Mediation (statistics)

src: i.ytimg.com

In statistics, the mediation model is one that seeks to identify and explain the mechanism or process underlying the observed relationship between independent variables and dependent variables through the inclusion of a third hypothetical variable, known as the mediator variable (also mediation variables , intermediate variables , or mixed variables ). Instead of a direct causal relationship between independent variables and dependent variables, the mediation model proposes that the independent variable influences the mediator variable (unobservable), which in turn affects the dependent variable. Thus, the mediator variable serves to clarify the nature of the relationship between independent and dependent variables.

Mediation analysis is used to understand the known relationship by exploring the underlying mechanism or process in which one variable affects other variables through the mediator variable. Mediation analysis facilitates a better understanding of the relationship between independent and dependent variables when the variables do not seem to have a definite connection. They are studied by means of operational definitions and have no separate existence.

Video Mediation (statistics)

Baron and Kenny (1986) steps for mediation

Baron and Kenny (1986) set out several requirements that must be met to form a true mediation relationship. They are described below using real-world examples. See the diagram above for a visual representation of the overall mediation relationship to be explained.

Step 1:

Regres the dependent variable on the independent variable to confirm that the independent variable is a significant predictor of the dependent variable.

Variabel independen ${\ displaystyle \ to}  ÂÂ$ variabel dependen

${\ displaystyle Y = \ beta_ {10} \ beta_ {11} X \ varepsilon _ {1}}  ÂÂ$

• ? ₁₁ signifikan

Step 2:

Regress mediator on independent variables to confirm that independent variables are significant predictors of mediators. If the mediator is not related to the independent variable, then it is not possible to mediate anything.

Variabel independen ${\ displaystyle \ to}  ÂÂ$ mediator

${\ displaystyle Me = \ beta_ {20} \ beta_ {21} X \ varepsilon _ {2}}  ÂÂ$

• ? ₂₁ signifikan

Langkah 3:

Regreskan dependent variable dependatory pada kedua mediator dan variable independen untuk mengkonfirmasi bahwa mediator adalah prediktor signifikan dari variable dependen, dan variable independen signifikan sebelumnya pada Langkah # 1 sekarang sangat berkurang, jika tidak tidak signifikan.

• ? ₃₂ signifikan
• ? ₃₁ harus lebih kecil dalam nilai absolut daripada efek mediasi asli (< ₁₁ di atas)

Contoh

The following example, taken from Howell (2009), explains each step of the Baron and Kenny requirements to further understand how mediation effects are characterized. Step 1 and step 2 use simple regression analysis, while step 3 uses multiple regression analysis.

Step 1:

How you are cared for (ie, independent variables) predict how confident you are about taking care of your own children (ie, the dependent variable).

Bagaimana Anda diasuh ${\ displaystyle \ to}  ÂÂ$ percaya diri dalam kemampuan menjadi orang tua sendiri.

Step 2:

How you are cared for (ie, independent variables) predicts your feelings about competence and self-esteem (ie, mediators).

Bagaimana Anda diasuh ${\ displaystyle \ to}  ÂÂ$ Perasaan kompetensi dan harga diri.

Step 3:

Your sense of competence and self-esteem (eg, mediator) predicts how confident you are about taking care of your own children (ie, the dependent variable), while controlling how you are cared for (ie, independent variables).

Such findings will lead to conclusions implying that your feelings about competence and self-esteem mediate the relationship between how you are cared for and how confident you are about caring for your own children.

Note: If step 1 does not produce significant results, one may still have reason to move to step 2. Sometimes there is actually a significant relationship between independent and dependent variables but because of the small sample size, or other foreign factors, there can be no enough power to predict the actual effects there (See Shrout & Bolger, 2002 for more info).

Maps Mediation (statistics)

Live versus indirect effects

In the diagram shown above, the indirect effect is the product of the path coefficients of "A" and "B". The immediate effect is the "C" coefficient. Direct effects measure the extent to which the dependent variable changes when the independent variable increases by one unit and the mediator variable remains unchanged. Instead, the indirect effect measures the extent to which the dependent variable changes when the independent variable is fixed and the mediator variable changes with the amount that will change if the independent variable increases by one unit. In a linear system, the total effect is equal to the sum of the direct and indirect effects ( C 'AB in the above model). In the nonlinear model, the total effect is generally not equal to the sum of the direct and indirect effects, but the modified combination of the two.

src: pediatrics.aappublications.org

Full partial versus partial

The mediator variable can take into account all or a part of the observed relationship between two variables.

Full Mediation

The maximum proof for mediation, also called full mediation, will occur if the inclusion of mediation variables drops the relationship between the independent variable and the dependent variable (see path c in the diagram above) to zero. This is rare, if ever, happens. The most likely occurrence is that c is a weaker path, but remains significant with the entry of mediation effects.

Partial Mediation

The partial mediation states that the mediating variable contributes some, but not all, the relationships between independent variables and dependent variables. Partial mediation implies that there is not only a significant relationship between the mediator and the dependent variable, but also some direct relationship between independent and dependent variables.

In order for full or partial mediation to be made, the reduction of variation described by the independent variable should be significant as determined by one of several tests, such as the Sobel test. The effect of the independent variable on the dependent variable can become insignificant when the mediator is introduced only because the number of trivial variants is described (ie, incorrect mediation). Thus, it is important to point out significant reductions in the variants described by independent variables before confirming either full or partial mediation. It is possible to have a statistically significant indirect effect in the absence of total effect. This can be explained by the existence of several mediation paths that cancel out each other, and become apparent when one of the revocation mediators is controlled. This implies that the terms 'partial' and 'full' mediation should always be interpreted relative to the set of variables present in the model. In all cases, the "fix variable" operation must be distinguished from "controlling variables," which have been used incorrectly in the literature. The first stands for physical improvement, while the latter stands for conditioning, adjusting, or adding a regression model. These two concepts coincide only when all the error terms (not shown in the diagram) are statistically uncorrelated. When errors correlate, adjustments should be made to neutralize the correlation before starting the mediation analysis (see Bayesian Networks).

src: i.ytimg.com

Sobel Test

As mentioned above, the Sobel test is performed to determine whether the relationship between independent variables and dependent variables has decreased significantly after the inclusion of mediator variables. In other words, this test assesses whether the mediation effect is significant. It examines the relationship between independent variables and dependent variables compared to the relationship between independent variables and dependent variables including mediation factors.

The Sobel test is more accurate than the Baron and Kenny steps described above; However, it has a low statistical power. Thus, large sample sizes are required to have sufficient strength to detect significant effects. This is because the main assumption of the Sobel test is the assumption of normality. Since the Sobel test evaluates the samples given on the normal distribution, small sample sizes and the slope of the sampling distribution can be a problem (see normal distribution for more details). Thus, the rule of thumb as suggested by MacKinnon et al. (2002) is that a sample size of 1000 is required to detect small effects, sufficient sample size 100 in detecting moderate effects, and a sample size of 50 is required to detect large effects.

src: bmjopen.bmj.com

The bootstrap Preacher and Hayes (2004) methods

The bootstrapping method gives some advantages to the Sobel test, especially the power boost. The Preacher and Hayes Bootstrapping method is a non-parametric test (See Non-parametric statistics for discussion of non parametric tests and their strengths). Thus, the bootstrap method does not violate the assumption of normality and is therefore recommended for small sample sizes. Bootstrapping involves repeated random observations with replacements from the data set to calculate the desired stats in each resample. Over hundreds, or thousands, of bootstrap samples provide approximate sampling distributions of interesting statistics. Hayes offers macro & lt; http://www.afhayes.com/> which calculates direct bootstrapping in SPSS, a computer program used for statistical analysis. This method provides an approximate point and confidence interval by which one can assess the significance or insignificance of mediation effects. The dot estimate reveals an average above the number of bootstrap samples and if zero does not fall between the confidence intervals generated from the bootstrapping method, one can confidently conclude that there is a significant mediation effect to report.

src: i.ytimg.com

Significance of mediation

As outlined above, there are several different options to choose from to evaluate the mediation model.

Bootstrapping is the most popular method for testing mediation because it does not require the assumption of normality to be met, and can therefore be used effectively with smaller sample sizes ( N Ã, & lt; Ã, 25). However, mediation continues most often determined by using Baron and Kenny logic or the Sobel test. It is increasingly difficult to publish pure mediation tests based on Baron and Kenny methods or tests that make distribution assumptions such as the Sobel test. Thus, it is important to consider your options while choosing the tests to be performed.

src: jaha.ahajournals.org

Approach to mediation

While the concept of mediation as defined in psychology is theoretically interesting, the methods used to study mediation are empirically challenged by statisticians and epidemiologists and are formally interpreted.

(1) Chain-causal experimental chain design

The causal-causal chain design is used when the proposed mediator is manipulated experimentally. Such design implies that one manipulates some third-controlled variables that they have reason to believe can be the underlying mechanism of a given relationship.

(2) Mediation-measurement design

A mediation-measurement design can be conceptualized as a statistical approach. Such design implies that one measures the proposed intervention variable and then uses statistical analysis to establish mediation. This approach does not involve the manipulation of hypothesized mediation variables, but involves measurement only.

src: i.ytimg.com

Criticism on mediation measurement

An experimental approach to mediation should be done with caution. First, it is important to have strong theoretical support for exploratory investigation of potential mediation variables. Criticism of a mediation approach lies in the ability to manipulate and measure mediation variables. Thus, one must be able to manipulate the proposed mediator in an acceptable and ethical manner. Thus, one must be able to measure the intervention process without disrupting the outcome. The mediator should also be able to specify the validity of the construct manipulation. One of the most common criticisms of a mediation-measurement approach is that it is ultimately a correlational design. Consequently, it is possible that some other third variable, independent of the proposed mediator, may be responsible for the proposed effect. However, researchers have worked hard to provide counter-evidence for this humiliation. In particular, the following counter arguments have been put forward:

(1) Temporal precedent. For example, if the independent variable precedes the dependent variable in time, it will provide evidence that indicates a directional, and potentially causal, relationship from the independent variable to the dependent variable.

(2) Nonspuriousness and/or no confounds. For example, should someone identify another third variable and prove that they do not change the relationship between the independent variable and the dependent variable he/she will have a stronger argument for the mediation effect. See the other 3 variables below.

Mediation can be a very useful and powerful statistical test, but must be used correctly. It is important that the measures used to assess the mediator and the dependent variables are theoretically different and that independent variables and mediators can not interact. There should be an interaction between independent variables and mediators who will have reason to investigate moderation.

src: www.frontiersin.org

Other third variables

(1) Scaffolding:

Another model that is often tested is the model in which the competing variables in the model are alternative potential mediators or the unmeasured causes of the dependent variable. Additional variables in the causal model may obscure or confuse the relationship between independent and dependent variables. Potential confounders are variables that may have a causal impact on both the independent variable and the dependent variable. They include common sources of measurement error (as discussed above) as well as other influences shared by both independent and dependent variables.

In an experimental study, there are particular concerns about aspects of manipulation or experimental setting that may lead to a study effect, rather than a motivating theoretical factor. Each of these problems can result in a false relationship between independent and dependent variables being measured. Ignoring confounding variables can bias the empirical estimates of the causal effects of independent variables.

(2) Suppression:

The suppression variable increases the predictive validity of other variables by including them in the regression equation. For example, a higher intelligence score ( X ) causes a decrease in errors made while working on the assembly line ( Y ). However, increased intelligence ( X ) may cause an increase in errors made on the assembly line ( Y ) since it may also be associated with increased boredom at work ( Z ) thereby introducing an element of carelessness that results in a higher percentage of error at work. The suppressor variable will lead to an increase in the relationship between two variables.

In general, the removal of suppressors or confounders will lead to an overestimation or too high estimate of the effect of X on Y , thereby reducing or artificially artificially the amount of relationship between two variables.

(3) Moderator:

Another important third variable is the moderator. Moderator is a variable that can make the relationship between two variables become stronger or weaker. These variables further characterize the interaction in the regression by affecting the direction and/or strength of the relationship between X and Y . Moderate relationships can be considered as interactions. This occurs when the relationship between variables A and B depends on level C. See moderation for further discussion.

src: images.slideplayer.com

Mediated moderation

Mediation and moderation can occur simultaneously in the statistical model. It is possible to mediate moderation moderate.

Moderate mediation is when the effect of the A treatment on the mediator and/or partial effects B on the dependent variable depends on the other variable levels (moderator). Basically, in mediation moderated, mediation is first set, and then investigates whether the mediation effect that describes the relationship between independent variables and dependent variables is moderated by different levels of other variables (ie, moderators). This definition has been outlined by Muller, Judd, and Yzerbyt (2005) and Preacher, Rucker, and Hayes (2007).

Moderate mediation model

There are five moderated mediation models, as illustrated in the diagram below.

1. In the first model, the independent variable also moderates the relationship between the mediator and the dependent variable.
2. A second moderation mediation model that may involve new variables that moderate the relationship between independent variables and mediators (path A ).
3. The third moderation mediation model involves a new moderator variable that moderates the relationship between the mediator and the dependent variable (path B ).
4. Moderate mediation can also occur when a moderating variable affects the relationship between independent variables and mediators (path A ) and the relationship between the mediator and the dependent variable ( B path).
5. The fifth and final moderation moderation model may involve two new moderator variables, which moderate the path A and the other moderates the path B .

src: i.ytimg.com

Mediation moderation

Mediation moderation is a variant of moderation and mediation. This is where initially there is overall moderation and the direct effect of moderator variables on mediated results. The main difference between moderation mediation and moderation mediation is that for the first there is initial moderation (overall) and the effect is mediated and for the latter there is no moderation but the effect of either treatment on mediators (pathway A ) is moderated or effects mediator on result (path B ) is moderated.

To define moderation mediation, it must first define moderation, which means that the direction and/or strength of the relationship between independent and dependent variables (path C ) differ depending on the level of one-third variable (moderator variable). Researchers then look for the existence of mediated moderation when they have a theoretical reason to believe that there is a fourth variable acting as a mechanism or process that causes the relationship between independent variables and moderators (path A ) or between the moderator and the dependent variable (path C ).

Contoh

Here is a published example of moderation mediated in psychological research. Participants are presented with an initial stimulus (prime) that makes them think about morality or make them think about strength. They then participate in the Game of Rescue Dilemma (PDG), where participants pretend that they and their partners in crime have been arrested, and they must decide whether to remain loyal to their spouse or to compete with their partner and work with the parties authorized. The researchers found that prosocial individuals are influenced by morality and perhaps prime numbers, whereas individual proism is not. Thus, the social value orientation (proself vs. prosocial) moderates the relationship between prime (independent variables: morality vs strength) and behavior chosen in PDG (dependent variable: competitive vs cooperative).

The researchers then sought the presence of moderated mediation effects. Regression analysis reveals that prime types (morality vs. possible) mediate the moderating relationship of participant's social value orientation to PDG behavior. Prospective participants who experience moral priorities are expected to partner with them, so they choose to work together on their own. Prospective participants who are likely to expect their partners to compete with them, which makes them more likely to compete with their partners and cooperate with the authorities. Conversely, participants with a pro-social value orientation always act in a competitive manner.

src: slideplayer.com

Regression equations for mediation mediation moderation and moderation

Muller, Judd, and Yzerbyt (2005) outline three basic models that underlie moderation mediation and mediation moderation. Mo represents the moderator variable, Me represents the mediator variable (s), and ? _i represents the measurement error of each regression equation.

Langkah 1 : Moderasi hubungan antara variabel independen (X) dan variabel dependen (Y), juga disebut efek perlakuan keseluruhan (path C dalam diagram).

${\ displaystyle Y = \ beta_ {40} \ beta_ {41} X \ beta_ {42} Mo \ beta {43} XMo \ varepsilon4 } Â Â$

• Untuk menetapkan keseluruhan moderasi, ? ₄₃ berat regresi harus significal (langkah pertama untuk menetapkan moderasi yang dimediasi).
• Menetapkan mediasi moderasi mensyaratkan bahwa tidak ada efek moderasi, sehingga ? ₄₃ bobot regresi tidak boleh significant.

Langkah 2 : Moderasi hubungan antara variabel independen dan mediator (jalur A ).

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; ÂM\; Â\; Â\; Â\; Â\; Â\; Â\; ÂeÂ\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â50Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â51Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂX\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â52Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; ÂM\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âor\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â53Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂX\; Â\; Â\; Â\; Â\; Â\; Â\; ÂM\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âor\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â5Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â}Â\; Â\; ÂÂ\; Â\; Â\{\backslash \; displaystyle\; Me\; =\; \backslash \; beta\_\; \{50\}\; \backslash \; beta\_\; \{51\}\; X\; \backslash \; beta\_\; \{52\}\; Mo\; \backslash \; beta\; 53\; \{X\}\; X\; \backslash \; varepsilon5\; \}\; Â\; Â$

• Jika ? ₅₃ bobot regresi signifik, moderator mempengaruhi hubungan antara variabel independen dan mediator.

Langkah 3 : Moderasi dari kedua hubungan antara variabel independen dan dependen (jalur A ) dan hubungan antara mediator dan variabel dependen (jalur B ).

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂYÂ\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â60Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â61Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂX\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â62Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; ÂM\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âor\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â63Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂX\; Â\; Â\; Â\; Â\; Â\; Â\; ÂM\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âor\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â64Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; ÂM\; Â\; Â\; Â\; Â\; Â\; Â\; ÂeÂ\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â65Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; ÂM\; Â\; Â\; Â\; Â\; Â\; Â\; ÂeÂ\; Â\; Â\; Â\; Â\; Â\; ÂM\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âor\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â6Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â}Â\; Â\; ÂÂ\; Â\; Â\{\backslash \; displaystyle\; Y\; =\; \backslash \; beta\_\; \{60\}\; \backslash \; beta\_\; \{61\}\; X\; \backslash \; beta\_\; \{62\}\; Mo\; \backslash \; beta\; 63\; \{XMo\; \backslash \; beta\_64\}\; Me\; \backslash \; beta\_\; \{65\}\; MeMo\; \backslash \; varepsilon\; 6\}\; Â\; Â$

• Jika keduanya ? ₅₃ pada langkah 2 dan ? ₆₃ per langkah 3 significant, moderator mempengaruhi hubungan antara variabel independen dan mediator (path A ).
• Jika keduanya ? ₅₃ pada langkah 2 dan ? ₆₅ per langkah 3 significant, moderator mempengaruhi hubungan antara mediator dan variabel dependen (path B ).
• Salah satu atau kedua kondisi di at mungkin benar.

src: i.ytimg.com

Analisis mediasi kausal

Memperbaiki dibandingkan pengondondian

Mediation analysis quantifies the extent to which a variable participates in the transmittance of change from one cause to its effect. This is inherently a causal idea, since it can not be defined in statistical terms. Traditionally, however, most of the mediation analysis has been done within the limits of linear regression, with statistical terminology masking the causal character of the relationships involved. This leads to difficulties, biases, and limitations that have been mitigated by the modern method of causal analysis, based on causal and counterfactual logic.

The source of this difficulty lies in defining mediation in terms of changes caused by the addition of the third variable into the regression equation. Such statistical changes are epiphenomena that sometimes accompany mediation but, in general, fail to capture the causal relationships that mediation analysis aims to measure.

The basic premise of a causal approach is that it is not always appropriate to "control" the mediator when we attempt to estimate the direct effect of X on Y (see Figure above ). The classical reason for "controlling" for M "is that, if we succeed in preventing M from changing, then whatever changes we measure in Y are only caused by variations in X and we are justified later in proclaiming the observed effect as "direct effect X on Y ." Unfortunately, "controlling for M "physically does not prevent M/i from changing, it only narrows the analyst's attention to cases with the value of M .In addition, the language of probability theory has no notation for expressing the idea of" preventing M from changing "or" physically holding M constant. "The only possibility the carrier provides is" Conditioning "which is what we do when we" control "for < i> M , or add M as a deceiver in the equation for Y The result is, instead of holding M "constant say pad a M = m ) and compare Y for units under X Ã, = Ã, 1 'to the under X = 0, we allow M to vary but ignore all units except where M reaches M Ã, = m . These two operations are essentially different, and produce different results, except in the case that no variables are omitted.

To illustrate, assume that the error terms M and Y are correlated. In such conditions, the structural coefficients of B and A (between M and Y and between Y and X ) can no longer be estimated by the Y regression on X and M . In fact, the slope of the regression may not be zero even when C is zero. This has two consequences. First, a new strategy should be designed to estimate the structural coefficients of A, B and C . Second, the basic definition of direct and indirect effects must go beyond regression analysis, and should invoke operations that mimic "fixing M ", rather than "conditioning on M ."

Definition

Seperti operator, dilambangkan lakukan ( M Â = Â m ), didefinisikan dalam Pearl (1994) dan beroperasi dengan menghapus persamaan M dan menggantinya dengan konstanta m . Misalnya, jika model mediasi dasar terdiri dari persamaan:

${\ displaystyle X = f (\ varepsilon _ {1}), ~~ M = g (X, \ varepsilon _ {2}), ~~ Y = h (X , M, \ varepsilon _ {3}),}  ÂÂ$

kemudian setelah menerapkan operator lakukan ( M Ã, = Ã, m ) model menjadi:

${\ displaystyle X = f (\ varepsilon1), ~~ M = m, ~~ Y = h (X, m, \ varepsilon _ {3})} Â Â$

dan setelah menerapkan operator lakukan ( X Ã, = Ã, x ) model menjadi:

${\ displaystyle X = x, M = g (x, \ varepsilon2), Y = h (x, M, \ varepsilon3 })} Â Â$

where are the functions f and g , as well as the distribution of the error term? ₁ and? ₃ remains unchanged. If we further change the variable names M and Y resulting from doing ( X Ã, = Ã, x ) as M ( x ) and Y ( x ), respectively, we obtained what came to be known as "potential outcomes" or "structural counterfactuals". These new variables provide an easy notation for determining direct and indirect effects. In particular, four types of securities have been established for the transition from X Ã, = Ã,0 to X Ã, = Ã, 1:

(A) Total Efek -

${\ displaystyle TE = E [Y (1) -Y (0)]} Â Â$

(b) Efek langsung terkendali -

${\ displaystyle CDE (m) = E [Y (1, m) -Y (0, m)]} Â Â$

(c) Efek langsung alami -

${\ displaystyle NDE = E [Y (1, M (0)) - Y (0, M (0))]} Â Â$

(d) Efek tidak langsung alami

${\ displaystyle NIE = E [Y (0, M (1)) - Y (0, M (0))]} Â Â$

Where E [] stands for expectation of overtaken error terms.

These effects have the following interpretations:

• TE measures the expected improvement in the Y result as X changes from X = 0 to X Ã, = 1 , while the mediator is allowed to track changes in X as dictated by the function M = g (X ,? ₂ ) .
• The CDE measures the expected improvement in the Y result as X changes from X = 0 to X = 1 , while the mediator is set at a predetermined level M = m uniformly across the population
• NDE measures expected improvement in Y as X changes from X = 0 to X = 1, when setting the mediator variable to whatever value will get below X = 0, that is, before the change.
• NIE measures the expected improvement in Y when X is maintained constant, at X = 0, and < i> M changes to whatever value it has (for each individual) under X = 1.
• The difference TE-NDE measures the extent to which mediation is required to explain its effect, while NIE measures the extent to which mediation is sufficient > to keep it.

The controlled version of the indirect effect does not exist because there is no way to disable the direct effect by fixing the variable to a constant.

Menurut has defined it, it's totally rich in the sebagai penjumlahan

${\ displaystyle TE = NDE-NIE_ {r}} Â Â$

where NIE _r stands for reverse transition, from X Ã, = Ã,1 to X = 0 ; it becomes additive in a linear system, where a transitional reversal requires a reversal of the mark.

The power of this definition lies in its announcement; they apply to models with arbitrary nonlinear interactions, arbitrary dependencies between interruptions, and both continuous and categorical variables.

Mediation formula

Dalam analisis linear, semua efek ditentukan oleh jumlah produk dari koefisien struktural, pemberian

${\ displaystyle {\ begin {aligned} TE & amp; = C AB \\ CDE (m) & amp; = NDE = C, {\ teks {independen}} m \ \ NIE & amp; = AB. \ End {aligned}}}  ÂÂ$

The last two equations are called Mediation Formulas and have been the target estimates in many mediation studies. They provide free distribution expressions for both direct and indirect effects and show that, despite the arbitrary nature of the distribution of fault and function f , g , and h , the mediation effect can still be estimated from the data using regression. Moderate mediation analysis and mediator mediator fall as a special case of causal mediation analysis, and the mediation formula identifies how various interaction coefficients contribute to the necessary and adequate mediation components..

Example

Asumsikan model mengambil bentuk

Source of the article : Wikipedia