Weighted Likelihood Perturbations for Subject Influence

Why Perturb Subject Contributions?

A fitted selection model combines an outcome distribution with a model for the probability of each response pattern conditional on the outcomes. A subject's observed-data likelihood contribution therefore reflects both parts of the model, including integration over outcomes that were not observed. To assess how the fit responds to a subject's influence, one can multiply that subject's entire contribution by a weight near one. This changes its relative contribution to the fitting criterion without changing the response pattern, the observed values, or the model for selection.

The distinction is important. A likelihood weight is a controlled perturbation of the objective function; it is not a new value of a missing outcome and it is not a direct change to a subject's probability of being observed. The perturbation is useful only when its unit and construction are clear: here, one contribution corresponds to one independent subject's observed record.

Observed-Data Contributions and Weights

Let $Y_i=(Y_i^{obs},Y_i^{mis})$ denote the complete outcome vector for subject $i$, split into observed and missing components, and let $r_i$ record the subject's response pattern. Write $x_i$ for the subject's covariates. A selection model specifies an outcome density $f(y_i\mid x_i;\theta)$ and a response-pattern probability $g(r_i\mid y_i,x_i;\psi)$. The combined parameter is $\eta=(\theta,\psi)$. For independent subjects, the observed-data likelihood contribution is

L_i(\eta) = \int f(y_i^{obs},y_i^{mis}\mid x_i;\theta)\, g(r_i\mid y_i^{obs},y_i^{mis},x_i;\psi)\, dy_i^{mis}.

The integral is over values compatible with the subject's observed outcomes. If missing outcomes are discrete, the integral is replaced by a sum. If there are no missing outcomes, the contribution is the joint density or mass evaluated at the complete outcome and observed response pattern. In all cases, the response-pattern term remains part of the observed-data contribution. This construction does not assume that the missingness mechanism can be ignored.

Definition: For positive observed-data contributions $L_i(\eta)$, the subject-weighted log likelihood is $$ \ell(\eta;\omega)=\sum_{i=1}^{n}\omega_i\ell_i(\eta), \qquad \ell_i(\eta)=\log L_i(\eta), $$ where each likelihood weight $\omega_i$ is positive. The unperturbed fit has $\omega_i=1$ for every subject. A subject-specific perturbation changes one or more weights while leaving each $L_i(\eta)$'s selection-model construction unchanged.

For a single subject $j$, a simple path is $\omega_j(a)=1+a$, with $\omega_i(a)=1$ for $i\ne j$. Its baseline is $a=0$, and it is valid for $a>-1$, so that the perturbed weight stays positive. Alternatively, $\omega_j(a)=\exp(a)$ is positive for every real $a$. More generally, for a direction $d=(d_1,\ldots,d_n)$, the linear path $\omega_i(a)=1+a d_i$ is valid on any interval where all its weights remain positive. A path is a way to define the perturbation, not an assumption that the changed weights describe a new sampling process.

Worked Example: Reweighting One Incomplete Subject

Suppose a fitted selection model gives three hypothetical observed-data likelihood contributions at a fixed parameter value: $L_1=0.12$, $L_2=0.08$, and $L_3=0.20$. These are likelihood factors from the model, not necessarily probabilities of the observed records. Subject 2 has missing outcomes, and its contribution includes the integration over those outcomes. Consider the fixed-parameter log likelihood and then increase only subject 2's weight to $1.2$.

The unweighted log likelihood is

\ell =\log(0.12)+\log(0.08)+\log(0.20) \approx -2.1203-2.5257-1.6094 =-6.2554.

With $\omega_2=1.2$, the weighted criterion is

\ell(\omega) =\log(0.12)+1.2\log(0.08)+\log(0.20) \approx -2.1203-3.0308-1.6094 =-6.7605.

Thus the criterion decreases by about $0.5051$ at this fixed parameter value: the change is $0.2\log(0.08)$. No response probability was modified and no missing outcome was filled in. The entire observed-data contribution for subject 2 was given 20% more weight.

Why the Perturbation Is Smooth

The smoothness of the weighted criterion follows directly from its definition. Suppose the $L_i(\eta)$ are positive and the subject log contributions are differentiable in $\eta$. Along a linear weight path $\omega_i(a)=1+a d_i$, while holding $\eta$ fixed, the criterion is

\ell(\eta;\omega(a)) =\sum_i (1+a d_i)\ell_i(\eta) =\ell(\eta;\mathbf{1})+a\sum_i d_i\ell_i(\eta).

Consequently, its derivative with respect to the path parameter is

\left.\frac{\partial\ell(\eta;\omega(a))}{\partial a}\right|_{a=0} =\sum_i d_i\ell_i(\eta).

This derivative describes how the objective changes at a fixed parameter value. It is not the change in the fitted parameter; that requires studying how the maximizing parameter moves and is a separate calculation. For a single-subject path, $d_j=1$ and all other $d_i=0$, so the fixed-parameter derivative is simply $\ell_j(\eta)$. For an exponential path $\omega_i(a)=\exp(a d_i)$, the derivative at zero is the same, because $\left.\frac{d}{da}\exp(a d_i)\right|_{a=0}=d_i$.

The regularity conditions behind this construction are modest but consequential: the observed-data contributions must be finite and positive at the parameter values considered, the log contributions must be defined, and the chosen weights must remain positive. Smooth dependence of $L_i$ on $\eta$ is needed for smooth optimization in the model parameters. In likelihoods defined by integrals, differentiating with respect to $\eta$ may also require conditions that permit differentiation under the integral sign. Numerical integration should be sufficiently accurate that apparent changes along the weight path are not merely integration error.

Worked Example: A Linear Weight Path and Its Change

For a hypothetical incomplete subject $j$, suppose the integrated observed-data contribution at a fixed parameter value is $L_j=0.30$. Suppose the unperturbed log likelihood for all subjects is $-10.00$. Use the linear path $\omega_j(a)=1+a$ and set $a=0.25$, leaving every other weight at one.

The subject's log contribution is $\ell_j=\log(0.30)\approx-1.2040$. The perturbed weight is $1.25$, which is positive, and the change in the criterion is

\ell(\eta;\omega(a))-\ell(\eta;\mathbf{1}) =a\ell_j =0.25\log(0.30) \approx 0.25(-1.2040) =-0.3010.

The perturbed criterion at this same parameter value is therefore $-10.3010$. Its derivative with respect to $a$ is $\log(0.30)\approx-1.2040$ throughout this linear path. The path remains valid for $a>-1$; a value such as $a=-1.1$ would give weight $-0.1$, which is not an admissible likelihood weight.

Keeping the Selection Model Intact

The subject weight multiplies the log of the whole observed-data contribution. It does not belong inside just the outcome density, just the response-pattern probability, or just the integral over missing values. In general, weighting only one factor would define a different criterion and could distort the selection-model structure. The intended perturbation is

\omega_i \log\!\left\{ \int f(y_i^{obs},y_i^{mis}\mid x_i;\theta) g(r_i\mid y_i^{obs},y_i^{mis},x_i;\psi)\,dy_i^{mis} \right\},

not a replacement of $g$ by a newly parameterized response model. The integral is formed first to obtain the subject's observed-data contribution; then its log is weighted. This order matters because the log of an integral is not generally the integral of a log.

Worked Example: Reweighting Is Not Changing Selection Probabilities

Consider two hypothetical subjects, one with incomplete outcomes and one with a complete record. At a fixed fitted parameter value, suppose their correctly integrated observed-data contributions are $L_A=0.24$ and $L_B=0.40$. Upweight subject A by 20%, so $\omega_A=1.2$ and $\omega_B=1$. The change in the objective is

\Delta\ell =(1.2-1)\log(0.24) =0.2(-1.4271) =-0.2854.

This operation leaves both subjects' model-based contributions at $0.24$ and $0.40$, respectively; only the coefficient on $\log L_A$ changes. Now suppose, instead, that changing a selection-model parameter led to hypothetical recalculated contributions $L_A=0.20$ and $L_B=0.43$. The change in the unweighted objective would be

\log\left(\frac{0.20}{0.24}\right) +\log\left(\frac{0.43}{0.40}\right) \approx -0.1823+0.0723 =-0.1100.

That second calculation changes the model-based contributions themselves, and it changes them for both subjects. It is not the same perturbation as upweighting A. The numerical values are hypothetical illustrations of the distinction; in an actual fit, each recalculated contribution would have to follow from the specified selection model and its integration over missing outcomes.

Diagnostics, Alternatives, and Limitations

A weighted path is interpretable only if its definition and numerical behavior are checked. Before using the perturbed criterion, verify that the unit is the intended subject, every weight remains positive over the stated range, and the contribution being weighted is the observed-data contribution for the subject's full response pattern. Then check that the optimizer converges from suitable starting values, that the fitted parameters are not at an unintended boundary, and that numerical integration accuracy is adequate at each fit. In particular, a change in the estimate that disappears when integration tolerances are tightened is not reliable evidence of subject influence.

Some common failure modes are conceptual rather than computational:

Weighting only the outcome part. This fails to perturb the whole observed-data record under the selection model. Construct the integrated joint contribution first, then weight its log.
Allowing a nonpositive weight. A zero weight removes a contribution, and a negative weight is not a positive likelihood weight. Restrict a linear path to an interval where all weights are positive, or use a positive path such as an exponential.
Confusing an objective change with a parameter change. The fixed-parameter derivative above does not tell how far the estimate moves. Those are distinct quantities.
Treating a fractional weight as an ordinary new observation. Unless weights represent actual frequency replication under an appropriate sampling scheme, the weighted criterion is a perturbed or pseudo-likelihood, not the ordinary likelihood for the original independent sample.

The main alternative benchmark is exact case deletion, which sets one subject's contribution aside and refits without it. That is a finite change, not a smooth perturbation around weight one; it can be useful as a direct benchmark but may require a separate optimization. A subject-weight path instead retains the contribution and varies its weight continuously near one. Changing a selection-model parameter is another kind of analysis: it changes the response mechanism and its implications for observed-data contributions, rather than adjusting one subject's relative contribution. These approaches answer different questions and should not be described as interchangeable.

The construction also relies on the chosen observational unit. If records are dependent within clusters, treating each record as an independent subject contribution may not represent the stipulated likelihood. The weight should correspond to the unit whose contribution enters the correctly specified likelihood, such as a whole independent cluster when that is the sampling unit. Finally, a local weight perturbation does not resolve model misspecification, weak identification, or uncertainty about the unobserved outcomes. It is a diagnostic perturbation within the specified selection model, not a remedy for those limitations.

Key takeaway: Start from each subject's observed-data contribution under the full selection model, take its log, and multiply that log contribution by a positive weight near one. This gives a smooth, clearly defined perturbation while preserving the model's outcome and selection structure.

Check Your Understanding

Answer each question, then compare with the solutions below.

For a single subject $j$, write a linear weight path around the ordinary likelihood and state its positivity condition.
If $\ell_j=-0.8$ and the subject's weight increases from $1$ to $1.1$, what is the change in the objective at fixed parameters?
Why is multiplying only the selection probability by a subject weight not the same as the defined subject-weight perturbation?
What sampling feature might require weighting a cluster contribution rather than an individual-record contribution?

Solutions. (1) Set $\omega_j(a)=1+a$, keeping other weights at one; it is positive when $a>-1$. (2) The change is $(1.1-1)\ell_j=0.1(-0.8)=-0.08$. (3) The defined perturbation multiplies the log of the entire integrated observed-data contribution, including both the outcome and response-pattern components; weighting only one component changes the criterion's construction. (4) Dependence among records within a cluster can mean the independent sampling unit, and therefore the appropriate likelihood contribution, is the cluster rather than each record.