中介變項統計分析

國 立 交 通 大 學

統計學研究所

碩士論文

中介變項統計分析

Statistical Mediation Analysis

研 究 生：陳昱均

指導教授：陳鄰安 教授

中介變項統計分析

Statistical Mediation Analysis

研 究 生：陳昱均 Student: Yu-Chun Chen

指導教授：陳鄰安 博士 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 2014

Hsinchu, Taiwan, Republic of China

中華民國一百零三年六月

I

研究生：陳昱均

指導老師：陳鄰安 博士

國立交通大學統計學研究所碩士班

摘要

我們觀察到，傳統中介變項的分析中的直接影響力和中

介質有關，而我們認為的直接影響力是不經由中介質所產生

的。而後，我們提出了一個新的中介變相分析，此分析中的

影響力可分解出直接影響與間接影響，其中新的直接影響是

不透過中介值的。我們新的假設檢定力相較於傳統 Baron 和

Kenny 的假設檢定力是有明顯改善。最後我們將中介變項的

分析和交互作用做聯結。

關鍵字：中介變項；中介值；分解效應；交互作用。

II

Statistical Mediation Analysis

Student: Yu-Chun Chen

Advisor: Dr. Lin-An Chen

Institute of Statistics

National Chiao Tung University

Abstract

We observe that the classical mediation analysis gives the direct

effect involving the inputs of mediated variable leading that the direct

effect and indirect effect do not serve their roles appropriately. We then

propose a new mediation analysis with interaction identification for

determining a clean direct effect for defining the total effect and then the

effect decomposition. Power comparison between the Baron and Kenny’

s test and a new test based on our approach for indirect effect detection

has been done and their corresponding efficiencies for detection are

displayed. A new indirect effect proportion then is proposed for further

investigation.

Key words:

Mediation; Mediator; Effect Decomposition; Interaction.

  III   感謝陳鄰安老師，讓我在大三的時候重新認識統計學，統計不再是高中時死 記公式，因為陳鄰安老師教學方式善用例子帶入理論，讓學生可以清楚了解與應 用，這讓我對統計學產生了濃厚的興趣，也很幸運地錄取了交大統計學研究所就 讀。 在研究所學習以及論文我要再次感謝陳鄰安老師，謝謝老師循序漸進地指導 及糾正，當我碰到問題時給我提點，幫我一起找出問題的癥結處進而感改善。除 了在專業統計領域方面給了我很多的意見，也時常和我分享他的人生經驗，在我 對未來彷徨時給了我很多建議和方向，當碰到社會議題和運動時，時常會在街頭 和論壇看到老師正義的身影。我也要謝謝口試委員許文郁老師、蕭金福老師和洪 慧念老師給我在論文上的指正，使我的論文更加完善。 接著更要感謝統研所的同學及學長姐，在課業上，大家一起討論作業、準備 考試報告，感謝書卷多多(育駿)熱心毫不保留地教我們。除了課業，研究所的生 活娛樂也是重要的一部分，感謝揪團王小紅(姿琪)安排出遊，感謝一起跟團的夥 伴們明燕、佑珊、多牛、翔之、奇煒、民翰，有了大家的笑聲旅行更加精采，感 謝皇儲濛濛亮(青樺)不吝嗇地和我們一起嬉笑玩耍。感謝兩位班代不辭辛勞的處 理了很多班上的瑣事，感謝小強(家瓏)一路陪伴我，感謝所有的統研所的同學有 大家的歡笑和崩潰聲生活不再那麼無趣。 最後感謝我的家人，讓我可以全心投入研究所學習，尊重我在求學過程中的 選擇，謝謝家人的支持才有現在的我。 陳昱均 謹誌于 國立交通學統計學研究所 中華民國一零三年六月

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.

Statistical theory for Classical Effect Decomposition Methods

2.1. Verification of An New Baron and Kenny’s Conditions···3

2.2. Statistical Properties for Estimators of Baron and Kenny’s effects···9

3.

A New Meditation Analysis ···11

4.

Statistical Interaction Identification·· ···16

5.

Concluding Remarks· ···19

References ···20

!

!

!

!

!

!

!

Table 1. Power performance for refined and classical indirect effect detection ····7

Table 2. Total and indirect effect···8

Table 3. Total effect ratio and indirect effects of new decompositions ···14

Table 4. Power performance for B-K condition test and test for presence of clean

indirect effect···16

Table 5. Full effect decomposition···19

!

!

Statistical Mediation Analysis

Abstract

We observe that the classical mediation analysis gives the direct effect involv-ing the inputs of mediated variable leadinvolv-ing that the direct effect and indirect effect do not serve their roles appropriately. We then propose a new mediation analysis with an interaction identification for determining a clean direct effect for defining the total effect and then the effect decomposition. Power comparison between the Baron and Kenny’s test and a new test based on our approach for indirect effect detection has been done and their corresponding efficiencies for detection are displayed. A new indirect effect proportion then is proposed for further investigation.

1. Introduction

Since Woodworth (1928), effect decomposition (mediation analysis), with de-compose the total effect of an exposure (independent) variable on the response (effect) variable into the effect that go directly (direct effect) and the effect that is influenced by a mediator variable (indirect effect), has been extensively stud-ied and used in psychological science for more than 80 years. It is now also very popular in social science (Geneletti (2007)) and fast growing in medical and epi-demiology studies relevant to the design of clinical and public health interventions (Laan and Petersen (2008) and Richiardi, Bellocco and Zugna (2013)).

In many studies of estimating and testing the mediation effects, Baron and Kenny (1986) proposed the causal steps regression approach that requires con-ditions for establishing mediation. Because of its simplicity for understanding and implementing, this approach is very influential and widely used. Once these conditions are satisfied, it needs to quantify the indirect effect to be tested for significance (Sobel (1982)). There are criticisms for this traditional approach such as bias effect estimates and without natural extension to non-linear models are observed (Robins and Greenland (1992) and Richiardi, Bellocco and Zugna (2013)). In a Monte Caro simulation by MacKinnon et al. (2002), it is ob-served that the condition requiring that response and exposure variables has to

be correlated is not correct indicating that this approach may miss some true mediation effects. This problem has also been tackled extensively in the causal inference literature by using counterfactual framework (Robins and Greenland (1992), Pearl (2001), Robins (2003) and van der Lann and Petersen (2008)). This approach is also debatable for that it involves many untestable assumptions (Geneletti (2007)).

Although considerable effects has been devoted to the possibility of undesired behavior in statistical inference for Baron and Kenny’s conditions and induced inferences methods for indirect effect detection, there is no satisfactory alterna-tive mediation analysis technique for use. From our view, the disadvantages are resulted from the fact that we have not known enough to the unknown mech-anism of cuasual relationship for creation of these effects. We propose a para-metric study of mediation analysis by introducing the underlying distribution of involved random variables into the regression framework, not been treated in this field, that allows us to structure analysis in two parts. First, for this tra-ditional mediation analysis, the correct conditions for presence of indirect effect can be drawn theoretically. We observe that significance for correlation between exposure and mediator variables is the only condition to be satisfied when the underlying distribution is true. This indicates that conditions for this presence must be case by case and conservative approach of maximizing the number of conditions in the classical one is not surprised in sacrificing its power perfor-mance. We then propose a corrected version of Baron and Kenny’s approach for effect decomposition and inference methods development. As a consequence of parametric study, the unknown total, direct and indirect effects can be estimated with best asymptotically normal estimators and tests for statistical hypotheses of these effects can be developed with the derived asymptotic distributions.

Second and most importantly, the traditional approach interprets the expo-sure coefficient for (multiple) regression model with conditioning on values of exposure and mediator variables as a direct effect. But this parametric approach shows that the size of this exposure coefficient is dependent on the true values of the distributional parameters of mediator variable indicating that statistical mechanism for mediation not only through variable’s given values but also its

3

parameters. This traditionally unaware unclean direct effect then induce the indirect effect also unclean. We then develop a refined version of Baron and Kenny’s approach for developing conditions for clean indirect effect and use them to construct inference methods for this indirect effect.

2. Statistical Theory for Classical Effect Decomposition Methods 2.1. Verification of A New Baron and Kenny’s Conditions

Suppose that we have response (effect) variable Y , exposure (independent) variable X and mediator (intermediate) variable M .

The paths that exposure variable X and mediation variable M affect the response variable Y can be described in the following figure.

Effect decomposition (mediation analysis) involves to identify the total effect of X on Y , the part of total effect because X influences M which in turn in-fluences Y (indirect effect) and the effect of X unexplained by this variables M (direct effect). The direct and indirect effect together form the total effect of X on Y . The approach of Baron and Kenny (1986) for mediation analysis is most widely-used that considers a series of tests for regression coefficients of all paths (regression models) for inferencing the existence of indirect effect that is summarized by Howell (2009) as follows:

Step 1. Test hypothesis H0a : β1a = 0 vs H1a : β1a ̸= 0 for significance of the

simple linear regression model

y(x) = β0a+ β1ax + ϵa (2.1)

describing the path (X → Y ) requiring H0a to be rejected to confirm that X is

Step 2. Test hypothesis H0b : β1b = 0 vs H1b : β1b ̸= 0 for significance of the

simple linear regression model

m(x) = β0b+ β1bx + ϵb. (2.2)

describing the path (X _{→ M → Y ) requiring H}0b to be rejected to confirm that

X is a significant predictor of the mediator M .

Step 3. Consider the following multiple linear regression model

y(x, m) = β0c+ β1cx + β2cm + ϵc. (2.3)

Performing the test for hypothesis H0c : β2c = 0 vs H1c : β2c ̸= 0 requiring

H0c to be rejected to confirm that the partial effect of M must be significant.

The effect relationships among variables following this series of tests may be explained in the followings (Howell (2009) and Hayes (2009)):

(a) If it shows significant evidence to reject H0a in step 1, it defines β1a as total

effect for possible decomposition into direct and indirect effects.

(b) If one or more hypothesis in steps 1 - 3 are not rejected, researchers usually conclude that indirect effect does not exist.

(c) If three hypothesis in steps 1 - 3 are rejected, indirect effect exists. If hypoth-esis H0c : β1c = 0 vs H1c : β1c ̸= 0 is not rejected, there is complete mediation

and if it is rejected, there is partial mediation.

We consider here the problem that how many conditions is required for pres-ence of indirect effect. With decision error generated when a hypothesis is tested, the more tests in order to claim an indirect effect, the more errors to be generated. This situation of lower power (Fritz and MacKinnon (2007) and MacKinnon et al. (2002)) could be even worse when number of exposures or mediators increases. Our concern is correct since as observed by MacKinnon et al. (2002) with Monte Carol simulation that the condition that Y and X has to be correlated is not correct.

Within the framework of series of regression models (2.1)-(2.3), it is seen the following coefficient decomposition

5

holds when the underlying distribution of variables Y, X and M are jointly nor-mal. Then, following the path analysis, usage of models leads to the following effects identification (decomposition):

Total effect: TBK = β1c+ β1bβ2c

Direct effect: DBK = β1c (2.5)

Indirect effect: IDBK = β1bβ2c

satisfying that the sum of direct effect and indirect effect is equal to the total effect.

Now suppose that Y, X and M has a joint normal distribution as ⎛ ⎝XY M ⎞ ⎠ ∼ N3( ⎛ ⎝µµyx µm ⎞ ⎠ , ⎛ ⎝ σ 2 y σyx σym σxy σx2 σxm σmy σmx σ2m ⎞ ⎠). (2.6)

for verification of Baron and Kenny’s conditions. Denote β0(θ) = µy − (σyxσm2 − σymσxm)µx σ2 xσm2 − σxm2 + (σyxσxm− σymσ 2 x)µm σ2 xσ2m− σxm2 β1(θ) = σyxσm2 − σymσxm σ2 xσm2 − σ2xm , β2(θ) = σymσx2− σyxσxm σ2 xσm2 − σxm2 where θ = (µy, µx, µm, σ2y, σx2, σ2m, σyx, σym, σxm).

The following theorem gives parametrized regression parameters of models (2.1)-(2.3).

Theorem 2.1. Suppose that the underlying distribution is normal.

(a) The regression parameters for regression function of Y given X = x of (2.1) includes β0a= µy − σyx σ2 x µx, β1a = σyx σ2 x and ϵa ∼ N(0, σa2) where σa2 = σ2y − σ2yx σ2 x .

(b) The regression parameters for regression function of M given X = x of (2.2) includes β0b = µm− σmx σ2 x µx, β1b = σmx σ2 x

and ϵb ∼ N(0, σ2b) where σ2b = σm2 − σ

2 mx

σ2 x .

(c) The regression function for Y given X = x and M = m is

µ(x, m; θ) = β0(θ) + β1(θ)x + β2(θ)m (2.7)

Hence β0c = β0(θ), β1c = β1(θ), β2c = β2(θ). and ϵc ∼ N(0, σc2) where σ2c =

σ_y2− (σyx, σym) % σ2 x σxm σmx σ2m &−1% σyx σym & .

Proof. The result in (c) are induced from Chen et al. (2013) and the others are trivial. !

We now are ready to give a theoretical verification of Baron and Kenny’s conditions for existence of indirect effect and mediation.

Theorem 2.2. Suppose that the underlying distribution is normal. The effects can be decomposed following the Baron and Kenny (1986)’s approach as

Total effect (β1c+ β1bβ2c) : TBK = (σymσx2− σyxσxm) σ2 xσm2 − σxm2 σxm σ2 x + (σyxσ 2 m− σymσxm) σ2 xσ2m− σxm2 =σyx σ2 x Direct effect (β1c) : DBK = σyxσm2 − σymσxm σ2 xσm2 − σ2xm (2.8) Indirect effect (β1bβ2c) : IDBK = (σymσx2− σyxσxm) σ2 xσm2 − σxm2 σxm σ2 x

We first examine the Baron and Kenny’s condition for presence of indirect effect.

Theorem 2.3. The indirect effect under the Baron and Kenny’s conditions are:

IDBK = ⎧ ⎨ ⎩ σymσxm σ2 xσm2−σ2xm if σyx = 0 0 if σxm = 0 σyxσ2m σ2 xσm2−σ2xm if σym = 0

Proof. It is straight forward but careful re-arrangements. !

This shows that the existence of partial mediation or complete mediation (Howell (2009)) does not require all hypothesis in three steps to be rejected.

7

This also confirms the observation of MacKinnon et al. (2002) that Y and X are not necessary to be associated but in addition that Y and M are not necessary for presence of indirect effect. Combining the results in (a) and (c), a new Baron and Kenny’s rule for indirect effect identification is:

Indirect effect IDBKexists if H0b for model (2.2) is rejected. (2.9)

We consider (µy, µx, µm) = (2, 3, 3) and σ2y = σx2 = σm2 = 2 to conduct a

Monte Carlo simulation to evaluate the powers of the classical Baron and Kenny’s three conditions test and the above new test for claiming an indirect effect. The simulated results are displayed in Table 1.

Table 1. Power performance for refined and classical indirect effect detection σxm σyx = σym B-K Revised B-K σxm = 0 0 0.0480 0.050 σxm = 0.2 0.3 0.049 0.081 0.7 0.060 0.081 0.9 0.076 0.081 σxm = 0.8 0.3 0.055 0.611 0.7 0.127 0.611 0.9 0.250 0.611 σxm = 1.0 0.3 0.055 0.825 0.7 0.131 0.827 0.9 0.256 0.826 σxm = 1.2 0.3 0.055 0.955 0.7 0.122 0.955 0.9 0.228 0.955 From our investigation, expressing effect decomposition in terms of distribu-tional parameters is desired for each specific underlying distribution for develop-ing correct conditions for improvdevelop-ing power performance for claimdevelop-ing an indirect effect.

When both direct and indirect effects are identified, a measure of the propor-tion mediated is sometimes calculated as the ratio of the indirect effect to the total effect (Ditlevsen, et al. (2005) and Hafeman (2009)). This measure in some

sense captures how important the pathway through the intermediate is in ex-plaining the actual operation of the effect of the exposure on the outcome. This implicitly assumed that all effects are positive values. In the following table, we present the total and indirect effect when the underlying distribution is normal. Table 2. Total and indirect effect

(σym, σyx) TBK IDBK σxm= 0.2 (0.6, 0.8) 0.4 0.026 (0.8, 0.2) 0.1 0.039 (0.8, 0.4) 0.2 0.038 (0.8, 0.8) 0.4 0.036 σxm= 0.6 (0.8, 0.2) 0.1 0.122 (0.8, 0.4) 0.2 0.112 (0.8, 0.6) 0.3 0.102 σxm= 0.8 (0.2, 0.6) 0.3 −0.010 (0.2, 0.8) 0.4 _−0.029 (0.6, 0.2) 0.1 0.124 (0.8, 0.2) 0.1 0.171 σxm=−0.2 (0.6, 0.6) 0.3 _−0.033 (0.6, 0.8) 0.4 −0.034 (0.8, 0.6) 0.3 _−0.043 (0.8, 0.8) 0.4 −0.044 σxm=−0.8 (0.2, 0.2) 0.1 −0.067 (0.4, 0.4) 0.2 _−0.133 (0.6, 0.6) 0.3 −0.200 (0.8, 0.8) 0.4 _−0.267 We have two comments:

(a) Indirect effect could be negative value when direct effect is larger than the total effect. Then the ratio of indirect effect is also negative when β1b is small or

direct effect is large enough.

(b) Indirect effect could have value larger than the total effect such that the ratio of indirect effect is larger than one when β1c < 0.

9

(2009)) if there is nonzero indirect effect. It is seen that it is not always dangerous for presence of mediation effect, depending on its sign.

Definition 2.4. We say that there is synergistic mediation effect if IDBK > 0

and antagonistic mediation effect if IDBK < 0.

2.2. Statistical Properties for Estimators of Baron and Kenny’s Effects Effects of (2.5) are generally estimated by least squares estimators of the corresponding parameters and hypothesis testing for effect parameters are done by assuming that the parameter estimators are normally distributed to develop the scale estimator of the effect estimator. For example, to deal with hypothesis of presence of indirect effect as H0 : β1bβ2c= 0, researchers have done (Sobel (1982),

Aroian (1944) and Goodman (1960)) to develop the asymptotic distribution of the product of two normal distributions of least squares estimators of β1b and

β2c. With our approach scale estimator of any effect estimator or their functions

are much easier to develop.

We denote the sample means (¯y, ¯x, ¯m)′ = _n1 *_i=1n (yi, xi, mi)′ and the sample

covariance matrix ⎛ ⎝ s 2 y syx sym sxy s2x sxm smy smx s2m ⎞ ⎠ = 1 n−1 *n i=1 ⎛ ⎝ xyii− ¯y− ¯x mi− ¯m ⎞ ⎠ ⎛ ⎝ xyii− ¯y− ¯x mi− ¯m ⎞ ⎠ ′ . Let ˆθ be the maximum likelihood estimator of parameters θ. The maximum likelihood estimators of direct and indirect effect are, respectively,

ˆ DBK,mle = syxs2m− symsxm s2 xs2m− s2xm ˆ IDBK,mle= (syms2x− syxsxm) s2 xs2m− s2xm sxm s2 x

The following theorem states the asymptotic distributional theory for the max-imum likelihood estimators.

Theorem 2.5. (a) We have n1/2_{( ˆD}

BK,mle − DBK) convergent in

distribu-tion to a normal distribudistribu-tion N (0, Σd) with asymptotic covariance matrix Σd = ∂DBK

∂θ′ Vθ∂D

′ BK

∂θ , where Vθ = −[E(∂2logφN(X, Y )/∂θ∂θ′)]−1 is the Cramer-Rao’s

lower bound for θ and φN(Y, X, M ) is the probability density function of the

nor-mal distribution for Y, X and M . Hence ˆDBK,mle(x) forms a best asymptotically

(b) We have n1/2( ˆIDBK,mle− IDBK) convergent in distribution to a normal

dis-tribution N (0, Σid) with asymptotic covariance matrix Σid = ∂ID_∂θBK′ Vθ∂ID

′ BK

∂θ .

Hence ˆIDBK,mleforms a best asymptotically normal estimator of unknown IDBK.

Optimal properties of the least squares estimators for direct and indirect effects are implied from the following theorem when normality assumption holds. Theorem 2.6. Let ˆβ0b and ˆβ1b be the least squares estimators of β0b and β1b.

Then, under the normality assumption, ˆDBK = ˆβ1c and ˆIDBK = ˆβ1bβˆ2c and

then they are also best asymptotically normal.

Proof. The least squares estimators _{{ ˆ}β0c, ˆβ1c, ˆβ2c} for regression model of (2.3)

are also maximum likelihood estimators for this regression model summing that ϵc is normal of zero mean and constant variance. On the other hand, the

max-imum likelihood estimator ˆθmle for distribution of Y, X and M in (2.6) makes

{ ˆβ0(θ), ˆβ1(θ), ˆβ2(θ)} = {β0(ˆθmle), β1(ˆθmle), β2(ˆθmle)} the maximum likelihood

estimator of {β0(θ), β1(θ), β2(θ)} for regression model (2.7) that has a normal

error variable. This indicates that _{{ ˆ}β0(θ), ˆβ1(θ), ˆβ2(θ)} and { ˆβ0c, ˆβ1c, ˆβ2c} are

identical which further implies that ˆβ1c = ˆβ1(θ) (direct effect) and ˆβ2c = ˆβ2(θ).

Analogous discussion for model of M given X = x can be done to finish the theorem. !

We have several comments from the above theorem:

(a) The least squares estimator of the product of coefficients to be best asymp-totically holds only occasionally. If the distribution of underlying distribution of variables involved is no-longer normal, the theory may be different. However, the optimality properties always hold for ˆDBK and ˆIDBK if they are derived from

the process stated in this paper for any underlying distribution.

(b) The test statistics based on least squares estimators of coefficients such as the commonly used one of Sobel (1982) and some others as Aroian (1944) and Good-man (1960) all assume that this product of least squares estimators is asymptoti-cally normal. However our theory verified that this is certain when the underlying distribution is normal but not certain for other situations.

11 H0 : IDBK(x) = 0 is: rejecting H0 if |n 1/2_IDˆ BK| + ˆ Σid ≥ tα γ = IDBK TBK

3. A New Mediation Analysis

Is the classical specification of total, direct and indirect effects statistical ap-propriate? When the variables Y, X and M follows the normal distribution (2.6), the direct effect and total effect are respectively as

DBK = σyxσm2 − σymσxm σ2 xσm2 − σxm2 and TBK = σyx σ2 x (3.1) If there is a unit change in X, it is expected that the direct effect measures only the direct (no-mediation) impact on response Y , unfortunately, the magnitude of this impact DBK involves distribution parameters {σym, σxm, σm2 } related to

mediator M indicating that this effect is mixed with effect of association between X and M . On the other hand, the total effect is supposed to contain effect of X mediated and not-mediated by variable M . Unfortunately this magnitude influ-enced by X TBK dose not involve distributional parameters related to mediator

M indicating absence of mediation. This classical effect specification does not give precise direct and indirect effects. This also indicates that the commonly used test statistics of Sobel (1982), Aroian (1944) and Goodman (1960) and many others may lead to in-correct conclusion for indirect effect. Beneficial from parametrization of regression function, uncontroversial specification of indirect effect can be specified with correct derivation of statistical interaction.

We assume that Y, X and M have a joint distribution with probability density function f (y, x, m, θ) where θ is vector of unknown parameters. We further as-sume that the parameter vector θ my be partitioned based on association between variables into no-association vectors _{θy, θx, θm}, two-variables association

vec-tors {θyx, θym, θxm} and three-variables association vector {θyxm} where θzj:j∈A

for parameters partition work for most interesting multivariate distributions. For examples, if the joint distribution of Y, X and M is multivariate normal or mul-tivariate t distribution, the parameter vector θ is vector of population mean vector

⎛ ⎝µµyx

µm

⎞

⎠ and covariance matrix ⎛ ⎝ σ 2 y σyx σym σxy σx2 σxm σmy σmx σ2m ⎞

⎠. Then we have the

no-association parameter vectors {θy, θx, θm} = {

% µy σ2 y & , % µx σ2 x & , % µm σ2 m & }, two-variables association parameter vectors{θyx, θym, θxm} = {{σyx}, {σym}, {σxm}}

and empty set for three-variables association parameters.

Let µ(x, m; θ) be the conditional mean of Y given X = x and M = m. We consider additivity of control values X = x and M = m for specification of total effects of explanatory and mediation variables.

Definition 3.1 We say that regression function is total effect decomposable if the regression function is

µ(x, m; θ) = g(θ) + Tx(θ)x + Tm(θ)m. (3.2)

In this case, we say that Tx(θ) and Tm(θ) are, respectively, the total effects of

exposure and mediator variables.

A common feature of decomposition in linear model of effects contributed by explanatory variables is that their combined effect (sum of separated effects) has to be the conditional mean of response variable Y . For example, the group mean in analysis of variance is the sum of main effects and interactions and, in multiple linear regression model, the regression function is the sum of univariate terms x1 and x2 (for main effects) and product term x1x2 (for interaction). The

specification of total effects in (3.2) does confirm this expectation. But the Baron and Kenny’s framework does not indicates this expectation.

If the total effect Tx(θ) involves distributional parameters of mediator M , it

contains effect mediated by M requiring for disentangling the interrelationships between variables from total effect decomposable regression function of (3.2). Denoting parameter sets θyx = {θx, θyx} and θym = {θm, θym}, obviously θyx

and θym respectively are sets of distributional parameters that are considered

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An application for interaction identification in Chen et al. (2013) is to used for direct and indirect effects identification.

Definition 3.2. (a)Suppose that there exists a minimal function of parameters G(θ), denoting its dimension as c, such that the regression function given G(θ) = 0c can be decomposed as

µ(x, m, θ_{|G(θ) = 0}c) = g(θ) + Dx(θyx)x + Dm(θym)m. (3.3)

We say that µ(x, m, θ_{|G(θ) = 0}c) is the no-interaction regression function

and G(θ) is the intercorrelation parameter set. We call Dx(θyx) and Dm(θym),

respectively, the direct effects of exposure and mediator variables.

(b) We call IDx = Tx− Dx(θyx) and IDm= Tm− Dm(θym) the indirect effects

of X and M .

(c) We say that there is no mediation effect if Tx = Dx(θyx) leading to IDx = 0.

The intercorrelation parameter set contributes the indirect effects of exposure X and mediator M . But IDx is pure indirect effect of exposure X.

Now, consider the normal distribution for variables Y, X and M of (2.6). Theorem 3.3. (a) Under the normality assumption, we have the total effects as

Tx = σyxσm2 − σymσxm σ2 xσm2 − σxm2 and Tm = σymσ2x− σyxσxm σ2 xσ2m− σxm2

Renoting Tx = T (θ), hence, Tx is the total effect.

(b) The ineraction parameter is G(θ) ={σxm} and the induced regression

func-tion is µ(x, m, ; θ) = µy+ σyx σ2 x (x− µx) + σym σ2 m (m− µm)

indicating that the direct effect is Dx = σyx σ2 x and Dm= σym σ2 m (3.4) which is a function of θyx only and the indirect effects are IDx = Tx− Dx with

IDx = σyxσm2 − σymσxm σ2 xσm2 − σ2xm − σyx σ2 x and IDm = σymσx2− σyxσxm σ2 xσm2 − σ2xm − σym σ2 m

Proof. The conditional mean of M given X = x is µ(x, θ) = µm+ σ_σxm2

x (x− µx)

that gives (a). !

This direct effect Dx = σyx/σx2 is identical to β1a of (2.1) that makes sense

for it measures the effect of X on Y in the environment that mediator is not involved. Similarly, Dm = σym/σm2 is identical to the regression coefficient β1d

in model y = β0d+ β1dm + ϵc that measures the effect of mediator M on Y in

the environment that X is not involved.

We denote the total effect as a ratio between the new total effect and classical total effect in the following

rt =

TBK

Tx

.

Table 3. Total effect ratio and indirect effects of new decompositions (σym, σyx) rt IDx σxm = 0.6 (0.4, 0.4) 0.765 _−0.046 (0.4, 0.6) 0.87 −0.036 (0.4, 0.8) 0.932 _−0.026 (0.6, 0.2) 0.1 −0.089 (0.6, 0.4) 0.6 _−0.079 (0.6, 0.6) 0.76 −0.069 σxm = 0.8 (0.6, 0.4) 0.45 −0.104 (0.6, 0.6) 0.7 _−0.085 (0.6, 0.8) 0.832 −0.066 (0.8, 0.2) 0.235 _−0.171 (0.8, 0.6) 0.553 −0.133

The total, direct and indirect effects adjusted with interaction are generally varying in the underlying distribution. The derived indirect effect allows us to es-tablish new Baron and Kenny’s conditions of mediation analysis for identification of clean indirect effect.

Theorem 3.4. Baron and Kenny’s conditions are true for σyx = 0 and σxm = 0.

In specific, we have the followings:

(a) When σyx = 0 we have IDx =−_σ2σymσxm xσ2m−σ2xm.

15 (c) When σym = 0 we have IDx = σyx( σ2_m σ2 xσm2 − σxm2 − 1 σ2 x )

This study gives a refined two steps tests of Baron and Kenny’s approach to identify the clean indirect effect as follows:

If hypotheses for model (2.2) is rejected, indirect effect exists. (3.5) This refined direct and indirect effects are no longer equal to β1c and β1bβ2c,

respectively. Hence, their asymptotic distributions are varied. The maximum likelihood estimators of direct and indirect effect may be defined as:

ˆ Dx = syx s2 x ˆ IDx = syx( s2_m s2 xs2m− s2xm − 1 s2 x ) Theorem 3.5. (a) We have n1/2_{( ˆD}

x−Dx) convergent in distribution to a normal

distribution N (0, Σd) with asymptotic covariance matrix Σd = ∂D_∂θ′xVθ∂D_∂θx, where

Vθ =−[E(∂2logφN(X, Y )/∂θ∂θ′)]−1 is the Cramer-Rao’s lower bound for θ and

φN(Y, X, M ) is the probability density function of the normal distribution for

Y, X and M . Hence ˆDxforms a best asymptotically normal estimator of unknown

Dx.

(b) We have n1/2_{( ˆID}

x−IDx) convergent in distribution to a normal distribution

N (0, Σid) with asymptotic covariance matrix Σid = ∂ID_∂θ′xVθ∂ID_∂θx. Hence ˆIDx

forms a best asymptotically normal estimator of unknown IDx.

Let ˆΣid be the maximum likelihood estimator of Σid. A new test for presence

of indirect effect based on estimator ˆIDx is as follows;

rejecting H0 if √n| ˆIDx|/

, ˆ

Σid ≥ t (3.6)

where t is the threshold assuring the size of the test is the significance level α. We define the efficiencies of the test based on Baron and Kerry’s condition and new test as

Ef fBK =

Power for B-K test

max{Powers for B-K test and new test}, Ef fN ew =

Power for new test

In the following, we display the efficiencies with α = 0.05.

Table 4. Power performance for B-K condition test and test for presence of clean indirect effect

σxm σyx = σym Ef fBK Ef fN ew σxm=−0.5 0.3 0.283 1 0.5 0.052 1 0.7 0.049 1 0.9 0.269 1 σxm=−1 0.3 0.953 1 0.5 0.874 1 0.7 0.846 1 0.9 0.835 1 σxm= 0.5 0.3 0.493 1 0.5 0.377 1 0.7 0.335 1 0.9 0.311 1 σxm= 1 0.3 1 0.174 0.5 1 0.320 0.7 1 0.440 0.9 1 0.549

Verify the situations that the indirect effect is negative showing that the direct effect needs not be smaller than the total effect.

4. Statistical Interaction Identification

It is a consensus that the statistical interaction represents the effect of inter-correlation between explanatory variables on the conditional mean of Y given the values of explanatory variables. Suppose that Y, X and M be respectively the response, explanatory and mediation variable. We denote µ(x, m; θ) the con-ditional mean of Y given values X = x and M = m. The direct effect, indirect

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effect and interaction effect must characterize are generally defined in the follow-ing way.

Definition 4.1. We consider X and M all explanatory variables.

(a) The direct effect of an explanatory variable is the change of mean function that is not mediated by the other one variable when this explanatory variable increases one unit and the other variable is held fixed.

(b) The indirect effect of an explanatory variable is the change of mean function that is purely mediated by other one variable when this variable increases one unit and the other variable is held fixed.

(c) The interaction effect is the the change of mean function when the explanatory variables X and M both increase one unit simultaneously.

How to measure these effects? The literature has a consistent treatment in a wider range of effect decomposition. Summarizing from the direct effect of Baron and Kenny (1986) and the second order interaction of Mullahy (1999) and Ai and Norton (2003), the direct effect and interaction are defined from derivatives of the mean function as follows:

(a) The direct effect of variable X: ∂µ_∂x. (b) The direct effect of variable M : _∂m∂µ.

(c) The interaction effect of variables X and M : _∂x∂m∂2µ . Example 1. (a) Suppose that we have a regression model

y = βa0+ βa1x + βa2m + ϵa.

We have the direct effects Dx = βa1 and Dm = βa2 and interaction effect

IAxm= 0.

(b) Suppose that we have a regression model

y = βb0 + βb1x + βb2m + βb3xm + ϵb.

We have the direct effects Dx = βb1 and Dm = βb2 and interaction effect

IAxm= βb3.

(c) Suppose that we have a regression model

We have the direct effects Dx = βc1 + 2βc2x and Dm = βc3 + 3βc4m and

interaction effect IAxm = βc5+ 4βc6xm.

The controversy of this classical way to characterize the interaction is that it is model dependent - various interaction to be specified when different models are applied.

Suppose now that Y, X and M have a joint normal distribution of (2.6). The regression model then is

y = β0(θ) + β1(θ)x + β2(θ)m + ϵ (4.1) where β0(θ) = µy − (σyxσm2 − σymσxm)µx σ2 xσm2 − σxm2 + (σyxσxm− σymσ 2 x)µm σ2 xσ2m− σxm2 β1(θ) = σyxσ 2 m− σymσxm σ2 xσm2 − σ2xm , β2(θ) = σymσ 2 x− σyxσxm σ2 xσm2 − σxm2

This is not model dependence since the regression model is solving determined by the analyzing distribution.

The classical effect decomposition method considers, the direct effects as Dx =

β1(θ) and Dm = β2(θ) indicating that direct effect is not appropriate since

variable X’s direct effect involves the effect mediated by variable M due to the effect that involves M ’s parameters. Similarly variable M ’s direct effect involves the effect mediated by variable X.

Definition 4.2. The interaction effect of X and M is defined as the sum of indirect effects of X and M .

Theorem 4.3. When the normality assumption is made, the effects decomposi-tion is: g(θ) = µy Dx(θyx) = σyx σ2 x , Dm(θym) = σym σ2 m IDx(θ) = (σyxσ2m− σymσxm) σ2 xσ2m− σ2xm − σyx σ2 x IDm(θ) = (σymσx2− σy1σxm) σ2 xσ2m− σxm2 − σym σ2 x

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Hence the size of interaction effect is IDx(θ) + IDm(θ)

Table 5. Full effect decomposition

Effect classified Effect quantity

Combined effect µ(x, m; θ) = g(θ) + Tx(θ)x + Tm(θ)m X’s direct effect Dx(θyx) X’s Total effect Tx(θ) M ’s direct effect Dm(θym) M ’s Total effect Tm(θ) No Interaction µ(x, m, θ|G(θ) = 0c) = g(θ) +Dx(θyx)x + Dm(θym)m X’s indirect effect IDx = Tx(θ)− Dx(θyx) M ’s indirect effect IDm = Tm(θ)− Dm(θym) Interaction IAxm= IDx+ IDm

The full decomposition model with two direct effects, two driving interactions and compounding interaction is formulated as

Y =g(θ) + Dx(θyx)x + Dm(θym)m

+ IDx(θ)x + IDm(θ)m + ϵ (4.2)

where ϵ has a distribution with mean zero and variance var(Y|X = x, M = m). 5. Concluding Remarks

In this paper, we have shown that the classical effect decomposition method is not statistically correct due to the fact that the derived direct effect involves the effect mediated by other variables. Our approach of effect decomposition not only provide clean effects but there effects can also be estimated this BAN estimation.

There are several related topics to be studied. First, a real data analysis to compare the direct and indirect effects of classical and new versions is desired. Second, the new concept of interaction is interesting but that requires further investigation.

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