Intuition
Forward Marginal Effects (FMEs) are probably the most intuitive way
to interpret feature effects in supervised ML models. Remember how we
interpret the beta coefficient of a numerical feature in a linear
regression model \(\mathbb{E}[Y] = \beta_{0} +
\beta_{1}x_{1} + \ldots + \beta_{p}x_{p}\):
If \(x_{j}\) increases by one unit,
the predicted target variable increases by \(\beta_{j}\).
FMEs make use of this instinct and apply it straightforwardly to any
model.
In short, FMEs are the answer to the following question:
What is the change in the predicted target variable if we
change the value of the feature by \(h\) units?
A few examples: What is the change in predicted blood pressure if
a patients’ weight increases by \(h\) = 1 kg? What is the change in
predicted life satisfaction if a person’s monthly income increases
by \(h\) = 1,000 US
dollars? Per default, \(h\) will
be 1. However, \(h\) can be chosen to
match the desired scale of interpretation.
Compute Effects
The big advantage of FMEs is that they are very simple. The FME is
defined observation-wise, i.e., it is computed separately for each
observation in the data. Often, we are more interested in estimating a
global effect, so we do the following:
1. Compute the FME for each observation in the
data
2. Compute the Average Marginal Effect (AME)
Numerical Features
Univariate Feature Effects
For a given observation \(i\) and
step size \(h_{j}\), the FME of a
single numerical feature \(x_{j}\) is
computed as:
\(\textrm{FME}_{\mathbf{x}^{(i)}, \, h_{j}} =
\widehat{f}(x_{1}^{(i)},\, \ldots, \, x_{j}^{(i)}+h_{j},\, \ldots, \,
x_{p}^{(i)})-\widehat{f}(\mathbf{x}^{(i)})\)
As can be seen from the formula, the FME is simply the difference in
predictions between the original observation \(x^{(i)}\) and the changed observation \((x_{1}^{(i)},\, \ldots, \, x_{j}^{(i)}+h_{j},\,
\ldots, \, x_{p}^{(i)})\), where \(h_{j}\) is added to the feature \(x_{j}\).
Bivariate Feature Effects
This is just the extension of the univariate FME to two features
\(x_{j}, x_{k}\) that are affected
simultaneously by a step. Therefore, the step size becomes a vector
\(\mathbf{h} = (h_{j}, h_{k})\), where
\(h_{j}\) denotes the change in \(x_{j}\) and \(h_{k}\) the change in \(x_{k}\):
\(\textrm{FME}_{\mathbf{x}^{(i)}, \,
\mathbf{h}} = \widehat{f}(x_{1}^{(i)},\, \ldots, \, x_{j}^{(i)}+h_{j},\,
\ldots, \, x_{k}^{(i)}+h_{k}, \, \ldots,
x_{p}^{(i)})-\widehat{f}(\mathbf{x}^{(i)})\)
Categorical Features
Equivalent to the step size \(h_{j}\) of a numerical feature, we select
the category of interest \(c_{h}\) for
a categorical feature \(x_{j}\). For a
given observation \(i\) and category
\(c_{h}\), the FME is:
\(\textrm{FME}_{\mathbf{x}^{(i)}, \, c_{h}} =
\widehat{f}(c_{h},\,
\mathbf{x}_{-j}^{(i)})-\widehat{f}(\mathbf{x}^{(i)}), \, \, \, \, \, \,
\, \,x_{j} \neq c_{h}\)
where we simply change the categorical feature to \(c_{h}\), leave all other features \(\mathbf{x}_{-j}^{(i)}\) unchanged, and
compare the predicted value of this changed observation to the predicted
value of the unchanged observation. Obviously, we can only compute this
for observations where the original category is not the category of
interest \(x_{j} \neq c_{h}\). See here
for an example.
Average Marginal Effects (AME)
The AME is the mean of every observation’s FME as a global estimate
for the feature effect:
\(\textrm{AME} = \frac{1}{n}\sum_{i =
1}^{n}{\, \textrm{FME}_{\mathbf{x}^{(i)}, \, h_{j}}}\)
Therefore, the AME is the expected difference in the predicted target
variable if the feature \(x_{j}\) is
changed by \(h_{j}\) units. For \(h_{j}\) = 1, this corresponds to the way we
interpret the coefficient \(\beta_{j}\)
of a linear regression model. However, be careful: the choice of \(h_{j}\) can have a strong effect on the
estimated FMEs and AME for non-linear prediction
functions, auch as random forests or gradient-boosted trees.
Why we need FMEs
Marginal effects (ME) are already a widely used concept to interpret
statistical models. However, we believe they are ill-suited to
interpret feature effects in most ML models. Here, we explain why you
should abandon MEs in favor of FMEs:
In most implementations (e.g., Leeper’s margins package), MEs are
computed as numerical approximation of the partial derivative of the
prediction function w.r.t. the feature \(x_{j}\). In other words, they compute a
finite difference quotient, similar to this:
\(\textrm{dME}_{\mathbf{x}^{(i)}, \, j} =
\cfrac{\widehat{f}(x_{1}^{(i)}, \, \ldots,\, x_{j}^{(i)}+h,\, \ldots, \,
x_{p}^{(i)})-\widehat{f}(\mathbf{x}^{(i)})}{h}\)
where \(h\) typically is very
small (e.g. 10\({}^{-7}\)). As is
explained here, these
derivative-based MEs (dME) have a number of shortcomings:
Number 1: The formula above computes an estimate for
the partial derivative, i.e., the tangent of the prediction function at
point \(\mathbf{x}^{(i)}\). The default
way to interpret this is to say: if \(x_{j}\) increases by one unit, the
predicted target variable can be expected to increase by \(\textrm{ME}_{\mathbf{x}^{(i)}, \, j}\).
Unconsciously, we use a unit change (\(h\) = 1) to interpret the computed ME even
though we computed an instantaneous rate of change. For
non-linear prediction functions, this can lead to substantial
misinterpretations. The image below illustrates this:
The yellow line is the prediction function, the grey line is the
tangent at point \(x\) = 0.5. If
interpreted with a unit change, the dME is subject to an error, due to
the non-linearity of the prediction function. The FME, however,
corresponds to the true change in prediction along the secant (green
line) between \(x\) = 0.5 and \(x\) = 1.5. This is simply by way of design
of the FME, as it describes exactly our intuition of interpreting
partial derivatives.
Number 2: In general, dMEs are ill-suited to
describe models based on piecewise constant prediction functions (e.g.,
CART, random forests or gradient-boosted trees), as most observations
are located on piecewise constant parts of the prediction function where
the derivative equals zero. In contrast, FMEs allow for the choice of a
sensible step size \(h\) that is large
enough to traverse a jump discontinuity, as can be seen in the example
below: At \(x\) = -2.5 (green), the dME
is zero. Using the FME with \(h\) = 1,
we get the red point with a different (lower) function value. Here, the
FME is negative, corresponding to what happens when \(x\) = -2.5 increases by one unit.
In a way, the FME is the much smarter, little brother of the
dME:
- it describes the exact behavior of the prediction function
- its design corresponds to the way we naturally want to interpret a
partial derivative
- through the flexible choice of \(h\), it can be tailored to answer the
desired research question
- it is conceptually simple: researchers can discuss the
interpretations because they understand how they are computed