Special Topic: Logistic Regression for Binary outcomes

1
Special Topic Logistic Regression for Binary
outcomes

• The dependent variable is often binary such as
  whether a person litters or not, uses drugs or
  not, dead or alive, diseased 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.

2
Logistic Regression Model for Equations 1 and 2

• Standard logistic regression model, where Y
  depends on X, ?1 is the intercept and t codes the
  relation between X and Y.
• logit PrY1X ?1 tX (1)
• Standard logistic regression model, where Y
  depends on X and M, ?2 is the intercept, t' codes
  the relation between X and Y adjusted for M and ?
  codes the relation between M and Y, adjusted for
  X.
• logit PrY1X,M ?2 t'X ?M (2)
• .

3
Logistic Regression Model for latent variable Y

• Y ?1 tX e1 (1)
• Y ?2 t'X ?M e2 (2)
• The unobserved latent variable Y is linearly
  related to X and then to both X and M, e1 and e2
  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 gt 0. The same model applies for
  the probit with the errors having a standard
  normal distribution rather than a standard
  logistic distribution.

4
Equation 3

• M ?3 aX e3 (3)
• M is a continuous variable so ordinary least
  squares regression is used to estimate this model
  where ?3 is the intercept, a represents the
  relation between X and Y, and e3 is residual
  variability.

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Logistic Regression Model for latent variable Y

• t - t' Difference in coefficients. The
  coefficients are from separate logistic
  regression equations.
• a? Product of coefficients. The ?
  coefficient is from a logistic regression model
  and a 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 t - t'
  and
• a?.
• All possible combinations of a, ? and t' 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 t - t' and
• a?
• X-axis is the true value of the b coefficient in
  the continuous variable mediation model. It is
  indicated by ?C
• Later plots will show the same information for
  expected values after standardization.

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Plot of true values of a? and t - t' as a
function of true mediated effect and true value
of ?C.
8
Plot of true proportion mediated as a function of
true value of ?C.
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a? and t - t' are not equal in Logistic and
Probit Regression

• The two estimators, a? and t - t' 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 B2/3 (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

10
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 t 2sX2 ?2/3 and divide the t coefficient
  and standard error by sY from this equation.
• s2Y t'2sX2 ? 2sM2 2 t' ?sXM ? 2/3 and
  divide the t' and ? coefficients and standard
  errors by sY from this equation.
• where sX2 is the variance of the X variable, sM2
  is the variance of the M variable, and sXM is the
  covariance of the X and M variables.
• The a 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 s233X 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.

13
Plot of true values of a? and t - t' as a
function ?C, after standardization.
14
Plot of true values of proportion mediated as a
function of ?C, after standardization.
15
The estimated mediated effect, t - t', as a
function of ?C for a.14.
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Summary and Future Directions

• 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.
• 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.
• 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..
• The standardization approaches should apply to
  other examples of the Generalized Linear model
  such as the Poisson and survival analysis model.