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Lyytinen & Gaskin Mediation and Multi-group Analyses

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Mediation In an intervening variable model, variable X, is postulated to exert an effect on an outcome variable, Y, through one or more intervening variables called mediators (M) mediational models advance an X M Y causal sequence, and seek to illustrate the mechanisms through which X and Y are related. (Mathieu & Taylor) X X Y Y M M 2 2

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Why Mediation? Seeking a more accurate explanation of the causal effect the antecedent (predictor) has on the DV (criterion, outcome) focus on mechanisms that make causal chain possible Missing variables in the causal chain Intelligence Performance Intelligence Work Effectiveness Performance 3 3

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Conditions for mediation 4 4 (1) justify the causal order of variables including temporal precedence; (2) reasonably exclude the influence of outside factors; (3) demonstrate acceptable construct validity of their measures; (4) articulate, a priori, the nature of the intervening effects that they anticipate; and (5) obtain a pattern of effects that are consistent with their anticipated relationships while also disconfirming alternative hypotheses through statistical tests.

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Conditions for mediation 5 5 Inferences of mediation are founded first and foremost in terms of theory, research design, and the construct validity of measures employed, and second in terms of statistical evidence of relationships. Mediation analysis requires: 1) inferences concerning mediational X M Y relationships hinge on the validity of the assertion that the relationships depicted unfold in that sequence (Stone-Romero & Rosopa, 2004). As with SEM, multiple qualitatively different models can be fit equally well to the same covariance matrix. Using the exact same data, one could as easily confirm a Y M X mediational chain as one can an X M Y sequence (MacCallum, Wegener, Uchino, & Fabrigar, 1993).

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Conditions for mediation 6 6 2) experimental designs is to isolate and test, as best as possible, X Y relationships from competing sources of influence. In mediational designs, however, this focus is extended to a three phase X M Y causal sequence requiring random assignments to both X and M and related treatments Because researchers may not be able to randomly assign participants to conditions, the causal sequence of X M Y is vulnerable to any selection related threats to internal validity (Cook & Campbell, 1979; Shadish et al., 2002). To the extent that individuals status on a mediator or criterion variable may alter their likelihood of experiencing a treatment, the implied causal sequence may also be compromised. For example, consider a typical: training self-efficacy performance, mediational chain. If participation in training is voluntary, and more efficacious people are more likely to seek training, then the true sequence of events may well be self-efficacy training performance. If higher performing employees develop greater self-efficacy (Bandura, 1986), then the sequence could actually be performance efficacy training. If efficacy and performance levels remain fairly stable over time, one could easily misconstrue and find substantial support for the training efficacy performance sequence when the very reverse is actually occurring. (Mathieu and Taylor 2006)

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Conditions of mediation 7 7 It is a hallmark of good theories that they articulate the how and why variables are ordered in a particular way (e.g., Sutton & Staw, 1995; Whetten, 1989). This is perhaps the only basis for advancing a particular causal order in non-experimental studies with simultaneous measurement of the antecedent, mediator, and criterion variables (i.e., classic cross-sectional designs). Implicitly, mediational designs advance a time-based model of events whereby X occurs before M which in turn occurs before Y. It is the temporal relationships of the underlying phenomena that are at issue, not necessarily the timing of measurements In other words, in mediation analyses, omitted variables represent a significant threat to validity of the X M relationship if they are related both to the antecedent and to the mediator, and have a unique influence on the mediator. Likewise omitted variables (and related paths) may lead to conclude falsely that no direct effect X Y exists, while in fact it holds in the population

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Importance of theory Cause and effect Performance Self-efficacy Training Self-efficacy Training Performance Training Self-efficacy Performance Training Performance Self-efficacy 8 8

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Types of Mediation X X M M Y Y Indirect Effect M M X X Y Y Partial Mediation X X Y Y M M Full Mediation Significant Path Insignificant Path 9 9

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More complex mediation structures 10 X X M1M1 M1M1 Y Y Chain Model M2M2 M2M2 M3M3 M3M3 X X M1M1 M1M1 Y Y M2M2 M2M2 M3M3 M3M3 Parallel Model

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Hypothesizing Mediation All types of mediation need to be explicitly and with good theoretical reasons and logic hypothesized before testing them Indirect Effect You still need to assume and test that X has an indirect effect on Y, though there is no effect in path X Y X has an indirect, positive effect on Y, through M. Partial or Full M partially/fully mediates the effect of X on Y. The effect of X on Y is partially/fully mediated by M. The effect of X on Y is partially/fully mediated by M 1, M 2, & M 3. 11

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Statistical evidence of relationships. 12 Each type of mediation needs to be backed by appropriate statistical analysis Sometimes the analysis can be based on OLS, but in most cases it needs to be backed by SEM based path analysis There are four types of analyses to detect presence of mediation relationships 1. Causal steps approach (Baron-Kenny 1986) (tests for significance of different paths) 2. Difference in coefficients (evaluates the changes in betas/coefficients and their significance when new paths are added to the model) 3. Product of effect approach (tests for indirect effects a*b- this always needs to be tested or evaluated using bootstrapping) 4. Sometimes evaluating differences in R squares

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Statistical evidence of relationships 13 Convergent validity is critical for mediation tests as this forms the basis for reliability especially poor reliability of mediator as to the extent that a mediator is measured with less than perfect reliability, the M Y relationship would likely be underestimated, whereas the X Y would likely be overestimated when the antecedent and mediator are considered simultaneously (see Baron & Kenny 1986) Discriminant validity must be gauged in the context of the larger nomological network within which the relationships being considered are believed to reside. Discriminant validity does not imply that measures of different constructs are uncorrelated the issue is whether measures of different variables are so highly correlated as to raise questions about whether they are assessing different constructs. It is incumbent on researchers to demonstrate that their measures of X, M, and Y evidence acceptable discriminant validity before any mediational tests are justified.

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Statistical evidence of relationships 14

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Statistical evidence of the relationships 15 In simple partial mediation mx is the coefficient for X for predicting M, and ym.x and yx.m are the coefficients predicting Y from both M and X, respectively. Here yx.m is the direct effect of X, whereas the product mx * ym quantifies the indirect effect of X on Y through M. If all variables are observed then yx = yx.m + mx * ym or mx * ym = yx - yx.m Indirect effect is the amount by which two cases who differ by one unit of X are expected to differ on Y through Xs effects on M, which in turn affects Y Direct effect part of the effect of X on Y that is independent of the pathway through M Similar logic can be applied to more complex situations

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What would be the paths here? 16

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Statistical analysis 17 The testing of the existence of the mediational effect depends on the type of indirect effect The lack of direct effect X Y ( yx is either zero or not significant) is not a demonstration of the lack of mediated effect Therefore three different situations prevail (in this order) 1. The presence of a indirect effect ( mx * ym is significant) 2. The presence of full mediation ( yx is significant but yx.m is not) 3. The presence of partial mediation ( yx is significant and yx.m is non zero and significant)

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Testing for indirect effect 18

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Testing for full mediation 19

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Testing for partial mediation 20

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Observations of statistical analysis 21 The key is to test for the presence of a significant indirect effect just demonstrating the significant of paths yx, yx.m, mx.y, and mx is not enough One reason is that Type I testing of statistical significance of paths is not based on inferences on indirect effects as products of effects and their quantities Can be done either using Sobel test (see e.g. www.quantpsy.org) or bootstrappingwww.quantpsy.org Sobel tests assumes normality of product terms and relatively large sample sizes (>200) Lacks power with small sample sizes or if the distribution is not normal

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Bootstrapping 22 Bootstrapping (available in most statistical packages, or there is additional code to accomplish it for most software packages) Samples the distribution of the indirect effect by treating the obtained sample of size n as a representation of the population as a minitiature and then resampling randomly the sample with replacement so that a sample size n is built by sampling cases from the original sample by allowing any case once drawn to be thrown back to be redrawn as the resample of size n is constructed mx and ym and their product is estimated for each sample recorded The process is repeated for k times where k is large (>1000) Hence we have k estimates of the indirect effect and the distribution functions as an empirical approximation of the sampling distribution of the indirect effect when taking the sample of size n from the original population Specific upper and lower bound for confidence intervals are established to find i th lowest and j st largest value in the ordered rank of value estimates to reject the null hypothesis that the indirect effect is zero with e.g. 95 level of confidence

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Observations of statistical analysis 23 In full and partial mediation bivariate X Y (assessed via correlation r YX or coefficient yx ) must be nonzero in the population if the effects of X on Y are mediated by M Hence establishing a significant bivariate is conditional on sample size For example Assume that N=100 and sample correlations r XM =.30 and r MY =.30 and both would be significant at p