Testing and Interpreting Mediational and Moderational Models in Logistic Regression

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Testing and Interpreting Mediational and
Moderational Models in Logistic Regression

• A practical talk
• Kevin M. King, M.A.

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Goals

• Provide a basic understanding of logistic
  regression
• Demonstrate how to compute standardized
  coefficients using Excel to compute the indirect
  and direct effect for mediational models in
  logistic regression (e.g. MacKinnon Dwyer,
  1993)
• Demonstrate how to use Excel to create graphs of
  logistic regression interactions

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A caveat

• I am not a quantitative expert.
• Im just a guy whos worked with these models
  quite a bit and has some shortcuts to share.
• To really understand whats going on, you really
  need to understand logistic regression, which you
  wont find here. Take a course or read a book.
• A great (and cheap) resource for learning
  logistic regression is the Sage Primer by Pampel
  (2000)

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Prediction of Dichotomous Variables

• Many variables are dichotomous (e.g. death,
  psychological diagnosis, participation in an
  intervention) and very relevant.
• It is desirable to apply the same modeling
  framework that we use to predict continuous
  outcomes to dichotomous outcomes

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Difficulties

• We often think in terms of continuous prediction
• i.e.
• How does a personality lead to drug use disorder
• How is income related to participation
• Yet we cant talk about 1 to 1 relations, where
  5,000 more of income leads to 1 more
  participation, or 1 higher impulsivity leads to 1
  more drug use disorder

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Continuous Distribution
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Risk and Odds

• We must translate our thinking to binary outcomes
  and talk about risk and odds, to describe the
  probability of being in one state or another.
• These probabilities are not distributed linearly
• E.g.
• relation of of pubs to tenure
• Relation of of alcoholic relatives to alcoholism

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Binary Distributions
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Probability Curve (Sigmoid Curve)
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Estimation of Logistic Regression-and
implications for mediation-

• Logistic Regression uses a transformation of the
  probability curve, called the logit (logged
  odds), which linearizes the probability curve.
• Logitln( Pi /(1-Pi) ), Odds Pi /(1-Pi),
  Probability Pi
• But since we observe the actual presence or
  absence, rather than its probability, we cant
  use OLS procedures
• Error term is non-normally distributed
• Error variances are non-equal across values of IV
• (i.e. distribution is inherently nonlinear)

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Probability vs. Odds vs. Logit

• Probability Pi chance of occurrence
• Odds Pi /(1-Pi) odds of occurring vs. not (e.g.
  men are twice as likely to be diagnosed with
  alcoholism)
• Logitln( Pi /(1-Pi) ) log of the odds,
  otherwise un-interpretable
• Standardizing variables (Z-scores) by dividing by
  their SE helps interpretation

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Maximum Likelihood Estimation

• Maximum likelihood estimation (MLE) is used to
  estimate these unobserved logged odds.
• An estimation procedure that tries to best
  reproduce the covariance matrix of the data given
  the model
• In predicting binary outcomes, the initial best
  guess is the proportion in the population.
• Each explanatory variable that is added to the
  model improves model fit and heightens the chance
  of accurate prediction of binary class membership
  (e.g. correctly predicting diagnosis).
• Because Y is unobserved in logistic regression,
  its variance is also unobserved. In order to
  estimate the model, the variance of the residual
  is fixed to p2/3 in logistic regression.
• The scale in logistic regression depends on the
  extent of prediction that depends on the
  variables in the model (MacKinnon Dwyer, 1993,
  p. 150)
• Thus the coefficients in each model are scaled
  according to the explanatory power of the other
  coefficients in the model and p2/3.

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Mediational ModelsApplication to Logistic
Regression

• Mediation is where a predictor has an indirect
  effect on an outcome through a third variable.
  This indirect effect accounts for some to all of
  the main effect of the predictor on the outcome
  (see Baron and Kenny, 1988 )
• Mediation can be tested through measuring the
  impact of the mediator on t (e.g. t-t) or
  through testing the significance of the indirect
  effect (ab)
• See Shrout Bolger, 2002 and MacKinnon, Krull
  Lockwood, 2000 for good discussions of these
  methods)

Mediator
a
b
Predictor
Outcome
t
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The Rub

• In logistic regression, a and b and t and t
  dont come from the same equations, which means
  that when the outcome or mediator is binary they
  do not have the same scale.
• Thus one cannot compute the mediation effect
  either through ab or t-t

Mediator
a
b
Predictor
Outcome
t
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The Solution

• The solution is to standardize the coefficients,
  as MacKinnon and Dwyer (1993) recommend.
• Because the variance of the outcome is dependent
  on the variables in the model plus p2/3, we can
  estimate the variance of the outcome and use it
  to standardize the coefficients.

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The Formula

• Variance of Outcome
• s2(O) b2s2(M) t2 s2(P) 2bts(PM) p2/3
• This means
• Variance of outcome (coefficient for Mediator
  squaredvariance of the Mediator) (coefficient
  of Predictor squared variance of Predictor)
  (2coeff. of med. coeff. of pred. covariance
  of P and M) pi squared/3.
• This can be expanded to include any number of
  covariates. Each variable must be included both
  as its coeffcient squared by its variance and a
  term for its coefficient times each other
  coefficient in the mode times their covariance
• To standardized coefficents
• b b/ s2(O)

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An Example The mediating effect of behavioral
undercontrol on the relation between parental
alcoholism and drug use disorder (from King
Chassin, 2004)
Table 2 Logistic Regression Predicting Drug
Diagnosis from Parental Alcoholism and Behavioral
Undercontrol
Note.plt.001
Undercontrol
0.27
0.61
Parent Alc.
Offspring Drug Disorder
0.54
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Standardization of coefficients

• Using the coefficients and the variance-covariance
  matrix of the variables in the equation, we can
  easily fill in the values of the formula.
• Steps
• Get the variance-covariance matrix for all
  variables in the model, paste into Excel
• Make a table of all coefficients and SEs from
  model results
• Using the table of coefficients, make a
  variance-covariance like table (where the
  on-diagonal is the coefficient squared and the
  off diagonal is 2ab)
• Combine the variance-covariance table and the new
  table of coefficients my multiplying matching
  cells
• Sum the new combined table and add p2/3.
• Use this outcome variance to standardize b and t
  by dividing each coefficient by the outcome
  variance

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Standardized Model
Undercontrol
0.27
0. 15
Parent Alc.
Offspring Drug Disorder
0.12
Proportion Mediatedb/(tb)
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Moderation in Logistic Regression Interpreting
Coefficients and Graphing Interactions

• Testing interactions in logistic regression is
  similar to OLS regression methods, in that one
  includes an interaction term in the model
  predicting a binary outcome.
• Interactions can also be probed using Aiken
  Wests method (test at 1 SD above and below the
  mean).
• Centering is just as important as in OLS
  regression, and standardizing variables will also
  aid in interpretation of coeffcients
• Present model shows maternal support moderating
  the relation between behavioral undercontrol and
  risk for young adult drug use disorder.

Support
-0. 61
0.40
Undercontrol
Offspring Drug Disorder
0.85
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Interpreting Coefficients and Graphing
Interactions An Example (from King Chassin,
2004)

• An Example Behavioral undercontrols effect on
  drug use disorder is moderated by parental
  support.

Table 5 Logistic Regression Predicting Drug
Diagnosis From Behavioral Undercontrol and
Parental Support
Note. B the unstandardized logistic regression
coefficient. plt.05, plt.01, plt.001
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The OLS Extension

• Probe the interaction at 1 SD above and below the
  mean of the moderated variable to obtain
  coefficients and intercepts.
• Plot these coefficients across a range of data
  points of the moderated variable (remember,
  youve standardized your predictors for easy
  interpretation).

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The Logistic Twist

• The previous graph is in terms of the logit. Its
  good for helping us understand the nature of the
  interaction (in this case protective but
  reactive)
• However, it fails to give us a sense of whats
  really happening in terms of how the predicted
  probabilities differ across levels of the
  moderator
• Thus, we need to transform the coefficients to
  the odds or probability to create interpretable
  graphs
• Pe(logit)/(1e(logit)), Oddse(logit)

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Odds and Probability Metric
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Interpreting with REAL values

• While we may see the shape of the probability or
  odds function in the above graphs, note that they
  extend out to 6.5 SD above the mean for
  undercontrol!
• Its important to display your interactions where
  there is real data.
• To do this, you can run your moderational model
  in SPSS and save out the predicted probabilities
  for each participant as a variable. See code
  below for an example.

LOGISTIC REGRESSION VARc4drugdx /METHODENTER
rgrp paranti rgen zc3age /METHODENTER zunder
zc3ss /METHODENTER unbyks /CRITERIA PIN(.05)
POUT(.10) ITERATE(20) CUT(.5) /save pred
/CLASSPLOT.
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Putting it all together

• Take the predicted probabilities from SPSS and
  move them next to the participants scores on the
  moderated variable (e.g. undercontrol).
• Select both columns, copy and past into Excel.
• Select the predicted probabilities and the model
  implied probabilities and graph
• I use a scatter plot in Excel. Using the chart
  wizard
• For the X values of the predicted probabilities,
  select the actual values of the moderated
  variable
• For the model implied, select the column of
  values used to make the simple slope graph (e.g.
  -1.5 SD to 1.5 SD, etc)

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