Consider the following model:

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(
𝑥
)
=
𝑓
1
(
𝑥
1
,
𝑥
2
)
+
𝑓
2
(
𝑥
3
)
+
𝑓
3
(
𝑥
4
)

True or false: this model is additive in the predictors.

Question

Consider the following model:

𝑓
(
𝑥
)
=
𝑓
1
(
𝑥
1
,
𝑥
2
)
+
𝑓
2
(
𝑥
3
)
+
𝑓
3
(
𝑥
4
)

True or false:  this model is additive in the predictors.

BlackTom AI · Accepted Answer

Here is a detailed analysis of the statement and each aspect of the model.
First, recall what 'additive in the predictors' usually means: a model is additive if the effect of each predictor (or block of predictors) enters the outcome additively, typically as a sum of functions that depend on individual predictors separately, e.g., f(x) = g1(x1) + g2(x2) + g3(x3) ... where each term depends on a single predictor (or on predefined, non-overlapping groups) without interactions between predictors.
Now, examine the given model: f(x) = f1(x1, x2) + f2(x3) + f3(x4).
- The term f2(x3) depends additively on x3 alone, and f3(x4) depends additively on x4 alone, which by itself aligns with additivity for those predictors.
- However, the crucial point is f1(x1, x2). This term is a function of both x1 and x2 together, i.e., it captures any interaction or joint effect between x1 and x2. Because x1 and x2 are not entering the model in separate additive terms, their combined influence is not additive with respect to the predictors.
- If the model intended to be additive in the sense of individual predictors, we would expect something like g1(x1) + g2(x2) + g3(x3) + g4(x4) or at least a decomposition where each predictor appears in its own separate additive component. The presence of f1 that includes both x1 and x2 breaks that. 
- Therefore, while the model is additive at the level of the decomposition into components (f1, f2, f3), it is not additive in the individual predictors x1 and x2 because their effects are entangled within f1.
Consequently, the statement 'this model is additive in the predictors' is False.
If someone interpreted 'additive in the predictors' as 'sum of functions of blocks of predictors,' one might argue it is additive in blocks {x1, x2}, {x3}, {x4}; but standard phrasing focusing on individual predictors without interactions would still label it not additive in the single-predictor sense due to f1(x1, x2).

Consider the following model: 𝑓 ( 𝑥 ) = 𝑓 1 ( 𝑥 1 , 𝑥 2 ) + 𝑓 2 ( 𝑥 3 ) + 𝑓 3 ( 𝑥 4 ) True or false: this model is additive in the predictors. 判断题

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