GENERAL ARTICLE Measuring Causality∗ The Science of Cause and Effect Aditi Kathpalia and Nithin Nagaraj Determining and measuring cause-effect relationships is fun- damental to most scientific studies of natural phenomena. The notion of causation is distinctly different from correlation which only looks at the association of trends or patterns in measure- ments. In this article, we review different notions of causality and focus especially on measuring causality from time-series data. Causality testing finds numerous applications in diverse Aditi Kathpalia is currently a PhD Scholar, Consciousness disciplines such as neuroscience, econometrics, climatology, Studies Programme, National physics, and artificial intelligence. Institute of Advanced Studies, IISc Campus, Bengaluru. Her research interests include causality testing and its 1. Introduction applications, chaos and information theory. Most studies in natural as well as social sciences are centred around the theme of determining cause-effect relationships be- tween processes or events. Such studies have been conducted since the early 20th century. While some studies are observa- Nithin Nagaraj is currently tional, others involve experiments to understand the nature of de- Associate Professor, pendencies. Examples of observational studies involve, studying Consciousness Studies the particle size and fertility of the soil, availability of water, dis- Programme, National Institute of Advanced Studies, eases or pests in a particular place in order to study their effect IISc Campus, Bengaluru. His on crop yield, or observing the death rates of smoking vs non- research areas include smoking people to determine its influence on mortality. On the complexity theories of other hand, an example of experimental study would be studying consciousness, chaos, information theory and a diseased group of people who are being administered medica- causality testing. tion to check its efficacy against a control group of people being administered a similar dose of placebo drug. ∗Vol.26, No.2, DOI: https://doi.org/10.1007/s12045-021-1119-y RESONANCE | February 2021 191 GENERAL ARTICLE Three Types of Statistical Causality Keywords Causality, correlation, ladder of Cox and Wermuth have given three notions (levels) of statisti- causation, Granger causality, model-based causality. cal causality based on existing approaches for estimating causal- ity [1]. The zero-level view of causality is basically a statisti- cal association, i.e. non-independence with the cause happen- The zero-level view of ing before the effect. This association cannot be done away causality is basically a with by conditioning on other features or variables of the system statistical association, that could be potential causes for the perceived effect. For exam- i.e. non-independence with the cause happening ple, when looking at the causal influence that greenhouse gases before the effect. in the atmosphere have on increasing the temperature of Earth’s surface, other features such as solar output which are also poten- tial causes of the effect in question need to be conditioned. Only then can greenhouse gases be said to have an effect on Earth’s temperature. In mathematical terms, it is a dependence based on a multiple-regression like analysis that cannot be explained by other appropriate explanatory variables. This type was studied by Good [2, 3] and Suppes [4]. In a time-series context, it was for- malized as Wiener–Granger causality by Granger [5] and later, formulated in a more general context by Schweder [6] and Aalen [7]. In the first-level view of In the first-level view of causality, the aim is to compare the out- causality, the aim is to compare the outcomes comes arising under different interventions, given two or more arising under different (possible) interventions in a system. For example, take the case interventions, given two of two medical interventions, D1 and D0—a treatment drug and or more (possible) control respectively, only one of which can be given to a particu- interventions in a system. lar patient. The outcome observed with D1 use is compared with the outcome that would have been observed on that patient had D0 been used, other things being equal. If there is evidence that use of D1 instead of D0 causes a change in outcome, then it can be said that D1 causes that change. The key principles of such kind of experimental design for randomized control trials were developed mainly at Rothamsted [8, 9, 10, 11]. This way of infer- ring causation may have an objective of decision-making or may require conducting a controlled experiment, although that is not always the case. For example, when trying to check if an anoma- lous gene is the cause of a particular disease, the intervention as 192 RESONANCE | February 2021 GENERAL ARTICLE between the abnormal and normal version of the gene is hypothet- ical (since explicit intervention is not possible) and also no imme- diate decision-making process is generally involved. Rubin [12] adapted the notions of causality to observational studies using a representation similar to Fisher’s. The definition of causality in the above discussed first-level view is explicitly comparative and has been the most widely used in scientific studies. Suppose that preliminary analysis in a scientific context has es- tablished a pattern of associations/ dependencies or have pro- vided a good amount of evidence of first- or zero-level causal- ity. Second-level causality is used for explaining how these Second-level causality is dependencies arose or what underlying generating process was used for explaining what involved for the causal relationships observed. On several occa- underlying generating process was involved for sions, this will require incorporating information from previous the causal relationships studies in the field or by doing laboratory experiments. Attempts observed. in this regard started with graphical representations of causal path diagrams by Sewall Wright [13, 14] and were later promoted by Cochran [15]. Currently, non-parametric structural equa- tion models (NPSEMs) [16] which provide a very general data- generating mechanism suitable for encoding causation, dominate the field. Each of the above types for determining causality has its pros and cons and their use depends on the motive and the nature of the study. While first-level causal estimation, that mostly in- volves randomization experiments, may make the conclusions of the study more secure, it fails to reveal the biological, psycholog- ical, or physical processes working behind the effect observed. On the other hand, zero-level causality suffers from the criticism that there is no intervention involved to observe the causal effect of doing something on the system. The second-level of causality requires field knowledge and cannot be solely data-driven. While it is useful to know all these notions of causality, for the rest of this article, we will mostly deal with causality as estimated from collected time-series measurements where it is not possible to intervene on the experimental setup. RESONANCE | February 2021 193 GENERAL ARTICLE 2. Correlation and Causation We have often heard the saying ‘correlation does not imply cau- sation’. But even to this date, there are several scientific stud- ies which make erroneous conclusions regarding a variable being a cause of another, merely based on observed correlation value. Thus it becomes necessary to clarify the meaning and usage of these two terms. Correlation is a statistical technique which tells how strongly are Correlation is a a pair of variables linearly related and change together. It does statistical technique not tell us the ‘why’ and ‘how’ behind the relationship, but it which tells how strongly just tells that a mathematical relationship exists. For example, are a pair of variables linearly related and Pearson’s correlation coefficient for a pair of random variables change together. (X, Y) is given as: E[(X − µX)(Y − µY )] ρX,Y = , (1) σXσY where, the numerator is the covariance of variables X, Y and σX, σY are the standard deviations of X and Y respectively. E is the expectation and µX,µY are the means of X and Y respectively. Note that: −1 ≤ ρX,Y ≤ +1 and is always symmetric ρX,Y = ρY,X. The closer the magnitude is to 1, the stronger is the relationship between the variables. Figure 1 illustrates two signals with pos- itive, negative, and zero correlation. An example of a positive correlation would be between the temperature in a region, and the sale of coolers—as temperature increases (decreases), sale of coolers also increases (decreases). However, as temperature in- creases (decreases), the sale of heaters decreases (increases), in- A variable X can be said dicating a negative correlation. An example of zero correlation to bea cause of another would be between the amount of tea consumed by an individual variable Y “if it makes a and his/her level of intelligence. difference to Y, and the difference X makes is a In contrast, causation indicates that one event is a result of the difference from what occurrence of another event. A variable X can be said to be a would have happened without it.” cause of another variable Y, “if it makes a difference to Y and the difference X makes must be a difference from what would have happened without it”. This definition is adapted from the 194 RESONANCE | February 2021 GENERAL ARTICLE Figure 1. Positive, nega- tive and zero correlation. definition of a ‘cause’ given by philosopher David Lewis [17]. As discussed in the previous section, there are several means of estimating causality. Unlike correlation, causation is asymmetric. Interestingly, for conventional statistics, causation was a non-scientific concept, and as per the ideas prevalent in the late 19th and early 20th century, all analysis could be reduced to correlation. Since correlation got rigorously mathematically defined first (when sci- entist Galton was in search of a tool for causation) and causation seemed to be only a limited category of correlation, the latter be- came the central tool.