Having fitted a GAM or other model containing penalised splines, we
often want to evaluate the model at some pre-specified values of the
covariates. For more complex models, this will typically involve holding
some covariates at fixed, representative values while visualising the
change in the response or effect of a smooth over supplied values of one
or more other covariates. The values of the covariates at which we
evaluate a smooth or a model are called a *data slice*^{1}.

This article will explain how to create data slices with {gratia} and
its `data_slice()`

function, and how to use them to visualise
features of your fitted GAMs.

We’ll need the following packages for this article

```
library("mgcv")
#> Loading required package: nlme
#> This is mgcv 1.9-1. For overview type 'help("mgcv-package")'.
```

```
library("gratia")
library("dplyr")
#>
#> Attaching package: 'dplyr'
#> The following object is masked from 'package:nlme':
#>
#> collapse
#> The following objects are masked from 'package:stats':
#>
#> filter, lag
#> The following objects are masked from 'package:base':
#>
#> intersect, setdiff, setequal, union
```

The first example uses a small data set from an experimental study of
the cold tolerance of the grass *Echinochloa crusgalli*. The data
are in data frame `CO2`

and provided with the {datasets}
package that ships with R.

```
## data load and prep
data(CO2, package = "datasets")
plant <- CO2 |>
as_tibble() |>
rename(plant = Plant, type = Type, treatment = Treatment) |>
mutate(plant = factor(plant, ordered = FALSE))
```

```
plant_ylab <- expression(CO[2] ~ uptake ~ (mu * mol ~ m^{-3})) # nolint: brace_linter.
plant_xlab <- expression(CO[2] ~ concentration ~ (mL ~ L^{-1})) # nolint: brace_linter.
plant |>
ggplot(aes(x = conc, y = uptake, group = plant, colour = treatment)) +
geom_point() +
geom_line() +
facet_wrap(~type) +
labs(y = plant_ylab, x = plant_xlab, colour = "Treatment")
```

One way to model these data is to allow for different smooths for all
combinations of the `treatment`

and `type`

covariates

```
plant <- plant |>
mutate(tt = fct_cross(treatment, type))
m_plant <- gam(uptake ~ treatment * type + s(conc, by = tt, k = 6) +
s(plant, bs = "re"),
data = plant, method = "REML", family = Gamma(link = "log")
)
overview(m_plant)
#>
#> Generalized Additive Model with 8 terms
#>
#> term type k edf statistic p.value
#> <chr> <chr> <dbl> <dbl> <dbl> <chr>
#> 1 treatment parametric NA 1 1.59 0.2124864
#> 2 type parametric NA 1 11.2 0.0014830
#> 3 treatment:type parametric NA 1 7.45 0.0085489
#> 4 s(conc):ttnonchilled:Quebec TPRS 5 4.72 69.7 < 0.001
#> 5 s(conc):ttchilled:Quebec TPRS 5 4.71 86.5 < 0.001
#> 6 s(conc):ttnonchilled:Mississippi TPRS 5 4.62 74.1 < 0.001
#> 7 s(conc):ttchilled:Mississippi TPRS 5 4.39 25.3 < 0.001
#> 8 s(plant) Random effect 12 7.40 12.8 < 0.001
```

We can look at the fitted smooths using `draw()`

We might want to compare model fitted values for the treatment for
each of the types (origins), ignoring the random effect component. For
this we want to evaluate the model at a range of values of covariate
`conc`

for some combinations of the other factors. This is a
data slice through the covariate space, which we can create using
`data_slice()`

. To create a data slice for `conc`

for the `Quebec`

`type`

in the
`chilled`

`treatment`

we would use

```
ds1 <- data_slice(m_plant,
conc = evenly(conc, n = 100),
type = level(type, "Quebec"), treatment = level(treatment, "chilled")
)
ds1
#> # A tibble: 100 × 5
#> conc type treatment tt plant
#> <dbl> <fct> <fct> <fct> <fct>
#> 1 95 Quebec chilled nonchilled:Quebec Qn1
#> 2 104. Quebec chilled nonchilled:Quebec Qn1
#> 3 113. Quebec chilled nonchilled:Quebec Qn1
#> 4 122. Quebec chilled nonchilled:Quebec Qn1
#> 5 132. Quebec chilled nonchilled:Quebec Qn1
#> 6 141. Quebec chilled nonchilled:Quebec Qn1
#> 7 150. Quebec chilled nonchilled:Quebec Qn1
#> 8 159. Quebec chilled nonchilled:Quebec Qn1
#> 9 168. Quebec chilled nonchilled:Quebec Qn1
#> 10 177. Quebec chilled nonchilled:Quebec Qn1
#> # ℹ 90 more rows
```

Notice how `data_slice()`

has filled in something for the
remaining covariates that we didn’t mention? In this case,
`data_slice()`

doesn’t know how `tt`

was created,
so it has chosen the modal level for the `tt`

factor, which
is not the correct choice in this case. Instead, we need to specify the
correct level explicitly for `tt`

```
ds1 <- data_slice(m_plant,
conc = evenly(conc, n = 100),
treatment = level(treatment, "chilled"), type = level(type, "Quebec"),
tt = level(tt, "chilled:Quebec")
)
ds1
#> # A tibble: 100 × 5
#> conc treatment type tt plant
#> <dbl> <fct> <fct> <fct> <fct>
#> 1 95 chilled Quebec chilled:Quebec Qn1
#> 2 104. chilled Quebec chilled:Quebec Qn1
#> 3 113. chilled Quebec chilled:Quebec Qn1
#> 4 122. chilled Quebec chilled:Quebec Qn1
#> 5 132. chilled Quebec chilled:Quebec Qn1
#> 6 141. chilled Quebec chilled:Quebec Qn1
#> 7 150. chilled Quebec chilled:Quebec Qn1
#> 8 159. chilled Quebec chilled:Quebec Qn1
#> 9 168. chilled Quebec chilled:Quebec Qn1
#> 10 177. chilled Quebec chilled:Quebec Qn1
#> # ℹ 90 more rows
```

Having created the data slice, we can predict from the model using
the combination of covariate values specified in our slice. We could use
`predict.gam()`

for this, but the
`fitted_values()`

function in {gratia} is easier to use,
especially for non-Gaussian models

```
fv1 <- fitted_values(m_plant, data = ds1, scale = "response", exclude = "s(plant)")
fv1
#> # A tibble: 100 × 10
#> .row conc treatment type tt plant .fitted .se .lower_ci .upper_ci
#> <int> <dbl> <fct> <fct> <fct> <fct> <dbl> <dbl> <dbl> <dbl>
#> 1 1 95 chilled Quebec chille… Qn1 13.0 0.0783 11.2 15.2
#> 2 2 104. chilled Quebec chille… Qn1 14.1 0.0757 12.1 16.3
#> 3 3 113. chilled Quebec chille… Qn1 15.2 0.0737 13.1 17.5
#> 4 4 122. chilled Quebec chille… Qn1 16.3 0.0722 14.2 18.8
#> 5 5 132. chilled Quebec chille… Qn1 17.6 0.0714 15.3 20.2
#> 6 6 141. chilled Quebec chille… Qn1 18.9 0.0711 16.4 21.7
#> 7 7 150. chilled Quebec chille… Qn1 20.2 0.0712 17.6 23.3
#> 8 8 159. chilled Quebec chille… Qn1 21.6 0.0716 18.8 24.9
#> 9 9 168. chilled Quebec chille… Qn1 23.0 0.0721 20.0 26.5
#> 10 10 177. chilled Quebec chille… Qn1 24.4 0.0726 21.2 28.1
#> # ℹ 90 more rows
```

Notice how we excluded the random effect term; even though we had to
specify something for the `plant`

covariate we can ignore
this term in the model using the `exclude`

argument.
`fitted_values()`

creates the credible interval on the scale
of the link function and then back-transforms to the response scale when
`scale = "response"`

, which is also the default.

Plotting the fitted values for the data slice now only requires some simple {ggplot2} knowledge

```
fv1 |>
ggplot(aes(x = conc, y = .fitted)) +
geom_point(
data = plant |>
filter(type == "Quebec", treatment == "chilled"),
mapping = aes(y = uptake),
alpha = 0.8, colour = "steelblue"
) +
geom_ribbon(aes(ymin = .lower_ci, ymax = .upper_ci), alpha = 0.2) +
geom_line() +
labs(
x = plant_xlab, y = plant_ylab,
title = expression(Estimated ~ CO[2] ~ uptake),
subtitle = "Chilled plants of the Quebec type"
)
```

Next, let’s compare the fitted effects of the treatment in the Mississippi origin plants

```
ds2 <- data_slice(m_plant,
conc = evenly(conc, n = 100),
treatment = evenly(treatment), type = level(type, "Mississippi")
) |>
mutate(tt = fct_cross(treatment, type, keep_empty = TRUE))
ds2
#> # A tibble: 200 × 5
#> conc treatment type tt plant
#> <dbl> <fct> <fct> <fct> <fct>
#> 1 95 nonchilled Mississippi nonchilled:Mississippi Qn1
#> 2 95 chilled Mississippi chilled:Mississippi Qn1
#> 3 104. nonchilled Mississippi nonchilled:Mississippi Qn1
#> 4 104. chilled Mississippi chilled:Mississippi Qn1
#> 5 113. nonchilled Mississippi nonchilled:Mississippi Qn1
#> 6 113. chilled Mississippi chilled:Mississippi Qn1
#> 7 122. nonchilled Mississippi nonchilled:Mississippi Qn1
#> 8 122. chilled Mississippi chilled:Mississippi Qn1
#> 9 132. nonchilled Mississippi nonchilled:Mississippi Qn1
#> 10 132. chilled Mississippi chilled:Mississippi Qn1
#> # ℹ 190 more rows
```

Here, we replaced the automatically-generated `tt`

variable with the correctly specified call to `fct_cross()`

,
retaining the levels of the `type`

and `treatment`

factors. This insures that we have the correct combinations
corresponding to the `treatment`

and `type`

factors but also that we preserve the original levels of the
`tt`

covariate we created.

We can again visualise the fitted values for this data slice

```
fitted_values(m_plant,
data = ds2, scale = "response",
exclude = "s(plant)"
) |>
ggplot(aes(x = conc, y = .fitted, group = treatment)) +
geom_point(
data = plant |> filter(type == "Mississippi"),
mapping = aes(y = uptake, colour = treatment),
alpha = 0.8
) +
geom_ribbon(aes(ymin = .lower_ci, ymax = .upper_ci, fill = treatment),
alpha = 0.2
) +
geom_line(aes(colour = treatment)) +
labs(
x = plant_xlab, y = plant_ylab,
title = expression(Estimated ~ CO[2] ~ uptake),
subtitle = "Comparison of treatment in plants of the Mississippi type",
colour = "Treatment", fill = "Treatment"
)
```

When we were creating our data slices, we used some helper functions to specify covariate values for the slice. {gratia} provides several such helper functions:

`evenly(x, n = 100)`

— creates`n`

evenly spaced values over the range of the covariate,`evenly(x, by = 5`

— creates evenly spaced values over the range of the covariate in increments of 5,`evenly(x, ..., lower = 5, upper = 10)`

— either of the two uses of`evenly()`

shown above will use the lower and upper limits of the vector`x`

. Arguments`lower`

and`upper`

can be used to change one or both of the upper or lower bounds.`evenly(fct)`

— produces a new factor containing each level of the specified factor`fct`

just once,`ref_level(fct)`

— creates a new factor containing just the reference level of the specified factor covariate`fct`

, and`level(fct, "level")`

— creates a factor with requested`"level"`

from the factor`fct`

.

In all cases involving factors, the helper functions set the levels
of the factor to match those in the original model fit^{2}.

The second argument to `data_slice()`

is
`...`

The `...`

argument allows you to provide expressions to
create the covariate values you want for your data slice. Expressions
passed to `...`

are evaluated within the model frame of the
fitted model (argument `object`

) or in `data`

(if
supplied). You are not restricted either to using only the helper
functions provide by {gratia}; any R function could be used as long as
it makes sense in the context of the model frame, and it returns
something that can be combined using
`tidyr::expand_grid()`

.

In the second example, I’ll use the bivariate example data set from
{mgcv} but fit a tensor product of covariates `x`

and
`z`

```
# simulate data from the bivariate surface
df <- data_sim("eg2", n = 1000, scale = 0.25, seed = 2)
# fit the GAM
m_biv <- gam(y ~ te(x, z), data = df, method = "REML")
```

The aim of the example will be to create a univariate data slice
through the 2D smooth at user-specified values of `x`

while
holding `z`

at one or more fixed values. We could visualise
the effect at the smooth level, using `smooth_estimates()`

,
or at the response level, as we did above using
`fitted_values()`

.

`smooth_estimates()`

We begin by creating a slice through the data space. We also create a label at this point for a nice axis label.

```
ds3 <- data_slice(m_biv,
x = evenly(x, n = 100),
z = quantile(z, probs = 0.25)
)
z_val <- with(ds3, round(quantile(z, probs = 0.25), 2))
ylab <- bquote(hat(f)(x, .(z_val)))
```

Then we evaluate the smooth at the desired values and add a confidence interval

```
sm <- smooth_estimates(m_biv, smooth = "te(x,z)", data = ds3) |>
add_confint()
#> Warning: The `smooth` argument of `smooth_estimates()` is deprecated as of gratia
#> 0.8.9.9.
#> ℹ Please use the `select` argument instead.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.
```

```
sm
#> # A tibble: 100 × 9
#> .smooth .type .by .estimate .se x z .lower_ci .upper_ci
#> <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 te(x,z) Tensor prod… <NA> 0.103 0.0583 6.63e-4 0.245 -0.0107 0.218
#> 2 te(x,z) Tensor prod… <NA> 0.122 0.0548 1.08e-2 0.245 0.0148 0.230
#> 3 te(x,z) Tensor prod… <NA> 0.141 0.0514 2.08e-2 0.245 0.0400 0.242
#> 4 te(x,z) Tensor prod… <NA> 0.159 0.0482 3.09e-2 0.245 0.0648 0.254
#> 5 te(x,z) Tensor prod… <NA> 0.177 0.0451 4.10e-2 0.245 0.0890 0.266
#> 6 te(x,z) Tensor prod… <NA> 0.195 0.0422 5.11e-2 0.245 0.113 0.278
#> 7 te(x,z) Tensor prod… <NA> 0.213 0.0396 6.12e-2 0.245 0.135 0.291
#> 8 te(x,z) Tensor prod… <NA> 0.230 0.0372 7.13e-2 0.245 0.157 0.303
#> 9 te(x,z) Tensor prod… <NA> 0.247 0.0351 8.14e-2 0.245 0.178 0.316
#> 10 te(x,z) Tensor prod… <NA> 0.263 0.0333 9.14e-2 0.245 0.198 0.328
#> # ℹ 90 more rows
```

We can plot `sm`

using {ggplot2}

```
sm |>
ggplot(aes(x = x, y = .estimate)) +
geom_ribbon(aes(ymin = .lower_ci, ymax = .upper_ci), alpha = 0.2) +
geom_line() +
labs(
title = "Evaluation of smooth te(x,z) at fixed z",
y = ylab
)
```

Note that the above interval is not the Marra and Wood (2012) interval. It doesn’t include the uncertainty from the model constant term at the moment, but unless the smooth is very close to linear that shouldn’t make too much difference.

This extends to multiple slices by asking for several discrete
`z`

```
ds4 <- data_slice(m_biv,
x = evenly(x, n = 100),
z = round(quantile(z, probs = c(0.25, 0.5, 0.75)), 2)
)
sm <- smooth_estimates(m_biv, smooth = "te(x,z)", data = ds4) |>
add_confint() |>
mutate(fz = factor(z))
sm |>
ggplot(aes(x = x, y = .estimate, colour = fz, group = fz)) +
geom_ribbon(aes(ymin = .lower_ci, ymax = .upper_ci, fill = fz, colour = NULL),
alpha = 0.2
) +
geom_line() +
labs(
title = "Evaluation of smooth te(x,z) at fixed z",
y = expression(hat(f)(x, z)), colour = "z", fill = "z"
)
```

`fitted_samples()`

If you want to evaluate the surface over `x`

at fixed
`z`

conditional upon other values of other covariates (model
predicted or fitted values) then `fitted_samples()`

is a tidy
wrapper to `predict.gam()`

.

For single `z`

we have

```
fitted_values(m_biv, data = ds3) |> # default is response scale, not link
ggplot(aes(x = x, y = .fitted)) +
geom_ribbon(aes(ymin = .lower_ci, ymax = .upper_ci), alpha = 0.2) +
geom_line() +
labs(
title = "Fitted values from model",
y = expression(hat(y))
)
```

And for the multiple `z`

we have

```
fitted_values(m_biv, data = ds4) |>
mutate(fz = factor(z)) |>
ggplot(aes(x = x, y = .fitted, colour = fz, group = fz)) +
geom_ribbon(aes(ymin = .lower_ci, ymax = .upper_ci, fill = fz, colour = NULL),
alpha = 0.2
) +
geom_line() +
labs(
title = "Fitted values from model",
y = expression(hat(y)), colour = "z", fill = "z"
)
```

where the only difference here is that now the model constant is included as well as its uncertainty.