smmargins 0.1.0

Marginal effects for StatsModels

smmargins — Stata-style margins for StatsModels

A small module that fills in the marginal-effects gaps in StatsModels: adjusted predictions and marginal effects at user-specified covariate profiles, with delta-method standard errors, for any fitted model that exposes params, cov_params(), and a predict(params, exog) method.

Math

The delta method: for a parameter vector $\hat\beta$ with estimated covariance $\widehat V$, and a (possibly vector-valued) statistic $g(\beta)$,

$$ \widehat{\mathrm{Var}}\bigl[g(\hat\beta)\bigr] ;\approx; G,\widehat V,G^\top, \qquad G = \left.\frac{\partial g}{\partial \beta}\right|_{\hat\beta}. $$

This is exactly the approach documented in Stata's margins-delta-method FAQ. The cases in Richard Williams' Margins01 notes all fit the $g(\beta)$ schema:

Statistic	$g(\beta)$
AAP	$\frac{1}{n}\sum_i f(x_i^\top\beta)$
APM	$f(\bar x^\top\beta)$
APR	$\frac{1}{n}\sum_i f(x_i^\top\beta)$ with some $x$ fixed
AME	$\frac{1}{n}\sum_i \partial f(x_i^\top\beta)/\partial x_{ij}$
MEM	$\partial f(\bar x^\top\beta)/\partial x_j$
MER	AME / AAP with some $x$ held at representative values
Contrast	$E[f(\cdot)\mid x_j=\text{level}] - E[f(\cdot)\mid x_j=\text{ref}]$

where $f$ is the response-scale map (identity for OLS, inverse link for GLMs, etc.). Every statistic above is a linear combination of mean-predictions $\frac{1}{n}\sum_i f(x_i^\top\beta)$ on different design matrices, so the single Jacobian primitive needed is

$$ \frac{\partial}{\partial\beta},\frac{1}{n}\sum_i f(x_i^\top\beta) ;=; \frac{1}{n}\sum_i f'(x_i^\top\beta),x_i . $$

The module computes $f'$ analytically when the model exposes it (any GLM via family.link.inverse_deriv — Logit, Probit, Poisson, NegBin, Gamma, Gaussian — plus OLS/WLS/GLS via the identity link), and falls back to central finite differences when it doesn't. See design choice 4 below.

Why patsy

When we want the marginal effect of x1 and the formula is y ~ x1 + I(x1**2) + x1:x2 + C(group), we can't just nudge one column of the design matrix — x1 enters three columns. What we can nudge is the x1 column of the original data frame, and then ask patsy to rebuild the design matrix using the stored DesignInfo:

patsy.dmatrix(design_info, perturbed_frame, return_type="matrix")

That preserves polynomial terms, interactions, splines (bs(x, df=4)), and categorical contrasts automatically. It's also the right abstraction for "hold age=45" or "set group='b'" — you mutate the data frame, not the design matrix.

API

from smmargins import Margins

M = Margins(fit)                 # fit is a statsmodels results object

# --- adjusted predictions ---
M.predict()                                       # AAP             (at="overall")
M.predict(at="mean")                              # APM             (at="mean")
M.predict(at="median")                            # APM at medians
M.predict(at="zero")                              # APM at zero
M.predict(atexog={"x1": [0, 1, 2]})               # APR (others at observed values)
M.predict(atexog={"x1": [0, 1, 2]}, at="mean")    # APR holding others at means

# --- marginal effects ---
M.dydx("x1")                                      # AME (continuous auto-detected)
M.dydx(["x1", "x2"])                              # multiple variables stacked jointly
M.dydx("*")                                       # all RHS columns
M.dydx("x1", at="mean")                           # MEM (factors at proportions)
M.dydx("x1", at="median")                         # ME at medians
M.dydx("x1", atexog={"x2": [0, 1]})               # MER
M.dydx("kids", count=True)                        # unit increment x -> x+1 (for integers)
M.dydx("group")                                   # discrete contrasts vs reference level
M.dydx("group", reference="b")                    # discrete contrasts vs "b"

# --- elasticities (continuous only) ---
M.dydx("x1", method="eyex")                       # full elasticity:    ey/ex = (dy/dx)·x/y
M.dydx("x1", method="dyex")                       # semi-elasticity:    dy/ex = (dy/dx)·x
M.dydx("x1", method="eydx")                       # semi-elasticity:    ey/dx = (dy/dx)/y
# Combine with at=/atexog= as usual; raises on discrete variables.

# --- factor handling at evaluation points ---
M.predict(at="mean")                              # factor dummies at observed proportions (Stata default)
M.predict(at="mean", factor_stat="mode")          # factors held at modal level
M.predict(at="zero", factor_stat="mode")          # numerics at 0, factors at mode

The at= parameter mirrors statsmodels' get_margeff(at=...): "overall" (default), "mean", "median", "zero". The atexog= dict fixes specific variables at specific values (Stata's margins, at(...)); unlike statsmodels' atexog, it's keyed by variable name rather than column index.

Formula vs. Raw Exog Mode

Margins supports models fit without formulas (e.g. sm.OLS(y, X).fit()). In this "raw mode", variable names are taken from model.exog_names (often x1, x2, ... or const).

Important Limitation: In raw mode, Margins does not know about relationships between columns in the design matrix. If you manually included an interaction column (e.g. X["x1_x2"] = X["x1"] * X["x2"]), perturbing x1 for a marginal effect will not automatically update x1_x2. This will lead to incorrect marginal effects for that variable. If your model has interactions or transformations, you should fit it using a formula to ensure Margins can correctly rebuild the design matrix.

Each call returns a MarginsResult with .estimate, .se, .vcov, .ci_lower, .ci_upper, .pvalue, plus .summary() returning a DataFrame. Use use_t=True on the Margins constructor if you want t-distribution inference (reads results.df_resid).

M = Margins(fit, analytic=True)   # default: analytic outer Jacobian when possible
M = Margins(fit, analytic=False)  # force central finite differences everywhere

Design choices worth knowing

1. at="mean" averages the design matrix by default. This matches Stata's margins, atmeans and statsmodels' get_margeff(at='mean'): numeric columns become their sample mean, factor dummies become their observed proportions (a "person" who is 0.33 female). The test suite verifies a match to statsmodels.get_margeff(at='mean') to machine precision. Pass factor_stat="mode" for the modal-individual variant (numerics at their mean, factors at their modal level) — gives a "typical individual" rather than a fictional fractional one. factor_stat="zero" puts factors at their reference level instead. Williams (Margins01) notes that atmeans-on-dummies is usually a worse choice than AME regardless of which convention you pick, so prefer AME for factor-heavy models.

2. dydx is response-scale by default. For GLMs this means $\partial\hat\mu / \partial x$, not $\partial\hat\eta / \partial x$. This matches Stata's default and is almost always what's wanted.

3. Discrete vs continuous is auto-detected, based on dtype and number of unique values. Override with discrete=True/False if needed.

4. Outer Jacobian is analytic when possible, FD otherwise. Every statistic in this module is a linear combination of mean-predictions $\frac{1}{n}\sum_i f(x_i^\top\beta)$, whose gradient is $\frac{1}{n}\sum_i f'(x_i^\top\beta),x_i$. We obtain $f'$ analytically for any GLM via family.link.inverse_deriv (Logit, Probit, Poisson, NegBin, Gamma, Gaussian) and via the identity link for OLS/WLS/GLS. For continuous AME we still perturb the data column with central differences — patsy doesn't expose symbolic derivatives of basis transforms (I(x**2), bs(...), cr(...), interactions) — but that's a single 2-rebuild step, independent of $p$. When family.link.inverse_deriv is missing (custom Link subclass), an offset/exposure is present (so $\eta\neq X\beta$), or the model is neither GLM nor a linear-regression model, the entire Jacobian falls back to central finite differences with $h = \epsilon^{1/3}\cdot\max(|x|, 1)$ — the truncation-vs-rounding sweet spot. Stata's margins uses FD throughout; our analytic path removes $p$ forward predict calls per statistic without changing any answer to within FD tolerance. Pass analytic=False to force FD on every model.

5. Everything goes through model.predict(params, exog). That way the inverse link, offsets, exposures, and any other model-specific wrinkles handled by StatsModels come along for the ride for free.

Verification

test_margins.py checks the module against:

• An OLS with polynomial + interaction, where the analytic AME is $\beta_1 + 2\beta_2,\overline{x_1} + \beta_4,\overline{x_2}$ and its SE is $\sqrt{g^\top V g}$. Matches to machine precision.
• Logit AME, compared against statsmodels.get_margeff(at='overall'). Matches to $10^{-5}$ on estimate and SE.
• Logit MEM, compared against get_margeff(at='mean') on a formula with a C(grp) factor. Matches to $10^{-5}$ on estimate and SE under the default factor_stat="mean". A separate hand-computed check covers the factor_stat="mode" path.
• Poisson AAP with hand-derived $g$ gradient. Matches exactly; and sanity-checks that AAP equals the sample mean for canonical-link Poisson.
• Analytic-vs-FD parity matrix. Every public API call (AAP, APM, APR, AME, MEM, MER — continuous and discrete) is run twice on each of OLS+poly, Logit+C(grp), and Poisson, with analytic=True and analytic=False, and the two paths must agree on both estimate and SE. This guards against the analytic chain-rule formula drifting from the FD answer that the other tests pin to Stata / get_margeff.
• Elasticities (eyex, dyex, eydx) matched against statsmodels.get_margeff(method=...) for both AME-style (at='overall') and MEM-style (at='mean'), on Logit and Poisson, to $10^{-5}$ on estimates and $10^{-3}$ on SEs. Plus a hand check that Poisson MEM $\mathrm{ey/ex}(x_1)=\beta_1,\overline{x_1}$ for the canonical-link case. Elasticities on a discrete variable raise.

Difference-in-differences

Two small additions turn the module into a full DiD estimator:

• MarginsResult.contrast(c, labels=None) forms any linear combination of the estimates, using $c^\top V_m c$ directly on the already-computed joint covariance. Exact under the same delta-method approximation; no extra differentiation.
• Margins.did(group, condition, ...) sets up the 2×2 grid and returns a DiDResult bundling the four cell predictions, the two simple effects, and the DiD — all sharing the same joint covariance (so you can test them jointly through result.joint.vcov).

M = Margins(fit)
res = M.did("group", "preexist_Y",
            group_levels=["A", "B"], condition_levels=[0, 1],
            atexog={"age": 60, "female": 0})  # optional: fix covariates
print(res)          # shows cells + simple effects + DiD
res.did.estimate    # the DiD point estimate
res.did.ci_lower    # lower 95% CI
res.cells.vcov      # 4x4 joint covariance of the cell predictions

Why this matters for nonlinear models (Ai & Norton, 2003). In a logit, the coefficient on the group × condition interaction is on the log-odds scale. On the probability scale the DiD is a nonlinear function of every parameter and every covariate profile — you cannot read it off the interaction coefficient. test_did.py verifies:

• On OLS, did() returns exactly the interaction coefficient with exactly its SE (matches to 1e-8).
• On logit, did() returns something very different from the interaction coefficient (e.g. −0.676 on log-odds vs −0.147 on probability in one test run), and matches a by-hand four-cell computation to 1e-10.

Files

• smmargins.py — the module.
• test_margins.py — correctness tests for predictions and marginal effects.
• test_did.py — correctness tests for DiD and contrasts.
• demo_margins.py — Williams-style walkthrough on a simulated dataset.
• demo_did.py — healthcare-style 2×2 DiD walkthrough.