scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods.

01 Jan 2004-Multivariate Behavioral Research (Lawrence Erlbaum Associates, Inc.)-Vol. 39, Iss: 1, pp 99-128
TL;DR: Two alternatives for improving the performance of confidence limits for the indirect effect are evaluated: a method based on the distribution of the product of two normal random variables, and resampling methods.
Abstract: The most commonly used method to test an indirect effect is to divide the estimate of the indirect effect by its standard error and compare the resulting z statistic with a critical value from the standard normal distribution. Confidence limits for the indirect effect are also typically based on critical values from the standard normal distribution. This article uses a simulation study to demonstrate that confidence limits are imbalanced because the distribution of the indirect effect is normal only in special cases. Two alternatives for improving the performance of confidence limits for the indirect effect are evaluated: (a) a method based on the distribution of the product of two normal random variables, and (b) resampling methods. In Study 1, confidence limits based on the distribution of the product are more accurate than methods based on an assumed normal distribution but confidence limits are still imbalanced. Study 2 demonstrates that more accurate confidence limits are obtained using resampling methods, with the bias-corrected bootstrap the best method overall.

Summary (3 min read)

Estimation of the Indirect Effect and Standard Error

  • The indirect effect model is shown in Figure 1 and is summarized in the three equations described below (see also Allison, 1995a and MacKinnon & Dwyer, 1993) .
  • The residuals have expected values of zero.
  • Two extensive simulation studies (MacKinnon et al., 1995; Stone & Sobel, 1990) showed an imbalance in the number of times a true value fell to the left or right of the confidence limits.

The Distribution of the Product

  • The assumption that the indirect effect divided by its standard error has a normal sampling distribution is incorrect in some situations.
  • In these situations, the confidence limits calculated using Equation 5 will be incorrect.
  • Because the indirect effect is the product of regression estimates which are normally distributed asymptotically (Hanushek & Jackson, 1977) , an alternative method for testing indirect effects can be developed based on the distribution of the product of two normally distributed random variables (Aroian, 1947; Craig, 1936; Springer, 1979) .

Simulation Description

  • The SAS ® (1989) programming language was used to conduct the statistical simulations.
  • Third, one thousand replications were conducted for each of the 40 combinations of sample size and parameters.
  • Fourth, for each of the 40,000 (4 combinations of sample size times 10 parameter value combinations times 1000 replications) different data sets, six resampling methods were applied.

Confidence Limits

  • Sample values were inserted in Equation 5to obtain upper and lower confidence limits for the z test.
  • For the bootstrap methods, the confidence limits were obtained from the bootstrap distribution.
  • Confidence limits for the indirect effect were calculated for 80%, 90%, and 95% intervals.
  • Table 4 summarizes the performance of the confidence limits by showing the number of times that the observed percentage was outside the robustness interval for each of the three confidence intervals and nine methods 3 .
  • All of these methods were considerably better than the jackknife (59 times) which only had slightly better performance than the traditional z test (61 times).

Results

  • Confidence limits were calculated for the z and M tests.
  • The proportions of times that true values of the indirect effect fell to the left and right of the confidence limits are shown in Tables 1 and 2 .
  • The distribution of the product, M, has more balanced confidence limits because it incorporates the skewness and kurtosis of the product distribution.
  • Most importantly, note that the confidence limits based on the distribution of the product are nearly always as close or closer to the expected Type I error rate of .025 than the traditional test.

Discussion

  • The confidence limits for the indirect effect based on the distribution of the product were more accurate than the confidence limits based on the normal distribution assumption.
  • The Type I error rates based on the confidence limits did not exceed nominal rates using this method for any combination of parameter values both in Study 1 1 and in MacKinnon et al. (2002) .
  • The proportions outside the confidence limits for the product distribution were often less than the expected values for small effect sizes and small sample sizes (i.e., small values of ␦ ␣ and ␦ ␤ ).
  • One possible explanation for this discrepancy is that the appropriate comparison distribution is the product of two t distributions rather than two normal distributions.
  • The fourth group consists of the bias-corrected bootstrap which had slightly more power than methods in the second category and had the most accurate confidence intervals.

Study 2

  • In Study 1, although confidence limits for the indirect effect were more accurate when the distribution of the product was taken into account, there were still cases where the number of times that the true value was outside the range of the confidence limits was smaller than expected.
  • Several researchers have suggested that resampling methods such as the jackknife and the bootstrap may provide more accurate tests of the indirect effect (Bollen & Stine, 1990; Lockwood & MacKinnon, 1998; Shrout & Bolger, 2002) .
  • Bollen and Stine (1990) found that bootstrap confidence limits for the indirect effect were asymmetric.
  • Most recently, Shrout and Bolger (2002) recommended bootstrap methods to assess mediation for small to moderate sample sizes.

Type I Error Rates and Statistical Power

  • The observed Type I error rates and statistical power were also computed for each method.
  • An effect was considered statistically significant if zero was not included in the confidence interval.
  • For Type I error rates, the liberal Bradley (1978) robustness interval was also computed and Type I error rates outside the interval are indicated by an asterisk in the Tables.

Single Sample Methods

  • The calculation of the M test confidence limits was also the same as reported in Study 1 with one minor exception.
  • The critical values for the M test confidence limits in Study 2 come from an augmented table for the 95% confidence limits.
  • These additional values were obtained with a FORTRAN algorithm written by Alan Miller which is a minor modification of the method in Meeker and Escobar (1994) and is available at http://users.bigpond.net.au/amiller (file name: fnprod.f90).
  • This method is called the empirical-M method in this article.
  • The values were standardized so that they could be used for any sample size.

Resampling Methods

  • Six resampling methods were evaluated in this study: jackknife, percentile bootstrap, bias-corrected bootstrap, bootstrap-t, bootstrap-Q, and Monte Carlo.
  • All of the methods adjust for nonnormal distributions although the bootstrap-Q and the bias-corrected bootstrap may be especially appropriate for severely nonnormal data (Chernick, 1999; Manly, 1997) .
  • The jackknife estimate is the average estimate across the N jackknife samples.
  • The basic bootstrap confidence limits were obtained with the percentile method as described by Efron and Tibshirani (1993) .
  • It requires the standard error of the parameter estimate for each bootstrap sample which is the sampling standard deviation of the bootstrap sample.

Example

  • The following example illustrates the methods used in this article with data from the Adolescents Training and Learning to Avoid Steroids program.
  • The data for this simplified example were from 861 cases (from 15 treatment schools and 16 control schools) with complete data on three variables, X-exposure to the program or not, X M -perceived severity of anabolic steroid use, and Y-nutrition behaviors.
  • Confidence limits for the indirect effect were computed for three single sample tests: the traditional z, the M test, and the empirical-M test, as well as six resampling methods: the jackknife, percentile bootstrap, bootstrap-t, bootstrap-Q, bias-corrected bootstrap, and the Monte Carlo method.
  • These values were used in Equations 6 and 7 to find the upper and lower M test confidence limits.

General Discussion

  • The purpose of this article was to evaluate two alternatives to improve confidence limit coverage for the indirect effect.
  • In Study 2, resampling methods had better performance than the method based on the normal distribution, with the exception of the jackknife.
  • There are limitations to the use of resampling methods, however.
  • The practical implication of the results of this article is that the traditional z test confidence limits can be substantially improved by using a method such as the M test that incorporates the distribution of the product of two normal random variables.

Did you find this useful? Give us your feedback

Content maybe subject to copyright    Report

MULTIVARIATE BEHAVIORAL RESEARCH 99
Multivariate Behavioral Research, 39 (1), 99-128
Copyright © 2004, Lawrence Erlbaum Associates, Inc.
Confidence Limits for the Indirect Effect:
Distribution of the Product and Resampling Methods
David P. MacKinnon, Chondra M. Lockwood, and Jason Williams
Arizona State University
The most commonly used method to test an indirect effect is to divide the estimate of the
indirect effect by its standard error and compare the resulting z statistic with a critical value
from the standard normal distribution. Confidence limits for the indirect effect are also
typically based on critical values from the standard normal distribution. This article uses
a simulation study to demonstrate that confidence limits are imbalanced because the
distribution of the indirect effect is normal only in special cases. Two alternatives for
improving the performance of confidence limits for the indirect effect are evaluated: (a) a
method based on the distribution of the product of two normal random variables, and (b)
resampling methods. In Study 1, confidence limits based on the distribution of the product
are more accurate than methods based on an assumed normal distribution but confidence
limits are still imbalanced. Study 2 demonstrates that more accurate confidence limits are
obtained using resampling methods, with the bias-corrected bootstrap the best method
overall.
An indirect effect implies a causal hypothesis whereby an independent
variable causes a mediating variable which, in turn, causes a dependent
variable (Sobel, 1990). Hypotheses regarding indirect or mediated effects
are implicit in social science theories (Alwin & Hauser, 1975; Baron &
Kenny, 1986; Hyman, 1955; Sobel, 1982). Examples of indirect effect
hypotheses are that attitudes affect intentions which then affect behavior
(Ajzen & Fishbein, 1980), that poverty reduces local social ties which
increases assault and burglary rates (Warner & Rountree, 1997), that social
status has an indirect effect on depression through changes in social stress
(Turner, Wheaton, & Lloyd, 1995), and that father’s education affects
offspring education which then affects offspring income (Duncan,
Featherman, & Duncan, 1972).
This research was supported by the National Institute on Drug Abuse grant number 1
R01 DA09757. We acknowledge the contributions of Ghulam Warsi and Jeanne Hoffman
to the work described in this article. We thank William Meeker, Leona Aiken, Michael
Sobel, Steve West, and Jenn-Yun Tein for comments on an earlier version of this
manuscript.
Correspondence concerning this article should be addressed to David P. MacKinnon,
Department of Psychology, Arizona State University, Tempe, AZ 85287-1104.

D. MacKinnon, C. Lockwood, and J. Williams
100 MULTIVARIATE BEHAVIORAL RESEARCH
Analysis of indirect effects is also important for experimental studies of
social policy interventions. Substance abuse prevention programs, for
example, are designed to change mediating variables such as social bonding
(Hawkins, Catalano, & Miller, 1992) and social influence (Bandura, 1977)
which are hypothesized to be causally related to drug abuse (see also Hansen
& Graham, 1991, and Tobler, 1986, for more examples). In these contexts,
the randomization of participants to treatment conditions and the knowledge
that the treatment precedes both the mediating variable and the dependent
variable in time strengthen the causal inferences that may be drawn about
the indirect effects of the intervention (Holland, 1988; Sobel, 1998). In these
experimental studies, analysis of indirect effects (also called mediation
analysis) provides a check on whether the manipulation changed the
variables it was designed to change, tests theory by providing information on
the process through which the experiment changed the dependent variable,
and generates information that may improve programs (MacKinnon, 1994;
West & Aiken, 1997). Thus, the accuracy of confidence limits for the
indirect effect is important for both basic and applied researchers in several
substantive areas of social science (Allison, 1995a; Bollen & Stine, 1990;
Sobel, 1982).
Prior research provides much information on the relative performance of
various methods for conducting significance tests for indirect effects
(MacKinnon, Lockwood, Hoffman, West, & Sheets, 2002), but very little
information about confidence limits. Confidence limit estimation has been
advocated for several reasons, including that it forces researchers to
consider the size of an effect in addition to making a binary decision regarding
significance and that the width of the interval provides a clearer
understanding of variability in the size of the effect (Harlow, Mulaik, &
Steiger, 1997; Krantz, 1999). The purpose of this article is to explain why
the traditional method used to test the significance of the indirect effect
based on the assumption of the z distribution has statistical power and Type
I error rates that are too low and imbalanced confidence limits. Two
alternatives to address the problem are evaluated in this article, one based
on the distribution of the product of two normal random variables and another
based on resampling methods. First, the equations used to estimate the
indirect effect and its standard error are described, followed by evidence that
traditional confidence limits for the indirect effect are imbalanced. Next, an
overview of the distribution of the product is given with a description of how
this distribution explains inaccuracies in the traditional test of the indirect
effect. In Study 1, confidence limits for the traditional and distribution of the
product methods are compared in a statistical simulation. In Study 2, a
simulation study compares the distribution of the product method evaluated

D. MacKinnon, C. Lockwood, and J. Williams
MULTIVARIATE BEHAVIORAL RESEARCH 101
in Study 1 with resampling methods which should also adjust for the
nonnormal distribution of the indirect effect.
Estimation of the Indirect Effect and Standard Error
The indirect effect model is shown in Figure 1 and is summarized in the
three equations described below (see also Allison, 1995a and MacKinnon &
Dwyer, 1993). We focus on a recursive model with a single indirect effect
and ordinary regression models in order to more clearly describe the
approach.
(1) Y
O
=
01
+ X + ε
1
(2) Y
O
=
02
+ X + X
M
+ ε
2
(3) X
M
=
03
+ X + ε
3
In these equations, Y
O
is the dependent variable, X
is the independent variable,
X
M
is the mediating variable, codes the relation between the independent
variable and the dependent variable,  codes the relation between the
independent variable and the dependent variable adjusted for the effects of
the mediating variable, codes the relation between the independent
variable and the mediating variable, and codes the relation between the
mediating variable and the dependent variable adjusted for the independent
variable. The residuals are coded by ε
1
, ε
2
, and ε
3
and the intercepts are
coded by
01
,
02
, and
03
in Equations 1, 2, and 3, respectively. The residuals
have expected values of zero.
In the first regression equation, the dependent variable (Y
O
) is regressed
on only the independent variable (X). In the second regression equation, the
dependent variable (Y
O
) is regressed on both the independent variable (X)
and the mediating variable (X
M
). The indirect effect equals the difference
in the estimated independent variable coefficients
()
ˆˆ
JJ
in the two
regression equations (Judd & Kenny, 1981).
A second method to calculate the indirect effect is illustrated in Figure 1.
First, the coefficient relating the mediating variable to the dependent variable
is estimated (
ˆ
>
) in Equation 2 above. Second, the coefficient relating the
independent variable to the mediating variable is estimated (
ˆ
=
) in Equation
3. The product of these two estimates (
ˆ
=
ˆ
>
) is the estimated indirect effect.
The estimated coefficient relating the independent variable to the dependent
variable adjusted for the mediating variable (
ˆ
J
) is the estimate of the direct
effect. The
ˆˆ
JJ
and
ˆ
=
ˆ
>
estimators of the indirect effect are equivalent

D. MacKinnon, C. Lockwood, and J. Williams
102 MULTIVARIATE BEHAVIORAL RESEARCH
Figure 1
The Indirect Effect Model

D. MacKinnon, C. Lockwood, and J. Williams
MULTIVARIATE BEHAVIORAL RESEARCH 103
in ordinary least squares regression (MacKinnon, Warsi, & Dwyer, 1995).
Additional assumptions of the
ˆ
=
ˆ
>
estimator of the indirect effect from
Equations 2 and 3 have been outlined (James & Brett, 1984; McDonald,
1997). These assumptions include no measurement error in variables (Hoyle
& Kenny, 1999), the causal relations of X to M to Y are correct (McDonald,
1997), no omitted variables (McDonald, 1997), and a zero interaction of X
and X
M
(Judd & Kenny, 1981). The same assumptions are made for the
indirect effect model examined in this article.
Although there are several estimators of the variance of the indirect
effect (see MacKinnon et al., 2002), the most commonly used estimator was
derived by Sobel (1982; 1986). This formula (Equation 4), based on the
multivariate delta method, is used to calculate the standard error of the
indirect effect in statistical software packages, including EQS (Bentler,
1997), LISREL (Jöreskog & Sörbom, 1993), and LINCS (Schoenberg &
Arminger, 1996), and is based on the estimates
ˆ
=
and
ˆ
>
, and the estimated
standard errors,
ˆ
ˆ
=
I
and
ˆ
ˆ
>
I
. Allison (1995a) used a reduced form
parameterization of the indirect effect model to derive the same standard
error formula in Equation 4. The formula assumes that and are
independent (Sobel, 1987). This variance estimator can be used to calculate
standard errors and confidence limits for the indirect effect. MacKinnon and
Dwyer (1993) and MacKinnon et al. (1995, 2002) found evidence that the
multivariate delta method standard error had the least bias of several
formulas for the standard error of the indirect effect.
(4)
22222
ˆˆ
ˆ
ˆ
ˆ
ˆˆˆ ˆ
=
=> >
I=I>I
=+
For nonzero values of both and, simulation studies suggest that the variance
estimator has relative bias less than 5% for sample sizes of 100 or more in a single
indirect effect model (MacKinnon et al., 1995) and for sample sizes of 200 or
more in a recursive model with seven total indirect effects (Stone & Sobel, 1990).
In many studies, the indirect effect is divided by its standard error and
the resulting ratio is then compared to the standard normal distribution to test
its significance, z =
ˆ
ˆ
ˆ
ˆˆ
/
=>
=> I
(Bollen & Stine, 1990; MacKinnon et al., 1991;
Wolchik, Ruehlman, Braver, & Sandler, 1989). Confidence limits for the
indirect effect lead to the same conclusion with regard to the null hypothesis.
Confidence limits are constructed using Equation 5,
(5)
ˆ
1/2
ˆ
ˆ
ˆˆ
*
z
L
=>
=> I
±
where z
1 - /2
is the value on the standard normal distribution corresponding
to the desired Type I error rate, .

Citations
More filters
Journal ArticleDOI
TL;DR: An overview of simple and multiple mediation is provided and three approaches that can be used to investigate indirect processes, as well as methods for contrasting two or more mediators within a single model are explored.
Abstract: Hypotheses involving mediation are common in the behavioral sciences. Mediation exists when a predictor affects a dependent variable indirectly through at least one intervening variable, or mediator. Methods to assess mediation involving multiple simultaneous mediators have received little attention in the methodological literature despite a clear need. We provide an overview of simple and multiple mediation and explore three approaches that can be used to investigate indirect processes, as well as methods for contrasting two or more mediators within a single model. We present an illustrative example, assessing and contrasting potential mediators of the relationship between the helpfulness of socialization agents and job satisfaction. We also provide SAS and SPSS macros, as well as Mplus and LISREL syntax, to facilitate the use of these methods in applications.

25,799 citations


Cites background or methods from "Confidence Limits for the Indirect ..."

  • ...…bases inference on a mathematical derivation of the distribution of the product of two normally distributed variables (Aroian, 1947; Craig, 1936; MacKinnon et al., 2004; Springer, 1979) and thus acknowledges the skew of the distribution of products rather than imposing the assumption of…...

    [...]

  • ...In extensive sets of simulations, MacKinnon et al. (2002; MacKinnon et al., 2004) examined the performance of these methods (among others) to assess their Type I error rates and power....

    [...]

  • ...For details of its application to simple mediation models, see Bollen and Stine (1990), Lockwood and MacKinnon (1998), MacKinnon et al. (2004), Shrout and Bolger (2002), and Preacher and Hayes (2004, 2008)....

    [...]

Journal ArticleDOI
TL;DR: This article disentangle conflicting definitions of moderated mediation and describes approaches for estimating and testing a variety of hypotheses involving conditional indirect effects, showing that the indirect effect of intrinsic student interest on mathematics performance through teacher perceptions of talent is moderated by student math self-concept.
Abstract: This article provides researchers with a guide to properly construe and conduct analyses of conditional indirect effects, commonly known as moderated mediation effects. We disentangle conflicting definitions of moderated mediation and describe approaches for estimating and testing a variety of hypotheses involving conditional indirect effects. We introduce standard errors for hypothesis testing and construction of confidence intervals in large samples but advocate that researchers use bootstrapping whenever possible. We also describe methods for probing significant conditional indirect effects by employing direct extensions of the simple slopes method and Johnson-Neyman technique for probing significant interactions. Finally, we provide an SPSS macro to facilitate the implementation of the recommended asymptotic and bootstrapping methods. We illustrate the application of these methods with an example drawn from the Michigan Study of Adolescent Life Transitions, showing that the indirect effect of intrinsic student interest on mathematics performance through teacher perceptions of talent is moderated by student math self-concept.

7,973 citations


Cites background or methods from "Confidence Limits for the Indirect ..."

  • ...The distribution of the product strategy is probably the most accurate analytic method available for determining the significance of, and confidence intervals (CIs) for, a1b1 in simple mediation models (MacKinnon et al., 2004)....

    [...]

  • ...Some research has been undertaken with respect to the power of tests of simple mediation (MacKinnon et al., 2004), and our own simulation addressed power under a set of limited conditions....

    [...]

  • ...A growing literature now advocates the use of bootstrapping for assessing indirect effects (Bollen & Stine, 1990; Lockwood & MacKinnon, 1998; MacKinnon et al., 2004; Preacher & Hayes, 2004; Shrout & Bolger, 2002)....

    [...]

  • ...MacKinnon et al. (2004) showed that such corrections can improve CIs and inferences when used in the context of simple mediation models....

    [...]

  • ...Bootstrapping has been advocated as an alternative to normal-theory tests of mediation (Lockwood & MacKinnon, 1998; MacKinnon et al., 2004; Preacher & Hayes, 2004; Shrout & Bolger, 2002)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the authors focus on communication processes and understand how messages have an effect on some outcome of focus in a focus-based focus-oriented focus-set problem, which is the goal of most communication researchers.
Abstract: Understanding communication processes is the goal of most communication researchers. Rarely are we satisfied merely ascertaining whether messages have an effect on some outcome of focus in a specif...

7,914 citations


Cites methods from "Confidence Limits for the Indirect ..."

  • ...Simulation research shows that bootstrapping is one of the more valid and powerful methods for testing intervening variable effects (MacKinnon et al., 2004; Williams & MacKinnon, 2008) and, for this reason alone, it should be the method of choice....

    [...]

Journal ArticleDOI
TL;DR: A general analytical framework for combining moderation and mediation that integrates moderated regression analysis and path analysis is presented that clarifies how moderator variables influence the paths that constitute the direct, indirect, and total effects of mediated models.
Abstract: Studies that combine moderation and mediation are prevalent in basic and applied psychology research. Typically, these studies are framed in terms of moderated mediation or mediated moderation, both of which involve similar analytical approaches. Unfortunately, these approaches have important shortcomings that conceal the nature of the moderated and the mediated effects under investigation. This article presents a general analytical framework for combining moderation and mediation that integrates moderated regression analysis and path analysis. This framework clarifies how moderator variables influence the paths that constitute the direct, indirect, and total effects of mediated models. The authors empirically illustrate this framework and give step-by-step instructions for estimation and interpretation. They summarize the advantages of their framework over current approaches, explain how it subsumes moderated mediation and mediated moderation, and describe how it can accommodate additional moderator and mediator variables, curvilinear relationships, and structural equation models with latent variables.

3,624 citations


Cites methods from "Confidence Limits for the Indirect ..."

  • ...The bootstrap has been used to test indirect effects in mediated models ( MacKinnon, Lockwood, & Williams, 2004; Shrout & Bolger, 2002) and can be extended to models that combine mediation and moderation, as we later illustrate....

    [...]

Journal ArticleDOI
TL;DR: The necessary sample sizes for six of the most common and the most recommended tests of mediation for various combinations of parameters are presented to provide a guide for researchers when designing studies or applying for grants.
Abstract: Mediation models are widely used, and there are many tests of the mediated effect. One of the most common questions that researchers have when planning mediation studies is, "How many subjects do I need to achieve adequate power when testing for mediation?" This article presents the necessary sample sizes for six of the most common and the most recommended tests of mediation for various combinations of parameters, to provide a guide for researchers when designing studies or applying for grants.

3,165 citations


Cites background or methods from "Confidence Limits for the Indirect ..."

  • ...A word of caution is needed for the bias-corrected bootstrap test, however, as it has been found to have larger-than-normal Type I error rates in certain conditions (see MacKinnon et al., 2004, for more information)....

    [...]

  • ...Aword of caution is needed for the bias-corrected bootstrap test, however, as it has been found to have larger-than-normal Type I error rates in certain conditions (see MacKinnon et al., 2004, for more information)....

    [...]

  • ...Initial sample sizes were estimated using results from MacKinnon et al. (2002, 2004) ....

    [...]

  • ...MacKinnon et al. (2004) compared the confidence limits for the indirect effect, ab, from MacKinnon and Lockwood’s (2001) asymmetric confidence-interval test with more traditional symmetric confidence intervals and with confidence intervals from six resampling methods....

    [...]

  • ... MacKinnon et al. (2004) compared the confidence limits for the indirect effect, ab, from MacKinnon and Lockwood’s (2001) asymmetric confidence-interval test with more traditional symmetric confidence intervals and with confidence intervals from six resampling methods....

    [...]

References
More filters
Book
01 Dec 1969
TL;DR: The concepts of power analysis are discussed in this paper, where Chi-square Tests for Goodness of Fit and Contingency Tables, t-Test for Means, and Sign Test are used.
Abstract: Contents: Prefaces. The Concepts of Power Analysis. The t-Test for Means. The Significance of a Product Moment rs (subscript s). Differences Between Correlation Coefficients. The Test That a Proportion is .50 and the Sign Test. Differences Between Proportions. Chi-Square Tests for Goodness of Fit and Contingency Tables. The Analysis of Variance and Covariance. Multiple Regression and Correlation Analysis. Set Correlation and Multivariate Methods. Some Issues in Power Analysis. Computational Procedures.

115,069 citations

Journal ArticleDOI
TL;DR: This article seeks to make theorists and researchers aware of the importance of not using the terms moderator and mediator interchangeably by carefully elaborating the many ways in which moderators and mediators differ, and delineates the conceptual and strategic implications of making use of such distinctions with regard to a wide range of phenomena.
Abstract: In this article, we attempt to distinguish between the properties of moderator and mediator variables at a number of levels. First, we seek to make theorists and researchers aware of the importance of not using the terms moderator and mediator interchangeably by carefully elaborating, both conceptually and strategically, the many ways in which moderators and mediators differ. We then go beyond this largely pedagogical function and delineate the conceptual and strategic implications of making use of such distinctions with regard to a wide range of phenomena, including control and stress, attitudes, and personality traits. We also provide a specific compendium of analytic procedures appropriate for making the most effective use of the moderator and mediator distinction, both separately and in terms of a broader causal system that includes both moderators and mediators.

80,095 citations


"Confidence Limits for the Indirect ..." refers background or methods in this paper

  • ...These include the steps mentioned in Baron and Kenny (1986) and Judd and Kenny (1981) and the joint significance test of and described in MacKinnon et al. (2002), which do not include explicit methods to compute confidence limits....

    [...]

  • ...Hypotheses regarding indirect or mediated effects are implicit in social science theories (Alwin & Hauser, 1975; Baron & Kenny, 1986; Hyman, 1955; Sobel, 1982)....

    [...]

Book
01 Jan 1993
TL;DR: This article presents bootstrap methods for estimation, using simple arguments, with Minitab macros for implementing these methods, as well as some examples of how these methods could be used for estimation purposes.
Abstract: This article presents bootstrap methods for estimation, using simple arguments. Minitab macros for implementing these methods are given.

37,183 citations

Book
17 Mar 1980
TL;DR: In this paper, the author explains "theory and reasoned action" model and then applies the model to various cases in attitude courses, such as self-defense and self-care.
Abstract: Core text in attitude courses. Explains "theory and reasoned action" model and then applies the model to various cases.

26,683 citations


"Confidence Limits for the Indirect ..." refers background in this paper

  • ...Examples of indirect effect hypotheses are that attitudes affect intentions which then affect behavior (Ajzen & Fishbein, 1980), that poverty reduces local social ties which increases assault and burglary rates (Warner & Rountree, 1997), that social status has an indirect effect on depression…...

    [...]

Journal ArticleDOI
TL;DR: In this article, an exploración de the avances contemporaneos en la teoria del aprendizaje social, con especial enfasis en los importantes roles que cumplen los procesos cognitivos, indirectos, and autoregulatorios.
Abstract: Una exploracion de los avances contemporaneos en la teoria del aprendizaje social, con especial enfasis en los importantes roles que cumplen los procesos cognitivos, indirectos, y autoregulatorios.

20,904 citations