Military interventions and occupations underlie this thread - despite the apparent plethora of medical procedures, cats in hats and persistent bathtub rings. So, to go to the former (and not confronting the latter), here's another resource to review.

2009 Vernetti (MAJ USA), Three's Company: The Efficacy of Third-Party Intervention in Support of Counterinsurgency.

Its BLUF is in this graphic:

Intervention Success.jpg

as explained by the author:

Analysis

Taken together, these results confirm that third-party intervention on behalf of a counterinsurgent decreases the likelihood of a successful outcome. Five of the seven hypotheses are confirmed with Hypothesis #5 and #6 being found not valid (see Table 3).

The results indicate that the occurrence of an early intervention or the occurrence of an early termination of an intervention do not significantly affect the chances for counterinsurgent success.

The results also indicate that third party interventions, military deployments, military occupations, and interventions involving democracies all decrease the likelihood of a successful conflict outcome.

Interventions involving an “indirect” approach to counterinsurgency represent the most promising possibility for third-party intervention with the results indicating that an intervener that participates in this type of counterinsurgent effort has a significantly better chance of bringing a successful outcome.
What is an "indirect" approach, as opposed to a "direct" approach, will be a point of controversy. At least it requires definition in the context of reality - as opposed to "translating" and "interpreting" metaphors and analogies.

Here's his definition of the chart's variables:

INTERVENTION variable: tests for the occurrence of third party intervention in the form of military occupation, military intervention, or other military aid in support of the counterinsurgent forces. The study also includes the suppression of colonial rebellions as interventions if the colonial power deployed additional troops from outside of the colony in orderbto support the counterinsurgent. The variable titled INTERVENTION is coded “1” if an outside country or colonial power provided assistance to the counterinsurgent during the conflict. The dataset includes 59 conflicts that involved third party intervention.

MILITARY variable: refers to the type of military intervention. Specifically, cases are coded “1” if the intervention involved the deployment combat units to assist incumbent government forces. The dataset includes 50 conflicts that involved direct military interventions.

OCCUPY variable: denotes conflicts involving military occupation. Specifically, cases are coded “1” if the intervention involved the deployment combat forces across international boundaries to establish effective control over a territory to which it had previously enjoyed no sovereign title. This includes cases of colonial rebellions or where the intervening power set up a new government after occupation. The dataset includes 30 conflicts that involved military occupations.

STRATEGY variable: used to code counterinsurgent strategy. Specifically, the study uses Nagl’s binary categorization of counterinsurgency strategy [82] [82 Nagl, Counterinsurgency Lessons from Malaya and Vietnam: Learning to Eat Soup with a Knife, 27.] with the STRATEGY variable coded “1” for the “indirect” approach, characterized by counterinsurgent strategies that concentrated on winning support among the population. Cases are coded “0” for the “direct” approach to counterinsurgency, characterized by attempts to achieve victory through the destruction of the insurgency’s armed forces. The coding is based off of the RAND “89 Insurgents” dataset’s evaluation of counterinsurgent competency. The RAND study presents a list of capabilities relevant to conducting an effective indirect counterinsurgency campaign [83 & 84] [83 & 84. Gompert and Gordon, War by Other Means: Building Complete and Balanced Capabilities for Counterinsurgency, xxxiii & 389]; and the variable coding comes from RAND analysts’ evaluation of counterinsurgency competency in 63 insurgency based conflicts. Specifically, the coding represents a subjective analyst evaluation of how well a counterinsurgent or intervening military demonstrated an ability to plan and carry out military operations relevant to a population-centric approach to counterinsurgency.

DEMOCRACY variable: coded to reflect the intervener’s form of government at the time of the intervention. The study codes the intervening state’s regime using Polity2 values from the Polity IV Project dataset. The Polity2 rating is a 21-point scaled composite index of regime type that ranges from highly autocratic (-10) to highly democratic (+10). The DEMOCRACY variable is coded “1” for states with a Polity2 rating 6 or higher. In cases where RAND rated government and intervener with separate competencies, the intervener competency coding was used. Sixty three conflicts are coded for STRATEGY with thirty of these involving third-party interventions for the counterinsurgent.
...
[91] Results for the DEMOCRACY sample are included because the Chi Square probability is very close to the 0.05 Alpha level probability threshold, but they are annotated to show that the results reflect a lower statistical significance (0.076).
The author's conclusion:

Part 5: Conclusion

Does third party military intervention help or hurt an incumbent government during an insurgency?

This study attempted to answer this question by testing prevailing military theories of counterinsurgency in the context of third party intervention using basic tests for statistical significance and bivariate contingency. The results show that intervention on behalf of a counterinsurgent decreases the likelihood of a successful government outcome, and specifically, interventions in general, interventions involving the deployment of combat forces, interventions involving military occupation, and interventions by democratic states decrease the likelihood of counterinsurgent success.

Early intervention, meaning the commitment of third-party support within the first year of conflict, does not appear to have a significant effect on counterinsurgency success. Likewise, the decision to end an intervention early does not appear to significantly alter the chance of counterinsurgent failure.

Interventions in support of an “indirect” approach to counterinsurgencies are the only cases that exhibit a significant improvement for the chances of successful outcome.

In addition, conflicts involving intervention demonstrated longer average duration for losses and shorter durations for successful outcomes. If one accepts conflict duration as a proxy for conflict costliness, then these results indicate that intervention to support a counterinsurgent provides cheaper victories but more costly losses.

The implications for policymakers are twofold. First, the results show that the decision of whether to intervene involves a risky tradeoff. An intervening state may be able to significantly decrease the duration of a successful conflict if it is willing to accept the poorer odds of success. More specifically, intervention provides an opportunity to realize a quicker victory for a besieged government, but the intervening state must be willing to gamble with lower odds of winning and a higher cost for defeat.
All of this should be very "old hat" - e.g., Jack McCuen in the 1960s; but I guess these things have to be periodically "discovered".

Regards

Mike