11-Replication.qmd

# 先行研究の再現 {-}


[訳者のGitHubのスクリプトのフォルダ](https://github.com/ShoFujihara/Association_Models_Japanese/tree/main/scripts) にRのスクリプトファイルを保存しているのでそれも活用してください．

```{r}
#| message: false
library(tidyverse)
library(gnm)
library(broom)
library(vcdExtra)
```


## Duncan (1979)

@duncan1979 の結果を再現する．まずはデータを作成する．

```{r}
# Data (p. 795)
Freq_duncan_1979 <- c(50, 19, 26, 8, 7, 11, 6, 2,
                      16, 40, 34, 18, 11, 20, 8, 3,
                      12, 35, 65, 66, 35, 88, 23, 21,
                      11, 20, 58, 110, 40, 183, 64, 32,
                      2, 8, 12, 23, 25, 46, 28, 12,
                      12, 28, 102, 162, 90, 554, 230, 177,
                      0, 6, 19, 40, 21, 158, 143, 71,
                      0, 3, 14, 32, 15, 126, 91, 106
                      )

d_duncan <- tibble(Freq = Freq_duncan_1979,
                   O = gl(n = 8, k = 8, length = 8 * 8),
                   D = gl(n = 8, k = 1, length = 8 * 8))

# データの完成
d_duncan

# Row and column scores
d_duncan <- d_duncan |> 
  mutate(U = as.numeric(O),
         V = as.numeric(D))

# 確認
d_duncan |> count(O, U)
d_duncan |> count(D, V)

# Diagonal parameters
d_duncan <- d_duncan |> 
  mutate(Diag = case_when(U == V ~ U, 
                          .default = 0))
# 確認
d_duncan |> xtabs(Diag ~ O + D, data = _)
# factorにする
d_duncan <- d_duncan |> mutate(Diag = factor(Diag))

# weight 
d_duncan <- d_duncan |> mutate(W = ifelse(Diag != 0, 0, 1))

# weight変数がどのようになっているのかを確認
d_duncan |> xtabs(W ~ O + D, data = _)
d_duncan |> uncount(weights = W) |> 
  xtabs(Freq ~ O + D, data = _)
# データの完成
d_duncan
```

このデータについて`gnm`を用いたモデルの推定を行う．`glm`でもよい．結果は`summary`でみることができるが，`tidy`や`glance`でも確認でき，これらを使ったほうが表の整理やその後の`ggplot`による図の作成が簡単である．`tidy`は`gnm`をサポートの対象外としているが，`glm`とほぼ同じ構造なのではじめに警告がでるものの，結果は問題なく出力される．

```{r}
# (1) Independence
fit_d_1 <- d_duncan |> 
  gnm(Freq ~ O + D, data = _, family = poisson)
summary(fit_d_1)
tidy(fit_d_1)
glance(fit_d_1)

# (2) Row effects
fit_d_2 <- d_duncan |> 
  gnm(Freq ~ O + D + O:V, data = _, family = poisson)
summary(fit_d_2)
tidy(fit_d_2)
glance(fit_d_2)

# (3) Quasi independence, diagonal omitted
fit_d_3 <- d_duncan |> gnm(Freq ~ O + D, data = _, weights = W, family = poisson)
summary(fit_d_3)
tidy(fit_d_3)
glance(fit_d_3)

# (3) Quasi independence, diagonal fitted
fit_d_3b <- d_duncan |> gnm(Freq ~ O + D + Diag, data = _, family = poisson)
summary(fit_d_3b)
tidy(fit_d_3b)
glance(fit_d_3b)

# (4) Uniform association, diagonal omitted
fit_d_4 <- d_duncan |> gnm(Freq ~  O + D + U:V, data = _, weights = W, family = poisson)
summary(fit_d_4)
tidy(fit_d_4)
glance(fit_d_4)

# log-odds b_hat
fit_d_4$coefficients["U:V"] |> exp()

fit_d_4b <- d_duncan |> gnm(Freq ~  O + D + U:V + Diag, data = _, family = poisson)
summary(fit_d_4b)
tidy(fit_d_4b)
glance(fit_d_4b)

# log-odds b_hat
fit_d_4b$coefficients["U:V"] |> exp()

# (5) Row effects, diagonal omitted*
fit_d_5 <- d_duncan |> gnm(Freq ~ O + D + O:V, data = _, weights = W, family = poisson)
summary(fit_d_5)
tidy(fit_d_5)
glance(fit_d_5)

# (5) Row effects, diagonal fitted
fit_d_5b <- d_duncan |> gnm(Freq ~ O + D + O:V + Diag, data = _, family = poisson)
summary(fit_d_5b)
tidy(fit_d_5b)
glance(fit_d_5b)
```


```{r}
# Tabel 1
d_duncan_fitted <- 
  d_duncan |> 
  mutate(fitted_1 = predict(fit_d_1, type = "response"),
         fitted_2 = predict(fit_d_2, type = "response"),
         fitted_3 = predict(fit_d_3, type = "response"),
         fitted_4 = predict(fit_d_4, type = "response"),
         fitted_5 = predict(fit_d_5, type = "response"))

tab_obs <- d_duncan_fitted |> xtabs(Freq ~ O + D, data = _)
tab_fitted_1 <- d_duncan_fitted |> xtabs(fitted_1 ~ O + D, data = _)
tab_fitted_2 <- d_duncan_fitted |> xtabs(fitted_2 ~ O + D, data = _)
tab_fitted_3 <- d_duncan_fitted |> xtabs(fitted_3 ~ O + D, data = _)
tab_fitted_4 <- d_duncan_fitted |> xtabs(fitted_4 ~ O + D, data = _)
tab_fitted_5 <- d_duncan_fitted |> xtabs(fitted_5 ~ O + D, data = _)

# Observed Counts
tab_obs
# Fitted Counts, Model (4)
tab_fitted_4 |> round(1)
# Fitted Counts, Model (5)
tab_fitted_5 |> round(1)

d_1 <- (tab_obs/tab_fitted_1) |> diag()
d_2 <- (tab_obs/tab_fitted_2) |> diag()
d_3 <- (tab_obs/tab_fitted_3) |> diag()
d_4 <- (tab_obs/tab_fitted_4) |> diag()
d_5 <- (tab_obs/tab_fitted_5) |> diag()

# Diagonal Ratios
occ <- c("I","II","III","IV","Va","Vb","VI","VII")
tibble(occ, d_1, d_2, d_3, d_4, d_5)

# Table 2
Model <- c("(1) Independence",
           "(2) Row effects",
           "(3) Quasi independence, diagonal omitted",
           "(4) Uniform association, diagonal omitted",
           "(5) Row effects, diagonal omitted")

# 複数のモデルの適合度をglanceで示してまとめる
list(fit_d_1, fit_d_2, fit_d_3, fit_d_4, fit_d_5) |> 
  map_dfr(glance) |> 
  mutate(Model, .before = 1) |> 
  dplyr::select(Model, L2 = deviance, df = df.residual)

# Figure 2
b_4 <- exp(fit_d_4$coefficients["U:V"] * (7:0))
b_5 <- fit_d_5$coefficients[grep(":V", names(fit_d_5$coefficients))]
b_5["O8:V"] <- 0
b_5 <- exp(b_5 * (-1)) 

d_duncan_b_4 <- tibble(y = b_4, Model = "Model 4", x = gl(8, 1))
d_duncan_b_5 <- tibble(y = b_5, Model = "Model 5", x = gl(8, 1))
d <- bind_rows(d_duncan_b_4, d_duncan_b_5)

d |> 
  mutate(x = factor(x, levels = 8:1)) |>
  ggplot(aes(x = x, y = y, 
             group = Model, 
             linetype = Model,
             color = Model)) + 
  geom_line() + 
  ylim(1,3) + 
  scale_linetype_manual(values = c("longdash","solid")) +
  labs(x = "FATHER'S OCCUPATION (i)", y = "b_i/b_8") +
  theme_classic() + 
  ggthemes::scale_color_colorblind()

# mosaic plot
mosaic(fit_d_1, ~ O + D, residuals_type = "rstandard")
mosaic(fit_d_2, ~ O + D, residuals_type = "rstandard")
mosaic(fit_d_3b, ~ O + D, residuals_type = "rstandard")
mosaic(fit_d_4b, ~ O + D, residuals_type = "rstandard")
mosaic(fit_d_5b, ~ O + D, residuals_type = "rstandard")
```


## Yamaguchi (1987)

@yamaguchi1987 の結果を再現する．なおデータは`vcdExtra`パッケージの`Yamaguchi87`からも入手できる．

```{r}
# Data
Freq_yamaguchi_1987 <- c(1275, 364, 274, 272, 17,
                         1055, 597, 394, 443, 31,
                         1043, 587, 1045, 951, 47,
                         1159, 791, 1323, 2046, 52,
                         666, 496, 1031, 1632, 646,
                         
                         474, 129, 87, 124, 11,
                         300, 218, 171, 220, 8,
                         438, 254, 669, 703, 16,
                         601, 388, 932, 1789, 37,
                         76, 56, 125, 295, 191,
                         
                         127, 101, 24, 30, 12,
                         86, 207, 64, 61, 13,
                         43, 73, 122, 60, 13,
                         35, 51, 62, 66, 11,
                         109, 206, 184, 253, 325)
d_yamaguchi <- tibble(O = gl(5, k = 5, length = 3 * 5 * 5),
                      D = gl(5, k = 1, length = 3 * 5 * 5),
                      C = gl(3, k = 5 * 5, length = 3 * 5 * 5),
                      Freq = Freq_yamaguchi_1987)
d_yamaguchi

# Diagonal parameters
d_yamaguchi <- d_yamaguchi |> 
  mutate(Diag = case_when(O == D ~ O, .default = "0") |> factor())

# Row and column scores
d_yamaguchi <- d_yamaguchi |> 
  mutate(U = as.numeric(O),
         V = as.numeric(D))

# Country dummy
d_yamaguchi <- d_yamaguchi |> 
  mutate(C1 = case_match(C,
                         c("2","3") ~ 0,
                         "1" ~ 1),
         C2 = case_match(C,
                         c("1","3") ~ 0,
                         "2" ~ 1),
         C3 = case_match(C,
                         c("1","2") ~ 0,
                         "3" ~ 1))
  
# Off diagonal cells (Diag*C)
# No association between R and C, given L
fit_y_1 <- d_yamaguchi |> glm(Freq ~ O*C + D*C + Diag*C, data = _, family = poisson)
glance(fit_y_1)

# U0-Cross-nationally homogeneous uniform association
fit_y_2 <- d_yamaguchi |> glm(Freq ~ O*C + D*C + Diag*C + U:V, data = _, family = poisson)
glance(fit_y_2)
b_2 <- fit_y_2$coefficients[grep("U:V", names(fit_y_2$coefficients))]
b_2

# U1-Cross-nationally uniform uniform association
fit_y_3 <- d_yamaguchi |> glm(Freq ~ O*C + D*C + Diag*C + U:V*C, data = _, family = poisson)
glance(fit_y_3)
b_3 <- fit_y_3$coefficients[grep("U:V", names(fit_y_3$coefficients))]
b_3

# U2-Cross-nationally uniform uniform association
fit_y_4 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + U:V + U:V:C3, data = _, family = poisson)
glance(fit_y_4)
b_4 <- fit_y_4$coefficients[grep("U:V", names(fit_y_4$coefficients))]
b_4
      
# R0-Cross-nationally homogeneous row effect association
fit_y_5 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + O:V, data = _, family = poisson)
glance(fit_y_5)

# R1-Cross-nationally uniform row effect association
fit_y_6 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + O:V + U:V*C, data = _, family = poisson)
glance(fit_y_6)
b_6 <- fit_y_6$coefficients[grep("V:U", names(fit_y_6$coefficients))]
b_6

# R2-Cross-nationally uniform row effect association
fit_y_7 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + O:V + U:V:C3, data = _, family = poisson)
glance(fit_y_7)
b_7 <- fit_y_7$coefficients[grep("V:U", names(fit_y_7$coefficients))]
b_7

# C0-Cross-nationally homogeneous column effect association
fit_y_8 <- d_yamaguchi |>
  glm(Freq ~ O*C + D*C + Diag*C + U:D, data = _, family = poisson)
glance(fit_y_8)

# C1-Cross-nationally uniform column effect association
fit_y_9 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + U:D + U:V*C, data = _, family = poisson)
glance(fit_y_9)
b_9 <- fit_y_9$coefficients[grep("U:V", names(fit_y_9$coefficients))]
b_9

# C2-Cross-nationally uniform column effect association
fit_y_10 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + U:D + U:V:C3, data = _, family = poisson)
glance(fit_y_10)
b_10 <- fit_y_10$coefficients[grep("U:V", names(fit_y_10$coefficients))]
b_10

# H0-Cross-nationally homogeneous homogeneous row & column effect association
fit_y_11 <- d_yamaguchi |> 
  gnm(Freq ~ O*C + D*C + Diag*C + MultHomog(O, D), 
                               data = _, family = poisson)
glance(fit_y_11)

# H1-Cross-nationally uniform homogeneous row & column effect association
fit_y_12 <- d_yamaguchi |> 
  gnm(Freq ~ O*C + D*C + Diag*C + MultHomog(O, D) + U:V:C2 + U:V:C3, 
                               data = _, family = poisson)
glance(fit_y_12)
b_12 <- fit_y_12$coefficients[grep("U:V", names(fit_y_12$coefficients))]
b_12

# H2-Cross-nationally uniform homogeneous row & column effect association
fit_y_13 <- d_yamaguchi |> gnm(Freq ~ O*C + D*C + Diag*C + MultHomog(O, D) + U:V:C3, 
                               data = _, family = poisson)
glance(fit_y_13)
b_13 <- fit_y_13$coefficients[grep("U:V", names(fit_y_13$coefficients))]
b_13

# R+C0 -Cross-nationally uniform row & column effect association
fit_y_14 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + U:D + O:V, data = _, family = poisson)
glance(fit_y_14)

# R+C1 -Cross-nationally uniform row & column effect association
fit_y_15 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + U:D + O:V + U:V:C2 + U:V:C3, data = _, family = poisson)
glance(fit_y_15)
b_15 <- fit_y_15$coefficients[grep("U:V", names(fit_y_15$coefficients))]
b_15

# R+C2-Cross-nationally uniform row & column effect association
fit_y_16 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + U:D + O:V + U:V:C3, data = _, family = poisson)
glance(fit_y_16)
b_16 <- fit_y_16$coefficients[grep("U:V", names(fit_y_16$coefficients))]
b_16

d_yamaguchi <- d_yamaguchi |> 
  mutate(SYM = c(rep(c(0, 1, 2, 3, 4,
                       1, 0, 5, 6, 7, 
                       2, 5, 0, 8, 9,
                       3, 6, 8, 0, 10,
                       4, 7, 9, 10, 0), 3)) |>  factor())

d_yamaguchi |> dplyr::select(O, D, C, SYM) |> unique()

# QS0-Cross-nationally homogeneous quasi-symmetry
fit_y_17 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + SYM, data = _, family = poisson)
glance(fit_y_17)
tidy(fit_y_17) |> print(n = Inf)

# QS1-Cross-nationally uniform quasi-symmetry
fit_y_18 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + SYM + U:V:C2 + U:V:C3, data = _, family = poisson)
glance(fit_y_18)
b_18 <- fit_y_18$coefficients[grep("U:V", names(fit_y_18$coefficients))]
b_18

# QS2-Cross-nationally uniform quasi-symmetry
fit_y_19 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + SYM + U:V:C3, data = _, family = poisson)
glance(fit_y_19)
b_19 <- fit_y_19$coefficients[grep("U:V", names(fit_y_19$coefficients))]
b_19

# FI0-Cross-nationallyhomogeneous full two-way interaction
fit_y_20 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + O:D, data = _, family = poisson)
glance(fit_y_20)

# FI1-Cross-nationally uniform full two-way interaction
fit_y_21 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + O:D + U:V:C2 + U:V:C3, data = _, family = poisson)
glance(fit_y_21)
b_21 <- fit_y_21$coefficients[grep("U:V", names(fit_y_21$coefficients))]
b_21

# FI2-Cross-nationally uniform full two-way interaction
fit_y_22 <- d_yamaguchi |> 
  glm(Freq ~ O*C + D*C + Diag*C + O:D + U:V:C3, data = _, family = poisson)
glance(fit_y_22)
b_22 <- fit_y_22$coefficients[grep("U:V", names(fit_y_22$coefficients))]
b_22
```

```{r}
Model <- c("No association between R and C, given L",
           "U0-Cross-nationally homogeneous uniform association",
           "U1-Cross-nationally uniform uniform association",
           "U2-Cross-nationally uniform uniform association",
           "R0-Cross-nationally homogeneous row effect association",
           "R1-Cross-nationally uniform row effect association",
           "R2-Cross-nationally uniform row effect association",
           "C0-Cross-nationally homogeneous column effect association",
           "C1-Cross-nationally uniform column effect association",
           "C2-Cross-nationally uniform column effect association",
           "H0-Cross-nationally homogeneous homogeneous row & column effect association",
           "H1-Cross-nationally uniform homogeneousrow & column effect association",
           "H2-Cross-nationally uniform homogeneousrow & column effect association",
           "R+C0 -Cross-nationally uniform row & column effect association",
           "R+C1 -Cross-nationally uniform row & column effect association",
           "R+C2-Cross-nationally uniform row & column effect association",
           "QS0-Cross-nationally homogeneous quasi-symmetry",
           "QS1-Cross-nationally uniform quasi-symmetry",
           "QS2-Cross-nationally uniform quasi-symmetry",
           "FI0-Cross-nationallyhomogeneous full two-way interaction",
           "FI1-Cross-nationally uniform full two-way interaction",
           "FI2-Cross-nationally uniform full two-way interaction")

list(fit_y_1, fit_y_2, fit_y_3, fit_y_4, fit_y_5, fit_y_6, fit_y_7, fit_y_8, fit_y_9, fit_y_10,
     fit_y_11, fit_y_12, fit_y_13, fit_y_14, fit_y_15, fit_y_16, fit_y_17, fit_y_18, fit_y_19,
     fit_y_20, fit_y_21, fit_y_22) |> 
  map_dfr(glance) |> 
  mutate(Model = Model, .before = 1) |> 
  dplyr::select(-c(2:3)) |> 
  print(n = Inf)
```

H1とH2が微妙に異なるが誤差の範囲か．また，FI1の$\beta_1$については符号が @yamaguchi1987 のものとは異なっているがこちらが正しいか．

```{r}
# R+C2: No uniform difference between any two
fit_y_23 <- d_yamaguchi |> 
  gnm(Freq ~ O*C + D*C + Diag*C + U:D + O:V + U:V:C3, 
      data = _, family = poisson)
glance(fit_y_23)
tidy(fit_y_23) |> print(n = Inf)
# R+C3: Uniform difference between G.B. and U.S.
```


## Xie (1992)

@xie1992 の結果を再現する．


```{r}
# Data
d_xie <- d_yamaguchi
# 確認
d_xie
```



```{r}
# Off diagonal cells (Diag*C)
# Null association between R and C, given L
fit_x_1 <- d_xie |> glm(Freq ~ O*C + D*C + Diag*C, data = _, family = poisson)
glance(fit_x_1)

# R0 Cross-nationally homogeneous row effect association
fit_x_2 <- d_xie |> glm(Freq ~ O*C + D*C + Diag*C + O:V, data = _, family = poisson)
glance(fit_x_2)

# R1 Cross-nationally uniform row effect association
fit_x_3 <- d_xie |> glm(Freq ~ O*C + D*C + Diag*C + O:V + U:V*C, data = _, family = poisson)
glance(fit_x_3)

# Rx Cross-nationally log-multiplicative row effect association
fit_x_4 <- d_xie |> gnm(Freq ~ O*C + D*C + Diag*C + Mult(C,O,V), data = _, family = poisson)
glance(fit_x_4)

# C0-Cross-nationally homogeneous column effect association
fit_x_5 <- d_xie |> glm(Freq ~ O*C + D*C + Diag*C + U:D, data = _, family = poisson)
glance(fit_x_5)

# C1-Cross-nationally uniform column effect association
fit_x_6 <- d_xie |> glm(Freq ~ O*C + D*C + Diag*C + U:D + U:V*C, data = _, family = poisson)
glance(fit_x_6)

# Cx-Cross-nationally log-multiplicative column effect association
fit_x_7 <- d_xie |> gnm(Freq ~ O*C + D*C + Diag*C + Mult(C,U,D), data = _, family = poisson)
glance(fit_x_7)

# (R+C) 0 - Cross-nationally homogeneous row and column effects association I
fit_x_8 <- d_xie |> gnm(Freq ~ O*C + D*C + Diag*C  + O:V + U:D, data = _, family = poisson)
glance(fit_x_8)

# (R+C) u - Cross-nationally uniform row and column effects association I
fit_x_9 <- d_xie |> gnm(Freq ~ O*C + D*C + Diag*C  + O:V + U:D + U:V*C, data = _, family = poisson)
glance(fit_x_9)

# (R+C) x - Cross-nationally log-multiplicative row and column effects association I
fit_x_10 <- d_xie |> gnm(Freq ~ O*C + D*C + Diag*C + Mult(C,O:V+U:D), data = _, family = poisson)
glance(fit_x_10)

# RC0 Cross-nationally homogeneous row and column effects association
fit_x_11 <- d_xie |> gnm(Freq ~ O*C + D*C + Diag*C + Mult(O,D), data = _, family = poisson)
glance(fit_x_11)

# RCx Cross-nationally log-multiplicative row and column effects association
fit_x_12 <- d_xie |> gnm(Freq ~ O*C + D*C + Diag*C + Mult(Exp(C),O,D), data = _, family = poisson)
glance(fit_x_12)

# FI0 Cross-nationally homogeneous full two-way R and C interaction
fit_x_13 <- d_xie |> gnm(Freq ~ O*C + D*C + Diag*C + Mult(1,O*D), data = _, family = poisson)
glance(fit_x_13)

# FIu Cross-nationally uniform full two-way R and C interaction
fit_x_14 <- d_xie |> gnm(Freq ~ O*C + D*C + Diag*C + Mult(1,O*D) + U:V*C, data = _, family = poisson)
glance(fit_x_14)

# FIx Cross-nationally log-multiplicative full two-way R and C interaction
fit_x_15 <- d_xie |> gnm(Freq ~ O*C + D*C + Diag*C + Mult(Exp(C),O*D), data = _, family = poisson)
glance(fit_x_15)
```

```{r}
Model <- c("Null association between R and C, given L",
           "R0 Cross-nationally homogeneous row effect association",
           "R1 Cross-nationally uniform row effect association",
           "Rx Cross-nationally log-multiplicative row effect association",
           "C0-Cross-nationally homogeneous column effect association",
           "C1-Cross-nationally uniform column effect association",
           "Cx-Cross-nationally log-multiplicative column effect association",
           "(R+C) 0 - Cross-nationally homogeneous row and column effects association I",
           "(R+C) u - Cross-nationally uniform row and column effects association I",
           "(R+C) x - Cross-nationally log-multiplicative row and column effects association I",
           "RC0 Cross-nationally homogeneous row and column effects association",
           "RCx Cross-nationally log-multiplicative row and column effects association",
           "FI0 Cross-nationally homogeneous full two-way R and C interaction",
           "FIu Cross-nationally uniform full two-way R and C interaction",
           "FIx Cross-nationally log-multiplicative full two-way R and C interaction")

list(fit_x_1, fit_x_2, fit_x_3, fit_x_4, fit_x_5, 
     fit_x_6, fit_x_7, fit_x_8, fit_x_9, fit_x_10,
     fit_x_11, fit_x_12, fit_x_13, fit_x_14, fit_x_15) |> 
  map_dfr(glance) |> 
  mutate(Model = Model, .before = 1) |> 
  dplyr::select(-c(2:3)) |> 
  print(n = Inf)
```





## 参考文献 {-}