answers to even

Chapter 1

PAGE 35

These websites contain information for mediation analysis as well as downloadable software to conduct mediation analysis.

David Kenny

http://davidakenny.net/kenny.htmThis site has general information on mediation.

David MacKinnon

http://www.public.asu.edu/~davidpm/ripl/This site has mediation information and computer programs.Kristopher Preacher

http://www.psych.ku.edu/preacher/This site has computer programs.

Comments and Short Answers to Even Numbered Questions for each ChapterChapter 1.Comments: Chapter 1 is dense but should provide an introduction to mediation relations and how they differ from other relations like moderation and confounding. A purpose of the chapter is to provide a variety of examples of the mediation model so that the reader appreciates mediation for more than just the role of mediation in their particular field. A good student activity would be to take one of the examples described in this chapter and learn more about it by reading some of the research cited in the chapter.

1.2. If the disease was transmitted by person to person contact, the mediators targeted would be variables that would reduce contact among persons. Some possible mediators are quarantining of infected individuals in hospitals, reducing all physical contact among persons, and screening of persons for disease before they are contagious.

1.4. The four-way interaction is an example of a relation among four variables, whereby a three-way interaction differs across levels of the fourth variable. A mediation relation may change across levels of a fourth variable such as a moderator. A moderator effect may be mediated such that the mediator explains the relation of the moderator to the outcome.

1.6. There are thousands of articles with the words moderator or mediator in the title. 1.8. a. Simon (1954) describes a negative correlation between marital status and candy consumption, a negative correlation between candy consumption and age, and a positive correlation between martial status and age. When age is held constant, there is a nearly zero correlation between marital status and candy consumption because the correlation between marital status (independent variable) and candy consumption (dependent variable) is spurious, because both variables are related to age (confounder). It does not make sense that age could be a mediator, i.e., that marital status causes age.b. Simon (1954) reports a positive correlation between percentage of employees who are married and absenteeism. When housework is held constant, the correlation was nearly zero. Amount of housework is a mediator such that marriage (independent variable) leads to housework and more housework leads to more absenteeism (dependent variable). Simon notes that common sense was used to conclude that the third variable was a confounder in 1.8a and a mediator in part 1.8b, even though similar statistical information was obtained.

1.10. More intelligent workers tend to get bored easily and produce less in the assembly line so boredom may be considered a mediator. It is also likely that intelligent workers will, in general, be more productive so there is both an increase in productivity and a decrease in productivity through boredom--an inconsistent mediation model. Adding boredom to the equation will increase the magnitude of the relation between intelligence and productivity, a suppression relation.1.12. Note that the test of statistical significance of the overall relation may be underpowered so that there may actually be a significant relation between the independent and the dependent variable in the population. Example 1.10 provides a possible example where there is a zero relation between two variables that is explained when a mediator is included in the analysis. There are many possible examples of opposing mediation relations that have an overall nearly zero relation. For example, as a sports team gets more successful, the success increases confidence which leads to better performance but the increased performance leads opposing teams trying harder to win which reduces subsequent performance. Incarceration of criminals may reduce recidivism to crime through rehabilitative activities but may also increase recidivism by putting persons in an environment where the norm is to commit crimes and inmates may learn new criminal activities. Similarly, interventions that select high risk adolescents for special programs may have beneficial effects but may also introduce adolescents to a less healthy social norm. A program to promote safe sex behaviors may have beneficial effects on risky sexual behavior but may also increase interest in sexual activity. Other examples are described in Chapter 5.Chapter 2.Comments: A goal of chapter 2 is to provide a wide range of examples of mediation relations. Another goal is to provide background for mediation in prevention and treatment studies. A good student activity would be to summarize mediation relations in the students area of interest. There is another interesting example of mediation in social psychology that I almost used in this chapter. Isen and Levin (Isen, A. M. & Levin, P. F. (1972). Effect of feeling good on helping: Cookies and kindness. Journal of Personality and Social Psychology, 21, 384-388) describe two studies that suggest that an intervention (X) that makes people feel good (M) leads to more helping behavior (Y). The intervention used to make people feel good was giving them cookies or having subjects find a dime in the coin return of the public telephone. 2.2. Action theory refers to the actions taken to change mediators in the X to M part of the mediation model. Action theory is most relevant when X codes an intervention or manipulation because typically actions are taken to change mediators. Action theory when X is not an intervention (or does not represent assignment to a group) is not as clear as when X is an intervention. In this case X does not represent manipulation or actions to change M. In this case the action theory refers to how the X variable may change the mediator. Conceptual theory refers to the theory for how the mediators are related to the dependent variable. This is typically how theory is conceived in terms of how existing variables are related to the dependent variable.

2.4. In drug prevention, interventions to change norms are delivered which are expected to change norms to be less tolerant of drugs and the norm change leads to reduced drug use.

2.6. In a program to reduce dating violence, Foshee et al. (1998) targeted changing norms, decreasing gender stereotyping and improving conflict management. 2.8. As described in section 2.10.3, there are many ways that mediators for drug prevention programs may be selected including a review of prior mediation analyses, theories of drug use such as social learning theory, and empirical studies of variables that correlate with drug use.

2.10a. Seedlings are most likely a surrogate because presumably more seedlings leads to more potato yield because there are more plants. And it is likely that the number of seedlings completely mediated fertilizer effects on potato yield. On the other hand, the number of seedlings may be considered a measure of how much fertilizer was delivered and more fertilizer would then lead to more potatoesso in this case seedlings may be a mediator.

b. Norms is most likely a mediator because the relation of norms and assault is not as direct as for most surrogates. It would be difficult to defend norms as a surrogate such that the intervention effect on norms is the same as the intervention effect on assault. It is also unlikely that norms completely mediate the relation of socioeconomic status on assault. There are many other predictors of assault. c. Carotene is likely a mediator as the amount of alfalfa eaten by cows increases carotene consumed which increases Vitamin A in butter produced from the cows milk. On the other hand, there may be a large relation between carotene and Vitamin A in butter so that a measure of carotene consumed could be used in place of amount of Vitamin A in butter made from the cows milk. More evidence for a surrogate is that the relation between alfalfa consumed and Vitamin A in milk is likely to be entirely through carotene in cows milk. d. Fighting may be more accurately considered as a mediator rather than a surrogate. It is a surrogate in that fighting is probably closely related to adult incarceration. On the other hand, many more persons in fights in sixth grade are not incarcerated as an adult. It is also likely that there are many influences of an intervention on adult incarceration besides fighting in sixth grade. Persons incarcerated for non violent crimes as well as violent crimes that may be related to fighting in the sixth grade. This example illustrates how surrogates tend to be strongly related to the outcome variable and conceptually closer to the dependent variable.

Chapter 3

Comments: A useful exercise is to prove that ab = c - c, either algebraically based on c = c + ab or based on writing the formula for coefficients (i.e., c) as in MacKinnon, Warsi, & Dwyer (1995, page 45-46) and shown below. (1)c = Cov(Y,X)/Var(X)

: from Equation 3.1

(2)Cov(Y,X) = c' Var(X) + b Cov(M,X) : taking covariance of Y

(from Equation 3.2) and X

(3)c = b Cov(M,X)/Var(X) + c' : substitute (2) into (1)

(4)c = ab + c'

: a = Cov (M,X)/Var(X)

from Equation 3.3

(5)c - c' = abPlots of the mediated related relations are described in more detail in Fritz & MacKinnon (2007), A graphical representation of the mediated effect. Behavior Research Methods. Sample SAS, SPSS, and R programs to make the plots described in the article are given at http://www.public.asu.edu/~davidpm/ripl/mediate.htm. Note that the plots are difficult for many to understand especially Figure 3.5 because it includes so much information. An entire class period may be needed to clearly describe the plot. Describing the plots in Figures 3.3 and 3.4 first provides a simpler introduction to the information in more complicated plots that show the mediated effect. Note that the water consumption example was created during a hot summer day in Phoenix, Arizona.

There is a lot of information in this chapter. There will likely be a tendency to get bogged down in the assumptions described in this chapter. Keep in mind that models and data to address assumptions are covered in later chapters. For example, time ordered relations from X to M to Y are a critical assumption of the mediation model and longitudinal data can provide insight into these longitudinal relations as described in chapter 8. But understanding material in chapter 8 requires understanding material in earlier chapters. The purpose of this chapter is not to advocate cross-sectional studies but understanding this material is required before applying more complicated models.

3.2.a.

EMBED Equation.3 b. Equation 3.6/3.7 = .02467; Equation 3.8 = .02490; Equation 3.9/3.10 = .02473; Equation 3.11/= 3.12 = .02461

c. LCL = .12124; UCL = .21795

d. Test 1, Assess X(Y: , there is a significant program effect on the dependent variable.

Test 2, Assess X(M: , there is a significant program effect on the mediating variable.

Test 3, Assess M1(Y: , there is a significant relation of the mediating variable to the dependent variable adjusted for the independent variable.

Test 4, Assess X(Y: , there is a significant relation of the independent variable to the dependent variable adjusted for the mediating variable. Because this path is statistically significant there is not complete mediation. This significant direct effect coefficient would not satisfy the causal step criteria outlined in Judd and Kenny (1981b) but would satisfy criteria in Baron and Kenny (1986) because the total effect is larger than the direct effect. Note that James, Mulaik, & Brett (2006) confirmatory approach described in section 3.13 would lead to the conclusion that this is a partial mediation model because the direct effect is statistically significant.e. The results are consistent with a mediation process by which the program changed peers as an information source for nutrition behavior which in turn changed nutrition behavior.

Here is SAS code used to compute quantities for question 3.2.

data a;input c sec cp secp b seb a sea mse ssx N;ta=a/sea;tb=b/seb;tc=c/sec;tcp=cp/secp;varab=a*a*seb*seb+b*b*sea*sea;se36=sqrt(varab);se37=(a*b*(sqrt(ta*ta+ tb*tb)))/(ta*tb);se38=sqrt(sec*sec+secp*secp-2*(mse/(N*ssx)));se39=sqrt(varab+sea*sea*seb*seb);se310=(a*b*(sqrt(ta*ta+tb*tb+1)))/(ta*tb);se311=sqrt(varab-sea*sea*seb*seb);se312=(a*b*(sqrt(ta*ta+tb*tb-1)))/(ta*tb);ccp=c-cp;ab=a*b;*Normal theory confidence limits with first order standard error;ucl=ab+1.96*se36; lcl=ab-1.96*se36;*Causal Steps are the same as the t-tests for regression estimates, c a, b, and cp;cards;.3552 .0631 .1856 .0642 .1711 .0195 .9912 .0896 1.1289 .2459 1227;proc print; var ccp ab;proc print;var se36 se37 se38 se39 se310 se311 se312;proc print;var ab se36 lcl ucl;proc print; var tc ta tb tcp ccp;run;3.4. The water consumption data are simulated data so it is a unique situation where the population model is known. In a real study, population values are not known. Assumptions are briefly addressed here for this example. For mediation analysis of real data, consideration of assumptions can be very complicated including investigation of assumptions that can be tested as well as some assumptions that can only be addressed theoretically or in a program of research. Statistical assumptions of the mediation regression analysis include: (1) Correct functional form. Linear relations are assumed. Figures 3.3, 3.4, and 3.5 suggest that linear relations are appropriate. The usual methods of investigating nonlinear relations could be applied here including plots of residuals as a function of independent variables. A test of the XM interaction was nonsignificant (as addressed in Chapter 10) so it appears that this interaction is not substantial in the study. If the interaction was statistically significant then the size of the mediated effect is likely to depend on the value of X and contrast methods described in Chapter 10 would be applicable. (2) No omitted influences. Additional variables could be incorporated in the analysis such as an additional mediator or covariate in the analysis. It may also be sensible to consider how large a correlation an omitted variable would have to be to substantially reduce the mediated effect. (3) Accurate Measurement. The measures are valid and reliable. Measures like temperature and water consumed are likely to be accurately measured. Self-reported thirst may be more problematic but this could be improved with a latent variable model described in Chapter 7. No measures of reliability or validity are given for these examples but psychometric studies are important for a real study to obtain test-retest correlations and other measures of validity and reliability. To some extent, the measures must be reasonably accurate to obtain a significant mediated effect because error in the mediator tends to reduce power to detect a mediated effect. (4) Well-behaved residuals. Plots of residuals in the equations ought to resemble a normal distribution as the residuals for the water consumption example. Inferential assumptions are often more complicated than the assumptions of the regression analysis because they require information from other studies and in some cases there is not sufficient information to address the assumption. There is much overlap between these assumptions and the statistical assumptions described above. (5) Temporal Precedence. The random assignment of participants to temperature ensures that this variable comes before self-reported thirst and water consumed. The timing of the measures in the study was also such that temperature, thirst, and water consumed were measured in a temporal sequence. (6) Micro Versus Macro Mediation Chain. It is possible to provide more measures of the entire mediation chain including measures of blood volume, actual work during the session, etc. that would provide a more detailed micromediational chain. (7) Measurement timing. It is assumed that the actual timing of measures was adequate to capture the mediated effect. It is important in other studies to consider if the mediated effect would be present with other times of data collection. (8) Normally distributed X, M, and Y. If X is binary then the methods described in this chapter are accurate. Measures of skewness and kurtosis indicate a normal distribution. Plots of residuals and actual distributions suggest that the variables are normally distributed. In chapter 12, resampling methods that do not make as many statistical assumptions led to the same research conclusions as described in this chapter. (9) Normally distributed product of coefficients. Confidence limits based on the distribution of the product and resampling methods led to the same conclusions as the analysis. The confidence limits from the distribution of the product and resampling methods are asymmetric consistent with the often nonnormal distribution of the product of regression coefficients. (10) Omitted influences. This issue is the same statistical assumption as discussed for the regression analysis but here the focus is on the extent to which inference regarding the mediated effect is accurate and refers to substantive aspects of this assumption. If other data were available it may be worth investigating whether the mediated effect differs by subgroups, such as the fitness of participants in the study (fit versus normal moderator for this example is described in Chapter 10). There may also be other, more important, mediators of this relation. For the water consumption example, perhaps actual blood volume is a better mediating measure. (11). Causal inference. Because temperature is randomized, the relation of temperature to self-reported thirst and the relation of temperature to water consumed may be considered a causal effect (with some assumptions discussed in Chapter 13). The relation self-reported thirst to water consumption is a reasonable assumption as a causal relation based on other research. The size of the relation of temperature to water consumption adjusted for self-reported thirst may be more difficult to defend as a causal effect. (12). Theoretical versus Empirical Mediator. One benefit of the water consumption study is that future studies can be designed based on these results to replicate and extend the mediated effect such that temperature affects self-reported thirst and this affects water consumed. These future studies help provide more evidence for a theoretical mediator. 3.6.

Chapter 4Comments: There is a lot of information in this chapter. There are many unsolved issues discussed in the chapter. For example, the ideal effect size measure for mediation models is not yet determined. This chapter may be of most interest to more technically inclined readers. The Monte Carlo simulation program can be used for a wide variety of simulation studies. Note that the original version of this program was written around 1987 and parts of if have been used in many simulations studies both by me and my collaborators but also by other researchers. The program can be adapted for longitudinal data, categorical data analysis, and missing data, for example. A simulation program for covariance structure models is described in MacKinnon, D. P. (1992). Statistical simulations in CALIS. In Sugi 17. Proceedings of the Seventeenth Annual SAS Users Group International Conference 1992, 1199-1203. Note that simulation studies can also be conducted in SPSS and major covariance structure analysis programs. 4.2. Taking absolute values of the coefficients before computing the proportion mediated measure changes the meaning of the proportion measure. Now all relations in the model are positive so the total effect is the sum of the absolute value of all of the effects in the model. The new measure is the proportion of the total relations between X and Y that is attributable to the mediator. One of the reasons that absolute values are taken is to ensure that the proportion will not be less than zero or greater than 1. The measure is subject to sampling variability as the regular proportion measure so it requires large sample sizes or effect sizes for it to be a stable measure.4.4. The population correlation between X and Y for a equal to .5, b equal to .1 and error variances equal to 1, rXY = Cov[X,Y]/(Var[X] Var[Y] ), is equal to .04969. The required sample size to have .8 power to detect a correlation this size is approximately 3172 participants. The number of required participants is very large because the relation between X and Y (first step in the Baron and Kenny causal steps test) is equal to the product of the a and b relations which can be very small. 4.6. For each case, all possible pairs of elements in the formula for the variable are written out, values that are 0 by definition are deleted, and terms are then combined and simplified to give the following variances of X, M and Y.

Cov [X, X] = Var [X] = Cov (X, X ) = 2X

Cov [M, M]= Var [M]= Cov (aX + e3, aX + e3) = a 2 2X + 2e3

Cov [Y, Y] = Var [Y] = Cov(bM + c'X + e2, bM + c'X + e2)

= b 2 (a2 2X + 2e3) + 2bc'a 2X + c'2 2X + 2e2

4.8. The product of standardized regression coefficients for and for the analysis of standardized data equal to (rXM(rMY-rXYrXM)/(1-rXM2) with partial derivatives of (rXM2rMY + rMY -2 rXMrXY)/((1-rXM2)2), -rXM2/(1-rXM2), and rXM/(1-rXM2), for rXM, rXY, and rMY, respectively. The covariance matrix among the correlations is pre and post multiplied by the vector of partial derivatives to obtain the multivariate standard error of the product of and for standardized data as described in Bobko and Reick (1980). The covariance matrix among the correlations is given in Olkin & Siotani (1976) and a SAS program to compute the standard errors of functions of correlation coefficients adapted from Appendix B of MacKinnon et al., (2002) is shown below. For the water consumption data the t-test for the product of standardized regression coefficients is equal to 2.246. The t-test for the raw minus partial correlation is also included in the program and is equal to 1.997 for the water consumption data. data a; input rxm rxy rmy nobs;

/*revised program from Appendix B, MacKinnon et al., 2002 Psychological Methods, 7, 83-104;*note r corresponds to correlation;*x, m, and y represent the independent, mediating, and dependent variables, respectively;*variance of the correlations from Olkin & Siotani 1976 and Olkin & Finn (1995);*/vrxy=((1-rxy*rxy)*(1-rxy*rxy))/nobs;

vrmy=((1-rmy*rmy)*(1-rmy*rmy))/nobs;

vrxm=((1-rxm*rxm)*(1-rxm*rxm))/nobs;

*covariances among the correlations from Olkin & Siotani 1976 and Olkin & Finn (1995)t;crxyrmy=(.5*(2*rxm-rxy*rmy)*(1-rxm*rxm-rxy*rxy-rmy*rmy)+rxm*rxm*rxm)/nobs;

crxyrxm=(.5*(2*rmy-rxy*rxm)*(1-rxy*rxy-rmy*rmy-rxm*rxm)+rmy*rmy*rmy)/nobs;

crmyrxm=(.5*(2*rxy-rmy*rxm)*(1-rxm*rxm-rxy*rxy-rmy*rmy)+rxy*rxy*rxy)/nobs;

*olkin and finn (1995);*partial correlation or correlation with mediating variable removed;rxyi=(rxy-rmy*rxm)/sqrt((1-rmy*rmy)*(1-rxm*rxm));

*difference between simple and partial correlations;diff=rxy-rxyi;

*partial derivatives of the difference between simple and partial correlations;opd1=1-(1/(sqrt(1-rmy*rmy)*sqrt(1-rxm*rxm)));

opd2=(rxm-rxy*rmy)/((sqrt(1-rxm*rxm))*(1-rmy*rmy)**(1.5));

opd3=(rmy-rxm*rxy)/((sqrt(1-rmy*rmy))*(1-rxm*rxm)**(1.5));

ovar=opd1*opd1*vrxy+opd2*opd1*crxyrmy+opd3*opd1*crxyrxm+opd1*opd2*crxyrmy

+opd2*opd2*vrmy+opd2*opd3*crmyrxm+opd1*opd3*crxyrxm+opd2*opd3*crmyrxm

+opd3*opd3*vrxm;

ose=sqrt(ovar);

zolkin=diff/ose;

polkin=1-probnorm(zolkin);

*bobko & rieck (1980) product of standardized a and b;bobko=rxm*(rmy-rxy*rxm)/(1-rxm**2);

*partial derivatives for product of standardized a and b;bpd1=((rxm*rxm*rmy+rmy-2*rxm*rxy)/(1-rxm*rxm)**2);

bpd2=(-(rxm*rxm)/(1-rxm*rxm));

bpd3=(rxm/(1-rxm*rxm));

bobkovar=((bpd1**2)*vrxm)+((bpd2**2)*vrxy)+((bpd3**2)*vrmy)+

(2*bpd1*bpd2*crxyrxm)+

(2*bpd2*bpd3*crxyrmy)+(2*bpd1*bpd3*crmyrxm);

bobkose=sqrt(bobkovar);

zbobko=bobko/bobkose;

pbobko=1-probnorm(zbobko);

cards;

.371 .361 .490 50

;

proc print;

var diff ose zolkin polkin bobko bobkose zbobko pbobko;

run;4.10. The derivation of the asymptotic covariance among parameter estimates described in Bollen (1989, p.107-109) is applied to the mediation model described in the text to show that the covariance between , the estimates of a and b parameters, equals zero.

Model:Y = c X + bM + 1

M = aX + 2

X, 2 and 3 are pairwise independent.

Let 2X = c1, 21 = c2, 22 = c3 and normal distributions.

The likelihood function for the estimation of the parameters a, b and c ( = a, b, c) is:

FML = log () + tr (S-1()) - logS - (p + q) where

()=

c1 ac1 abc1 + cc1

ac1

a2c1 + c3

a2bc1 + acc1 + bc3

abc1+cc1 a2bc1+acc1+bc3 b2(a2c1 + c3)+ 2abcc1+c2c1+ c2

= c1c2c3 and Var (X) COV (X,M)COV (X,Y)

S = COV (M,X) Var (M)COV (M,Y) COV (Y,X) COV (Y,M) Var(Y)

Is defined as the matrix of sample variances and covariances (Note that this is different from the text where these quantities (e.g., COV(X,M)) are population values.

tr (S-1 ()) = Sum of elements on the main diagonal of S-1 ()

= (VarX) (a2c1c2 + c2c1c3 + c2c3)

+ [COV (X,M)] (bcc1c3 ac1c2) [COV (Y,X)] cc1c3+ [COV (M,X)] (bdc1c3 ac1c2)

+ (Var M) (b2c1c3 ac1c2) + [COV (M,Y)] (-bc1c3)

+ [COV (Y,X)] (-cc1c3)

+ [COV (M,Y)] (-bc1c3) + (Var Y) (c1c3)

log S and (p + q) are constants.

Taking partial derivatives of FML and setting them equal to zero we get the estimates of a, b and c(c1, c2, c3 are assumed known). The inverse of the matrix of the second derivatives of FML gives the variances and covariances of estimates of a, b and c respectively. The only elements of FML that depend on a, b and c are the elements of tr(S-1 ()). By inspection we can see that in tr(S-1 ()) there are no cross terms like ab or ac. So when we take second partial derivatives we get = 0 and = 0. This implies that the estimate of a is uncorrelated with the estimates of b and c.

Chapter 5Comments: One important point to emphasize in this chapter is that with multiple mediators it is often unclear if the mediators should be treated as if they are all in a vertical line between X and Y as in the two mediator model in this chapter or if there are relations among the multiple mediators such as X to M1 to M2 to Y. In this way, students will start to think of the complexity of models discussed in later chapters. The true model for the relations among multiple mediators is complicated and will likely require experimental studies to manipulate different mediators in order to understand how the mediators are related. The single mediator model often has X representing a randomized intervention so that it is only the M to Y relation that may be reversed (i.e., Y to M) or correlational. With multiple mediators there are many possible patterns among the mediators and Y, even when X represents a randomized study. Theory and prior research are required to specify complex mediation models because the number of possible relations among variables is very large. This following paper describes useful description of methods for multiple mediator models: Brown, R. L. (1997). Assessing specific mediational effects in complex theoretical models. Structural Equation Modeling, 4(2), 142-156.

5.2. MacKinnon et al., (2001) investigated the mediation relations in a program to prevent anabolic steroid use among high school football players. There were three dependent variables, intent to use anabolic steroids, nutrition behaviors, and strength training self-efficacy. Twelve mediators were evaluated: Knowledge of the effects of anabolic steroids, perceived coach tolerance of steroids, team as an information source, peers as an information source, ability to turn down drug offers, perceived peer tolerance of drug use, normative beliefs about anabolic steroid use, perceived severity of anabolic steroid use, perceived susceptibility to anabolic steroid use, belief in media advertisements, reasons for using anabolic steroids, reasons for not using anabolic steroids. The mediators were evaluated one mediator at a time and also together in a multiple mediator model. The models were estimated using a covariance structure model and the estimates of the mediated effects were tested for each mediator using methods described in Chapter 5.Here is a SAS program to compute the quantities in the text of Chapter 5 and exercise 5.1.

/* SAS program to compute quantities in Chapter 5 (text example and

exercise 5.1: MacKinnon, D. P. (2007) Introduction to Statistical

Mediation Analysis. Mahwah, NJ: Erlbaum.). Note that there are two sets of coefficients for the

expectancy example; one set has four decimal places and the other set

of coefficients uses all of the decimal places in the SAS output. With

more decimal places c-c' is equal to ab to more than four decimal

places.

*/Data a;

Input c sc cp scp b1 sb1 b2 sb2 #2 a1 sa1 a2 sa2 mse varx sb1b2 N;

*Compute t-values for coefficients;ta1=a1/sa1;ta2=a2/sa2;tb1=b1/sb1;tb2=b2/sb2;tc=c/sc;tcp=cp/scp;

*Computes mediated effects and variances of coefficients;a1b1=a1*b1; a2b2=a2*b2; ccp=c-cp;totab=a1b1+a2b2;

vara1=sa1*sa1;vara2=sa2*sa2;varb1=sb1*sb1; varb2=sb2*sb2;

*Equation 5.5;vara1b1=vara1*b1*b1+varb1*a1*a1;vara2b2=vara2*b2*b2+varb2*a2*a2;

sa1b1=sqrt(vara1b1);sa2b2=sqrt(vara2b2);ta1b1=a1b1/sa1b1;ta2b2=a2b2/sa2b2;

*Equation 5.6;sa1b1t=(a1b1*sqrt(ta1*ta1+tb1*tb1))/(ta1*tb1);

sa2b2t=(a2b2*sqrt(ta2*ta2+tb2*tb2))/(ta2*tb2);

*Equation 5.7;Stot57=sqrt(vara1*b1*b1+varb1*a1*a1+vara2*b2*b2+varb2*a2*a2+2*a1*a2*sb1b2);

*Equation 5.8;Stot58=sqrt(vara1b1+vara2b2+2*a1*a2*sb1b2);ttot=totab/stot58;

/*General formula for standard error of the total mediated that uses

covariances. Most of these covariances are zero with real data. For

the multiple regression analysis in chapter 5, only sb1b2 is

nonzero as in Equation 5.7 and 5.8;

Stotcov=sqrt(vara1b1+vara2b2+2*a1*a2*sb1b2+2*b1*b2*sa1a2+

2*b1*a1*sa1b1+2*b1*a2*sa1b2+2*a1*b2*sa2b1+2*a2*b2*sb2a2);

ttotcov=totab/stotcov;

*/*Equation 5.9;covccp=mse**2/(N*varx);rccp=covccp/(sc*scp);

Stot59c=sqrt(sc*sc+scp*scp-2*covccp);

Stot59r=sqrt(sc*sc+scp*scp-2*rccp*sc*scp); tdiff=(c-cp)/stot59c;

*Equation 5.10;Stot510=sqrt(vara1*b1*b1+varb1*a1*a1+vara2*b2*b2+varb2*a2*a2-2*a1*a2*sb1b2);

*Equation 5.11;Stot511=sqrt(vara1b1+vara2b2-2*a1*a2*sb1b2);tequal=(a1b1-a2b2)/stot511;

/*General formula for standard error of the difference between two

mediated effects. Most of these covariances are zero with real data.

For the multiple regression analysis in chapter 5, only sb1b2 is

nonzero as in Equations 5.10 and 5.11;

Stotcov=sqrt(vara1b1+vara2b2-2*a1*a2*sb1b2-2*b1*b2*sa1a2+

2*b1*a1*sa1b1-2*b1*a2*sa1b2-2*a1*b2*sa2b1+2*a2*b2*sb2a2);

ttotcov=totab/stotcov;

*/*Confidence Limits;lcla1b1=a1b1-1.96*sa1b1;ucla1b1=a1b1+1.96*sa1b1;

lcla2b2=a2b2-1.96*sa2b2;ucla2b2=a2b2+1.96*sa2b2;

*c sc cp scp b1 sb1 b2 sb2 a1 sa1 a2 sa2 #2 mse varx sb1b2 N;Cards;

-.0014 .0603 .0044 .0635 -.4830 .0647 .3365 .0562

.3441 .0471 .0542 .0129 10.1913 125.5615 .004 300

.7078 .1734 .1122 .2073 .5690 .1568 .5297 .1696

.8401 .1580 .2219 .1460 8.3879 84.87 .0079 40

.707760 .17342970 .112152 .20731147 .569029 .15681205 .529720 .16963747

.840138 .15795758 .221903 .14601522 8.3879 84.87 .0079 40

;

Proc print;

Run;

Here is a SAS program to compute quantities for Exercise 5.3.

/* SAS program to compute quantities in Chapter 5 Exercise 5.3.,

MacKinnon, D. P. (2007) Introduction to Statistical

Mediation Analysis. Mahwah: NJ: Erlbaum.

*/Data a;

Input c sc cp scp b1 sb1 b2 sb2

#2 b3 sb3 b4 sb4 #3 a1 sa1 a2 sa2 a3 sa3 a4 sa4

#4 mse sb1b2 N;

*Compute t-values for coefficients;ta1=a1/sa1;ta2=a2/sa2;tb1=b1/sb1;tb2=b2/sb2;tc=c/sc;tcp=cp/scp;

ta3=a3/sa3;ta4=a4/sa4;tb3=b3/sb3;tb4=b4/sb4;

*Computes mediated effects and variances of coefficients;a1b1=a1*b1; a2b2=a2*b2; a3b3=a3*b3; a4b4=a4*b4; ccp=c-cp;totab=a1b1+a2b2+a3b3+a4b4;

vara1=sa1*sa1;vara2=sa2*sa2;vara3=sa3*sa3;vara4=sa4*sa4;

varb1=sb1*sb1; varb2=sb2*sb2;varb3=sb3*sb3; varb4=sb4*sb4;

*Equation 5.5;vara1b1=vara1*b1*b1+varb1*a1*a1;vara2b2=vara2*b2*b2+varb2*a2*a2;

vara3b3=vara3*b3*b3+varb3*a3*a3;vara4b4=vara4*b4*b4+varb4*a4*a4;



*Equation 5.6;sa1b1t=(a1b1*sqrt(ta1*ta1+tb1*tb1))/(ta1*tb1);




*Equation 5.10;da1b1a2b2=a1b1-a2b2;

Stot510=sqrt(vara1*b1*b1+varb1*a1*a1+vara2*b2*b2+varb2*a2*a2-2*a1*a2*sb1b2);

*Equation 5.11;Stot511=sqrt(vara1b1+vara2b2-2*a1*a2*sb1b2);tequal=(a1b1-a2b2)/stot511;

*Confidence Limits;lcla1b1=a1b1-1.96*sa1b1;ucla1b1=a1b1+1.96*sa1b1;




* c sc cp scp b1 sb1 b2 sb2 #2

b3 sb3 b4 sb4 #3 a1 sa1 a2 sa2 a3 sa3 a4 sa4

#4 mse sb1b2 N;Cards;

-.095112 .05284463 -.279107 .05449605 .281868 .05323348 .059703 .05495897

.335794 .05427165 .114041 .05350677

.237149 .04621743 .131117 .04478027 .242897 .04526981 .243409 .04589909

1.03824 .004 400

;

Proc print;

Run;Chapter 6Comments: The material in Chapter 6 is more complicated than the first five chapters. Ideally the reader of Chapter 6 has exposure to structural equation models and matrices. If the reader does not have this prior exposure, it makes sense to skip the sections of this chapter dealing with deriving standard errors and instead focus on the examples including the two factor model and the socioeconomic status. Note that describing these more complicated models is usually easier to understand if done in regression form as the examples in Mplus and EQS. The problem with the regression form of specifying these models is that the role of error variances and residuals can be difficult to describe. One benefit of this chapter is that often students have their own data or can use published data sets like the one in exercise 6.4. In this way, the chapter can be made more compelling by having the reader use their own data.

The following paper describes a simulation study of different approaches to investigate three path mediation effects including Equation 6.23. Taylor, A. B., MacKinnon, D. P., & Tien, J.-Y. (in press). Tests of the three-path mediated effect. Organizational Research Methods.6.2a. The equations in matrix form:

ADVANCE \u 266.2b. The parameter estimates from LISREL are below:

LISREL Estimates (Maximum Likelihood)

BETA

GRADES EDUCEXPE OCCUASPI

-------- -------- --------

GRADES - - - - - -

EDUCEXPE 0.405150 - - - -

(0.032273)

12.554022

OCCUASPI 0.157912 0.549593 - -

(0.036799) (0.037601)

4.291224 14.616388

GAMMA

INTELLIG SIBLINGS FATHEDUC FATHOCCU

-------- -------- -------- --------

GRADES 0.525902 -0.029942 0.118966 0.040603

(0.030646) (0.029630) (0.037600) (0.037135)

17.160610 -1.010527 3.163956 1.093375

EDUCEXPE 0.160270 -0.111779 0.172719 0.151852

(0.032147) (0.026414) (0.033716) (0.033108)

4.985583 -4.231814 5.122768 4.586508

OCCUASPI -0.039405 -0.018825 -0.041333 0.099577

(0.033907) (0.027737) (0.035593) (0.034836)

-1.162145 -0.678684 -1.161246 2.858434

PHI


-------- -------- -------- --------

INTELLIG 1.000000

(0.051232)

19.519221

SIBLINGS -0.100000 1.000000

(0.036407) (0.051232)

-2.746735 19.519221

FATHEDUC 0.277000 -0.152000 1.000000

(0.037590) (0.036642) (0.051232)

7.368924 -4.148214 19.519221

FATHOCCU 0.250000 -0.108000 0.611000 1.000000

(0.037341) (0.036437) (0.042453) (0.051232)

6.695038 -2.964033 14.392382 19.519221

PSI

Note: This matrix is diagonal.


-------- -------- --------

0.650995 0.516652 0.556617

(0.033351) (0.026469) (0.028516)

19.519221 19.519221 19.519221

6.2c. There are many possible indirect effects. Here are three of them:

Intelligence to grades to occupational aspirations= .08297

Applying Equation 6.16:

EMBED Equation.3

= (.525402)2(.036799)2 +(.157912)2 (.0306462)+ 2 (.525402) (.157912) (.000000) = .0199312

= 4.16

Normal theory LCL=.0439, UCL = .1220 using Equations 3.4 and 3.5 and distribution of the product LCL= .0446, UCL = .1228 using Equations 4.37 and 4.38 and PRODCLIN.

Intelligence to educational expectations to occupational aspirations= .08808Applying Equation 6.16:

= (.160270)2(.037601)2 +(.549593)2 (.0321472)+ 2 (.160270) (.549543) (.00000) = .0186672

= 4.72

Normal theory LCL=.0515, UCL = .1247 using Equations 3.4 and 3.5 and distribution of the product LCL= .0525, UCL = .1257 using Equations 4.37 and 4.38 and PRODCLIN.

Applying Equation 6.16:

Number of siblings to educational expectations to occupational aspirations= -.061443

= (-.111779)2(.037601)2 +(.549593)2 (.0264142)+ 2 (-.111779) (.549593) (.000000) = .0151132

= -4.06

Normal theory LCL=-.0911, UCL = -.0318 using Equations 3.4 and 3.5 and distribution of the product LCL= -.0918, UCL =-.0325 using Equations 4.37 and 4.38 and PRODCLIN.

All three indirect effects tested were statistically significant. The indirect effects are consistent with intelligence leading to improved grades which leads to greater occupational aspirations and intelligence leads to greater educational expectations which leads to greater occupational aspirations. As in the achievement example described in the chapter, number of siblings is negatively associated with educational aspirations which are, in turn, positively related to occupational aspirations. The two indirect effects from intelligence to occupational aspirations, and, were tested for equality using Equation 6.18. Only the correlation (covariance) between and was nonzero. All other correlations involved in the formula were zero as shown below in a section of the covariance and correlation matrix among parameter estimates. Note, for example, the covariance, -.000573, between BE 3,1 and BE 3,2 is used in the calculation of the standard error comparing two mediated effects.

Part of the Covariance Matrix of Parameter Estimates

BE 2,1 BE 3,1 BE 3,2 GA 1,1 GA 1,2 GA 1,3

-------- -------- -------- -------- -------- --------

BE 2,1 0.001042

BE 3,1 0.000000 0.001354

BE 3,2 0.000000 -0.000573 0.001414

GA 1,1 0.000000 0.000000 0.000000 0.000939

GA 1,2 0.000000 0.000000 0.000000 0.000054 0.000878

GA 1,3 0.000000 0.000000 0.000000 -0.000179 0.000110 0.001414

GA 1,4 0.000000 0.000000 0.000000 -0.000120 0.000014 -0.000807

GA 2,1 -0.000548 0.000000 0.000000 0.000000 0.000000 0.000000

GA 2,2 0.000031 0.000000 0.000000 0.000000 0.000000 0.000000

Part of the Correlation Matrix of Parameter Estimates

BE 2,1 BE 3,1 BE 3,2 GA 1,1 GA 1,2 GA 1,3

-------- -------- -------- -------- -------- --------

BE 2,1 1.000000

BE 3,1 0.000000 1.000000

BE 3,2 0.000000 -0.413983 1.000000

GA 1,1 0.000000 0.000000 0.000000 1.000000

GA 1,2 0.000000 0.000000 0.000000 0.059255 1.000000

GA 1,3 0.000000 0.000000 0.000000 -0.155187 0.098581 1.000000

GA 1,4 0.000000 0.000000 0.000000 -0.105199 0.012960 -0.578140

GA 2,1 -0.527960 0.000000 0.000000 0.000000 0.000000 0.000000

GA 2,2 0.036583 0.000000 0.000000 0.000000 0.000000 0.000000

= .018667 + .015113 - (-.11178)(.16027) (-.00057282) = .025482The difference between the two mediated effects equals -.005116 with a standard error of .02548 so it does not appear that the equality of the two indirect effects can be rejected because the difference between the two mediated effects is about one fifth of the size of its standard error. Of course there are many other potential tests of equal mediated effects. Below are total indirect effects from the LISREL output. Note that many of the total indirect effects are statistically significant. In general there are substantial indirect effects of intelligence and fathers occupation on both educational experience and occupational aspirations. Number of siblings and fathers occupation had significant indirect effects on occupational aspirations but not educational expectations. And grades had a significant indirect effect on occupational aspirations. In general background variables had substantial relations with occupational aspirations. Kerckhoff (1974) and subsequent researchers present interpretations of these results including some discussion of the results that may or may not be consistent in other generations. AMBITION AND ATTAINMENT KERCKHOFF

Indirect Effects of X on Y


-------- -------- -------- --------

GRADES - - - - - - - -

EDUCEXPE 0.213069 -0.012131 0.048199 0.016450

(0.021029) (0.012043) (0.015710) (0.015102)

10.132186 -1.007269 3.068019 1.089252

OCCUASPI 0.288231 -0.072828 0.140201 0.098909

(0.026605) (0.018876) (0.024530) (0.023765)

10.833758 -3.858306 5.715393 4.161984

Indirect Effects of Y on Y


-------- -------- --------

GRADES - - - - - -

EDUCEXPE - - - - - -

OCCUASPI 0.222668 - - - -

(0.023381)

9.523449

6.4. There are three indirect effects of number of siblings on respondents income: through respondent education, (3-> 1 -> 3,= -.0456 (se=.0090, t = -5.05), respondent occupation, (3 -> 2 -> 3, =-.0326 (se=.0085, t= -3.65), and through respondent education to respondent occupation, (3 -> 1 -> 2 -> 3, = -.0703 (se=.0073, t=-9.62). Equation 6.23 was used to compute the standard error of the three-path mediated effect and Equation 6.16 was used to compute the standard error of the two-path mediated effects. All the indirect effects are much larger than their standard errors with relatively small confidence intervals, at least in part because of the large sample size. The results are consistent with a model whereby more siblings are associated with less respondent income because of less education and a lower SES occupation and also through the association that less education leads to a lower SES occupation. It is important to keep in mind that these relations are descriptive and may not reflect a causal relation among variables. The results do provide the basis for a future study, perhaps a randomized experiment that may address the descriptive relations observed among these variables. The order of parameters in the columns of the V and V matrices in Sobel (1987, p. 169) was 11, 12, 13, 21, 21, 22, 23, 31, 32, 31, 32, 33. The columns of the V and V matrices in Sobel (1986, p. 173) was 11, 12, 13, 21, 22, 23, 31, 32, 33, 21, 31, 32. The order of the parameters differs so the elements in the columns differ, making the V and V matrices different in the two articles. Note also that the order of the elements in the covariance matrix among the parameters must correspond to the different ordering of parameters. The rows of the V and V matrices correspond to all the elements in the and matrices, respectively. The matrices are different because the order of parameters is different.

6.6. Sobel (1982; 1986) and Stone and Sobel (1990) investigated the total indirect effects rather than specific indirect effects. Some specific indirect effects were investigated because there was only one mediator between the independent and dependent variable so the total indirect effect is the same as the specific indirect effect. Bollen (1987) investigated specific indirect effects but not for the achievement model discussed in this chapter. Stone and Sobel (1990) suggest sample sizes of 200 or more for recursive models and 400 or more for nonrecursive models based on the two models simulated in that paper. The achievement model discussed in this chapter was the recursive model studied in Stone and Sobel (1990). One of the total indirect effects studied in the Stone and Sobel article was the indirect effect of fathers occupation to respondents education to respondents occupational status, 1221. In this case the total indirect effect is the same as the specific indirect effect. Stone and Sobel conducted a simulation study where the population values for the achievement model discussed in this chapter were the estimates in earlier analyses of the model. A total of 500 replications at sample sizes of 50, 100, 200, 400, and 800 were used to ascertain how close observed estimates in the 500 replications for each sample size were to the population values. Table 1 in Stone and Sobel presents results on the relative bias of the indirect effect estimate and standard error as a function of sample size. Relative bias is the difference between the estimate and the population value divided by the population value. Relative bias was one measure used to evaluate the statistical performance of the total indirect effect and standard error estimators. For the indirect effect estimate of fathers occupation to respondents education to respondents occupational status, the relative bias was always less than .05 and was less than .01 at a sample size of 400 or greater. Similarly, the standard error for this total indirect effect had less than or equal to .05 relative bias at sample sizes of 100 or greater and relative bias less than .01 at a sample size of 200 or greater. Chapter 7

Comments: The three factor mediation model example described in the text had a nonzero covariance between the GA(1,1) and BE(2,1) coefficients in the mediated effect. That is, the coefficient, GA(1,1),was correlated with the coefficient, BE(2,1). As a result this correlation (actually the covariance) was included in the standard error of the mediated effect. The EQS program output in the CD accompanying this book has the correlation among the estimates. For the LISREL program with this book, add pc to get the covariance and correlation matrix among the parameter estimates and nd=6 to get numbers to 6 decimal places in the output. The LISREL output line should be OU MI RS EF MR SS SC pc nd=6. The section of the LISREL output with the covariance and correlation among the parameter estimates matrices is shown below. In the output, BE(2,1) corresponds to thecoefficient, GA(1,1) corresponds to the coefficient and GA(2,1) corresponds to the coefficient. The correlation between BE(2,1) and GA(2,1) is equal to .051518 and the covariance is equal .000231. Equation 6.16 in the text is given below with the covariance term added to incorporate the covariance between the coefficients where corresponds to in the equation and corresponds to in the equation.

EMBED Equation.3

= (-.41458)2(.048115)2 +(.26629)2 (.0929932)+ 2 (-.41458) (.26629) (.051518) = 0009602

so the standard error equals the square root of .0009602 which is .030988. The normal theory confidence limits are LCL = -.17114 and UCL = -.049665. Using the PRODCLIN program and the following input, -.41458 .092993 .26629 .048115 .051518 .05 for the following quantities:

coefficient, standard error of the coefficient,coefficient, standard error of the coefficient, correlation between and, and .05 for 95% confidence limits. The distribution of the product confidence limits are LCL = -.17114 and UCL = -.049665, which differ slightly from the normal theory confidence limits. Note that in most situations theandcoefficients from multiple regression are uncorrelated.

Selected LISREL Output for covariance and correlations among parameter estimates Covariance Matrix of Parameter Estimates

LY 2,1 LY 3,1 LY 5,2 LY 6,2 LX 2,1 LX 3,1

-------- -------- -------- -------- -------- --------

LY 2,1 0.005905

LY 3,1 0.003709 0.006753

LY 5,2 0.000000 0.000000 0.004714

LY 6,2 0.000000 0.000000 0.003361 0.005108

LX 2,1 0.000000 0.000000 0.000000 0.000000 0.063004

LX 3,1 0.000000 0.000000 0.000000 0.000000 0.037509 0.045651

BE 2,1 0.000818 0.000895 -0.000699 -0.000624 0.000000 0.000000

GA 1,1 0.001265 0.001586 0.000000 0.000000 -0.011984 -0.010174

GA 2,1 0.000002 -0.000051 -0.000004 -0.000003 0.000040 0.000034

PH 1,1 0.000000 0.000000 0.000000 0.000000 -0.027605 -0.023540

PS 1,1 -0.005462 -0.006410 0.000000 0.000000 0.000032 0.000045

PS 2,2 -0.000001 0.000029 -0.003639 -0.003462 0.000000 0.000000

TE 1,1 0.001223 0.001920 0.000000 0.000000 0.000000 0.000000

TE 2,2 -0.001574 0.001230 0.000000 0.000000 0.000000 0.000000

TE 3,3 0.000434 -0.002896 0.000000 0.000000 0.000000 0.000000

TE 4,4 0.000000 0.000000 0.000595 0.000313 0.000000 0.000000

TE 5,5 0.000000 0.000000 -0.001252 0.000269 0.000000 0.000000

TE 6,6 0.000000 0.000000 0.000921 -0.000673 0.000000 0.000000

TD 1,1 0.000000 0.000000 0.000000 0.000000 0.006766 0.005853

TD 2,2 0.000000 0.000000 0.000000 0.000000 -0.014395 0.009638

TD 3,3 0.000000 0.000000 0.000000 0.000000 0.007895 -0.009042

Covariance Matrix of Parameter Estimates

BE 2,1 GA 1,1 GA 2,1 PH 1,1 PS 1,1 PS 2,2

-------- -------- -------- -------- -------- --------

BE 2,1 0.002315

GA 1,1 0.000231 0.008648

GA 2,1 0.001054 -0.000264 0.004574

PH 1,1 0.000000 0.006509 -0.000022 0.015707

PS 1,1 -0.001267 -0.001211 -0.000093 0.000002 0.011585

PS 2,2 0.000420 0.000005 -0.000066 0.000000 -0.000002 0.005528

TE 1,1 0.000284 0.000468 -0.000007 0.000000 -0.001959 0.000004

TE 2,2 0.000003 0.000060 -0.000015 0.000000 -0.000139 0.000008

TE 3,3 -0.000026 -0.000474 0.000115 0.000000 0.001105 -0.000066

TE 4,4 -0.000083 0.000000 0.000000 0.000000 0.000000 -0.000402

TE 5,5 0.000132 0.000000 0.000001 0.000000 0.000000 0.000370

TE 6,6 -0.000075 0.000000 0.000000 0.000000 0.000000 -0.000210

TD 1,1 0.000000 -0.001561 0.000005 -0.003562 -0.000002 0.000000

TD 2,2 0.000000 0.000006 0.000000 0.000348 -0.000057 0.000000

TD 3,3 0.000000 0.000007 0.000000 0.000406 -0.000067 0.000000


TE 1,1 TE 2,2 TE 3,3 TE 4,4 TE 5,5 TE 6,6

-------- -------- -------- -------- -------- --------

TE 1,1 0.006577

TE 2,2 0.000162 0.006538

TE 3,3 -0.001292 -0.002777 0.006101

TE 4,4 0.000000 0.000000 0.000000 0.001910

TE 5,5 0.000000 0.000000 0.000000 -0.000438 0.002181

TE 6,6 0.000000 0.000000 0.000000 0.000249 -0.001630 0.003327

TD 1,1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

TD 2,2 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

TD 3,3 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000


TD 1,1 TD 2,2 TD 3,3

-------- -------- --------

TD 1,1 0.042451

TD 2,2 -0.000348 0.025794

TD 3,3 -0.000406 -0.014313 0.013286

Chapter 7 Three Factor Latent Variable Model

Correlation Matrix of Parameter Estimates

LY 2,1 LY 3,1 LY 5,2 LY 6,2 LX 2,1 LX 3,1

-------- -------- -------- -------- -------- --------

LY 2,1 1.000000

LY 3,1 0.587342 1.000000

LY 5,2 0.000000 0.000000 1.000000

LY 6,2 0.000000 0.000000 0.684983 1.000000

LX 2,1 0.000000 0.000000 0.000000 0.000000 1.000000

LX 3,1 0.000000 0.000000 0.000000 0.000000 0.699387 1.000000

BE 2,1 0.221285 0.226413 -0.211465 -0.181495 -0.000010 -0.000016

GA 1,1 0.177061 0.207521 0.000000 0.000000 -0.513427 -0.512048

GA 2,1 0.000436 -0.009204 -0.000784 -0.000673 0.002361 0.002352

PH 1,1 0.000000 0.000000 0.000000 0.000000 -0.877494 -0.879082

PS 1,1 -0.660405 -0.724697 0.000000 0.000000 0.001171 0.001958

PS 2,2 -0.000227 0.004796 -0.712822 -0.651540 0.000000 0.000000

TE 1,1 0.196257 0.288075 0.000000 0.000000 0.000000 0.000000

TE 2,2 -0.253329 0.185110 0.000000 0.000000 0.000000 0.000000

TE 3,3 0.072273 -0.451195 0.000000 0.000000 0.000000 0.000000

TE 4,4 0.000000 0.000000 0.198372 0.100306 0.000000 0.000000

TE 5,5 0.000000 0.000000 -0.390617 0.080542 0.000000 0.000000

TE 6,6 0.000000 0.000000 0.232584 -0.163248 0.000000 0.000000

TD 1,1 0.000000 0.000000 0.000000 0.000000 0.130834 0.132963

TD 2,2 0.000000 0.000000 0.000000 0.000000 -0.357080 0.280850

TD 3,3 0.000000 0.000000 0.000000 0.000000 0.272865 -0.367152


BE 2,1 GA 1,1 GA 2,1 PH 1,1 PS 1,1 PS 2,2

-------- -------- -------- -------- -------- --------

BE 2,1 1.000000

GA 1,1 0.051518 1.000000

GA 2,1 0.323864 -0.041930 1.000000

PH 1,1 -0.000001 0.558490 -0.002571 1.000000

PS 1,1 -0.244623 -0.121008 -0.012805 0.000121 1.000000

PS 2,2 0.117320 0.000749 -0.013093 0.000000 -0.000229 1.000000

TE 1,1 0.072871 0.062061 -0.001231 0.000000 -0.224452 0.000641

TE 2,2 0.000831 0.007917 -0.002654 0.000000 -0.015959 0.001383

TE 3,3 -0.006843 -0.065222 0.021863 0.000000 0.131471 -0.011393

TE 4,4 -0.039493 0.000000 -0.000146 0.000000 0.000000 -0.123846

TE 5,5 0.058569 0.000000 0.000217 0.000000 0.000000 0.106634

TE 6,6 -0.026914 0.000000 -0.000100 0.000000 0.000000 -0.049001

TD 1,1 0.000001 -0.081466 0.000375 -0.137937 -0.000073 0.000000

TD 2,2 0.000028 0.000427 0.000006 0.017309 -0.003319 0.000000

TD 3,3 0.000045 0.000694 0.000010 0.028108 -0.005390 0.000000


TE 1,1 TE 2,2 TE 3,3 TE 4,4 TE 5,5 TE 6,6

-------- -------- -------- -------- -------- --------

TE 1,1 1.000000

TE 2,2 0.024751 1.000000

TE 3,3 -0.203898 -0.439695 1.000000

TE 4,4 0.000000 0.000000 0.000000 1.000000

TE 5,5 0.000000 0.000000 0.000000 -0.214756 1.000000

TE 6,6 0.000000 0.000000 0.000000 0.098686 -0.605027 1.000000

TD 1,1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

TD 2,2 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

TD 3,3 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000


TD 1,1 TD 2,2 TD 3,3

-------- -------- --------

TD 1,1 1.000000

TD 2,2 -0.010529 1.000000

TD 3,3 -0.017098 -0.773138 1.000000

Here is a short SAS program to compute the standard error.

data a;

input a sea b seb rab;

ab=a*b;

sab=sqrt(a*a*seb*seb+b*b*sea*sea);

sabr=sqrt(a*a*seb*seb+b*b*sea*sea+2*a*b*(rab*sea*seb));

tsab=ab/sab;tsabr=ab/sabr;

covab=rab*sea*seb;sab=sqrt(a*a*seb*seb+b*b+sea*sea+2*a*b*(rab*sea*seb));+

lcl=ab-1.96*sabr;

ucl=ab+1.96*sabr;

cards;

-.414584 .092993 .266294 .048115 .0515

;

proc print;

run;7.2 Matrices for the two mediator model with three indicators of X, M1, M2, and Y. x, , and are the same as on page 179.

7.4. More measures of the same latent variable will improve reliability of the measure, all else being equal. The Spearman-Brown formula can be used to calculate reliability of a scale if items are added or removed (see Allen & Yen, 1979; Crocker & Algina, 1986; Lord & Novick, 1968). 7.6. Binary X, four indicators of the mediator and four indicators of the dependent variable. There is the indirect effect of to 1 to 2. If X is a binary variable coding exposure to a randomized intervention, then the mediated effect refers to the relation of X to the latent dependent variable through the latent mediating variable. There is also the indirect effect of 1 to 2 to Y, which would reflect the relation of the latent mediating variable to the latent dependent variable to each of the indicators (Y variables) of the latent dependent variable. The same matrices are needed for this situation as in the answer to exercise 7.1 but with new x and matrices shown below to reflect that there is one X variable. Typically in this model x11 will be fixed to 1 and 11 would be fixed to 0 to reflect that there is one measure of with perfect reliability. As described in section 7.12, an adjustment for measurement error may be obtained by fixing 11to 1 minus reliability times the variance of X.

Chapter 8Comment: It is difficult to give general guidelines for the estimation of longitudinal mediation models because of different hypotheses about growth over time and the specifics of research areas. The models in this chapter provide a starting point for models relevant for research. Note also that the longitudinal mediation models are first described in this chapter or have only recently been proposed. Future simulation studies and application of the models to real data should clarify the best set of models and model comparisons for your research. Note that the data sets used for illustration contain a subset of the participants in the original study. An additional exercise for this chapter would be to further analyze these data to find a good model. The data for these models are real so the data have the usual idiosyncrasies and challenges of applying models to real data. One of these idiosyncrasies is in the development of individual models for media and nutrition for the LGC model, the second slope for nutrition behavior is very small and may not be justified as a predictor. I kept the second slope in the model as it codes the delivery of the program. Readers may want to consider alternative longitudinal models for these data. Note that a program to estimate a latent difference score model is included on the CD accompanying this book (Chap8_LDS.mplus.inp).A good additional reading for this chapter is Maxwell, S. E. & Cole, D. A. (2007). Bias in cross-sectional analyses of longitudinal mediation. Psychological Methods, 12, 23-44.

This interesting article demonstrates the bias possible in cross-sectional mediation analysis. The authors do this by comparing the cross sectional indirect effect to the longitudinal indirect effect. One interesting task would be to apply the calculations in this article to the examples in this chapter and your own longitudinal data. It is also helpful to consider whether the longitudinal mediation relation described in the article is really the mediation relation that researchers wish to investigate. It is also useful to consider the points of this article for the case where X represents randomization to conditions. It would also be interesting to think of situations where cross-sectional mediation may be more important than longitudinal mediation.8.4. The autoregressive Model II gives the same chi-square as the autoregressive model described in the book because the contemporaneous mediated effects are equivalent to allowing correlated residuals for the mediator and dependent variable at each wave for this model. Adding the three additional parameters (paths from intentions to severity at each measurement after baseline) in Model III for these data significantly improves the fit of the model (103.257 - 83.694 = 19.563 with three degrees of freedom). However, the RMSEA is .117. When allowing errors in adjacent measures of intent and severity after the first wave, contemporaneous mediation, and correlated residuals at each wave, the model fit improves dramatically 8.75 (p = .1139) with a difference of 74.944 on seven degrees of freedom which is highly significant. The RMSEA is also acceptable, .042. However, there are several ambiguities in the model including that some relations between perceived severity and intention to use steroids becomes positive. Perhaps a latent growth curve model or a latent difference score model would be more appropriate for these data. It may also be helpful to construct a measurement model for intentions and perceived severity that would include the four measures of each construct as indicators of a latent construct. 8.6. The equations for the latent growth curve models would be the same as 8.21 to 8.29 with t or time equal to 5. The autoregressive equations would just include lag 1 relations among variables as in the autoregressive model equations described in the chapter.

Chapter 9Comments: This chapter only touches on the possibilities for multilevel mediation models. These models have many applications. Multilevel mediation models have only recently been applied so their usefulness is to be determined. The ability to investigate mediational processes at different levels is an exciting aspect of these models. There is plenty of research on methodology possible in this area as well. Effect size measures, ways to choose multilevel versus other longitudinal models, and models at three or more levels are just a few of the possibilities for future development. Note that Keenan Pituch and colleagues have conducted simulation studies of mediated effect standard errors in multilevel mediation models (Pituch, K. A., Stapleton, L. M., & Kang, J.-Y. (2006). A comparison of single sample and bootstrap methods to assess mediation in cluster randomized trials. Multivariate Behavioral Research, 41, 367-400).9.2. Equations for random slopes are described in Equations 9.15, 9.16, 9.17, 9.18, and 9.19. Include two more Level 2 predictors, one for the intercept in the Level 1 Equation 9.15 for M and one for the Level 1 Equation 9.16 for Y to model random intercepts. 9.4. The Sampson, Raudenbush, and Earls (1997) article modeled for data from 343 neighborhoods in Chicago, Illinois. There are many possible answers to this question just based on the description in Section 9.5. In the article, the first level of analysis was within person and represented the responses of each person to several questions, the second level of analysis was a model for each persons average response to the questions within neighborhoods, and the third level had a model where the mean of each neighborhood was predicted. The article examined the relation of social composition of the neighborhood (concentrated disadvantage, immigrant concentration, and residential instability) to the mediator of collective efficacy to violence (perceived violence, violent victimization, and homicide). A difference in coefficients method was used to assess mediation such that collective efficacy was added to the model that had concentrated disadvantage, immigrant concentration, and residential instability as predictors of violence. When collective efficacy was added to the model the coefficients for the three social composition variables were reduced consistent with a mediation effect. It is not exactly clear how the significance test was conducted although the Clogg, Petkova, and Haritou (1995) article was cited and this method consists of testing whether the coefficient relating the mediator to the dependent variable is statistically significant. It would be helpful to test whether the product of the coefficient relating social composition to collective efficacy and the coefficient relating collective efficacy to violence was statistically significant using product of coefficient methods described in this book. The test could be conducted at the neighborhood or level 3 of the multilevel model. More cities could be added to form an additional level of analysis; for example if the same data were obtained from 20 cities in the United States. Another level could be longitudinal data from the neighborhoods so that there were four waves of data for example within each neighborhood. There are many other levels that could be investigated. See the following article for a fascinating review of neighborhood effects: Sampson, R. J., Morenoff, J. D., & Gannon-Rowley, T. (2002). Assessing Neighborhood effects: Social processes and new directions in research. Annual Review of Sociology, 28, 443-478. See page 467-8 for the importance of indirect effects in this research.Chapter 10Comments: Models with mediation and moderation are an active area of methodological development and application. These models are important because these ideas most clearly match many hypotheses in psychology and other fields. A recent paper by Edward also describes moderation and mediation models (Edwards, J. R. & Lambert, L. S. (2007). Methods for Integrating Moderation and Mediation: A General Analytical Framework Using Moderated Path Analysis. Psychological Methods, 12, 1-22.) describes a method similar to the one in this chapter.

10.2. An important part of investigating mediation and moderation is to clearly specify effects to be tested prior to statistical analysis either based on theory or prior empirical work. Tests of moderation are also conducted to investigate whether a mediation process does not differ across groups to help provide evidence for generalizability of the results across groups. It is often helpful to test each moderator and mediator relation separately in order to understand the patterns in the data. Models with moderation and mediation are complex and often require detailed thinking to understand them. Plots of relations are also helpful. After these first tests, then models with both moderation and mediation are estimated including the models in this chapter. Note that sometimes large sample sizes are needed to detect effects with these models.

10.4. There are several options for the analysis of these data. Equations 10.15 and 10.16 were expanded to model two moderators and their interaction effects. Note that there are only 25 participants in each group defined by fit and recorded groups making power to detect effects low. There is an interaction effect on both the mediator and dependent variable where the fitness group effect depends on whether the participant was recorded or not as shown in the means below. For fit persons, more water was consumed if recorded but less water was consumed if not recorded. Similarly, more self-reported thirst was reported by fit persons if recorded but less thirst was reported by fit persons if not recorded.

Means for M and Y water consumption study

Y M

Not Recorded Normal 2.92 2.92

Not Recorded Fit 3.56 3.72

Recorded Normal 3.56 3.20

Recorded Fit

2.482.68

The results are consistent with more fit persons being more sensitive to their own thirst and drinking more water in response. The recording factor did not interact with the mediator or the fit moderator suggesting that being recorded did not alter the mediation relations. However sample size was very small. Below are the values of the mediation relations for each of the four groups defined by whether the experiment was recorded or not and whether the participants were in the fit or normal groups. For the most part the coefficients are comparable across groups with the exception of thecoefficient which appears to differ from the other groups. One very tentative interpretation of these results is that when fit persons are recorded some of them want to prove how fit they are by not consuming as much water thereby reducing the relation between self-reported thirst and water consumed.

se

se

se

seNot Recorded Normal .3527 .1910 .2387 .1945 .4456 .2563 .2558 .1490

Not Recorded Fit .3507 .1910 .0905 .2250 .3534 .1824 .7361 .2063

Recorded Normal .2808 .1934 .1041 .1886 .4397 .1772 .4 .2057

Recorded Fit

-.1066 .2783 -.0986 .3130 -.0144 .2374 .5492 .2499

Chapter 11Comments: Many readers from psychology may not be familiar with methods for categorical outcomes such as logistic or probit regression. A brief description of the odds ratio and predicted logits and odds are described to provide some preliminary information for readers unfamiliar with logistic regression. Survival analysis is another important methodology for categorical variables. It is likely that the same issues described in this chapter are also present in survival analysis. Here are some other recent papers on mediation with categorical dependent variables. Huang, B., Sivaganesan, S., Succop, P., Goodman, E. (2004). Statistical assessment of mediational effects for logistic mediational models Statistics in Medicine, 23, 2713-2728. and Li, Y., Schneider, J. A., & Bennett, D. A. (2007). Estimation of the mediation effect with a binary mediator. Statistics In Medicine, 26, 3398-3414.

11.2. = -3.985 (se=.755), odds ratio =2.7360, LCL =1.8787 UCL = 3.9846For each day of eating grilled meat per week, the person was 2.7360 times more likely to get pancreatic cancer.

= 1.050 (se = .229), odds ratio =2.8574, LCL = 1.8251,UCL = 4.4734For each day of eating grilled meat per week, the person was 3.8574 times more likely to get pancreatic cancer adjusted for the relation of blood fats to pancreatic cancer.

= 1.737 (se = .276), odds ratio = 5.6820, LCL = 3.3112,UCL = 9.7501For each one unit increase in blood fats a person is 5.6820 times more likely to get pancreatic cancer, adjusted for days of eating grilled meat.

= .2149 (se = .075), ,LCL = .0684, UCL = .3613For each day of eating grilled meat per week, persons had an increase of .2149 units of fat in their blood. 11.4. The mediated effect from logistic regression analysis of the data in 11.3 was equal to .1487 with a standard error of .04977, z = 2.9877 (LCL =.0512 UCL = .2462). With Mplus the mediated effect estimate was .088 with a standard error of .030 (z = 2.935) and LCL = .030 and UCL = .146. The z values are very close but the estimate and confidence limits differ because the Mplus coefficients are standardized. It is the standardization that makes Mplus the program of choice for mediation analysis for models with categorical and continuous measures. The asymmetric confidence intervals from the distribution of the product were LCL = .0609 and UCL = .2554. Using Mplus to obtain the percentile bootstrap asymmetric confidence limits discussed in Chapter 12 yields LCL = .036 and UCL = .155 . Note that the model indirect statement is used to obtain the estimate of the mediated effect and its standard error and the cinterval(symmetric) command generates confidence limits. The !bootstrap = 1000 (the ! indicates a comment in Mplus so this command was ignored in the analysis below) and cinterval(bootstrap) program commands are necessary for the bootstrap analysis.

title: Exercise 11.3

Data:

File is "c:\Bookfinal\Chap11_ex11.3.txt";

Variable:

names are id x m y;

usevariables are x m y;

categorical y;

analysis:

type is general;

!bootstrap=1000;

estimator is wlsmv;

iterations=1000;

convergence=.00005;

model:

Y on x m;

m on x;

model indirect:

y ind x;

output: samp mod standardized cinterval(symmetric)

Selected Mplus output

MODEL RESULTS

Estimates S.E. Est./S.E. Std StdYX

Y ON

X 0.085 0.072 1.169 0.085 0.099

M 0.261 0.066 3.926 0.261 0.300

M ON

X 0.337 0.073 4.618 0.337 0.344

Residual Variances

M 1.218 0.203 6.004 1.218 0.882

TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS

Estimates S.E. Est./S.E. Std StdYX

Effects from X to Y

Total 0.173 0.069 2.488 0.173 0.202

Total indirect 0.088 0.030 2.969 0.088 0.103

Specific indirect

Y

M

X 0.088 0.030 2.969 0.088 0.103

Direct

Y

X 0.085 0.072 1.169 0.085 0.099

CONFIDENCE INTERVALS OF MODEL RESULTS

Lower .5% Lower 2.5% Estimates Upper 2.5% Upper .5%

Y ON

X -0.102 -0.057 0.085 0.227 0.271

M 0.090 0.131 0.261 0.391 0.432

M ON

X 0.149 0.194 0.337 0.481 0.526

Residual Variances

M 0.695 0.820 1.218 1.615 1.740

TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS

Lower .5% Lower 2.5% Estimates Upper 2.5% Upper .5%

Effects from X to Y

Total -0.006 0.037 0.173 0.309 0.352

Total indirect 0.012 0.030 0.088 0.146 0.164

Specific indirect

Y

M

X 0.012 0.030 0.088 0.146 0.164

Direct

Y

X -0.102 -0.057 0.085 0.227 0.271

Chapter 12Comments: Over the last decade or so, resampling methods have been incorporated in many structural equation modeling programs including EQS and Mplus. In many of these programs, conducting a resampling analysis is as simple as including a keyword and the number of resamples. Resampling methods are a good general purpose approach to confidence limit estimation for mediation effects in simple and complex mediation models. It is important to note that the resampling methods yield more accurate confidence limits but the interpretational difficulties of mediation relations remain including temporal precedence, omitted variables, and measurement error. Note that a few simple keywords will allow for bootstrap estimation of all the data in this book that include individual data including the models with moderation and mediation, categorical dependent variable models, and structural equation models. The effects of non-normality on estimation of mediation is described in Finch, J., West, S., and MacKinnon, D. P. (1997) Effects on non-normality on mediated effect estimates. Structural Equation Modeling, 4, 87-107. Note also that there can be problems with the bootstrap if the process generating the data is not the same as the way the bootstrap samples are taken. The following article describes this issue for measures of fit in structural equation modeling: See Bollen, K. A. and Stine, R. A. (1992). Bootstrapping goodness-of-fit measures in structural equation models. Sociological Methods & Research, 21, 205-229.12.2. Approximate tests are used rather than exact randomization tests because of the sheer number of data sets that would need to be analyzed with an exact randomization test. Instead, an approximate randomization test takes a random sample of all the possible data sets.

12.4. It is not possible to conduct a typical resampling analysis for the achievement model described in Chapter 6 because the raw data are not available. If the raw data were available, a resampling method could be used by adding the bootstrap= 1000 command in the Analysis section and Cinterval(Bootstrap) Cinterval(bcbootstrap) in the Output section to get bootstrap and bias corrected bootstrap confidence intervals. 12.6. Fishers quote is important at several different levels. It is interesting that Fisher used the permutation test as a check on his t-test suggesting that Fisher himself thought the test the most accurate. In some respects, he is correct that a resampling test shows ignorance about the underlying distribution and other detailed aspects of a variable and area of research. However, as described in this chapter, there are instances where a resampling method is the method of choice if an analytical solution for a statistic is not known or the distribution of a statistic is known to be nonnormal, for example. Fishers other fascinating comment is about trusting persons who are experts in a substantive field rather than statisticians. I think this is probably true if the statistician is not acquainted with a field of substantive research. The ideal situation occurs when the substantive researcher is knowledgeable about statistical methods and the statistician is well acquainted with a substantive area of research. If this is not possible, the statistician and the substantive research must keep an open mind and be aware that statistical models are only abstractions and science requires rigorous tests of substantive ideas.

Chapter 13Comment: Causal inference in mediation models is an important area for future research, especially to identify ways to test the assumptions of the models and sensitivity to violation of assumptions. Small group activities may be very helpful in the classroom. Often the persons most adamant about causal inference approaches and very critical of mediation analysis are technical researchers who have not had much exposure to having to make decisions based on actual data. This is sensible given the difficult mathematical aspects of determining a cause. However, making a decision based on data entails taking into account the practical strengths and limitations of a research study in a way to make the best possible decision. Other researchers want to claim causal mediation relations without considering the sometimes insurmountable assumptions as described in this book. These two views suggest a series of small group activities that may help convey these issues. A group activity in a course of mediation analysis could require small groups to discuss two alternative approaches to causal inference. One approach argues that studies without randomization should not be conducted and that mediation analysis should not be attempted because of the difficulties associated with identifying true mediators. Resources may be better spent on other research projects where randomization is possible. Another point of argument could be that humans will never know true causes so it is pointless to make the establishment of causal inference as the most important aspect of research. Is causal inference needed to make sound decisions about research? This latter approach appears to be the one favored by eminent statistician, Karl Pearson, who considered it best to focus on the values of statistics and not focus on underlying causes as humans were not especially capable of identifying causes. Here is a quote from Pearsons famous book on the scientific method, Science in no case can demonstrate any inherent necessity in a sequence, nor prove with absolute certainty that it must be repeated (Pearson p. 113, Grammar of Science, 1957; originally published in 1892). There are several outstanding articles that I was not able to include in the book. Here are some them. They could be assigned as additional reading for this chapter. Kaufman, J. S., MacLehose, R. F., & Kaufman, S. (2004). A further critique of the analytical strategy of adjusting for covariates to identify biologic mediation. Epidemiologic Perspectives & Innovations, 1, 4. Available on-line at http://www.epi-perspectives.com/content/1/1/4

Judea Pearls papers are available at http://bayes.cs.ucla.edu/jp_home.html

Pearl, J. (June, 2001). Direct and Indirect Effects. UCLA Cognitive Systems Laboratory, Technical Report (R-273), In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, San Francisco, CA: Morgan Kaufmann, 411-420, 2001.The articles by James Robins are available at the website, http://biosun1.harvard.edu/~robins/Robins J. M. (1989). The control of confounding by intermediate variables. Statistics in Medicine, 8:679-701.

Robins, J. M., Hernn M., & Brumback B. (2000). Marginal structural models and causal inference in epidemiology. Epidemiology, 11(5):550-560.

Robins, J.M. (2003). Semantics of causal DAG models and the identification of direct and indirect effects. In: Highly Structured Stochastic Systems, Eds by P. Green, N. Hjort and S. Richardson, Oxford University Press.

Steyer, R. (2005). Analyzing individual and average causal effects via structural equation models. Methodology, 1, 39-54.

Ten Have, T., Elliott, M., Zanutto, E., & Datto, C. (2004). Causal models for randomized encouragement trials in treating primary care depression. Journal of the American Statistical Association, 99, 8-16.Ten Have, T., Joffe, M., Lynch, K., Maisto, S., Brown, G., & Beck, A. (2007), Causal mediation analyses with rank preserving models, Biometrics, 63, 926934.The Kaufman et al. (2004) article is critical of the investigation of direct and indirect effects based on causal inference for mediation models. It is one of the most accessible articles describing the limitations of studies of direct and indirect effects. In general the approach in this book is based on a program of research and distillation of information from many sources while the focus of this criticism is the difficulty of making causal inferences from a single study. Information on mediation relations is a goal of many research projects so the idea is to obtain as much clear information as possible. In many cases, avoiding a decision about data is not possible and a decision must be made. Second, mediation relations are indeed unknown and potentially very complicated. They force deeper statistical methods and deeper thinking. That is why the investigation of mediation relations is so interesting. 13.2. Holland applied the Rubin causal model to the case of mediation using an encouragement design. He made several important observations regarding the extent to which the usual approaches to investigating mediation are limited. His first contribution was a new notation for the causal structure relating the three variables in the single mediator model based on counterfactual conditions. He showed how this notation was different from the structural equation notation and he used this to demonstrate several challenging aspects of causal inference from the single mediator model. In particular, with X representing random assignment, the c and a coefficients reflect causal relations but the b and c coefficients do not because c and a are relations estimated between a variable randomized to participants and a dependent variable. With some assumptions, including the exclusion restriction that all of the effect of X on Y is through M, Holland demonstrated that b may be treated as a causal effect as well. 13.4. Satisfying the assumptions of instrumental variable methods is one of the challenges of the analysis. It is ideal to obtain some information from the data set regarding these assumptions and to conduct programs of research so that the reasonableness of the assumptions can be more accurately addressed.

a. X is exogenous is ensured if X reflects random assignment. There are certainly research situations where instrumental variables would be very useful but X is not a measure of random assignment. In this case some information must be used to satisfy this assumption.b. Effect of the instrument is large. This may be difficult to obtain in practice but makes sense given that the relation between the instrument and mediator must be substantial for prediction to be accurate.

c. SUTVA is probably realistic in many situations whereby the potential outcomes for a participant do not depend on their assignment.

d. Monotonicity is generally reasonable as the relation of X to M is likely to be the same for all participants (but see answer to Exercise 13.6 below).

e. Exclusion restriction is unrealistic in behavioral research but may be sensible in some situations and it may be possible to design a research study so that the exclusion restriction holds.

The extent to which violation of these assumptions affects research conclusions and estimates of effects is a critical area of future research. It is likely that these assumptions are violated in actual research studies. The important question is how much the assumption must be violated to invalidate a research conclusion.

13.6. Defiers define a unique group of persons who do the opposite of what they are assigned, i.e. get the treatment if in the control group and get the control if in the treatment group. It is probably unlikely that there are many persons like this in a real research project. One example of this situation may occur where there is reactance among subjects. For example, adolescents who feel that experimenters are reducing their freedom attempt to regain their freedom by doing the opposite of what the experimenter wishes. Another possible example may occur in situations where persons in the control group resent that treatment group participants got the treatment and find ways to get the treatment themselves (called resentful demoralization by Cook and Campbell, 1979). Alternatively, persons in the treatment

answers to even

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