86MEDIATION AND MODERATION ANAL YSIS
党风廉政总结
4.2Estimation of the Direct,Indirect,and Total
日本签证要求Effects of X
云南的风景When empirically testing a causal process that involves a mediation component,of primary interest is the estimation and interpretation of the direct and indirect effects along with inferential tests thereof.To derive the effects,one must also estimate the constituent components of the indirect effect,meaning the effect of X on M as well as the effect of M on Y,although the constituent components of the indirect effect are not of primary interest in modern mediation analysis.Many analysts will often estimate the total effect of X on Y as well,although doing so is not required for the purpo of interpretation.The total effect will be defined later.
The simple mediation model reprented in the form of a statistical diagram can be found in Figure4.2.Notice comparing Figures4.1and4.2 that there is little difference between the conceptual and statistical diagrams reprenting a simple mediation model.As there are two conquent variables in this diagram,two linear models are required,one for each conquent.This statistical diagram reprents two equations:
M=i1+aX+e M(4.1)
Y=i2+c′X+bM+e Y(4.2) where i1and i2are regression intercepts or constants,e M and e Y are errors in the estimation of M and Y,respectively,and a,b,and c′are the regression coefficients given to the antecedent variables in the model in the estimation of the conquents.1The coefficients of the model are treated as estimates of the putative causal influences of each variable in the system on others,and the analytical goal is to estimate the coefficients, piece them together,and interpret.The coefficients can be estimated by conducting two OLS regression analys using the procedures that come with SPSS,SAS,and other statistical packages,using a structural equation modeling program such as LISREL,AMOS,Mplus,EQS,or through the u of PROCESS mentionedfirst in Chapter1and illustrated in the next ction. In a simple mediation model,it generally makes no difference,although without additional computational aides,OLS regression procedures that come with most statistical packages will not get you all the information you will need to conduct some of the more preferred inferential tests described later in this Chapter.For now,we can talk about the coefficients and effects in the model without concerning ourlves with the specifics of the method ud to estimate them.
狮子图片卡通
1To simplify mathematical expressions,from this point forward I drop the j subscript indexing ca nu
mber for measured variables,estimated values of variables,and residuals
The Simple Mediation Model 87
e M
回族建筑
FIGURE 4.2.A statistical diagram of the simple mediation model.
The Direct Effect of X on Y
In equation 4.2,c ′estimates the direct e ffect of X on Y .A generic interpretation of the direct e ffect is that two cas who di ffer by one unit on X but who are equal on M are estimated to di ffer by c ′units on Y .More formally,
虚线c ′=ˆY |(X =x ,M =m )−ˆY |(X =x −1,M =m )(4.3)
where m is any value of M ,|means conditioned on or given ,and the hat over Y means estimated or expected from the model.In other words,for two cas with M =m ,c ′is the estimated value of Y given X =x minus the estimated value of Y given X =x −1.As can be determined looking at equation 4.3,the sign of c ′tells whether the ca one unit higher on X is estimated to be higher (c ′=+)or lower (c ′=−)on Y .So a positive direct e ffect means that the ca higher in X is estimated to be higher on Y ,whereas a negative direct e ffect means that the ca higher in X is estimated to be lower on Y .In the special ca where X is dichotomous with the two values of X di ffering by a single unit (e.g.,X =1and X =0),ˆY
can be interpreted as a group mean,so c ′=Y |(X =x ,M =m )−Y |(X =x −1,M =m ),meaning c ′estimates the di fference between the groups means holding M constant.This is equivalent to what in analysis of covariance terms is called an adjusted mean di fference .
The Indirect Effect of X on Y
Before defining the indirect e ffect,it is first necessary to discuss what a and b estimate.In this model,a quantifies how two cas who di ffer by one unit
88MEDIATION AND MODERATION ANAL YSIS
on X are estimated to differ on M,with the sign determining whether the ca higher in X is estimated to be higher(+)or lower(−)on M.That is
a=ˆM|(X=x)−ˆM|(X=x−1)(4.4) When X is a dichotomous variable coded by a unit difference,a in equation 4.1reprents the difference between the two group means on M:a=M| (X=x)−M|(X=x−1).
The b coefficient from equation4.2has an interpretation analogous to c′,except with M as the antecedent.Two cas who differ by one unit on M but who are equal on X are estimated to differ by b units on Y.As with a and c′,the sign of b determines whether the ca higher on M is estimated as higher(+)or lower(−)on Y:
b=ˆY|(M=m,X=x)−ˆY|(M=m−1,X=x)(4.5) The indirect effect of X on Y is the product of a and b.For instance,if a=0.500and b=1.300,then the indirect effect of X on Y through M is ab=0.650.The indirect effect tells us that two cas that differ by one unit on X are estimated to differ by ab units on Y as a result of the effect of X on M which,in turn,affects Y.The indirect effect will be positive(meaning the ca higher in X is estimated to be higher on Y)if a and b are both positive or both negative,whereas it will be negative(meaning the ca higher in X is estimated to be lower on Y)if either a or b but not
both is negative.
Although one can interpret the indirect effect without considering the signs of a and b,doing so can be dangerous,becau the sign of ab is determined by two different configurations of the signs of a and b.A certain theory you are testing might predict ab to be positive becau,according to the process the theory explains,a and b should both be positive.But what if after estimation,a and b turned out to be negative?This would yield a positive indirect effect as predicted,yet this pattern of results for a and b are exactly opposite to what the theory predicts,and this should cast some doubt on whether the theory is adequately describing the process generating your data.
The Total Effect of X on Y
The direct and indirect effects perfectly partition how differences in X map on to differences in Y,the so-called total effect of X,denoted here as c.The total effect c quantifies how much two cas who differ by one unit on X are estimated to differ on Y.That is
c=ˆY|(X=x)−ˆY|(X=x−1)(4.6)
The Simple Mediation Model89 In a simple mediation model,c can be derived by estimating Y from X alone:
Y=i3+cX+e Y∗(4.7) When X is a dichotomous variable coded by a single unit difference,c is the difference between the group means on Y:c=Y|(X=x)−Y|(X=x−1). Regardless of whether X is dichotomous,the total effect of X on Y is equal to the sum of the direct and indirect effects of X:
c=c′+ab(4.8) This relationship can be rewritten as ab=c−c′,which provides another definition of the indirect effect.The indirect effect is the difference between the total effect of X on Y and the effect of X on Y controlling for M,the direct effect.
That the total effect of X is the sum of the direct and indirect effects can be illustrated by substituting equation4.1into equation4.2,thereby expressing Y as a function of only X
Y=i2+b(i1+aX+e M)+c′X+e Y(4.9) which can be equivalently written as
Y=(i2+bi1)+(ab+c′)X+(e Y+be M)(4.10) Although it may not look obvious,equation4.10is a simple linear function of X,just as is equation4.7.In fact,equation4.10and4.7are identical if you make the following substitutions:c=ab+c′,i3=i2+bi1,and e Y∗=e Y+be M.
So ab+c′,has the same interpretation as c.The sum of the direct and indirect effects quantifies how two much cas that differ by a unit on X are estimated to differ on Y.
4.3Example with Dichotomous X:The Influence of
Presumed Media Influence
To illustrate the estimation of direct and indirect effects in a simple mediation model,I u data from a study conducted in Israel by Tal-Or, Cohen,Tsfati,and Gunther(2010).The datafile is named PMI.The participants in this study(43male and80female students studying political science or communication at a large university in Israel)were given one of two newspaper articles to read that described an economic crisis that purportedly may affect the price and supply of sugar in Israel. Approximately half of the participants(n=58)were given an article
90MEDIATION AND MODERATION ANAL YSIS
TABLE4.1.Descriptive Statistics for Presumed Media Influence Study
Y M Y
REACTION PMI adjusted
Front page(X=1)Mean 3.746 5.853 3.616
SD 1.452 1.267
Interior page(X=0)Mean 3.250 5.377 3.362
锦字取名SD 1.608 1.338
鲁滨逊漂流记的作者是谁Mean 3.484 5.602
SD 1.550 1.321
they were told would be appearing on the front page of a major Israeli newspaper(henceforth referred to as the front page condition).The remaining participants(n=65)were given the same article but they were told it would appear in the middle of an economic supplement of this newspaper(referred to here as the internal page condition).Which of the two articles any participant read was determined by random assignment. In all other respects,the participants in the study were treated equivalently, the instructions they were given were the same,and all measurement procedures were the identical in both experimental conditions.
After the participants read the article,they were asked a number of questions about their reactions to the story.Some questions asked participants how soon they planned on buying sugar and how much
they intended to buy.Their respons were aggregated to form an intention to buy sugar measure(REACTION in the datafile),such that higher scores reflected greater intention to buy sugar(soon and in larger quantities). They were also asked questions ud to quantify how much they believed that others in the community would be prompted to buy sugar as a result of exposure to the article,a measure referred to as presumed media influence (PMI in the datafile).
Tal-Or et al.(2010)reasoned that relative to an article buried in the interior of a newspaper,an article published on the front page of a major newspaper would prompt a belief that others are likely to be influenced by the possibility of a shortage and so would go out and buy sugar.This belief that others were going to respond in this way would,in turn,prompt the participant to believe he or she should also go out and buy sugar. That is,people would u their beliefs about how others would respond
