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⁹⁰ Reiter and Stam, Democracies at War. We focus on politically relevant triads to avoid inflating our sample by including third parties for which there is only a small probability of joining. Politically relevant triads account for about 40 per cent of all triads. In the web appendix we report the results from the full sample of potential joiners. Our findings are similar regardless of whether our sample includes all triads or only politically relevant triads.

⁹¹ If there were multiple initial belligerents (that is, more than one initiator and/or more than one target), then we created a set of initiator-target dyads. For each initiator-target dyad we included all potential joiners that met the two conditions that define a politically relevant triad.

⁹² We lose few joining cases by limiting the set of all potential joiners to those that are politically relevant. We lose fifty-four cases of joining: twenty-four on the initiator and thirty on the target. This suggests that contiguous states and/or major powers account for most of the joining.

⁹³ In addition, when using the year as the temporal unit, care has to be taken to ensure that independent variables, such as alliances, are actually operative during the MID. For example, when the temporal unit is the year, it is necessary to ensure that an alliance between a third party and an initial belligerent was in effect during the MID and did not end before the MID began or start after the MID ended.

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¹⁰⁶ In all of these operationalizations, if a MID lasts past December 31 of a given year, then a state's CINC score can change, which means that a potential joiner's capability contribution to an initial belligerent can change as well.

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¹¹¹ Correlates of War Project State System Membership List.

¹¹² We also estimated these models on three additional samples: (1) all triads, (2) all triads without World Wars I and II and 3) politically relevant triads without World Wars I and II. We note the differences in the results for our risk specific models in the politically relevant triads sample and the three other samples.

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¹¹⁴ Formally, the percentage change in the hazard ratio is: $$$ \ % \rDelta {{{\rm{h}}}_{\rm{i}}}{\rm{(t)}} = ({{e}^{({{{\rm{X}}}_{\rm{i}}}{\rm{ - }}{{{\rm{X}}}_{\rm{j}}}{\rm{)(}}{{\beta }_{\rm{1}}} + {{\beta }_{\rm{2}}}{\rm{ln(t))}}}} {\rm{ - }}1)\times 100 $$$. See Licht (2011). The summary statistics reveal that the capability variable is skewed, with most values clustering near zero. As a result, a percentage change in the hazard ratio that is calculated, for example, using the mean value and one standard deviation above the mean value, is not meaningful because the percentage of third parties that have that value of capability is quite small. For example, in the Join Either Initiator or Target model, one standard deviation above the mean is 0.28, and only 12 per cent of potential joiners have a value above that number. Additionally, one standard deviation below the mean is −0.1, which is nonsensical since a third party's capability contribution is always greater than zero. Thus in order to calculate the substantive effect of capability on the risk of joining we used the first, second and third quartiles. In the Join Either Initiator or Target model, the value that cuts off the first quartile is 0.001, the value that cuts off the second (or median) quartile is 0.007 and the value that cuts off the third quartile is 0.05. Therefore we calculate the percentage change in the hazard ratio if a third party's capability increases from the first to the second quartile (an increase of 0.006) and from the second to the third quartile (an increase of 0.043). In the Join Initiator model, the value that cuts off the first quartile is 0.0001, the value that cuts off the second quartile is 0.0015 and the value that cuts off the third quartile is 0.016. An increase in a third party's capability from the first to the second quartile is 0.0014 and from the second to the third quartile is 0.015. In the Join Target model, the value that cuts off the first quartile is 0.0002, the value that cuts off the second quartile is 0.003 and the value that cuts off the third quartile is 0.03. An increase in a third party's capability from the first to the second quartile is 0.003 and from the second to the third quartile is 0.027.

¹¹⁵ In the Join Initiator model, the initial effect of capability is positive in all four samples but is only statistically significant in the politically relevant triads sample with World Wars I and II. The effect of capability does not significantly decrease over time in any of the samples except the all triads sample without the world wars, in which the effect increases over time but not significantly. In the Join Target model, the effect of capability is initially positive and statistically significant, and significantly decreases over time in all four samples.

¹¹⁶ Altfeld and Bueno de Mesquita, ‘Choosing Sides in Wars’; Bremer, ‘National Capabilities and War Proneness’; Huth, Extended Deterrence.

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¹¹⁹ In both the Join Initiator and Join Target models, the initial effect of geographic proximity is positive and statistically significant, and significantly decreases over time in all four samples except the politically relevant triads samples without the world wars. In the politically relevant triads sample without the world wars, the effect of geographic proximity is positive and statistically significant but does not change over time in either the Join Initiator or the Join Target model.

¹²⁰ The formula for the percentage change in the hazard ratio is the same as for continuous variables (see fn. 114), where X_i = 1 and X_j = 0.

¹²¹ Siverson and Starr, The Diffusion of War; see also Melin and Koch, ‘Jumping into the Fray’.

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¹²³ Raknerud and Hegre, ‘The Hazard of War’; Werner and Lemke, ‘Opposites Do Not Attract’.

¹²⁴ Mousseau, Peace in Anarchy; Reiter and Stam, Democracies at War; Lake, ‘Powerful Pacifists’; Choi, ‘Democratic Synergy’.

¹²⁵ In the Join Initiator model, the initial effect of autocracy is positive and statistically significant, and significantly decreases over time in all four samples. In the Join Target model, the initial effect of autocracy is negative and not statistically significant in both samples with the world wars, but positive in both samples without the world wars. However, the effect of autocracy is not statistically significant in any of the samples. The initial effect of autocracy increases over time but is not significant in any of the samples except the politically relevant triads sample without the world wars, in which the effect is negative but not significant.

¹²⁶ Raknerud and Hegre, ‘The Hazard of War’; Corbetta, ‘Determinants of Third Parties’ Intervention and Alignment Choices’; Werner and Lemke, ‘Opposites Do Not Attract’.

¹²⁷ Leeds et al., ‘Alliance Treaty Obligations and Provisions’, p. 240.

¹²⁸ Siverson and Starr, The Diffusion of War; Melin and Koch, ‘Jumping into the Fray’.

¹²⁹ Leeds et al., ‘Alliance Treaty Obligations and Provisions’.

¹³⁰ Corbetta, ‘Determinants of Third Parties’ Intervention and Alignment Choices’; Leeds, ‘Alliance Reliability in Times of War’; Leeds, ‘Do Alliances Deter Aggression?’; Smith, ‘To Intervene or Not to Intervene’.

¹³¹ Siverson and Starr, The Diffusion of War.

¹³² Diehl and Goertz, War and Peace in International Rivalry.

¹³³ Haldi, Why Wars Widen. In the Join Initiator model, the initial effect of rivalry is positive and statistically significant in all four samples. However, the effect of rivalry significantly decreases over time in all of the samples except the politically relevant triads sample that includes World Wars I and II. Since rivalry did not violate the proportional hazards assumption in that sample, we did not include an interaction with the natural log of time. In the Join Target model, the effect of rivalry is positive and statistically significant in both politically relevant triads samples, but is positive and not statistically significant in both ‘all triads’ samples. In addition, the effect of rivalry changes inconsistently over time across the four samples; it is positive in the all triads sample, positive and significant in the all triads sample without the world wars, was not included in the politically relevant triads sample without the world wars (because it did not violate the proportional hazards assumption), and is negative and not significant in the politically relevant triads sample without the world wars.

¹³⁴ In the Join Initiator model, the initial effect of previous major power joining is positive but not statistically significant in any of the samples except the politically relevant triads sample without World Wars I and II, in which the effect is negative but not statistically significant. The effect of previous major power joining increases over time in all of the samples, but is not statistically significant. In the Join Target model, the initial effect of previous major power joining is positive and statistically significant and significantly decreases over time in all four samples.

¹³⁵ Yamamoto and Bremer, ‘Wider Wars and Restless Nights’.

¹³⁶ Corbetta, ‘Determinants of Third Parties’ Intervention and Alignment Choices’; Corbetta and Dixon, ‘Danger Beyond Dyads’.