Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Multiple Mediation Analysis For General Models
-with Application to Explore Racial Disparity in
Breast Cancer Survival Analysis
Solution to the
Motivating
Example
Conclusion
Qingzhao Yu
Reference
Joint Work with Ms. Ying Fan and Dr. Xiaocheng Wu
Louisiana Tumor Registry, LSUHSC
June 5th, 2012 NAACCR Annual Conference
1 / 42
Outline
Motivating
Example
1
Motivating Example
2
Mediation Analysis
Concept
Literature Review
The Proposed Method
3
Solution to the Motivating Example
4
Conclusion
5
Reference
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
2 / 42
Motivating Example
Motivating
Example
Some facts about female breast cancer:
The most common cancer and the second leading cause of
cancer death among American women.
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
3 / 42
Motivating Example
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Some facts about female breast cancer:
The most common cancer and the second leading cause of
cancer death among American women.
Significant racial disparity in mortality between Whites and
African American females.
Solution to the
Motivating
Example
Conclusion
Reference
http://apps.nccd.cdc.gov/uscs/cancersbyraceandethnicity.aspx
4 / 42
Motivating Example
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Question: How to efficiently reduce the racial disparity in female
breast cancer survival rate?
Goal: Explore the racial disparity in breast cancer survival.
Reference
5 / 42
Racial disparity in breast cancer survival
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
race
Breast cancer
survival rate
Reference
6 / 42
Racial disparity in breast cancer survival
Age at diagnosis
Motivating
Example
SES
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Insurance
Marital status
Breast cancer
survival rate
race
Conclusion
Reference
Stage at diagnosis
Tumor grade
Treatment
ER/PR receptors
.
.
.
7 / 42
Racial disparity in breast cancer survival
Age at diagnosis
Motivating
Example
SES
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Insurance
Marital status
Breast cancer
survival rate
race
Conclusion
Reference
Stage at diagnosis
Tumor grade
Treatment
ER/PR receptors
.
.
.
8 / 42
Racial disparity in breast cancer survival
Age at diagnosis
Motivating
Example
SES
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Insurance
Marital status
Breast cancer
survival rate
race
Conclusion
Reference
Stage at diagnosis
Tumor grade
Treatment
ER/PR receptors
.
.
.
9 / 42
Racial disparity in breast cancer survival
Age at diagnosis
Motivating
Example
SES
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Insurance
Marital status
Breast cancer
survival rate
race
Conclusion
Reference
Stage at diagnosis
Tumor grade
Treatment
ER/PR receptors
.
.
.
10 / 42
The concept of mediation
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Definition
Mediation effect refers to the effect transmitted by an intervening
variable to an observed relationship between a predictor and
dependent variable of interest.
Solution to the
Motivating
Example
Conclusion
Reference
11 / 42
The concept of mediation
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
Definition
Mediation effect refers to the effect transmitted by an intervening
variable to an observed relationship between a predictor and
dependent variable of interest.
Application in Disciplines:
Social Science
Prevention Studies
Behavior Research
Epidemiological studies
Genetics Epidemiology
12 / 42
Linear regression models
Motivating
Example
𝑀
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
𝛼1
X
X
𝑐1
𝑐2
𝛽1
Y
Y
Figure: Mediation Diagram
Reference

 M = αX + 1 ; (1)
Y = βM + c1 X + 2 ; (2)

Y = c2 X + 3 .(3)
13 / 42
Coefficients difference and product methods
Coefficients difference method: c2 − c1
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
14 / 42
Coefficients difference and product methods
Coefficients difference method: c2 − c1
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Limitation:
Model (2) and (3) are assumed to be true simultaneously;
For binary response with logistic regression: scales for
coefficients are different when different subsets of variables are
used as predictors;
Multiple mediators: cannot differentiate mediation effects from
multiple mediators.
Reference
15 / 42
Coefficients difference and product methods
Coefficients difference method: c2 − c1
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Limitation:
Model (2) and (3) are assumed to be true simultaneously;
For binary response with logistic regression: scales for
coefficients are different when different subsets of variables are
used as predictors;
Multiple mediators: cannot differentiate mediation effects from
multiple mediators.
Reference
Coefficients product method: α · β
Limitation:
Hard to explain when the predictive model is not linear
regression.
16 / 42
Coefficients difference and product methods
Coefficients difference method: c2 − c1
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Limitation:
Model (2) and (3) are assumed to be true simultaneously;
For binary response with logistic regression: scales for
coefficients are different when different subsets of variables are
used as predictors;
Multiple mediators: cannot differentiate mediation effects from
multiple mediators.
Reference
Coefficients product method: α · β
Limitation:
Hard to explain when the predictive model is not linear
regression.
Property: When Y and M are continuous and linear regression
models are fitted for the relationships, c2 − c1 = α · β.
17 / 42
Counterfactual framework
Donald B. Rubin (1974)
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
X (i): treatment for subject i, control(X (i) = 0) and treatment
(X (i) = 1).
YX (i): potential post-treatment outcome if subject i is treated
with X (0 or 1). Usually, only one of the responses, Y1 (i) and
Y0 (i), is observed.
Y1 (i) − Y0 (i): causal effect of treatment on the response
variable for subject i.
E (Y1 ) − E (Y0 ): average causal effect
MX (i): potential M when subject i is exposed to treatment X .
M1 (i) or M0 (i) is observed if subject i is actually assigned to
treatment or control group.
18 / 42
Counterfactual framework
The potential outcome depends not only on the exposure variable but
also on the mediator.
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Yx,m (i): potential outcome of subject i for given x and m.
Y0,m0 (i); Y0,m1 (i), Y1,m0 (i), Y1,m1 (i)
E (Y1,m1 − Y0,m0 ): Total Effect
E (Y1,m0 − Y0,m0 ): Natural Direct Effect(Pearl, 2001)
E (Y1,m1 − Y0,m1 ): alternative definition
E (Y1,m1 − Y1,m0 )
E (Y0,m1 − Y0,m0 ): Indirect Effect
Reference
19 / 42
Counterfactual framework
The potential outcome depends not only on the exposure variable but
also on the mediator.
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Yx,m (i): potential outcome of subject i for given x and m.
Y0,m0 (i); Y0,m1 (i), Y1,m0 (i), Y1,m1 (i)
E (Y1,m1 − Y0,m0 ): Total Effect
E (Y1,m0 − Y0,m0 ): Natural Direct Effect(Pearl, 2001)
E (Y1,m1 − Y0,m1 ): alternative definition
E (Y1,m1 − Y1,m0 )
E (Y0,m1 − Y0,m0 ): Indirect Effect
Reference
Limitation:
Assumption: E (Y1,m0 − Y0,m0 ) = E (Y1,m1 − Y0,m1 )
Difficult to differentiate indirect effects from multiple mediators.
20 / 42
Counterfactual framework
The potential outcome depends not only on the exposure variable but
also on the mediator.
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Yx,m (i): potential outcome of subject i for given x and m.
Y0,m0 (i); Y0,m1 (i), Y1,m0 (i), Y1,m1 (i)
E (Y1,m1 − Y0,m0 ): Total Effect
E (Y1,m0 − Y0,m0 ): Natural Direct Effect(Pearl, 2001)
E (Y1,m1 − Y0,m1 ): alternative definition
E (Y1,m1 − Y1,m0 )
E (Y0,m1 − Y0,m0 ): Indirect Effect
Reference
Limitation:
Assumption: E (Y1,m0 − Y0,m0 ) = E (Y1,m1 − Y0,m1 )
Difficult to differentiate indirect effects from multiple mediators.
Common limitation: Only suitable for binary exposure variable
21 / 42
Challenge
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
Recall the motivating example
Challenge
Various types of mediators
Differentiate indirect effect from each mediator
Compare the indirect effects conveyed by mediators that
contribute to the racial disparity
Potential nonlinear relationship and interactions among X , Ms,
and Y
22 / 42
Notations and Definitions
Notations
Motivating
Example
𝑀1
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
X
Solution to the
Motivating
Example
Conclusion
Reference
Figure:
𝑀2
⋮
𝑀𝑝
Y
Z
Multiple Mediators Mediation Diagram
Definitions:
Total effect
23 / 42
Notations and Definitions
Notations
Motivating
Example
𝑀1
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
X
Solution to the
Motivating
Example
Conclusion
Reference
Figure:
𝑀2
⋮
𝑀𝑝
Y
Z
Multiple Mediators Mediation Diagram
Definitions:
Total effect
Direct effect not from M1
24 / 42
Notations and Definitions
Notations
Motivating
Example
𝑀1
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
X
Solution to the
Motivating
Example
Conclusion
Reference
Figure:
𝑀2
⋮
𝑀𝑝
Y
Z
Multiple Mediators Mediation Diagram
Definitions:
Total effect
Direct effect not from M1
Indirect effect from M1
25 / 42
Properties
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Under the linear regressions setting, we get the same results as
the product method.
Solution to the
Motivating
Example
Conclusion
Reference
26 / 42
Properties
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
Under the linear regressions setting, we get the same results as
the product method.
In logistic regression with binary mediator,recall the natural
direct effect:
ζ̄(0) = E (Y1,m0 − Y0,m0 ) or ζ̄(1) = E (Y1,m1 − Y0,m1 )b
The relationship between direct effect and natural direct effect:
DE=P(X = 0) · ζ̄(0) + P(X = 1) · ζ̄(1)
27 / 42
To measure uncertainties
Motivating
Example
Mediation
Analysis
Delta Method
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
28 / 42
To measure uncertainties
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Delta Method
Bootstrap Method
1
Solution to the
Motivating
Example
2
Conclusion
3
Reference
4
Randomly draw a sample of n observations from original data of
size N with replacement;
Estimate DE , IE , and TE ;
Repeat last two steps B times. Obtain a set of estimates for
each quantity;
Obtain empirical variances of mediation effects and α2 th and
(1 − α2 )th percentiles.
29 / 42
LA Breast Cancer Data
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
1473 non-Hispanic White or African American female patients
diagnosed with malignant breast cancer in 2004 in LA collected
by Louisiana Tumor Registry.
Followed up for five years.
Solution to the
Motivating
Example
Conclusion
Reference
30 / 42
LA Breast Cancer Data
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
1473 non-Hispanic White or African American female patients
diagnosed with malignant breast cancer in 2004 in LA collected
by Louisiana Tumor Registry.
Followed up for five years.
Exclude:
Lost follow up within three years (20, 1.4%);
Death due to causes other than breast cancer (157, 10.7%).
1293 patients were included.
31 / 42
LA Breast Cancer Data
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
1473 non-Hispanic White or African American female patients
diagnosed with malignant breast cancer in 2004 in LA collected
by Louisiana Tumor Registry.
Followed up for five years.
Exclude:
Lost follow up within three years (20, 1.4%);
Death due to causes other than breast cancer (157, 10.7%).
1293 patients were included.
The odds of dying within three years for blacks is significantly
higher than whites (173 death, OR=2.03, CI:[1.47, 2.81])
32 / 42
Variable Description(1)
Explanatory variable: racial indicator
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
Response variable: alive (0) or Dead (1) at the end of the 3rd year of diagnosis
Third variable considered:
Census Track SES variables poverty (> 20% versus < 20% of persons with an income below the
federal poverty level)
education (> 25% versus < 25% of adults (25 years and older) with
less than a high school education)
residence area (grouped using Beale codes: 100% rural; urban-rural
mix; 100% urban)
workclass (> 66% of persons ages 16 and over who are unemployed
versus < 66%)
insurance (no insurance; Medicaid; Medicare and public; private insurance)
marital status (single-never married; married; separated; widowed; divorced;
unknown)
age at diagnosis
33 / 42
Variable Description(2)
stage (regional; distance; localized)
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
grade (moderately differentiate; poorly/undifferentiated; well
differentiate; unknown)
tumor size (< 1cm; 1.1 ∼ 2cm; 2.1 ∼ 3cm; > 3cm; unknown)
comorbidity (mild; moderate; severe; none; unknown)
surgery (mastectomy, lumpectomy, no surgery)
radiation (not administered; administered)
chemotherapy (not administered, administered)
Reference
hormonal therapy (not administered; administered)
ER/PR receptor (either is positive; both are negative; unknown).
34 / 42
Variable Description(2)
stage (regional; distance; localized)
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
grade (moderately differentiate; poorly/undifferentiated; well
differentiate; unknown)
tumor size (< 1cm; 1.1 ∼ 2cm; 2.1 ∼ 3cm; > 3cm; unknown)
comorbidity (mild; moderate; severe; none; unknown)
surgery (mastectomy, lumpectomy, no surgery)
radiation (not administered; administered)
chemotherapy (not administered, administered)
Reference
hormonal therapy (not administered; administered)
ER/PR receptor (either is positive; both are negative; unknown).
Eligibility for Mediator:
Significantly associated with race;
Significantly relates to vital status controlling for race.
35 / 42
Data Analysis Results
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
Table: Indirect Effects(IE) and Relative Effects(RE)
Mediator
IE[1]
95% CI for IE[2]
RE
95% CI for RE[2]
Stage
0.276
(0.127,0.488)
28.14
(0.122,0.767)
0.275
(0.074,0.43)
28.05
(0.058,0.708)
Insurance
ER/PR
0.181
(0.068,0.332)
18.46
(0.055,0.56)
Grade
0.158
(0.027,0.379)
16.09
(0.024,0.476)
0.145
(0.067,0.333)
14.75
(0.063,0.504)
Surgery
Tumor Size
0.135
(0.025,0.417)
13.77
(0.022,0.561)
Hormonal Therapy
0.114
(0.02,0.253)
11.57
(0.016,0.402)
Age
-0.113
(-0.301,-0.034)
-11.5
(-0.432,-0.031)
-0.074
(-0.392,0.156)
-7.51
(-0.739,0.159)
Marital Status
Comorbidity
-0.002
(-0.15,0.109)
-0.17
(-0.205,0.136)
[1] After considering all indirect effects through mediators, direct effect of race
on mortality is -0.115 with 95% CI:[-0.713,0.526].
[2] 95% confidence interval is 0.025 and 0.975 percentiles of the distribution
of statistics obtained by bootstrap with 1000 repetitions.
36 / 42
Racial disparity in breast cancer survival
Age at diagnosis
(-11.5%)
Motivating
Example
Insurance
(28.05%)
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Stage at diagnosis
(28.14%)
Tumor size
(13.77%)
race
Conclusion
Reference
Breast cancer
survival rate
Tumor grade
(16.09%)
Surgery (14.75%)
Hormonal
Therapy (11.57%)
ER/PR receptors
(18.46%)
37 / 42
Racial disparity in breast cancer survival
Age at diagnosis
(-11.5%)
Motivating
Example
Insurance
(28.05%)
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
Stage at diagnosis
(28.14%)
Tumor size
(13.77%)
race
-.116 (not significant)
Breast cancer
survival rate
Tumor grade
(16.09%)
Surgery (14.75%)
Hormonal
Therapy (11.57%)
ER/PR receptors
(18.46%)
38 / 42
Conclusion
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
The proposed method can deal with continuous, binary or
categorical variables
We can separate mediation effects from each mediator
With the proposed method, we can use predictive models other
than linear regression models
We will extend the method to survival analysis and multilevel
analysis
39 / 42
Acknowledgment
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
Louisiana Tumor Registry
Dr. Xiaocheng Wu
Ms. Ying Fan
Dr. Vivien Chen
Data used for this study was from the CDC-NPCR funded
Breast and Prostate Cancer Data Quality and Patterns of Care
Study (grant number: 1 U01 DP000253-01).
40 / 42
Reference
Motivating
Example
Mediation
Analysis
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
Imai Kosuke(2010a). “A general approach to causal mediation analysis.”Psychological
Methods. Vol.15, No.4, 309-334.
Imai Kosuke(2010b). “Identification, Inference and sensitivity analysis of causal
mediation effects.”Statistical Science. Vol. 25, No. 1, 51-71.
Baron RM, Kenny DA(1986). “The moderator-mediator variable distinction in social
psychological research: conceptual, strategic, and statistical considerations.”J. Pers
Soc Psychol. 51(6): 1173-1182.
Alwin, D. F., Hauser, R. M. (1975). “The decomposition of effects in path
analysis.”American Sociological Review 40: 37-47.
Judd, C. M., Kenny, D. A. (1981). David P. MacKinnon, J. Dwyer(1995). “Process
Analysis: Estimating mediation in treatment evaluations.”Evaluation Review. 5(5),
602-619.
David P. MacKinnon, James H. Dwyer(1993). “Estimating mediated effects in
prevention studies.”Evaluation review. Vol.17, No.2, 144-158.
Donald B. Rubin(1974). “Estimating causal effects of treatments in randomized and
nonrandomized studies.”Journal of Educational Psychology. Vol.66, No.5, 688-701.
Paul W. Holland(1986). “Statistics and Causal inference.”J. of the American
Statistical Association. Vol.81, No.396, pp. 945-960.
Pear J(2001). “Direct and indirect effects. In:”Proceedings of the Seventeenth
Conference on Uncertainty and Artificial Intelligence. San Francisco, CA: Morgan
Kaufmann; 2001: 411-420.
Maya L. Petersen, Sandra E. Sinisi, and Mark J. van der Laan(2006). “Estimation of
direct causal effects.”Epidemiology. Vol. 17, No.3.
David MacKinnon, Jennifer Krull(2000). “Equivalence of the mediation, confounding
and suppression effect.”Prevention Science. Vol.1, No.4.
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Motivating
Example
Mediation
Analysis
Question
Concept
Literature
Review
The Proposed
Method
Solution to the
Motivating
Example
Conclusion
Reference
42 / 42