warn about the delta method for non-linear expressions

Fixes #821.

R/hypotheses.R

@@ -13,6 +13,8 @@
 #' 
 #' Warning #2: For hypothesis tests on objects produced by the `marginaleffects` package, it is safer to use the `hypothesis` argument of the original function.  Using `hypotheses()` may not work in certain environments, in lists, or when working programmatically with *apply style functions.
 #' 
+#' Warning #3: The tests assume that the `hypothesis` expression is (approximately) normally distributed, which for non-linear functions of the parameters may not be realistic. More reliable confidence intervals can be obtained using the \code{inferences()} function with `method = "boot"`.
+#' 
 #' @inheritParams comparisons
 #' @param model Model object or object generated by the `comparisons()`, `slopes()`, `predictions()`, or `marginal_means()` functions.
 #' @param FUN `NULL` or function. 

book/articles/hypothesis.qmd

@@ -226,6 +226,20 @@ slopes(
     hypothesis = "exp(am) - 2 * vs = -400")
 ```
 
+But note that the p values and confidence intervals are calculated using the delta method and are thus based on the assumption that the `hypotheses` expression is approximately normally distributed. For (very) non-linear functions of the parameters, this is not realistic, and we get p values with incorrect error rates and confidence intervals with incorrect coverage probabilities. For such hypotheses, it’s better to calculate the confidence intervals using the bootstrap (see [`inferences`](reference/inferences.html) for details):
+
+```{r}
+set.seed(1234)
+slopes(
+    mod,
+    newdata = "mean",
+    hypothesis = "exp(am) - 2 * vs = -400") |>
+  inferences(method = "boot")
+```
+
+While the confidence interval from the delta method is symmetric (equal to the estimate ± 1.96 times the standard error), the more reliable confidence interval from the bootstrap is (here) highly skewed.
+
+
 ### Adjusted Predictions
 
 Now consider the case of adjusted predictions:

book/articles/uncertainty.qmd

@@ -38,7 +38,7 @@ Scroll down this page to the Numerical Derivatives section to see a detailed exp
 
 ## Standard errors and intervals for `slopes()` and `comparisons()`
 
-All standard errors for the `slopes()` and `comparisons()` functions are computed using the delta method, as described above.
+All standard errors for the `slopes()` and `comparisons()` functions are computed using the delta method, as described above. The confidence intervals are calculated as estimate ± `qnorm((1 - conf_level) / 2)` standard errors (e.g., for 95% confidence intervals, estimate ± 1.96 standard errors) and assume that the (transformed) estimates are normally distributed.
 
 ## Standard errors and intervals for `marginal_means()` and `predictions()`