## Causation 1b: Measuring causal effects

In the previous post, we defined a causal effect and learned the difference between individual vs average causal effects. Let’s refresh the concepts. An average causal effect is present if the average potential outcome of a group of individuals under one action is not equal to the average potential outcome under another action.

For example, Table 1 in the previous example on average causal effects showed the potential outcomes (if they were sick or not sick) of a group of 20 people depending on their actions (if they ate or did not eat leftovers). The Table showed:

• if all 20 people ate leftovers, 50% of people got sick.
• if all 20 people did not eat leftovers, 50% of people still got sick.

Since the average potential outcome of the 20 people under one action (ate leftovers) is equal to the average potential outcome under another action (did not eat leftovers), in that example, there was no average causal effect of eating leftovers on health outcomes.

On the other hand,

• if all 20 people ate leftovers and 70% of people got sick.
• if all 20 people did not eat leftovers, 50% of people still got sick.

Now, the average potential outcome of the 20 people under one action (ate leftovers) is not equal to the average potential outcome under another action (did not eat leftovers). So there is an average causal effect of eating leftovers on health outcomes in this example.

In general terms, regardless of binary or continuous outcomes, we say that the expected potential outcome Y under action a=1 is not equal to the expected potential outcome Y under action a=0: $E[Y^{a=1}] \neq E[Y^{a=0}]$.

What are different ways to measure causal effects? How might random variability affect these measures?

### Measures of causal effects

#### Binary outcomes

Risk difference. This is the difference between the probability of outcome under action 1, and the probability of outcome under action 0. This is a causal effect measure on an additive scale, and is used to show e.g. the absolute number of cases of disease that are changed by treatment.

If there is no causal effect, $Prob[Y^{a=1} = 1] - Prob[Y^{a=0} = 1] = 0$

Risk ratio. This is the ratio between the probability of outcome under action 1, and the probability of outcome under action 0. This is a causal effect measure on a multiplicative scale, and is used to show e.g. how many more times that treatment, relative to no treatment, changes disease risk.

If there is no causal effect, $\frac {Prob[Y^{a=1} = 1]} {Prob[Y^{a=0} = 1]} = 1$

Odds ratio. This is the ratio between the odds of the outcome under action 1, and the odds of the outcome under action 0. This is a causal effect measure on a multiplicative scale, using the relationship between odds and probabilities. Odds and odds ratios are attractive because of their mathematical properties.

If there is no causal effect, $\frac {Odds[Y^{a=1} = 1]} {Odds[Y^{a=0} = 1]} = \frac {\frac{Prob[Y^{a=1} = 1]}{1 - Prob[Y^{a=1} = 1]}} {\frac{Prob[Y^{a=0} = 1]}{1 - Prob[Y^{a=0} = 1]}} = 1$

#### Continuous outcomes

Mean difference. This is the difference between the mean of outcomes under action 1, and the mean of outcomes under action 0. This is a causal effect measure on an additive scale.

If there is no causal effect, $Mean[Y^{a=1} = 1] - Mean[Y^{a=0} = 1] = 0$

Overall, choosing to use a measure on the additive or multiplicative scale depends on whether the aim is to show absolute (additive) or relative (multiplicative) effects.

### How does random variability influence measures of causal effects? A theoretical overview

Ultimately, the aim is to determine the average causal effect in a population of individuals. However, it is often only feasible to measure an average causal effect in a sample of the population. Since one sample can differ from another, sampling variability is a source of random variability in measures of causal effects.

Another source of random variability arises if counterfactual or potential outcomes are not necessarily fixed/determined in advance. For instance, in our previous example on individual causal effects, Peter would have gotten sick from eating leftovers, or would have remained well if he didn’t. Here, Peter has 100% chance of getting sick from eating leftovers, and 0% change of getting sick if he didn’t. That is, the outcome (getting sick) is fixed or fully determined by the exposure/action (eating leftovers).

But suppose Peter has 90% change of getting sick from eating leftovers and 10% chance of getting sick if he didn’t. Now, the counterfactual outcomes (getting sick or remaining well) are probabilistic/non-determined, since the probability of the outcome under the exposure is neither 0 nor 1. In other words, it is not possible to observe counterfactual outcomes with certainty. (This is the basis of probabilistic/stochastic Bayesian statistics.) These probabilities of outcomes vary across individuals, since different people are susceptible or resilient to exposures in different ways.

Therefore, random variability in average causal effects arises from at least two sources: sampling variability, non-deterministic counterfactuals, or both. For simplicity, we will assume that counterfactual outcomes are deterministic/fully-determined, and were obtained using data from large samples that approximate the population consistently.

### Summary

Causal effects can be measured for binary and continuous outcomes. Choosing a measure on either scale depends on whether the analysis aims to show absolute or relative differences.

Sampling variability, non-determined counterfactuals, or both contribute to variability in causal effects. It is helpful to keep this in mind, even though we assume non-determined counterfactuals and sufficiently large sample sizes give consistent estimators.

### Reference

Hernán MA, Robins JM (2020). Causal Inference: What If. Chp 1.3-1.4. Boca Raton: Chapman & Hall/CRC.