special topic: logistic regression for binary outcomes
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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.  PowerPoint PPT PresentationTRANSCRIPT
Special Topic: Logistic Regression for Binary outcomesThe 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.
Logistic Regression Model for Equations 1 and 2Standard logistic regression model, where Y depends on X, 1 is the intercept and codes the relation between X and Y.logit Pr{Y=1X} =1 + X (1)Standard logistic regression model, where Y depends on X and M, 2 is the intercept, codes the relation between X and Y adjusted for M and codes the relation between M and Y, adjusted for X.logit Pr{Y=1X,M} = 2 + X + M (2)
Logistic Regression Model for latent variable Y*Y* = 1 + X + 1 (1)Y* = 2 + X + M + 2 (2)The unobserved latent variable Y* is linearly related to X and then to both X and M, 1 and 2 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.
Equation 3M = 3 + X + 3 (3)M is a continuous variable so ordinary least squares regression is used to estimate this model where 3 is the intercept, represents the relation between X and Y, and 3 is residual variability.
Logistic Regression Model for latent variable Y*  Difference in coefficients. The coefficients are from separate logistic regression equations. Product of coefficients. The coefficient is from a logistic regression model and 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.
What is the in the next plot?Expected logistic regression coefficients based on Haggstrom (1983) are used to compute  and .All possible combinations of , and values for small (2% variance explained), medium (13%), large (26%), and very large (40%) effects (4 X 4 X 4 = 64)Yaxis is the expected value for  and Xaxis is the true value of the b coefficient in the continuous variable mediation model. It is indicated by C
Plot of true values of and  as a function of true mediated effect and true value of C.
Chart1
0.11987433370.1105121968
0.33864277840.2568982182
0.51694608480.3294832979
0.86802549560.4039127943

c
Mediated Effect
TRUEBMEANS
HB Standardization
_TYPE__FREQ_c//(+)1(/)/1(/)
1400000.140.11987433370.11051219680.33331692730.32927920390.3213190460.11680623090.33078088110.3265775869
1400000.390.33864277840.25689821820.49310513970.45264680360.40525619180.31488833240.46522228090.4392280106
1400000.590.51694608480.32948329790.61248960410.51503909180.42589516260.46977456650.53944086470.4945893575
14000010.86802549560.40391279430.86334060180.60.42776648670.78548548250.63916387170.5759176539
TRUEBMEANS
00
00
00
00

c
Mediated Effect
Fig2: True values of mediated effect estimates: Logistic regression
BMEANS
000
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1(/)
c
Proportion Mediated
Fig3: True values of proportion mediated: Logistic regression
fig8
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HB tt'
c
Mediated Effect
Fig4: True values of mediated effect estimates: Logistic regression with HB standardization of t
000
000
000
000
HB ab/t
/(+)
HB 1(t'/t)
c
Proportion Mediated
Fig5: True values of proportion mediated: Logistic regression with HB standardization of t
HB Standardization
_TYPE__FREQ_c//(+)1(/)/1(/)
1400000.140.12467492240.10787716810.2044677448425340040927.3720.18385117920.11927830390.20030663750.1932706962
1400000.390.3535059810.25346487390.451877549815396197751.52860.3471532650.32285228190.41931624550.3877639116
1400000.590.54530845730.32094263250.61043416132.21759028910.39088144920.48278129750.52727644310.4704341142
14000010.95543571610.35905596610.92574261640.61802433830.38602575660.82063248530.65174093070.5627688948
&A
Page &P
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c
Mediated Effect
Fig6: Mediated effect estimates: Logistic regression
000
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/
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1(/)
c
Proportion Mediated
Proportion Mediated: Logistic Regression
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/
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1(/)
Bc
Proportion Mediated
Proportion Mediated: Logistic Regression with HB standardization
000
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HB ab/t
/(+)
HB 1(t't)
Bc
Proportion Mediated
Proportion Mediated: Logistic Regression with HB standardization of t
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c
Mediated Effect
Fig7: Mediated effect estimates: Logistic regression with HB standardization
betataup
=.14=.39=.59=1
0.140.02938289740.02088001770.01190978360.0009841766
0.390.06330254250.01850115090.00914542270.0730105895
0.590.07503924150.00003200970.06355714440.1809276938
10.07709570140.07850741870.20046131810.4457443602
0.02938289740.02088001770.01190978360.0009841766
0.06330254250.01850115090.00914542270.0730105895
0.07503924150.00003200970.06355714440.1809276938
0.07709570140.07850741870.20046131810.4457443602
=.14
=.39
=.59
=1
c
Mediated Effect
Fig8: Estimated mediated effect tt' as function of bc and t'c: Logistic regression
Plot of true proportion mediated as a function of true value of C.
Chart2
0.33331692730.32927920390.321319046
0.49310513970.45264680360.4052561918
0.61248960410.51503909180.4258951626
0.86334060180.60.4277664867
/
/(+)
1(/)
c
Proportion Mediated
TRUEBMEANS
HB Standardization
_TYPE__FREQ_c//(+)1(/)/1(/)
1400000.140.11987433370.11051219680.33331692730.32927920390.3213190460.11680623090.33078088110.3265775869
1400000.390.33864277840.25689821820.49310513970.45264680360.40525619180.31488833240.46522228090.4392280106
1400000.590.51694608480.32948329790.61248960410.51503909180.42589516260.46977456650.53944086470.4945893575
14000010.86802549560.40391279430.86334060180.60.42776648670.78548548250.63916387170.5759176539
TRUEBMEANS
00
00
00
00

c
Mediated Effect
Fig2: True values of mediated effect estimates: Logistic regression
BMEANS
000
000
000
000
/
/(+)
1(/)
c
Proportion Mediated
Fig3: True values of proportion mediated: Logistic regression
fig8
00
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00
HB tt'
c
Mediated Effect
Fig4: True values of mediated effect estimates: Logistic regression with HB standardization of t
000
000
000
000
HB ab/t
/(+)
HB 1(t'/t)
c
Proportion Mediated
Fig5: True values of proportion mediated: Logistic regression with HB standardization of t
HB Standardization
_TYPE__FREQ_c//(+)1(/)/1(/)
1400000.140.12467492240.10787716810.2044677448425340040927.3720.18385117920.11927830390.20030663750.1932706962
1400000.390.3535059810.25346487390.451877549815396197751.52860.3471532650.32285228190.41931624550.3877639116
1400000.590.54530845730.32094263250.61043416132.21759028910.39088144920.48278129750.52727644310.4704341142
14000010.95543571610.35905596610.92574261640.61802433830.38602575660.82063248530.65174093070.5627688948
&A
Page &P
00
00
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00

c
Mediated Effect
Fig6: Mediated effect estimates: Logistic regression
000
000
000
000
/
/(+)
1(/)
c
Proportion Mediated
Proportion Mediated: Logistic Regression
000
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1(/)
Bc
Proportion Mediated
Proportion Mediated: Logistic Regression with HB standardization
000
000
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HB ab/t
/(+)
HB 1(t't)
Bc
Proportion Mediated
Proportion Mediated: Logistic Regression with HB standardization of t
00
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00

c
Mediated Effect
Fig7: Mediated effect estimates: Logistic regression with HB standardization
betataup
=.14=.39=.59=1
0.140.02938289740.02088001770.01190978360.0009841766
0.390.06330254250.01850115090.00914542270.0730105895
0.590.07503924150.00003200970.06355714440.1809276938
10.07709570140.07850741870.20046131810.4457443602
0.02938289740.02088001770.01190978360.0009841766
0.06330254250.01850115090.00914542270.0730105895
0.07503924150.00003200970.06355714440.1809276938
0.07709570140.07850741870.20046131810.4457443602
=.14
=.39
=.59
=1
c