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Introduction to Statistical Mediation

Introduction to Statistical Mediation. David P. MacKinnon Arizona State University Center for AIDS Prevention Studies, UCSF, June 12-13, 2007 Brown, Cheong, Fairchild, Fritz, Lockwood, Morgan-Lopez, Taylor, Tein, Williams, West, Wang, Yoon. Undergraduate Social Psychology Class

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Introduction to Statistical Mediation

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  1. Introduction to Statistical Mediation David P. MacKinnon Arizona State University Center for AIDS Prevention Studies, UCSF, June 12-13, 2007 Brown, Cheong, Fairchild, Fritz, Lockwood, Morgan-Lopez, Taylor, Tein, Williams, West, Wang, Yoon Undergraduate Social Psychology Class Graduate School UCLA Quantitative Psychology Drug Prevention Research at USC Support from the National Institute on Drug Abuse http://www.public.asu.edu/~davidpm/ MacKinnon, D. P. (2007) Introduction to Statistical Mediation Analysis, Mahwah, NJ: Erlbaum.

  2. Goals of CAPS Presentation • Describe many mediating variable examples. • Describe reasons for mediation analysis--it can help improve prevention programs and reduce their cost. It is also useful for testing theories. • Describe the latest methods to assess mediation. • Describe limitations of mediation analysis. • Describe experimental as well as non-experimental designs to investigate mediating variables.

  3. Overview of Presentation • Mediation Examples and Definition • Statistical Mediation Analysis • New tests for Mediation • Limitations of Statistical Mediation Analysis • Designs to address limitations of Mediation Analysis • Summary and Future Directions

  4. Psychology Example Stimulus: Multiply 24 and 16 Organism:You Response: Your Answer Organism as a Black Box Stimulus>Organism >Response (SOR) theory whereby the effect of a Stimulus on a Response depends on mechanisms in the organism (Woodworth, 1928). These mediating mechanisms translate the Stimulus to the Response. SOR theory is ubiquitous in psychology.

  5. Mediation Statements • If norms become less tolerant about smoking then smoking will decrease. • If you increase positive parental communication then there will be reduced symptoms among children of divorce. • If children are successful at school they will be less anti-social. • If unemployed persons can maintain their self-esteem they will be more likely to be reemployed. • If pregnant women know the risk of alcohol use for the fetus then they will not drink alcohol during pregnancy.

  6. Mediator Definition and Examples A variable that is intermediate in the causal process relating an independent to a dependent variable. Attitudes cause intentions which then cause behavior (Azjen & Fishbein, 1980) Prevention programs change norms which promote healthy behavior (Judd & Kenny, 1981) Exposure to an argument affects agreement with the argument which affects behavior (McGuire, 1968)

  7. More Mediation Examples PPsychotherapy induces catharsis, insight, and other mediators which lead to a better outcome (Freedheim & Russ, 1981) PPsychotherapy changes attributional style which reduces depression (Hollon, Evans, & DeRubies, 1991) PParenting programs reduce parents’ negative discipline which reduces symptoms among children with ADHD (Hinshaw, 2002).

  8. CAPS Mediation Examples Social problem solving affects psychological health which affects adherence to HIV medications (Johnson et al., 2006) Girl/boy friend in 7th grade affects peer norms about sexual behavior which affects sexual behavior in 9th grade (VanOss et al., 2006) Condom promotion program changes attitudes about sexual enjoyment from condoms which changes condom use (Choi et al., 2007). Affective regulation affects stimulant use and nonadherence to medications which affects viral load (Carrico et al., 2007).

  9. Mediation Analysis in Treatment and Prevention Research • Mediation is important for prevention and treatment research. Practical implications include reduced cost and more effective treatments. • Mediation analysis is based on theory for the processes underlying treatments. Action theory corresponds to how the treatment will affect mediators—the X to M relation. Conceptual Theory focuses on how the mediators are related to the outcome variables—the M to Y relation (Chen, 1990, Lipsey, 1993).

  10. Questions about mediators for treatment and prevention. • Are these the right mediators? Are they causally related to the outcome? Is self-esteem causally related to symptoms? Conceptual Theory • Can these mediators be changed? Can personality be changed? Action Theory • Will the change in these mediators that we can muster with our treatment be sufficient to lead to desired change in the outcome? Do we have the resources to change self-esteem in four sessions? Both Action and Conceptual Theory.

  11. Quotes about mediation analysis In the absence of a concern for such mediating or intervening mechanisms, one ends up with facts, but with incomplete understanding (Rosenberg, 1968, p. 63). .. much of what social psychologists do is attempt to understand how internal processes mediate the effect of the situation on behavior (Kenny, Kashy, & Bolger, 1998, p. 259).

  12. More Quotes Nursing “.. Should consider hypotheses about mediators …. that could provide additional information about why an observed phenomenon occurs” (Bennett, 2000). Children’s programs “.. Including even one mediator ….. in a program theory and testing it with the evaluation .. will yield more fruit….” (Petrosino, 2000) Child mental health “rapid progress … depends on efforts to identify … mediators of treatment outcome. We recommend randomized clinical trials routinely include and report such analyses” (Kraemer et al., 2002).

  13. “Everyone talks about the weather but nobody does anything about it.” (Mark Twain)

  14. Mediation Examples • Residential instability reduced collective efficacy which increased violence (neighborhoods, Sampson et al., 1997) • Anabolic prevention program affects norms regarding healthy behavior which reduced intentions to use steroids (Krull & MacKinnon, 1999; 2001). • Alcohol prevention program affected norms which reduced alcohol use, (Komro et al., 2001)

  15. Mediation is important because … Central questions in many fields are about mediating processes. Important for basic research on mechanisms of effects. Critical for applied research, especially prevention and treatment. Many interesting statistical and mathematical issues.

  16. 2, 3, or 4, variable effects Two variables: X Y, Y  X , X  Y are reciprocally related. Measures of effect include the correlation, covariance, regression coefficient, odds ratio, mean difference. Three variables: X M Y, XY M, YXM, and all combinations of reciprocal relations. Special names for third-variable effects, confounder, mediator, moderator/interaction. Four variables: many possible relations among variables, e.g., XZMY

  17. Mediator versus Confounder • Confounder is a variable related to two variables of interest that falsely obscures or accentuates the relation between them (Meinert & Tonascia, 1986). • The definition below is also true of a confounder because a confounder also accounts for the relation but it is not intermediate in a causal sequence. • In general, a mediator is a variable that accounts for all or part of the relation between a predictor and an outcome (Baron & Kenny, 1986, p.1176).

  18. Mediator versus Moderator • Moderator is a variable that affects the strength of the relation between two variables. The variable is not intermediate in the causal sequence so it is not a mediator. • Moderator is usually an interaction, the relation between X and Y depends on a third variable. There are other more detailed definitions of a moderator.

  19. Other names for Variables in the Mediation Model Antecedent to Mediating to Consequent (James & Brett, 1984). Initial to Mediator to Outcome (Kenny, Kashy & Bolger, 1998). Program to surrogate endpoint to ultimate endpoint (Prentice, 1989). Independent to Mediating to Dependent used in this presentation.

  20. Three ways to specify a model • Verbal description: A variable M is intermediate in the causal sequence relating X to Y. • Diagram • Equations

  21. Mediation Regression Equations -Start here with the simplest mediation model with one mediator. -Tests of mediation use information from some or all of three equations -The coefficients in the equations may be obtained using methods such as ordinary least squares regression, covariance structure analysis, or logistic regression.

  22. Single Mediator Model MEDIATOR M a b INDEPENDENT VARIABLE DEPENDENT VARIABLE c’ X Y

  23. Relation of X to Y MEDIATOR M INDEPENDENT VARIABLE DEPENDENT VARIABLE c X Y • The independent variable is related to the dependent variable: • Y = i1 + cX + e1

  24. Relation of X to M MEDIATOR M a INDEPENDENT VARIABLE DEPENDENT VARIABLE X Y 2. The independent variable is related to the potential mediator: M = i2 + aX + e2

  25. Relation of X and M to Y MEDIATOR M a b INDEPENDENT VARIABLE DEPENDENT VARIABLE c’ X Y 3. The mediator is related to the dependent variable controlling for exposure to the independent variable: Y = i3+ c’X + bM + e3

  26. Mediated Effect Measures Mediated effect=ab Standard error= Mediated effect=ab=c-c’ (MacKinnon et al., 1995) Direct effect=c’ Total effect=ab+c’=c Test for significant mediation: z’= Compare to empirical distribution of the mediated effect ab

  27. Assumptions I • For each method of estimating the mediated effect based on Equations 1 and 3 (c-c’) or Equations 2 and 3(ab): • Predictor variables are uncorrelated with the error in each equation. • Errors are uncorrelated across equations. • Predictor variables in one equation are uncorrelated with the error in other equation. • Reliable and valid measures • No omitted influences. • Normally distributed variables

  28. Assumptions II • Data are a random sample from the population of interest. • Coefficients, a, b, c’ reflect true causal relations and the correct functional form. • Mediation chain is correct: Temporal ordering is correct X before M before Y. Any mediation model is part of a longer mediation chain. The researcher decides what part of the micromediational chain to examine. • Homogeneous effects across subgroups: The relation from X to M and from M to Y are homogeneous across subgroups or other characteristics of participants in the study. Routine to test XM interaction in Equation 3. This means there are not moderator effects.

  29. Three Major Types of Single Sample Tests for the Mediation Effect • (1) Causal Steps: Series of tests described in Baron and Kenny (1986) for example. • (2) Difference in Coefficients: c-c’, e.g., from Clogg et al. (1992) • (3) Product of Coefficients: ab, e.g., from Sobel (1982) • See MacKinnon et al., Psychological Methods (2002) for a review and comparison of single sample tests

  30. Causal Steps Tests of Mediation • Judd & Kenny (1981), 3 Steps plus Step 4 c’ is nonsignificant • Baron & Kenny (1986), 3 Steps plus Step 4 drop from c to c’ • Test of whether the a and b paths are statistically significant (MacKinnon et al., 2002).

  31. Difference in Coefficients Significance test: tN-2= (c-c’)/sc-c’ • General formula for s2c-c’ : s2c-c’=s2c+ s2c’-2scc’ • Clogg, Petkova, and Shihadeh (1992) s2c-c’=(sc’|rxm|)2

  32. Product of Coefficients Formulas for the variance of ab • Multivariate delta variance: Sobel (1982), Folmer (1981) s2ab=a2s2b+ b2s2a • Exact variance: Aroian (1944) s2ab=a2s2b+ b2s2a+s2as2b • Unbiased variance: Goodman (1960) s2ab=a2s2b+ b2s2a-s2as2b • Test based on the distribution of the product of two random variables using critical values from Meeker et al. (1988) using a program called PRODCLIN.

  33. Empirical Sample size estimates for .8 power to detect the mediated effect • Test S-S S-M S-L M-S M-M M-L L-S L-M L-L • Baron/Kenny 20886 3039 1561 2682 397 204 1184 175 92 • (τ’ = 0) • a & b Joint 530 403 403 405 74 58 405 59 36 • Delta 667 422 412 421 90 66 410 67 42 • PRODCLIN 539 401 402 404 74 57 404 58 35 Note: Table entries are based on empirical simulation so they are not exact. Fritz & MacKinnon (2007).

  34. Reasons for Differences Among Methods • Requirement for significant total effect, c, and requirement that c’ is nonsignificant reduces accuracy of causal steps methods. • Assumption that the mediated effect divided by its standard error has a normal distribution is incorrect for some values. • Mediation is a test of two paths corresponding to a and b paths.

  35. Distribution of the Product • The mediated effect is the product of two coefficients a and b. The distribution of the product has a normal distribution only in special cases. • At low values of a and b, the distribution has excess kurtosis and skewness, e.g. when a and b are both zero, kurtosis is 6. It is not surprising that the confidence limits are inaccurate if the distribution is assumed to be normal. • One solution is to use the distribution of the product in statistical tests and confidence limits.

  36. PRODCLIN (distribution of the PRODuct Confidence Limits for the INdirect effect) • MacKinnon, Fritz, Williams, and Lockwood, (In Press, Behavior Research Methods) describes program to compute critical values for the distribution of the product. • Web location includes programs in SAS, SPSS, and R that access a FORTRAN program. http://www.public.asu.edu/~davidpm/ripl/Prodclin/ • Input a, sa, b, sb, correlation between a and b, and Type I error rate. Output includes the input values and normal and distribution of the product confidence limits.

  37. Critical Values for Distribution of the Product • Because the distribution of the product is not normal, there are different critical values for the distribution for each value of a/sa and b/sb. • The critical values are -1.96 and +1.96 for the 95% confidence interval from the normal distribution. There are different upper and lower critical values for the distribution of the product. Confidence limits and significance tests are more accurate using the critical values from the distribution of the product (MacKinnon et al. 2004).

  38. Example Calculations using the Distribution of the Product • For example, a = .3386, sa = .1224, b= .4510, sb = .1460. Enter these values in the PRODCLIN program. • PRODCLIN returns the critical value for the 2.5% percentile, Mlower =-1.6175 and Mupper = 2.2540 the critical value for the 97.5% percentile. • Use the critical values to calculate upper and lower confidence limits. • LCL= ab + Mupper sab= .1527 +(-1.6175) (.0741) UCL= ab + Mlower sab= .1527 + (2.2540 )(.0741) • Asymmetric Confidence Limits are (.0329, .3197)

  39. Resampling Methods • -Another good option for data that do not have a normal distribution is resampling methods (MacKinnon et al. 2004). • -Bootstrap method for mediated effects was described by Bollen & Stine (1991), Lockwood & MacKinnon (1998), MacKinnon et al., (2004) and Shrout & Bolger (2002) -Purpose is to use the data itself to form a distribution of a statistic (Manly, 1997). Does not make as many assumptions and can handle nonnormal distributions. -The value of a statistic in the observed sample is compared to the distribution of the statistic formed by resampling from the data a large number of times.

  40. Bootstrap Test for Mediation -Estimate the mediated effect in the sample. -Make a new data set by sampling N subjects data with replacement and estimating the mediated effect in each of a large number (1000) of bootstrap samples. -Determine significance level by locating the mediated effect for the observed sample in the distribution of the bootstrap sample. Find 2.5% and 97.5% values for confidence interval. -Bias-corrected bootstrap makes a correction for the difference between the observed and average bootstrapped mediated effect.

  41. Statistical Mediation Tests Summary Three general types of tests, causal steps, difference in coefficients, and product of coefficients. Tests differ substantially in Type I error and statistical power. Requirement of significant X to Y relation and assumed normal distribution of the mediated effect reduces power. Best tests are based on the distribution of the product and resampling methods.

  42. Quotes about mediation analysis In the absence of a concern for such mediating or intervening mechanisms, one ends up with facts, but with incomplete understanding (Rosenberg, 1968, p. 63). .. much of what social psychologists do is attempt to understand how internal processes mediate the effect of the situation on behavior (Kenny, Kashy, & Bolger, 1998, p. 259).

  43. Reasons for Mediation analysis in prevention research. 1. Manipulation check. Did the program change the mediators it was designed to change? 2. Program Improvement. What do the program effects on mediators suggest about program improvements? 3. Measurement Improvement. Is a lack of program effects due to poor measurement? 4. Delayed effects. Will program effects on the dependent variable emerge later? 5. Test the process of mediation. Was the theory-based prediction of mediation correct? 6. Practical implications. Can the program be redesigned to cost less and be more efficient?

  44. Interpretation of Mediation Results in prevention research. • Program effect on mediator but not outcome. The mediator may not be causally related to the outcome. Lack of power or insufficient measurement—explanations for all null effects below. • Program effect on the outcome but not the mediator. The program did not affect the intended mediator. Other constructs were mediators. • No program effects on the outcome or the mediator. Program was ineffective, lack of statistical power. • Program effects on the mediator and the outcome but nonsignificant mediation. The mediator may not be causally related to the outcome. • Program effects on the mediator and the outcome and significant mediation. Program was effective and there is evidence for the hypothesized mediating mechanism.

  45. Causal Inference for Mediation The Rubin Causal Model (RCM, Rubin, 1974) describes a general way to interpret evidence for causal relations, developed to interpret non-experimental as well as experimental research. It is a solution not a problem. Helpful because the RCM clearly displays limits and strengths of models, including mediation.

  46. Counterfactual Counterfactual is central to modern causal inference. The counterfactual refers to conditions in which a participant could serve, not just the condition that they did serve in. For example, for a participant in the treatment group, the counterfactual is the same participant in the control group. For a participant in the control group, the counterfactual is the same participant in the treatment group.

  47. Why b and c’ do not reflect a causal relation? Because M is not under experimental control, and M is both a dependent and independent variable, b and c’ do not necessarily represent causal effects. Need: The relation between M and Y for participants in the treatment group if they were in the control group; the relation between M and Y for control participants if they instead were in the treatment group. Coefficients b and c’ are not clearly causal effects, because M is not randomly assigned making the counterfactuals for these relations complicated.

  48. Causal inference for mediation -Counterfactual idea helps organize causal inference and highlights ambiguity regarding interpretation of c’ and b coefficients as causal effects. -In treatment and prevention, the M to Y, b, relation is based on prior research and theory. It is all we consider known. -Do we need to know the true causal structure to make good decisions based on research? Is a descriptive model sufficient? -Can we ever know the true causal relation among variables? “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).

  49. Improving Mediation Inference using the Rubin Causal Model Statistical approaches to improving causal inference from a mediation study: (I). Instrumental Variable Methods, Holland 1988; Sobel 2006. (II). Principal Stratification and latent classes; Frangakis & Rubin, 2002; Jo, 2006. Both approaches use aspects of the data such as no direct effect or stratifications of types of participants, such as compliers, never compliers etc. to improve inference regarding b and c’.

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