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Approaches to Improving Causal

Inference from Mediation Analysis David P. MacKinnon, Arizona State University

Pennsylvania State University February 27, 2013

Background Traditional Mediation Methods Modern Causal Inference Methods Example Traditional and Causal Inference Methods Summary *Supported by National Institute on Drug Abuse and collaborations…

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Mediating Variable 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) Increasing exercise skills increases self-efficacy

which increases physical activity (Bandura, 1977) Exposure to an argument affects agreement with the

argument which affects behavior (McGuire, 1968)

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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

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Single Mediator Model

MEDIATOR

M

INDEPENDENT VARIABLE

X Y

DEPENDENT VARIABLE

a b

c’

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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.

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S-O-R Mediator Model

Mental and other Processes

M

Stimulus

X Y

Response

a b

c’

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Three-variable Models Complex even though only one more variable is added to a two-variable model. Next complex model after randomized X to Y relation. Often includes hidden assumptions about how variables are related. Usually only one variable is randomized. Special names for third-variable effects: mediator, confounder, covariate, moderator.

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Mediator Definitions A mediator is a variable in a chain whereby an

independent variable causes the mediator which in turn causes the outcome variable (Sobel, 1990) The generative mechanism through which the

focal independent variable is able to influence the dependent variable (Baron & Kenny, 1986) A variable that occurs in a causal pathway from

an independent variable to a dependent variable. It causes variation in the dependent variable and itself is caused to vary by the independent variable (Last, 1988)

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Applications

Two overlapping applications of mediation analysis: (1) Mediation for Explanation (2) Mediation by Design

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Mediation for Explanation • Observed relation and try to explain it. • Elaboration method described by Lazarsfeld

and colleagues (1955; Hyman, 1955) where third variables are included in an analysis to see if/how the observed relation changes.

• Replication (Covariate) • Explanation (Confounder) • Intervening variable (Mediator) • Specification (Moderator)

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Mediation by Design

• Select mediating variables that are causally related to an outcome variable.

• Intervention is designed to change these mediators.

• If mediators are causally related to the outcome, then an intervention that changes the mediator will change the outcome.

• Common in applied research like prevention and treatment.

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Mediation in Intervention Research Theory

• Mediation is important for intervention science. Practical implications include reduced cost and more effective interventions if the mediators of programs are identified. Mediation analysis is an ideal way to test theory.

• A theory based approach focuses on the processes underlying interventions. Mediators play a primary role. Action theory corresponds to how the program will affect mediators. Conceptual Theory focuses on how the mediators are related to the dependent variables (Chen, 1990, Lipsey, 1993).

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Intervention Mediation Model

MEDIATORS

M1, M2, M3, …

INTEREVENTION PROGRAM

X Y

OUTCOMES

Action theory

If the mediators selected are causally related to Y, then changing the mediators will change Y.

Conceptual Theory

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Mediation Regression Equations

For the single mediator model information from some or all of three functional equations are used. Now parametric and nonparametric

approaches. Regression equations are shown next. Can

also be more generally described as Expectations.

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Equation 1

MEDIATOR

M

INDEPENDENT VARIABLE

X Y

DEPENDENT VARIABLE

c

1. The independent variable is related to the dependent variable:

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Equation 2

MEDIATOR

M

INDEPENDENT VARIABLE

X Y

DEPENDENT VARIABLE

2. The independent variable is related to the potential mediator:

a

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Equation 3

MEDIATOR

M

INDEPENDENT VARIABLE

X Y

DEPENDENT VARIABLE

a

3. The mediator is related to the dependent variable controlling for exposure to the independent variable:

b

c’

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Mediated Effect Measures

Indirect Effect = Mediated effect = ab = c-c’

Direct effect= c’ Total effect= ab+c’=c

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Social Science Equations with Covariate C.

E[Y|X=x, C=c] = i1+ c X + c2 C E[Y|X=x, M=m, C=c] = i2+ c’ X + b M + c3 C E[M|X=x, C=c] = i3+ a X + a2 C With XM interaction E[Y|X=x, M=m, C=c] = i4+ c’ X + b M + h XM + c4 C

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Identification Assumptions 1. No unmeasured X to Y confounders given covariates. 2. No unmeasured M to Y confounders given covariates. 3. No unmeasured X to M confounders given covariates. 4. There is no effect of X that confounds the M to Y relation. VanderWeele & VanSteelandt (2009)

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Assumptions Reliable and valid measures. 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. No moderator effects. The relation from X to M

and from M to Y are homogeneous across subgroups or other participant characteristics.

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Significance Testing and Confidence Limits

Product of coefficients estimation, ab, of the mediated effect and standard error is the most general approach with best statistical properties for the linear model given assumptions. Best tests are the Joint Significance, Distribution of the Product, and Bootstrap for confidence limit estimation and significance testing again under model assumptions.

For nonlinear models and/or violation of model assumptions, the usual estimators are not necessarily accurate. New developments based on potential outcome approaches provide more accurate estimators (Robins & Greenland, 1992; Pearl, 2001).

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Causal Inference in Mediation Assumptions of true causal relations and self-

contained/comprehensive model. Blalock’s (1979) presidential address states that ~50 variables are involved in sociological phenomenon;Weinstein’s comprehensive vs limited health psychology models. How many variables are relevant in a research area?

Problem with mediation analysis because M is not randomly assigned but is self-selected (Holland, 1988; Judd & Kenny, 1981; James, 1980; James & Brett, 1984; MacKinnon, 2008).

Active research area. Frangakis & Rubin, (2002); Pearl (2009), Geneletti (2007), Imai et al., 2010, VanderWeele, 2010….

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Modern Causal Inference in Mediation

Introduction of counterfactual/potential outcome model illustrates problems with the causal interpretation of results from mediation analysis.

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.

Problem with mediation analysis because M is not randomly assigned but is self-selected.

This approach has led to several innovative methods.

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Randomized Two Group Design Unit (individual level) have observed data for one of

two conditions but not the other—the fundamental problem of causal inference.

Randomization of a large number of persons solves the fundamental problem of causal inference. The average in each group can be compared and it is an estimator of a causal effect. It is called an average causal effect (ACE).

Randomization does not solve the problem for mediation because some potential outcomes never exist.

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For a participant in the treatment group, their mediator value if they were in the control group is not observable because they are in the treatment group. That is, we will never get the value M for the control group if instead they were in the treatment group. We will never get the value of M for the treatment group, if instead they were in the control group because they are in the treatment group. One solution is the natural indirect effect is to use the value of M in the control group as the counterfactual to compare the value of M in the treatment group (Robins & Greenland, 1992; Pearl, 2001). effect.

Problem with Mediation Analysis

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Analytical Causal Inference Methods Natural Indirect Effects (Pearl 2001; Robins & Greenland,

1992, Imai et al., 2010; VanderWeele, 2010) Inverse Probability Weighting (Coffman, 2011; Robins,

Hernán & Brumbeck, 2000) Principal Stratification (Frangakis & Rubin, 2002; Jo, 2008) Instrumental Variable Methods (Angrist et al., 1996;

Holland, 1988; MacKinnon, 2008; Sobel, 2007) G-estimation (ten Have et al., 2007; Robins, 1989) Decision Theory (Didelez et al., 2006, Geneletti, 2007) Stochastic Processes (Steyer,2012; Mayer et al., 2012) Focus on dealing with confounding. Active area of research …

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Which method is best? What are the best methods to find mediation when it

exists and not find it when it does not exist? How can we assess the accuracy of mediation methods? (1) Mathematics (2) Statistical simulations (3) Use by Applied Researchers (4) Applications to known mediation theory examples. This

is what is considered in this talk.

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Memory Experiment • It is well known in human memory research that making an

image increases word recall more than just repeating the words over and over (Craik, 2002; Paivo, 1971; Richardson, 1998)

• This is a situation where the mediation process is assumed known– Manipulation to increase imagery (X) increases imagery (M) which increases word recall (Y).

• Participants hear a list of 20 words with 10 seconds between the words.

• Participants are randomly assigned to either make images of the words or repeat the words.

• A way to validate mediation methods.

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Rehearsal Manipulation

Subjects in the Primary Rehearsal Condition P condition As you hear each word, repeat it over and over until you hear the next word. Subjects in the Secondary Rehearsal Condition S Condition As you hear each word, make an image out of the word and the other words that you hear. For example, if you heard the words camel and woman, you would imagine a woman riding a camel.

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Experiment Scoring

• I is answer to, “I made mental images of the words.”

• R is answer to, “I repeated the words over and over.”

• Scale 1-Not at all to 9-Absolutely

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Experiment Order of Words Read Nice, Rock, Chair, Box, Calm, Think, Jump, Car,

Happy, Flag, Star, Study, Talk, Clock, Stick, Caution, Rope, Send, Pole, Quick

Concrete Rock, Chair, Box, Car, Flag, Star, Clock, Stick, Rope,

Pole Not Concrete Nice, Calm, Think, Jump, Happy, Study, Talk,

Caution, Send, Quick

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Data are… ID number X primary or second rehearsal condition M self-reported imagery score R self-reported repetition score Y total number of words recalled Some data…. ID X M R Y 1 0 3 6 9 2 0 5 7 6 3 1 6 2 12 …

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Memory Experiment Mediator Model

Imagery

M

Rehearsal Group

X Y

Words Recalled

a b

c’

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Memory Experiment: Imagery Mediator

Imagery M

Group

X Y

Recall

3.558 (.674) .614 (.218)

.569 (1.319)

Mediated effect = 2.185* ( =.880) N=39, average M =6.38 (SD=2.93) Y =13.62 (SD=4.11)

bas ˆˆba ˆˆ

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Memory Experiment: Repetition Mediator

Repetition M

Group

X Y

Recall

-2.90* (.532) .050 (.300)

2.662+ (1.371)

Mediated effect = -.145 ( =.871) bas ˆˆba ˆˆ

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Covariates • Adjusted for Verbal GRE, Quantitative GRE, and Sex. • Minimal change in coefficients. Imagery Mediated effect = 2.177* (.944) Repetition Mediated effect = -.360 (.762) Evidence for Specificity of mediation because effect was observed for imagery but not for repetition. But there could be other confounders… Note: Interaction of X and M was nonsignificant.

Two Mediator Model Straightforward extension of the single mediator

case to include both mediators in the same model. The product of coefficients methods is

straightforward for models with multiple mediators but difference and causal step methods can work, somewhat. Note the covariance between the two mediators,

estimated in the next figure, is not shown to make the diagram easier to read.

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Two Mediator Model

Imagery

M1

X Y Repetition

M2 -2.91** (.532)

INDEPENDENT VARIABLE

Recall

3.51** (.673)

.379 (.304)

1.32 (1.44)

.702** (.239)

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Mediated Effect Measures

11ˆˆ ba

11ˆˆ ba

22ˆˆ ba

= (3.505) (.702) = 2.46 for mediation through imagery and

= (-2.91) (.379) = -1.101 for mediation through repetition. The total mediated effect of ( 2.46) plus (-1.101) equals 1.359 which is equal to .

The mediated effect (s = .953 ) was statistically significant (z = 2.58 ) and the mediated effect (s = .912) was not (z = -1.21).

Imagery LCL=.593, UCL=4.04 and Repetition LCL=-2.89, UCL = .686

With covariates:

Mediated effect through imagery = 2.666* (1.016)

Mediated effect through repetition = -1.444 (.775)

11ˆˆ ba

22ˆˆ ba

11ˆˆ ba11

ˆˆ ba22

ˆˆ ba22

ˆˆ ba22

ˆˆ ba

'ˆˆ cc −

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Followup Experiments Another question is, “What is the best next study or studies to

conduct after a statistical mediation analysis to test mediation theory.”

1) Designs to address Consistency of the mediation relation. 2) Designs to address Specificity of the mediation relation.

MacKinnon, 2008; MacKinnon & Pirlott, 2010; 2012; related to

pattern matching (Shadish et al., 2002; Mark, 1986)

More detailed and different manipulations of imagery with corresponding effects.

Different types of concrete words that differ in ease of making images. Persons differ in imagery. 41

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Replication Studies • Conduct replications of the same experiment. • Investigate the consistency of results across different

research participants. • Conduct integrated data analysis, meta analysis. • Analogous to Moderation of a mediated effect. The

moderator is research study. It is a nominal moderator. • Data from eight different memory studies conducted

with different groups of participants.

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Path Model for Mediation in Two Studies.

X M Y

astudy1 bstudy1

c’study1 Study 1

X M Y

astudy2 bstudy2

c’study2

Study 2

Mediated effect:

astd1bstd1

Mediated effect:

astd2bstd2

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Estimates of a, b, ab, and N

Obs a sa b sb ab N 1 2.54500* 0.84180 0.69580* 0.15350 1.77100* 45 2 3.92420* 0.77710 0.36685 0.19522 1.43820 43 3 3.23813* 0.50524 0.65524* 0.12466 2.12175* 79 4 5.28333* 0.79539 0.49360 0.38804 2.60785 22 5 3.88333* 0.76949 0.71480* 0.20622 2.77580* 35 6 3.74561* 0.47772 0.38900* 0.15988 1.45704* 77 7 5.74825* 0.60274 0.31689 0.35127 1.82156 24 8 3.55833* 0.68940 0.61406 * 0.22613 2.18503* 44 Ave 3.99 0.6824 0.5308 0.2256 2.022 44.4 *p<.05; complete cases Average standardized effect size was .596. Using average of a, sea, b, seb for distribution of the product, LCL=.338 and UCL=4.196

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Causal Approaches • Inverse Probability Weighting (Coffman, 2011; Robins et

al., 2000) • Causal estimators and delta standard errors (Valeri and

VanderWeele, 2013) • Plots of sensitivity to confounders. Also in traditional

mediation analysis but emphasized more in modern causal inference approaches.

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Causal Estimators

Natural Indirect Effect formulas (Pearl, 2001) and standard errors derived via the multivariate delta method. Analysis with the same three covariates.

Variable Indirect Effect (se) LCL UCL Imagery 2.177 (1.023) .171 4.18 Repeat -.360 ( .824) -1.984 1.263 See Valeri & VanderWeele (2013).

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Inverse Probability Weighting (IPW)

Method to adjust results for confounders. Assumes no unmeasured confounding. Weights observations as a way to deal with

confounding, missing data etc. Here weights are used to adjust for confounding of

the M to Y relation; weights can differ for each person.

Uses GENMOD to conduct weighted regression analysis.

See Robins, Hernan, & Brumback (2000) and also Coffman (2011).

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Inverse Probability Weighting Results for Imagery

from proc genmod Standard 95% Confidence Parameter Estimate Error Limits Z Pr > |Z| Intercept 9.6814 1.3600 7.0158 12.3469 7.12 <.0001 exposure 1.7479 1.4291 -1.0530 4.5488 1.22 0.2213 mediator 0.5005 0.2434 0.0235 0.9775 2.06 0.0397 Weights ranged from .81 to 1.34 Imagery mediated effect = 1.90; LCL=.087 UCL=4.051 Repetition mediated effect = -.138; LCL = -1.664 UCL = 1.356 Coefficient was .5233 in unweighted analysis

Sensitivity Analysis for Confounding Can be confounders of X to M and M to Y relation. Even

with randomization of X, there are possible confounders of the M to Y relation.

Two types of confounding: (1) Measured and (2) Unmeasured. If measures of confounders are available, you can make statistical adjustments. What about confounders that you have not measured?

Purpose of plots of confounder bias is to investigate how much results would change for different effects of a confounder. Methods refer to one confounder but it could be viewed as a number of confounders with a single effect.

New area…

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Directed Acyclic Graph

X Y

U

M

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Sensitivity Analysis for Confounding How will results change with confounding of the M to Y

relation? X is randomized. VanderWeele (2010), confounder effect on Y and difference in

proportions of the confounder between groups at level of M. Imai et al. (2010), confounder effect as the correlation between

error terms. Adaptation of Left Out Variables Error (LOVE; Mauro, 1990)

based on the correlation of a confounder with Y and the correlation of a confounder with M (also in Imai et al. but as r-squared).

Cox, et al. under review has examples and SAS and R programs to make these plots.

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Bias Correction for Memory Data X was primary or secondary rehearsal, M was self-

reported Imagery on a 1 to 9 scale and Y was the number of words recalled. γ -Corresponds to a binary confounder effect on the

number of words recalled. Perhaps a confounder like knowing mnemonic techniques such as the pegword. How large could that effect be? 6 words, 8 maximum. δ-And how should the percentage of persons with versus

without this skill differ between groups at levels of M, e.g., .05, .0, .2 maximum ? With randomization the percentages should not be very different across groups.

(VanderWeele, 2010) 52

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• Bias Plot Word Experiment

Gamma

0

2

4

6

8

10

12

14

16

18

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delta

0.0 0.1 0.2 0.3 0.4 0.5

Mediated Effect Sensitivity to Confounding VanderWeele(2010)

beta is relation of confounder U to Y delta is difference proportion in U between groups

ab=2.185

rabnew 0

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Imai et al., 2010 Based on Pearl’s natural and controlled effects describes

a method to conduct sensitivity analysis and has an R program to make all the calculations. Possible to call R from SAS but you need SAS 9.3.

The correlation between the error in the equation predicting M and the error in equation predicting Y is the measure of confounder bias.

An omitted variable that predicts both M and Y will lead to a correlation between these residuals.

Program gives the correlation that makes = δ(t)=0. Dotted line is observed value of ; ρ is the correlation

between residuals.

ba ˆˆba ˆˆ

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Left Out Variable Error Plots This method is based on calculating how different

correlations for U with Y and U and M lead to different values for the mediated effect.

The plot on the next page shows values where the indirect effect will be zero for certain values for rMU and rYU.

Adapted from Mauro (1980) based on early work on

consideration of confounding variables (James, 1980) Still a work in progress...

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Memory Data

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Summary Mediation analysis is central to many disciplines because

it provides information about the causal mediating mechanism between two variables.

Mediation analysis is used to extract the most information from a research study when measures of M are available in addition to measures of X and Y.

Programs of research address the many aspects of a mediation relation with a variety of information, clinical observations, qualitative data, experiments, and detailed consideration of assumptions.

Causal methods provide a general way to approach mediation analysis for linear and nonlinear models including sensitivity analysis.

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Question How do we know that our statistical methods give the

correct answers?

What other substantive mediation effects are considered known? Cognitive Dissonance,…..

When can we assume that there are no unmeasured confounders?

A man who carries a cat by the tail learns something he can learn in no other way. Mark Twain

A researcher who applies new statistical methods to real data learns something she can learn in no other way.

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Hypothesized Effects of Penn State Mediation Analysis Presentation

Penn State

Presentation

Entertainment and Mediation

Analysis

Causal Challenges of Mediation

Analysis

Traditional Methods

Causal Mediation Analysis

Memory Example

For Validation

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Thank You