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    This term is the interaction between X and the proposed moderating variable. Thus, for a response Y and two variables x 1 and moderating variable x 2 ,:.

    In this case, the role of x 2 as a moderating variable is accomplished by evaluating b 3 , the parameter estimate for the interaction term.

    However, the new interaction term will be correlated with the two main effects terms used to calculate it.

    This is the problem of multicollinearity in moderated regression. Multicollinearity tends to cause coefficients to be estimated with higher standard errors and hence greater uncertainty.

    Mean-centering subtracting raw scores from the mean has been suggested as a remedy for multicollinearity. However, mean-centering is unnecessary in any regression analysis, as one uses a correlation matrix and the data are already centered after calculating correlations.

    Correlations are derived from the cross-product of two standard scores Z-scores or statistical moments hence the name: Much Ado About Nothing".

    Like simple main effect analysis in ANOVA, in post-hoc probing of interactions in regression, we are examining the simple slope of one independent variable at the specific values of the other independent variable.

    Below is an example of probing two-way interactions. If both of the independent variables are categorical variables, we can analyze the results of the regression for one independent variable at a specific level of the other independent variable.

    Then the interaction effect shows whether the effect of condition on the dependent variable Y is different for European Americans and East Asians and whether the effect of ethnic status is different for the two conditions.

    The coefficient of A shows the ethnicity effect on Y for the control condition, while the coefficient of B shows the effect of imposing the experimental condition for European American participants.

    If the first independent variable is a categorical variable e. However, a zero score on the Satisfaction With Life Scale is meaningless as the range of the score is from 7 to This is where centering comes in.

    When the analysis is run again, b 1 now represents the difference between males and females at the mean level of the SWLS score of the sample.

    Then one can explore the effects of gender on the dependent variable Y at high, moderate, and low levels of the SWLS score.

    As with two categorical independent variables, b 2 represents the effect of the SWLS score on the dependent variable for females.

    By reverse coding the gender variable, one can get the effect of the SWLS score on the dependent variable for males. When treating categorical variables such as ethnic groups and experimental treatments as independent variables in moderated regression, one needs to code the variables so that each code variable represents a specific setting of the categorical variable.

    There are three basic ways of coding: Dummy-variable coding, Effects coding, and Contrast coding. Below is an introduction to these coding systems.

    Dummy coding is used when one has a reference group or one condition in particular e. In this case, the intercept is the mean of the reference group, and each of the unstandardized regression coefficients is the difference in the dependent variable between one of the treatment groups and the mean of the reference group or control group.

    This coding system is similar to ANOVA analysis, and is appropriate when researchers have a specific reference group and want to compare each of the other groups with it.

    Effects coding is used when one does not have a particular comparison or control group and does not have any planned orthogonal contrasts. The intercept is the grand mean the mean of all the conditions.

    The regression coefficient is the difference between the mean of one group and the mean of all the group means e. This coding system is appropriate when the groups represent natural categories.

    Contrast coding is used when one has a series of orthogonal contrasts or group comparisons that are to be investigated.

    In this case, the intercept is the unweighted mean of the individual group means. The unstandardized regression coefficient represents the difference between the unweighted mean of the means of one group A and the unweighted mean of another group B , where A and B are two sets of groups in the contrast.

    This coding system is appropriate when researchers have an a priori hypothesis concerning the specific differences among the group means.

    If both of the independent variables are continuous, it is helpful for interpretation to either center or standardize the independent variables, X and Z.

    Centering involves subtracting the overall sample mean score from the original score; standardizing does the same followed by dividing by the overall sample standard deviation.

    By centering or standardizing the independent variables, the coefficient of X or Z can be interpreted as the effect of that variable on Y at the mean level of the other independent variable.

    To probe the interaction effect, it is often helpful to plot the effect of X on Y at low and high values of Z some people prefer to also plot the effect at moderate values of Z , but this is not necessary.

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    The intercept is the grand mean the mean of all the conditions. The regression coefficient is the difference between the mean of one group and the mean of all the group means e.

    This coding system is appropriate when the groups represent natural categories. Contrast coding is used when one has a series of orthogonal contrasts or group comparisons that are to be investigated.

    In this case, the intercept is the unweighted mean of the individual group means. The unstandardized regression coefficient represents the difference between the unweighted mean of the means of one group A and the unweighted mean of another group B , where A and B are two sets of groups in the contrast.

    This coding system is appropriate when researchers have an a priori hypothesis concerning the specific differences among the group means.

    If both of the independent variables are continuous, it is helpful for interpretation to either center or standardize the independent variables, X and Z.

    Centering involves subtracting the overall sample mean score from the original score; standardizing does the same followed by dividing by the overall sample standard deviation.

    By centering or standardizing the independent variables, the coefficient of X or Z can be interpreted as the effect of that variable on Y at the mean level of the other independent variable.

    To probe the interaction effect, it is often helpful to plot the effect of X on Y at low and high values of Z some people prefer to also plot the effect at moderate values of Z , but this is not necessary.

    Often values of Z that are one standard deviation above and below the mean are chosen for this, but any sensible values can be used and in some cases there are more meaningful values to choose.

    The plot is usually drawn by evaluating the values of Y for high and low values of both X and Z , and creating two lines to represent the effect of X on Y at the two values of Z.

    Various internet-based tools exist to help researchers plot and interpret such two-way interactions. The principles for two-way interactions apply when we want to explore three-way or higher-level interactions.

    For instance, if we have a three-way interaction between A , B , and C , the regression equation will be as follows:. It is worth noting that the reliability of the higher-order terms depends on the reliability of the lower-order terms.

    For example, if the reliability for variable A is. In this case, low reliability of the interaction term leads to low power; therefore, we may not be able to find the interaction effects between A and B that actually exist.

    The solution for this problem is to use highly reliable measures for each independent variable. If this is the case, it is worth testing a nonlinear regression model by adding nonlinear terms in individual variables into the moderated regression analysis to see if the interactions remain significant.

    From Wikipedia, the free encyclopedia. Aiken; West, Stephen H. Multiple regression testing and interpretation 1.

    Moderation in management research: What, why, when and how. Journal of Business and Psychology. Mean arithmetic geometric harmonic Median Mode. Central limit theorem Moments Skewness Kurtosis L-moments.

    Grouped data Frequency distribution Contingency table. Pearson product-moment correlation Rank correlation Spearman's rho Kendall's tau Partial correlation Scatter plot.

    Sampling stratified cluster Standard error Opinion poll Questionnaire. Observational study Natural experiment Quasi-experiment. Z -test normal Student's t -test F -test.

    Bayesian probability prior posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator. Pearson product-moment Partial correlation Confounding variable Coefficient of determination.

    Simple linear regression Ordinary least squares General linear model Bayesian regression. Regression Manova Principal components Canonical correlation Discriminant analysis Cluster analysis Classification Structural equation model Factor analysis Multivariate distributions Elliptical distributions Normal.

    Spectral density estimation Fourier analysis Wavelet Whittle likelihood. Cartography Environmental statistics Geographic information system Geostatistics Kriging.

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    Als "Düsseldorfer Jong" sozusagen ein da- Heimspiel. Sandra war 12 und fand es total lustig. Ich war 14 Jahre alt, als ich nach Nordrhein-Westfalen kam.

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