GAMS

Posted: Nov 11, 2018 | Updated: Sep 16, 2024
Category:
statistics
    by Sir Panda
Comments
   : 6 Minutes
   : 3567 Words
   : 18 Scripts 
   : 1 Figures  
library(mgcv) # Fit and interrogate GAMs
library(tidyverse) # Tidy and flexible data manipulation
library(marginaleffects) # Compute conditional and marginal effects
library(ggplot2) # Flexible plotting
library(patchwork) # Combining ggplot objects


plant <- CO2 |>
  as_tibble() |>
  rename(plant = Plant, type = Type, treatment = Treatment) |>
  mutate(plant = factor(plant, ordered = FALSE))
model_1 <- gam(
  uptake ~ treatment * type +
    s(plant, bs = "re") +
    s(conc, by = treatment, k = 7),
  data = plant,
  method = "REML",
  family = Gamma(link = "log")
)
summary(model_1)
#> 
#> Family: Gamma 
#> Link function: log 
#> 
#> Formula:
#> uptake ~ treatment * type + s(plant, bs = "re") + s(conc, by = treatment, 
#>     k = 7)
#> 
#> Parametric coefficients:
#>                                  Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)                        3.5161     0.0638   55.14  < 2e-16 ***
#> treatmentchilled                  -0.1132     0.0902   -1.26  0.21399    
#> typeMississippi                   -0.3120     0.0902   -3.46  0.00098 ***
#> treatmentchilled:typeMississippi  -0.3604     0.1275   -2.83  0.00631 ** 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Approximate significance of smooth terms:
#>                              edf Ref.df     F p-value    
#> s(plant)                    7.03   8.00  8.02  <2e-16 ***
#> s(conc):treatmentnonchilled 5.19   5.68 83.89  <2e-16 ***
#> s(conc):treatmentchilled    5.01   5.55 58.97  <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> R-sq.(adj) =  0.957   Deviance explained = 95.5%
#> -REML = 239.08  Scale est. = 0.010327  n = 84
coef(model_1)
#>                      (Intercept)                 treatmentchilled                  typeMississippi 
#>                          3.51609                         -0.11321                         -0.31196 
#> treatmentchilled:typeMississippi                       s(plant).1                       s(plant).2 
#>                         -0.36041                         -0.04123                         -0.01529 
#>                       s(plant).3                       s(plant).4                       s(plant).5 
#>                          0.05652                         -0.04010                          0.02693 
#>                       s(plant).6                       s(plant).7                       s(plant).8 
#>                          0.01317                         -0.05593                          0.04989 
#>                       s(plant).9                      s(plant).10                      s(plant).11 
#>                          0.00604                         -0.21558                          0.09712 
#>                      s(plant).12    s(conc):treatmentnonchilled.1    s(conc):treatmentnonchilled.2 
#>                          0.11846                          5.23232                          1.13324 
#>    s(conc):treatmentnonchilled.3    s(conc):treatmentnonchilled.4    s(conc):treatmentnonchilled.5 
#>                         -1.90483                          0.12919                          0.23606 
#>    s(conc):treatmentnonchilled.6       s(conc):treatmentchilled.1       s(conc):treatmentchilled.2 
#>                          1.20260                          3.69059                          2.11750 
#>       s(conc):treatmentchilled.3       s(conc):treatmentchilled.4       s(conc):treatmentchilled.5 
#>                         -2.36770                          0.06347                          0.23654 
#>       s(conc):treatmentchilled.6 
#>                          0.96775
plot(model_1, select = 2, shade = TRUE)
abline(h = 0, lty = "dashed")

plot_predictions(model_1,
  condition = "conc",
  type = "link"
) +
  labs(
    y = "Linear predictor (link scale)",
    title = "Average smooth effect of concentration",
    subtitle = "Aggregated across treatments and types"
  )

plot_predictions(model_1,
  condition = c("conc", "treatment", "type"),
  type = "link"
) +
  labs(
    y = "Linear predictor (link scale)",
    title = "Average smooth effect of concentration",
    subtitle = "Per treatment, per type"
  )

plot_slopes(model_1,
  variables = "conc",
  condition = c("conc", "treatment"),
  type = "link"
) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  labs(
    y = "1st derivative of the linear predictor",
    title = "Conditional slopes of the concentration effect",
    subtitle = "Per treatment, per type"
  )

plot_predictions(model_1,
  condition = "conc",
  type = "response", points = 0.5,
  rug = TRUE
) +
  labs(
    y = "Expected response",
    title = "Average smooth effect of concentration",
    subtitle = "Aggregated across treatments and types"
  )

plot_predictions(model_1,
  condition = c("conc", "treatment", "type"),
  type = "response", points = 0.5,
  rug = TRUE
) +
  labs(
    y = "Expected response",
    title = "Average smooth effect of concentration",
    subtitle = "Per treatment, per type"
  )

plot_slopes(model_1,
  variables = "conc",
  condition = c("conc", "treatment", "type"),
  type = "response"
) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  labs(
    y = "1st derivative of the expected response",
    title = "Conditional slopes of the concentration effect",
    subtitle = "Per treatment, per type"
  )

Xp <- predict(model_1, type = "lpmatrix")
dim(Xp)
#> [1] 84 28
colnames(Xp)
#>  [1] "(Intercept)"                      "treatmentchilled"                 "typeMississippi"                 
#>  [4] "treatmentchilled:typeMississippi" "s(plant).1"                       "s(plant).2"                      
#>  [7] "s(plant).3"                       "s(plant).4"                       "s(plant).5"                      
#> [10] "s(plant).6"                       "s(plant).7"                       "s(plant).8"                      
#> [13] "s(plant).9"                       "s(plant).10"                      "s(plant).11"                     
#> [16] "s(plant).12"                      "s(conc):treatmentnonchilled.1"    "s(conc):treatmentnonchilled.2"   
#> [19] "s(conc):treatmentnonchilled.3"    "s(conc):treatmentnonchilled.4"    "s(conc):treatmentnonchilled.5"   
#> [22] "s(conc):treatmentnonchilled.6"    "s(conc):treatmentchilled.1"       "s(conc):treatmentchilled.2"      
#> [25] "s(conc):treatmentchilled.3"       "s(conc):treatmentchilled.4"       "s(conc):treatmentchilled.5"      
#> [28] "s(conc):treatmentchilled.6"
beta <- coef(model_1)
all.equal(names(beta), colnames(Xp))
#> [1] TRUE
preds <- as.vector(Xp %*% beta)
all.equal(fitted(model_1), exp(preds))
#> [1] TRUE
newXp <- predict(model_1,
  type = "lpmatrix",
  newdata = data.frame(
    plant = "Qn1",
    treatment = "nonchilled",
    type = "Mississippi",
    conc = 278
  )
)
dim(newXp)
#> [1]  1 28
exp(newXp %*% beta)
#>   [,1]
#> 1 27.6
conc_seq <- seq(from = min(plant$conc), max(plant$conc), length.out = 500)
newdat <- data.frame(
  conc = conc_seq,
  plant = "Qn1",
  treatment = "nonchilled",
  type = "Mississippi"
)
newXp <- predict(model_1, type = "lpmatrix", newdata = newdat)
conc_coefs <- model_1$smooth[[2]]$first.para:model_1$smooth[[2]]$last.para
conc_coefs
#> [1] 17 18 19 20 21 22

Trace plot of imputed datasets.

Python

model_2 <- glm(
  uptake ~ treatment * type +
    poly(conc, 4) * treatment + plant,
  data = plant,
  family = Gamma(link = "log")
)
plot_predictions(model_1,
  condition = c("conc", "treatment", "type"),
  type = "response", points = 0.5,
  rug = TRUE
) +
  labs(
    y = "Expected response",
    title = "Average smooth effect of concentration",
    subtitle = "Per treatment, per type"
  )

plot_predictions(model_2,
  condition = c("conc", "treatment", "type"),
  type = "response", points = 0.5,
  rug = TRUE
) +
  labs(
    y = "Expected response",
    title = "Average smooth effect of concentration",
    subtitle = "Per treatment, per type"
  )
#> Warning: Model matrix is rank deficient. Some variance-covariance parameters are missing.

R

avg_comparisons(model_1,
  newdata = datagrid(
    conc = conc_seq,
    treatment = unique,
    type = unique
  ),
  variables = "treatment",
  by = "type"
)
#> 
#>       Term        type Estimate Std. Error     z Pr(>|z|)    S 2.5 % 97.5 %
#>  treatment Quebec         -4.76       3.04 -1.57    0.117  3.1 -10.7   1.19
#>  treatment Mississippi   -10.71       2.20 -4.87   <0.001 19.8 -15.0  -6.40
#> 
#> Type:  response 
#> Comparison: mean(chilled) - mean(nonchilled)
#> Columns: term, contrast, type, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
avg_comparisons(model_1,
  newdata = datagrid(
    conc = conc_seq,
    treatment = unique,
    type = unique
  ),
  variables = "treatment",
  by = c("conc", "type")
)
#> 
#>       Term   conc        type Estimate Std. Error      z Pr(>|z|)   S  2.5 % 97.5 %
#>  treatment   95.0 Quebec         0.413       1.60  0.258  0.79644 0.3  -2.73   3.55
#>  treatment   95.0 Mississippi   -3.114       1.02 -3.057  0.00224 8.8  -5.11  -1.12
#>  treatment   96.8 Quebec         0.373       1.61  0.232  0.81662 0.3  -2.78   3.53
#>  treatment   96.8 Mississippi   -3.183       1.03 -3.102  0.00192 9.0  -5.19  -1.17
#>  treatment   98.6 Quebec         0.332       1.62  0.205  0.83731 0.3  -2.84   3.51
#> --- 990 rows omitted. See ?print.marginaleffects --- 
#>  treatment  996.4 Mississippi  -11.426       2.75 -4.156  < 0.001 14.9 -16.81  -6.04
#>  treatment  998.2 Quebec        -4.436       3.97 -1.119  0.26334  1.9 -12.21   3.34
#>  treatment  998.2 Mississippi  -11.427       2.76 -4.147  < 0.001 14.9 -16.83  -6.03
#>  treatment 1000.0 Quebec        -4.435       3.98 -1.115  0.26467  1.9 -12.23   3.36
#>  treatment 1000.0 Mississippi  -11.428       2.76 -4.138  < 0.001 14.8 -16.84  -6.02
#> Type:  response 
#> Comparison: mean(chilled) - mean(nonchilled)
#> Columns: term, contrast, conc, type, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
plot_comparisons(model_1,
  newdata = datagrid(
    conc = conc_seq,
    treatment = unique,
    type = unique
  ),
  variables = "treatment",
  by = c("conc", "type"),
  type = "link"
) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  labs(
    y = "Estimated difference",
    title = "Difference between treatment levels",
    subtitle = "Chilled - nonchilled, per type"
  )

Stata

max_growth <- function(hi, lo, x) {
  dydx <- (hi - lo) / 1e-6
  dydx_max <- max(dydx)
  x[dydx == dydx_max][1]
}

comparisons(model_1,
  newdata = datagrid(
    conc = conc_seq,
    treatment = unique,
    type = unique
  ),
  variables = list("conc" = 1e-6),
  vcov = FALSE,
  by = "treatment",
  comparison = max_growth
)
#> 
#>   treatment Estimate
#>  nonchilled      157
#>  chilled         151
#> 
#> Term: conc
#> Type:  response 
#> Comparison: +1e-06
#> Columns: term, contrast, treatment, estimate
hypotheses(slopes(model_1,
  newdata = datagrid(
    conc = conc_seq,
    treatment = unique,
    type = unique
  ),
  variables = "conc",
  by = c("conc", "treatment"),
  type = "link"
)) %>%
  dplyr::filter(p.value <= 0.05)
#> 
#>  Term  conc  treatment Estimate Std. Error    z Pr(>|z|)    S    2.5 %  97.5 %
#>  conc  95.0 nonchilled  0.00809   0.000876 9.23   <0.001 65.0 6.37e-03 0.00980
#>  conc  95.0 chilled     0.00647   0.000828 7.81   <0.001 47.3 4.85e-03 0.00809
#>  conc  96.8 nonchilled  0.00809   0.000878 9.21   <0.001 64.8 6.37e-03 0.00981
#>  conc  96.8 chilled     0.00647   0.000829 7.80   <0.001 47.2 4.84e-03 0.00809
#>  conc  98.6 nonchilled  0.00808   0.000874 9.25   <0.001 65.2 6.37e-03 0.00980
#> --- 196 rows omitted. See ?print.marginaleffects --- 
#>  conc 278.2 nonchilled  0.00127   0.000507 2.51   0.0122 6.4 2.78e-04 0.00227
#>  conc 280.0 nonchilled  0.00122   0.000521 2.35   0.0190 5.7 2.01e-04 0.00224
#>  conc 281.8 nonchilled  0.00117   0.000522 2.25   0.0246 5.3 1.50e-04 0.00220
#>  conc 283.6 nonchilled  0.00113   0.000536 2.10   0.0357 4.8 7.50e-05 0.00218
#>  conc 285.4 nonchilled  0.00108   0.000546 1.98   0.0482 4.4 8.71e-06 0.00215
#> Type:  link 
#> Comparison: mean(dY/dX)
#> Columns: term, contrast, conc, treatment, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted

Computational Environments

Overall Platforms

sessioninfo::session_info(info = c("platform", "external"))
#> Error in info != "auto" && info != "all": 'length = 2' in coercion to 'logical(1)'

R Packages

sessioninfo::session_info(info = "packages")
#> ═ Session info ═════════════════════════════════════════════════════════════════════════════════════════════════════════════════
#> ─ Packages ─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
#>  package           * version    date (UTC) lib source
#>  abind               1.4-8      2024-09-12 [1] CRAN (R 4.4.1)
#>  Amelia            * 1.8.2      2024-04-23 [1] CRAN (R 4.4.0)
#>  arm                 1.14-4     2024-04-01 [1] CRAN (R 4.4.0)
#>  arrayhelpers        1.1-0      2020-02-04 [1] CRAN (R 4.4.0)
#>  askpass             1.2.0      2023-09-03 [1] CRAN (R 4.4.0)
#>  backports           1.5.0      2024-05-23 [1] CRAN (R 4.4.0)
#>  base64enc           0.1-3      2015-07-28 [1] CRAN (R 4.4.0)
#>  bayesplot         * 1.11.1     2024-02-15 [1] CRAN (R 4.4.0)
#>  bcaboot             0.2-3      2021-05-09 [1] CRAN (R 4.4.0)
#>  bitops              1.0-8      2024-07-29 [1] CRAN (R 4.4.0)
#>  blogdown          * 1.19.1     2024-08-17 [1] Github (rstudio/blogdown@c6e73fb)
#>  bookdown            0.40       2024-07-02 [1] CRAN (R 4.4.0)
#>  boot              * 1.3-31     2024-08-28 [1] CRAN (R 4.4.1)
#>  bootImpute        * 1.2.1      2023-06-01 [1] CRAN (R 4.4.0)
#>  bridgesampling      1.1-2      2021-04-16 [1] CRAN (R 4.4.0)
#>  brms              * 2.21.0     2024-03-20 [1] CRAN (R 4.4.0)
#>  Brobdingnag         1.2-9      2022-10-19 [1] CRAN (R 4.4.0)
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#>  car               * 3.1-2      2023-03-30 [1] CRAN (R 4.4.0)
#>  carData           * 3.0-5      2022-01-06 [1] CRAN (R 4.4.0)
#>  caTools             1.18.3     2024-09-04 [1] CRAN (R 4.4.1)
#>  checkmate         * 2.3.2      2024-07-29 [1] CRAN (R 4.4.0)
#>  class               7.3-22     2023-05-03 [1] CRAN (R 4.4.0)
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