应用多元统计笔记之统计距离

（最近在看图书馆借的《实用多元统计分析》（Ａ约翰逊，迪安　Ｗ威克恩），觉得其用线性代数的方式解释统计的方式很受启发，再加上是借的书，所以希望作些笔记，算是总结以下自己读书的收获，准备用专题的方式把书中的一些概念记下来，同时参考一些其他资料。这也利于以后查阅）

统计距离（statistical distance）是多元统计的基础概念。它是为了解释变异差异和引入相关概念而发展起来的。试想我们测量两个变量，得到一系列数值对，每个变量的数值均值为0。两个变量的测量相互独立。而我们假定变量X1的变异量比变量X2要大。那么在以二维坐标图上（X1为x 轴，X2为y轴）表现为一个形状类似扁长椭圆的散点图。因为x轴的变异量大于y轴（因为X1变异量大于X2），所以在统计上认为将两个坐标赋予相同的权重并不合适。因此我们引进了一个标准化后的坐标 \inline$x_{1}^{*}=x_{1}/\sqrt{s_{11}}$ 和 \inline$x_{2}^{*}=x_{2}/\sqrt{s_{22}}$ 从而定义了统计距离为：
\inline$d(O,P)=\sqrt{(x_{1}^{*})^{2}+(x_{1}^{*})^{2}}$ $=\sqrt{\frac{x_{1}^{2}}{s_{11}}+\frac{x_{2}^{2}}{s_{22}}}$
(引自理查德　Ａ约翰逊，迪安　Ｗ威克恩（２00８）,实用多元统计分析（影印版），清华大学出版社，以下同上）

因此统计距离是对欧式距离的推广（二维是 \inline$d(O,P)=\sqrt{(x_{1})^{2}+(x_{2})^{2}}$ ），当变量间变异量一致时，我们用欧式距离来计算距离，当变异量不一致时我们考虑统计距离。
当统计距离为常量c时， 有以下方程
\frac{x_{1}^{2}}{s_{11}}+\frac{x_{2}^{2}}{s_{22}}=c^{2}$

该方程是一个中心在原点的椭圆方程，与x轴的交点为 \inline$-c\sqrt{s_{11}}$和\inline$c\sqrt{s_{11}}$与y轴的交点为\inline$-c\sqrt{s_{22}}$和 \inline$c\sqrt{s_{22}}$

如果对统计距离在进行推广对任何一个点P(X1,X2)到固定点Y(Y1,Y2)的统计距离是：
\inline$d(O,P)=\sqrt{\frac{(x_{1}-y_{1})^{2}}{s_{11}}+\frac{(x_{2}-y_{2})^{2}}{s_{22}}}$
统计距离还可以推广到啊更高的维度

以上的统计距离公式考虑的是变量之间相互独立的情形，因此有必要考虑变量之间相关的情形，所以引进了旋转角度的方法，如果把相关变量的散点图的坐标轴旋转至符合独立变量的情形那么可以得出
\inline$d(O,P)=\sqrt{\frac{(x_{1}^{'})^{2}}{s_{11}^{'}}+\frac{(x_{2}^{'})^{2}}{s_{22}^{1}}}$
x1’和x2’分别为 \inline$x_{1}^{'}=x_{1}cos(\theta)+x_{2}sin(\theta)$ 和 \inline$x_{2}^{'}=-x_{1}sin(\theta)+x_{2}cos(\theta)$

最终我们得出：
\inline$d(O,P)=\sqrt{a_{11}x_{1}^{2}+2a_{12}x_{1}x_{2}+a_{22}x_{2}^{2}}$

其中a11,a12,a22是由角度θ决定的

同样对于P(X1,X2)到固定点Y(Y1,Y2)统计距离为
\inline$d(O,P)=\sqrt{a_{11}(x_{1}-y_{2})^{2}+2a_{12}(x_{1}-y_{1})(x_{2}-y_{2})+a_{22}(x_{2}-y_{2})^{2}}$
同时可以把统计距离推广到更高维的地方。

要指出的是k距离并不只是和椭圆或圆联系在一起，任何距离测量只需符合以下特征：
d（P,Q）=d(Q,P)
d(P,Q)>0 如果P！＝Q
d(P,Q)=0,如果P=Q
d(P,Q)<=d(P,R)+d(R,Q) (三角不等式)

对于统计距离的解释可以根据以下定理： \inline$(x-\mu)\sum^{-1}(x-\mu)$ 为 \inline$\chi_{p}^{2}$ 分布，其中 \inline$\chi_{p}^{2}$ 为自由度为p的卡方分布，\inline$(x-\mu)\sum^{-1}(x-\mu)$ 为多元正态分布的形式

根据卡方分布的定义我们可以得出
\inline$(x-\mu)\sum^{-1}(x-\mu)$$=Z_{1}^{2}+Z_{2}^{2}+........+Z_{P}^{2}$

其中 \inline$Z=\sum^{-\frac{1}{2}}(X-\mu)$$是\inline$N_{p}(0,I_{p})$ 的正态分布

所以统计距离平方可以转化为p个 独立的标准正态分布变量的平方和，即把标准正态分布变量数值看成坐标值。其中 \inline$(x-\mu)\sum^{-1}(x-\mu)$ 也称作Mahalanobis距离（由P. C. Mahalanobisi在1936年提出）。Mahalanobis距离可以检测线性回归模型中的异常值。

统计距离在聚类分析中有着更广泛的应用，聚类分析有通过距离测度来获得相似性测度的方法，即通过统计距离的远近来判断案例（在坐标系中视作点）是否相似，属于同一类别（这也是统计距离的初衷所在）。所用的距离测度有：

欧式距离


（引自郭志刚《社会统计分析方法——SPSS软件应用》p122）


绝对值距离


（引自郭志刚《社会统计分析方法——SPSS软件应用》p122）

Minkowski距离

（引自郭志刚《社会统计分析方法——SPSS软件应用》p122）

Mahalanobis距离

（引自郭志刚《社会统计分析方法——SPSS软件应用》p122）


统计距离此外还有Cook距离 （http://en.wikipedia.org/wiki/Cook%27s_distance） 、Hellinger 距离 （http://en.wikipedia.org/wiki/Hellinger_distance）和Bhatta charyya距离 （http://en.wikipedia.org/wiki/Bhattacharyya_distance）等

参考文献
1.郭志刚（1999）郭志刚《社会统计分析方法——SPSS软件应用》，人民大学出版社
2.理查德　Ａ约翰逊，迪安　Ｗ威克恩（２00８）,实用多元统计分析（影印版），清华大学出版社
3.维基有关statistical distance的相关条目

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
bv
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.

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

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

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