廣義Neyman-Rubin的因果模型在評估迴歸上交互作用的應用

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(1)國立交通大學統計學研究所碩士論文. 廣義Neyman-Rubin的因果模型在評估迴歸上交互作用的應用 Generalized Neyman-Rubin’s causal model for Regression Interaction Assessment. 研究生：莊揚凱指導教授：陳鄰安. 教授. 中華民國一百零二年六月.

(2) 廣義 Neyman-Rubin 的因果模型在評估迴歸上交互作用的應用 Generalized Neyman-Rubin’s causal model for Regression Interaction Assessment. 研究生：莊揚凱. Student：Yang-Kai Chuang. 指導教授：陳鄰安. Advisor：Dr. Lin-An Chen. 國立交通大學統計學研究所碩士論文. A Thesis Submitted to Institute of Statistics College of Science National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Master in Statistics June 2013 Hsinchu, Taiwan, Republic of China. 中華民國一百零二年六月.

(3) 廣義 Neyman-Rubin 的因果模型在評估迴歸上交互作用的應用研究生：莊揚凱. 指導教授：陳鄰安博士. 國立交通大學統計學研究所. 摘要. 儘管在經濟，社會和健康科學上，藉由插入一個分析模型的相乘項來檢定交互作用影響的表現是非常常見的，但交互作用是否存在決定於模型型態為一直被批評的地方(Greenland (2009) and Mauderly and Samet (2009))。有些文章努力解決這個爭議性問題但卻導致於更複雜且不清楚的交互作用定義。這讓評估統計交互作用更加困難(Greenland (1980))。我們提出一個有系統的定義介紹，方法和定理將相互關係(結合)參數融入在廣義 Neyman-Rubin 的因果模型。這項創舉帶來許多的優點： (a) 此方法允許我們定義和測量關於未知統計上交互作用的統計推論的相互關係影響。 (b) 對於統計上交互作用的統計推論全都可從分佈參數的估計理論來建構。 (c) 此因果模型測量一個明確且模型獨立能避免插入爭論的相互關係影響。 (d) 廣義 Neyman-Rubin 的因果分析理論擴展到對於 probit 迴歸的統計交互作用評估。. 關鍵字：因果推論；內部相互關係；迴歸分析；統計交互作用。 I.

(4) Generalized Neyman-Rubin’s causal model for Regression Interaction Assessment Student：Yang-Kai Chuang. Advisor：Dr. Lin-An Chen. Institute of Statistics National Chiao Tung University Abstract Although the insertion of product terms into analytical to test for presence of interaction effect is very common in economic, social and health sciences, it has long been criticized for that existence of interaction is model dependent (Greenland (2009) and Mauderly and Samet (2009)). The efforts for resolving this criticism leads to multiple but ambiguous definitions of statistical interaction resulting in assessing various but unknown versions of effect (Greenland (2009)). We report that a systematic introduction of definitions, methods and theorems to fit the intercorrelation (association) parameter into a generalized Neyman-Rubin’s causal model brings interesting advantages: (a) This approach allows us to define and measure a clean effect of intercorrelation for statistical inferences of unknown statistical interaction. (b) Statistical inferences for statistical interaction all can be constructed from the estimation theory of the distributional parameters. (c) This causal model measures an unambiguous but also model independent effect of intercorrelation that avoids the controversy of insertion. (d) The theory of the generalized Neyman-Rubin’s causality is extended to statistical interaction assessment for probit regression. Key words: Causal inference; intercorrelation; regression analysis; statistical interaction. II.

(5) 誌謝在碩士這兩年中，我最感謝的是我的指導教授陳鄰安老師，這一年來跟老師一起做研究，老師總是詳細地教導我，當我論文遇到問題時，老師也很耐心地引導我找出疑點並進行改善。我覺得跟老師做研究，學到的不僅僅是學術上的知識，更讓我學習到如何在遇到問題時，找出解決問題的方法與態度，想必這在往後的工作以及待人處世上有深深的影響。也要謝謝我的口試委員許文郁老師、蕭金福老師及洪慧念老師，老師們給予我的建議，使我的論文更加完整。再來我要感謝交大統研所碩士班的所有同學，沒有你們，我的碩士生活不會如此精采。有好多我們一起做過的事，現在還是歷歷在目，不論是班遊、統研盃及研究室發生的點點滴滴，都讓我的碩士的生活增添了許多歡樂的回憶，謝謝你們這一群好同學好朋友。最後我要感謝的是我的家人，從小到大不辭辛勞的栽培我，也都尊重我在我求學過程中所做的選擇，有家人的陪伴讓我順利完成學生生活。. 莊揚凱. 謹誌於. 國立交通大學統計學研究所中華民國一百零二年六月. III.

(6) 目錄中文摘要…………….………………………………………………………….…...…I 英文摘要……………………………….…..…………………....……………………II 誌謝………………………..………………………………...…...…………………..III 目錄………………………………………………………….…………………….…IV 表目錄…………………………………………………….…..………………………V 1. Introduction………………….…………………………………………………….1 2. Parametrized Regression for Effect Assessment………………….……………….3 2.1. A Normal Regression Model……………………….………………………....3 2.2. Can Classical Interaction Detection Methods Deal with Normal Data?...........4 2.3. Quantities to be Explained ………………….……………………...…………6 2.4. Motivation of Causality Analysis for Statistical Interaction.………………....7 3. The Neyman-Rubin’s Causal Model for Interaction Analysis .………………...…8 4. Statistical Inferences for Neyman-Rubin’s Causal Effect of Intercorrelation……12 5. The Neyman-Rubin’s Causal effect of Intercorrelation for Binary Variable…….16 6. Concluding Remarks……………………………………….…………………….18. 7. Appendix……………………………….……………………………………...…19 References………….…………………………………………………...………..23. IV.

(7) 表目錄. Table 1. Effects of intercorrelation on some outcomes quantities………......................8 Table 2. Power performance for interaction detection by Test 1……………………..12 Table 3. Power performance for interaction detection by Test 2……………………..13 Table 4. Power performance for statistical interaction for conditional variance……..14 Table 5. Power performance for statistical interaction for regression quantile………14 Table 6. Predicted mean sales for first four territories……………………………….15 Table 7. p-value of observation points……………………………………………….16. V.

(8) Generalized Neyman-Rubin's causal model for Regression Interaction Assessment Abstract. Although the insertion of product terms into analytical model to test for presence of interaction eect is very common in economic, social and health sciences, it has long been criticized for that existence of interaction is model dependent (Greenland (2009) and Mauderly and Samet (2009)). The eorts for resolving this criticism leads to multiple but ambiguous de nitions of statistical interaction resulting in assessing various but unknown versions of eect (Greenland (2009)). We report that a systematic introduction of de nitions, methods and theorems to t the intercorrelation (association) parameter into a generalized Neyman-Rubin's causal model brings interesting advantages: (a) This approach allows us to de ne and measure a clean eect of intercorrelation for statistical inferences of unknown statistical interaction. (b) Statistical inferences for statistical interaction all can be constructed from the estimation theory of the distributional parameters. (c) This causal model measures an unambiguous but also model independent eect of intercorrelation that avoids the controversy of insertion. (d) The theory of the generalized Neyman-Rubin's causality analysis is extended to statistical interaction assessment for probit regression. Key words: Causal inference intercorrelation regression analysis statistical interaction.. 1. Introduction. The notions of "interaction" is common in researches of business, economics, education, sociology, health science and many others but it has long been in literature with controversies surrounding concept of interaction (Greenland (1993), Rothman, Greenland and Walker (1980) and Suhnel (1992)). The classical regression interaction assessment often inserts product terms in causal (regression) model as y = g(0 1 x1 2 x2 12x1 x2 ) + (1.1) Typeset by. 1. AMS-TEX.

(9) 2. to test hypothesis H0 : 12 = 0 for the presence of interaction. It has made the controversy that the presence or absence of interaction is entirely dependent on the form of the regression model been choosed for the same data, interaction may appear to be present when using one regression model but absent when another regression model is applied (Mauderly and Samet (2009) and Rothman, Greenland and Walker (1980)). Some eorts for resolving this controversy are done, for examples, by Mullahy (1999) and Ai and Norton (2003) for cross-derivative method measuring so-called the second order interaction and VanderWeele (2009) and VanderWeele and Robin (2008) for sucient cause interaction in biologic approach. Although the these approaches can measure eect other than product terms but they and some others measure dierent versions of interaction that can not avoid the concern of Greenland (1993) and Greenland (2009) that makes the users confused for how much and in what direction we can learn from a data set. A correct resolution of this controversy requires a mechanism that can accurately measure the causal eect of consensus intercorrelation (association) cause in random world (Rothman, Greenland and Walker (1980) and Ai and Norton (2003)). The approach of Chen et al. (2013) extends the concept of biologic isobole (Loewe (1928, 1953)) to de ne an unknown statistical isobole for inferences of statistical interaction. While this is interesting that allows us to detect if the statistical interaction is present or not, it can not measures the size of statistical interaction and, isobole is not popularly applied in social and economic sciences. Fitting the statistical interaction into a causal model can completely solve this controversy. The Neyman-Rubin framework (Neyman (1990) and Rubin (1974)) of causality analysis discovering causal relationships between outcome variable and causal variables has become increasingly popular in applied research (Holland (1986), Rubin (2006) and Sekhon (2008)). This interesting approach, also called the dierence in dierence method, is a popular tool for evaluating the eects of policy interventions in economics and biology (Abadie (2005) for a review). This article attempts to formulate its generalization so that regression interaction analysis can be done with this.

(10) 3. common framework with expectation of making this advance of interaction assessment more accessible to the general research community. We derive a Neyman-Rubin's causal model from the joint distribution of outcome and cause variables forcing the regression function to have distributional parameters involved that leads to several important advantages: (a) The novel parametrization of imposing distributional parameters in regression model builds a bridge between the Neyman-Rubin's causal model and the parameter of intercorrelation between causes (explanatory variables) allowing us to measure clean causal eect of intercorrelation. (b) This approach is not model dependent that avoids the controversy of insertion of product terms in model. (c) This success in assessment of statistical interaction by NeymanRubin's causality analysis may be applied to statistical interaction for other models where we interpret this for probit regression model.. 2. Parametrized Regression for Eect Assessment 2.1. A Normal Regression Model. We rst clarify two concepts of our interest when we have random variables y x1 and x2 and our interest is the eect of some causes on variable y. Statistically causal study consider a comparison between the eect (outcome) of variable y on taking a treatment x1 = x1b relative to the eect of taking another treatment x1 = x1a holding all other factors unchanged. On the other hand, statistically interaction assessment consider a comparison between the eect of variable y when there is no intercorrelation between variables x1 and x2 and the eect of variable y when the intercorrelation is present. Two closely concepts have not been uni ed in analysis. We propose to assess statistical interaction via a generalized causal model. A proper perspective in a theoretical analysis of causality requires method to assess eect of the cause to understand why the values of a quantity to be explained is aected with causes (Holland (1986)) while looking for the cause of an eect such as approach of sucient cause approach of Rothman (1976) and Vanderweele and Robins (2008) is not of this kind. For references of general causal analysis and modeling, see Vanderweele and Rubin (2008).

(11) 4. and Sekhon (2008) and for review, see Holland (1986), Heckman (2008) and Rothman and Greenland (1998). Suppose that variables y x1 and x2 have a joint distribution. We propose to study causal eect from a regression model that is parametrized from this joint distribution. Here we consider joint normal distribution as. 0y1 0 1 0 2 1 y1 y2 @ x1 A N3(@ y1 A @ 1yy 12 12 A): x2. 2. (2.1). 2y 21 . 2 2. for interpretation. The conditional expectation of y given (x1 x2) under normality assumption is. 2 ;1 x ; 1 1 norm (x1 x2) = y + (y1 y2) 1 122 : 21 2. x2 ; 2. Setting a xed vector of distributional parameetrs as 0 = (y 1 2 y2 12 22 y1 y2 12), a parametrized regression model is stated in the following theorem.. Theorem 2.1. The regression model under the normal distribution of (2.1) is. y(x1 x2 ) = 0 () + 1 ()x1 + 2 ()x2 + . (2.2). where () = (0() 1() 2())0 with. ; y2 1 )2 0 () = y ; (y122;2 ;y2212)1 + (y112 22 ; 2 1 2 12 1 2 12 2; 1 () = y1 22 2 ; y22 12 1 2 12 2 2 () = y2 21 2;;y1212 2. 1 2. 2. 12. where error variable has the normal distribution N (0 y2jx1x2 ()) with. 2 ;1 y1 : yjx x () = y ; (y1 y2) 1 122 2. 1. 2. 2. 21 2. y 2. 2.2. Can Classical Interaction Detection Methods Deal with Normal Data?.

(12) 5. Parametrized normal regression model provides important messages for veri cation of existent interaction detection methods. Three methods are considered here. Applied econometrics researchers commonly infer about the presence or absence of statistical interaction via testing a hypothesis for existence of something about interaction. The most popular one is assuming the following regression model. y = 0 + 1 x1 + 2 x2 + 12 x1 x2 + to test hypothesis H0 : 12 = 0 for existence of product term interaction. It is argued that power of developed statistical tests for the presence of interaction is remarkably low (Geenland (2009) and Mauderly and Samet (2009)) which is not surprised from our derivation (2.2) that the true regression model does not include any product term even the trivariate data is drawn from a normal distribution. An eort in econometrics for avoiding this model dependence disadvantage is verifying the presence or absence of 2 x1 x2 ) . This approach is not applicable too second-order interaction @ E@x(y1j@x 2 for normal data since its true regression model virtually does not have this interaction. Hence, these two popularly methods can measure the product term interaction and second order interaction that can not measure a pure eect of intercorrelation. Besides the above, departure from additivity is another popularly used test for biologic synergism eect. Following Greenland (1993), the eects of two explanatory variables are de ned as mean dierences in the absence of the other variable as. Ex1 = cm(x1 0 ) ; cm(0 0 ) = 1 ()x1 Ex2 = cm(0 x2 ) ; cm(0 0 ) = 2 ()x2 and the combined (total) eect as the sum of separate eects is. Ex1x2 = cm(x1 x2 ) ; cm(0 0 ) = 1 ()x1 + 2 ()x2: The test by Prentice and Kalbeisch (1988) de ning synergistic eect if the combined eect Ex1x2 is greater than the sum of separate eects Ex1 + Ex2.

(13) 6. shows no synergism since Ex1 x2 = Ex1 + Ex2 . This criterion of interaction as departure from additivity limits its application since it can not detect the eect of intercorrelation even the data is observed from a normal distribution.. 2.3. Quantities to be Explained. To forestall confusion that the reader of the literature on causality encounters unclearly terminologies, we provide step by step the de nitions. Beni ted from parametrization, a framework of causal model can assess effects on the internal variable y of various causes of interest not restricted to external variables x1 and x2 . That is, the potential causes in this causal model includes elements in the following set: Potential causes: x1 x2 :. (2.3). where the unobservable external (error) variable is not considered in this paper. We need to specify outcomes of interest. The classical Neyman-Rubin's causality analysis de nes the comparison of eects at treatments x1 = x1b and x1 = x1a as the dierence of two potential outcomes as. y(x1b x2 ) ; y(x1a x2 ):. (2.4). This causal inference is a missing data problem because we cannot observe two outcome variables in (2.4) at the same time (Holland (1986)). Outcome quantities to be explained other than the response variable exist in literature, for examples, a utility function R(y) as subjective evaluation of outcome in economic approach (Heckman (2008)) and biology approach (Greenland (1993)), conditional mean by Holland (1986) and variation eect y2jx1 x2 (x1 x2 ) in social science (Russo (2011)). A framework broadens the range of quantity to be explained is available. De nition 2.2. Any quantity (x1 x2 ) that characterizes the regression model (2.2) is called an outcome quantity.. Example 1. Some interests of outcome quantity are:.

(14) 7. (1) Outcome variable: y(x1 x2 ) measuring the outcome of possible experiment (2) Conditional mean: cm(x1 x2 ) = 0 ()+1 ()x1+2 ()x2 = (1 x1 x2) () measuring the central tendency of the regression model (1986)) 2(Holland ; 1 12 y 1 2 1 (3) Conditional variance: cv () = y ; (y1 y2) 2 y 2 12 2 measuring the conditional variation of the regression model (applied in social science (Russo (2011))) (4) Regression parameters: () p (5) Regression quantile: ( ) = (0 () + z cv () 1 () 2())0 satisfying = P (Y (1 x1 x2) ( )jx1 x2) for (x1 x2)0 2 R2 de ned by Koenker and Bassett (1978) (6) Conditional quantile: cq (x1 x2 ) = (1 x1 x2) ( ) (7) Reference charts: rc = f(1 x1 x2) ( ) : 2 (0 1) (x1 x2)0 2 R2g (8) Conditional signal-to-noise ratio at (x1 x2): sn() = cmp(x1 x2 ) : cv () One can measure the eect of causes on the central tendency of outcome's distribution by conditional mean, but it does not provide a complete picture of this distribution. When one is interested in the distributional extreme behavior, the conditional quantile of (6) above is desired for investigation.. 2.4. Motivation of Causality Analysis for Statistical Interaction. Causal comparisons for interaction assessment entail contrasts between outcomes in states of presence or absence of the intercorrelation between variables x1 and x2 that can be answered from an extension of the classical causality model of Rubin (1974) and Holland and Rubin (1980). The generalized Neyman-Rubin's causal model for interaction assessment then considers the dierence. (x1 x2 j12) ; (x1 x2 j12 = 0) holding all other factors including variables x1 and x2 and parameters ; f12g unchanged. To see if intercorrelation parameter 12 causes eect on.

(15) 8. the de ned outcome quantities, we let 0 = (1 1 1 2 2 2 0:7 0:7 12) and compute some true values of outcome quantities under = 0:5 and 12 = 0 for veri cation. The results are displayed in Table 1.. Table 1. Eects of intercorrelation on some outcome quantities. Eect quantity 12 = 0 12 = 0:5 cv () 1:51 1:61 () (0:3 0:35 0:35) (0:44 0:28 0:28) ( ) = 0:1 ;1:27 ;1:18. = 0:2 ;0:73 ;0:62. = 0:3 ;0:34 ;0:22. = 0:4 ;0:01 0:11. = 0:5 0:30 0:44 The second and third elements for ( ) and () are identical in either case of 12 = 0 and 12 = 0:5. The fact that cv (j12 = 0:5) = 1:61 that is not equal to cv (j12 = 0) = 1:51 reveals that eect of intercorrelation does exist on the mean of outcome variable. This statistical interaction theoretically can not be detected by the test for synergism of Prentice and Kalbeisch (1988). Its eect on regression parameters and regression quantiles give the same conclusion. The comparison results shown in Table 1 also indicate that the approach of Neyman-Rubin causal model is appropriate for assessment of regression interaction.. 3. The Neyman-Rubin's Causal Model for Interaction Analysis. The classical versions giving regression interactions falling short of formalism necessary for rigorous logical analysis. The usage of Neyman-Rubin's causal model of two treatment levels matches to measure clean eect of intercorrelation for de ning statistical interaction.. De nition 3.1. (a) We de ne the following dierence + (x1 x2 ) = (x1 x2 j12) ; (x1 x2 j12 = 0) (x1 x2)0 2 R2 as the Neyman-Rubin's causal eect of intercorrelation for outcome quantity (x1 x2 ) where is vector true parameters..

(16) 9. (b) We say that Rubin's statistical interaction for outcome quantity (x1 x2 ) exists if there are (x1 x2) such that its causal eect of intercorrelation + (x1 x2 ) is not zero (vector). Here (x1 x2 j12 = 0) represents the no-interaction response surface for outcome quantity (x1 x2 ). Unlike many causal models in statistics are incomplete guides to interpreting data or for suggesting answers to particular policy questions (Heckman (2008)), this causal model clearly specify the mechanism determing how hypothetical interventions are implemented. We explore the interaction assessment in detail while the others can be done analogously. 2 , We denote 0 = 1222 ; 12 2 2 0+ () = 12(;y1212 + y2 1 ) 1 + 12(y1 22; y212 ) 2 1 0 2 0 2 2 1+ () = ; 12 (;y1212 + y21 ) 2+ () = ; 12 (y122 ; y2 12) : 1 0 2 0 This help in formulating the excess eects of several interaction quantities.. Theorem 3.2. The Neyman-Rubin's causal eect of intercorrelation for. some outcome quantities are: + (x x ) = (1 x x ) + ( ). (1) Conditional mean at (x1 x2): cm 1 2 1 2 + + + (2) Regression parameters: + () = (0 () 1 () 2 ())0. + (x x ) (3) Outcome variable: y+ (x1 x2 ) = cm 1 2 2 ;1 2 2 y1 y2 y1 . + (4) Conditional variance: cv () = 12 + 22 ;(y1 y2) 1 122 y2 p12 2 p (5) Regression quantile: + ( ) = (0+ ()+z ( cv (); cv (j12 = 0)) 1+ () 2+())0 (6) Conditional quantile: cq+ (x1 x2 ) = (1 x1 x2) + ( ) (7) Reference charts: rc+ () = f(1 x1 x2) + ( ) : 2 (0 1) (x1 x2)0 2 R2g. Proof. It is seen that y+ (x1 x2 ) = y(x1 x2 j12) ; y(x1 x2 j12 = 0). Then regression model (2.2) indicates that y+ (x1 x2 ) = (1 x1 x2)( (j12); + (x x ). The others are straight forward. (j12 = 0)) = cm 1 2 The parametrization for a Neyman-Rubin's causal model is novel that generates several advantages:.

(17) 10. (a) It makes no controversy of model dependence occurred in classical statistical interaction assessment (Mantel et al. (1977)). (b) The approach of causality analysis for interaction assessment resolves the concern by Greenland (1993) to measure clear eect of intercorrelation that makes the users unconfused for how much and in what direction we can learn from a data set. (c) It creates a framework for eect of the cause to be shown with proper perspective in theoretical analysis of causality (Holland (1986)). (d) The Neyman-Rubin's causal eect of intercorrelation model then is. y+ (x1 x2 ) = y(x1b x2 j12) ; y(x1a x2 j12 = 0). (3.1). where, like the classical Neyman-Rubin model, we can only measure one observation, here is y(x1b x2 j12) = y(x1b x2 ). Interestingly the parametrization leads the causality analysis requiring only estimators of distributional parameters . This makes this causality analysis much more easier than the classical Neyman-Rubin's causality analysis. For statistical inferences of unknown Neyman-Rubin's causal eect, we assume that we have a random sample (yi x10i x2i1)0 i = 1::: n from 0 y distribu1 y i P tional model (2.1), we denote sample means @ x1 A = n1 ni=1 @ x1i A, sam0 s2 s s 1 x2 0 y ; y x12i0 y ; y 10 i i y y1 y2 P ple covariance matrix @ s1y s21 s12 A = n;1 1 ni=1 @ x1i ; x1 A @ x1i ; x1 A . x2i ; x2 x2i ; x2 s2y s21 s22 2 2 2 0 ^ The mle of is mle = (y x1 x2 sy s1 s2 sy1 sy2 s12) . We de ne statistic ^+ (x1 x2 ) = + (x1 x2 ^mle) as the mle of Neyman-Rubin's causal eect of intercorrelation + (x1 x2 ). The asymptitic theory of this mle is a direct result of the asymptotic theory of the mle of .. Theorem 3.3. The random quantity n1=2(^ + (x1 x2 ); +(x1 x2 )) con-. x1 x2 ) V @ (x1 x2 ) where verges to N (0 ()) with () (x1 x2) = @ (@ @ + @ () is the partial derivative of + () with respect to and V is the @ Cramer-Rao lower bound of the regression parameters . +. +. 0.

(18) 11. We further denote 2 2 ^0+ = s12(;sy1ss212s + sy2 s1) x1 + s12(sy1ss22 s; sy2 s12) x2 1 0 2 0 2 2 ^1+ = ; s12(;sy1ss212s + sy2s1) ^2+ = ; s12 (sy1ss22s; sy2s12) : 1 0 2 0 We then have mles of the Neyman-Rubin's causal eects for some outcome quantities (parameters): + (x x ) = (1 x x ) ^+ (). (1) Conditional mean at (x1 x2): ^cm 1 2 1 2 (2) Regression parameters: ^+ () = (^0+ () ^1+() ^2+())0. + (x x ) (3) Outcome variable: y^+ () = ^cm 1 2 s2 s ;1 s 2 sy1 s2y2 y1 . + (4) Conditional variance: ^cv () = s21 + s22 ;(sy1 sy2) s 1 s122 s 12 y2 2 p p + + ^ ^ (5) Regression quantile: ( ) = (0 ()+z ( ^cv (j12); ^cv (j12 = 0)) ^1+ () ^2+())0. (6) Conditional quantile at (x1 x2): ^cq+ (x1 x2) = (1 x1 x2)^+ ( ) (7) Reference charts: ^rc+ = f(1 x1 x2)^+ ( ) : 2 (0 1) (x1 x2)0 2 R2 g. Here the Neyman-Rubin's causal eects for outcome variable and conditional mean are identical and their estimators also have the same asymptotic distribution. Advanced statistical inferences for Neyman-Rubin's causal ef+ fect + (x1 x2 ) can be developed when partial derivative @ (x@1 x2 ) is derived. The partial derivatives of causal eect for regression parameters and it for conditional variance help the formulation of asymptotic distribution of mle's of some causal eect estimators. For designing tests and evaluations of them, we list the derived matrices of partial derivatives in Appendix: 0 + 1 @ 0 () @ + (a) Partial derivative of + ( 12) w.r.t. 0: @ @() = B @ @ @1++() CA a 3 9 @ 2 () @ 0. 0. 0 0. matrix (see (a1) of Appendix A). + () cv (b) Partial derivative of cv+ () w.r.t. 0 : @@ (see (b1) of Appendix B). + (c) Partial derivative of Conditional quantile w.r.t. 0: @ @() (see (c1) of Appendix C) + For partial derivative of conditional mean w.r.t. 0 , it is @cm (@x1 x2 ) = + (1 x1 x2) @ @() 0. 0. 0. 0.

(19) 12. 4. Statistical Inferences for Neyman-Rubin's Causal Eect of Intercorrelation. Assessment of statistical interaction on any outcome quantity (x1 x2 ) can be done by statistical inferences for its Neyman-Rubin's causal eect + (x1 x2 ). We consider a simulation with signi cance level = 0:05 to verify the power of the mle based test. Suppose that we have test statistic T for testing hypothesis H0 : + (x1 x2 ) = 0 vs H1 : + (x1 x2 ) 6= 0 and T j represents its value at j th replication. We then de ne the power function as m X 1 p = m I (T j t ) j =1. where t is the simulated constant so that the probabilities at various designs to be close to . In our studies, we let m = 10 000 and choose = 0:05. + (x x ) = First we consider conditional mean by testing hypothesis H0 : cm 1 2 + (x x ) ^ cm @ 1 2 + ^ and V = V^mle be mle's of 0 vs H1 : cm(x1 x2 ) 6= 0. We let @ + (x x ) @cm 1 2 and V . A test for this hypothesis is de ned below: @ 0. 0. Test 1: rejecting H0 if. 1=2 + q @^ (nx xj ^cm)) (x1@^ x2)(xj x )) t : + cm. 1. @. 0. 2. V^. + cm. 1. @. 2. The simulated powers when (x1 x2) = (2 2) are displayed in Table 2.. Table 2. Power performance for interaction detection by Test 1 n n = 30 n = 50 n = 100 12 = 0 0:048 0:05 0:052 12 = ;0:2 0:081 0:101 0:165 12 = ;0:5 0:299 0:46 0:733 12 = ;0:8 0:701 0:882 0:988 12 = 0:2 0:078 0:101 0:161 12 = 0:5 0:196 0:361 0:683 12 = 0:8 0:391 0:669 0:967 This test involves asymptotic variance estimate of interaction that is inuence by all parameters. We may consider the partial inuence of it inuenced by only covariance 12..

(20) 13. We consider the test constructing the test statistic based on the sacle of variance change due to parameter 12 that de nes the following test: Test 2: rejecting H0 if. 1=2 + r ^ n j ^cm(x1^ x2)j. + (x x ) + (x x ) @cm 1 2 1 2 V^ @cm@ @12 12. t. 0. ^ cm (x1 x2 ) ^ (j12 ) @ (j12 ) where @ = (1 x1 x2) @ @12 @12 with @12 displaying in Appendix (a2). Hopefully this tset is more sensitive in detection a change in covariane. +. +. +. Table 3. Power performance for interaction detection by Test 2. n n = 30 n = 50 n = 100 12 = 0 0:049 0:052 0:050 12 = ;0:2 0:080 0:110 0:169 12 = ;0:5 0:233 0:429 0:719 12 = ;0:8 0:428 0:815 0:988 12 = 0:2 0:077 0:108 0:167 12 = 0:5 0:234 0:420 0:710 12 = 0:8 0:434 0:817 0:987 It shows that the test considering partial derivative with respect to 12 only does improve the power a bit. We may also interest in assessment of statistical interaction for the average conditional mean on a region A for variables (x1 x2) as cm(Aj) = +. Z. A. + cm (x1 x2 )f12(x1 x2 )dx1dx2:. One interesting unknown quantity to be veri ed is the averaging Neyman+ + (R2 j ) which can be shown to be the Rubin's causal eect cmave = cm Neyman-Rubin's causal eect for conditional mean at mean vector (1 2) + + ( ). We would not go further to investigate it in as cmave = cm 1 2 simulation but we will study it in data analysis. We now consider a test for hypothesis of Neyman-Rubin's causal eect for conditional variance. For testing hypothesis H0 : cv+ () = 0 vs H1 : cv+ () 6= 0 for existence of statistical interaction for conditional variance, we.

(21) 14. de ne ^cv+ () = cv+ (^mle) and the test is de ned as rejecting H0 if. pnj ^+ ()j. q @^. cv + () V^ @ ^cv @ @ +( ) cv. t. 0. Table 4. Power performance for statistical interction for conditional variance. n n = 30 n = 50 n = 100 12 = 0 0:051 0:056 0:049 12 = ;0:2 0:116 0:165 0:247 12 = ;0:5 0:378 0:592 0:844 12 = ;0:8 0:749 0:932 0:997 We next consider the test for detection of statistical interaction on the regression quantile ( ). We test hypothesis H0 : + ( ) = 0 vs H1 : + ( ) 6= 0 by setting the following test: ^+ ^+ ( ) ;1 ^+ rejecting H0 if n^+ ( )0( @ @( 0 ) V^ @ @ ) ( ) t. Table 5. Power performance for statistical interction for regression quantile. n n = 30 n = 50 n = 100. = 0:7 12 = 0 0:055 0:048 0:048 12 = ;0:2 0:070 0:093 0:160 12 = ;0:5 0:267 0:440 0:718 12 = ;0:8 0:655 0:855 0:989. = 0:8 12 = 0 0:053 0:049 0:054 12 = ;0:2 0:069 0:088 0:145 12 = ;0:5 0:253 0:417 0:724 12 = ;0:8 0:662 0:861 0:992. = 0:9 12 = ;0:2 0:056 0:073 0:143 12 = ;0:5 0:207 0:396 0:704 12 = ;0:8 0:615 0:853 0:988 The power performance of assessing statistical interactions for conditional variance and regression quantile by testing for hypotheses for their corresponding Neyman-Rubin's causal eects shows these tests are satisfactory..

(22) 15. We consider a real data analysis for further interpretation. The sales data of size 25 from 25 territories for 1994 is available in Dielman (1996) that includes the sales (response variable) y (in US$1000), the amount the company spent on advertising (explanatory variables x1 ) and the total amount of bonuses paid (x2). The estimated regression model by least squares method computed in Dielman (1996) is. y^ = ;516:4 + 2:47x1 + 1:86x2 with R2 = 85:5%. Management department of the Meddicop Company concerned if the cause of bonuses (x2) paid in 1994 to salesmen is related to the response of sales. We further evaluate if correlation plays a role of aecting the sale outcomes (y) by computing the estimated sales ^cm(x1 x2 ) = (1 x1 x2) (^mle j12) for (x1 x2) in territories 1, 2, 3 and 4 (T1 T2 T3 and T4 ) based on mle's of ^ except that covariance 12 is replaced by some speci ed values including its s12. The estimated mean sales (^ cm(x1 x2 )) are listed in Table 6.. Table 6. Predicted mean sales for rst four territories (x1 x2) = 12 (;1:89 ;1:59) (;1:43 ;1:40) (;1:35 ;0:17) (;0:892 0:522) 12 = 0 673:935 792:929 969:53 1153:362 12 = ;0:2 529:926 674:309 913:38 1146:779 12 = ;0:5 94:089 318:129 729:952 1109:164 12 = ;0:8 ;1655:61 ;1107:133 ;28:724 928:929 12 = 0:2 768:354 871:874 1000:722 1150:307 12 = 0:5 857:058 950:334 1009:448 1120:455 12 = 0:8 887:623 1000:22 902:930 966:567 s12 = 0:419 837:937 932:435 1012:292 1133:086 The predicted mean sales with 12 = 0 present the outcome results of nointercorrelation. Then the predicted mean sales for 12 6= 0, in terms of (x1 x2), are dierent from it for 12 = 0 and regression parameters () for 12 6= 0 are also dierent vectors from it of 12 = 0. These results show that intercorrelation does making inuence on conditional mean and regression parameters. We then are appropriate to consider them as potential outcome quantities..

(23) 16. Table 7. p-values of observation points Obs # 1 2 3 4 5 6 7 8 9 10 11 12 13. Z-test 0:019 0:021 0:109 0:402 0:381 0:020 0:032 0:108 0:030 0:359 0:036 0:750 0:024. t-test 0:028 0:030 0:122 0:410 0:390 0:028 0:043 0:121 0:040 0:368 0:046 0:753 0:034. Obs # 14 15 16 17 18 19 20 21 22 23 24 25. Z-test 0:028 0:028 0:818 0:895 0:215 0:024 0:080 0:430 0:053 0:075 0:055 0:429. t-test 0:038 0:038 0:820 0:896 0:227 0:034 0:093 0:438 0:065 0:088 0:067 0:436. 5. The Neyman-Rubin's Causal eect of Intercorrelation for Binary Variable. Ai and Norton (2003) considered the second-order interaction veri cation for probit and logic regression models. It is interesting to see if we can assess eect of intercorrelation for binary response variable. It is very common that the categorical dependent variable is observed from categorization of a continuous explanatory variable while Prock et al. (2004) reported that 84% of epidemiological papers from leading journals made categorization of continuous variables. This categorization is widespread from epidemiology to other areas such as psychology (MacCallum, et al. (2002)) and marketing (Irwin and McClelland (2003)). With categorization, we are allowed to apply parametrization to assess statistical interaction for categorical dependent variables. Again, we let Y X1 and X2 be continuous random variables with a joint distribution. One categorization is to set a binary variable I (Y ) with outcome quantity as the regression function of the conditional mean de ned as p(x1 x2) = E (I (Y )jX1 = x1 X2 = x2), a probability as a function of (x1 x2). For assessment of statistical interaction, most applied scientists consider the model-dependent outcome quantity in the framework of logistic.

(24) 17. regression as. plog (x1 x2) = 1 + e;( 0 + 1 x11+ 2 x2 + 12 x1 x2 ). (5.1). or of probit regression as. pprob (x1 x2) = !(0 + 1 x1 + 2 x2 + 12x1 x2 ). (5.2). where !(:) is the distribution function of the standard normal distribution and consider a test for hypothesis H0 : 12 = 0. Now, we also consider that these random variables follow the normal distribution of (2.1). We then easily obtain the following theorem.. Theorem 5.1. The outcome quantity of conditional mean for binary variable I (Y ) under normality assumption (2.1) is p (x x ) = !( ; (1p x1 x2) () ) (5.3) cat. 1. 2. cv (). called the probit outcome quantity, where regression coecients 0 () 1() and 2 () and error conditional variance y2jx1 x2 are denoted in Theorem 2.1. The true outcome quantity under normality assumption in (5.3) indicates that either probit one with product term of (5.2) and logistic one are all inappropriate. Following Theorem 3.3, we have the following theorem.. Theorem 5.2. The Neyman-Rubin's causal eect of intercorrelation for probit outcome quantity pcat (x1 x2) is. p+cat (x1 x2) = !( ; (1p x1 x2) (j12) ) ; !( ; (1p x1 x2) (j12 = 0) ): cv (j12) cv (j12 = 0) The mle of the Neyman-Rubin's causal eect of intercorrelation is ^ ^ p^+cat (x1 x2) = !( ; (1p x1 x2) (j12) ) ; !( ; (1p x1 x2) (j12 = 0) ): ^cv (j12) ^cv (j12 = 0) that leads to the following theorem..

(25) 18. Theorem 5.3. The random quantity n1=2(^p+cat (x1 x2) ; p+cat (x1 x2)) con+ + (x1 x2 ) verges to N (0 @pcat@(x1 x2 ) V @pcat@ ) with 0. @p+cat(x1 x2) = ;( ; (1p x1 x2) (j12) ) @0 p (j )(1 x x ) @ (j12 cv) (+j( 12;) (1 x x )(j )) 1 @cv (j12) cv 12 1 2 1 2 12 2 @ @ cv (j12) + ( ; (1p x1 x2) (j12 = 0) ) (j12 = 0) p (j = cv0)(1 j12 =0) + ( ; (1 x x ) (j = 0)) 1 @cv (j12 =0) x1 x2) @ (@ 1 2 12 cv 12 2 @ cv (j12 = 0) 0. 0. 0. (j12 ) where @cv@ and @cv (@j12 =0) are displayed in Appendix (b2) and (b3) j12 ) and @ (j12 =0) are displayed in Appendix (a3) and (a4). and @ (@ @ Dierent underlying distribution or binary variable results dierent outcome quantity and their corresponding Neyman-Rubin's causal eect of intercorrelation. We would not go further on it. 0. 0. 0. 0. 6. Concluding Remarks. Whether or not eects of explanatory variables are intercorrelated are frequently assessed with ambiguous and controversial concept of statistical interaction give the practitioners limited and confused view of the nature of interaction in statistical world. We attempt here to elucidate some of the controversial issues surrounding the concept of statistical interaction with systematic introduction of de nitions, methods and theorems to build the Neyman-Rubin's causality analysis for assessment of interaction eect of intercorrelation. The parametrization of constructing regression model formulated from a multivariate distribution brings a theoretical foundation in connecting a causal model with the interaction cause of intercorrelation parameter that allows us to measure the eect of intercorrelation. Hopefully this would be recognized to have made permanent contribution in assessment of statistical interaction. We have several further conclusions: (a) Suppose that the conditional mean of the response variable y given. 0.

(26) 19. X1 = x1 and X2 = x2 is an exponential function as ()x1x2 g (x1 x2) = expf0 () + 1 ()x1 + 2 ()x2 + 12 () are function of where regression parameters j () j = 0 1 2 and 12 distributional parameters . Let us evaluate the approach of Ai and Norton (2003) considering the cross derivative as. @ 2 (x1 x2) = (x x )" () + ( () + ()x )( () + ()x )] 1 2 2 1 12 1 12 2 12 @x1@x2 for assessing statistical interaction. This interaction parameter involves complicated function of distributional parameters that is entirely dependent on intercorrelation parameter only if all regression parameters depend on covariance between X1 and X2 are zeros. This is generally not true. (b) Any utility function g(Y ) that its explicit form of conditional mean g (x1 x2 ) = E (g(Y )jX1 = x1 X2 = x2 ) available serves an interaction parameter for assessing its statistical interaction. Simpler form of other categorization such as I (Y ) I ( 1 Y 2) and I (Y 1 or 2 ) are candidates. (c) (Prediction interval) Again, we are interesting in the prediction of response y0 when X1 = x10 and X2 = x20 are speci ed. The interest of interaction parameter is the socalled naive coverage interval (norm (x10 x20) ; z yjx1 x2 norm(x10 x20) + z yjx1 x2 ).. 7. Appendix Appendix A. (a1) The partial derivatives for elements of + () w.r.t. :. @0+ () = @0 ( 0 ;0;1 (y122 ; y212 ) + y121 0;1 (y112 + y212 ) + y222 0 d15 d16 d17 d18 d19 ).

(27) 20. where. d15 = ;0;1 y2 2 + 0;2 "(y122 ; y2 12)221 + (;y112 + y212)222 ] ; y141 1 y 2 ; 1 ; 2 d16 = ;0 y1 1 + 0 "(y122 ; y2 12)121 + (;y112 + y212)122 ] ; 42 2 2 1 ; 1 ; 1 2 2 d17 = 0 ";2 1 + 12 2] + 2 d18 = 0 "121 ; 1 2] + 2 1. 2. d19 = ;1 (y21 + y1 2 ) ; 2;2"(y12 ; y2 12)121 + (;y112 + y22)122 ] 0. 0. 2. 1. @1+ () = ( 0 0 0 0 ;;2 (y12 ; y2 12)2 + y41 d26 d27 d28 d29 ) 2 2 0 1 @0 where d26 = 0;1y1 ; 0;2 (y122 ; y2 12)12 d27 = 0;122 ; 1;2 d28 = ;0;1 12 d29 = ;0;1y2 + 20;2(y122 ; y2 12)12 @2+ () = ( 0 0 0 0 d d ;;1 ;12 ; ;2 d ) 35 36 12 39 1 0 0 2 @0 where d35 = 0;1 y2 + 0;2(y1 12 ; y212)22 d36 = 0;2(y1 12 ; y212)12 + y2 2;2 d39 = ;0;1 y1 ; 20;2 (y112 ; y2 12)12 + + ( ) w.r.t. : @ (j12 ) = (a2) The partial derivatives of elements of 12 @12 0 0 @ 0+() 1 08 @12 BB 0 @ 1+() CC @ 08 @+12 A 008 @ @2 12() 0G 1 j12 ) = @ G1 A (a3) The partial derivatives of elements of () w.r.t. : @ (@ 2 G3 where 0. G1 = ( 1 ; y1 22;0y2 12 ; y2 12;0y1 12 0 g5 g6 g7 g8 g9 ) G2 = ( 0 0 0 0 h5 h6 202 ; 120 h9 ) G3 = ( 0 0 0 0 k5 k6 ; 120 102 k9 ).

(28) 21. with 2 2 2 2 g5 = ; y2 2 + (y1 2 ; y2212)1 2 + (y21 ; y2112 )22 0. 0. 0. 2 2 2 2 g6 = ; y1 1 + (y1 2 ; y2212)1 1 + (y21 ; y2112 )21 0 0 0 2 2 g7 = 122; 2 1 g8 = 121; 1 2 0 0 2 2 g9 = y21 + y12 ; 2 (y12 ; y212 )112 +2 (y2 1 ; y112)2 12 0 0 2 ) 2 2 2 ( ; ( ; h5 = y2 12 2 y1 2 2 h6 = y1 + y2 12 2y12 )1 0 0 0 2 h9 = ; y2 + 2 (y1 2 ;2y2 12)12 0 0 2 2 2 2 ( ; k5 = y2 + y1 12 2 y21 )2 k6 = (y1 12 ;2y21 )1 0 0 0 2 k9 = ; y1 + 2 (y21 ;2y112 )12 0 0 @ (j12 =0) 0(a4)H The 1 partial derivatives of elements of () w.r.t. 0 if 12 = 0: @ = @ H12 A where H3 H1 =( 1 ; y121 ; y222 0 y114 1 y224 2 ; 112 ; 222 0 ) H2 =( 0 0 0 0 ; y141 0 114 0 0 ) H3 =( 0 0 0 0 0 ; y242 0 124 0 ) 0. Appendix B. (b1) The partial derivative of the causal eect for conditional variance + () 2 y2 12 2y2 ; cv w.r.t. 0 , cv+ (), is @@ = (0 0 0 0 t5 t6 21y21 ; 2y1 2 +2 0 22 2 2y1 12 ;2y2 1 t9) where 0 2 2 2 2 + 2y1 y212 22 ; y221222 t5 = ; y41 + y2 + y1 2 02 0 1 2 2 2 2 2 + 2y1y212 12 ; y2214 t6 = ; y42 ; y1 + y1 1 2 02 0 2 2 ; 2 2 2 2y2122 12 + 4y1y2 12 2 y 1 y 2 y2 1 12 t9 = ; ; 2 0 0 0.

(29) 22. (j12 ) = ( 0 0 0 1 u5 u6 u7 u8 u9 ) (b2) The partial derivative of cv () w.r.t. 0: @cv@ with 0. 2 2 2 ; 2y1 y212 22 + y221222 u5 = ; y2 + y1 2 02 0 2 2 2 2 ; 2y1y212 12 + y2214 u6 = ; y1 + y1 1 2 02 0 2 2 u7 = 2 y2 12; y12 u8 = 2 y1 12; y21 0 0 2 2 2 + 2 2 ; 2y1 y212 y2 1 12 u9 = 2 y1 y2 ; 2 y1 2 12 2 0 0 2 2 (b3) @cv (@j12 =0) = ( 0 0 0 1 y141 y242 ;2 y121 ;2 y222 0 ) 0. Appendix C (c1) @ @) = @ @ +(. +( ). 0. 0. E = (0 0 0. 0E1 + z @ 009 A where 009. (. 1 1 2 (cv (j12 ))1=2. ; (cv (j121 =0))1=2 ) e5 e6 e7 e8 e9 ). with 2 y21 )2 ; e5 = 21 ( ((y 12(;jy))2112 =2 14( cv (j12 = 0))1=2 ) cv 12 y22 ; y2 12)2 ; e6 = 21 ( ((y 112 24( cv (j12 = 0))1=2 ) cv (j12))1=2 2 e7 = y(2 12(;jy))11=22 + 2( (jy1 = 0))1=2 0. cv. 12. 1. cv. 12. e8 = y(1 12(;j )) + 2( (jy2 = 0))1=2 0 cv 12 2 cv 2 e9 = (y212 ;( (j)( y))11=122 ; y2 1 ) : y2 12 1=2 12 y1 22 cv. 12. A 0 36 (c2) Cramer-Rao lower bound of the regression parameters : V = 063. B.

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