Table 4 The alternative forms for the OTR (overall treatment ratio) in the case of three interventions (A = augmentation, V = ventouse, C = cesarean section)

Form 1

Treatment ratio for treated women. Letting N₀ denote the number of women who did not receive any treatments, S = (N − N₀)/N is the proportion of women (0 ≤ S ≤ 1) who received some treatment, or treated proportion. The OTR can then be defined as:

\( H=\frac{E}{N}=\frac{N-{N}_0}{N}\bullet \frac{E}{N-{N}_0}=S\bullet {H}_T \)

i.e., as the product of S times the average number of treatments (H_T) among treated women.

Form 2

OTR as the sum of type-specific treatment ratios. A shortcoming of OTR is that it does not distinguish between different types of interventions i.e., the same OTR can be obtained with either a large number of augmentations or a large number of cesareans sections, possibly with different health outcomes. Letting E_A, E_V, E_C (E_A + E_V + E_C = E)   denote the number of interventions of type A,V,C respectively, it holds:

\( H=\frac{E_A+{E}_V+{E}_C}{N} \)= H_A + H_V + H_C

where H_A, H_V, H_C are the type-specific treatment ratios (STR) i.e., the proportions of women treated by augmentation, ventouse, and cesarean section, respectively.

Form 3

OTR as the average of the distribution of the number of treatments. Let N₀, N₁, N₂, N₃ (N₀ + N₁ + N₂ + N₃ = N) denote the number of women who received i = 0,1,2,3 treatments respectively, and f₀ = N₀/N, f₁ = N₁/N, etc the corresponding proportions. Then:

\( H=\frac{E}{N}=\frac{1\bullet {N}_1+2\bullet {N}_2+3\bullet {N}_3}{N} \)= 1 ∙ f₁ + 2 ∙ f₂ + 3 ∙ f₃

In this form H is the average of the statistical distribution of the number of treatments women received.

Form 4

OTR as the average of the distribution of the number of treatments along the different intervention paths in Fig. 1a. Let N(x, y, z) and f(x, y, z) = N(x, y, z)/N, respectively denote the number and the proportion of women who followed path (x,y,z), therefore receiving (x + y + z) treatments. Then H is simply the average of the number of treatments along each path weighted by the proportion of women who followed that path:

\( H=\sum \limits_{x=0}^1\sum \limits_{y=0}^1\sum \limits_{z=0}^1\left(x+y+z\right)\bullet f\left(x,y,z\right) \)

By aggregating paths with the same number of treatments Form 4 collapses into Form 3.

Form 5

OTR as a function of the progression proportions. Under the assumptions of Fig. 1b, the STRs can be expressed in terms of the progression proportions (PP), which specify the proportions of women progressing to a further treatment from current treatment. For example the STR for ventouse H_V  is given by the sum of the PP p_OV of women who entered labour (O) and progressed directly to ventouse (V) (the direct path OV in Fig. 1b), therefore receiving exactly one treatment, plus the proportion of women who experienced augmentation before ventouse (receiving exactly two treatments), which can be factored as the product p_OAp_AV of the PP to augmentation p_OA times the PP p_AV from augmentation to ventouse (the two-steps path from O to A and from A to V). By this reasoning we can write STRs H_A, H_V, H_C as:

H_A = p_OA

H_V = p_OV + p_OAp_AV

H_C = p_OC + p_OAp_AC + p_OVp_VC + p_OAp_AVp_VC

Recalling form 2, the OTR can then be represented as:

H = p_OA + p_OV + p_OAp_AV + p_OC + p_OAp_AC + p_OVp_VC + p_OAp_AVp_VC