causality and statistics

In methodology and philosophy, causality has been discussed by experts for a long time. However, in statistics, the avoidance to talk causality became a common trend among early statisticians before 1970s. Since correlation doesn’t equal to causality, the interpretations of causality in statistics have been doubted in academic fields. Because of the wide application of experimental methods in scientific research, especially for the establishment of well-designed randomized experiments, has induced statisticians to seek for the statistical form of causality. Rubin developed the first causal model for analyzing the relationship between cause and effect. He hypothesized that the potential outcomes would be produced through the manipulation of experimental conditions. By comparing the outcomes for the same unit, we will measure the causal effect. The framework of Rubin causal model has been constructed through the introduction of the concept “counterfactual assumption” which means we assume how the world will behave if event X is absent. The concept “counterfactual assumption” reminds me of an article named “culture and cause: American and Chinese attributions for social and physical events” by Kaiping Peng and Morris (1994) . In one study of the article, they applied the counterfactual fact judgment to investigate the causal inferences of Americans and Chinese on the murderer Lu Gang. Nowadays counterfactual assumption has been a powerful tool in the research of causality, but its limitations have been revealed because of Lord’s paradox which will be discussed in next article.

By contrast to studying the cause of give effect, Rubin pay his attention to the causal effect, which is defined by him below:

“intuitively, the causal effect of one treatment, E, over another, C, for a particular unit and an interval of time from t₁ to t₂ is the difference between what would have happened at time t₂ if the unit had been exposed to E initiated at t₁ and what would have happened at t₂ if the unit had been exposed to C initiated at t₁: 'If an hour ago I had taken two aspirins instead of just a glass of water, my headache would now be gone,' or because an hour ago I took two aspirins instead of just a glass of water, my headache is now gone.' Our definition of the causal effect of the E versus C treatment will reflect this intuitive meaning.” (Rubin, Donald B., 1974)

According to the definition, Rubin (1982) applied some symbols in his analysis of causal effects to represent the elements of causal effect. Among them P represents the population of unit; t represents experimental manipulation on different levels; c represents experimental manipulation on level t; S represents the associated indicator of experimental manipulation; G represents a subpopulation indicator variable; X represents a concomitant variable; Y represents an outcome variable. In terms of those symbols, Rubin can discuss the causal effect in more accurate mathematical and statistical language.

In Rubin causal model, several assumptions should be noticed, which are temporal stability, causal transience, unit homogeneity, independence and the constant effect (Holland, 1986) . Temporal stability means the causal effect of treatment doesn’t depend on when we start the treatment, that is to say the response of unit is constant through the time. Causal transience means the causal effect wouldn’t last long to affect the measurement next time. Unit homogeneity plays a important role since it constructs the validity of experiment in laboratory. It assumes causal effect happen in one unit can produce a same effect in other units. Without that assumption, it is impossible for us to convince the outcomes acquired from the laboratories. However, unit homogeneity assumption is never testified effectively while it creates a necessary condition to make experiment reliable. There is a close association between the assumption of independence and randomization. The assumption of independence means the cause we select to be exposed is independent of other variables through the correct randomization. The constant effect means the effect on every unit is same, which makes the additivity of experimental effect possible.

Those assumptions are necessary for the solution of fundamental problem of causal inference, which is “It is impossible to observe the value of \inline $Y_{t}(u)$ and \inline $Y_{c}(u)$ on the same unit and, therefore, it is impossible to observe the effect of t on u” (Holland, 1986) . The possibility of causal inference would be endangered by the fundamental problem. There are two solutions for the fundamental problem of causal inference: scientific solution and statistical solution. The scientific solution is based on the homogeneity or invariance assumption, which makes scientists believe the causal effects happened at different times are same while the assumption is never proved. The statistical solution is based on the concept of average causal effect, which extends the causal effect on a specific unit to the general unit population and can be calculated by statistical method.

As we can see, time plays an important role in the causal model, the equality of the same causal effects happened at different times guarantees the effectiveness of causal effects. Time will eliminate the causal effects produced previously and create conditions for next one. Time will change the units without experimental treatment. So how to separate the interference of time from the causal inference is a serious problem.

From the view of Rubin causal model and its extension, cause can only be revealed from the treatment in experiment, where experiment can be generalized as any manipulation to the event. In article “causal inference and statistics”, Holland believes that the concepts of causality gotten from the operation in experiment and research in observational research are the same. (It makes me produce a question that whether thought experiment is a experiment). He compared three use of the word “cause”: the attributes one has possesses, voluntary activity and activity that is imposed on one. The first can not be the cause in experiment. The author wrote a short but meaningful sentence: every thing has a cause (law of causality), but not every thing can be a cause.

In Rubin causal model the causal inference is discussed within the experimental framework. It is an advantage, however, in my opinion, is also a limit, for it is difficult to interpret the causal relationship in non-experimental situation. I have doubts on whether experiment (even general one) can embody the entire causal relationships we talk. I remember Andrew Gelman said in his article "Foreign aid and military intervention, or Statistical modeling, causal inference, and social science" in his blog, " To me, a laboratory evokes images of test tubes and scientific experiments, whereas for me (and, I think, for most quantitative social scientists), the world is something that we gather data on and learn about rather than directly manipulate."

Reference

Holland (1986), causal inference and statistics, Journal of the American Statistical Association, Vol. 81, No. 396 pp. 945- 960

the article about Rubin causal model in wikipedia

http://en.wikipedia.org/wiki/Rubin_causal_model

Holland, Paul W. Rubin, Donald B.(1982), On Lord's Paradox. Program Statistics Research

Morris M W,Peng K P.(1994) Culture and cause:American and Chinese attributions for social and physical events.Journal of Personality and Social Psychology, 67, 949-971

Rubin, Donald B. (1974) Estimating causal effects of treatments in randomized and non-randomized studies . Journal of Educational Psychology 66:688-701