Before answering both of your questions, it's important to address the purpose of using ti() (and te() for that matter). Both smoothing functions are used when you're including interaction terms with different scales or units in your model.
On that note, whenever you're trying to examine the effects of an explanatory variable (x1) on y across different values of another explanatory variable (x2), you need to make sure that they're on the same scale.
- Is it correct using ti() (i.e. tensor product) when there is no interaction in the formula?
It's not appropriate to use the ti() function when there is no interaction in the formula. The purpose of ti() is to separate the interactions from individual univariate effects. In order to accomplish this, you use regular smoothing terms (s()) for each variable, and then ti() for each interaction.
It would look something like this: mod_ti <- gam(y ~ s(x1) + s(x2) + ti(x1, x2)) Where the effects of x1 and x2 are separately smoothed, and the interaction between both on different scales is a separate term contained in ti().
- Can I compare directly (e.g. using AIC for example) models fitted with ti() and models fitted with s()?
Sure you can compare them, but again, if your interaction terms are on different scales, you shouldn't fit the interaction with s(). You should either use te() or ti() (the choice depends on whether or not you're including the separate effects of the interaction terms in your model).
In these models, the linear predictor is a latent variable, with estimated thresholds $t_i$ that mark the transitions between levels of the ordered categorical response. The plots you show in the question are the smooth contributions of the four variables to the linear predictor, thresholds along which demarcate the categories.
The figure below illustrates this for a linear predictor comprised of a smooth function of a single continuous variable for a four category response.
The "effect" of body size on the linear predictor is smooth as shown by the solid black line and the grey confidence interval. By definition in the ocat family, the first threshold, $t_1$ is always at -1, which in the figure is the boundary between least concern and vulnerable. Two additional thresholds are estimated for the boundaries between the further categories.
The summary() method will print out the thresholds (-1 plus the other estimated ones). For the example you quoted this is:
> summary(b)
Family: Ordered Categorical(-1,0.07,5.15)
Link function: identity
Formula:
y ~ s(x0) + s(x1) + s(x2) + s(x3)
Parametric coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.1221 0.1319 0.926 0.354
Approximate significance of smooth terms:
edf Ref.df Chi.sq p-value
s(x0) 3.317 4.116 21.623 0.000263 ***
s(x1) 3.115 3.871 188.368 < 2e-16 ***
s(x2) 7.814 8.616 402.300 < 2e-16 ***
s(x3) 1.593 1.970 0.936 0.640434
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Deviance explained = 57.7%
-REML = 283.82 Scale est. = 1 n = 400
or these can be extracted via
> b$family$getTheta(TRUE) ## the estimated cut points
[1] -1.00000000 0.07295739 5.14663505
Looking at your lower-left of the four smoothers from the example, we would interpret this as showing that for $x_2$ as $x_2$ increases from low to medium values we would, conditional upon the values of the other covariates tend to see an increase in the probability that an observation is from one of the categories above the baseline one. But this effect is diminished at higher values of $x_2$. For $x_1$, we see a roughly linear effect of increased probability of belonging to higher order categories as the values of $x_1$ increases, with the effect being more rapid for $x_1 \geq \sim 0.5$.
I have a more complete worked example in some course materials that I prepared with David Miller and Eric Pedersen, which you can find here. You can find the course website/slides/data on github here with the raw files here.
The figure above was prepared by Eric for those workshop materials.
Best Answer
To get the actual values directly, {mgcv} has this hidden functionality of an extractor function buried in the
familyobject of the fitted model. If the model has some additional parameters like the scaled t or negative binomial (nb()) families, there will be a functiongetThetain thefamily.These are not typically well documented in the {mgcv} help, unfortunately. Usually what is returned by
getTheta()will be on the scales used for actual model fitting. To get them back on a more useful scale (like the $\nu$ and $\sigma$ parameters displayed in the output fromsummary())getTheta()typically has atransargument:which produces: