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
#> 0.96775plot(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] TRUEpreds <- 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 27.6conc_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
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, predictedComputational 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 ─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
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#> ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────References
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