如何计算二次项的'economic significance'？含二次项的logistic回归substantive显著性分析

Great question—this is such a common roadblock when working with polynomial predictors, since the marginal effect of your core variable isn’t constant anymore (it shifts depending on where the variable sits in its distribution). Let’s walk through how to adapt your simulation approach to account for both the linear and quadratic terms.

First, Clarify the Model Form #

Let’s start by writing out your logistic regression explicitly to ground the math:

logit(P(Y=1)) = β₀ + β₁X + β₂X² + ΣβₖZₖ

Where:

• X is your core predictor (with linear and quadratic terms)
• Zₖ are your control variables
• β coefficients are the estimated parameters from your model

The key thing here is that the marginal effect of X on the probability of Y=1 is not just β₁—it’s dP/dX = β₁ + 2β₂X. That means the impact of a one-standard-deviation change in X depends entirely on the starting value of X.

Step-by-Step Simulation Approach #

Here’s how to adjust your existing workflow to handle the quadratic term:

1. Gather Key Inputs #

• Extract all estimated coefficients (β₀, β₁, β₂, and all βₖ) from your regression output.
• Calculate your sample’s descriptive stats for X: mean (X_mean), standard deviation (X_sd), and any percentiles you care about (e.g., median, Q1, Q3, min/max).
• For control variables, you can use their sample means (Z̄ₖ) to hold them constant at typical values, or use each observation’s actual values for a more granular average effect.

2. Choose Benchmark Values for X #

Since the effect changes with X, you shouldn’t just use a single value. Pick meaningful points across X’s distribution to show how the impact varies:

• Mean of X
• Median of X
• 25th and 75th percentiles (Q1, Q3)
• Min/max (if those values are theoretically or practically relevant)

3. Compute Probability Changes for Each Benchmark #

For each starting value X₀:
a. Calculate the predicted probability when X = X₀:

P₀ = 1 / (1 + exp(-(β₀ + β₁*X₀ + β₂*(X₀)² + Σβₖ*Z̄ₖ)))

b. Calculate the predicted probability when X increases by one standard deviation (X = X₀ + X_sd):

P₁ = 1 / (1 + exp(-(β₀ + β₁*(X₀+X_sd) + β₂*(X₀+X_sd)² + Σβₖ*Z̄ₖ)))

c. The substantive effect is the difference: ΔP = P₁ - P₀

If you want a more robust estimate, use bootstrapping:

• Randomly resample your dataset with replacement, refit the logistic regression each time, and recalculate ΔP for each benchmark.
• After 1000+ iterations, you can report the median ΔP and its 95% confidence interval.

4. Present Your Results #

Instead of a single number, highlight how the effect changes with X. For example:

When X is at its mean (X=4.2), a one-standard-deviation increase (ΔX=1.8) is associated with a 7.3% rise in the probability of Y=1. At the 75th percentile of X (X=5.6), that same increase only leads to a 3.1% rise. At the 25th percentile (X=2.7), the increase is linked to a 10.2% jump.

You can also visualize this: plot X on the x-axis and ΔP (one-SD change effect) on the y-axis to show the downward/upward trend from the quadratic term.

Quick Notes to Keep in Mind #

• If you want to test the effect of a decrease in X, swap to X₀ - X_sd and calculate P₀ - P₁.
• Instead of using control variable means, you can compute ΔP for every observation in your sample, then take the average. This gives you the average marginal effect (AME) across all cases, which is another common measure of substantive significance.
• Double-check if your quadratic term is statistically significant—if it’s not, you might revisit whether it belongs in the model, but don’t drop it just for significance if your theory supports it.

内容的提问来源于stack exchange，提问作者Emma