Optimal weighting for estimating generalized average treatment effects

2.1 Identification of GATE by weighting

In this section, we provide a general formulation of the weights that make τ ˆ W unbiased for GATE for any V 1 : n . To do so, we impose the assumption of consistency, noninterference, ignorable treatment assignment, and ignorable sample assignment [20]. Consistency states that the observed outcome corresponds to the potential outcome of the treatment applied to that unit, and noninterference reflects the fact that units potential outcomes are not affected by how the treatment or intervention has been allocated. Consistency together with noninterference are also known as stable unit treatment value assumption (SUTVA) [20]. Ignorable treatment assignment (also called unconfoundeness, no unmeasured confounding, or exchangeability) states that the potential outcome, Y i ( t ) , is conditionally independent of the treatment assignment mechanism given covariates. Similarly, ignorable sample assignment states that the potential outcome, Y i ( t ) , is conditionally independent of the sampling assignment mechanism, i.e., being labeled, given covariates. Thus, while the 1-treated, 0-treated, and unlabeled units can differ systematically in their distribution of X , Y ( 0 ) , Y ( 1 ) (despite units being marginally iid), we assume that, conditioned on X , the distributions of the potential outcomes are the same. Treatment and selection overlap state that the propensity of being treated and of being selected conditional on confounders is bounded away from zero and one and away from zero, respectively, ensuring we can always (eventually) find representative units in both treatment groups [20,22]. We formalize these assumptions as follow,

Letting H 1 : n = { X 1 : n , T 1 : n , S 1 : n } , we additionally assume,

Assumption 2.5 requires W 1 : n and V 1 : n to be functions of H 1 : n . We can easily relax this assumption to only require that W 1 : n and V 1 : n are independent of all else given H 1 : n , that is, that any residual randomness is independent of units’ idiosyncracies as they pertain to outcomes, and our results easily extend. However, all of our examples actually have W 1 : n and V 1 : n being functions of H 1 : n , which simplifies the presentation.

In the next lemma, we define the generalized IPW weights, W 1 : n IPW , and show that τ ˆ W IPW , the weighted estimator in equation (2.2) weighted by W 1 : n IPW , is unbiased for GATE.

This is a well-known result for SATE [13] and TATE [22,1]. If we assume appropriate bounds on the norms of V and the variances of Y 1 : n , it is easy to additionally see that τ ˆ W IPW has diminishing variance and is therefore also consistent. We show examples of generalized IPW, W i IPW , in Table 1. Note that when V i is a deterministic function of X 1 : n , T 1 : n , S 1 : n , as is usually the case, E [ V i ∣ X 1 : n , T − i , T i = t , S − i , S i = s ] just means plug in X 1 : n , T 1 : n , S 1 : n into this function except change T i to t and S i to s . Notice that ϕ ′ ( x ) does not in fact appear in Table 1; this is because it is only relevant when V i depends on treatment assignment T i for unlabeled units, which, while permissible in our framework, is an uncommon choice. When V i does not depend on T i for unlabeled units, it does not need to be observed or even exist as it appears nowhere else. Note also that we do not need overlap conditions on ϕ ′ ( x ) . Moreover, we do not actually need to know or estimate any of ϕ ( x ) , ϕ ′ ( x ) , ψ ( x ) in our method, which we present in the next section.

In the next section, we introduce KOM for estimating GATE, which, instead of plugging estimated propensities into the weighted estimator, provides weights that minimizes the CMSE of τ ˆ W for GATE. By doing so, the proposed methodology optimally minimizes the bias with respect to GATE while simultaneously controlling precision. We further consider simultaneously choosing V to minimize the worst-case CMSE to obtain KOWATE.

3.1 Decomposing the CMSE of τ ˆ W

Recall that, in Section 2, we defined g t ( X ) = E [ Y i ( t ) ∣ X i ] = E [ Y i ∣ X i , T i = t , S i = 1 ] . Further define ε i t = Y i ( t ) − g t ( X i ) and σ i t 2 = Var ( Y i ( t ) ∣ X i , S i = 1 ) = E [ ε t i 2 ∣ X i , S i = 1 ] , for t ∈ 0 , 1 , and σ i 2 = T i σ i 1 2 + ( 1 − T i ) σ i 0 2 . We then define, for each function f , the f -moment imbalance between the weighted t -treated labeled sample and the V -weighted total sample,

where I [ T i = t ] is equal to 1 if T i = t and 0 otherwise. In the following theorem, we show that the CMSE of τ ˆ W can be decomposed into the squared such imbalances in the conditional expectations of the potential outcomes.

In the next section, we show how to find weights that minimize equation (3.1). The main challenge in this task is that the functions g t , on which this quantity depends, are unknown.

3.2 Worst-case CMSE

To overcome the issue that we do not know the g t -functions that the CMSE of τ ˆ W depends, we will guard against any possible realizations of the unknown functions. Specifically, since the CMSE of τ ˆ W scales quadratically homogeneously with g 0 and g 1 , we consider its magnitude with respect to that of g 0 and g 1 . We therefore need to define a magnitude. We choose the following,^[2]

where ‖ ⋅ ‖ t are some extended seminorm on functions from the space of confounders to the space of outcomes. We discuss a specific choice of such extended seminorms in Theorem 3.2. Given this magnitude, we can define the worst-case squared bias as follows:

where

is the worst-case imbalance in the g t -moment between the weighted t -treated group and the V -weighted sample over all g t functions in the unit ball of ‖ ⋅ ‖ t .

There are many possible ways to choose the seminorm ‖ ⋅ ‖ t . RKHS norms is one rich family of choices that offer a flexible and general modeling framework, encompassing both parametric families and nonparametric universal approximators. As we will see, RKHS norms also give rise to an optimization problem that is amenable to both theoretical analysis and to the computational solution, with the choice of particular RKHS only affecting a matrix parameter fed into the solver.

Given a positive semidefinite (PSD) kernel K t ( x , x ′ ) , if we choose the corresponding RKHS (a Hilbert space of functions with continuous evaluations, which is associated with the reproducing kernel K t ) to specify the norm, we can show that the worst-case imbalance can be expressed as a convex-quadratic function in W 1 : n .

Based on Theorem 3.2, letting the RKHS given by the kernel K t specify the norm, both the worst-case bias and the worst-case CMSE of τ ˆ W are convex-quadratic functions in W 1 : n . Specifically, we define the worst-case CMSE as follows:

Note that there is freedom to scale this objective. In particular, λ 0 , λ 1 represent a ratio between variance and norm of g rather than variance bounds in the sense that for any hypothetical norm bound M > 0 , we have that the worst-case CMSE over g -functions with squared norms bounded by M and over residual variances bounded by λ t M is just a scaling of the aforementioned objective: M C ( W 1 : n , V 1 : n , λ 0 , λ 1 ) = sup ‖ g ‖ 2 ≤ M , σ i t 2 ≤ λ t M E [ ( τ ˆ W − τ V ) 2 ∣ H 1 : n ] . Since we will minimize this objective, it does not matter what positive constant we scale it by; all that matters is the trade-off rate between the bias and variance terms. For simplicity, we use within-treatment-group equal variance weights, λ 0 , λ 1 . More generally, we can use any positive definite matrix Λ to penalize the variances as W 1 : n T I S Λ I S W 1 : n . In the next section, we show how to minimize the worst-case CMSE of τ ˆ W in estimating GATE, C ( W 1 : n , V 1 : n , λ 0 , λ 1 ) , by using off-the-shelf solvers for quadratic optimization.

3.3 Minimizing the worst-case CMSE

In the previous two sections, we showed that the CMSE of τ ˆ W in estimating GATE can be decomposed in squared bias plus its variance. We also showed that, since the bias depends on unknown conditional expectations, by guarding against any possible realizations of these unknown functions, embedded in an RKHS given by the kernel K t , the worst-case CMSE of τ ˆ W can be expressed as a convex-quadratic function in W 1 : n . Here, we use quadratic programming to obtain the weights W 1 : n that minimizes the worst-case CMSE of τ ˆ W . When interested in estimating, for example, SATE, and TATE, the set of weights V 1 : n is fixed, i.e., all V i are given, known scalars. We show the corresponding convex-quadratic optimization problem when the set of weights V 1 : n is fixed in the next section. In addition, given the flexibility of the proposed methodology, we can also let V 1 : n be variable and let it be chosen by the solver in such a way that the worst-case CMSE of τ ˆ W is minimized. We show this in Section 3.3.2.

3.3.2 Variable V 1 : n

We can also let V 1 : n be variable. Instead of being given set values, we are given a feasible set V . We assume that V ⊂ { V 1 : n ∈ R n : V i ≥ 0 ∀ i , ∑ i n V i = 1 } and that V is a polytope (expressed by linear constraints). To simultaneously find the GATE, subject to V 1 : n ∈ V , that is most easily estimable and the weights W 1 : n to estimate this GATE, we propose to solve the following optimization problem:

(3.5) min V 1 : n ∈ V W 1 : n ∈ W C ( W 1 : n , V 1 : n , λ 0 , λ 1 ) .

When V = { V 1 : n } is a singleton, this optimization problem is the same as that in equation (3.3). Again, we can show that the optimization problem (3.5) reduces to a linearly-constrained convex-quadratic optimization problem:

(3.6) min V 1 : n ∈ V , W 1 : n ≥ 0 , W 1 : n T I S I 1 e n = n , W 1 : n T I 0 e n = n ( W 1 : n , V 1 : n ) T ∑ t ∈ { 0 , 1 } I S I t K t I t I S + λ t I S I t − I S I t K t − K t I t I S K t ( W 1 : n , V 1 : n ) .

The solution to the optimization problem (3.6) provides both weights V 1 : n ∗ that define a GATE of interest and the weights W 1 : n ∗ to estimate it. The weights V 1 : n are chosen to allow for minimal CMSE. That is, it focuses on the subpopulation where the average effect on which is easiest to estimate by KOM. We discuss this further in this study.

When we use V = { V 1 : n ∈ R n : V i ≥ 0 ∀ i , ∑ i n V i = 1 } , we term the resulting GATE estimand Kernel optimal weighted ATE (KOWATE). We can also construct other causal estimands by choosing different V . For instance, we may restrict to an unweighted subsample as in the OSATE of [8] by choosing V = { V 1 : n ∈ { 0 , 1 / n ′ } n : ∑ i n V i = 1 } , where n ′ is a chosen subsample size. We refer to this as Kernel optimal SATE (KOSATE). Table 2 summarizes these causal estimands. KOWATE and KOSATE can be seen as an extension of OWATE [9] and OSATE [8], where the average treatment effect is weighted by or restricted by overlap in covariate distributions to make the estimation easier. It is worth noticing that, other causal estimands can be easily constructed by plugging the set of overlap or truncated weights of [8,9] as fixed V 1 : n in the optimization problem.

Table 2

Summary of causal estimands, the corresponding of GATE weights V , and the set of optimal KOM weights W i KOM

Estimand	V	Type	W i KOM
SATE	{ S 1 : n }	Fixed ( ∣ V ∣ = 1 )	W i ∗ from (3.3)
SATT	T 1 : n ∑ j = 1 n T j	Fixed ( ∣ V ∣ = 1 )	W i ∗ from (3.3)
TATE	U 1 : n ∣ U ∣	Fixed ( ∣ V ∣ = 1 )	W i ∗ from (3.3)
OWATE	S i ϕ ( X 1 : n ) ( 1 − ϕ ( X 1 : n ) ) n O	Fixed ( ∣ V ∣ = 1 )	W i ∗ from (3.3)
KOWATE	{ V 1 : n ∈ R ≥ 0 n : ∑ i = 1 n V i = 1 }	Variable	W i ∗ from (3.5)
KOSATE	{ V 1 : n ∈ { 0 , 1 / n ′ } n : ∑ i = 1 n V i = 1 }	Variable	W i ∗ from (3.5)

What target populations are KOWATE and KOSATE choosing? The idea is to pick the subpopulation that is easiest to estimate by KOM. This subpopulation will emphasize areas with better overlap, where overlap is characterized in terms of worst-case moment imbalances as defined by the kernels, rather than in terms of (unknown) propensity scores.

We illustrate this in a simple simulated example described in Figure 1. Specifically, Figure 1 shows scatterplots between two confounders, one on the vertical axis and one on the horizontal axis, weighted by the weights V 1 : n , obtained when targeting SATE (first column of Figure 1, KOSATE (second top panel), KOWATE (third top panel), OSATE (second bottom panel), and OWATE (third bottom panel). The histograms on the top and right axes represent the distributions of the confounders across treated (dark-gray) and control (light-gray). The data was generated to exhibit practical positivity violations and we provide more details on the data generation in the simulation section (Section 4).

Figure 1

Weigths V 1 : n : Scatterplots between two confounders, confounder 1 in the X -axis and confounder 2 in the Y -axis, weighted by the set of weights V 1 : n , obtained when targeting SATE (first top and bottom panels), KOSATE (second top panel), KOWATE (third top panel), OSATE (second bottom panel), and OWATE (third bottom panel). The histograms on the top and right axes represent the distributions of the confounders across treated (dark-gray) and control (light-gray).

When targeting SATE, we consider a fixed V 1 : n that is equal to 1 for all units in the sample. On the other hand, all of KOSATE, OSATE, KOWATE, and OWATE focus on the area of confounders with high overlap. In practice, we find that this translate to better performance, as shown in Table 3 in our case study. KOSATE and OSATE do this while restricting to either including or excluding samples, as can be seen by the two point sizes in Figure 1. KOWATE and OWATE consider a range of weights, as can be seen by the variable point sizes. Visually, the weights that define the GATE for KOSATE and OSATE are similar as they both focus on the area of overlap; the same for KOWATE and OWATE. The differences are that KOWATE and KOSATE guard against possible misspecification of propensity models and that they target the CMSE of the estimator itself, rather than the asymptotic variance, and therefore they account for the desired precision of the KOM estimate that will be applied. We provide a deeper study of this in Section S3.1 of the Supplementary Material, where we consider the effects of misspecification as well as how the weights W 1 : n differ as well between the methods.

Table 3

The effect of fusion-plus-laminectomy on ODI

	SATE	KOSATE	KOWATE	Unadjusted
τ ˆ W (SE)	1.33 (3.98)	2.54 (2.56)	3.03 (2.38)	5.09* (2.31)

∗ indicates statistical significance at the 0.05 level.

3.4 Consistency

In this section, we study the consistency of the proposed weighted estimator with respect to the true causal estimand GATE (for V 1 : n both fixed and variables).

The aforementioned theorem shows that for any GATE estimand, under appropriate assumptions, the KOM estimate is root- n consistent.

The assumption about the kernel can be automatically satisfied by using a bounded kernel, such as the Gaussian or Matern kernels. The assumption of ‖ g ‖ < ∞ requires no model misspecification. We can relax this assumption if we use a universal kernel, such as Gaussian, but the rate may deteriorate from O p ( 1 / n ) to o p ( 1 ) as we need to include a vanishing approximation term. For brevity, we omit the details.

To apply Theorem 3.3 to the case of variable V 1 : n , note that the solution V 1 : n ∗ of problem (3.5) is a function of H 1 : n and is therefore honest (satisfies Assumption 2.5) and that, given these V 1 : n = V 1 : n ∗ , the solution W 1 : n ∗ to problem (3.3) is the exactly same as that in problem (3.5), as it can just be viewed as a nested minimization problem (once in V 1 : n and once in W 1 : n ). So to apply Theorem 3.3 to the case of variable weights, we need only guarantee that ∑ i = 1 n E [ V i 2 ] = O ( 1 / n ) (we already constrain ∑ i = 1 n V i = 1 ). We can either take that as an assumption, or we can enforce it in the construction of V by including a bound on each V i ≤ v ¯ / n for some v ¯ > 0 . In practice, we find that this is not necessary.

4.1 Setup

We considered a sample size of n = 400 . We computed the potential outcomes Y ( 1 ) , Y ( 0 ) from the following models: Y i ( 0 ) = 3 ( X i , 1 + X i , 2 ) + N ( 0 , 1 ) , Y i ( 1 ) = Y i ( 0 ) + δ , where δ = 4 . We computed the observed outcome as Y i = Y i ( 0 ) ( 1 − T i ) + Y i ( 1 ) T i , where T i ∼ binom ( π i ) , π i = ( 1 + exp ( − α ( − 1.5 + 1.5 X i , 1 + 1.5 X i , 2 ) ) ) − 1 , X k , i ∼ N ( 0.5 , 1 ) , k = 1 , 2 . We generated Z 1 = X 2 / exp ( X 1 ) and Z 2 = log ( ∣ X 2 ∣ ) and considered a convex combination between the correct variables ( X 1 , X 2 ) and the misspecified variables ( Z 1 , Z 2 ) , X ˜ 1 = ξ X 1 + ( 1 − ξ ) Z 1 and X ˜ 2 = ξ X 2 + ( 1 − ξ ) Z 2 . The true GATE was computed as τ V = ∑ i = 1 n V i ( Y i ( 1 ) − Y i ( 0 ) ) . We considered S i = 1 for all units in the sample. We consider several V i , namely, (a) KOWATE; (b) KOSATE, where we set n ′ = n T equal to the number of units chosen by OSATE, (c) SATE, (d) OWATE, and (e) OSATE with α = 0.1 . For KOWATE and KOSATE, we consider the KOM weights given by problem (3.5). For SATE, we consider several estimates: (a) KOM as in the optimization problem (3.3); (b) IPW; and (c) using outcome regression modeling. For OWATE and OSATE, we use the estimated propensity to define V 1 : n . To estimate OSATE, we computed the set of truncated weights setting α = 0.1 . We used a product of polynomial-degree-2 kernels for KOM. We modeled the propensity score, ϕ ( X i ) , by using a polynomial-degree-4 logistic regression and we used a polynomial-degree-4 regression model for outcome regression modeling. We evaluated the performance of the proposed method across levels of practical positivity violation and misspecification as described in Sections 4.1.1 and 4.1.2, respectively. In practice, the solver sometimes fails due to the quadratic objective being numerically nonPSD (despite being PSD in theory). We found that this can be fixed by adding 1 0 − 8 I to the quadratic objective matrix to inflate its spectrum slightly. If this also fails, when using a product of polynomial-degree-2 kernels, we rerun the same optimization problem by considering a product of a polynomial-degree-2 and degree-1 kernels. If the latter also failed, then we considered a product of polynomial-degree-1 kernels. We estimated the estimand of interest by plugging in the set of obtained weights into the weighted estimator τ ˆ W . We used scikit-learn (through the R package reticulate) to tune the hyperparameters and the R interface of Gurobi to obtain the set of KOM weights. More specifically, within scikit-learn, we used the GaussianProcessRegressor function with the kernel as the parameter (as described in Section S3.2 of the Supplementary Materials). Other parameters, such as alpha, and optimizer, among others were set to default. More details can be found in the scikit-learn’s https://scikit-learn.org/stable/modules/generated/sklearn.gaussian_process.GaussianProcessRegressor.htmlUser guide. We also included a simple difference in means (Crude) estimator to have a sense of the scale of improvement in absolute bias and RMSE.

4.2 Results

In this section, we discuss the results of our simulation study. In summary, KOM outperformed IPW, overlap, truncated weights, and outcome regression modeling with respect to absolute bias and RMSE in estimating GATE across levels of practical positivity violation under both moderate and strong misspecification.

4.2.1 Results across levels of practical positivity violations and model misspecification

Reference [19] presented KOM for SATE. The authors showed that KOM outperformed IPW, truncated IPW, propensity score matching, regression adjustment, CBPS, and SBW with respect to bias and MSE across most of the considered levels of practical positivity violation and considered scenarios. In addition, the authors showed that KOM for SATE outperformed the other methods especially under strong practical positivity violation. Figure 2 shows the absolute bias (left panels) and RMSE (right panels) of SATE estimated by using KOM (KOM-SATE; solid-black), KOSATE by using KOM (solid-dark-gray), KOWATE estimated by using KOM (solid-light-gray), SATE estimated by using IPW (long-dashed-black), OSATE estimated by using truncated weights (long-dashed-dark-gray), OWATE estimated by using overlap weights (long-dashed-light-gray), and SATE estimated by using outcome regression modeling (OM; dotted-black), and a simple mean difference (Crude; dotted-light-gray), with estimated optimal λ 0 , λ 1 (i.e., λ t = σ t 2 γ t 2 ). The top panels of Figure 2 show absolute bias and RMSE across levels of practical positivity violations under strong misspecification, while the middle and bottom panels under moderate misspecification and correct specification, respectively. In summary, KOM showed a consistently low absolute bias and RMSE across all the considered scenarios, outperforming the other methods especially under strong practical positivity violation and strong misspecification (top-right panel). KOM matched the performance of the other methods with respect to absolute bias and RMSE under weak levels of practical positivity violations and correct model misspecification. We obtained similar results when λ 0 = λ 1 = 0 (Figure S4). IPW exhibited extremely high absolute bias and RMSE under moderate to strong practical positivity violations and moderate to strong model misspecification. Under moderate and strong misspecification, OM resulted in even higher absolute bias and RMSE across all levels of practical positivity violations that the results are outside the plot region in the top and middle panels of Figures 2 and 4. In Figure 1, we showed that the weights obtained by using KOSATE and KOWATE select a population where there is more overlap compared with that of SATE. In our simulation results, this can be seen in the right bottom panel of Figure 2, where the RMSE obtained by using SATE is higher than that of KOWATE and KOSATE with similar absolute bias. To further analyze this behaviour, we evaluated the distribution of the maximum value of weights obtained using SATE, KOWATE, and KOSATE. While SATE obtained a median maximum value of weight around 27, KOWATE and KOSATE obtained values of 7 and 9, respectively. Consequently, the standard error computed from simulations for SATE, KOWATE, and KOSATE was around 0.29, 0.12, and 0.13, respectively (Figure S11 shows the distribution of the maximum value of the weights computed by using SATE, KOWATE, and KOSATE). We provide additional simulations results and some considerations about standard error estimation and coverage in Sections S4.1 and S4.2, respectively, in the Supplementary Material.

Figure 2

(Estimated optimal λ 0 , λ 1 ): Absolute bias (left panels) and RMSE (right panels) of SATE estimated by using KOM (solid-black) (which we refer to as KOM-SATE), KOSATE by using KOM (solid-dark-gray), KOWATE estimated by using KOM (solid-light-gray), SATE estimated by using IPW (long-dashed-black), OSATE estimated by using truncated weights (long-dashed-dark-gray), OWATE estimated by using overlap weights (long-dashed-light-gray), SATE estimated by using outcome regression modeling (dotted-black) (which we refer to as OM), and a simple difference in means (Crude) when increasing the strength of practical positivity violation under strong misspecification (top panels), moderate misspecification (middle panels), and correct (and overparametrized) specification (bottom panels), n = 400 .

5.1 The effect of fusion-plus-laminectomy on ODI

In this section, we apply KOM in the evaluation of two spine surgical interventions, laminectomy alone versus fusion-plus-laminectomy, on the Oswestry Disability Index (ODI), among patients with lumbar stenosis or lumbar spondylolisthesis. Briefly, lumbar stenosis is caused by the narrowing of the space around the spinal cord in the lumbar spine [23]. Lumbar spondylolisthesis is caused by the slippage of one vertebra on another. These pathologies lead to low back and leg pain, ultimately limiting the quality of life of those patients affected by them [24]. In case these pathologies are not anymore controlled by medications or physical therapy, surgical interventions may be needed. Typically, patients with lumbar stenosis are treated with laminectomy alone, while those with lumbar spondylolistheses with fusion-plus-laminectomy [23,25,26]. In addition, laminectomy alone is done to patients with leg pain, while fusion-plus-laminectomy to patients with mechanical back pain [23]. This surgical practice leads to a practical positivity violation.

Differently from other medical areas where randomized controlled trials are the gold standard to evaluate interventions, the use of randomized controlled trials to evaluate surgical interventions is rare. This is due to practical and methodological issues [27]. Lately, a number of large real-world observational datasets have collected information about surgical interventions and outcomes. However, these datasets are purely observational and confounding must be carefully taken into account. Furthermore, the assumption of the correct model specification is hardly ever met. To overcome these challenges, in this section, we evaluate the effect of fusion-plus-laminectomy on ODI by estimating SATE, KOWATE, and KOSATE using KOM.

5.1.3 Results

In this section, we present the results of our analysis. Previous randomized trials showed no statistically significant difference between laminectomy alone versus fusion-plus-laminectomy on ODI [31,32]. The proposed methodology consistently showed similar results to those of [31,32]. Specifically, Table 3 shows point estimates and standard errors with respect to SATE, KOWATE, and KOSATE. While the unadjusted method, i.e., naive method regressing only the treatment on the outcome, shows a significant effect of fusion-plus-laminectomy on ODI, adjusted estimates from SATE, KOWATE, and KOSATE show a non statistically significant effect of it. Standard errors are lower for KOWATE and KOSATE compared to SATE. Figure 3 shows the covariate balance with respect to SATE (top panel), KOSATE (middle panel), and KOWATE (lower panel). The black dots show the level of balance after weighting, while the light-gray dots show the unadjusted balance. KOWATE provides the lowest covariate balance compared with SATE and KOSATE. Finally, on the basis of the results obtained by applying KOM, we conclude that fusion-plus-laminectomy has no statistically significant effect on ODI.

Figure 3

Covariate balance with respect to SATE (top panel), KOSATE (middle panel), and KOWATE (lower panel). The black dots reflect the level of balance after weighting for SATE, KOSATE, and KOWATE weights, while the light-gray dots show the unadjusted balance.