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.

《女士饮茶》的读书笔记

最近一两天把李老师热情推荐的八卦书《女士饮茶》给读完了，虽然大多时候是当八卦读，但是感觉收获确实很多，至少从统计史的角度认识了统计学的意义。下面是一些自己感兴趣的片段的记录，算是纪念这本书的阅读历程吧。

奇人奇事

1．皮尔逊读的是政治学博士，研究的是科学哲学问题。这位统计学的奠基者创造了统计学，也创造了统计的哲学思考了吧。也许统计学就是对数据的哲学思考。从皮尔逊的身上我们可以看到统计学和政治学、科学哲学如何紧密联系起来的。

2．皮尔逊对头骨和骨骼的统计学分析用来解决历史上的难题。

3．从达尔文到高尔顿我们可以看到生物学与行为科学对早期统计的影响。回归趋势、正态分布、相关，这些都是高尔顿的贡献。我在想生物学为基础的统计和政治社会学为基础的统计如何融合为一起的，从生物进化论到社会进化论？这种影响对日后的统计学发展又会有什么影响？

4．《生物统计》是高尔顿创立的第一个重要的统计杂志，高尔顿的经费是如此充足，以至于杂志都为全彩照片，这让我想到自己刚看到《科学·科学美国人中文版》时的感受了，不知道《生物统计》那时侯售价多少，个人是否订的起？现在拍卖的价格又是多少？

5．戈塞特用自己化学的背景进入酿酒企业，用自己的数学背景研究统计学问题，随后他的统计学研究文章成了企业重要资产，他还升上了企业管理层，真是一个工作、研究两不误的好典型。T-分布起源于戈塞特以学生（student）为笔名发表的文章，戈塞特当年的谨慎弄的我们现在刚学统计时对t-分布的意思摸不着头脑。

6．费歇尔，务过农，当过兵、教过书，就差没做过工了。他的近视造成他极强几何直觉能力，但他的几何直觉能力又让他不被人理解，塞翁失马，焉知非福。

7．费歇尔从农田里收获了统计学的果实：随机化、方差分析、自由度。费歇尔大概用统计学描述了农业收成的谱写了一曲华丽的乐章。

8．世态炎凉：艾森哈特想去拜访老年的皮尔逊，被同事所阻止：看那个老家伙有什么收获？于是那个老家伙只能孤零零地住在远离两个系（优生学系和生物统计学系）和生物统计研究所的办公室，庆幸的是他的儿子当了生物统计系的系主任，只是不知道英国人是否有中国人的想法。

9．皮尔逊的代表作：《科学的法则》 费歇尔的代表作：《研究工作者的统计方法》

10．布利斯研究杀虫剂提出概率单位分析，在列宁格勒布利斯和克格勃的对话让我感到做一个单纯的人真好。度过了大清洗的残酷和卫国战争的血腥，布利斯迎来了温馨的秋季，秋季虽然美好但很多人和事都凋零了。

11．奈曼通过勒贝格的书进入了数学殿堂，可是亲身接触勒贝格时被他的傲慢所伤害了。后来为了弥补所带来的伤害，奈曼成了平易近人的老师。奈曼把文章写的非常的简单、自然，这是学术文章的最高境界吧。

趣事：奈曼在国际会议上宣讲一篇法语文章时，已经做好准备迎接费歇尔刻薄而强力的批评，结果费歇尔表现的非常平静，原因是他不会讲法语。

12．柯尔莫戈洛夫，一个天才加全才的数学家，他在莫斯科大学数学系从没考过试，原因是他写出了14篇独创性的论文来代替14门基础课的考试（当然他后来承认有一篇的结果其实是错的）。他70多岁的时候还爬山、海洋探险、滑雪、和教皇讨论宗教史，背普希金的诗歌。

13．饼图是护士行业的传奇女性南丁格尔发明的，大概是为了和愚蠢无知的军事将领打交道时更好的说明自己的观点吧。她的女儿大卫是皮尔逊晚年的研究生，不过相当怕皮尔逊。因为费歇尔对女生不屑一顾，所以大卫想向费歇尔提问时总是请旁边的男生代劳。大卫后来写了一本组合数学，很经典的著作。

14．皮特曼解决了非参数检验中关于检验效力和检验范围的问题，他是一个墨尔本大学的数学本科学生，但当他担任统计学教授，他还没正式接触过统计学的理论。

15．大萧条中大学生找不到合适的工作，结果他们去了劳工部和商务部，从实际问题中提出了抽样理论。

16．斯内德克建立了美国第一个统计系。

17．古德说，2的开方是无理数的发现“如果是当今的大人物所为，我会觉得很平常，但在两千五百年前是个惊人之举”，因为他在10岁时就通过心算发现了这一点。古德对偶然出现的数字巧合很感兴趣，所以后来他的书有了哲学的意味。

18．迪亚科尼斯14岁时离家出走跟着一个魔术师学魔术，24岁时回来开始念纽约市立学院的成人教育班。他为了看懂一本研究生概率数学教材而想上大学。

19．博克斯因为看过费歇尔《研究工作者的统计方法》，所以被军方送去学统计。战后他报考伦敦大学的统计学研究生时，向皮尔逊大谈费歇尔的理论，结果皮尔逊静静听完后同意了他的申请，但希望他会知道统计界除了费歇尔还有别人存在。（皮尔逊真是一个厚道人！）后来博克斯还娶了费歇尔的女儿。

20．戴明提出用统计的方法进行质量的管理，启发了日本产业界人士，结果重塑了日本产业界，促进了日本经济的崛起。可惜的是墙内开花墙外香美国人很长时间不买他的帐。戴明嘲笑假设检验的广泛应用，提出有意义的是差异大小程度而不是差异显著性水平。

统计技术思想

1．费歇尔关于吸烟是否制癌的讨论，开始了统计学中什么是因果关系的讨论。实验设计能否揭示出因果关系。

2．皮尔逊用分布的概念重塑了人们对世界的理解，世界不再是精确的公式所简单描述的，而是随机图景中的概率分布。

分布的四个参数：平均数、标准差、对称性、峰度

如果一个分布接近正态分布，它只需考虑前两个参数：平均数和标准差

3．皮尔逊和皮尔逊个人矛盾演化为两种统计学派的矛盾，皮尔逊认为统计分布是真实分布的描述，费歇尔认为统计分布只是用来估计真实分布的参数。皮尔逊用假设检验，费歇尔用显著性检验。“知道如何设计实验，这个实验就几乎一定能给出一个显著性的结果”

4．中心极限定理：一个被广泛应用但到20世纪30年代还未证明的定理。

要证明中心极限定义，必须先证明符合Lindeberg-Levy 条件，要证明符合Llindeberg-Levy条件，先证明是U-统计量（可惜的是书中还是没告诉我什么是Lindeberg-Levy条件和U-统计量）也许这就是八卦书的无奈吧。

5．拟合度检验：混沌理论的一个缺陷：没有给出数据绘出的图形和混沌理论预测图形的拟合度，拟合度是皮尔逊的贡献，他最早提出了卡方检验。不过费歇尔批评他比较两种比例时得出的参数值弄错了。

6．概率论，柯尔莫戈洛夫解决了概率论的基础，把求解概率和求一个不规则图形的面积联系起来，把概率论和数学测试理论，随机过程联系起来。（逆命题我曾经在一本书中见到，如果要求一个不规则图形的面积，就把分布在图形内的随机点的数量和另一个已知面积的规则图形中的随机点的数量进行比例关系）

7．极大似然法，费歇尔提出，用迭代算法计算来逼近一致而有效的统计量。（书中就到这里，我还是不能理解，于是又查了一本比较普及的书极大似然法就是让概率密度函数的值尽可能最大化，用来使数据尽可能的符合特定的参数估计。具体怎么计算等我好好看百科全书吧：））

8．置信区间上奈曼提出的，置信区间中提出的概率是统计学家使用某种方法从长期以来看作出正确陈述的频率。

9．非参数检验：从几何的角度看就是将观测数据的散点图和纯随机分布的预期图形进行比较。

10．贝叶斯发现条件概率的公式是内部对称的，那么条件概率公式倒过来代表什么？用分布估计参数变到由参数估计分布？贝叶斯学派的两个取向：通过积累数据获得分布的信息，主观概率（先验概率和后验概率）

11．快速傅立叶变换：一种以电脑为基础的数据分析方法，可以用来分析一长串相互关联因素的影响结果，一种向临近频率借力的方法。

12．鞅（martingale）：如果数列满足变异是有界的和下一个数字的最佳估计值是他的前一个数值时，这样的数列称作鞅。鞅这类数列将趋向于正态分布。如果能把长时间得到的数据看成是鞅，那么我们就可以解决生存分析之类的问题。

13．格利文科－坎泰利引理：如果有一些数，我们对它们的概率一无所知，那么我们可以构造一个非参数分布，尽管构造的数学函数结构不雅观，我们还是可以通过增大观测值的数量使经验分布函数越来越逼近真实分布函数。

14．Bootstrap:通过对自身样本进行有放回的全抽样，从而提升数据自身的模拟过程。

核密度回归估计（kernel density-based regression estimation）：一种有关运算密集（在数学上对重复抽样方法的推广）的程序，由两个参数确定：核（kernel）和带宽（bandwidth）

统计与世界观

1．统计革命的基本观点：科学真实的主体是数字的分布，这个分布可以通过参数来描述。

2．在当今统计应用急剧发展的世界，统计学家已经失去对统计的控制权，各种各样的统计方法是从在应用行业使用统计的人手中直接得出。

3．坎利夫女士的忠告：人类本身在偏爱方面充满了变异，在产品制造和质量管理等方面要要注意这些变异。统计学家的工作就是阅读数据，并质疑它为什么这样。将数学问题尽可能的用数学模型的方式表达出来，会促进科学家充分了解会出现什么问题。“如果我们一本正经地对一个不懂统计的男人或女人说“P 值小于0.001”意味着什么，我们就不会成功，所以，我们必须用他们的语言来解释我们的发现，以增强说服工作的效力。”

统计学的三个哲学问题：

4．可以用统计模型做决策么？L·乔纳森·科恩的彩票悖论和无票入场者悖论

5．当概率应用于现实社会时，其含义是什么？

柯尔莫哥洛将概率定义为一个抽象空间里对一事件集合的一种测量。但我们如何来确定现实中的抽象空间。威廉·S·戈塞特试图为一个设计好的试验描述其事件空间。而第二种方法是用样本调查理论来选择抽样的方法来尽可能的使样本反映总体。

6．人们真的懂得什么是概率么？

凯恩斯在博士论文《关于概率的讨论》中指出概率的含义的结论取决于人类对不确定性量化的愿望，是一个和人的主观判断和文化背景有关的概念。（牛人的一篇牛文，有想看的冲动）凯恩斯的观念是进行统计决策的基础，影响到了卡尼曼等人关于主观概率和启发式的研究（李老师推荐了很多他的文章）

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

一篇据说充满统计谬误的例子

从博文（http://www.stat.columbia.edu/~cook/movabletype/archives/2005/03/lowess_is_great.html）的一段留言中举了一篇包括生态谬误等一系列统计谬误的文章，留言具体如下：

Further Googling of Martin Voracek would have turned up this gem: National intelligence and suicide rate: an ecological study of 85 countries.

Now here's an article that has it all in terms of statistical fallacies. He attempts to argue that intelligence is a causal factor for suicidality using the following methodology: Each country in the world is assigned an IQ and this national IQ is correlated with national reported suicide rates. It's really amazing - he's managed to incorporate the ecological fallacy, reporting bias, selection bias, profound confusion about the definition of IQ (a measure of intelligence relative to a typical individual), unmeasured confounders, terrible measurement methodology and just a generally goofy scientific approach, all in one bogus study! There should be an award for this.

Posted by: js at December 20, 2005 10:43 AM.

还没有细看这篇文章，等有机会好好琢磨一下这个反面教材。

找到一个感兴趣的BLOG

最近找到一个和自己的兴趣有点相投的博客。博客将心理、社会和复杂性方法大杂烩一样的融合在一起，企图用复杂性科学的研究理念和方法将心理和社会串接起来，进行全面地思考和操作。我觉得这是一个很有前瞻性和值得进行的研究视角。

博客的文章比较通俗活泼，易于理解。我可以从中获得很多启发。特别是里面一些别具心思的图片来衬托说明的原理，让人拍案叫绝。更重要的是我意识到有人和自己在想相似的东西，这是一件让人觉得幸福的事情。

顺便我可以学习一下博客上的英文写作：）

博客地址：http://encefalus.com/

The attempt to understand LOESS

I first look up some English papers and literatures on LOESS, which confuse me quite a lot. Then I find an article written by Xie Yihui in Chinese (http://cos.name/2008/11/lowess-to-explore-bivariate-correlation-by-yihui/) which concisely and clearly introduces the main ideas of LOESS. It is helpful for me to further understand the details of the English introduction to loess.

Local weighted polynomial regression (LOESS) is a statistical method which aims at proportionally generating regression lines from a localized and limited data, which are integrated into a curve to show the whole trend of data without losing the important details that a small proportion of data may reflect. The method, I believe, is similar to the concept of approximation in calculus in mathematics, constructs a linear equation within the vicinity of a given point.

LOESS is a good adjustment for classical statistical methods (e.g. linear least square) to the flexibility of modern procedures. The algorithm is determined by two key parameters: the bandwidth which controls the smoothing property of curve and the degree of local polynomials which reflects the accurateness and complex of imitation。.(e.g. the first degree is linear, the second degree is quadratic.

Although higher degree of local polynomials can fit the empirical data better, it also makes the calculation expand to quite a big one and go against the sprit of LOESS. Therefore, selecting an available degree is very important for an effective LOESS. A general polynomial of degree p is calculated as the first picture below indicates. The second picture shows formula for calculating coefficient a (i) which is used to interpolate the local regression value at x. In the second formula, X is a Design Matrix; W is a Diagonal Matrix ; Y is simply the y value of the data.

(quoted from http://voteforamerica.net/Docs/Local%20Regression.pdf)

The method to select the bandwidth is Mean integrated Square error which is considered a reliable method for the selection of optimal bandwidth.

As it is said in the Wikipedia (http://en.wikipedia.org/wiki/Local_regression) , the advantages of LOESS are various: first, it is unnecessary to produce certain function to fit the model to all the data. Second, it is flexible to imitate some kinds of data that have no certain mathematical model. However, some disadvantages still exist. Effective LOESS needs a large sample of data. Besides, it is difficult for us to extract a concrete mathematical formula from LOESS, so the possibility of extending it to other instances is no obviously be limited.

Loess technique can be very useful to various fields. In the website below, loess technique is applied to reflect the trend of American election poll and attempt to predict the ultimate outcome of election.

(http://voteforamerica.net/editorials/Comments.aspx?ArticleId=28&ArticleName=Electoral+Projections+Using+LOESS)

In addition, LOESS can be used in neurocognitive science as a useful smoothing technique of data.

Reference:

谢益辉：用局部加权回归散点平滑法观察二维变量之间的关系

http://cos.name/2008/11/lowess-to-explore-bivariate-correlation-by-yihui/

NIST Engineering Statistics Handbook Section on LOESS

http://www.itl.nist.gov/div898/handbook/pmd/section1/pmd144.htm

apply LOESS to describe the trend of American election.

http://voteforamerica.net/polls.aspx

local regression

http://voteforamerica.net/Docs/Local%20Regression.pdf

the article about local regression in wikipedia

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

经济学人笔下的Aul Lauterbur

今天把经济学人的这篇老文章找出来，算是一半庆祝今天的开博，一半怀念过去的时光吧。

现代科学的历史是奠基在被拒绝的论文的基础上的，这是开头Aul Lautebur开的一个笑话，也是对70年代碰壁岁月的无奈解嘲。这种无奈在f-mri日受追捧的现在是多么的令人感慨。但我还是怀念那个岁月，Aul Lautebur可以饶有兴趣拿青辣椒和女儿海滩拣的蛤蜊做二维呈像，像淘气的孩子一样对新奇的事物进行试验、玩耍。比较现在用f-mri做实验，一篇论 文要10万的元的现状，科学的乐趣也许就在发明最初的纯真岁月吧。

科学界在纠缠了20多年后，终于在2003年把诺贝尔奖颁给了他，而4年之后他就去世了。也许迟到的荣誉对他并不意味着什么，科学的快乐在于过程，而我相信他享受到了。

纪念这位慈祥的老人。

Aul Lauterbur, father of MRI, died on March 27th, aged 77

THE whole history of modern science, Paul Lauterbur once joked, might be written on the basis of papers turned down by academic journals. His own experience was a case in point. In 1971 he sent a paper to Nature; it was rejected. The Nature folk were especially unimpressed by the fuzziness of the pictures that accompanied the piece. Never mind that they showed the difference between heavy water (with deuterium atoms) and ordinary water (with hydrogen atoms) in a way that no image had done before. Never mind that nuclear magnetic resonance (NMR) had been used for the first time to make those images, and could henceforth be used, with just a little development, to make non-invasive pictures of brains and spinal cords. Never mind that this technique, in 2003, was to win Mr Lauterbur a joint share in the Nobel prize for medicine. It did not yet look professional enough.

Nor did some other aspects of Mr Lauterbur's work. His core discovery, of how to get spatial information about atoms in a magnetic field, was scribbled on a paper napkin over dinner in a Big Boy restaurant in Pittsburgh, between two bites of a hamburger. His early exploration of this idea, building up two-dimensional images of the soft interiors of organisms by spectroscopy, was performed on green peppers or on clams which his small daughter collected on the beach. His speciality for a long time, before he had developed the technology he needed, was fuzziness--or rather, that interesting gradation of shading, pinpointed by hydrogen atoms, that showed where the water content of the cells was changing, and tissues were becoming diseased.

There was always something serendipitous, even wild, about Paul Lauterbur's approach to science. As a boy in the Ohio countryside he trespassed widely in search of terrapins, fish and birds; as a teenager he built his own lab in the basement of his house, entranced by the strange vials in his chemistry set and by the stink of burning sulphur. His greatest joy, he reported, was to be left alone to explore the world or to experiment. His chemistry teacher at school was understanding, allowing him to lark around with apparatus, just within the limit of danger and expulsion, at the back of the class. His army superiors were kind when he was drafted in the 1950s, letting him spend his time setting up and running an early NMR machine; by the end of his service, when his colleagues had nothing but a cropped head to show for it, he had produced four scientific papers. Academic authorities, first at the State University of New York at Stony Brook and then at the University of Illinois at Urbana-Champaign, were aware of his low boredom threshold and let him rove between his native chemistry, physics and medicine, knowing that if this professor was left to himself he might well produce something extraordinary.

What Mr Lauterbur did was not, in its essence, brand new. The fact that the nuclei of atoms were magnetic, acting like tiny compass needles, had been discovered in the 1940s. When those nuclei were aligned in a strong magnetic field and bombarded with radio waves, they would send back radio signals of the internal structure of substances; this had been observed in 1952, and NMR machines had been built to exploit it. But Mr Lauterbur was the first to introduce gradients, or variations, into the magnetic field, allowing him clearly to track where atoms were and to take "slices" of what he was observing in two dimensions. He began with rubber and silica, then progressed to bivalves and mice. By taking many slices, he could then build a three-dimensional image of organs and other soft tissue that could not be seen by X-rays.

Mr Lauterbur had a grand name for his discovery: "zeugmatography", from the Greek zeugma, or yoke, since he had linked together both chemical and spatial information. The name did not catch on and nor, for some time, did the NMR machine. The first company that produced them, of which Mr Lauterbur briefly became president in 1971, almost went bust. Radiography departments in hospitals clung rigidly to X-rays, dangerous though they were, for making diagnoses. "Nuclear magnetic resonance" proved a terrifying concept to patients, who thought they were about to undergo a micro-version of Hiroshima. But Mr Lauterbur, undaunted, continued to work to improve the technique and went round the world promoting it. By the time he died, more than 22,000 magnetic resonance imaging (MRI) machines were in use, and more than 60m scans were being carried out each year.

What the machines see now is almost incredible. A tiny ripple of water in the brain, tracked by hydrogen atoms, that shows the passage of a stroke; a patch of inflammation in the spinal cord, indicating multiple sclerosis; the narrowing of a blood vessel, and constriction of the flow, that presage heart disease. Mr Lauterbur believed MRI could get better and better, until the living body could actually be watched at work. He believed, too, that it might begin to throw up clues about the origins of life; and that in time that fascinating fuzziness, too, would become as clear as day.

mediation analysis and cultural psychology

Mediation mechanism is defined as how the addition of the third variable influences the relationship between two variables. In the simplest mediation model, X causes the Y, then Y causes the Z, where Y is perceived as a mediated factor in a directional causal chain. Experimental approaches and statistical approaches are applied in mediation analysis. Randomized experimental design is used in the investigation of mediation. Controlling the individual differences and experimental mechanism, the difference of means is interpreted as the outcome of the manipulation of mediator. Since the experiment is the unique method to explore the causal relationship, the mediational process is confirmed by two randomized experiments. Despite the robust power to explain the mediation process, the difficulty of extending the results to other situations has limited the application of experimental approaches to mediation. The manipulation and measurement of mediator are not both considered by the experimental approaches, thus the statistical analysis in necessary to investigate the mediation. Causal steps, differences in coefficient, product of coefficient are three major approaches of statistical analysis, where causal steps is widely used one which first establishes the linear equation between the independent variables and dependent variables, then establishes two equations to explore the relationships between the mediator and two variables respectively. In the process of establishing the three equations, the coefficient relating the independent variables and dependent variables in the context of mediation variable must be bigger than the coefficient of the direct relationship between independent and dependent variables. The mediated effect is calculated through two ways: aˆ *bˆ and cˆ−cˆ, the rationale behind the first method the author (David P. MacKinnon, et.al, 2007) emphasis is “that mediation depends on the extent to which the program changes the mediator, a, and the extent to which the mediator affects the outcome variable, b”. the first is involves the reduction of effects between independent variables on the dependent variables within and without the mediator.

The picture is clearly illustrate the components of mediated effect

(quoted from David P. MacKinnon, Amanda J. Fairchild, and Matthew S. Fritz(2007))

The traditional calculation of limit confidence based on the normal distribution of mediated effect is inaccurate, which is gradually replaced by the bootstrap analysis, which is a non-parametric method of effect-size estimation and test hypothesis and will overcome the power problem, where small samples are also available to be tested.

Beyond the single model of mediation analysis, multiple mediators, multi-level mediation level and longitudinal mediation analysis have been developed quickly. Although the mediation analysis has a continued progress, the doubts still exist in the fundamental assumption of mediation analysis. How regression model reflects the causal relations among variables? The alternative possibilities need more information to exclude while mediation analysis never provides. The principle stratification of possible relations of variables is suggested to take as a promising alternative.

The application of mediation analysis in psychology has a long history(Dov Cohen, 2007). In Hebb’s stimulus-response model, mediated mechanism such as cognitive or neural processing mechanism is considered as bridges between the stimulus reception and behavioral response. Besides, psychological factors mediating how social context influences individual action play important roles in the field of social psychology. Psychological factors mediating the cultural and psychology is an indispensable issue in cultural psychology. How mediation mechanism effects in the interaction of culture and individuals? Cultural properties such as individualism-collectivism, dispositional measures, the judgment and reason of individual, choice-making perception, goals, social beliefs have been considered as the alternative explanation of mediator, however a large number of hypothesized mediators have not been proved. As Dov Cohen(2007) pointed out, the overemphasis of individual differences in cultural psychology will produce a risk to “reduce culture to individual difference or inside-the-head variable that neglects how situation ,practice and institutional arrangement afford certain type of behavior”. How culture outside the individual play the roles will be overlooked.

In order to solve the paradox, Hong(2003) presented a view in his bicultural model that there can be more than one cultural constructs which can be coexisting disregarding their contrastive properties. Cultural activation as a effective tool can be applied to direct people to behave in different cultural situation. Therefore, the previous absolute and dichotomous view on cultural syndrome of people from different cultural atmosphere has been challenged powerfully. In another way a dynamic viewpoint is introduced to replace the dispositional one. Reasoning styles rather than differences of personalities and preferences act as a key mediation mechanism.
The measurement of mediator also becomes the focus of criticism, since questionnaire is the most popular way. However, the questionnaire is not a valid and sensitive indicator to cross-cultural differences. Therefore a new measurement method is necessary to seek for the mediator better.

Although mediation analysis is very useful in the interpretation of psychological mechanism in cultural psychology, a lot of problems can’t be solved through it, like the problem of biased sampling and the problem of excluding other cultural syndrome.



References:
David P. MacKinnon, Amanda J. Fairchild, and Matthew S. Fritz(2007)，mediation analysis Annual Review. Of Psychology.. 58:593–614

Hazel Rose Markus and MarYam G. Hamedan (2007)i, Sociocultural Psychology: The Dynamic Interdependence among Self Systems and Social Systems, hand book of cultural psychology, The Guilford Press A Division of Guilford Publications, Inc.

Dov Cohen (2007 ), Methods in Cultural Psychology, handbook of cultural psychology The Guilford Press A Division of Guilford Publications, Inc.

Hong, Y., Benet-Martinez, V., Chiu, C., & Morris, M.W. (2003). Boundaries of cultural influence: Construct activation as a mechanism for cultural differences in social perception. Journal of Cross-Cultural Psychology, 34, 453–464.

Briley, D. A., Morris, M. W., & Simonson, I. (2000). Reasons as carriers of culture: Dynamic versus dispositional models of cultural influence on decision making. Journal of Consumer Research, 27, 157–178.

SPSS and SAS procedures for estimating indirect effects in simple mediation models
Behavior Research Methods(2004), Instruments, & Computers, 36 (4), 717-731