Link Misspecification and Quasi-Likelihood Targets

Why the Link Matters

A quasi-score combines a proposed conditional mean with a variance function to form an estimating equation. If the conditional-mean link is wrong, the equation does not generally target a coefficient that reproduces the true conditional means. Instead, it targets a value that balances the residuals according to the model’s link and working variance. This distinction matters: a covariance estimator can describe uncertainty around that limiting value, but it cannot make the value equal the scientific target.

Let $Y$ be a response and $Z$ a vector of covariates. Write the true conditional mean as $m_0(Z)=E[Y\mid Z]$. Suppose the working model proposes $\mu_\beta(Z)=h(Z^T\beta)$, where $h$ is the inverse link and $\beta$ belongs to a parameter space in which the mean and variance function are defined. The working variance is $\phi V(\mu_\beta(Z))$, with $V(\mu)>0$ on the relevant range. The factor $\phi$ is common to the score and does not affect its roots.

Definition: For one observation, define $D_\beta(Z)=\partial\mu_\beta(Z)/\partial\beta$. The quasi-score contribution, with the common dispersion factor omitted, is $$ \psi(Y,Z;\beta)=\frac{D_\beta(Z)}{V(\mu_\beta(Z))}\{Y-\mu_\beta(Z)\}. $$ A pseudo-true coefficient $\beta^*$ is a parameter value solving the population estimating equation $E[\psi(Y,Z;\beta^*)]=0$. It is the target induced by the specified estimating equation; it need not reproduce $m_0(Z)$.

Since the covariates are observed, conditioning on $Z$ gives the key population equation:

E[\psi(Y,Z;\beta)] = E\left[ \frac{D_\beta(Z)}{V(\mu_\beta(Z))} \{m_0(Z)-\mu_\beta(Z)\} \right].

Thus the target depends on the link through both $\mu_\beta(Z)$ and its derivative $D_\beta(Z)$, and on the working variance through its weight. The equation is vector-valued when $\beta$ has multiple components. A root need not exist or be unique without additional assumptions; when discussing a particular pseudo-true value, existence and uniqueness are understood or stated for that model.

The Population Target and the Correctly Specified Case

Result: Suppose $E[\|\psi(Y,Z;\beta)\|]$ is finite for the parameter values under consideration. If there is a $\beta_0$ such that $m_0(Z)=\mu_{\beta_0}(Z)$ almost surely, then $E[\psi(Y,Z;\beta_0)]=0$, provided the score is defined there. If the mean model is misspecified, the target is instead any root of the weighted residual equation above; it generally depends on the working link and variance.

Proof. By conditional expectation, $E[Y-\mu_{\beta_0}(Z)\mid Z]=m_0(Z)-\mu_{\beta_0}(Z)=0$ almost surely. The factor $D_{\beta_0}(Z)/V(\mu_{\beta_0}(Z))$ is a function of $Z$, so

E[\psi(Y,Z;\beta_0)] = E\left[ \frac{D_{\beta_0}(Z)}{V(\mu_{\beta_0}(Z))} E\{Y-\mu_{\beta_0}(Z)\mid Z\} \right] =0.

Under misspecification, the conditional residual $m_0(Z)-\mu_\beta(Z)$ is not generally zero. The estimating equation only requires its weighted average, after multiplication by $D_\beta(Z)$, to vanish. This proves the stated distinction. $\square$

In the identity-link case, a familiar projection result makes the target especially transparent. Let $\mu_\beta(Z)=Z^T\beta$, use constant working variance, and assume $E[Y^2]<\infty$, $E[\|Z\|^2]<\infty$, and that $M=E[ZZ^T]$ is nonsingular. These moment conditions ensure that the expected squared-error objective is finite and its relevant cross-moments exist.

Result: Under those assumptions, the population score root is $$ \beta^*=M^{-1}E[ZY], $$ and it uniquely minimizes $E[(Y-Z^T\beta)^2]$ over $\beta\in\mathbb{R}^p$.

Derivation. With constant working variance, $D_\beta(Z)=Z$, so the population score equation is $E[Z(Y-Z^T\beta)]=0$, or $E[ZY]-M\beta=0$. Nonsingularity gives the stated root. To verify the minimization claim, define $Q(\beta)=E[(Y-Z^T\beta)^2]$. The assumed second moments make $Q(\beta)$ finite. For any vector $a$, expansion gives

Q(\beta^*+a) =Q(\beta^*)-2a^T E[Z(Y-Z^T\beta^*)]+a^TMa =Q(\beta^*)+a^TMa.

Because $M$ is positive definite, $a^TMa>0$ for every nonzero $a$. Hence $\beta^*$ is the unique minimizer. $\square$

Worked Examples: Different Links, Different Targets

Worked Example: Identity Link for a Nonlinear Poisson Mean

Let $X$ equal 1 or 2 with probability $1/2$ each, and suppose $Y\mid X=x\sim\operatorname{Poisson}(e^x)$. Fit the one-parameter identity-link mean $\mu_\beta(X)=\beta X$ using constant working variance. The score contribution is $X(Y-\beta X)$, so the population target is

\beta^* =\frac{E[XY]}{E[X^2]} =\frac{\tfrac12(e+2e^2)}{\tfrac12(1^2+2^2)} =\frac{e+2e^2}{5} \approx 3.4993.

The true conditional means are $e\approx2.7183$ at $X=1$ and $e^2\approx7.3891$ at $X=2$. At the target, the fitted means are $\beta^*\approx3.4993$ and $2\beta^*\approx6.9986$. They do not equal the true means; the coefficient balances the residuals through the identity-link score.

The sandwich calculation also illustrates what uncertainty is being estimated. Here $\psi=X(Y-\beta X)$, and the negative expected derivative at the target is $J=E[X^2]=5/2$. The score variance is

B=E[X^2(Y-\beta^*X)^2] =\frac12\left[e+(e-\beta^*)^2 +4\left\{e^2+(e^2-2\beta^*)^2\right\}\right] \approx16.747.

In this calculation, $E[(Y-c)^2\mid X=x]=e^x+(e^x-c)^2$, since a Poisson variable has conditional variance equal to its mean. Therefore the asymptotic variance of the estimator around $\beta^*$ is $B/(nJ^2)\approx16.747/[n(5/2)^2]\approx2.680/n$. This is uncertainty around $3.4993$, not a correction that moves the target to the true mean relationship.

Worked Example: Comparing Identity and Log Links

Let $X$ again equal 1 or 2 with equal probabilities, but now let $Y\mid X=1\sim\operatorname{Poisson}(1)$ and $Y\mid X=2\sim\operatorname{Poisson}(9)$. Use constant working variance in both fits. With the identity mean $\mu_\beta(X)=\beta X$, the score is $X(Y-\beta X)$, so

0=E[X(Y-\beta X)] =\frac12\{(1-\beta)+2(9-2\beta)\} =\frac12(19-5\beta), \qquad \beta^*=3.8.

The fitted means are therefore 3.8 and 7.6, rather than the true means 1 and 9. Now use the log link $\mu_\beta(X)=e^{\beta X}$ with the same constant working variance. Its derivative is $Xe^{\beta X}$, so its score is $Xe^{\beta X}(Y-e^{\beta X})$. Put $t=e^\beta>0$. The population equation becomes

0=\frac12\{t(1-t)+2t^2(9-t^2)\} =\frac{t}{2}(1+17t-2t^3).

Since $t>0$, the relevant equation is $1+17t-2t^3=0$. Its derivative is $17-6t^2$: the polynomial first increases from 1 and then decreases to negative infinity, so it has exactly one positive root. Numerically that root is $t\approx2.9445$, giving $\beta^*=\log(t)\approx1.0799$. The fitted means are $t\approx2.9445$ and $t^2\approx8.6698$. Even with the same observations and constant working variance, changing the link changes the score equation and its target.

Worked Example: A Saturated Group-Mean Model

Suppose a covariate defines two groups, each with positive probability, and the true conditional means are 2 and 5. Fit a separate mean parameter for each group. For either group $j$, let its fitted mean be $\mu_j=h(\eta_j)$, with a differentiable link whose inverse derivative is nonzero at the relevant parameter values. Under a positive working variance $V(\mu_j)$, the population score for that group is proportional to

\frac{h'(\eta_j)}{V(\mu_j)}(m_j-\mu_j)=0.

The nonzero derivative and positive variance imply $\mu_j=m_j$. Thus both an identity link and a log link recover group means 2 and 5 when each group has its own unrestricted mean parameter and those means are in the link’s range. This is not a contradiction: link misspecification changes targets when it constrains the mean pattern, but a saturated group-mean model can reproduce the true means directly.

Sandwich Covariance Around a Misspecified Target

Suppose $(Y_i,Z_i)$, $i=1,\ldots,n$, are independent and identically distributed, and the estimator solves $\sum_{i=1}^n\psi(Y_i,Z_i;\hat\beta)=0$. The following result describes its first-order sampling variation around a pseudo-true root; it does not assert that the root is scientifically correct.

Theorem: Let $\beta^*$ be an interior root of $E[\psi(Y,Z;\beta)]=0$, and suppose it is the uniquely identified target in a neighborhood. Assume the estimator is consistent for $\beta^*$; the score is differentiable there; the sample derivative converges in probability to its expectation; the Taylor expansion remainder is negligible; $E[\|\psi(Y,Z;\beta^*)\|^2]<\infty$; and $$ J=-E\left[\frac{\partial\psi(Y,Z;\beta^*)}{\partial\beta^T}\right] $$ is nonsingular. Then, with $B=E[\psi(Y,Z;\beta^*)\psi(Y,Z;\beta^*)^T]$, $$ \sqrt{n}(\hat\beta-\beta^*)\xrightarrow{d} N\left(0,\;J^{-1}B J^{-T}\right). $$

Proof. Write $P_n\psi(\beta)=n^{-1}\sum_{i=1}^n\psi(Y_i,Z_i;\beta)$. A Taylor expansion of the estimating equation around $\beta^*$, together with consistency and the derivative assumptions, gives

0=P_n\psi(\beta^*)-J(\hat\beta-\beta^*)+o_p(n^{-1/2}).

Because $E[\psi(Y,Z;\beta^*)]=0$, the multivariate central limit theorem applies to the independent score contributions and yields

\sqrt{n}\,P_n\psi(\beta^*)\xrightarrow{d}N(0,B).

Rearranging the Taylor expansion and multiplying by $\sqrt n$ gives $\sqrt n(\hat\beta-\beta^*)=J^{-1}\sqrt nP_n\psi(\beta^*)+o_p(1)$. The limiting covariance is therefore $J^{-1}B J^{-T}$, as claimed. $\square$

The sandwich uses the variability of the actual score contributions through $B$ and the sensitivity of the estimating equation through $J$. It remains useful under misspecification because it can estimate sampling covariance around $\beta^*$, subject to the theorem’s assumptions. But if $\beta^*\ne\beta_0$, where $\beta_0$ is the coefficient or functional of scientific interest, then $\hat\beta$ converges to the wrong target for that purpose. An interval centered at $\hat\beta$ whose width shrinks at rate $n^{-1/2}$ cannot cover a distinct fixed target with probability approaching the nominal level; its coverage approaches zero under the stated convergence.

Common Pitfall and Practical Interpretation

Wrong reasoning: “The mean link is misspecified, but I used a sandwich standard error, so inference is robust to the misspecification.” This confuses two separate questions. The sandwich addresses the covariance of the estimator under the estimating equation; it does not alter the equation’s population root.

Correction: First identify the estimand and ask whether the working mean model represents it. Then derive the population score equation and determine what coefficient it targets. Only after that should covariance estimation be considered. A robust covariance estimate can protect standard errors against certain variance-model errors while leaving link-induced target bias intact. If the target is not the scientific quantity, remedies must address the mean model or estimand itself, rather than only the standard error.

The key diagnostic is the conditional residual equation $E[D_\beta(Z)\{m_0(Z)-\mu_\beta(Z)\}/V(\mu_\beta(Z))]=0$. It displays both the balance being imposed and the way the chosen link and variance weight that balance. Under a correct conditional mean, the residual vanishes given the covariates. Under misspecification, only the weighted average vanishes.

Check Your Understanding

Use the population equation and the distinction between target and covariance to answer these questions.

Write the population quasi-score equation in terms of the true conditional mean $m_0(Z)$ and the working mean $\mu_\beta(Z)$.
Why can two links with the same working variance produce different pseudo-true coefficients?
For an identity-link model with constant working variance, what moment assumptions ensure the squared-error objective is finite, and what is its minimizing coefficient?
In the two-group saturated example, why can both links recover the true group means despite a misspecified link elsewhere?
What do $J$ and $B$ represent in the sandwich covariance, and why does that covariance not remove a fixed difference between $\beta^*$ and the scientific target?