Confidence Intervals and Tests for Blip Parameters

Inference for Blip Parameters

A fitted structural nested mean model (SNMM) produces estimates of the blip parameters, but an estimate alone does not show how precisely the data determine an effect. Confidence intervals and hypothesis tests use the sampling variability of the g-estimate to express that uncertainty. For example, a blip parameter indexed by treatment time can quantify the effect of treatment at one decision point, conditional on the model’s parameterization. A joint test can ask whether a collection of such treatment-time effects is zero.

The procedures below assume a correctly specified set of g-estimating equations and the regularity conditions needed for their asymptotic normality. They do not repair a poorly identified parameter or a misspecified blip model. The earlier material on asymptotic variance of g-estimates supplies the central input: a consistent estimate of the estimator’s covariance on the correct scale.

Wald Intervals for Single Parameters and Contrasts

Suppose \(\hat\beta\) is consistent for \(\beta_0\), and \(\sqrt n(\hat\beta-\beta_0)\xrightarrow{d}N(0,V)\), where \(V\) is positive definite. Assume also that \(n\widehat{\operatorname{Var}}(\hat\beta)\xrightarrow{p}V\). For the \(j\)th coefficient, the estimated standard error is \(s_j=\sqrt{\widehat{\operatorname{Var}}(\hat\beta)_{jj}}\). An approximate two-sided \(100(1-\alpha)\%\) Wald interval is \[ \hat\beta_j \mathbin{\pm} z_{1-\alpha/2}s_j, \] where \(z_{1-\alpha/2}\) is the corresponding standard normal quantile.

More generally, for a scalar linear contrast \(a^T\beta\), with fixed vector \(a\), estimate it by \(a^T\hat\beta\). Its estimated variance is \(a^T\widehat{\operatorname{Var}}(\hat\beta)a\), so its interval is \[ a^T\hat\beta \mathbin{\pm} z_{1-\alpha/2} \sqrt{a^T\widehat{\operatorname{Var}}(\hat\beta)a}. \] This is useful when the scientific target is a sum or difference of blip parameters rather than one coefficient on its own.

Wald Tests of Joint Restrictions

Let the null hypothesis impose \(q\) linear restrictions \(H_0:R\beta=r\), where \(R\) is a \(q\times p\) matrix of rank \(q\), and \(r\) is a fixed \(q\)-vector. Assume the asymptotic normality above, that \(RVR^T\) is positive definite, and that the covariance estimator satisfies \(n\widehat{\operatorname{Var}}(\hat\beta)\xrightarrow{p}V\). The Wald statistic is

Under \(H_0\), \(W\xrightarrow{d}\chi^2_q\). The rank condition ensures that the restrictions are linearly independent; positive definiteness ensures that their asymptotic covariance can be inverted.

Proof. Under the null, \(R\hat\beta-r=R(\hat\beta-\beta_0)\). Therefore \[ \sqrt n(R\hat\beta-r)\xrightarrow{d}N(0,RVR^T). \] The covariance assumption gives \[ nR\widehat{\operatorname{Var}}(\hat\beta)R^T \xrightarrow{p}RVR^T. \] Since \(RVR^T\) is positive definite, the continuous mapping theorem permits inversion of this matrix with probability tending to one, and its inverse converges in probability to \((RVR^T)^{-1}\). The statistic \(W\) is the quadratic form of the asymptotically normal vector \(\sqrt n(R\hat\beta-r)\) using this consistently estimated covariance. By Slutsky’s theorem, its limit is the squared Mahalanobis length of a \(q\)-dimensional standard normal vector, which has a \(\chi^2_q\) distribution. \(\square\)

Score-Type Tests From G-Estimating Equations

A score-type test can be useful when the null model is fitted directly and the unrestricted estimate is inconvenient to compute. Its construction can be stated for general estimating equations, not just likelihood scores. Let \(g_i(\beta)\) be a \(p\)-vector of individual g-estimating functions, with \(E[g_i(\beta_0)]=0\), and define \(U_n(\beta)=\sum_{i=1}^n g_i(\beta)\). Assume independent, identically distributed observations, finite second moments, differentiability in a neighborhood of \(\beta_0\), a nonsingular matrix \(A=-E[\partial g_i(\beta_0)/\partial\beta^T]\), and the usual conditions for a consistent root and a central limit theorem. Let \(B=E[g_i(\beta_0)g_i(\beta_0)^T]\). Then the asymptotic covariance of \(\hat\beta\) is \(V=A^{-1}BA^{-T}\).

Partition \(\beta=(\gamma^T,\eta^T)^T\), where \(\gamma\) is \(q\)-dimensional and the null is \(H_0:\gamma=\gamma_0\). The remaining parameters \(\eta\) are nuisance parameters. Fit the restricted model by solving \(U_{\eta,n}(\gamma_0,\tilde\eta)=0\). Write \(A_{\gamma\eta}\) and \(A_{\eta\eta}\) for the corresponding blocks of \(A\), and assume \(A_{\eta\eta}\) is nonsingular. The adjusted score vector is \[ S_n^*=n^{-1/2}\{U_{\gamma,n}(\gamma_0,\tilde\eta) -A_{\gamma\eta}A_{\eta\eta}^{-1}U_{\eta,n}(\gamma_0,\tilde\eta)\}. \] At an exact restricted solution, the second term is zero. Its presence shows the adjustment that removes first-order nuisance-parameter variation and is helpful when estimating the quantities from approximate roots. Let \(C=(I_q,\,-A_{\gamma\eta}A_{\eta\eta}^{-1})\), and define \(\Omega=CBC^T\). If \(\Omega\) is positive definite, the score-type statistic is \(S_n^{*T}\hat\Omega^{-1}S_n^*\), where \(\hat\Omega\) consistently estimates \(\Omega\) under the null.

Derivation. Put \(S_n=n^{-1/2}U_n(\beta_0)\). The central limit theorem gives \(S_n\xrightarrow{d}N(0,B)\). A first-order expansion of the nuisance estimating equation at \((\gamma_0,\eta_0)\) gives \[ 0=n^{-1/2}U_{\eta,n}(\gamma_0,\tilde\eta) =S_{\eta,n}-A_{\eta\eta}\sqrt n(\tilde\eta-\eta_0)+o_p(1). \] Consequently, \(\sqrt n(\tilde\eta-\eta_0)=A_{\eta\eta}^{-1}S_{\eta,n}+o_p(1)\). Expanding the \(\gamma\) equations at the same point yields \[ n^{-1/2}U_{\gamma,n}(\gamma_0,\tilde\eta) =S_{\gamma,n}-A_{\gamma\eta}A_{\eta\eta}^{-1}S_{\eta,n}+o_p(1) =CS_n+o_p(1). \] Thus the adjusted score converges to \(N(0,CBC^T)=N(0,\Omega)\). Consistency of \(\hat\Omega\), its positive definiteness, and Slutsky’s theorem give the chi-square limit for the quadratic form, as in the Wald proof. \(\square\)

For likelihood scores under standard likelihood regularity conditions, the corresponding score test is commonly called the Rao score test. For g-estimating equations, the sandwich covariance \(CBC^T\) matters: treating the estimating functions as likelihood scores without justification can give the wrong reference variance.

Choosing and Checking an Inference Procedure

Wald and score-type procedures have the same large-sample reference distribution under their respective assumptions, but they use the data differently. Wald inference uses the unrestricted estimate and its covariance; the score-type procedure fits the null and measures departure from it. A likelihood-ratio test is another familiar alternative when a full likelihood is available and its regularity conditions hold. G-estimation is often based on moment equations rather than a full likelihood, so the robust Wald and score-type constructions above are directly applicable without asserting a likelihood model.

Both procedures are asymptotic. Sparse treatment histories, weakly informative treatment-residual instruments, or near-singular covariance matrices can make their finite-sample approximations unstable. A nearly singular restriction covariance can yield a large Wald statistic from small numerical perturbations; the score calculation can likewise be unstable if \(A_{\eta\eta}\) or \(\Omega\) is poorly conditioned. Check that the estimated covariance is symmetric and positive definite for the tested restrictions, that the nuisance fit converged, and that standard errors are plausible on the parameter scale. These checks do not establish the causal assumptions or blip-function specification.

A second interpretive pitfall is to infer that a non-significant joint test proves every treatment-time effect is zero. The test only indicates that the data do not provide sufficient evidence against the specified joint restriction at the chosen level; its power depends on sample size, covariance, and identification.