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## Confidence limits for the indirect effect: Distribution of the product and resampling methods (2004)

Venue: | Multivariate Behavioral Research |

Citations: | 228 - 3 self |

### Citations

10859 | Statistical power analysis for the behavioral sciences (2nd ed - Cohen - 1988 |

5256 | The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations
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Citation Context ...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 whi... |

4960 | An Introduction to the Bootstrap. - Efron, Tibshirani - 1993 |

2794 |
Social Learning Theory.
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Citation Context ...ial 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 c... |

2693 |
Understanding attitudes and predicting social behavior
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Citation Context ... 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... |

770 | Asymptotic confidence intervals for indirect effects in structural equation models
- Sobel
- 1982
(Show Context)
Citation Context ... 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 b... |

733 |
Randomization, bootstrap and Monte Carlo methods
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- 2007
(Show Context)
Citation Context ... the method based on a normal distribution for the indirect effect. Resampling approaches may yield more accurate confidence limits and also provide a test of the significance of the indirect effect (=-=Manly, 1997-=-; Noreen, 1989). The bootstrap resampling procedure is one method that may provide more accurate confidence limits (Bollen & Stine, 1990), but Type I error rates based on the standard percentile boots... |

693 | Risk and protective factors for alcohol and other drug problems in adolescence and early adulthood: Implications for substance abuse prevention
- Hawkins, Catalano, et al.
- 1992
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Citation Context ...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 randomi... |

630 | 2002. Mediation in experimental and non-experimental studies: New procedures and recommendations
- Shrout, Bolger
(Show Context)
Citation Context ...al 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. Lockwood and MacKinnon (1998) also obtained asymmetric confidence limits and presented a comp... |

215 |
Statistical methods for social scientists
- Hanushek, Jackson
- 1977
(Show Context)
Citation Context ...hese 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)... |

194 |
Data Analysis and Regression: A Second Course in Statistics
- Mosteller, Tukey
- 1977
(Show Context)
Citation Context ...or nonnormal distributions although the bootstrap-Q and the bias-corrected bootstrap may be especially appropriate for severely nonnormal data (Chernick, 1999; Manly, 1997). Jackknife. The jackknife (=-=Mosteller & Tukey, 1977-=-) is one of the first resampling methods described in the research literature. For a sample size N, there are N jackknife samples (8) s jackknife N −1 = Σ⎡u −u N ⎣ () i (.) ⎤ ⎦ 2 formed by removing on... |

193 | Process analysis: Estimating mediation in treatment evaluations
- Judd, Kenny
- 1981
(Show Context)
Citation Context ...dependent variable (X) O and the mediating variable (X ). The indirect effect equals the difference M in the estimated independent variable coefficients ( tˆ − t′ ˆ ) 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 b) in Equation 2 above... |

175 | Estimating mediated effects in prevention studies
- MacKinnon, Dwyer
- 1993
(Show Context)
Citation Context ...irect 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 ... |

163 |
Computer-intensive Methods for Testing Hypotheses: An Introduction
- Noreen
- 1989
(Show Context)
Citation Context ...ased on a normal distribution for the indirect effect. Resampling approaches may yield more accurate confidence limits and also provide a test of the significance of the indirect effect (Manly, 1997; =-=Noreen, 1989-=-). The bootstrap resampling procedure is one method that may provide more accurate confidence limits (Bollen & Stine, 1990), but Type I error rates based on the standard percentile bootstrap approach ... |

145 |
Mediators, Moderator, and Test for Mediation
- R, Brett
- 1984
(Show Context)
Citation Context ...and J. Williams in ordinary least squares regression (MacKinnon, Warsi, & Dwyer, 1995). Additional assumptions of the â ˆ b 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, ... |

133 | The decomposition of effects in path analysis - Alwin, Hauser - 1975 |

130 |
A simulation study of mediated effect measures
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- 1995
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Citation Context ...101D. MacKinnon, C. Lockwood, and J. Williams Figure 1 The Indirect Effect Model 102 MULTIVARIATE BEHAVIORAL RESEARCHD. MacKinnon, C. Lockwood, and J. Williams in ordinary least squares regression (=-=MacKinnon, Warsi, & Dwyer, 1995-=-). Additional assumptions of the â ˆ b estimator of the indirect effect from Equations 2 and 3 have been outlined (James & Brett, 1984; McDonald, 1997). These assumptions include no measurement error ... |

123 |
Socioeconomic Background and Achievement
- Featherman, Duncan
- 1972
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Citation Context ...us 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 art... |

119 |
Causal inference, path analysis, and recursive structural equations models
- Holland
- 1988
(Show Context)
Citation Context ...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 chang... |

118 |
Bootstrap Methods: A Practitioner’s Guide
- Chernick
- 1999
(Show Context)
Citation Context ...ootstrap-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). Jackknife. The jackknife (Mosteller & Tukey, 1977) is one of the first resampling methods described in the research literature. For a sample size N, there are N jackknife samples (8) s... |

104 |
The Algebra of Random Variables
- Springer
- 1979
(Show Context)
Citation Context ...& 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-=-). The product of two normal random variables is not normally distributed (Lomnicki, 1967; Springer & Thompson, 1966). In the null case where both random variables have means equal to zero, the distri... |

95 |
Direct and Indirect Effects: Classical and Bootstrap Estimates of Variability." Sociological Methodology
- Bollen, Stine
- 1990
(Show Context)
Citation Context ...est & 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, & Sh... |

86 |
The epidemiology of social stress
- Turner, Wheaton, et al.
- 1995
(Show Context)
Citation Context ... 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... |

84 |
Meta-analysis of 143 adolescent drug prevention programs: Quantitative outcomes result of program participants compared to a control or comparison group
- Tobler
- 1986
(Show Context)
Citation Context ...iables 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 ... |

80 |
Preventing alcohol, marijuana and cigarette use among adolescents: peer pressure training versus establishing conservative norms
- Hansen, Graham
- 1991
(Show Context)
Citation Context ...ned 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 d... |

79 | Some New Results on Indirect Effects and their Standard Errors in Covariance Structure Models." Sociological Methodology - Sobel - 1986 |

71 | Robustness? British - Bradley - 1978 |

66 |
Direct and indirect effects in linear structural equation models
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Citation Context ...ˆ s ˆ bˆ . 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 ... |

62 |
What if there were no significance tests
- Harlow, Mulaik
- 1997
(Show Context)
Citation Context ...ider 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... |

56 |
The probability function of the product of two normally distributed variables
- Aroian
- 1947
(Show Context)
Citation Context ...d 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). The product of two normal random variables is not normally distributed (Lomnicki, 1967; Springer & Thompson, 1966). In the null case where both random variables have me... |

54 |
A comparison of methods to test the significance of the mediated effect
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(Show Context)
Citation Context ...ce (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 ad... |

52 |
Effect analysis and causation in linear structural equa- tion models." Psychometrika
- Sobel
- 1990
(Show Context)
Citation Context ...as-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 hypothes... |

52 | Local Social Ties in a Community and Crime Model: Questioning the Systemic
- Warner, Rountree
- 1997
(Show Context)
Citation Context ...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... |

39 | Analysis of mediating variables in prevention and intervention research
- MacKinnon
- 1994
(Show Context)
Citation Context ...ables 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, 1995... |

34 | Mediating mechanisms in a school-based drug prevention program: Firstyear effects of the Midwestern Prevention Project
- MacKinnon, Johnson, et al.
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(Show Context)
Citation Context ... 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 = ab ˆ ˆ / s ˆ ab ˆ ˆ (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 Equatio... |

32 |
The null hypothesis testing controversy in psychology
- Krantz
- 1999
(Show Context)
Citation Context ...ition 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 Typ... |

31 |
On the frequency function of xy
- CRAIG
- 1936
(Show Context)
Citation Context ...ly (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). The product of two normal random variables is not normally distributed (Lomnicki, 1967; Springer & Thompson, 1966). In the null case where both random variables have means equal to ... |

26 | Haldane’s lungs: A case study in path analysis. Multivariate Behavioral Research 32(1):1–38 - McDonald - 1997 |

25 |
The robustness of estimates total indirect effects in covariance structure models estimated by maximum likelihood . Psychometrika 55(2
- Sobel
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(Show Context)
Citation Context ...ias 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 = ab ˆ ˆ / s ˆ ab ˆ ˆ ... |

24 |
Causal inference in statistical models of the process of socioeconomic achievement
- Sobel
- 1998
(Show Context)
Citation Context ...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 theo... |

16 |
Towards understanding individual effects in multicomponent prevention programs: Design and analysis strategies
- West, Aiken
- 1997
(Show Context)
Citation Context ...gned 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, 1... |

15 |
Survey design and analysis: Principles, cases, and procedures
- Hyman
- 1955
(Show Context)
Citation Context ...riable 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 ... |

14 | Mathematical forms of the distribution of the product of two normal variables - Aroian, Taneja, et al. - 1978 |

14 | Bootstrapping the Standard Error of the Mediated Effect
- Lockwood, MacKinnon
- 1998
(Show Context)
Citation Context ...e value was more often to the right than to the left of the confidence interval. Asymmetric confidence intervals were also obtained in bootstrap analysis of the indirect effect (Bollen & Stine, 1990; =-=Lockwood & MacKinnon, 1998-=-). The implication of the imbalance is that there is less power than expected to detect a true indirect effect. Stone and Sobel (1990, p. 349) analytically demonstrated that at least part of the imbal... |

14 |
On the distribution of products of random variables
- Lomnicki
- 1967
(Show Context)
Citation Context ...ed on the distribution of the product of two normally distributed random variables (Aroian, 1947; Craig, 1936; Springer, 1979). The product of two normal random variables is not normally distributed (=-=Lomnicki, 1967-=-; Springer & Thompson, 1966). In the null case where both random variables have means equal to zero, the distribution is symmetric with kurtosis of six (Craig, 1936). When the product of the means of ... |

13 |
Statistical methods for analyzing collapsibility in regression models
- Clogg, Petkova, et al.
- 1992
(Show Context)
Citation Context ...ere are other methods to compute confidence limits for the indirect effect based on the standard error of the difference in coefficients, � – �� (e.g., Allison, 1995b ; Clogg, Petkova, & Cheng, 1995; =-=Clogg, Petkova, & Shihadeh, 1992-=-; Olkin & Finn, 1995) but these methods perform similar to the traditional z test described in this article, in part because a normal distribution for the indirect effect is assumed. There are several... |

11 | Mediating mechanisms in a program to reduce intentions to use steroids, improve exercise self-efficacy, and dietary behavior. Manuscript submitted for publication - MacKinnon, Goldberg, et al. - 1999 |

8 |
The distribution of independent random variables
- Springer, Thompson
- 1966
(Show Context)
Citation Context ...bution of the product of two normally distributed random variables (Aroian, 1947; Craig, 1936; Springer, 1979). The product of two normal random variables is not normally distributed (Lomnicki, 1967; =-=Springer & Thompson, 1966-=-). In the null case where both random variables have means equal to zero, the distribution is symmetric with kurtosis of six (Craig, 1936). When the product of the means of the two random variables is... |

7 |
The Impact of Random Predictors on Comparisons of Coefficients Between Models
- Allison
- 1995
(Show Context)
Citation Context ...cKinnon, 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, 1995-=-a; 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, Lockwo... |

7 |
Correlation redux
- Olkin, Finn
- 1995
(Show Context)
Citation Context ...b ˆ bˆ + ˆ 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 result... |

7 |
The bootstrap, the jackknife, and the randomization test: a sampling taxonomy
- Rodgers
- 1999
(Show Context)
Citation Context ...o assess mediation for small to moderate sample sizes. Resampling methods are generally considered the method of choice when the assumptions of classical statistical methods are not met (Manly, 1997; =-=Rodgers, 1999-=-), such as for the nonnormal distribution of the indirect effect. Thus the purpose of Study 2 was to determine whether jackknife, Monte Carlo, and bootstrap resampling methods yield more MULTIVARIATE ... |

6 |
Comment on ‘‘Bootstrapping confidence intervals’’ by T
- Gleser
- 1996
(Show Context)
Citation Context ...her small values of � � and � � , then the resampling methods are more accurate than single sample tests. A final limitation of resampling tests has been called the first law of statistical analysis (=-=Gleser, 1996-=-). The law requires that any two statisticians analyzing the same data set with the same methods should come to identical conclusions. With resampling methods (other than the jackknife), it is possibl... |

4 |
Reply to Allison: More on Comparing Regression Coefficients
- Clogg, Petkova, et al.
- 1995
(Show Context)
Citation Context ...o compute confidence limits. There are other methods to compute confidence limits for the indirect effect based on the standard error of the difference in coefficients, � – �� (e.g., Allison, 1995b ; =-=Clogg, Petkova, & Cheng, 1995-=-; Clogg, Petkova, & Shihadeh, 1992; Olkin & Finn, 1995) but these methods perform similar to the traditional z test described in this article, in part because a normal distribution for the indirect ef... |

3 | An algorithm to compute the CDF of the product of two normal random variables - Meeker, Escobar - 1994 |

2 |
Social support of children of divorce: Direct and stress-buffering effects
- Wolchik, Ruehlman, et al.
- 1989
(Show Context)
Citation Context ...ded by its standard error and the resulting ratio is then compared to the standard normal distribution to test its significance, z = ab ˆ ˆ / s ˆ ab ˆ ˆ (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) ab ˆ ± z1 −v /2* sˆ ab ˆ ˆ ˆ where ... |

1 |
Exact variance of indirect effects in recursive linear models
- Allison
- 1995
(Show Context)
Citation Context ...cKinnon, 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, 1995-=-a; 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, Lockwo... |

1 |
Amos 4.0 users’ guide version 3.6
- Arbuckle, Wothke
- 1999
(Show Context)
Citation Context ...ese include the lack of statistical software to conduct the analysis and the increased computational time required for the analysis. The bias-corrected bootstrap is included as an option in the AMOS (=-=Arbuckle & Wothke, 1999-=-) covariance structure analysis program. The EQS program and LISREL programs can be used to conduct resampling analyses but the user must write a separate program to compute the bootstrap method after... |

1 |
EQS for Windows (Version 5.6) [Computer program
- Bentler
- 1997
(Show Context)
Citation Context ... 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 â and ˆ b , and the estimated standard errors, s ˆ and aˆ s ˆ bˆ . Allison (1995a) used a re... |

1 | Sample size, reliability, and tests of statistical mediation - MacKinnon, Lockwood, et al. - 1999 |

1 |
Selected tables in mathematical statistics, volume VII: The product of two normally distributed random variables
- Meeker, Cornwell, et al.
- 1981
(Show Context)
Citation Context ...cal � values in the table assume that population values of �, �, � , and � are � � known, but the authors suggest that sample values can be used in place of the population values as an approximation (=-=Meeker et al., 1981-=-, p. 8). The frequency distribution function of the product of two standardized normal variables, presented in Meeker et al. (1981), can be used to compute the statistical power, Type I error, and con... |

1 | SAS (Version 6.12) [Computer program - Institute - 1989 |

1 |
LINCS (Version 2.0) [Computer program
- Schoenberg, Arminger
- 1996
(Show Context)
Citation Context ...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 â and ˆ b , and the estimated standard errors, s ˆ and aˆ s ˆ bˆ . Allison (1995a) used a reduced form parameterization of the indirect effect model to derive the same... |