income rises.

# statistical significance
z <- summary(mlogit)$coefficients/summary(mlogit)$standard.errors
z

##         (Intercept)    income    income2
## charter    2.709877  1.583910 -1.9813358
## pier       3.227442 -2.208939  1.5126387
## private    1.948037  1.186210 -0.5125039

# 2-tailed z test
p <- (1 - pnorm(abs(z), 0, 1)) * 2
p

##         (Intercept)     income    income2
## charter 0.006730807 0.11321416 0.04755363
## pier    0.001249021 0.02717891 0.13037147
## private 0.051410543 0.23553919 0.60829842

stargazer(mlogit, type = "text")

## 
## ===============================================
##                        Dependent variable:     
##                   -----------------------------
##                    charter    pier     private 
##                      (1)       (2)       (3)   
## -----------------------------------------------
## income              0.207   -0.317**    0.150  
##                    (0.131)   (0.144)   (0.126) 
##                                                
## income2           -0.022**    0.018    -0.005  
##                    (0.011)   (0.012)   (0.010) 
##                                                
## Constant          0.862***  1.101***   0.623*  
##                    (0.318)   (0.341)   (0.320) 
##                                                
## -----------------------------------------------
## Akaike Inf. Crit. 2,957.288 2,957.288 2,957.288
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01

# ********************************************
# The easy way to obtain marginal effects are
# presented in Exercise 1

这段 stargazer 的输出是对前面多项 Logit 模型（Multinomial Logit Model）的专业化整理。在学术论文或正式报告中，通常使用这种格式来呈现回归结果。

以下是对该表格的详细解析：

1.2.6 stargazer 代码解释

stargazer(...): 这是一个功能强大的 R 包，用于将回归模型转换为格式优美的表格。它支持将结果输出为 LaTeX、HTML 或纯文本。
mlogit: 传入的对象是我们之前拟合的多项 Logit 模型实例。
type = "text": 指定输出格式为纯文本（ASCII）。如果不指定，默认通常是 LaTeX 代码，适合直接复制到学术论文编辑器中。

1.2.7 参数解释

表格中的每一列（1、2、3）分别代表一个方程，描述了选择该类别相对于参照组（beach）的概率关系：

Dependent variable: 这里的因变量是 mode（钓鱼方式）。由于是多项模型，stargazer 将其拆分为三个对比：charter vs beach、pier vs beach 以及 private vs beach。
income & income2: 这些是自变量的系数估计值（Estimates）。
Standard Errors (括号中的数值): 位于系数下方的括号内。它们衡量了估计值的不确定性。标准误越小，估计越精确。
Constant: 对应于模型输出中的 (Intercept)，即当自变量为 0 时的对数几率。

1.2.8 输出结果深度解读

通过观察系数旁的星号，我们可以快速判断统计显著性：

Column (1) Charter:
- income 虽为正（0.207），但没有星号，说明在统计上不显著。
- income2 的系数为 -0.022**。两个星号表示在 $p < 0.05$ 水平下显著。这表明收入与选择包船（Charter）之间存在显著的负向二次项关系（即达到一定收入后，增加趋势减缓或下降）。
Column (2) Pier:
- income 的系数为 -0.317**。这说明收入水平对选择在码头钓鱼有显著的负面影响（相对于沙滩钓鱼）。收入越高，选择码头的概率显著降低。
Column (3) Private:
- income 和 income2 均没有星号。这意味着在当前模型设定下，没有足够的证据证明收入对选择私家船（Private）有显著的统计学影响。
Akaike Inf. Crit. (AIC):
- 三列下方显示的 AIC 数值相同（2,957.288），因为它们属于同一个整体模型。AIC 是评估模型优劣的指标，通常用于不同模型间的横向比较。

1.2.9 注释解释 (Note)

底部的注释定义了 P 值 (p-value) 的阈值，这是判断变量是否“重要”的学术标准：

*p < 0.1: 弱显著（$10\%$ 显著性水平）。
**p < 0.05: 显著（$5\%$ 显著性水平），通常是社会科学研究中接受的最低标准。
***p < 0.01: 极显著（$1\%$ 显著性水平）。

总结建议: 在回答考试问题时，应重点强调 Pier 组的 income 和 Charter 组的 income2 是模型中具有统计学意义的预测变量，而其他变量在统计上与参照组没有显著差异。

1.3 收入的平方项的解读

在经济学和统计学中，线性项（income）告诉我们一个趋势，而平方项（income2）告诉我们这个趋势的“弯曲程度”。

1.3.1 为什么要加平方项？

如果只有 income，模型会假设：你每多赚 1 块钱，想去钓鱼的欲望就会永远以同样的速度增加。但这不符合现实。

现实生活中，很多行为是“边际递减”或者“物极必反”的。平方项就是为了捕捉这种“曲线”关系。

1.3.2 两种常见的“弯曲”解释

你可以把 income2 想象成一个“调节器”，它决定了趋势线是会冲上云霄、慢慢变平，还是掉头向下。

1.3.2.1 1. 冲劲儿越来越小（倒 U 型 / 边际递减）

如果 income 是正的，income2 是负的（比如输出中的 Charter）：

直白解释：一开始你钱变多，你越来越想去包船（Charter）。但当你成了亿万富翁，收入再增加，你对包船的兴趣反而没那么大了（可能你去买私人飞机了，或者包船对你来说已经不再新鲜）。
图形：像一道彩虹（先升后降）。

1.3.2.2 2. 跌破底线后的平缓（U 型 / 触底反弹）

如果 income 是负的，income2 是正的（比如输出中的 Pier）：

直白解释：你稍微有点钱的时候，会非常嫌弃去码头（Pier）钓鱼，觉得那儿人多。所以收入增加一点，去码头的概率就暴跌。但当收入高到一定程度后，你已经基本不去码头了，再加钱，概率也不会再怎么跌了（已经到底了）。
图形：像一个山谷（先降后平或微升）。

1.3.3 结合你的数据来看

我们可以把这两个参数看作一对“搭档”：

对于 Charter（包船）：
- income (0.207) 是“油门”，income2 (-0.022) 是“刹车”。
- 这意味着：收入增加会推动你选包船，但随着你越来越富，这个推动力会越来越弱，甚至可能在某个收入点之后，你反而不想包船了。
对于 Pier（码头）：
- income (-0.317) 是“嫌弃”，income2 (0.018) 是“嫌弃到了极限”。
- 这意味着：你一有钱就想逃离码头。但因为有 income2 的正向抵消，说明这种“逃离”在收入很高时会变得不再那么剧烈。

1.3.4 总结一个考试万能金句

如果在考试中解释 income2，你可以这样写：

“The quadratic term (income2) allows the model to capture non-linear effects. It indicates whether the relationship between income and the choice probability is increasing at a decreasing rate (diminishing returns) or changing direction entirely at higher income levels.”

（平方项允许模型捕捉非线性效应。它展示了收入与选择概率之间的关系是在以递减的速度增长，还是在高收入水平下完全改变了方向。）

你觉得这种“油门与刹车”的比喻，是否让你对那个冷冰冰的二次项系数有了点画面感？

1.4 multinomial model with mlogit.data

# ------------------------------------------
# Exercise 1
# ------------------------------------------
# ------------------------------------------
# Exercise 3
# ------------------------------------------

# multinomial model
data("Fishing", package = "mlogit")
Fish <- mlogit.data(Fishing, shape="wide", choice="mode", varying=2:9)
Fish %>% as_tibble()

## # A tibble: 4,728 × 7
##    mode  income alt     price  catch  chid idx$chid $alt   
##    <lgl>  <dbl> <fct>   <dbl>  <dbl> <int>    <int> <fct>  
##  1 FALSE  7083. beach   158.  0.0678     1        1 beach  
##  2 FALSE  7083. boat    158.  0.260      1        1 boat   
##  3 TRUE   7083. charter 183.  0.539      1        1 charter
##  4 FALSE  7083. pier    158.  0.0503     1        1 pier   
##  5 FALSE  1250. beach    15.1 0.105      2        2 beach  
##  6 FALSE  1250. boat     10.5 0.157      2        2 boat   
##  7 TRUE   1250. charter  34.5 0.467      2        2 charter
##  8 FALSE  1250. pier     15.1 0.0451     2        2 pier   
##  9 FALSE  3750. beach   162.  0.533      3        3 beach  
## 10 TRUE   3750. boat     24.3 0.241      3        3 boat   
## # ℹ 4,718 more rows

## a pure "multinomial model"
options(scipen = 999)
mlogit1 = mlogit(mode ~ 0 | income, data = Fish)
summary(mlogit1)

## 
## Call:
## mlogit(formula = mode ~ 0 | income, data = Fish, method = "nr")
## 
## Frequencies of alternatives:choice
##   beach    boat charter    pier 
## 0.11337 0.35364 0.38240 0.15059 
## 
## nr method
## 4 iterations, 0h:0m:0s 
## g'(-H)^-1g = 8.32E-07 
## gradient close to zero 
## 
## Coefficients :
##                         Estimate   Std. Error z-value          Pr(>|z|)    
## (Intercept):boat     0.738920768  0.196730925  3.7560         0.0001727 ***
## (Intercept):charter  1.341291436  0.194516707  6.8955 0.000000000005367 ***
## (Intercept):pier     0.814150270  0.228631954  3.5610         0.0003695 ***
## income:boat          0.000091906  0.000040664  2.2602         0.0238116 *  
## income:charter      -0.000031640  0.000041846 -0.7561         0.4495908    
## income:pier         -0.000143403  0.000053288 -2.6911         0.0071223 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Log-Likelihood: -1477.2
## McFadden R^2:  0.013736 
## Likelihood ratio test : chisq = 41.145 (p.value = 0.0000000060931)

这段代码使用了 R 语言中专门用于离散选择分析的 mlogit 包。这与之前的 multinom 不同，mlogit 提供了更符合计量经济学标准的输出格式。

以下是对该模型及其输出的深度解析：

1.4.1 代码解释

mlogit(formula = mode ~ 0 | income, ...):
- mode: 因变量（选择的钓鱼方式）。
- 0 | income: 这是 mlogit 特有的公式语法。
  - 第一个部分（0）：表示没有备选项特有变量（Alternative-specific variables，如每种方式的价格或时间）。
  - 第二个部分（income）：表示存在个体特有变量（Individual-specific variables）。
- method = "nr": 指定使用 Newton-Raphson 算法进行数值优化。

1.4.2 参数解释

在 mlogit 中，所有的系数都是相对于基准组（beach）计算的。

(Intercept):[mode]:
- 这些是不同备选项的截距。
- 例如，(Intercept):charter 为 $1.3413$，表示在收入为 0 的情况下，选择 charter 相对于 beach 的基准对数几率。
income:[mode]:
- 这些系数表示收入对不同选择偏好的影响。
- income:boat ($9.1906 \times 10^{-5}$): 带有一个星号（*），在 $5\%$ 水平下显著。说明收入增加，选择 boat 的概率相对于 beach 显著增加。
- income:charter ($-3.1640 \times 10^{-5}$): P 值较大（0.449），不显著。说明在该模型设定下，收入对选择 charter 还是 beach 没有统计学上的显著差异。
- income:pier ($-1.4340 \times 10^{-4}$): 带有两个星号（**），非常显著且为负。说明收入越高，人们越不倾向于选择 pier。

1.4.3 输出结果解析

Frequencies of alternatives:
- 展示了样本中各选项的原始比例。例如，选择 charter 的人最多（$38.24\%$），选择 beach 的人最少（$11.33\%$）。
Scientific Notation (科学计数法):
- 注意输出中的 e-01 代表 $10^{-1}$，e-05 代表 $10^{-5}$。
- 比如 $9.1906 \times 10^{-5}$ 是一个很小的正数。
Log-Likelihood (-1477.2):
- 对数似然值。用于衡量模型对数据的拟合程度，通常用于计算似然比检验。
McFadden $R^2$ (0.013736):
- 伪 $R^2$。在离散选择模型中，这个值通常比较小。$0.013$ 说明收入这一个变量对解释钓鱼方式选择的贡献度有限。

1.4.4 注释与检验

Signif. codes (显著性代码):
- 这是 R 的标准标注。*** 代表极显著，$P < 0.001$；** 代表很显著，$P < 0.01$；* 代表显著，$P < 0.05$。
Likelihood ratio test (似然比检验):
- chisq = 41.145, p.value = 6.0931e-09。
- 结论：由于 P 值极小，我们==拒绝“收入没有影响”的原假设==。这说明将 income 引入模型是有统计学意义的，模型整体是有效的。
Optimization (优化信息):
- gradient close to zero: 表示算法已成功收敛，找到了似然函数的最大值点。

1.4.5 考试重点总结

在回答此类题目时，请务必提到：

Reference Group: 本模型的基准组是 beach。
Variable Type: income 被视为个体特有变量（==Individual-specific==），因此它在每个备选项方程中都有不同的系数。
Significant Predictors: 在该模型中，收入对 boat 和 pier 的选择有显著影响，但对 charter 的影响不显著。

stargazer(mlogit1, type = "text")

## 
## ===============================================
##                         Dependent variable:    
##                     ---------------------------
##                                mode            
## -----------------------------------------------
## (Intercept):boat             0.739***          
##                               (0.197)          
##                                                
## (Intercept):charter          1.341***          
##                               (0.195)          
##                                                
## (Intercept):pier             0.814***          
##                               (0.229)          
##                                                
## income:boat                  0.0001**          
##                              (0.00004)         
##                                                
## income:charter               -0.00003          
##                              (0.00004)         
##                                                
## income:pier                 -0.0001***         
##                              (0.0001)          
##                                                
## -----------------------------------------------
## Observations                   1,182           
## R2                             0.014           
## Log Likelihood              -1,477.151         
## LR Test                 41.145*** (df = 6)     
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01

1.4.6 参数解释

与 multinom 的分列展示不同，stargazer 在处理 mlogit 对象时，通常会将所有参数放在一个垂直列中，通过前缀区分：

(Intercept):[category]: 备选项的常数项（截距）。反映了在收入为 0 时，该选项相对于基准组（beach）的偏好。
income:[category]: 关键参数。反映了收入每增加一个单位，该选项相对于基准组的对数几率变化。
Standard Errors (括号中的数字): 下方的数字是标准误，用于计算显著性。

1.4.7 输出结果解析

系数的含义 (以 income 为例):
- income:boat (0.0001**): 系数为正，且有两个星号。这意味着随着收入增加，选择“船钓”的可能性相对于“岸钓”显著提高。
- income:pier (-0.0001***): 系数为负，且有三个星号。这意味着收入越高，选择“码头钓鱼”的可能性越低，且这种关联非常强（极显著）。
- income:charter (-0.00003): 没有星号。说明收入对选择“包船”的影响在统计上不显著（与 beach 相比没有显著差异）。
模型整体统计量:
- Observations (1,182): 样本量。共有 1182 个观察值。
- R2 (0.014): 这是 McFadden 伪 $R^2$。它衡量了模型解释力的提升。虽然数值看起来很小，但在离散选择模型中，大于 0.1 通常就被认为是很不错的拟合了。
- Log Likelihood (-1,477.151): 对数似然值，越接近 0 模型拟合越好。
- LR Test (41.145***): 似然比检验。三个星号表示该模型整体上显著优于“仅含截距”的空模型。

# compute a data.frame containing the mean value 
# of the covariates in the sample
z <- with(Fish, data.frame(income = 
                             tapply(income, index(mlogit1)$alt, mean)))
z

##           income
## beach   4099.337
## boat    4099.337
## charter 4099.337
## pier    4099.337

# compute the marginal effects
# impact of an addtional dollar
effects(mlogit1, covariate = "income", data = z)

##         beach          boat       charter          pier 
##  7.496226e-08  3.259851e-05 -1.201366e-05 -2.065981e-05

# impact of additional thousand dollars
1000*effects(mlogit1, covariate = "income",
             data = z)

##         beach          boat       charter          pier 
##  7.496226e-05  3.259851e-02 -1.201366e-02 -2.065981e-02

1.5 Hausman-McFadden 检验 IIA 假设

# independence from irrelevant alternatives assumption
mlogit1 = mlogit(mode ~ 0 | income, data = Fish, reflevel="beach")
mlogit2 = mlogit(mode ~ 0 | income, data = Fish, reflevel="beach", alt.subset=c("beach", "boat", "charter"))
mlogit3 = mlogit(mode ~ 0 | income, data = Fish, reflevel="beach", alt.subset=c("beach", "boat", "pier"))
mlogit4 = mlogit(mode ~ 0 | income, data = Fish, reflevel="beach", alt.subset=c("beach", "charter", "pier"))
mlogit5 = mlogit(mode ~ 0 | income, data = Fish, reflevel="pier")
mlogit6 = mlogit(mode ~ 0 | income, data = Fish, reflevel="pier", alt.subset=c("pier", "charter", "boat"))

# compute the test
hmftest(mlogit1, mlogit2)

## 
##  Hausman-McFadden test
## 
## data:  Fish
## chisq = 4.6145, df = 4, p-value = 0.3292
## alternative hypothesis: IIA is rejected

这段代码执行的是计量经济学中针对多项 Logit 模型（Multinomial Logit）最重要的诊断检验之一：Hausman-McFadden 检验。其目的是验证模型是否满足 IIA 假设（独立性无关备选项假设）。

1.5.1 代码解析

mlogit1: 这是全模型（Full Model）。它包含了所有可能的钓鱼方式（beach, boat, charter, pier）。我们将其作为基准。
mlogit2 和 mlogit3: 这是受限模型（Restricted Models）。
- 通过 alt.subset 参数，人为地删除了某些选项。
- 例如，mlogit2 只保留了 beach, boat, 和 charter，剔除了 pier。
hmftest(mlogit1, mlogit2): 这是核心函数。它对比了全模型和受限模型的系数。
- 逻辑：如果 IIA 假设成立，那么当你删掉一个选项（比如 pier）时，剩下选项（如 boat 和 beach）之间的相对偏好（系数）应该保持不变。

1.5.2 参数解释

reflevel = "beach": 明确指定“沙滩钓鱼”为对照组。在进行模型对比时，基准组必须保持一致，否则检验无意义。
alt.subset: 这是测试 IIA 的“手术刀”。它通过缩减备选集来观察模型参数是否稳健。
income: 依然作为个体特征变量。在 hmftest 中，程序会检查 income 对剩余选项的影响系数在两个模型间是否存在显著差异。

1.5.3 输出结果解析

这是考试中最关键的部分，你需要解读统计数据的含义：

chisq = 4.6145: 这是卡方统计量。它衡量了全模型系数与受限模型系数之间的“距离”。
df = 4: 自由度。通常对应于被比较的参数数量。
p-value = 0.3292: 这是最重要的判定指标。
- 在这里，$P = 0.3292 > 0.05$。
- 结论：我们无法拒绝原假设（$H_0$）。

1.5.4 注释与 IIA 假设解读

在报告或考试答案中，你应该这样描述这个结果：

什么是 IIA？ IIA (Independence of Irrelevant Alternatives) 假设认为，两个选项之间的相对概率不受第三个选项存在与否的影响（即经典的“红车/蓝车”问题）。
检验结论：由于 P 值（0.3292）显著大于常用的显著性水平（0.05），这表明：
1. 当你剔除一个钓鱼选项时，剩余选项的系数没有发生显著变化。
2. 模型满足 IIA 假设。
3. 这意味着使用多项 Logit 模型是合适的，不需要转而使用更复杂的模型（如嵌套 Logit 模型，Nested Logit）。
错误警示：输出中的 alternative hypothesis: IIA is rejected 只是告诉你备择假设是什么，并不代表结论是拒绝。实际结论取决于 P 值。在本例中，结论是 IIA Holds（IIA 成立）。

通俗比喻：这就像在问：“如果你在苹果和橘子之间选了苹果，那么当我把菜单上的香蕉拿走时，你会改变主意选橘子吗？”如果你的选择逻辑没变，就说明符合 IIA。本检验说明，钓鱼者在不同方式间的选择逻辑是相对独立的。

hmftest(mlogit1, mlogit3)

## 
##  Hausman-McFadden test
## 
## data:  Fish
## chisq = 14.701, df = 4, p-value = 0.005363
## alternative hypothesis: IIA is rejected

hmftest(mlogit1, mlogit4)

## 
##  Hausman-McFadden test
## 
## data:  Fish
## chisq = 0.095438, df = 4, p-value = 0.9989
## alternative hypothesis: IIA is rejected

hmftest(mlogit5, mlogit6)

## 
##  Hausman-McFadden test
## 
## data:  Fish
## chisq = 0.69605, df = 4, p-value = 0.9518
## alternative hypothesis: IIA is rejected

# !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
# difficult part here
# combine alternatives
# Wald test for combining alternatives
summary(mlogit1)

## 
## Call:
## mlogit(formula = mode ~ 0 | income, data = Fish, reflevel = "beach", 
##     method = "nr")
## 
## Frequencies of alternatives:choice
##   beach    boat charter    pier 
## 0.11337 0.35364 0.38240 0.15059 
## 
## nr method
## 4 iterations, 0h:0m:0s 
## g'(-H)^-1g = 8.32E-07 
## gradient close to zero 
## 
## Coefficients :
##                        Estimate  Std. Error z-value  Pr(>|z|)    
## (Intercept):boat     7.3892e-01  1.9673e-01  3.7560 0.0001727 ***
## (Intercept):charter  1.3413e+00  1.9452e-01  6.8955 5.367e-12 ***
## (Intercept):pier     8.1415e-01  2.2863e-01  3.5610 0.0003695 ***
## income:boat          9.1906e-05  4.0664e-05  2.2602 0.0238116 *  
## income:charter      -3.1640e-05  4.1846e-05 -0.7561 0.4495908    
## income:pier         -1.4340e-04  5.3288e-05 -2.6911 0.0071223 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Log-Likelihood: -1477.2
## McFadden R^2:  0.013736 
## Likelihood ratio test : chisq = 41.145 (p.value = 6.0931e-09)

coef(mlogit1)

##    (Intercept):boat (Intercept):charter    (Intercept):pier         income:boat 
##        7.389208e-01        1.341291e+00        8.141503e-01        9.190636e-05 
##      income:charter         income:pier 
##       -3.163988e-05       -1.434029e-04

vcov(mlogit1)

##                     (Intercept):boat (Intercept):charter (Intercept):pier
## (Intercept):boat        3.870306e-02        2.900879e-02     2.869945e-02
## (Intercept):charter     2.900879e-02        3.783675e-02     2.925105e-02
## (Intercept):pier        2.869945e-02        2.925105e-02     5.227257e-02
## income:boat            -6.895450e-06       -5.319592e-06    -5.248182e-06
## income:charter         -5.317341e-06       -7.021232e-06    -5.378442e-06
## income:pier            -5.232095e-06       -5.380896e-06    -1.050798e-05
##                       income:boat income:charter   income:pier
## (Intercept):boat    -6.895450e-06  -5.317341e-06 -5.232095e-06
## (Intercept):charter -5.319592e-06  -7.021232e-06 -5.380896e-06
## (Intercept):pier    -5.248182e-06  -5.378442e-06 -1.050798e-05
## income:boat          1.653540e-09   1.312826e-09  1.293147e-09
## income:charter       1.312826e-09   1.751113e-09  1.328287e-09
## income:pier          1.293147e-09   1.328287e-09  2.839655e-09

# let's test the hypothesis that private boat and charter boat 
# alternatives might be combined into one category
# i.e. non-constant variables' parameters in these categories
# are equal to themselves
#
# here: boat:beta_income=charter:beta_income
beta <- as.vector(t(coef(mlogit5))) 
beta

## [1] -0.8141502697 -0.0752295016  0.5271411661  0.0001434029  0.0002353093
## [6]  0.0001117630

A    <- c(0,0,0,1,0,-1)
A

## [1]  0  0  0  1  0 -1

# Wald test statistic
W.test = t(A %*% beta) %*% solve(A %*% vcov(mlogit1) %*% A) %*% A %*% beta
W.test

##           [,1]
## [1,] 0.5249786

# p-value from the asymptotic chi-square with 1-df
pchisq(W.test, 1, lower=FALSE)

##           [,1]
## [1,] 0.4687257

# we have to reject the null, that private boat and charter boat
# alternatives might be combined into one category

# let's test the hypothesis that private boat, charter boat and pier
# alternatives might be combined into one category
# i.e. non-constant variables' parameters in these categories
# are equal to themselves
#
# here: boat:beta_income=charter:beta_income=pier:beta_income
beta <- as.vector(t(coef(mlogit1))) 
A    <- rbind(c(0,0,0,1,-1,0), c(0,0,0,0,1,-1))

# Wald test statistic
W.test = t(A %*% beta) %*% solve(A %*% vcov(mlogit1) %*% t(A)) %*% A %*% beta
W.test

##          [,1]
## [1,] 37.48414

# p-value from the asymptotic chi-square with 1-df
pchisq(W.test, nrow(A), lower=FALSE)

##             [,1]
## [1,] 7.25141e-09

# Source:
# http://r.789695.n4.nabble.com/Multivariable-Wald-to-test-equality-of-multinomial-coefficients-td4725319.html
#
#
# the end of the difficult part
# !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

# 'impure' conditional logit
mlogit1 = mlogit(mode ~ price+catch | income, data = Fish)
summary(mlogit1)

## 
## Call:
## mlogit(formula = mode ~ price + catch | income, data = Fish, 
##     method = "nr")
## 
## Frequencies of alternatives:choice
##   beach    boat charter    pier 
## 0.11337 0.35364 0.38240 0.15059 
## 
## nr method
## 7 iterations, 0h:0m:0s 
## g'(-H)^-1g = 1.37E-05 
## successive function values within tolerance limits 
## 
## Coefficients :
##                        Estimate  Std. Error  z-value  Pr(>|z|)    
## (Intercept):boat     5.2728e-01  2.2279e-01   2.3667 0.0179485 *  
## (Intercept):charter  1.6944e+00  2.2405e-01   7.5624 3.952e-14 ***
## (Intercept):pier     7.7796e-01  2.2049e-01   3.5283 0.0004183 ***
## price               -2.5117e-02  1.7317e-03 -14.5042 < 2.2e-16 ***
## catch                3.5778e-01  1.0977e-01   3.2593 0.0011170 ** 
## income:boat          8.9440e-05  5.0067e-05   1.7864 0.0740345 .  
## income:charter      -3.3292e-05  5.0341e-05  -0.6613 0.5084031    
## income:pier         -1.2758e-04  5.0640e-05  -2.5193 0.0117582 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Log-Likelihood: -1215.1
## McFadden R^2:  0.18868 
## Likelihood ratio test : chisq = 565.17 (p.value = < 2.22e-16)

# marginal effects
z <- with(Fish, data.frame(price = tapply(price, index(mlogit1)$alt, mean),
                           catch = tapply(catch, index(mlogit1)$alt, mean),
                           income = tapply(income, index(mlogit1)$alt, mean)))
z

##             price     catch   income
## beach   103.42201 0.2410113 4099.337
## boat     55.25657 0.1712146 4099.337
## charter  84.37924 0.6293679 4099.337
## pier    103.42201 0.1622237 4099.337

# compute the marginal effects
# impact of an addtional dollar
effects(mlogit1, covariate = "income", data = z)

##         beach          boat       charter          pier 
## -7.214167e-07  3.176132e-05 -2.173391e-05 -9.305978e-06

# impact of additional thousand dollars
1000*effects(mlogit1, covariate = "income",
             data = z)

##         beach          boat       charter          pier 
## -0.0007214167  0.0317613158 -0.0217339147 -0.0093059782

# marginal effect for price
10*effects(mlogit1, covariate = "price", data = z)

##                 beach         boat      charter          pier
## beach   -0.0124912383  0.005531588  0.006091541  0.0008681094
## boat     0.0055315881 -0.061167595  0.048696270  0.0069397365
## charter  0.0060915419  0.048696271 -0.062430047  0.0076422349
## pier     0.0008681094  0.006939736  0.007642234 -0.0154500795

# marginal effect for catch rate
effects(mlogit1, covariate = "catch", data = z)

##                beach         boat      charter         pier
## beach    0.017793621 -0.007879681 -0.008677329 -0.001236612
## boat    -0.007879671  0.087132394 -0.069367164 -0.009885559
## charter -0.008677316 -0.069367154  0.088930726 -0.010886256
## pier    -0.001236612 -0.009885571 -0.010886272  0.022008455

# ------------------------------------------
# Exercise 2
# ------------------------------------------


# 'pure' conditional logit -- cola dataset
cola = read.csv(file='cola.csv', sep=",", header=TRUE)
cola %>% as_tibble()

## # A tibble: 1,822 × 13
##       id pepsi sevenup  coke pr_pepsi pr_7up pr_coke feat_pepsi feat_7up
##    <int> <int>   <int> <int>    <dbl>  <dbl>   <dbl>      <int>    <int>
##  1     1     0       0     1    1.79   1.79    1.79           0        0
##  2     2     0       0     1    1.79   1.79    0.890          0        0
##  3     3     0       0     1    1.41   0.840   0.890          0        0
##  4     4     0       0     1    1.79   1.79    1.33           0        0
##  5     5     0       0     1    1.79   1.79    1.79           0        0
##  6     6     0       0     1    0.990  1.79    1.79           1        0
##  7     7     0       0     1    0.770  1.79    1.79           1        0
##  8     8     0       0     1    1.33   0.990   1.79           1        0
##  9     9     0       0     1    1.79   1.79    0.990          0        0
## 10    10     0       0     1    1.79   1.79    1.29           0        0
## # ℹ 1,812 more rows
## # ℹ 4 more variables: feat_coke <int>, disp_pepsi <int>, disp_7up <int>,
## #   disp_coke <int>

# data preparation
cola$soda = 0
cola$soda[cola$pepsi==1] = 'pepsi'
cola$soda[cola$coke==1] = 'coke'
cola$soda[cola$sevenup==1] = 'sevenup'
cola %>% as_tibble()

## # A tibble: 1,822 × 14
##       id pepsi sevenup  coke pr_pepsi pr_7up pr_coke feat_pepsi feat_7up
##    <int> <int>   <int> <int>    <dbl>  <dbl>   <dbl>      <int>    <int>
##  1     1     0       0     1    1.79   1.79    1.79           0        0
##  2     2     0       0     1    1.79   1.79    0.890          0        0
##  3     3     0       0     1    1.41   0.840   0.890          0        0
##  4     4     0       0     1    1.79   1.79    1.33           0        0
##  5     5     0       0     1    1.79   1.79    1.79           0        0
##  6     6     0       0     1    0.990  1.79    1.79           1        0
##  7     7     0       0     1    0.770  1.79    1.79           1        0
##  8     8     0       0     1    1.33   0.990   1.79           1        0
##  9     9     0       0     1    1.79   1.79    0.990          0        0
## 10    10     0       0     1    1.79   1.79    1.29           0        0
## # ℹ 1,812 more rows
## # ℹ 5 more variables: feat_coke <int>, disp_pepsi <int>, disp_7up <int>,
## #   disp_coke <int>, soda <chr>

names(cola) = c("id","pepsi","sevenup","coke","price.pepsi","price.sevenup","price.coke",
                "feat.pepsi","feat.sevenup","feat.coke",
                "disp.pepsi","disp.sevenup","disp.coke","soda")
# names with "." are necessary for mlogit.data function

# mlogit.data
cola2 <- mlogit.data(cola, shape="wide", choice="soda", varying=5:13)
cola2 %>% as_tibble()

## # A tibble: 5,466 × 11
##       id pepsi sevenup  coke soda  alt     price  feat  disp  chid idx$chid
##    <int> <int>   <int> <int> <lgl> <fct>   <dbl> <int> <int> <int>    <int>
##  1     1     0       0     1 TRUE  coke    1.79      0     0     1        1
##  2     1     0       0     1 FALSE pepsi   1.79      0     0     1        1
##  3     1     0       0     1 FALSE sevenup 1.79      0     0     1        1
##  4     2     0       0     1 TRUE  coke    0.890     1     1     2        2
##  5     2     0       0     1 FALSE pepsi   1.79      0     0     2        2
##  6     2     0       0     1 FALSE sevenup 1.79      0     0     2        2
##  7     3     0       0     1 TRUE  coke    0.890     1     0     3        3
##  8     3     0       0     1 FALSE pepsi   1.41      0     0     3        3
##  9     3     0       0     1 FALSE sevenup 0.840     0     1     3        3
## 10     4     0       0     1 TRUE  coke    1.33      1     0     4        4
## # ℹ 5,456 more rows
## # ℹ 1 more variable: idx$alt <fct>

# varying - which variables are case-

# 'pure' conditional logit model
mlogit1 = mlogit(soda~feat+disp+price|0, data=cola2)
summary(mlogit1)

## 
## Call:
## mlogit(formula = soda ~ feat + disp + price | 0, data = cola2, 
##     method = "nr")
## 
## Frequencies of alternatives:choice
##    coke   pepsi sevenup 
## 0.27991 0.34577 0.37431 
## 
## nr method
## 4 iterations, 0h:0m:0s 
## g'(-H)^-1g = 0.000625 
## successive function values within tolerance limits 
## 
## Coefficients :
##        Estimate Std. Error z-value  Pr(>|z|)    
## feat  -0.010604   0.079937 -0.1326    0.8945    
## disp   0.462448   0.093048  4.9700 6.696e-07 ***
## price -1.744453   0.179932 -9.6950 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Log-Likelihood: -1822.2

# marginal effects
z <- with(cola2, data.frame(feat=tapply(feat, index(mlogit1)$alt, mean),
                            disp=tapply(disp, index(mlogit1)$alt, mean),
                            price=tapply(price, index(mlogit1)$alt, mean)))
z[,1:2] = 1
z

##         feat disp    price
## coke       1    1 1.210307
## pepsi      1    1 1.227453
## sevenup    1    1 1.117640

effects(mlogit1, covariate = "price", data = z)

##               coke      pepsi    sevenup
## coke    -0.3782432  0.1710623  0.2071809
## pepsi    0.1710623 -0.3721380  0.2010757
## sevenup  0.2071811  0.2010760 -0.4082570

# marginal effects calculated more "manually"
mlogit1$probabilities %>% as_tibble()

## # A tibble: 1,822 × 3
##      coke pepsi sevenup
##     <dbl> <dbl>   <dbl>
##  1 0.333  0.333  0.333 
##  2 0.791  0.105  0.105 
##  3 0.317  0.129  0.554 
##  4 0.525  0.238  0.238 
##  5 0.333  0.333  0.333 
##  6 0.120  0.760  0.120 
##  7 0.0884 0.823  0.0884
##  8 0.0916 0.321  0.587 
##  9 0.666  0.167  0.167 
## 10 0.542  0.229  0.229 
## # ℹ 1,812 more rows

cola3.1 = cola2
cola3.1$price[seq(from=1, to=5466, by=3)] = cola3.1$price[seq(from=1, to=5466, by=3)]+1
predict(mlogit1,newdata=cola3.1)

##             coke      pepsi    sevenup
## 1    0.080350098 0.45982495 0.45982495
## 2    0.397534545 0.30123273 0.30123273
## 3    0.074871195 0.17481183 0.75031698
## 4    0.161683466 0.41915827 0.41915827
## 5    0.080350098 0.45982495 0.45982495
## 6    0.023242707 0.84374468 0.13301262
## 7    0.016664757 0.88796671 0.09536853
## 8    0.017316378 0.34737241 0.63531121
## 9    0.258718204 0.37064090 0.37064090
## 10   0.171365972 0.41431701 0.41431701
## 11   0.013586410 0.49320679 0.49320679
## 12   0.013586410 0.49320679 0.49320679
## 13   0.013586410 0.49320679 0.49320679
## 14   0.161683466 0.41915827 0.41915827
## 15   0.161683466 0.41915827 0.41915827
## 16   0.080350098 0.45982495 0.45982495
## 17   0.080350098 0.45982495 0.45982495
## 18   0.080350098 0.45982495 0.45982495
## 19   0.024318666 0.48784067 0.48784067
## 20   0.080350098 0.45982495 0.45982495
## 21   0.037336968 0.74899223 0.21367080
## 22   0.152336024 0.11543298 0.73223100
## 23   0.030928745 0.79207323 0.17699802
## 24   0.030928745 0.79207323 0.17699802
## 25   0.234457995 0.38277100 0.38277100
## 26   0.015363358 0.19408232 0.79055432
## 27   0.015363358 0.19408232 0.79055432
## 28   0.074871195 0.17481183 0.75031698
## 29   0.074871195 0.17481183 0.75031698
## 30   0.042023678 0.51813971 0.43983661
## 31   0.030633788 0.11334576 0.85602045
## 32   0.030633788 0.11334576 0.85602045
## 33   0.030633788 0.11334576 0.85602045
## 34   0.045321179 0.18817732 0.76650150
## 35   0.080350098 0.45982495 0.45982495
## 36   0.080350098 0.45982495 0.45982495
## 37   0.144576208 0.45007517 0.40534862
## 38   0.144576208 0.45007517 0.40534862
## 39   0.144576208 0.45007517 0.40534862
## 40   0.161165757 0.68562363 0.15321061
## 41   0.027332103 0.21411558 0.75855232
## 42   0.080350098 0.45982495 0.45982495
## 43   0.064683226 0.57155109 0.36376569
## 44   0.064683226 0.57155109 0.36376569
## 45   0.075249898 0.63643821 0.28831189
## 46   0.307721115 0.41466243 0.27761646
## 47   0.093851947 0.53709290 0.36905516
## 48   0.149835952 0.45838271 0.39178134
## 49   0.064482485 0.26773666 0.66778085
## 50   0.042624328 0.41891081 0.53846486
## 51   0.042624328 0.41891081 0.53846486
## 52   0.045084381 0.56954223 0.38537339
## 53   0.198850184 0.47651135 0.32463847
## 54   0.080350098 0.45982495 0.45982495
## 55   0.080350098 0.45982495 0.45982495
## 56   0.051100997 0.65131354 0.29758546
## 57   0.087930616 0.40886289 0.50320649
## 58   0.087930616 0.40886289 0.50320649
## 59   0.068714472 0.55153165 0.37975388
## 60   0.068714472 0.55153165 0.37975388
## 61   0.029679628 0.59538339 0.37493698
## 62   0.029679628 0.59538339 0.37493698
## 63   0.034704650 0.75963721 0.20565814
## 64   0.034704650 0.75963721 0.20565814
## 65   0.017947594 0.33052816 0.65152424
## 66   0.088237918 0.50496511 0.40679697
## 67   0.088237918 0.50496511 0.40679697
## 68   0.083906161 0.52393846 0.39215538
## 69   0.083906161 0.52393846 0.39215538
## 70   0.043862762 0.55410975 0.40202749
## 71   0.043862762 0.55410975 0.40202749
## 72   0.043862762 0.40202749 0.55410975
## 73   0.080350098 0.45982495 0.45982495
## 74   0.080350098 0.45982495 0.45982495
## 75   0.080350098 0.45982495 0.45982495
## 76   0.070196109 0.63117892 0.29862497
## 77   0.070196109 0.63117892 0.29862497
## 78   0.070196109 0.63117892 0.29862497
## 79   0.033571229 0.77430842 0.19212035
## 80   0.122419908 0.67772002 0.19986007
## 81   0.017316378 0.34737241 0.63531121
## 82   0.070785699 0.18834289 0.74087142
## 83   0.033338273 0.24955202 0.71710971
## 84   0.051069920 0.24497444 0.70395564
## 85   0.051069920 0.24497444 0.70395564
## 86   0.008959086 0.11046293 0.88057799
## 87   0.021423942 0.08895400 0.88962206
## 88   0.008862530 0.12004989 0.87108758
## 89   0.096478131 0.55212193 0.35139994
## 90   0.080350098 0.45982495 0.45982495
## 91   0.080350098 0.45982495 0.45982495
## 92   0.076174689 0.47241772 0.45140759
## 93   0.036448738 0.46456145 0.49898981
## 94   0.052658258 0.67116169 0.27618005
## 95   0.107616154 0.61586226 0.27652159
## 96   0.079958161 0.45758199 0.46245985
## 97   0.084447375 0.48327272 0.43227991
## 98   0.040518070 0.24010820 0.71937373
## 99   0.041258422 0.46224254 0.49649904
## 100  0.096427620 0.31084352 0.59272886
## 101  0.031100335 0.59207284 0.37682683
## 102  0.039117426 0.74469826 0.21618432
## 103  0.039117426 0.74469826 0.21618432
## 104  0.073924951 0.43882722 0.48724783
## 105  0.078995895 0.46892894 0.45207516
## 106  0.102228604 0.58503056 0.31274084
## 107  0.080350098 0.45982495 0.45982495
## 108  0.080350098 0.45982495 0.45982495
## 109  0.161683466 0.41915827 0.41915827
## 110  0.161683466 0.41915827 0.41915827
## 111  0.397534545 0.30123273 0.30123273
## 112  0.234457995 0.38277100 0.38277100
## 113  0.074871195 0.17481183 0.75031698
## 114  0.065468744 0.28536061 0.64917065
## 115  0.161165757 0.68562363 0.15321061
## 116  0.027332103 0.21411558 0.75855232
## 117  0.064683226 0.57155109 0.36376569
## 118  0.093851947 0.53709290 0.36905516
## 119  0.068714472 0.55153165 0.37975388
## 120  0.034704650 0.75963721 0.20565814
## 121  0.088237918 0.50496511 0.40679697
## 122  0.083906161 0.52393846 0.39215538
## 123  0.080350098 0.45982495 0.45982495
## 124  0.053070209 0.47718881 0.46974098
## 125  0.111637164 0.24948927 0.63887356
## 126  0.080350098 0.45982495 0.45982495
## 127  0.039117426 0.74469826 0.21618432
## 128  0.078995895 0.46892894 0.45207516
## 129  0.148350805 0.31131469 0.54033451
## 130  0.044777257 0.36323254 0.59199020
## 131  0.032297254 0.37392551 0.59377723
## 132  0.043226451 0.24737494 0.70939861
## 133  0.051009348 0.29191465 0.65707600
## 134  0.051009348 0.29191465 0.65707600
## 135  0.079109587 0.14340341 0.77748700
## 136  0.125321577 0.71718628 0.15749214
## 137  0.030503563 0.17456481 0.79493163
## 138  0.026490229 0.15159743 0.82191234
## 139  0.080350098 0.45982495 0.45982495
## 140  0.080350098 0.45982495 0.45982495
## 141  0.022332765 0.48883362 0.48883362
## 142  0.030191536 0.17462099 0.79518747
## 143  0.017972690 0.10285353 0.87917378
## 144  0.023861930 0.14389309 0.83224498
## 145  0.017258152 0.13852126 0.84422059
## 146  0.028045691 0.61388173 0.35807258
## 147  0.073481623 0.42051826 0.50600012
## 148  0.108660605 0.27920537 0.61213403
## 149  0.054578689 0.31783761 0.62758371
## 150  0.180868788 0.27538062 0.54375060
## 151  0.048534146 0.38955568 0.56191017
## 152  0.041490455 0.39648890 0.56202064
## 153  0.063250024 0.36196520 0.57478478
## 154  0.062447480 0.56749166 0.37006086
## 155  0.080350098 0.45982495 0.45982495
## 156  0.073481623 0.42051826 0.50600012
## 157  0.073481623 0.42051826 0.50600012
## 158  0.147993927 0.42600304 0.42600304
## 159  0.032297254 0.37392551 0.59377723
## 160  0.094220810 0.45288959 0.45288959
## 161  0.050182882 0.17198279 0.77783432
## 162  0.080350098 0.45982495 0.45982495
## 163  0.079705140 0.46416086 0.45613400
## 164  0.080994153 0.46351073 0.45549512
## 165  0.037925668 0.59188287 0.37019146
## 166  0.073481624 0.42051827 0.50600011
## 167  0.069045717 0.46547714 0.46547714
## 168  0.068704802 0.46317883 0.46811637
## 169  0.094220810 0.45288959 0.45288959
## 170  0.034186067 0.23046807 0.73534587
## 171  0.077124669 0.22200467 0.70087066
## 172  0.054979957 0.22433368 0.72068636
## 173  0.054979957 0.22433368 0.72068636
## 174  0.086508274 0.56051364 0.35297808
## 175  0.067379745 0.27492831 0.65769195
## 176  0.046976233 0.31669397 0.63632979
## 177  0.079750478 0.27331452 0.64693500
## 178  0.037683003 0.44093049 0.52138651
## 179  0.029204270 0.52762850 0.44316723
## 180  0.147993927 0.42600304 0.42600304
## 181  0.077124669 0.22200467 0.70087066
## 182  0.022523165 0.49300121 0.48447562
## 183  0.079531142 0.14416757 0.77630129
## 184  0.030503563 0.79493163 0.17456481
## 185  0.023643804 0.51753047 0.45882573
## 186  0.023643804 0.51753047 0.45882573
## 187  0.054979957 0.22433368 0.72068636
## 188  0.219166487 0.39041676 0.39041676
## 189  0.112312544 0.40907323 0.47861422
## 190  0.219166487 0.39041676 0.39041676
## 191  0.057774742 0.19800102 0.74422424
## 192  0.055432934 0.31388378 0.63068329
## 193  0.077124669 0.22200467 0.70087066
## 194  0.096837052 0.55417596 0.34898699
## 195  0.026476377 0.48676181 0.48676181
## 196  0.080350098 0.45982495 0.45982495
## 197  0.048534146 0.38955568 0.56191017
## 198  0.048534146 0.38955568 0.56191017
## 199  0.054979957 0.22433368 0.72068636
## 200  0.045930481 0.36865756 0.58541196
## 201  0.038275552 0.25803765 0.70368680
## 202  0.094220810 0.45288959 0.45288959
## 203  0.086508274 0.56051364 0.35297808
## 204  0.219166487 0.39041676 0.39041676
## 205  0.161947958 0.45327638 0.38477566
## 206  0.250474164 0.37476292 0.37476292
## 207  0.024644545 0.44524873 0.53010673
## 208  0.109163604 0.44541820 0.44541820
## 209  0.109163604 0.44541820 0.44541820
## 210  0.093319743 0.45334013 0.45334013
## 211  0.097406418 0.55155468 0.35103890
## 212  0.057774742 0.19800102 0.74422424
## 213  0.057774742 0.19800102 0.74422424
## 214  0.103883548 0.18831162 0.70780483
## 215  0.056368924 0.22517466 0.71845641
## 216  0.147993927 0.42600304 0.42600304
## 217  0.054643635 0.63264354 0.31271283
## 218  0.080350098 0.45982495 0.45982495
## 219  0.026490229 0.15159743 0.82191234
## 220  0.080994153 0.46351073 0.45549512
## 221  0.022332765 0.48883362 0.48883362
## 222  0.096837052 0.55417596 0.34898699
## 223  0.123911221 0.16697364 0.70911514
## 224  0.097103121 0.29655217 0.60634471
## 225  0.080350098 0.45982495 0.45982495
## 226  0.080350098 0.45982495 0.45982495
## 227  0.080994153 0.46351073 0.45549512
## 228  0.080994153 0.46351073 0.45549512
## 229  0.108677872 0.62856815 0.26275398
## 230  0.108677872 0.62856815 0.26275398
## 231  0.048534146 0.38955568 0.56191017
## 232  0.054979957 0.22433368 0.72068636
## 233  0.067379745 0.27492831 0.65769195
## 234  0.067379745 0.27492831 0.65769195
## 235  0.161947958 0.45327638 0.38477566
## 236  0.250474164 0.37476292 0.37476292
## 237  0.032297254 0.37392551 0.59377723
## 238  0.219166487 0.39041676 0.39041676
## 239  0.038275552 0.25803765 0.70368680
## 240  0.103105588 0.18690140 0.70999301
## 241  0.147993927 0.42600304 0.42600304
## 242  0.040898225 0.43782899 0.52127278
## 243  0.133428744 0.28796543 0.57860582
## 244  0.024644545 0.44524873 0.53010673
## 245  0.109163604 0.44541820 0.44541820
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## 568  0.122419908 0.67772002 0.19986007
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## 572  0.046019941 0.56741246 0.38656760
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## 583  0.032297254 0.37392551 0.59377723
## 584  0.063419569 0.42754802 0.50903241
## 585  0.026490229 0.15159743 0.82191234
## 586  0.080350098 0.45982495 0.45982495
## 587  0.022332765 0.48883362 0.48883362
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## 589  0.013586410 0.49320679 0.49320679
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## 592  0.045321179 0.18817732 0.76650150
## 593  0.052658258 0.67116169 0.27618005
## 594  0.080350098 0.45982495 0.45982495
## 595  0.069045717 0.46547714 0.46547714
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## 598  0.030503563 0.79493163 0.17456481
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## 600  0.093319743 0.45334013 0.45334013
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## 603  0.096837052 0.55417596 0.34898699
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## 605  0.026476377 0.48676181 0.48676181
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## 607  0.079682824 0.44812046 0.47219671
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## 609  0.032297254 0.37392551 0.59377723
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## 618  0.054643635 0.63264354 0.31271283
## 619  0.074560574 0.52050269 0.40493674
## 620  0.080350098 0.45982495 0.45982495
## 621  0.080994153 0.46351073 0.45549512
## 622  0.081137141 0.45943143 0.45943143
## 623  0.081137141 0.45943143 0.45943143
## 624  0.108677872 0.62856815 0.26275398
## 625  0.022332765 0.48883362 0.48883362
## 626  0.067800321 0.54419419 0.38800549
## 627  0.080350098 0.45982495 0.45982495
## 628  0.173857308 0.41307135 0.41307135
## 629  0.030191536 0.17462099 0.79518747
## 630  0.062447480 0.56749166 0.37006086
## 631  0.077372214 0.44278321 0.47984457
## 632  0.048534146 0.38955568 0.56191017
## 633  0.034186067 0.23046807 0.73534587
## 634  0.055432934 0.31388378 0.63068329
## 635  0.022332765 0.48883362 0.48883362
## 636  0.068569197 0.40912708 0.52230372
## 637  0.073481623 0.42051826 0.50600012
## 638  0.066306551 0.37289495 0.56079850
## 639  0.149070113 0.35417891 0.49675098
## 640  0.085605386 0.21996457 0.69443004
## 641  0.044784487 0.69892430 0.25629122
## 642  0.035003823 0.48249809 0.48249809
## 643  0.051100997 0.65131354 0.29758546
## 644  0.080350098 0.45982495 0.45982495
## 645  0.052658258 0.67116169 0.27618005
## 646  0.080350098 0.45982495 0.45982495
## 647  0.028935604 0.53197410 0.43909029
## 648  0.079109587 0.14340341 0.77748700
## 649  0.080350098 0.45982495 0.45982495
## 650  0.079705140 0.46416086 0.45613400
## 651  0.030191536 0.17462099 0.79518747
## 652  0.250474164 0.37476292 0.37476292
## 653  0.067409269 0.54682314 0.38576759
## 654  0.040028517 0.22907391 0.73089758
## 655  0.103883548 0.18831162 0.70780483
## 656  0.028500818 0.22875968 0.74273951
## 657  0.028500818 0.22875968 0.74273951
## 658  0.066306551 0.37289495 0.56079850
## 659  0.248833130 0.37230758 0.37885929
## 660  0.035003823 0.48249809 0.48249809
## 661  0.031023699 0.17447115 0.79450515
## 662  0.073692887 0.30068770 0.62561941
## 663  0.080350098 0.45982495 0.45982495
## 664  0.080350098 0.45982495 0.45982495
## 665  0.096837052 0.34898699 0.55417596
## 666  0.034951107 0.20001707 0.76503183
## 667  0.028500818 0.22875968 0.74273951
## 668  0.063250024 0.36196520 0.57478478
## 669  0.026476377 0.48676181 0.48676181
## 670  0.079705140 0.46416086 0.45613400
## 671  0.027023125 0.15464707 0.81832981
## 672  0.080350098 0.45982495 0.45982495
## 673  0.073692887 0.30068770 0.62561941
## 674  0.028500818 0.22875968 0.74273951
## 675  0.073481623 0.42051826 0.50600012
## 676  0.058159490 0.47502772 0.46681279
## 677  0.080350098 0.45982495 0.45982495
## 678  0.051009348 0.29191465 0.65707600
## 679  0.080350098 0.45982495 0.45982495
## 680  0.123911221 0.16697364 0.70911514
## 681  0.097103121 0.29655217 0.60634471
## 682  0.097103121 0.29655217 0.60634471
## 683  0.080350098 0.45982495 0.45982495
## 684  0.080350098 0.45982495 0.45982495
## 685  0.080994153 0.46351073 0.45549512
## 686  0.080350098 0.45982495 0.45982495
## 687  0.081137141 0.45943143 0.45943143
## 688  0.191156318 0.56861675 0.24022693
## 689  0.035003823 0.48249809 0.48249809
## 690  0.022332765 0.48883362 0.48883362
## 691  0.073481623 0.42051826 0.50600012
## 692  0.073481623 0.42051826 0.50600012
## 693  0.149070113 0.35417891 0.49675098
## 694  0.044784487 0.69892430 0.25629122
## 695  0.063250024 0.36196520 0.57478478
## 696  0.063250024 0.36196520 0.57478478
## 697  0.117589807 0.20947104 0.67293916
## 698  0.117589807 0.20947104 0.67293916
## 699  0.080350098 0.45982495 0.45982495
## 700  0.080350098 0.45982495 0.45982495
## 701  0.081137141 0.45943143 0.45943143
## 702  0.081137141 0.45943143 0.45943143
## 703  0.250474164 0.37476292 0.37476292
## 704  0.250474164 0.37476292 0.37476292
## 705  0.079705140 0.46416086 0.45613400
## 706  0.032297254 0.37392551 0.59377723
## 707  0.086508274 0.56051364 0.35297808
## 708  0.048988513 0.28034988 0.67066161
## 709  0.075729367 0.61853272 0.30573792
## 710  0.125321577 0.71718628 0.15749214
## 711  0.030503563 0.17456481 0.79493163
## 712  0.026490229 0.15159743 0.82191234
## 713  0.080350098 0.45982495 0.45982495
## 714  0.080350098 0.45982495 0.45982495
## 715  0.081137141 0.45943143 0.45943143
## 716  0.081137141 0.45943143 0.45943143
## 717  0.022332765 0.48883362 0.48883362
## 718  0.022332765 0.48883362 0.48883362
## 719  0.191156318 0.56861675 0.24022693
## 720  0.030503563 0.79493163 0.17456481
## 721  0.030503563 0.17456481 0.79493163
## 722  0.096837052 0.34898699 0.55417596
## 723  0.022332765 0.48883362 0.48883362
## 724  0.035003823 0.48249809 0.48249809
## 725  0.035003823 0.48249809 0.48249809
## 726  0.035003823 0.48249809 0.48249809
## 727  0.033338273 0.24955202 0.71710971
## 728  0.080350098 0.45982495 0.45982495
## 729  0.064683226 0.57155109 0.36376569
## 730  0.155500811 0.32631900 0.51818019
## 731  0.044302270 0.35558885 0.60010888
## 732  0.048979298 0.19984931 0.75117139
## 733  0.056950957 0.27666336 0.66638568
## 734  0.105217079 0.18545367 0.70932925
## 735  0.150207069 0.42489647 0.42489647
## 736  0.063077290 0.42524052 0.51168219
## 737  0.030106486 0.32229987 0.64759365
## 738  0.103883548 0.18831162 0.70780483
## 739  0.147993927 0.42600304 0.42600304
## 740  0.040898225 0.43782899 0.52127278
## 741  0.029204270 0.52762850 0.44316723
## 742  0.125321577 0.71718628 0.15749214
## 743  0.030503563 0.79493163 0.17456481
## 744  0.035003823 0.48249809 0.48249809
## 745  0.117850189 0.20772055 0.67442926
## 746  0.075729367 0.61853272 0.30573792
## 747  0.097103121 0.29655217 0.60634471
## 748  0.080350098 0.45982495 0.45982495
## 749  0.081137141 0.45943143 0.45943143
## 750  0.081137141 0.45943143 0.45943143
## 751  0.250474164 0.37476292 0.37476292
## 752  0.068569197 0.40912708 0.52230372
## 753  0.081137141 0.45943143 0.45943143
## 754  0.109163604 0.44541820 0.44541820
## 755  0.028935604 0.53197410 0.43909029
## 756  0.079570029 0.46021499 0.46021499
## 757  0.066306551 0.37289495 0.56079850
## 758  0.026476377 0.48676181 0.48676181
## 759  0.080350098 0.45982495 0.45982495
## 760  0.073692887 0.30068770 0.62561941
## 761  0.044302270 0.35558885 0.60010888
## 762  0.173857308 0.41307135 0.41307135
## 763  0.112312544 0.40907323 0.47861422
## 764  0.161947958 0.45327638 0.38477566
## 765  0.063077290 0.42524052 0.51168219
## 766  0.155500811 0.32631900 0.51818019
## 767  0.051009348 0.29191465 0.65707600
## 768  0.022332765 0.48883362 0.48883362
## 769  0.080994153 0.46351073 0.45549512
## 770  0.080994153 0.46351073 0.45549512
## 771  0.080350098 0.45982495 0.45982495
## 772  0.108677872 0.62856815 0.26275398
## 773  0.030503563 0.79493163 0.17456481
## 774  0.035003823 0.48249809 0.48249809
## 775  0.034186067 0.23046807 0.73534587
## 776  0.028431485 0.23063585 0.74093267
## 777  0.044784487 0.69892430 0.25629122
## 778  0.044784487 0.69892430 0.25629122
## 779  0.048534146 0.38955568 0.56191017
## 780  0.107773524 0.26461010 0.62761638
## 781  0.107773524 0.26461010 0.62761638
## 782  0.051009348 0.29191465 0.65707600
## 783  0.079705140 0.46416086 0.45613400
## 784  0.023643804 0.51753047 0.45882573
## 785  0.079570029 0.46021499 0.46021499
## 786  0.067379745 0.27492831 0.65769195
## 787  0.093319743 0.45334013 0.45334013
## 788  0.068704802 0.46317883 0.46811637
## 789  0.068704802 0.46317883 0.46811637
## 790  0.046976233 0.31669397 0.63632979
## 791  0.046976233 0.31669397 0.63632979
## 792  0.079750478 0.27331452 0.64693500
## 793  0.074270222 0.50069855 0.42503123
## 794  0.074270222 0.50069855 0.42503123
## 795  0.030191536 0.17462099 0.79518747
## 796  0.105217079 0.18545367 0.70932925
## 797  0.103105588 0.18690140 0.70999301
## 798  0.068569197 0.40912708 0.52230372
## 799  0.068569197 0.40912708 0.52230372
## 800  0.066306551 0.37289495 0.56079850
## 801  0.147993927 0.42600304 0.42600304
## 802  0.079705140 0.46416086 0.45613400
## 803  0.035003823 0.48249809 0.48249809
## 804  0.048979298 0.19984931 0.75117139
## 805  0.029204270 0.52762850 0.44316723
## 806  0.080350098 0.45982495 0.45982495
## 807  0.080350098 0.45982495 0.45982495
## 808  0.067340523 0.39215582 0.54050365
## 809  0.085605386 0.21996457 0.69443004
## 810  0.058159490 0.47502772 0.46681279
## 811  0.030410146 0.17709271 0.79249714
## 812  0.026490229 0.15159743 0.82191234
## 813  0.073481623 0.42051826 0.50600012
## 814  0.217671003 0.38775275 0.39457625
## 815  0.079109587 0.14340341 0.77748700
## 816  0.027023125 0.15464707 0.81832981
## 817  0.123911221 0.16697364 0.70911514
## 818  0.125321577 0.71718628 0.15749214
## 819  0.030503563 0.17456481 0.79493163
## 820  0.028500818 0.22875968 0.74273951
## 821  0.073481623 0.42051826 0.50600012
## 822  0.073481623 0.42051826 0.50600012
## 823  0.080350098 0.45982495 0.45982495
## 824  0.080350098 0.45982495 0.45982495
## 825  0.080350098 0.45982495 0.45982495
## 826  0.079109587 0.14340341 0.77748700
## 827  0.058641224 0.47067939 0.47067939
## 828  0.058641224 0.47067939 0.47067939
## 829  0.046976233 0.31669397 0.63632979
## 830  0.029204270 0.52762850 0.44316723
## 831  0.080350098 0.45982495 0.45982495
## 832  0.080350098 0.45982495 0.45982495
## 833  0.217671003 0.38775275 0.39457625
## 834  0.123911221 0.16697364 0.70911514
## 835  0.080350098 0.45982495 0.45982495
## 836  0.075173286 0.49462746 0.43019926
## 837  0.022332765 0.48883362 0.48883362
## 838  0.022332765 0.48883362 0.48883362
## 839  0.073481623 0.42051826 0.50600012
## 840  0.073481623 0.42051826 0.50600012
## 841  0.058159490 0.47502772 0.46681279
## 842  0.059122446 0.47454205 0.46633551
## 843  0.034186067 0.23046807 0.73534587
## 844  0.039602217 0.42395480 0.53644298
## 845  0.216170009 0.38507892 0.39875107
## 846  0.032297254 0.37392551 0.59377723
## 847  0.117850189 0.20772055 0.67442926
## 848  0.043226451 0.24737494 0.70939861
## 849  0.026490229 0.15159743 0.82191234
## 850  0.081137141 0.45943143 0.45943143
## 851  0.080350098 0.45982495 0.45982495
## 852  0.080350098 0.45982495 0.45982495
## 853  0.030191536 0.17462099 0.79518747
## 854  0.030503563 0.17456481 0.79493163
## 855  0.096837052 0.34898699 0.55417596
## 856  0.073481623 0.42051826 0.50600012
## 857  0.073481623 0.42051826 0.50600012
## 858  0.063250024 0.36196520 0.57478478
## 859  0.048534146 0.38955568 0.56191017
## 860  0.044302270 0.35558885 0.60010888
## 861  0.048979298 0.19984931 0.75117139
## 862  0.105217079 0.18545367 0.70932925
## 863  0.096837052 0.55417596 0.34898699
## 864  0.086508274 0.56051364 0.35297808
## 865  0.086508274 0.56051364 0.35297808
## 866  0.109163604 0.44541820 0.44541820
## 867  0.155500811 0.32631900 0.51818019
## 868  0.109163604 0.44541820 0.44541820
## 869  0.109163604 0.44541820 0.44541820
## 870  0.044302270 0.35558885 0.60010888
## 871  0.048979298 0.19984931 0.75117139
## 872  0.057614909 0.27988879 0.66249630
## 873  0.080994153 0.46351073 0.45549512
## 874  0.081137141 0.45943143 0.45943143
## 875  0.108677872 0.62856815 0.26275398
## 876  0.035003823 0.48249809 0.48249809
## 877  0.147993927 0.42600304 0.42600304
## 878  0.103883548 0.18831162 0.70780483
## 879  0.127319742 0.43634013 0.43634013
## 880  0.031898454 0.48405077 0.48405077
## 881  0.069266780 0.53433553 0.39639769
## 882  0.023643804 0.51753047 0.45882573
## 883  0.035003823 0.48249809 0.48249809
## 884  0.079705140 0.46416086 0.45613400
## 885  0.045930481 0.36865756 0.58541196
## 886  0.045930481 0.58541196 0.36865756
## 887  0.045930481 0.58541196 0.36865756
## 888  0.109163604 0.44541820 0.44541820
## 889  0.044302270 0.35558885 0.60010888
## 890  0.027023125 0.15464707 0.81832981
## 891  0.069045717 0.46547714 0.46547714
## 892  0.069045717 0.46547714 0.46547714
## 893  0.093319743 0.45334013 0.45334013
## 894  0.061051470 0.35553168 0.58341685
## 895  0.093319743 0.45334013 0.45334013
## 896  0.093319743 0.45334013 0.45334013
## 897  0.095017685 0.26878015 0.63620216
## 898  0.097406418 0.55155468 0.35103890
## 899  0.067340523 0.39215582 0.54050365
## 900  0.028500818 0.22875968 0.74273951
## 901  0.028431485 0.23063585 0.74093267
## 902  0.147993927 0.42600304 0.42600304
## 903  0.103883548 0.18831162 0.70780483
## 904  0.031898454 0.48405077 0.48405077
## 905  0.125332291 0.20461476 0.67005295
## 906  0.080350098 0.45982495 0.45982495
## 907  0.080350098 0.45982495 0.45982495
## 908  0.087930616 0.40886289 0.50320649
## 909  0.026490229 0.15159743 0.82191234
## 910  0.191156318 0.56861675 0.24022693
## 911  0.034951107 0.20001707 0.76503183
## 912  0.056950957 0.27666336 0.66638568
## 913  0.074270222 0.50069855 0.42503123
## 914  0.074270222 0.50069855 0.42503123
## 915  0.096837052 0.55417596 0.34898699
## 916  0.030928745 0.79207323 0.17699802
## 917  0.065468744 0.28536061 0.64917065
## 918  0.307721115 0.41466243 0.27761646
## 919  0.149835952 0.45838271 0.39178134
## 920  0.051100997 0.65131354 0.29758546
## 921  0.240333639 0.40629432 0.35337204
## 922  0.031159736 0.18465115 0.78418912
## 923  0.074871195 0.17481183 0.75031698
## 924  0.040898225 0.43782899 0.52127278
## 925  0.107773524 0.26461010 0.62761638
## 926  0.109958814 0.23731271 0.65272847
## 927  0.219166487 0.39041676 0.39041676
## 928  0.219166487 0.39041676 0.39041676
## 929  0.125321577 0.71718628 0.15749214
## 930  0.125321577 0.71718628 0.15749214
## 931  0.080350098 0.45982495 0.45982495
## 932  0.080994153 0.46351073 0.45549512
## 933  0.080350098 0.45982495 0.45982495
## 934  0.081137141 0.45943143 0.45943143
## 935  0.081137141 0.45943143 0.45943143
## 936  0.108677872 0.62856815 0.26275398
## 937  0.108677872 0.62856815 0.26275398
## 938  0.022332765 0.48883362 0.48883362
## 939  0.250474164 0.37476292 0.37476292
## 940  0.191156318 0.56861675 0.24022693
## 941  0.250474164 0.37476292 0.37476292
## 942  0.125321577 0.71718628 0.15749214
## 943  0.125321577 0.71718628 0.15749214
## 944  0.096837052 0.34898699 0.55417596
## 945  0.096837052 0.34898699 0.55417596
## 946  0.035003823 0.48249809 0.48249809
## 947  0.035003823 0.48249809 0.48249809
## 948  0.034951107 0.20001707 0.76503183
## 949  0.073481623 0.42051826 0.50600012
## 950  0.149070113 0.35417891 0.49675098
## 951  0.028431485 0.23063585 0.74093267
## 952  0.063250024 0.36196520 0.57478478
## 953  0.030191536 0.17462099 0.79518747
## 954  0.028935604 0.53197410 0.43909029
## 955  0.079705140 0.46416086 0.45613400
## 956  0.026476377 0.48676181 0.48676181
## 957  0.026476377 0.48676181 0.48676181
## 958  0.079682824 0.44812046 0.47219671
## 959  0.032297254 0.37392551 0.59377723
## 960  0.109163604 0.44541820 0.44541820
## 961  0.217671003 0.38775275 0.39457625
## 962  0.217671003 0.38775275 0.39457625
## 963  0.081137141 0.45943143 0.45943143
## 964  0.075173286 0.49462746 0.43019926
## 965  0.250474164 0.37476292 0.37476292
## 966  0.030503563 0.79493163 0.17456481
## 967  0.067340523 0.39215582 0.54050365
## 968  0.032297254 0.37392551 0.59377723
## 969  0.035003823 0.48249809 0.48249809
## 970  0.092566460 0.52973635 0.37769719
## 971  0.149070113 0.35417891 0.49675098
## 972  0.248833130 0.37230758 0.37885929
## 973  0.080350098 0.45982495 0.45982495
## 974  0.121281048 0.18465568 0.69406327
## 975  0.067409257 0.38576752 0.54682322
## 976  0.069266780 0.53433553 0.39639769
## 977  0.058641224 0.47067939 0.47067939
## 978  0.058641224 0.47067939 0.47067939
## 979  0.080350098 0.45982495 0.45982495
## 980  0.058159490 0.47502772 0.46681279
## 981  0.038275552 0.25803765 0.70368680
## 982  0.107773524 0.26461010 0.62761638
## 983  0.079682824 0.44812046 0.47219671
## 984  0.066306551 0.37289495 0.56079850
## 985  0.028500818 0.22875968 0.74273951
## 986  0.030410146 0.17709271 0.79249714
## 987  0.026476377 0.48676181 0.48676181
## 988  0.080350098 0.45982495 0.45982495
## 989  0.044784487 0.69892430 0.25629122
## 990  0.030503563 0.17456481 0.79493163
## 991  0.026490229 0.15159743 0.82191234
## 992  0.080350098 0.45982495 0.45982495
## 993  0.026476377 0.48676181 0.48676181
## 994  0.079682824 0.44812046 0.47219671
## 995  0.080350098 0.45982495 0.45982495
## 996  0.217671003 0.38775275 0.39457625
## 997  0.027023125 0.15464707 0.81832981
## 998  0.080994153 0.46351073 0.45549512
## 999  0.031023699 0.17447115 0.79450515
## 1000 0.030503563 0.17456481 0.79493163
## 1001 0.035003823 0.48249809 0.48249809
## 1002 0.068569197 0.40912708 0.52230372
## 1003 0.067018592 0.38353184 0.54944957
## 1004 0.121281048 0.18465568 0.69406327
## 1005 0.048534146 0.38955568 0.56191017
## 1006 0.051009348 0.29191465 0.65707600
## 1007 0.051009348 0.29191465 0.65707600
## 1008 0.125321577 0.71718628 0.15749214
## 1009 0.026490229 0.15159743 0.82191234
## 1010 0.080350098 0.45982495 0.45982495
## 1011 0.081137141 0.45943143 0.45943143
## 1012 0.081137141 0.45943143 0.45943143
## 1013 0.108677872 0.62856815 0.26275398
## 1014 0.108677872 0.62856815 0.26275398
## 1015 0.108677872 0.62856815 0.26275398
## 1016 0.250474164 0.37476292 0.37476292
## 1017 0.250474164 0.37476292 0.37476292
## 1018 0.250474164 0.37476292 0.37476292
## 1019 0.030191536 0.17462099 0.79518747
## 1020 0.030191536 0.17462099 0.79518747
## 1021 0.048534146 0.38955568 0.56191017
## 1022 0.059922380 0.36134615 0.57873147
## 1023 0.022332765 0.48883362 0.48883362
## 1024 0.028500818 0.22875968 0.74273951
## 1025 0.028431485 0.23063585 0.74093267
## 1026 0.063250024 0.36196520 0.57478478
## 1027 0.030410146 0.17709271 0.79249714
## 1028 0.039828694 0.73224094 0.22793037
## 1029 0.109163604 0.44541820 0.44541820
## 1030 0.109163604 0.44541820 0.44541820
## 1031 0.069045717 0.46547714 0.46547714
## 1032 0.046976233 0.31669397 0.63632979
## 1033 0.054209625 0.58033192 0.36545846
## 1034 0.022332765 0.48883362 0.48883362
## 1035 0.034186067 0.23046807 0.73534587
## 1036 0.034186067 0.23046807 0.73534587
## 1037 0.080350098 0.45982495 0.45982495
## 1038 0.062447480 0.56749166 0.37006086
## 1039 0.094220810 0.45288959 0.45288959
## 1040 0.077372214 0.44278321 0.47984457
## 1041 0.077372214 0.44278321 0.47984457
## 1042 0.030106486 0.32229987 0.64759365
## 1043 0.103883548 0.18831162 0.70780483
## 1044 0.073481623 0.42051826 0.50600012
## 1045 0.041490455 0.39648890 0.56202064
## 1046 0.024644545 0.44524873 0.53010673
## 1047 0.103883548 0.18831162 0.70780483
## 1048 0.123911221 0.16697364 0.70911514
## 1049 0.077372214 0.44278321 0.47984457
## 1050 0.068569197 0.40912708 0.52230372
## 1051 0.250474164 0.37476292 0.37476292
## 1052 0.056368924 0.22517466 0.71845641
## 1053 0.046976233 0.31669397 0.63632979
## 1054 0.150207069 0.42489647 0.42489647
## 1055 0.109163604 0.44541820 0.44541820
## 1056 0.054979957 0.22433368 0.72068636
## 1057 0.117589807 0.20947104 0.67293916
## 1058 0.107773524 0.26461010 0.62761638
## 1059 0.096837052 0.55417596 0.34898699
## 1060 0.080350098 0.45982495 0.45982495
## 1061 0.081137141 0.45943143 0.45943143
## 1062 0.075173286 0.49462746 0.43019926
## 1063 0.048058928 0.38574138 0.56619969
## 1064 0.113362345 0.23789128 0.64874637
## 1065 0.028431485 0.23063585 0.74093267
## 1066 0.053813863 0.43193293 0.51425320
## 1067 0.155500811 0.32631900 0.51818019
## 1068 0.219166487 0.39041676 0.39041676
## 1069 0.123911221 0.16697364 0.70911514
## 1070 0.048058928 0.38574138 0.56619969
## 1071 0.035003823 0.48249809 0.48249809
## 1072 0.022332765 0.48883362 0.48883362
## 1073 0.080350098 0.45982495 0.45982495
## 1074 0.027023125 0.15464707 0.81832981
## 1075 0.066306551 0.37289495 0.56079850
## 1076 0.045667296 0.69298938 0.26134333
## 1077 0.079109587 0.14340341 0.77748700
## 1078 0.075173286 0.49462746 0.43019926
## 1079 0.096837052 0.34898699 0.55417596
## 1080 0.035003823 0.48249809 0.48249809
## 1081 0.068569197 0.40912708 0.52230372
## 1082 0.073481623 0.42051826 0.50600012
## 1083 0.092566460 0.52973635 0.37769719
## 1084 0.080350098 0.45982495 0.45982495
## 1085 0.058159490 0.47502772 0.46681279
## 1086 0.063250024 0.36196520 0.57478478
## 1087 0.105217079 0.18545367 0.70932925
## 1088 0.080350098 0.45982495 0.45982495
## 1089 0.039828694 0.73224094 0.22793037
## 1090 0.125321577 0.71718628 0.15749214
## 1091 0.030503563 0.17456481 0.79493163
## 1092 0.067340523 0.39215582 0.54050365
## 1093 0.096837052 0.34898699 0.55417596
## 1094 0.080350098 0.45982495 0.45982495
## 1095 0.035003823 0.48249809 0.48249809
## 1096 0.113362345 0.23789128 0.64874637
## 1097 0.028500818 0.22875968 0.74273951
## 1098 0.028431485 0.23063585 0.74093267
## 1099 0.048534146 0.38955568 0.56191017
## 1100 0.048534146 0.38955568 0.56191017
## 1101 0.109163604 0.44541820 0.44541820
## 1102 0.109163604 0.44541820 0.44541820
## 1103 0.086508274 0.56051364 0.35297808
## 1104 0.086508274 0.56051364 0.35297808
## 1105 0.045930481 0.36865756 0.58541196
## 1106 0.109163604 0.44541820 0.44541820
## 1107 0.109163604 0.44541820 0.44541820
## 1108 0.109163604 0.44541820 0.44541820
## 1109 0.058641224 0.47067939 0.47067939
## 1110 0.079570029 0.46021499 0.46021499
## 1111 0.079570029 0.46021499 0.46021499
## 1112 0.048979298 0.19984931 0.75117139
## 1113 0.048979298 0.19984931 0.75117139
## 1114 0.069045717 0.46547714 0.46547714
## 1115 0.068704802 0.46811637 0.46317883
## 1116 0.093319743 0.45334013 0.45334013
## 1117 0.093319743 0.45334013 0.45334013
## 1118 0.093319743 0.45334013 0.45334013
## 1119 0.046976233 0.31669397 0.63632979
## 1120 0.054209625 0.58033192 0.36545846
## 1121 0.079750478 0.27331452 0.64693500
## 1122 0.074270222 0.50069855 0.42503123
## 1123 0.173857308 0.41307135 0.41307135
## 1124 0.173857308 0.41307135 0.41307135
## 1125 0.161947958 0.45327638 0.38477566
## 1126 0.250474164 0.37476292 0.37476292
## 1127 0.150207069 0.42489647 0.42489647
## 1128 0.063077290 0.42524052 0.51168219
## 1129 0.094220810 0.45288959 0.45288959
## 1130 0.188097464 0.40595127 0.40595127
## 1131 0.034186067 0.23046807 0.73534587
## 1132 0.057774742 0.19800102 0.74422424
## 1133 0.039602217 0.42395480 0.53644298
## 1134 0.046976233 0.31669397 0.63632979
## 1135 0.056368924 0.22517466 0.71845641
## 1136 0.147993927 0.42600304 0.42600304
## 1137 0.150207069 0.42489647 0.42489647
## 1138 0.188097464 0.40595127 0.40595127
## 1139 0.077124669 0.22200467 0.70087066
## 1140 0.041490455 0.39648890 0.56202064
## 1141 0.103883548 0.18831162 0.70780483
## 1142 0.029204270 0.52762850 0.44316723
## 1143 0.121281048 0.18465568 0.69406327
## 1144 0.023643804 0.51753047 0.45882573
## 1145 0.079531142 0.14416757 0.77630129
## 1146 0.079705140 0.46416086 0.45613400
## 1147 0.051009348 0.29191465 0.65707600
## 1148 0.073481623 0.42051826 0.50600012
## 1149 0.073481623 0.42051826 0.50600012
## 1150 0.080350098 0.45982495 0.45982495
## 1151 0.048534146 0.38955568 0.56191017
## 1152 0.096837052 0.55417596 0.34898699
## 1153 0.096837052 0.55417596 0.34898699
## 1154 0.026476377 0.48676181 0.48676181
## 1155 0.117850189 0.20772055 0.67442926
## 1156 0.079705140 0.46416086 0.45613400
## 1157 0.125321577 0.71718628 0.15749214
## 1158 0.064291511 0.36156276 0.57414573
## 1159 0.096837052 0.55417596 0.34898699
## 1160 0.081648616 0.45917569 0.45917569
## 1161 0.027023125 0.15464707 0.81832981
## 1162 0.077372214 0.44278321 0.47984457
## 1163 0.027023125 0.15464707 0.81832981
## 1164 0.109958814 0.23731271 0.65272847
## 1165 0.079705140 0.46416086 0.45613400
## 1166 0.216170009 0.38507892 0.39875107
## 1167 0.032297254 0.37392551 0.59377723
## 1168 0.217671003 0.38775275 0.39457625
## 1169 0.027023125 0.15464707 0.81832981
## 1170 0.109958814 0.23731271 0.65272847
## 1171 0.080350098 0.45982495 0.45982495
## 1172 0.075729367 0.61853272 0.30573792
## 1173 0.074560574 0.52050269 0.40493674
## 1174 0.080350098 0.45982495 0.45982495
## 1175 0.080350098 0.45982495 0.45982495
## 1176 0.051895619 0.29698657 0.65111781
## 1177 0.052254491 0.29904031 0.64870519
## 1178 0.068569197 0.40912708 0.52230372
## 1179 0.024644545 0.44524873 0.53010673
## 1180 0.067340523 0.39215582 0.54050365
## 1181 0.043226451 0.24737494 0.70939861
## 1182 0.075173286 0.49462746 0.43019926
## 1183 0.080350098 0.45982495 0.45982495
## 1184 0.096837052 0.34898699 0.55417596
## 1185 0.096837052 0.34898699 0.55417596
## 1186 0.035003823 0.48249809 0.48249809
## 1187 0.061051470 0.35553168 0.58341685
## 1188 0.045930481 0.58541196 0.36865756
## 1189 0.155500811 0.32631900 0.51818019
## 1190 0.150207069 0.42489647 0.42489647
## 1191 0.103105588 0.18690140 0.70999301
## 1192 0.041490455 0.39648890 0.56202064
## 1193 0.217671003 0.38775275 0.39457625
## 1194 0.149070113 0.35417891 0.49675098
## 1195 0.080350098 0.45982495 0.45982495
## 1196 0.096837052 0.55417596 0.34898699
## 1197 0.079705140 0.46416086 0.45613400
## 1198 0.039828694 0.73224094 0.22793037
## 1199 0.027023125 0.15464707 0.81832981
## 1200 0.080350098 0.45982495 0.45982495
## 1201 0.081137141 0.45943143 0.45943143
## 1202 0.063419569 0.42754802 0.50903241
## 1203 0.022332765 0.48883362 0.48883362
## 1204 0.040898225 0.43782899 0.52127278
## 1205 0.079705140 0.46416086 0.45613400
## 1206 0.079682824 0.44812046 0.47219671
## 1207 0.109958814 0.23731271 0.65272847
## 1208 0.397534545 0.30123273 0.30123273
## 1209 0.080350098 0.45982495 0.45982495
## 1210 0.051009348 0.29191465 0.65707600
## 1211 0.022332765 0.48883362 0.48883362
## 1212 0.079705140 0.46416086 0.45613400
## 1213 0.022332765 0.48883362 0.48883362
## 1214 0.077124669 0.22200467 0.70087066
## 1215 0.096837052 0.55417596 0.34898699
## 1216 0.080350098 0.45982495 0.45982495
## 1217 0.080350098 0.45982495 0.45982495
## 1218 0.081137141 0.45943143 0.45943143
## 1219 0.105217079 0.18545367 0.70932925
## 1220 0.063077290 0.42524052 0.51168219
## 1221 0.022332765 0.48883362 0.48883362
## 1222 0.073481623 0.42051826 0.50600012
## 1223 0.080350098 0.45982495 0.45982495
## 1224 0.125321577 0.71718628 0.15749214
## 1225 0.125321577 0.71718628 0.15749214
## 1226 0.125321577 0.71718628 0.15749214
## 1227 0.081137141 0.45943143 0.45943143
## 1228 0.031023699 0.17447115 0.79450515
## 1229 0.030191536 0.17462099 0.79518747
## 1230 0.030191536 0.17462099 0.79518747
## 1231 0.030191536 0.17462099 0.79518747
## 1232 0.250474164 0.37476292 0.37476292
## 1233 0.073481623 0.42051826 0.50600012
## 1234 0.073481623 0.42051826 0.50600012
## 1235 0.080350098 0.45982495 0.45982495
## 1236 0.080350098 0.45982495 0.45982495
## 1237 0.051009348 0.29191465 0.65707600
## 1238 0.073692887 0.30068770 0.62561941
## 1239 0.113362345 0.23789128 0.64874637
## 1240 0.053813863 0.43193293 0.51425320
## 1241 0.044784487 0.69892430 0.25629122
## 1242 0.030410146 0.17709271 0.79249714
## 1243 0.086508274 0.56051364 0.35297808
## 1244 0.034951107 0.20001707 0.76503183
## 1245 0.066306551 0.37289495 0.56079850
## 1246 0.092566460 0.52973635 0.37769719
## 1247 0.149070113 0.35417891 0.49675098
## 1248 0.028431485 0.23063585 0.74093267
## 1249 0.248833130 0.37230758 0.37885929
## 1250 0.044784487 0.69892430 0.25629122
## 1251 0.030410146 0.17709271 0.79249714
## 1252 0.080350098 0.45982495 0.45982495
## 1253 0.117589807 0.20947104 0.67293916
## 1254 0.032297254 0.37392551 0.59377723
## 1255 0.039828694 0.73224094 0.22793037
## 1256 0.039828694 0.73224094 0.22793037
## 1257 0.054643635 0.63264354 0.31271283
## 1258 0.080350098 0.45982495 0.45982495
## 1259 0.026490229 0.15159743 0.82191234
## 1260 0.080994153 0.46351073 0.45549512
## 1261 0.081137141 0.45943143 0.45943143
## 1262 0.031023699 0.17447115 0.79450515
## 1263 0.030191536 0.17462099 0.79518747
## 1264 0.068569197 0.40912708 0.52230372
## 1265 0.068569197 0.40912708 0.52230372
## 1266 0.066306551 0.37289495 0.56079850
## 1267 0.067409257 0.38576752 0.54682322
## 1268 0.080350098 0.45982495 0.45982495
## 1269 0.080350098 0.45982495 0.45982495
## 1270 0.014402679 0.05422804 0.93136928
## 1271 0.028935604 0.53197410 0.43909029
## 1272 0.117850189 0.20772055 0.67442926
## 1273 0.043226451 0.24737494 0.70939861
## 1274 0.123911221 0.16697364 0.70911514
## 1275 0.026490229 0.15159743 0.82191234
## 1276 0.080350098 0.45982495 0.45982495
## 1277 0.081137141 0.45943143 0.45943143
## 1278 0.030191536 0.17462099 0.79518747
## 1279 0.030503563 0.17456481 0.79493163
## 1280 0.035003823 0.48249809 0.48249809
## 1281 0.034951107 0.20001707 0.76503183
## 1282 0.113362345 0.23789128 0.64874637
## 1283 0.035419706 0.20269875 0.76188154
## 1284 0.013586410 0.49320679 0.49320679
## 1285 0.161683466 0.41915827 0.41915827
## 1286 0.020026320 0.25298861 0.72698507
## 1287 0.161683466 0.41915827 0.41915827
## 1288 0.078864570 0.20445341 0.71668202
## 1289 0.080350098 0.45982495 0.45982495
## 1290 0.080350098 0.45982495 0.45982495
## 1291 0.080350098 0.45982495 0.45982495
## 1292 0.234457995 0.38277100 0.38277100
## 1293 0.024318666 0.48784067 0.48784067
## 1294 0.397534545 0.30123273 0.30123273
## 1295 0.037878388 0.48361129 0.47851032
## 1296 0.159366265 0.58045585 0.26017788
## 1297 0.152336024 0.11543298 0.73223100
## 1298 0.234457995 0.38277100 0.38277100
## 1299 0.042023678 0.51813971 0.43983661
## 1300 0.080350098 0.45982495 0.45982495
## 1301 0.161165757 0.68562363 0.15321061
## 1302 0.027332103 0.21411558 0.75855232
## 1303 0.034441443 0.19710038 0.76845818
## 1304 0.064683226 0.57155109 0.36376569
## 1305 0.182294844 0.29761058 0.52009458
## 1306 0.075249898 0.63643821 0.28831189
## 1307 0.149835952 0.45838271 0.39178134
## 1308 0.064482485 0.26773666 0.66778085
## 1309 0.049716726 0.43541667 0.51486660
## 1310 0.198850184 0.47651135 0.32463847
## 1311 0.068714472 0.55153165 0.37975388
## 1312 0.029679628 0.59538339 0.37493698
## 1313 0.240333639 0.40629432 0.35337204
## 1314 0.034704650 0.75963721 0.20565814
## 1315 0.017947594 0.33052816 0.65152424
## 1316 0.088237918 0.50496511 0.40679697
## 1317 0.043862762 0.55410975 0.40202749
## 1318 0.080350098 0.45982495 0.45982495
## 1319 0.106040736 0.28711276 0.60684651
## 1320 0.033571229 0.77430842 0.19212035
## 1321 0.017316378 0.34737241 0.63531121
## 1322 0.009522759 0.05449654 0.93598070
## 1323 0.043580992 0.59033859 0.36608042
## 1324 0.076174689 0.47241772 0.45140759
## 1325 0.053070209 0.47718881 0.46974098
## 1326 0.127916296 0.33161816 0.54046555
## 1327 0.080350098 0.45982495 0.45982495
## 1328 0.034905616 0.13142439 0.83367000
## 1329 0.258718204 0.37064090 0.37064090
## 1330 0.084447375 0.48327272 0.43227991
## 1331 0.079682824 0.44812046 0.47219671
## 1332 0.074391197 0.10657317 0.81903563
## 1333 0.080350098 0.45982495 0.45982495
## 1334 0.122618558 0.17566391 0.70171753
## 1335 0.042023678 0.51813971 0.43983661
## 1336 0.080350098 0.45982495 0.45982495
## 1337 0.080350098 0.45982495 0.45982495
## 1338 0.258718204 0.37064090 0.37064090
## 1339 0.087930616 0.40886289 0.50320649
## 1340 0.023242707 0.84374468 0.13301262
## 1341 0.063250024 0.36196520 0.57478478
## 1342 0.022019899 0.79935493 0.17862517
## 1343 0.033802882 0.19344604 0.77275108
## 1344 0.026310977 0.95512741 0.01856161
## 1345 0.293551731 0.35322413 0.35322413
## 1346 0.030503563 0.79493163 0.17456481
## 1347 0.125321577 0.71718628 0.15749214
## 1348 0.397534545 0.30123273 0.30123273
## 1349 0.397534545 0.30123273 0.30123273
## 1350 0.397534545 0.30123273 0.30123273
## 1351 0.037336968 0.74899223 0.21367080
## 1352 0.037336968 0.74899223 0.21367080
## 1353 0.397534545 0.30123273 0.30123273
## 1354 0.152336024 0.11543298 0.73223100
## 1355 0.152336024 0.11543298 0.73223100
## 1356 0.152336024 0.11543298 0.73223100
## 1357 0.152336024 0.11543298 0.73223100
## 1358 0.074871195 0.17481183 0.75031698
## 1359 0.074871195 0.17481183 0.75031698
## 1360 0.074871195 0.17481183 0.75031698
## 1361 0.065468744 0.28536061 0.64917065
## 1362 0.065468744 0.28536061 0.64917065
## 1363 0.030633788 0.11334576 0.85602045
## 1364 0.080350098 0.45982495 0.45982495
## 1365 0.094264980 0.24180082 0.66393420
## 1366 0.182294844 0.29761058 0.52009458
## 1367 0.046265732 0.77953750 0.17419677
## 1368 0.062712794 0.10959484 0.82769237
## 1369 0.149835952 0.45838271 0.39178134
## 1370 0.149835952 0.45838271 0.39178134
## 1371 0.146966447 0.58708093 0.26595262
## 1372 0.240333639 0.40629432 0.35337204
## 1373 0.088237918 0.50496511 0.40679697
## 1374 0.088237918 0.50496511 0.40679697
## 1375 0.070196109 0.63117892 0.29862497
## 1376 0.017316378 0.34737241 0.63531121
## 1377 0.080350098 0.45982495 0.45982495
## 1378 0.076174689 0.47241772 0.45140759
## 1379 0.080350098 0.45982495 0.45982495
## 1380 0.031159736 0.18465115 0.78418912
## 1381 0.217671003 0.38775275 0.39457625
## 1382 0.027023125 0.15464707 0.81832981
## 1383 0.026490229 0.15159743 0.82191234
## 1384 0.031023699 0.17447115 0.79450515
## 1385 0.030191536 0.17462099 0.79518747
## 1386 0.125321577 0.71718628 0.15749214
## 1387 0.248833130 0.37230758 0.37885929
## 1388 0.080350098 0.45982495 0.45982495
## 1389 0.121281048 0.18465568 0.69406327
## 1390 0.080350098 0.45982495 0.45982495
## 1391 0.258718204 0.37064090 0.37064090
## 1392 0.027332103 0.21411558 0.75855232
## 1393 0.080350098 0.45982495 0.45982495
## 1394 0.080350098 0.45982495 0.45982495
## 1395 0.356590393 0.32170480 0.32170480
## 1396 0.356590393 0.32170480 0.32170480
## 1397 0.439965201 0.28001740 0.28001740
## 1398 0.123763909 0.17543460 0.70080149
## 1399 0.224048813 0.32097337 0.45497782
## 1400 0.054209623 0.36545846 0.58033192
## 1401 0.059122446 0.47454205 0.46633551
## 1402 0.080350098 0.45982495 0.45982495
## 1403 0.080350098 0.45982495 0.45982495
## 1404 0.080350098 0.45982495 0.45982495
## 1405 0.017589275 0.22220190 0.76020883
## 1406 0.033802882 0.77275108 0.19344604
## 1407 0.258718204 0.37064090 0.37064090
## 1408 0.017188068 0.09836332 0.88444861
## 1409 0.054209623 0.36545846 0.58033192
## 1410 0.042023678 0.51813971 0.43983661
## 1411 0.017589275 0.22220190 0.76020883
## 1412 0.033802882 0.19344604 0.77275108
## 1413 0.078995895 0.46892894 0.45207516
## 1414 0.065468744 0.28536061 0.64917065
## 1415 0.161165757 0.68562363 0.15321061
## 1416 0.064683226 0.57155109 0.36376569
## 1417 0.051100997 0.65131354 0.29758546
## 1418 0.020026320 0.25298861 0.72698507
## 1419 0.152336024 0.11543298 0.73223100
## 1420 0.080350098 0.45982495 0.45982495
## 1421 0.080350098 0.45982495 0.45982495
## 1422 0.080350098 0.45982495 0.45982495
## 1423 0.051069920 0.24497444 0.70395564
## 1424 0.111637164 0.24948927 0.63887356
## 1425 0.080350098 0.45982495 0.45982495
## 1426 0.080350098 0.45982495 0.45982495
## 1427 0.080350098 0.45982495 0.45982495
## 1428 0.015363358 0.19408232 0.79055432
## 1429 0.074871195 0.17481183 0.75031698
## 1430 0.080350098 0.45982495 0.45982495
## 1431 0.080350098 0.45982495 0.45982495
## 1432 0.163954739 0.18556697 0.65047829
## 1433 0.028935604 0.53197410 0.43909029
## 1434 0.161683466 0.41915827 0.41915827
## 1435 0.161683466 0.41915827 0.41915827
## 1436 0.109163604 0.44541820 0.44541820
## 1437 0.109163604 0.44541820 0.44541820
## 1438 0.080350098 0.45982495 0.45982495
## 1439 0.080350098 0.45982495 0.45982495
## 1440 0.080350098 0.45982495 0.45982495
## 1441 0.080350098 0.45982495 0.45982495
## 1442 0.065468744 0.28536061 0.64917065
## 1443 0.065468744 0.28536061 0.64917065
## 1444 0.042023678 0.51813971 0.43983661
## 1445 0.044302270 0.35558885 0.60010888
## 1446 0.045321179 0.18817732 0.76650150
## 1447 0.094220810 0.45288959 0.45288959
## 1448 0.161165757 0.68562363 0.15321061
## 1449 0.161165757 0.68562363 0.15321061
## 1450 0.080350098 0.45982495 0.45982495
## 1451 0.149835952 0.45838271 0.39178134
## 1452 0.079750478 0.27331452 0.64693500
## 1453 0.038275552 0.25803765 0.70368680
## 1454 0.038275552 0.25803765 0.70368680
## 1455 0.030106486 0.32229987 0.64759365
## 1456 0.040028517 0.22907391 0.73089758
## 1457 0.039602217 0.42395480 0.53644298
## 1458 0.103883548 0.18831162 0.70780483
## 1459 0.046976233 0.31669397 0.63632979
## 1460 0.056368924 0.22517466 0.71845641
## 1461 0.056368924 0.22517466 0.71845641
## 1462 0.040898225 0.43782899 0.52127278
## 1463 0.045592238 0.51872162 0.43568614
## 1464 0.041490455 0.39648890 0.56202064
## 1465 0.024644545 0.44524873 0.53010673
## 1466 0.031159736 0.18465115 0.78418912
## 1467 0.024644545 0.44524873 0.53010673
## 1468 0.023643804 0.51753047 0.45882573
## 1469 0.023643804 0.51753047 0.45882573
## 1470 0.080350098 0.45982495 0.45982495
## 1471 0.042023678 0.51813971 0.43983661
## 1472 0.074871195 0.17481183 0.75031698
## 1473 0.104275293 0.15876368 0.73696103
## 1474 0.131148629 0.11831823 0.75053314
## 1475 0.045321179 0.18817732 0.76650150
## 1476 0.122618558 0.17566391 0.70171753
## 1477 0.080350098 0.45982495 0.45982495
## 1478 0.307721115 0.41466243 0.27761646
## 1479 0.080350098 0.45982495 0.45982495
## 1480 0.096837052 0.55417596 0.34898699
## 1481 0.123911221 0.16697364 0.70911514
## 1482 0.125321577 0.71718628 0.15749214
## 1483 0.067409257 0.38576752 0.54682322
## 1484 0.080350098 0.45982495 0.45982495
## 1485 0.080350098 0.45982495 0.45982495
## 1486 0.030633788 0.11334576 0.85602045
## 1487 0.080350098 0.45982495 0.45982495
## 1488 0.307721115 0.41466243 0.27761646
## 1489 0.080350098 0.45982495 0.45982495
## 1490 0.096837052 0.55417596 0.34898699
## 1491 0.234457995 0.38277100 0.38277100
## 1492 0.080350098 0.45982495 0.45982495
## 1493 0.051009348 0.29191465 0.65707600
## 1494 0.079109587 0.14340341 0.77748700
## 1495 0.144576208 0.45007517 0.40534862
## 1496 0.144576208 0.45007517 0.40534862
## 1497 0.075729367 0.61853272 0.30573792
## 1498 0.046265732 0.77953750 0.17419677
## 1499 0.149835952 0.45838271 0.39178134
## 1500 0.080350098 0.45982495 0.45982495
## 1501 0.080350098 0.45982495 0.45982495
## 1502 0.087930616 0.40886289 0.50320649
## 1503 0.067409269 0.54682314 0.38576759
## 1504 0.043862762 0.55410975 0.40202749
## 1505 0.035003823 0.48249809 0.48249809
## 1506 0.022332765 0.48883362 0.48883362
## 1507 0.028500818 0.22875968 0.74273951
## 1508 0.053813863 0.43193293 0.51425320
## 1509 0.017589275 0.22220190 0.76020883
## 1510 0.031159736 0.18465115 0.78418912
## 1511 0.127916296 0.33161816 0.54046555
## 1512 0.080350098 0.45982495 0.45982495
## 1513 0.051664705 0.65267019 0.29566511
## 1514 0.078864570 0.20445341 0.71668202
## 1515 0.080350098 0.45982495 0.45982495
## 1516 0.080350098 0.45982495 0.45982495
## 1517 0.234457995 0.38277100 0.38277100
## 1518 0.397534545 0.30123273 0.30123273
## 1519 0.037336968 0.74899223 0.21367080
## 1520 0.397534545 0.30123273 0.30123273
## 1521 0.397534545 0.30123273 0.30123273
## 1522 0.152336024 0.11543298 0.73223100
## 1523 0.234457995 0.38277100 0.38277100
## 1524 0.234457995 0.38277100 0.38277100
## 1525 0.074871195 0.17481183 0.75031698
## 1526 0.042023678 0.51813971 0.43983661
## 1527 0.030633788 0.11334576 0.85602045
## 1528 0.045321179 0.18817732 0.76650150
## 1529 0.080350098 0.45982495 0.45982495
## 1530 0.144576208 0.45007517 0.40534862
## 1531 0.027332103 0.21411558 0.75855232
## 1532 0.080350098 0.45982495 0.45982495
## 1533 0.046265732 0.77953750 0.17419677
## 1534 0.046265732 0.77953750 0.17419677
## 1535 0.125332291 0.20461476 0.67005295
## 1536 0.075249898 0.63643821 0.28831189
## 1537 0.062712794 0.10959484 0.82769237
## 1538 0.149835952 0.45838271 0.39178134
## 1539 0.149835952 0.45838271 0.39178134
## 1540 0.064482485 0.26773666 0.66778085
## 1541 0.198850184 0.47651135 0.32463847
## 1542 0.080350098 0.45982495 0.45982495
## 1543 0.088237918 0.50496511 0.40679697
## 1544 0.088237918 0.50496511 0.40679697
## 1545 0.083906161 0.52393846 0.39215538
## 1546 0.043862762 0.40202749 0.55410975
## 1547 0.080350098 0.45982495 0.45982495
## 1548 0.080350098 0.45982495 0.45982495
## 1549 0.033571229 0.77430842 0.19212035
## 1550 0.122419908 0.67772002 0.19986007
## 1551 0.034905616 0.13142439 0.83367000
## 1552 0.091955281 0.38180600 0.52623872
## 1553 0.080350098 0.45982495 0.45982495
## 1554 0.127916296 0.33161816 0.54046555
## 1555 0.127916296 0.33161816 0.54046555
## 1556 0.127916296 0.33161816 0.54046555
## 1557 0.080350098 0.45982495 0.45982495
## 1558 0.020026320 0.25298861 0.72698507
## 1559 0.025557537 0.14625985 0.82818261
## 1560 0.035419706 0.20269875 0.76188154
## 1561 0.161683466 0.41915827 0.41915827
## 1562 0.161683466 0.41915827 0.41915827
## 1563 0.161683466 0.41915827 0.41915827
## 1564 0.054979957 0.22433368 0.72068636
## 1565 0.080350098 0.45982495 0.45982495
## 1566 0.045667296 0.69298938 0.26134333
## 1567 0.109163604 0.44541820 0.44541820
## 1568 0.065468744 0.28536061 0.64917065
## 1569 0.042023678 0.51813971 0.43983661
## 1570 0.062712794 0.10959484 0.82769237
## 1571 0.022332765 0.48883362 0.48883362
## 1572 0.173857308 0.41307135 0.41307135
## 1573 0.087930616 0.40886289 0.50320649
## 1574 0.097406418 0.55155468 0.35103890
## 1575 0.030503563 0.79493163 0.17456481
## 1576 0.034704650 0.75963721 0.20565814
## 1577 0.017316378 0.34737241 0.63531121
## 1578 0.035003823 0.48249809 0.48249809
## 1579 0.050490842 0.19010488 0.75940428
## 1580 0.104275293 0.15876368 0.73696103
## 1581 0.104275293 0.15876368 0.73696103
## 1582 0.031898454 0.48405077 0.48405077
## 1583 0.031898454 0.48405077 0.48405077
## 1584 0.023643804 0.51753047 0.45882573
## 1585 0.024827496 0.83309051 0.14208200
## 1586 0.024827496 0.83309051 0.14208200
## 1587 0.293551731 0.35322413 0.35322413
## 1588 0.258718204 0.37064090 0.37064090
## 1589 0.074391197 0.10657317 0.81903563
## 1590 0.080350098 0.45982495 0.45982495
## 1591 0.258718204 0.37064090 0.37064090
## 1592 0.080350098 0.45982495 0.45982495
## 1593 0.022019899 0.79935493 0.17862517
## 1594 0.022019899 0.79935493 0.17862517
## 1595 0.013586410 0.49320679 0.49320679
## 1596 0.224048813 0.32097337 0.45497782
## 1597 0.067409262 0.38576755 0.54682319
## 1598 0.067409262 0.38576755 0.54682319
## 1599 0.031276881 0.71500541 0.25371771
## 1600 0.031276881 0.71500541 0.25371771
## 1601 0.080350098 0.45982495 0.45982495
## 1602 0.397534545 0.30123273 0.30123273
## 1603 0.062712794 0.10959484 0.82769237
## 1604 0.048988513 0.28034988 0.67066161
## 1605 0.080350098 0.45982495 0.45982495
## 1606 0.026490229 0.15159743 0.82191234
## 1607 0.080350098 0.45982495 0.45982495
## 1608 0.080350098 0.45982495 0.45982495
## 1609 0.080350098 0.45982495 0.45982495
## 1610 0.046265732 0.77953750 0.17419677
## 1611 0.075249898 0.63643821 0.28831189
## 1612 0.080350098 0.45982495 0.45982495
## 1613 0.074871195 0.17481183 0.75031698
## 1614 0.042023678 0.51813971 0.43983661
## 1615 0.144576208 0.45007517 0.40534862
## 1616 0.046265732 0.77953750 0.17419677
## 1617 0.051100997 0.65131354 0.29758546
## 1618 0.088237918 0.50496511 0.40679697
## 1619 0.043862762 0.40202749 0.55410975
## 1620 0.050490842 0.19010488 0.75940428
## 1621 0.021403643 0.48929818 0.48929818
## 1622 0.122618558 0.17566391 0.70171753
## 1623 0.080350098 0.45982495 0.45982495
## 1624 0.080350098 0.45982495 0.45982495
## 1625 0.024827496 0.83309051 0.14208200
## 1626 0.171365972 0.41431701 0.41431701
## 1627 0.161683466 0.41915827 0.41915827
## 1628 0.161683466 0.41915827 0.41915827
## 1629 0.080350098 0.45982495 0.45982495
## 1630 0.080350098 0.45982495 0.45982495
## 1631 0.074871195 0.17481183 0.75031698
## 1632 0.074871195 0.17481183 0.75031698
## 1633 0.065468744 0.28536061 0.64917065
## 1634 0.042023678 0.51813971 0.43983661
## 1635 0.042023678 0.51813971 0.43983661
## 1636 0.030633788 0.11334576 0.85602045
## 1637 0.017589275 0.22220190 0.76020883
## 1638 0.045321179 0.18817732 0.76650150
## 1639 0.080350098 0.45982495 0.45982495
## 1640 0.034441443 0.19710038 0.76845818
## 1641 0.182294844 0.29761058 0.52009458
## 1642 0.125332291 0.20461476 0.67005295
## 1643 0.093851947 0.53709290 0.36905516
## 1644 0.049716726 0.43541667 0.51486660
## 1645 0.042624328 0.41891081 0.53846486
## 1646 0.045084381 0.56954223 0.38537339
## 1647 0.045084381 0.56954223 0.38537339
## 1648 0.036100194 0.13126171 0.83263809
## 1649 0.080350098 0.45982495 0.45982495
## 1650 0.087930616 0.40886289 0.50320649
## 1651 0.068714472 0.55153165 0.37975388
## 1652 0.029679628 0.59538339 0.37493698
## 1653 0.240333639 0.40629432 0.35337204
## 1654 0.043862762 0.55410975 0.40202749
## 1655 0.091955281 0.38180600 0.52623872
## 1656 0.091955281 0.38180600 0.52623872
## 1657 0.091955281 0.38180600 0.52623872
## 1658 0.043580992 0.59033859 0.36608042
## 1659 0.052658258 0.67116169 0.27618005
## 1660 0.053070209 0.47718881 0.46974098
## 1661 0.053070209 0.47718881 0.46974098
## 1662 0.111637164 0.24948927 0.63887356
## 1663 0.080350098 0.45982495 0.45982495
## 1664 0.080350098 0.45982495 0.45982495
## 1665 0.080350098 0.45982495 0.45982495
## 1666 0.084447375 0.48327272 0.43227991
## 1667 0.031159736 0.18465115 0.78418912
## 1668 0.104275293 0.15876368 0.73696103
## 1669 0.065857452 0.73784005 0.19630250
## 1670 0.100539273 0.32409782 0.57536291
## 1671 0.039117426 0.74469826 0.21618432
## 1672 0.161683466 0.41915827 0.41915827
## 1673 0.080350098 0.45982495 0.45982495
## 1674 0.080350098 0.45982495 0.45982495
## 1675 0.080350098 0.45982495 0.45982495
## 1676 0.080350098 0.45982495 0.45982495
## 1677 0.064683226 0.57155109 0.36376569
## 1678 0.094264980 0.24180082 0.66393420
## 1679 0.046265732 0.77953750 0.17419677
## 1680 0.020491729 0.23562816 0.74388011
## 1681 0.307721115 0.41466243 0.27761646
## 1682 0.149835952 0.45838271 0.39178134
## 1683 0.146966447 0.58708093 0.26595262
## 1684 0.064482485 0.26773666 0.66778085
## 1685 0.064482485 0.26773666 0.66778085
## 1686 0.042624328 0.41891081 0.53846486
## 1687 0.198850184 0.47651135 0.32463847
## 1688 0.036100194 0.13126171 0.83263809
## 1689 0.051100997 0.65131354 0.29758546
## 1690 0.163954739 0.18556697 0.65047829
## 1691 0.080350098 0.45982495 0.45982495
## 1692 0.084447375 0.48327272 0.43227991
## 1693 0.039117426 0.74469826 0.21618432
## 1694 0.078995895 0.46892894 0.45207516
## 1695 0.080350098 0.45982495 0.45982495
## 1696 0.080350098 0.45982495 0.45982495
## 1697 0.117589807 0.20947104 0.67293916
## 1698 0.107773524 0.26461010 0.62761638
## 1699 0.107773524 0.26461010 0.62761638
## 1700 0.026476377 0.48676181 0.48676181
## 1701 0.080350098 0.45982495 0.45982495
## 1702 0.117850189 0.20772055 0.67442926
## 1703 0.074871195 0.17481183 0.75031698
## 1704 0.125321577 0.71718628 0.15749214
## 1705 0.146966447 0.58708093 0.26595262
## 1706 0.075173286 0.49462746 0.43019926
## 1707 0.084447375 0.48327272 0.43227991
## 1708 0.084447375 0.48327272 0.43227991
## 1709 0.024318666 0.48784067 0.48784067
## 1710 0.144576208 0.45007517 0.40534862
## 1711 0.080350098 0.45982495 0.45982495
## 1712 0.039117426 0.74469826 0.21618432
## 1713 0.051664705 0.65267019 0.29566511
## 1714 0.080350098 0.45982495 0.45982495
## 1715 0.024318666 0.48784067 0.48784067
## 1716 0.240333639 0.40629432 0.35337204
## 1717 0.080350098 0.45982495 0.45982495
## 1718 0.043862762 0.40202749 0.55410975
## 1719 0.017316378 0.34737241 0.63531121
## 1720 0.397534545 0.30123273 0.30123273
## 1721 0.037336968 0.74899223 0.21367080
## 1722 0.037878388 0.48361129 0.47851032
## 1723 0.152336024 0.11543298 0.73223100
## 1724 0.030928745 0.79207323 0.17699802
## 1725 0.065468744 0.28536061 0.64917065
## 1726 0.042023678 0.51813971 0.43983661
## 1727 0.042023678 0.51813971 0.43983661
## 1728 0.030633788 0.11334576 0.85602045
## 1729 0.017589275 0.22220190 0.76020883
## 1730 0.045321179 0.18817732 0.76650150
## 1731 0.080350098 0.45982495 0.45982495
## 1732 0.144576208 0.45007517 0.40534862
## 1733 0.161165757 0.68562363 0.15321061
## 1734 0.182294844 0.29761058 0.52009458
## 1735 0.020491729 0.23562816 0.74388011
## 1736 0.062712794 0.10959484 0.82769237
## 1737 0.080350098 0.45982495 0.45982495
## 1738 0.017589275 0.22220190 0.76020883
## 1739 0.013586410 0.49320679 0.49320679
## 1740 0.013586410 0.49320679 0.49320679
## 1741 0.080350098 0.45982495 0.45982495
## 1742 0.037336968 0.74899223 0.21367080
## 1743 0.033802882 0.77275108 0.19344604
## 1744 0.033802882 0.77275108 0.19344604
## 1745 0.015363358 0.19408232 0.79055432
## 1746 0.065468744 0.28536061 0.64917065
## 1747 0.065468744 0.28536061 0.64917065
## 1748 0.017589275 0.22220190 0.76020883
## 1749 0.144576208 0.45007517 0.40534862
## 1750 0.080350098 0.45982495 0.45982495
## 1751 0.080350098 0.45982495 0.45982495
## 1752 0.093851947 0.53709290 0.36905516
## 1753 0.163954739 0.18556697 0.65047829
## 1754 0.088237918 0.50496511 0.40679697
## 1755 0.080350098 0.45982495 0.45982495
## 1756 0.106040736 0.28711276 0.60684651
## 1757 0.258718204 0.37064090 0.37064090
## 1758 0.080350098 0.45982495 0.45982495
## 1759 0.080350098 0.45982495 0.45982495
## 1760 0.086508274 0.56051364 0.35297808
## 1761 0.144576208 0.45007517 0.40534862
## 1762 0.064482485 0.26773666 0.66778085
## 1763 0.225897666 0.32020859 0.45389375
## 1764 0.033338273 0.24955202 0.71710971
## 1765 0.020026320 0.25298861 0.72698507
## 1766 0.074871195 0.17481183 0.75031698
## 1767 0.030633788 0.11334576 0.85602045
## 1768 0.030633788 0.11334576 0.85602045
## 1769 0.030633788 0.11334576 0.85602045
## 1770 0.144576208 0.45007517 0.40534862
## 1771 0.161165757 0.68562363 0.15321061
## 1772 0.125321577 0.71718628 0.15749214
## 1773 0.080350098 0.45982495 0.45982495
## 1774 0.080350098 0.45982495 0.45982495
## 1775 0.051100997 0.65131354 0.29758546
## 1776 0.087930616 0.40886289 0.50320649
## 1777 0.087930616 0.40886289 0.50320649
## 1778 0.034704650 0.75963721 0.20565814
## 1779 0.017947594 0.33052816 0.65152424
## 1780 0.080350098 0.45982495 0.45982495
## 1781 0.033571229 0.77430842 0.19212035
## 1782 0.033338273 0.24955202 0.71710971
## 1783 0.046976233 0.31669397 0.63632979
## 1784 0.073481623 0.42051826 0.50600012
## 1785 0.073481623 0.42051826 0.50600012
## 1786 0.017589275 0.22220190 0.76020883
## 1787 0.397534545 0.30123273 0.30123273
## 1788 0.013586410 0.49320679 0.49320679
## 1789 0.037336968 0.74899223 0.21367080
## 1790 0.043226451 0.24737494 0.70939861
## 1791 0.030633788 0.11334576 0.85602045
## 1792 0.144576208 0.45007517 0.40534862
## 1793 0.161165757 0.68562363 0.15321061
## 1794 0.075729367 0.61853272 0.30573792
## 1795 0.061051470 0.35553168 0.58341685
## 1796 0.080350098 0.45982495 0.45982495
## 1797 0.080350098 0.45982495 0.45982495
## 1798 0.080350098 0.45982495 0.45982495
## 1799 0.080350098 0.45982495 0.45982495
## 1800 0.149835952 0.45838271 0.39178134
## 1801 0.146966447 0.58708093 0.26595262
## 1802 0.022332765 0.48883362 0.48883362
## 1803 0.051100997 0.65131354 0.29758546
## 1804 0.051100997 0.65131354 0.29758546
## 1805 0.017947594 0.33052816 0.65152424
## 1806 0.080350098 0.45982495 0.45982495
## 1807 0.070196109 0.63117892 0.29862497
## 1808 0.066306551 0.37289495 0.56079850
## 1809 0.066306551 0.37289495 0.56079850
## 1810 0.033802882 0.77275108 0.19344604
## 1811 0.152336024 0.11543298 0.73223100
## 1812 0.080350098 0.45982495 0.45982495
## 1813 0.064482485 0.26773666 0.66778085
## 1814 0.122419908 0.67772002 0.19986007
## 1815 0.078864570 0.20445341 0.71668202
## 1816 0.030928745 0.79207323 0.17699802
## 1817 0.080350098 0.45982495 0.45982495
## 1818 0.080350098 0.45982495 0.45982495
## 1819 0.050490842 0.19010488 0.75940428
## 1820 0.031159736 0.18465115 0.78418912
## 1821 0.048988513 0.28034988 0.67066161
## 1822 0.113362345 0.23789128 0.64874637

cola3.0 = cola2
cola3.0$price[seq(from=1, to=5466, by=3)] = z[1,3]
cola3.0$price[seq(from=2, to=5466, by=3)] = z[2,3]
cola3.0$price[seq(from=3, to=5466, by=3)] = z[3,3]
cola3.0$feat=0
cola3.0$disp=0
pr0 = predict(mlogit1,newdata=cola3.0)

cola3.1 = cola3.0
cola3.1$price[seq(from=1, to=5466, by=3)] = z[1,3]+1
cola3.1$price[seq(from=2, to=5466, by=3)] = z[1,3]
pr1 = predict(mlogit1,newdata=cola3.1)

(pr1-pr0)[1,]

##       coke      pepsi    sevenup 
## -0.2435138  0.1169981  0.1265157

# marginal effects for display for pepsi
cola4.0 = cola3.0
pr0 = predict(mlogit1,newdata=cola4.0)

cola4.1 = cola4.0
cola4.1$disp[seq(from=1, to=1822, by=3)] = 1
pr1 = predict(mlogit1,newdata=cola4.1)
(pr1-pr0)[1:5,]

##        coke       pepsi     sevenup
## 1 0.1074106 -0.04857698 -0.05883366
## 2 0.1074106 -0.04857698 -0.05883366
## 3 0.1074106 -0.04857698 -0.05883366
## 4 0.1074106 -0.04857698 -0.05883366
## 5 0.1074106 -0.04857698 -0.05883366

2 ——————————————

3 Exercise 4

4 ——————————————

5 ——————————————

6 Exercise 5

7 ——————————————

# ------------------------------------------
# Exercise 6
# ------------------------------------------

# read the data
dane = read.csv(file="fmld142_part.csv", header=TRUE, sep=",")
dane %>% as_tibble()

## # A tibble: 823 × 5
##    age_ref empltyp1         fam_size marital1 sex_ref
##      <int> <chr>               <int>    <int>   <int>
##  1      54 Private company         2        3       2
##  2      21 Private company         2        1       1
##  3      29 Private company         3        1       2
##  4      34 Private company         6        5       2
##  5      51 State government        2        1       2
##  6      29 Private company         4        1       2
##  7      68 Private company         1        5       2
##  8      24 Private company         5        5       2
##  9      64 Private company         3        1       2
## 10      35 Private company         2        5       2
## # ℹ 813 more rows

# multinom function from nnet library
mlogit = multinom(empltyp1~age_ref+as.factor(sex_ref)+fam_size+as.factor(marital1), 
                  data=dane)

## # weights:  45 (32 variable)
## initial  value 1324.567402 
## iter  10 value 896.692898
## iter  20 value 865.462171
## iter  30 value 865.063803
## final  value 865.063022 
## converged

summary(mlogit)

## Call:
## multinom(formula = empltyp1 ~ age_ref + as.factor(sex_ref) + 
##     fam_size + as.factor(marital1), data = dane)
## 
## Coefficients:
##                  (Intercept)     age_ref as.factor(sex_ref)2  fam_size
## Local government  -1.4281663 0.028335744           0.1844635 0.1287375
## Private company    0.7078712 0.017965616          -0.2257710 0.3064672
## Self-employed     -0.8744264 0.034059159          -0.3544452 0.2023819
## State government  -0.3797957 0.007126902           0.2602681 0.0380612
##                  as.factor(marital1)2 as.factor(marital1)3 as.factor(marital1)4
## Local government           0.01691011           -0.9233647            0.9262380
## Private company            1.16596620            0.3727020            0.8916766
## Self-employed              0.04659160           -0.3011707            0.1997522
## State government           0.38430583           -0.1977795           -0.1174740
##                  as.factor(marital1)5
## Local government            1.0334795
## Private company             1.4061073
## Self-employed               0.6074354
## State government            1.1062534
## 
## Std. Errors:
##                  (Intercept)    age_ref as.factor(sex_ref)2  fam_size
## Local government   1.0374140 0.01604022           0.4191594 0.1801395
## Private company    0.8221904 0.01321500           0.3353268 0.1473357
## Self-employed      0.9265579 0.01457842           0.3730280 0.1615470
## State government   1.0219440 0.01618219           0.4199156 0.1839811
##                  as.factor(marital1)2 as.factor(marital1)3 as.factor(marital1)4
## Local government             1.309139            0.6662405             1.204430
## Private company              1.088297            0.4566577             1.064979
## Self-employed                1.194192            0.5221873             1.195806
## State government             1.313952            0.6006915             1.455811
##                  as.factor(marital1)5
## Local government            0.6131294
## Private company             0.5234968
## Self-employed               0.5786374
## State government            0.6118367
## 
## Residual Deviance: 1730.126 
## AIC: 1794.126

stargazer(mlogit, type="text")

## 
## ====================================================================================
##                                            Dependent variable:                      
##                      ---------------------------------------------------------------
##                      Local government Private company Self-employed State government
##                            (1)              (2)            (3)            (4)       
## ------------------------------------------------------------------------------------
## age_ref                   0.028*           0.018         0.034**         0.007      
##                          (0.016)          (0.013)        (0.015)        (0.016)     
##                                                                                     
## as.factor(sex_ref)2       0.184           -0.226         -0.354          0.260      
##                          (0.419)          (0.335)        (0.373)        (0.420)     
##                                                                                     
## fam_size                  0.129           0.306**         0.202          0.038      
##                          (0.180)          (0.147)        (0.162)        (0.184)     
##                                                                                     
## as.factor(marital1)2      0.017            1.166          0.047          0.384      
##                          (1.309)          (1.088)        (1.194)        (1.314)     
##                                                                                     
## as.factor(marital1)3      -0.923           0.373         -0.301          -0.198     
##                          (0.666)          (0.457)        (0.522)        (0.601)     
##                                                                                     
## as.factor(marital1)4      0.926            0.892          0.200          -0.117     
##                          (1.204)          (1.065)        (1.196)        (1.456)     
##                                                                                     
## as.factor(marital1)5      1.033*         1.406***         0.607          1.106*     
##                          (0.613)          (0.523)        (0.579)        (0.612)     
##                                                                                     
## Constant                  -1.428           0.708         -0.874          -0.380     
##                          (1.037)          (0.822)        (0.927)        (1.022)     
##                                                                                     
## ------------------------------------------------------------------------------------
## Akaike Inf. Crit.       1,794.126        1,794.126      1,794.126      1,794.126    
## ====================================================================================
## Note:                                                    *p<0.1; **p<0.05; ***p<0.01

# preparing data for mlogit.data
table(dane$empltyp1)

## 
## Federal government   Local government    Private company      Self-employed 
##                 40                 60                545                118 
##   State government 
##                 60

dane_mlogit = mlogit.data(dane, choice="empltyp1", shape="wide", varying=NULL)
dane_mlogit %>% as_tibble()

## # A tibble: 4,115 × 8
##    age_ref empltyp1 fam_size marital1 sex_ref  chid alt                idx$chid
##      <int> <lgl>       <int>    <int>   <int> <dbl> <fct>                 <dbl>
##  1      54 FALSE           2        3       2     1 Federal government        1
##  2      54 FALSE           2        3       2     1 Local government          1
##  3      54 TRUE            2        3       2     1 Private company           1
##  4      54 FALSE           2        3       2     1 Self-employed             1
##  5      54 FALSE           2        3       2     1 State government          1
##  6      21 FALSE           2        1       1     2 Federal government        2
##  7      21 FALSE           2        1       1     2 Local government          2
##  8      21 TRUE            2        1       1     2 Private company           2
##  9      21 FALSE           2        1       1     2 Self-employed             2
## 10      21 FALSE           2        1       1     2 State government          2
## # ℹ 4,105 more rows
## # ℹ 1 more variable: idx$alt <fct>

# mlogit function from mlogit library
mlogit1 = mlogit(empltyp1~0|age_ref+as.factor(sex_ref)+fam_size+as.factor(marital1), data=dane_mlogit)
summary(mlogit1)

## 
## Call:
## mlogit(formula = empltyp1 ~ 0 | age_ref + as.factor(sex_ref) + 
##     fam_size + as.factor(marital1), data = dane_mlogit, method = "nr")
## 
## Frequencies of alternatives:choice
## Federal government   Local government    Private company      Self-employed 
##           0.048603           0.072904           0.662211           0.143378 
##   State government 
##           0.072904 
## 
## nr method
## 6 iterations, 0h:0m:0s 
## g'(-H)^-1g = 7.05E-08 
## gradient close to zero 
## 
## Coefficients :
##                                         Estimate Std. Error z-value Pr(>|z|)   
## (Intercept):Local government          -1.4271947  1.0374276 -1.3757 0.168913   
## (Intercept):Private company            0.7086517  0.8222150  0.8619 0.388753   
## (Intercept):Self-employed             -0.8735855  0.9265791 -0.9428 0.345780   
## (Intercept):State government          -0.3787420  1.0219562 -0.3706 0.710932   
## age_ref:Local government               0.0283223  0.0160400  1.7657 0.077442 . 
## age_ref:Private company                0.0179535  0.0132149  1.3586 0.174279   
## age_ref:Self-employed                  0.0340458  0.0145783  2.3354 0.019524 * 
## age_ref:State government               0.0071094  0.0161821  0.4393 0.660415   
## as.factor(sex_ref)2:Local government   0.1844409  0.4191608  0.4400 0.659920   
## as.factor(sex_ref)2:Private company   -0.2257793  0.3353310 -0.6733 0.500755   
## as.factor(sex_ref)2:Self-employed     -0.3544525  0.3730318 -0.9502 0.342014   
## as.factor(sex_ref)2:State government   0.2602585  0.4199177  0.6198 0.535400   
## fam_size:Local government              0.1286331  0.1801373  0.7141 0.475175   
## fam_size:Private company               0.3063941  0.1473317  2.0796 0.037560 * 
## fam_size:Self-employed                 0.2023018  0.1615435  1.2523 0.210459   
## fam_size:State government              0.0379830  0.1839773  0.2065 0.836436   
## as.factor(marital1)2:Local government  0.0188040  1.3098264  0.0144 0.988546   
## as.factor(marital1)2:Private company   1.1678662  1.0891347  1.0723 0.283591   
## as.factor(marital1)2:Self-employed     0.0486656  1.1949403  0.0407 0.967514   
## as.factor(marital1)2:State government  0.3869995  1.3145098  0.2944 0.768448   
## as.factor(marital1)3:Local government -0.9234328  0.6662348 -1.3860 0.165733   
## as.factor(marital1)3:Private company   0.3726670  0.4566588  0.8161 0.414458   
## as.factor(marital1)3:Self-employed    -0.3012248  0.5221896 -0.5768 0.564041   
## as.factor(marital1)3:State government -0.1979057  0.6006993 -0.3295 0.741809   
## as.factor(marital1)4:Local government  0.9255888  1.2043034  0.7686 0.442150   
## as.factor(marital1)4:Private company   0.8911899  1.0648173  0.8369 0.402625   
## as.factor(marital1)4:Self-employed     0.1995481  1.1956272  0.1669 0.867450   
## as.factor(marital1)4:State government -0.1181732  1.4557643 -0.0812 0.935302   
## as.factor(marital1)5:Local government  1.0330817  0.6131106  1.6850 0.091992 . 
## as.factor(marital1)5:Private company   1.4057687  0.5234762  2.6854 0.007243 **
## as.factor(marital1)5:Self-employed     0.6070936  0.5786192  1.0492 0.294081   
## as.factor(marital1)5:State government  1.1058654  0.6118180  1.8075 0.070683 . 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Log-Likelihood: -865.06
## McFadden R^2:  0.026945 
## Likelihood ratio test : chisq = 47.909 (p.value = 0.010961)

stargazer(mlogit1, type="text")

## 
## =================================================================
##                                           Dependent variable:    
##                                       ---------------------------
##                                                empltyp1          
## -----------------------------------------------------------------
## (Intercept):Local government                    -1.427           
##                                                 (1.037)          
##                                                                  
## (Intercept):Private company                      0.709           
##                                                 (0.822)          
##                                                                  
## (Intercept):Self-employed                       -0.874           
##                                                 (0.927)          
##                                                                  
## (Intercept):State government                    -0.379           
##                                                 (1.022)          
##                                                                  
## age_ref:Local government                        0.028*           
##                                                 (0.016)          
##                                                                  
## age_ref:Private company                          0.018           
##                                                 (0.013)          
##                                                                  
## age_ref:Self-employed                           0.034**          
##                                                 (0.015)          
##                                                                  
## age_ref:State government                         0.007           
##                                                 (0.016)          
##                                                                  
## as.factor(sex_ref)2:Local government             0.184           
##                                                 (0.419)          
##                                                                  
## as.factor(sex_ref)2:Private company             -0.226           
##                                                 (0.335)          
##                                                                  
## as.factor(sex_ref)2:Self-employed               -0.354           
##                                                 (0.373)          
##                                                                  
## as.factor(sex_ref)2:State government             0.260           
##                                                 (0.420)          
##                                                                  
## fam_size:Local government                        0.129           
##                                                 (0.180)          
##                                                                  
## fam_size:Private company                        0.306**          
##                                                 (0.147)          
##                                                                  
## fam_size:Self-employed                           0.202           
##                                                 (0.162)          
##                                                                  
## fam_size:State government                        0.038           
##                                                 (0.184)          
##                                                                  
## as.factor(marital1)2:Local government            0.019           
##                                                 (1.310)          
##                                                                  
## as.factor(marital1)2:Private company             1.168           
##                                                 (1.089)          
##                                                                  
## as.factor(marital1)2:Self-employed               0.049           
##                                                 (1.195)          
##                                                                  
## as.factor(marital1)2:State government            0.387           
##                                                 (1.315)          
##                                                                  
## as.factor(marital1)3:Local government           -0.923           
##                                                 (0.666)          
##                                                                  
## as.factor(marital1)3:Private company             0.373           
##                                                 (0.457)          
##                                                                  
## as.factor(marital1)3:Self-employed              -0.301           
##                                                 (0.522)          
##                                                                  
## as.factor(marital1)3:State government           -0.198           
##                                                 (0.601)          
##                                                                  
## as.factor(marital1)4:Local government            0.926           
##                                                 (1.204)          
##                                                                  
## as.factor(marital1)4:Private company             0.891           
##                                                 (1.065)          
##                                                                  
## as.factor(marital1)4:Self-employed               0.200           
##                                                 (1.196)          
##                                                                  
## as.factor(marital1)4:State government           -0.118           
##                                                 (1.456)          
##                                                                  
## as.factor(marital1)5:Local government           1.033*           
##                                                 (0.613)          
##                                                                  
## as.factor(marital1)5:Private company           1.406***          
##                                                 (0.523)          
##                                                                  
## as.factor(marital1)5:Self-employed               0.607           
##                                                 (0.579)          
##                                                                  
## as.factor(marital1)5:State government           1.106*           
##                                                 (0.612)          
##                                                                  
## -----------------------------------------------------------------
## Observations                                      823            
## R2                                               0.027           
## Log Likelihood                                 -865.063          
## LR Test                                   47.909** (df = 32)     
## =================================================================
## Note:                                 *p<0.1; **p<0.05; ***p<0.01

AE05 - UNORDER LOGIT