基于右删失数据的分位数回归结果，能否还原GLM的回归系数？

Great question—this is such a fascinating overlap between quantile regression (QR) and generalized linear models (GLMs), especially in that tricky right-censored data scenario where standard GLM estimators fail but QR stays consistent below the censoring threshold. Let’s break this down:

Short Answer: It Depends (But Yes, In Specific Cases) #

You can’t always directly map QR results to GLM coefficients, but if you know the GLM’s link and variance functions (which you stated you do), there are parametric scenarios where you can recover the target GLM coefficients using multiple quantile regression outputs.

When Recovery Works: Parametric GLMs with Known Distributions #

Let’s walk through a few common examples where this is feasible:

• Normal Linear GLM (Identity Link, Constant Variance): This is the simplest case. For a normal distribution, all quantile regression coefficients are identical across every τ (since the distribution is symmetric and location-scale). So any of your QR coefficient estimates will directly match the GLM’s coefficients.
• Logistic GLM (Logit Link): If your GLM assumes a logistic distribution for the response, the quantile function has a clear relationship to the GLM parameters. The τ-th quantile of Y given X is:

  Q_Y(τ | X) = Xβ + log(τ / (1 - τ))
  

  Notice that the slope coefficients for X are identical across all τ—only the intercept shifts with the quantile. So you can extract the slope terms directly from any QR run, and use differences in intercepts across multiple τ values to confirm consistency (or solve for any remaining parameters).
• Exponential GLM (Log Link): For an exponential GLM, the τ-th quantile follows:

  Q_Y(τ | X) = -log(1 - τ) * exp(Xβ)
  

  Taking the log of both sides gives a linear relationship: log(Q_Y(τ | X)) = Xβ + log(-log(1 - τ)). Here, running QR on the logged response (or using QR results to compute log-quantiles) will give you slope coefficients that match the GLM’s β—again, only the intercept varies with τ.

When Recovery Gets Tricky #

If your GLM doesn’t follow a parametric distribution with a clear quantile-to-mean mapping, or if you only have a single quantile’s QR results, recovery becomes much harder. QR targets specific quantiles, while GLMs target the mean (or another location parameter tied to the link function)—without a known distributional bridge between these two, there’s no direct way to translate coefficients.

Handling Right-Censored Data #

In your right-censored scenario, where QR is consistent below the censoring threshold but GLMs aren’t, you can leverage the above logic:

1. Focus on quantiles τ where the corresponding quantile Q_Y(τ | X) is below the censoring line (since those QR estimates are reliable).
2. Use multiple such τ values to collect QR coefficients.
3. Apply the distribution-specific quantile-to-GLM parameter relationship (like the ones above) to solve for the GLM’s β. For example, if you’re working with a logistic GLM, you can average the slope coefficients from multiple valid QR runs to get a robust estimate of the GLM’s slope parameters.

Practical Tips #

• Use multiple quantiles: The more valid QR estimates you have (across different τ below the censoring threshold), the more robust your GLM coefficient recovery will be—you can even set up a regression of QR intercepts against the quantile-specific constants (like log(τ/(1-τ)) for logistic) to solve for the GLM’s intercept and slopes.
• Validate your results: Once you’ve estimated the GLM coefficients, compute the predicted quantiles using the distribution’s quantile function and compare them to your QR results. If they line up, you’re on the right track.

内容的提问来源于stack exchange，提问作者andrewH