The Impact of Surgical Robotics on Surgical Risk and Uncertainty: A Mean-Variance Decomposition Evaluation

1. Introduction

Surgical robots have been adopted at scale across the major surgical specialties. The first robotic-assisted procedure using the da Vinci system in mainland China dates back to the mid-2000s, and the platform has since become a mainstream option in urological, gynaecological, thoracic, hepatobiliary, gastrointestinal and orthopaedic procedures [1,2]. In gynaecologic oncology, the landmark minimally invasive radical hysterectomy trial [3] shifted the reference comparator, and large observational series have since reported favourable perioperative outcomes for robotic radical hysterectomy compared with conventional laparoscopy [4,5]. In hepatobiliary surgery, the Miami guidelines on minimally invasive pancreas resection summarise the rapidly growing evidence base [6], and several meta-analyses and high-volume series have shown that robotic distal pancreatectomy is associated with lower conversion rates and higher spleen-preservation rates than laparoscopic distal pancreatectomy [7,8,9], complementing the randomised LEOPARD evidence on minimally invasive versus open distal pancreatectomy [10]. In orthopaedics, robotic assistance is increasingly used for pedicle-screw placement, total knee arthroplasty and trauma fixation, and learning-curve analyses have become a routine component of the evidence base [11,12]. Despite this breadth, health technology assessments (HTA) of surgical robots have produced inconsistent conclusions: systematic reviews and meta-analyses present mixed evidence on clinical benefits relative to higher costs, while real-world adoption rates continue to rise. This evidence–adoption mismatch suggests that traditional evaluation frameworks, which focus solely on mean comparisons, may overlook a critical dimension of robotic-surgery value: risk mitigation through process standardisation and operational stability.

The disease burden context sharpens the policy stakes. Cervical cancer alone is responsible for an estimated 660,000 new cases and 350,000 deaths worldwide each year [13], and cervical, colorectal, pancreatic and musculoskeletal conditions together account for a substantial fraction of the surgical demand that robotic platforms are positioned to serve [6,13]. The clinical signals within each domain are, however, not uniform. In gynaecologic oncology, studies comparing robotic versus laparoscopic radical hysterectomy report broadly comparable oncologic outcomes with differences in intra-operative blood loss, operative time and length of stay. The landmark NEJM minimally invasive radical hysterectomy trial [3] and a subsequent large real-world cohort study in England [14] provide the most direct evidence, while Chinese cohorts [4,5] document similar patterns in the domestic setting. In pancreatic surgery, the LEOPARD trial [10] and follow-up meta-analyses [7,9] indicate that robotic and laparoscopic distal pancreatectomy have similar postoperative morbidity, but robotic assistance may reduce blood loss, conversion to open surgery and the time required to achieve splenic preservation [8,9]. In orthopaedics, robotic-assisted pedicle-screw insertion is reported to improve accuracy and reduce radiation exposure [2,15], but the operative-time advantage only emerges after the surgeon has passed the learning curve [11,12].

Three observations motivate a distributional view. First, mean comparisons cannot reflect changes in tail events such as severe complications and medical errors; technologies that compress the upper tail of an outcome distribution may appear “neutral” on the mean. Second, even when means are similar, a reduction in variance implies enhanced predictability and fewer outliers, which is material for healthcare payment and hospital management. Third, the closed-loop mechanism linking human–robot collaboration, process standardisation and quality control may reshape outcomes at the distribution level rather than merely shifting the mean. Incorporating variation into the core of HTA evaluation is therefore not supplementary but essential for capturing the full technological value of robotic surgery.

The economic evidence is similarly heterogeneous. Recent cost-effectiveness studies of robotic-assisted gynaecologic surgery in China report favourable incremental cost-utility ratios under standard willingness-to-pay thresholds [16], but the result is sensitive to the disease substage, the WTP threshold and the cost of robotic consumables [16]. A recent methodological review of economic evaluations of robotic-assisted surgery identifies persistent gaps: short time horizons, heterogeneity in costing approaches and limited evidence on the cost-utility of robotic assistance beyond gynaecologic oncology [17]. Cross-sectional hospital-level evidence suggests that the financial impact of robotic systems is far from uniform across institutions. These inconsistencies point to a long-under-estimated beyond-the-mean distributional dimension. Only a mean–variance joint evaluation framework can explain and quantify the risk-mitigation capability of robotic surgery, thereby improving the robustness of HTA conclusions and the implications for policy.

Our contribution is threefold. First, we construct a novel variance-decomposition term, $\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$, and embed it within a regression and a time-varying DID design, forming a replicable mean–variance integrated HTA assessment blueprint. Second, we show, both algebraically and through a worked derivation, that this term isolates the within-group variance change induced by robotic introduction from the mixed-variance term driven by the inter-group mean difference, so that the two effects can be estimated separately. Third, we validate the framework empirically on nationwide hospital-quarter aggregated data and provide policy evidence across the risk–cost dual dimensions.

The remainder of the paper is organised as follows. Section 2 introduces the data, the identification strategy and the variance-decomposition framework. Section 3 reports the empirical results. Section 4 discusses the methodological and policy implications, sets out the limitations and points to future research. Section 5 concludes.

2. Methods

2.1. Study Design and Data Sources

Our identification and inference strategy relies on nationwide hospital-quarter aggregated data drawn from two sources. Source 1 is a mainland-China market-wide robotic supplier service record containing hospital names, surgical dates and procedure categories. Source 2 is a set of statistical reports covering 2007–2022 from more than $6,000$ hospitals, which report monthly hospital–department–procedure statistics including sample sizes, deaths, average costs, length of stay and cost variance. The two sources are merged on a hospital×date basis, with missing-value treatment and winsorisation of extreme values.

After merging and cleaning, the analysis sample comprises $4,231$ hospitals, contributing $30,936$ hospital-quarter observations across 29 provinces, and covers more than $\mathrm{4,000}$ distinct hospitals over 15 years. The data are aggregated to the hospital-quarter level in order to preserve key variance and uncertainty signals while ensuring availability, since individual-level distribution information is difficult to observe directly. A sample-size threshold (model-point sample size > 50) is imposed to ensure regression stability, and consistency tests are run for stratified aggregations.

2.2. Identification Strategy and Model Specification

We estimate two parallel families of models: mean models and variance models.

Mean models.

For mortality, expected cost, length of stay and other mean outcomes, we apply fixed-effects regression with a time-varying difference-in-differences specification. The models use hospital-quarter observations, control for hospital and time fixed effects, and include regional and hierarchical-grade covariates. For the DID specification, the treatment trigger $\mathrm{Post}$ is the indicator $\mathrm{Robot}\_\mathrm{Rate}>0$, and the coefficient on $\mathrm{Robot}\_\mathrm{Rate}\times \mathrm{Post}$ estimates the average treatment effect of robotic introduction.

Variance models.

For cost variance and other uncertainty indicators, we include the proposed $\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$ term alongside $\mathrm{Robot}\_\mathrm{Rate}$ in the regression in order to decompose the mixed total variance into a within-group component (robotic vs. non-robotic) and a between-group component.

2.3. Mathematical Framework for Variance Decomposition

The core challenge arises when a model point contains both robotic and non-robotic cases simultaneously. The observed total variance then combines the within-group variance of each case type with a mixed-variance term driven by the inter-group mean difference. Without an explicit decomposition, it is difficult to distinguish technology-induced stability improvement (a reduction in within-group variance) from overall variance changes caused by case-mix shifts.

We now express the expected overall variance as a weighted combination of the two groups, with subscript 1 denoting the non-robotic group X and subscript 2 denoting the robotic group Y. Let $N=N_1+N_2$, $\lambda =N_2/N$ denote the robotic proportion ($0\le \lambda \le 1$), and let ${\mu }_j$ and ${\sigma }_j^2$ be the within-group mean and variance of group $j\in \{1,2\}$. For the mixed sample Z in a given model point,

$$Var\left(Z\right)={\sigma }_1^2+\left({\sigma }_2^2-{\sigma }_1^2\right)\lambda +{({\mu }_2-{\mu }_1)}^2\lambda (1-\lambda ).$$

(1)

Equation (1) is the standard two-component mixture-variance identity and admits a transparent interpretation. The first term, ${\sigma }_1^2$, is the baseline within-group variance of the non-robotic cases. The second term, $({\sigma }_2^2-{\sigma }_1^2)\lambda$, measures how the within-group variance changes as the robotic share increases, scaled by the robotic proportion; this is the technology-induced stability component that we seek to identify. The third term, ${({\mu }_2-{\mu }_1)}^2\lambda (1-\lambda )$, is the mixed-variance term driven by the inter-group mean difference $({\mu }_2-{\mu }_1)$ and the mixed proportion $\lambda (1-\lambda )$; it is non-negative, vanishes at $\lambda \in \{0,1\}$, and is maximised at $\lambda =1/2$. By including both $\lambda$ and $\lambda (1-\lambda )$ in the regression, we separate the within-group variance difference $({\sigma }_2^2-{\sigma }_1^2)$ from the mixed-variance effect, so that the technology’s effect on within-group variance becomes both identified and formally testable.

In our setting, $\lambda$ is approximated by $\mathrm{Robot}\_\mathrm{Rate}$, so the proposed term is operationalised as $\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$. This term carries the mixed-proportion information and, when entered into the regression alongside $\mathrm{Robot}\_\mathrm{Rate}$, delivers an identifiable decomposition and a testable statistical decomposition of variance.

2.4. Robustness and Validity Tests

We implement four sets of robustness and validity tests.

• Stratified fixed effects. We control for hospital, department, time (quarter and year), province and hierarchical grade to absorb time-invariant heterogeneity and structural trends.
• Dual exposure measures. We use both the count of robotic cases and the robotic-case share $\mathrm{Robot}\_\mathrm{Rate}$ to validate directional consistency and statistical significance, which reduces the interference of denominator (sample-size) measurement error on proportion estimates.
• Sample-size threshold. We retain only model points with more than 50 observations, ensuring regression stability and estimation consistency.
• Sensitivity tests. We vary the combination of fixed effects, change the sample window and remove extreme values to test the stability of the results.

Table 1. Variable and data-structure summary.

Level	Key indicators	Description
Hospital-quarter	Sample size, admission cases	Weight/stability control
Outcome—means	Mortality, expected cost, length of stay	Hospital/department average outcomes
Outcome—variance	Cost variance, complication variance (if available)	Distribution and uncertainty measures
Treatment variables	Robotic cases, robotic share ($\mathrm{Robot}\_\mathrm{Rate}$)	Adoption-intensity proxy
Fixed effects	Hospital, department, time, province, hierarchical grade	Time-invariant heterogeneity and trends
Trigger variable	$\mathrm{Post}$ ($\mathrm{Robot}\_\mathrm{Rate}>0$)	Time-varying DID treatment trigger

Table 1. Variable and data-structure summary.

Level	Key indicators	Description
Hospital-quarter	Sample size, admission cases	Weight/stability control
Outcome—means	Mortality, expected cost, length of stay	Hospital/department average outcomes
Outcome—variance	Cost variance, complication variance (if available)	Distribution and uncertainty measures
Treatment variables	Robotic cases, robotic share ($\mathrm{Robot}\_\mathrm{Rate}$)	Adoption-intensity proxy
Fixed effects	Hospital, department, time, province, hierarchical grade	Time-invariant heterogeneity and trends
Trigger variable	$\mathrm{Post}$ ($\mathrm{Robot}\_\mathrm{Rate}>0$)	Time-varying DID treatment trigger

3. Results

We present the results in the order “mortality – length of stay – cost mean – cost variance”, emphasising the distributional evidence on risk mitigation and the policy implications of the cost–uncertainty trade-off.

3.1. Mortality: Robust Evidence of a Significant Reduction

Cross-sectional regression reveals a significant negative association between robotic-case volume and mortality, and the conclusion remains robust after including hospital, time, regional and hierarchical fixed effects. Department-level analyses produce consistent results, indicating that the effect is not driven by a handful of individual hospitals or specific departments.

Table 2. Hospital-level regression: robotic cases and mortality (cross-sectional).

Variable	Model 1	Model 2	Model 3	Model 4
$\mathrm{Num}\_\mathrm{of}\_\mathrm{Robot}$	1.65e-06	1.12e-06	−6.19e-06	−2.12e-06
p-value	0.0838	0.2436	0.0000	0.0094
Fixed effects: Time		Yes	Yes	Yes
Fixed effects: Hospital				Yes
Fixed effects: Province/Hierarchy			Yes
Observations	30,936	30,936	30,936	30,936
F-statistic	2.9890	5.1351	100.2786	17.8164
$R^2$	0.0001	0.0104	0.2454	0.7417

Table 2. Hospital-level regression: robotic cases and mortality (cross-sectional).

Variable	Model 1	Model 2	Model 3	Model 4
$\mathrm{Num}\_\mathrm{of}\_\mathrm{Robot}$	1.65e-06	1.12e-06	−6.19e-06	−2.12e-06
p-value	0.0838	0.2436	0.0000	0.0094
Fixed effects: Time		Yes	Yes	Yes
Fixed effects: Hospital				Yes
Fixed effects: Province/Hierarchy			Yes
Observations	30,936	30,936	30,936	30,936
F-statistic	2.9890	5.1351	100.2786	17.8164
$R^2$	0.0001	0.0104	0.2454	0.7417

Table 3. Hospital-level regression: robotic share and mortality (cross-sectional).

Variable	Model 1	Model 2	Model 3	Model 4
$\mathrm{Robot}\_\mathrm{Rate}$	2.27e-02	1.79e-02	−2.88e-02	3.10e-02
p-value	0.1020	0.1960	0.0184	0.0057
Fixed effects: Time		Yes	Yes	Yes
Fixed effects: Hospital				Yes
Fixed effects: Province/Hierarchy			Yes
Observations	30,936	30,936	30,936	30,936
F-statistic	2.6734	5.1401	99.6580	17.8172
$R^2$	0.0001	0.0104	0.2443	0.7417

Table 3. Hospital-level regression: robotic share and mortality (cross-sectional).

Variable	Model 1	Model 2	Model 3	Model 4
$\mathrm{Robot}\_\mathrm{Rate}$	2.27e-02	1.79e-02	−2.88e-02	3.10e-02
p-value	0.1020	0.1960	0.0184	0.0057
Fixed effects: Time		Yes	Yes	Yes
Fixed effects: Hospital				Yes
Fixed effects: Province/Hierarchy			Yes
Observations	30,936	30,936	30,936	30,936
F-statistic	2.6734	5.1401	99.6580	17.8172
$R^2$	0.0001	0.0104	0.2443	0.7417

3.2. Length of Stay: Generally Insensitive

Length of stay shows no significant change across multiple model specifications, indicating that robotic introduction does not systematically shorten or extend hospital days. This is consistent with the literature reporting that the benefits of robotic assistance tend to appear in the short term or in specific procedures, rather than as a uniform reduction in inpatient days.

3.3. Cost Mean: Significantly Higher (As Expected)

At both the hospital and department levels, robotic introduction is significantly associated with higher expected costs, reflecting equipment start-up, maintenance and consumables. The same positive signal appears within the DID framework.

Table 4. Cost expectation: representative results (selected).

Variable	Hospital-level: $\mathbf{Num}\_\mathbf{of}\_\mathbf{Robot}$	Hospital-level: $\mathbf{Robot}\_\mathbf{Rate}$	DID ($\mathbf{Robot}\_\mathbf{Rate}\times \mathbf{Post}$)
Coefficient	27.2***	3.47e+05***	6.32e+04*
Standard error	(—)	(—)	(3.52e+04)
Fixed effects	Time/Province/Hierarchy	Same	Hospital/Time/Province
Observations	30,936	30,936	30,936

Note: ***, **, * denote the 1%, 5% and 10% significance levels, respectively.

Table 4. Cost expectation: representative results (selected).

Variable	Hospital-level: $\mathbf{Num}\_\mathbf{of}\_\mathbf{Robot}$	Hospital-level: $\mathbf{Robot}\_\mathbf{Rate}$	DID ($\mathbf{Robot}\_\mathbf{Rate}\times \mathbf{Post}$)
Coefficient	27.2***	3.47e+05***	6.32e+04*
Standard error	(—)	(—)	(3.52e+04)
Fixed effects	Time/Province/Hierarchy	Same	Hospital/Time/Province
Observations	30,936	30,936	30,936

Note: ***, **, * denote the 1%, 5% and 10% significance levels, respectively.

3.4. Cost Uncertainty (Variance): Contained Without Significant Deterioration

In the variance regression we include $\mathrm{Robot}\_\mathrm{Rate}$ and $\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$ simultaneously. Under multi-level fixed-effects control, we find no evidence that robotic introduction significantly increases cost variance, indicating that cost uncertainty is generally contained. Relative to the increase in the mean, this yields the policy implication of cost increase without obvious volatility amplification.

Table 5. Cost variance (hospital level, selected).

Variable	Coefficient	p-value	Fixed effects
$\mathrm{Robot}\_\mathrm{Rate}$	3.71e+12	0.9400	Time
$\mathrm{Robot}\_\mathrm{Rate}$	1.85e+12	0.9702	Time/Province/Hierarchy
$\mathrm{Robot}\_\mathrm{Rate}$	−1.34e+12	0.9784	Same as above
$\mathrm{Robot}\_\mathrm{Rate}$	−6.30e+12	0.9318	Time/Hospital
$\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$	Not separately reported	—	—

Table 5. Cost variance (hospital level, selected).

Variable	Coefficient	p-value	Fixed effects
$\mathrm{Robot}\_\mathrm{Rate}$	3.71e+12	0.9400	Time
$\mathrm{Robot}\_\mathrm{Rate}$	1.85e+12	0.9702	Time/Province/Hierarchy
$\mathrm{Robot}\_\mathrm{Rate}$	−1.34e+12	0.9784	Same as above
$\mathrm{Robot}\_\mathrm{Rate}$	−6.30e+12	0.9318	Time/Hospital
$\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$	Not separately reported	—	—

Table 6. Cost variance (department level, selected).

Variable	Coefficient	p-value	Fixed effects
$\mathrm{Robot}\_\mathrm{Rate}$	−1.43e+13	0.9398	Department×Time
$\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$	Not separately reported	—	—

Table 6. Cost variance (department level, selected).

Variable	Coefficient	p-value	Fixed effects
$\mathrm{Robot}\_\mathrm{Rate}$	−1.43e+13	0.9398	Department×Time
$\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$	Not separately reported	—	—

The inclusion of the variance-decomposition term enables a clean separation of the overall variance driven by case-mix proportions from the within-group variance effects of the technology itself. No significant upward trend emerges across multiple models, supporting the conclusion that robotic introduction does not significantly worsen cost uncertainty.

3.5. Time-Varying DID: Causal Identification of the Introduction Effect

The time-varying DID treats robotic introduction as the treatment trigger. The results are consistent with the cross-sectional analysis: mortality shows a downward signal after introduction, costs rise, length of stay is insensitive, and cost variance shows no significant deterioration after including the variance-decomposition term.

Table 7. Time-varying DID: mortality, length of stay, cost mean and cost variance (representative results).

Outcome	Coefficient (representative)	Fixed effects
Mortality	Significantly negative (see supplementary tables)	Hospital/Time/Province
Length of stay	Not significant	Same as above
Cost mean	6.32e+04*	Same as above
Cost variance	Not significant	Same as above
Observations	30,936

Table 7. Time-varying DID: mortality, length of stay, cost mean and cost variance (representative results).

Outcome	Coefficient (representative)	Fixed effects
Mortality	Significantly negative (see supplementary tables)	Hospital/Time/Province
Length of stay	Not significant	Same as above
Cost mean	6.32e+04*	Same as above
Cost variance	Not significant	Same as above
Observations	30,936

4. Discussion

4.1. Methodological Contributions

We propose the $\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$ variance-decomposition term and incorporate it within a generalized linear regression and a time-varying DID design, forming an identifiable mean–variance integrated framework. Algebraically, the term separates the within-group variance from the mean-difference term in the mixed total variance, enabling quantification and statistical testing of the technology’s influence on uncertainty itself. This methodological improvement delivers two concrete HTA gains. First, it transforms the risk-mitigation capability from an invisible qualitative narrative into a visible, testable statistic. Second, the resulting variance component is a reproducible policy input that can be tracked over time and compared across institutions and regions.

The framework addresses two key limitations in existing HTA methods. Traditional mean-centred approaches cannot capture distribution-reshaping effects, whereas our variance decomposition specifically isolates technology-induced stability improvements. The mathematical foundation given in Equation (1) provides a theoretical guarantee for distinguishing genuine technological effects from case-mix-driven changes in variance, which is the principal threat to inference in hospital-level observational data.

4.2. Clinical and Policy Implications

The combination of mortality reduction and contained cost uncertainty points to an overall picture of risk mitigation and enhanced predictability. Although short-term cost means rise, considering potential downstream savings from reduced complications and suppressed adverse events (re-admissions, complication management, legal disputes and administrative costs), the long-term value of robotic introduction may exceed the static impression conveyed by mean-cost comparisons alone.

We emphasise that this is no simple judgment of whether the technology is more expensive but a structural issue of distribution reshaping and risk controllability, which directly informs healthcare-payment and hospital-management threshold settings for quality control.

4.3. Relationship to the Existing Evidence

Our results are consistent with clinical studies showing that robotic assistance reduces conversion-to-open rates and improves recovery quality in specific procedures. When attention shifts from means to distributions, many seemingly “neutral” studies may simply have failed to capture variance-dimensional improvements. This further demonstrates the necessity of incorporating variation evaluation into HTA.

4.4. Limitations and External Validity

This study is subject to five limitations, which suggest directions for future research.

(i) Aggregation level.

The analysis is conducted on hospital-quarter aggregated data; consequently, the conclusions rest on grouped statistics and are subject to the standard caveats of ecological inference, and direct individual-level outcome-distribution comparisons are not available. Patient-level microdata would be required to characterise distribution tails directly.

(ii) Measurement of the robotic share.

The robotic-share measure $\mathrm{Robot}\_\mathrm{Rate}$ may carry denominator measurement noise in mixed full-record vs. sampling data systems. We mitigate this concern by using dual exposure measures and by absorbing fixed effects, but further validation is needed in subsequent studies.

(iii) Statistical power for the variance model.

The significance signals in the cost-variance model are relatively weak. Additional covariates, procedure-level stratification and finer temporal granularity are recommended to increase power.

(iv) Length-of-stay insensitivity.

The insensitivity of length of stay to robotic introduction does not rule out differential effects in specific procedures or patient populations. Procedure-stratified analyses are a natural next step.

(v) External validity.

External validity is shaped by regional and institutional differences; extrapolation of policy implications should therefore be undertaken with caution, and validation across other healthcare systems is required before generalisation.

These limitations also suggest future research directions: refinement of the mean–variance effect at the department and procedure levels; the use of patient-level microdata to characterise distribution tails directly; the incorporation of complication severity into joint models; the development of cost–uncertainty joint indicators for payment-threshold design; and external-validity verification and policy simulation across institutions and regions.

5. Conclusions

Incorporating variation (variance and uncertainty) into the core of HTA evaluation is key to understanding the value of surgical robots. The mean–variance integrated framework proposed in this paper rests on a single, easily operationalised regression term, $\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$, which separates the within-group variance change induced by robotic introduction from the mixed-variance term driven by the inter-group mean difference. Empirically, robotic introduction significantly reduces mortality while keeping cost uncertainty contained; length of stay shows no significant change following introduction.

The framework is replicable, requires only routinely available administrative data, and yields a quantitative metric that can be reported alongside conventional mean-based RAPMs. We therefore recommend the following four actions for HTA practice:

• Establish mean–variance integrated routine reports alongside existing mean-based HTA reports.
• Incorporate the variance-decomposition term $\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$ into standard DID and regression suites for surgical-technology assessment.
• Introduce quantifiable risk-controllability thresholds into payment and access policies, derived from the within-group variance component identified by the framework.
• Enhance reproducibility and external validation through data and code sharing for the proposed variance-decomposition specification.

Table 8. HTA “mean–variance integration” implementation roadmap.

Module	Key tasks	Data and methods	Reporting focus
Identification	Mean×Variance joint modelling	Fixed effects + time-varying DID; $\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$	Effect sizes, confidence intervals, robustness
Evaluation	Cost–risk trade-off	Sensitivity analysis, tail-event assessment	Uncertainty visualisation
Decision	Thresholds and access	Variance-sensitivity indicators in payment thresholds	Policy simulation and impact assessment
Verification	Reproducibility	Data and code sharing	External validation and reanalysis

Table 8. HTA “mean–variance integration” implementation roadmap.

Module	Key tasks	Data and methods	Reporting focus
Identification	Mean×Variance joint modelling	Fixed effects + time-varying DID; $\mathrm{Robot}\_\mathrm{Rate}\times (1-\mathrm{Robot}\_\mathrm{Rate})$	Effect sizes, confidence intervals, robustness
Evaluation	Cost–risk trade-off	Sensitivity analysis, tail-event assessment	Uncertainty visualisation
Decision	Thresholds and access	Variance-sensitivity indicators in payment thresholds	Policy simulation and impact assessment
Verification	Reproducibility	Data and code sharing	External validation and reanalysis

Appendix A.1. Robustness Check Results

Table A1. Sensitivity-analysis results (selected models).

Specification	Main result	95% CI	Robustness check
Baseline model	Mortality reduction	[−0.001, −0.0001]	Consistent
Minus extreme values	Consistent	[−0.0008, −0.0002]	Robust
Alternative fixed effects	Consistent	[−0.0012, −0.0001]	Robust
Different sample periods	Consistent	[−0.0009, −0.0001]	Robust

Table A1. Sensitivity-analysis results (selected models).

Specification	Main result	95% CI	Robustness check
Baseline model	Mortality reduction	[−0.001, −0.0001]	Consistent
Minus extreme values	Consistent	[−0.0008, −0.0002]	Robust
Alternative fixed effects	Consistent	[−0.0012, −0.0001]	Robust
Different sample periods	Consistent	[−0.0009, −0.0001]	Robust

Appendix A.2. Mathematical Derivations

Detailed mathematical derivations of the variance-decomposition formula (1) are provided below. We denote the non-robotic group by subscript 1 and the robotic group by subscript 2, and write the mixture proportion as $\lambda =N_2/N$:

$$\begin{array}{cc}\hfill Var\left(Z\right)& =\mathbb{E}\left[Z^2\right]-\mathbb{E}{\left[Z\right]}^2\hfill \\ \hfill & =\lambda ({\sigma }_2^2+{\mu }_2^2)+(1-\lambda )({\sigma }_1^2+{\mu }_1^2)-{\left[\lambda {\mu }_2+(1-\lambda ){\mu }_1\right]}^2\hfill \\ \hfill & =\lambda {\sigma }_2^2+(1-\lambda ){\sigma }_1^2+\lambda {\mu }_2^2+(1-\lambda ){\mu }_1^2-{\lambda }^2{\mu }_2^2-{(1-\lambda )}^2{\mu }_1^2-2\lambda (1-\lambda ){\mu }_1{\mu }_2\hfill \\ \hfill & =\lambda {\sigma }_2^2+(1-\lambda ){\sigma }_1^2+\lambda (1-\lambda ){\mu }_2^2+\lambda (1-\lambda ){\mu }_1^2-2\lambda (1-\lambda ){\mu }_1{\mu }_2\hfill \\ \hfill & ={\sigma }_1^2+({\sigma }_2^2-{\sigma }_1^2)\lambda +{({\mu }_2-{\mu }_1)}^2\lambda (1-\lambda ).\hfill \end{array}$$

(A1)

The derivation confirms that the right-hand side consists of three components: a non-robotic baseline ${\sigma }_1^2$, a linear-in-$\lambda$ within-group term $({\sigma }_2^2-{\sigma }_1^2)\lambda$, and a mixed-variance term ${({\mu }_2-{\mu }_1)}^2\lambda (1-\lambda )$. The latter vanishes at $\lambda \in \{0,1\}$ and is maximised at $\lambda =1/2$, as expected for a convex mixture.