Summary

• Cost-effectiveness analyses (CEAs) of interventions accelerating animal welfare reforms usually estimate the increase in the welfare of the target animals (for example, hens in cages) based on the acceleration in years of the full implementation of the reform. This makes sense if each level of implementation of the reform is accelerated as much as its full implementation.
• However, there are many cases where the acceleration of the full implementation of the reform is not enough to determine the number of animals helped, or animal-years improved. I discuss some below.

Context

CEAs of interventions accelerating animal welfare reforms usually estimate the increase in the welfare of the target animals (for example, hens in cages) from one of the following:

• “Increase in welfare per animal helped”*“animals helped” = “increase in welfare per animal helped”*“initial number of animals raised per year in the baseline conditions”*“acceleration in years of the full implementation of the reform”.
• “Increase in welfare per animal-year improved”*“animal-years improved” = “increase in welfare per animal-year improved”*“initial population of animals (alive at any time) in the baseline conditions”*“acceleration in years of the full implementation of the reform”.

Using these formulas makes sense if each level of implementation of the reform is accelerated as much as its full implementation. I illustrate this case below. The full and dashed lines represent the (expected) population of animals in the baseline conditions without and with the intervention. Without the intervention, there would be no animals in the baseline conditions after $T$ years. With the intervention, there is an acceleration of not only that, but of the whole trajectory of the population by $\Delta t$ years. The animal-years improved are given by the area between the lines, which is “acceleration in years of the full implementation of the reform ($\Delta t$)”*“initial population of animals in the baseline conditions ($N$)”, as in the 2nd formula above.

However, there are many cases where the acceleration of the full implementation of the reform is not enough to determine the number of animals helped, or animal-years improved. I discuss some below.

Accelerating animal welfare reforms

I illustrate below a case where the intervention accelerates the full implementation by $\Delta t$ years (as above), but does not change when the population of animals in the baseline conditions starts to decrease. The animals-years improved are $0.5\ast \Delta t\ast N$, half of those improved in the case above.

Interventions often accelerate both the start of the implementation and its full achievement. I illustrate this case below. The start is accelerated by $\Delta t_1$, and the full achievement by $\Delta t_2$. The animal-years improved are $0.5\ast (\Delta t_1+\Delta t_2)\ast N$.

Generalising, for cases like the ones above, the animal-years improved are “mean acceleration over the population of animals in the baseline conditions”*N. Here is an example which I illustrated below. If the start of the implementation is accelerated by $\Delta t_1$, the completion of half of the implementation by $\Delta t_2$, and its full achievement by $\Delta t_3$, the animal-years improved are $(0.25\ast \Delta t_1+0.5\ast \Delta t_2+0.25\ast \Delta t_3)\ast N$. $\Delta t_2$ is weighted 2 times as heavily as each $\Delta t_1$ and $\Delta t_2$ because it affects the population both before and after that of half implementation, whereas $\Delta t_1$ only affects the population after the start of implementation, and $\Delta t_2$ only affects the population before full implementation.

The implementation above has 2 stages. The population of the animals in the baseline conditions with the intervention (dashed line) decreases at a given pace until half of the implementation, and then accelerates. For an implementation with k stages where the start of the 1st stage is accelerated by $\Delta t_0$, and the end of the last stage by $\Delta t_k$, the animal-years improved are $(\Delta t_0+2\ast \Delta t_1+2\ast \Delta t_2+\dots +2\ast \Delta t_{k-1}+\Delta t_k)/(2\ast k)\ast N$. This formula works even if the population of the animals in the baseline conditions without the intervention (full line) decreases at different paces in each stage. In practice, I think considering 1 or 2 stages (2 or 3 reference accelerations) is often the best trade-off between accuracy and complexity.

Flattening the growth of the population of animals in the baseline conditions

An intervention can be impactful even if the population of animals in the baseline conditions increases over the period affected by the intervention. I illustrate below a case where an intervention flattens the growth of the population of animals in the baseline conditions. It decreases their peak population by $\Delta N$, which is reached $\Delta t_1$ years after the intervention starts to have an effect, and it stops having an effect $\Delta t_2$ years after the peak. The animal-years improved are $0.5\ast (\Delta t_1+\Delta t_2)\ast \Delta N$.

An intervention can also accelerate the full implementation of the welfare reform after flattening the growth of the population of animals in the baseline conditions. I illustrate this case below. As above, the intervention decreases the peak population by $\Delta N$, which is reached $\Delta t_1$ years after the intervention starts to have an effect. In addition, it accelerates the full implementation of the reform by $\Delta t_2$ years, such that it happens $T-\Delta t_2$ years after the intervention starts to have an effect. The peak population would be N_p without the intervention. The animal-years improved are $0.5\ast T\ast \Delta N+0.5\ast \Delta t_2\ast (N_p-\Delta N)$. I obtained this from adding the 2 expressions below, but then realised it can be directly inferred from the drawing. The 1st term is the area of 2 triangles defined by the points ($0$, “initial population”), ($\Delta t_1$, $N_p$), ($T$, $0$), and ($\Delta t_1$, $N_p-\Delta N$). The 2nd term is the area of a triangle defined by the points ($\Delta t_1$, $N_p-\Delta N$), ($T$, $0$), and ($T-\Delta t_2$, $0$).