I was recently involved in a study published in the BMJ which found, using time series analysis, that the rise in use of e-cigarettes in England was associated with a rise in successful attempts to quit smoking. The question which arose was whether we could infer causality.

Causality (also referred to as causation or cause and effect) occurs when a process is responsible for a second process (or state). In other words, something is causal if it gives rise under certain conditions to something else.

Causality can only really be readily inferred from randomised controlled trials (RCTs) i.e where participants are randomly assigned to either the intervention or control group and strict procedures are put in place, including blinding of participants and research staff. Causality is difficult to infer from observational studies due to the inability to control both measured and unmeasured confounds.

There are ways to reduce these biases through statistical methods (e.g. regression adjustment, propensity score matching and multi-level modelling for cluster data) and study design (e.g. prospective data collection). In the BMJ paper we adjusted for population level policies which may have impacted on both the input and output series and removed underlying trends that may have created spurious associations. However, you can never be 100% confident that there is no residual confounding and that other biases haven't been introduced.

The English epidemiologist Sir Austin Bradford in 1965 devised a set of criteria for inferring causation from observational studies (these are now known as the Bradford Hill Criteria):

Is the association reported in the BMJ causal on the basis of these? The answer is perhaps so. The strength of the association was moderate (for every 10% increase in use of e-cigarettes prevalence of successful quit attempts increased by 0.58%), other observational studies and RCTs have found an association between use of e-cigarettes and quitting behaviour, I am not aware of other more plausible explanations for the association and the association appears to be temporal in nature (see discussion below on Granger causality).

Although these criteria are widely used they have also been heavily criticised. To be fair, Hill never intended them to be necessary nor sufficient to infer causation, but they have been taught in that way on both graduate and undergraduate courses.

Clive Granger argued that causality in economics could be tested for by measuring the ability to predict the future values of a time series using prior values of another time series. According to Granger, a series X causes Y if it can be shown that X values provide statistically significant information about future values of Y. The Granger causality test is now commonly applied during time series analysis if one is using ARIMAX modelling (the technique adopted in the BMJ paper). This is because ARIMAX has the assumption of weak exogeneity. That is, that Y can depend on lagged values of X but the reverse must not be true, i.e. X cannot depend on lagged values of Y. In other words, there should be no feedback from the output series. In our paper, this assumption was not violated, we found that prevalence of e-cigarette use was good at predicting prevalence of quit success, but that quit success was not good at predicting prevalence of e-cigarette use.

Does this imply causation? Yes, if we assume that for causality to be present one only needs to prove temporality. Inferring causality from temporality according to Hume (1748-1975) is a natural psychological phenomenon. Numerous studies have shown that people are more inclined to draw causal conclusions if Y occurs after x, and that causal inference and event timing are tightly coupled (see the work of Sloman, Rottman and Lagnado). Many, including Rothman and Greenland (1998), place a great deal of emphasis on temporality in assessing causality, seeing it as a necessity and possibly sufficient. For those who argue that temporality is not enough, the Granger test only gives “predictive” causality.

If we can infer predictive causality from our time series analysis, what does this mean? It means, that we can use the statistical models to make valid predictions about the pattern of time series data in the future. For example, if we were to hypothetically assume that e-cigarette use will increase to 50% among smokers, we could project the likely prevalence of successful quit attempts. It may be possible then to infer causation with greater confidence if our predicted value matches the actual value obtained (assuming the 50% prevalence occurs). However, a mediating variable cannot be ruled out. A rise in e-cigarette use may cause a change in time series m which itself causes a change in successful quitting activity. If this is the case, we don't have direct causality but mediated causality.

Causality (also referred to as causation or cause and effect) occurs when a process is responsible for a second process (or state). In other words, something is causal if it gives rise under certain conditions to something else.

Causality can only really be readily inferred from randomised controlled trials (RCTs) i.e where participants are randomly assigned to either the intervention or control group and strict procedures are put in place, including blinding of participants and research staff. Causality is difficult to infer from observational studies due to the inability to control both measured and unmeasured confounds.

There are ways to reduce these biases through statistical methods (e.g. regression adjustment, propensity score matching and multi-level modelling for cluster data) and study design (e.g. prospective data collection). In the BMJ paper we adjusted for population level policies which may have impacted on both the input and output series and removed underlying trends that may have created spurious associations. However, you can never be 100% confident that there is no residual confounding and that other biases haven't been introduced.

The English epidemiologist Sir Austin Bradford in 1965 devised a set of criteria for inferring causation from observational studies (these are now known as the Bradford Hill Criteria):

- Strength (effect size): the larger the association the more likely it is to be causal
- Consistency (reproducibility): if the observation is observed in different studies this strengthens the likelihood of an effect
- Specificity: the more specific an association, i.e. lack of another explanation, the bigger the probability of causality
- Temporality: the effect has to occur after the cause
- Biological gradient: greater exposure should lead to greater incidence of an effect
- Plausibility: the mechanism between cause and effect is plausible
- Coherence: similar findings in RCTs are found
- Experiment: experimental studies shows that removal of the input variable leads to removal of the output variable (or at least a decrease)
- Analogy: is there similar evidence from analogous situations?

Is the association reported in the BMJ causal on the basis of these? The answer is perhaps so. The strength of the association was moderate (for every 10% increase in use of e-cigarettes prevalence of successful quit attempts increased by 0.58%), other observational studies and RCTs have found an association between use of e-cigarettes and quitting behaviour, I am not aware of other more plausible explanations for the association and the association appears to be temporal in nature (see discussion below on Granger causality).

Although these criteria are widely used they have also been heavily criticised. To be fair, Hill never intended them to be necessary nor sufficient to infer causation, but they have been taught in that way on both graduate and undergraduate courses.

Clive Granger argued that causality in economics could be tested for by measuring the ability to predict the future values of a time series using prior values of another time series. According to Granger, a series X causes Y if it can be shown that X values provide statistically significant information about future values of Y. The Granger causality test is now commonly applied during time series analysis if one is using ARIMAX modelling (the technique adopted in the BMJ paper). This is because ARIMAX has the assumption of weak exogeneity. That is, that Y can depend on lagged values of X but the reverse must not be true, i.e. X cannot depend on lagged values of Y. In other words, there should be no feedback from the output series. In our paper, this assumption was not violated, we found that prevalence of e-cigarette use was good at predicting prevalence of quit success, but that quit success was not good at predicting prevalence of e-cigarette use.

Does this imply causation? Yes, if we assume that for causality to be present one only needs to prove temporality. Inferring causality from temporality according to Hume (1748-1975) is a natural psychological phenomenon. Numerous studies have shown that people are more inclined to draw causal conclusions if Y occurs after x, and that causal inference and event timing are tightly coupled (see the work of Sloman, Rottman and Lagnado). Many, including Rothman and Greenland (1998), place a great deal of emphasis on temporality in assessing causality, seeing it as a necessity and possibly sufficient. For those who argue that temporality is not enough, the Granger test only gives “predictive” causality.

If we can infer predictive causality from our time series analysis, what does this mean? It means, that we can use the statistical models to make valid predictions about the pattern of time series data in the future. For example, if we were to hypothetically assume that e-cigarette use will increase to 50% among smokers, we could project the likely prevalence of successful quit attempts. It may be possible then to infer causation with greater confidence if our predicted value matches the actual value obtained (assuming the 50% prevalence occurs). However, a mediating variable cannot be ruled out. A rise in e-cigarette use may cause a change in time series m which itself causes a change in successful quitting activity. If this is the case, we don't have direct causality but mediated causality.