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Special Topic: Logistic Regression for Binary outcomes
The dependent variable is often binary such as whether a person litters or not, used a condom or not, dead or alive, diseased or not, intercourse or not, or divorced or not.
In this case, logistic or probit regression is the method of choice because of violation of assumptions if ordinary least squares regression is used.
Estimates of the mediated effect using logistic and probit regression can be distorted using conventional procedures.
Here we examine binary or continuous X, continuous M, and binary Y.
MacKinnon et al., under review in Clinical Trials and MacKinnon et al., under review Psychological Methods.
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Logistic Regression Model for Equations 1 and 2
Standard logistic regression model, where Y depends on X, β1 is the intercept and τ codes the relation between X and Y.
logit Pr{Y=1|X} =β1 + τX (1)Standard logistic regression model, where Y depends
on X and M, β2 is the intercept, τ′ codes the relation between X and Y adjusted for M and β codes the relation between M and Y, adjusted for X.
logit Pr{Y=1|X,M} = β2 + τ′X + β M (2)
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Logistic Regression Model for latent variable Y*
Y* = β1 + τX + ε1 (1)Y* = β2 + τ′X + β M + ε2 (2)The unobserved latent variable Y* is linearly related
to X and then to both X and M, ε1 and ε2 represent residual variability and have a standard logistic distribution. The dichotomous Y is derived from Y* through the relation Y = 1 if and only if Y* > 0. The same model applies for the probit with the errors having a standard normal distribution rather than a standard logistic distribution.
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Equation 3
M = β3 + αX + ε3 (3)M is a continuous variable so ordinary least squares
regression is used to estimate this model where β 3 is the intercept, α represents the relation between X and Y, and ε3 is residual variability.
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Logistic Regression Model for latent variable Y*
τ - τ′ Difference in coefficients. The coefficients are from separate logistic regression equations.
αβ Product of coefficients. The β coefficient is from a logistic regression model and α is from an ordinary least squares regression model.
As will be shown, the difference in coefficient method can give distorted values for the mediated effect because of differences in the scale of separate logistic regression models. For both Equations 1 and 2, residual variability is fixed at 2/3 and fixed at 1 for probit regression.
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What is the in the next plot? Expected logistic regression coefficients based on
Haggstrom (1983) are used to compute τ - τ′ andα β.
All possible combinations of α, β and τ′ values for small (2% variance explained), medium (13%), large (26%),
and very large (40%) effects (4 X 4 X 4 = 64) Y-axis is the expected value for τ - τ′ and
α β X-axis is the true value of the b coefficient in the
continuous variable mediation model. It is indicated by βC
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Plot of true values of αβ and τ - τ′ as a function of true mediated effect
and true value of βC.
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Plot of true proportion mediated as a function of true value of βC.
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αβ/(αβ+τ΄)
1-(τ΄/τ)
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αβ and τ - τ′ are not equal in Logistic and Probit Regression
• The two estimators, α β and τ - τ′ are not identical in logistic or probit regression because, unlike ordinary least squares regression where the residual variance varies across equations, in logistic regression the residual variance is fixed to equal 2/3 (MacKinnon & Dwyer, 1993). So the logistic regression coefficients are a function of the relations among variables and the fixed value of the residual variance.
• There are solutions
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Solutions to mediation estimation in Logistic and Probit Regression
• Standardize the values of the coefficients. • One standardization method computes the variance of Y in
both equations and uses that to standardize values (MacKinnon & Dwyer, 1993; Winship & Mare, 1983).
• Another standardization method standardizes coefficients in Equation 2 to be in the same metric as Equation 1. To the best of our knowledge, this is a new method that is described below.
• Use a computer program such as Mplus that appropriately handles categorical variables in covariance structure models. I believe that this approach is similar to the first approach to standardization, i.e., the scale of the latent Y* is the same for all equations in a model.
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Standardizing across logistic regression equations
• Standardize the values of the coefficients in Equations 1 and 2 (see MacKinnon & Dwyer, 1993 and Winship & Mare, 1983).
• s2Y* = τ2sX
2 + 2/3 and divide the τ coefficient and standard error by sY*
from this equation.• s2
Y* = τ′2sX2 + β2sM
2 + 2 τ′ β sXM + 2/3 and divide the τ′ and β coefficients and standard errors by sY* from this equation.
• where sX2 is the variance of the X variable, sM
2 is the variance of the M variable, and sXM
is the covariance of the X and M variables.• The α parameter does not require rescaling if M is continuous. Note
that if probit regression is used the last term of the equations for s2Y*
should be 1 rather than 2/3.
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Standardizing Equation 2 to the metric of Equation 1
• The coefficients from Equation 2 are divided by the following quantity:
• where σ233·X is the residual variance in the regression
model for M predicted by X, i.e. Equation 3. The first term is replaced with 1 for probit regression.
22 2 2* 33.
ˆˆ ˆ3Y X
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Plot of true values of αβ and τ - τ′ as a function βC, after standardization.
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Plot of true values of proportion mediated as a function of βC, after standardization.
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ab/t
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Simulation Design
All possible combinations of α, β and τ′ effect size for small (2% variance explained), medium (13%), large
(26%), and very large (40%) effects.
6 Sample sizes, N= 50, 100, 200, 500, 1000, 5000
1000 Replications of each of the 4 X 4 X 4 X 6 = 384 generated data sets or 384,000 data sets.
Probit and Logistic Regression on the same data
Standardized and Unstandardized coefficients
Data were generated using standard normal deviate for the error term in Equation 2–which is the probit model.
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Simulation Outcomes
Estimates of α β and τ - τ′ before and after standardization for both probit and logistic regression.
Estimates of proportion mediated α β /(α β +τ′), 1-(τ′/τ), and
α β /τ before and after standardization for both probit and logistic regression
Measures of mean and average relative bias
Tables and plots
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The estimated mediated effect, τ - τ′, as a function of βC for α=.14.
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d E
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t
τ΄=.14
τ΄=.39
τ΄=.59
τ΄=1
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Power: Logistic Regression
Sample SizeSmall Effect Size 50 100 200 500 1000Delta Method .001 .005 .044 .367 .842
Joint Significance .012 .042 .142 .571 .905Asymmetric .012 .041 .143 .599 .921
*from MacKinnon, Yoon, & Lockwood (2003, SPR).
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Summary and Future Directions
1. Unlike the linear OLS model case, the difference in coefficients and product of coefficients estimators of the mediated effect are not equal. The difference in coefficients estimator is distorted, as shown with expected values and in the
simulation study. The same problem occurs for the proportion mediated measures.
2. Standardization of coefficients across equations solves the problem and removes distortion. Two approaches to standardization were mentioned, but the results for rescaling coefficients in Equation 2 to be in the same metric as those in Equation 1 were described. The other standardization method works in a similar manner.
3. The simplest approach is the product of coefficients estimator of the mediated effect, which does not require standardization. Researchers who prefer the logic of the difference in coefficients methods should standardize coefficients prior to
computing the mediated effect.. 4. The standardization approaches should apply to other examples of the
Generalized Linear model such as the Poisson and survival analysis model.
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Surrogate Endpoint Research I
• The length of time for a disease to occur and low incidence of the disease require very large sample sizes and long duration studies.
• Alternative is to find an outcome that can serve as a surrogate for the ultimate outcome. Here the mediator is called a surrogate or intermediate outcome.
• Surrogate endpoints are more frequent or more proximate to the prevention strategy.
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Examples of Surrogate endpoints
• Precancerous cells for colon cancer
• Cholesterol level for coronary heart disease.
• Bone density for osteoporosis
• Lymphocyte levels for HIV/AIDS
• Partial loss of vision for blindness
• Tumor size for breast cancer
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Surrogate endpoints Research II
• “Above all else, we believe that the issue of when and how to use surrogate endpoints is probably the pre-eminent contemporary problem in clinical trials methodology, so it merits much extensive scrutiny” (Begg & Leung, 2000, p. 27).
• A surrogate endpoint is a “response variable for which a test of the null hypothesis of no relationship to the treatment groups under comparison is also a valid test of the corresponding null hypothesis based on the true endpoint” (Prentice, 1989, p. 432)
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Micromediational chain• It is often not possible to study all steps in a mediation
chain, e.g., in a prevention program, to study each of six constructs in a theoretical chain from exposure to a component, comprehension, retention of the component’s message, short-term attitude change, long-term attitude change, and long-term refusal to use drugs.
• Cook and Campbell (1979) make a distinction between molar mediation where some steps are studied and micromediation where each link is measured. Kenny et al., (1998) make the distinction between proximal and distal mediators.
• Any mediation model is part of a longer mediation chain. The researcher decides what part of the micromediational chain to examine. Similar decisions must be made about outcomes.
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Possible Surrogate endpoints in Prevention
• Aggression at age 12 for incarceration at 24.
• Early onset gateway drug use for adult addiction and driving under the influence.
• Harming animals as a child for later assault.
• Social withdrawal at age 8 for adult depression.
• School dropout for adult unemployment.
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Prevention Mediators versus Surrogate Endpoints
• Many similarities between surrogates and mediators in prevention science, but…
• Theoretical causal connection between surrogate and outcome is often clearer than in prevention.
• In prevention, relation between mediator (surrogate) and outcome is weaker than in most areas of surrogate endpoint research.
• Surrogates are more likely to completely mediate effects of X on the outcome.
