漏掉了的一篇笔记

今天在清点读书笔记时，有一篇记着的总是找不到，结果发现在转移阵地时不慎遗弃了。于是赶紧把它拯救回来。虽然说只是对一篇文章粗浅的读书笔记，引文方面还不是很规范，但终究还是自己在读书完的的一点劳动。所以，人民（特指我）不会忘记你！

今天随便翻阅了《the sage handbook of quantitative methods in the social sciences》中Mulaik写的"Objectivity in Science and Structural Equation Model" 一章，作者从康德关于先验范畴的讨论爬到对科学和结构方程客观性的意义，使我印象比较深的是作者把客观性看作是将非经验的概念有效化的方式，换句话说就是内部一致性共识，既然客观性概念不再是客观的，那么因果关系的客观性也更加值得怀疑。读书笔记中阅读的文章就是在因果关系的主观和客观的解释及理论中摇摆。对我有感悟的是activity theory of causality， 它先是把因果看成三个层面的概念，一是自由意志观，相信我们的意识能自由主导我们的行为，诸如历史上的英雄决定成败之类的；二是物理因果观，在初始条件设定的情况下以一种机械的固定不变的方式运作，像机械钟的运动；心理层面的因果和物理层面的因果由于心身二元论的对立而无法融合在一起，因此人们又发明了第三种因果——逻辑因果，诸如充分、必要条件之类的。虽然逻辑因果可以用逻辑和数学的方式进行精确、严密的处理，但逻辑和数学被Mulaik指出终究是经验处理规范的隐喻。逻辑因果把物理事实用隐喻的方式抽象出来在心灵层面操作，这依旧无法解决两者的对立。可以说因果的客观性是无法被有效证实的，而activity theory of causality 更强调它对现实生活的指导意义，也就是说，提出的因果的关系不能仅仅停留在逻辑上的严密性和完备性，它必须是现实世界中可以直接操作，并对我们世界和社会的改善有着积极的影响。我想，因果是否仅仅是思维的经济法则，通过最小成本（事件在时空上的最短距离的组合）获得最大收益（直接预测和控制对我们生活有最大影响的事件的发生）。

也许我有机会要结合凯恩斯的经历思考一下经济学的本质了：）


Causal Inference from Philosophical and Methodological Perspective
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As it is often said, every why has a wherefore. It is natural for us to assume every event has a cause. However, understanding the causal relationship in scientific research is a troublesome one where different schools explain the causal relationship according their own perspectives. One of the earliest persons talking about causal inference is Hume who argued that people see some cases as causal events according to three points. The first point is the contiguity between the presumed cause and effect, the second temporal precedence between them and the third their constant conjunction. An instance taken by Hume is many glass balls colliding with each other. He argued that it was difficult for us to distinguish the cause and effect in such complex situation where the only principle we may rely was the closeness of events in space and time. However, are causalities people believe just the coincidences in space and time? Hume reached his conclusion in a sarcastic tongue: correlation means causal relationship. In other words, the psychological illustrations induce people believe the causal relationships exist in the world. Although there are many inconsistencies in fundamental points with Hume, the positivist also denied the importance of objective causal relationship in theory. Russel, a representative of positivist, argued that since the mathematical functions have sufficiently explain the relationships of variables in nature, is it necessary to remain a causal interpretation as a attractive but not useful method. Besides, Russel pointed out: contrast to the symmetrical properties of mathematical and physical laws, causality is unidirectional. It is strange from the view of mathematicians While Russel’s points seem too extreme, he successfully makes us focus on a fact that role played by causal inference in natural science is less important than in social science. The causal inference may be a delicate alternative as a more powerful analyzing tool.

Contrast with positivist’s prejudice to causality, essentialist believe the existence of underlying causes which hide behind the observational phenomenon. They attempt to seek the micromediational mechanisms which will offer the ultimate interpretation. Their perspectives are associated with reductionism. As Campbell and Stanley argued that experiment can probe but not prove the causal hypothesis, positivist’s operational definitions and strict manipulations don’t reflect the true meaning of cause, a deeper exploration is needed.

What’s the true meaning of causality? Different persons provide their own answer. Mill emphasized the temporal order and relevance of cause and effect as the indicator to the causality. In order to exclude the alternative possibilities, the joint method of agreement and difference is designed by Mill, who convincingly proved its effectiveness. Popper insisted falsification is a key method to test the available causal hypothesis.

While Mill’s methods are applied widely in the experimental design to explore the causes, the activity theory of causation raises a question: how can we test the causal relationship through observation. There are three levels of senses of causes according to the interpretation of activity theory of causation. Firstly, it is caused by “the free and deliberate act of a conscious and responsible agent”, where the free selections of individuals dominate the causal effect. The explanations of causes for the historical figures’ behaviors conform to that perspective. The second may call the mechanical causality such as mechanical clock’s work. The processions of events are fixed by the natural principles. The previous one naturally produces the next one. The third is the logical causation which fits the essentialists’ perspectives. It considers the sufficient and necessary condition of causation. Differed from the essentialists and positivists’ points, causal mechanisms are not seen so important in activity theory. Without understanding the concrete ways that causes influence effects, people are still able to predict and control effect in virtue of causation. According to the activity theory, the causal interpretation makes sense only when the causal factors can be manipulated reliably. A cause unable to help us improve the world is no useful in theory.

The evolutionary critical-Realist perspective believes while causal relationships are objective and existing beyond human conscious, our imperfect organ and intellectual limits us to perceive the accurate causal relationship. In another words, our causal perceptions are the product of million years of evolution. Survival values and logical meaning, which is ultimate goal of causality? Evolutionary perspective emphasizes the former. They try to look for the origin of the conception of causality from the biological evolution which make the causality appears not so

Until now, experiment is still a dominating method to confirm the causality. According to the logic of experiment, causal effects will be appeared through the manipulation of independent variables and the setting of the alternatives in the laboratory. However, the strictness of logic limits the causal exploration to the simple and effective part compared with the attempt to build the big and whole causal chains, which will reduce the real attractiveness of causation as the our key to understand the world.

Below is the eight assumptions about causal chains from the Cook and Campbell's article (Cook, T.D., & Campbell, D.T. 1979) . I hope they will be useful for someone:

1.Causal assertion are meaningful at the molar level even when the ultimate micromediation is not known

2.Molar causal laws, because they are contingent on many other conditions and causal laws, are fallible and hence probabilistic.

3.The effects in molar causal laws can be the result of multiple causes.

4.While it is easiest for molar causal laws to be detected in closed systems with controlled condition，field research involve mostly open systems.

5.Dependentable intermediate mediational units are involved in most strong molar laws.

6.Effect follow causes in time, even though they may be instantaneous at the level of ultimate micro mediation.

7.Some causal laws can be reversed, with cause and effect interchangeable

8.The paradigmatic assertion in causal relationships is that manipulation of a cause will result in the manipulation of an effect.


reference:

Cook, T.D., & Campbell, D.T. (1979). Causal inference and the language of experimentation. Quasi-Experimentation: Design and Analysis Issues for Field Settings. pg. 1-36. Boston: Houghton Mifflin Co.

causal faithfulness and d-separation

SEM and Neyman-Rubin potential-outcome model are considered as two major statistical tools for explaining the causal relationships of two variables. However, as Pearl（1997） says,SEM is used by many people and understood by few, while potential-outcome model is understood by few and used by fewer. People apply SEMs to solve the problems in their academic field without guaranteeing their power of causal interpretation. Therefore, as Pearl points out, “ the current dominating philosophy treats SEM as just a convenient way to encode density functions (in economics) or covariance information (in social science). ” So John Fox introduces a cynical view of SEMs in the appendix on structural equation model of his book（http://socserv.mcmaster.ca/jfox/Books/Companion/appendix-sems.pdf） that the popularity of SEMs may be attributed to their pretentious ability of causal interpretation of observational data although they’re not less problematic than other regression models, or they just translate the informal thinking of causal relationships into a formal data analysis.

However, Pearl（2000） believes there is causal interpretations for observational data from SEMs. He argues that almost all the academic educations and publications overlook the presuming condition raised by the fathers of SEM, that is in a equation like y=βx + ε, if we make sure that it is structural, β should be the unique causal connection between x and y, the statistical relationships of x and εcan not be changed because of the different interpretation of β. That is the condition called self-containment. In the language of graph favored by Pearl, it is also named d-separation.

d-separation is clarified by Pearl in his blog article “d-Separation Without Tears” （http://www.mii.ucla.edu/causality/?m=200001） as a criteria to determine whether a set X of variables is independent of a set Y of variables given a third set Z. If X and Y satisfy the conditions of d-separation, we can say X and Y is d-separated or disconnected. If we use the language of graph, x and y is d-connected if there is an unblocked path between them. In the graph \inline $x\rightarrow m\rightarrow o\rightarrow n\rightarrow y$ , x and y is d-connected conditioned on the set of Z, if X and Y have no collider-tree path that traverses the member of Z, we can say X and Y is d-separated by Z. if r and v are the members of Z like graph \inline $x\rightarrow m'\rightarrow o\rightarrow n'\rightarrow y$ , X and Y is d-separated by Z. however, if a collider is the member of conditioning set Z, like m’ and v’ in the paragraph below, s and y is d-connected because t has p’ as the member of z.

（quoted from Pearl’s blog article “d-Separation Without Tears” http://www.mii.ucla.edu/causality/?m=200001)

In another words, x and y is d-separated by z means there is the null partial correlation between x and y by z.

D-separation is the a key assumption for the causal faithfulness, which promises every conditional independence of variable in the causal relationship so that the causal structure can be testable. Therefore, d-separation is used to make the causal interpretation of SEMs more reliable. If we test the causality of structural equation model, we should carefully test the self-containment of equation by using d-separation to compute the conditional independence relations. Now I just primarily understand the conception of d-separation. A lot of efforts are still needed for me to study the statistical reading of causality by applying d-separation.

Reference:
d-separation, http://www.andrew.cmu.edu/user/scheines/tutor/d-sep.html#d-sepapplet2

Pearl’s blog article “d-Separation Without Tears” January 1, 2000

Judea Pearl (2000), Causality: Models of Reasoning and Inference, Cambridge University Press

John Fox, Web appendix to “An R and S-PLUS Companion to Applied Regression”, Structural-Equation Models http://socserv.mcmaster.ca/jfox/Books/Companion/appendix-sems.pdf

Pearl(1997), The New Challenge: From a Century of Statistics to the Age of Causation，Computing Science and Statistics，29，415--423

The interpretation of Lord’s paradox in Rubin causal model

Lord’s paradox was first raised by Lord in an article on “Psychological Bulletin” in 1967. It reveals a contrast of two statisticians’ conclusions based on the same set of data. Here are the Lord’s four examples to illustrate his points.

Example1: “A large university is interested in investigating the effects on the students of the diet provided in the university dining halls and any sex differences in these effects. Various types of data are gathered. In particular, the weight of each student at the time of his arrival in September and his weight the following June are recorded. ”(Lord,1967)

Example 2: “A group of underprivileged students is to be compared with a control group on freshman grade-point average than the control group. However, the underprivileged group started with a considerably lower mean aptitude score(x) than did the control group. Is the observed difference between groups on y attributable to initial differences on x? or shall we conclude that the two groups achieve differently even after allowing for initial differences in measured aptitude” (lord,1969)

Example 3 “suppose an agronomist is studying the yield of various varieties of corn. He plants 20 flower pots with seeds of a “white” variety. For simplicity of illustration. Suppose that he treat all 40 plants equally for several months, after which he finds that the white variety has yielded considerably more marketable grain than the black variety. However, it is a fact that black variety plants average only 6 feet high at flowering time: whereas white variety plants average 7 feet. He now asks the question, would the black variety process as much salable grain if conditions were adjusted so that it averaged 7 feet in height at flowering time?” ( Lord,1969)

Example 4: “consider the problem of evaluating federally funded special education programs. A group of disadvantaged children are pretested in September, then enrolled in a special program, and finally posttested in June. A control group of children are similarly pretested and posttested but not enrolled in the special program. Since the most disadvantaged children are selected for the special program, the control group will typically have higher pretest scores than the disadvantaged group”(Lord,1973)

Through analyzing four examples, it is easily found that the previous inconsistence on conditions of experimental group and control group make the statistical hypothesis difficult to balance them. Therefore, it is necessary to apply counterfactual thinking to modify the previous condition in theory. However, the question is whether it is available to such modifications.

Rubin(1982) investigated the Lord's paradox in the form of Rubin causal model. The first statistician assumes the causal effects in a form below.
\inline \emph{$D_{i}=E(Y_{t}-X\lyxmathsym{\textSFxi}G=i),$ i=1,2}
\inline \emph{$D=D_{1}-D_{2}$}
（quoted from Holland, Paul W. Rubin, Donald B,1982)

E represents the experimental effect, \inline \emph{$Y_{t}$ } represents the outcome variable in the experimental group, G represents the subpopulation indicator variable, X represents the concomitant variable. D represents causal effect.

The first statistician finds that there is no difference between the concomitant variable and outcome variable for both females and males.

Considering the previously differences of experimental group and control group before the treatment and attempting to use covariance to control it , the second statistician computes the causal effect in such a way below:
\inline\emph{$D_{i}=E(Y_{t}-X\lyxmathsym{\textSFxi}G=i),$ i=1,2}
\inline \emph{$D=D_{1}-D_{2}$}
(quoted from Holland, Paul W. Rubin, Donald B,1982)

If we convert the equation of causal effect into a regression form, we can get the equations below:
\inline \emph{$D_{i}=E(Y_{t}-X\lyxmathsym{\textSFxi}X,G=i)=a_{i}+bX$ i=1,2,}
\inline \emph{$D_{i}(x)=a_{i}+(b-1)x,$ i=1,2,}
(quoted from Holland, Paul W. Rubin, Donald B,1982)

In the equation, \inline $a_{i}$ represents the differences of previous conditions on experimental group and control group; b represents the influence of the causal effect in the experimental group. In Lord’s paradox, two statisticians have different underlying assumption on \inline $a_{i}$ and b. The first statistician assumes \inline $a_{i}$ is o and b is 1, while the second assumes b is same in two groups. Therefore, they make their conclusions according their different underlying assumptions.

From the view of Rubin causal model, there is a concomitant variable (X) closely associated with the outcome variable (Y) in the example of Lord’s paradox. The concomitant variables produce the previously unequal conditions and affect the following outcome variables.

Rubin pointed out that there are some underlying assumptions in the statistical hypothesis the statistician made such as which is not testified appropriately. These assumptions have influenced the conclusions the statisticians reach

Rubin identify three types of studies as descriptive studies, uncontrolled causal studies and the controlled causal studies. The descriptive studies have no experimental manipulation. The uncontrolled causal studies have experimental manipulations without strictly controlling relevant factors. All possible factors have been sufficiently considered by experimenter in the controlled causal studies. The first statistician uses the unconditional descriptive statement that control group and experimental group are equal before treatment. The second statistician uses conditional statement which considers the previous differences. If both statisticians use descriptive statements, they are both right. However, when the descriptive statements are converted into causal statement, neither of them is right.

If I apply the Rubin and Holland’s analysis to the example of Lord’s paradox in the Powerpoint （http://lixiaoxu.googlepages.com/08Dec2006.ALL.G.ppt）, I can assume P represents the students in the class; t represents the course students receive; G represents the genders of students; X represents the degree of confidence before the course; Y represents the degree of the confidence after the course. Then we can see two statisticians make the different conclusions. Because the first assumes average confidence gains for males and females are equal, and the second assumes the male students and female students have equal confidence before course. We can see that two statisticians respectively make the untestified underlying assumptions before raise their null hypothesis. Those two counterfactual thinking both go against the real causal chains. It is doubtful to simply alter the causal factor without careful consideration. It seems to be a paradox for me to setting the experimental condition according the hypothesized causal relationship to explore the real causal relationships.

Lord’s paradox reflects the influence of differences of statistical hypothesis on the ultimate conclusion made. Can statistical hypothesis recognize and eliminate the previously existed unequality? In another words, descriptive statement and causal statements are two different language systems. The free translation between them seems not a certain thing. Since hypothesis test can only fix two contrast propositions, it is necessary for us to understand the limits of statistical language.

Reference

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

Lord, F. M.(1969) Statistical adjustments when comparing preexisting groups. Pwchological. Bulletin, 72, 336- 337

Lord, F. M.(1967) A paradox in the interpretation of group comparisons. Psychological Bulletin, 68 , 304-305

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