letter

Merge branch 'main' of https://github.com/steno-aarhus/mediation-analysis-course

# Conflicts:
#	sessions/traditional-mediation-analysis.qmd

sessions/causal-mediation-analysis-estimation.qmd

@@ -1,7 +1,5 @@
 # Estimation of effects using causal mediation analysis
 
-\<\<\<\<\<\<\< style/apply-some-style-fixes
-
 ```{r setup}
 
 #| include: false
@@ -548,10 +546,10 @@ digraph {
 
 ### G-computation
 
-The G-computation algorithm was first introduced by Robins 1986 (refer)
-for estimating time-varying exposure causal in the presence of
-time-varying confounders of exposure effects. When estimating total
-effect, g-computation is generally equivalent to
+The G-computation algorithm was first introduced by Robins 1986
+@robins_new_1986 for estimating time-varying exposure causal in the
+presence of time-varying confounders of exposure effects. When
+estimating total effect, g-computation is generally equivalent to
 inverse-probability-of-treatment weighting (IPTW). But in
 high-dimensional settings, g-computation is more powerful. G-computation
 (using g-formula) could also provide an intuitive method for decomposing
@@ -670,4 +668,3 @@ res_gformula <- cmest(data = data, model = "gformula", outcome = "y", exposure =
 
 summary(res_gformula)
 ```
-

sessions/causal-mediation-analysis-introduction.qmd

@@ -270,6 +270,7 @@ digraph {
   edge [minlen = 2]
     A->M
     M->Y
+    A->Y
     W->A
     W->Y
     W->M
@@ -316,6 +317,7 @@ digraph {
   edge [minlen = 2]
     A->M
     M->Y
+    A->Y
     W1->A
     W1->Y
     W1->M
@@ -330,9 +332,10 @@ digraph {
 ```
 
 From the DAG rules, we have a special problem that we cannot solve with
-traditional regression approaches. If we adjust for W2 we open the
-backdoor path from $W1 \leftarrow W2 \rightarrow Y$. We will work on how
-to solve this problem later in the course.
+traditional regression approaches. If we adjust for W2 we open a
+backdoor path by adjusting for the collider
+$W1 \rightarrow W2 \leftarrow A$. We will work on how to solve this
+problem later in the course.
 
 ## Non-linearity and interactions
 
@@ -575,3 +578,4 @@ In addition to consistency there is the assumption of no carry-over
 effects.
 
 ## References
+

sessions/causal-mediation-analysis-survival-outcomes.qmd

@@ -1,6 +1,6 @@
 # Survival outcomes
 
-<!-- Teaching concepts - Jie (30 min) -->
+<!-- Jie (30 min) -->
 
 ::: callout-note
 ## Learning outcomes
@@ -9,6 +9,8 @@
     software
 :::
 
+## Time-to-event outcomes
+
 There are many studies conducting mediation analyses with
 **time-to-event outcomes**. Survival analysis allows investigators to
 study these important outcomes with appropriate consideration for
@@ -27,8 +29,9 @@ effect. They found the effect of SES on survival time was partially
 mediated by stage diagnosis, explaining 12% for lung cancer.
 
 ```{r echo=FALSE}
-# Creating The causal diagram for a mediation model
+
 library(DiagrammeR)
+
 grViz("
 digraph {
   graph []
@@ -44,23 +47,14 @@ digraph {
 }")
 ```
 
-### Conduct causal mediation analysis for time-to-event outcomes
-
-How to conduct causal mediation analysis for time-to-event outcomes? We
-introduced the difference and product methods for continuous and binary
-outcomes in previous session. It is tempting to run a linear regression
-model for the mediator and proportional hazard model for the outcome,
-then use product or difference method to estimate the direct effect and
-indirect effect.
-
-#### Product method for time-to-event outcomes
-
 The Cox proportional hazards model is commonly used for dealing with
 survival data in medical literature. Cox regression estimates the
 **hazard ratios** and the values are then used to determine the effect
 of the mediator variable between the exposure and the survival time of
 outcome.
 
+## Cox model for common outcomes
+
 Could we use the traditional approach for time-to-event outcomes? We
 have introduced the difference and product methods for continuous and
 binary outcomes in previous session. It is tempting to run a linear
@@ -69,58 +63,33 @@ outcome, then use product or difference method to estimate the direct
 effect and indirect effect.
 
 However, 'non-collaspsibility' is a problem of the hazard ratio as odds
-ratio (VanderWeele). Therefore, use of Cox PH regression to
+ratio @vanderweele_mediation_2016. Therefore, use of Cox regression to
 approximately estimate indirect effects via difference or product of
 coefficients rests on the assumption that the outcome is **rare**
-(VanderWeele).
+@vanderweele_mediation_2016.
 
 Where the outcome is common, measures of the indirect effect or
-proportion mediated will be incorrect. Tein and Mackinnon (2003)
-considered whether the product method and difference method yield
-comparable results with respect to time-to-event outcomes. They found
-that the methods coincides for the accelerated failure time model but
-not for the proportional hazards model.
+proportion mediated will be incorrect. Tein and Mackinnon considered
+whether the product method and difference method yield comparable
+results with respect to time-to-event outcomes @tein_estimating_2003.
+They found that the methods coincides for the accelerated failure time
+model but not for the proportional hazards model.
 
 To sum up, we can only use the traditional approaches for rare outcomes.
 Otherwise, we can use the product method to get an indication of whether
 there is mediation, but be aware that the estimate is not accurate.
 
 ### Causal mediation for time-to-event outcomes
 
-::: callout-note
-In earlier session,we have been familiar with the counterfactual
-concepts in causal mediation:
-
--   $Yi(a, m)$ is the outcome achieved for person i if, possibly
-    contrary to fact, exposure had been set to a and mediator to m.
-
--   $Mi(a)$ is the mediator achieved for person i if, possibly contrary
-    to fact, exposure had been set to a.
-
-One can combine the two counterfactuals, yielding so-called nested
-counterfactuals defined as
-
--   $Y(a, M (a\*)$.
-
-By introducing the nested counterfactual for a ≠ a\* we can give a
-precise mathematical definition of mediation.
-:::
-
 For a **survival outcome**, the outcome of interest will be survival
 time (SV).
 
 -   $SV (t) = P(V ≥ t)$ the survival function at time t
-
 -   $SV (t\|c)=P(V ≥ t\|c)$ the survival function conditional on
     covariates C
-
 -   $λV (t)$ : the hazard at time t
-
 -   $λV (t\|c)$: conditional hazard at time t
 
-<<<<<<< HEAD
-#### Definitions
-
 If we consider the survival functions for a time-to-event outcome T, we
 could decompose the survival function as follows:
 
@@ -132,6 +101,7 @@ the survival function scale.
 
 Similarly, we can demcompose the overal difference in hazards on the
 hazard scale:
+
 $λT_a(t) - λT_a*(t) = [λT_aM_a(t)-λT_aM_a*(t)] + [λT_aM_a*(t)-λT_a*M_a*(t)]$
 
 ### Assumptions of mediation analysis with a time-to-event outcome
@@ -140,26 +110,20 @@ Similar as our context, mediation analysis with a time-to-event outcome
 have to satisfy below assumptions:
 
 -   no unmeasured confounding of the exposure-outcome relation;
-
 -   no unmeasured confounding of the mediator-outcome relation;
-
 -   no unmeasured confounding of the exposure-mediation relation;
-
 -   no exposure induced mediation-outcome confounding
-
 -   Additionally, we assume that the mediator is measured for everyone
     before the outcome occurs.
 
 When the outcome is common, we can use weight approach (Lange 2012)
 
-=======
->>>>>>> f05dab6a872c949f856eb769c9c83d2ba3d9a9b4
-### Examples
+::: callout-note
 
-::: callout-note We will continue working on the obesity-CVD example in
-the Framingham dataset.
+We will continue working on the obesity-CVD example in the Framingham
+dataset.
 
-The outcome of interest is death from cardiovascular diseases (cvd). The
+The outcome of interest is death from cardiovascular diseases (CVD). The
 underlying time scale is in days, starting at participants entered the
 cohort. The exposure of interest is obesity status at baseline, where
 a=1 indicates obese, a=0 indicates non-obese. The mediator is blood
@@ -172,6 +136,7 @@ impact of obesity on CVD-related death (measured in years).
 
 ```{r}
 library(tidyverse)
+
 framingham <- read_csv(here::here("data/framingham_dataset.csv"))
 
 framingham <- framingham %>%
@@ -191,7 +156,7 @@ framingham <- framingham %>%
       bmi >= 25 ~ 1,
       TRUE ~ 0
     ),
-    y_time = y_time / 365.25
+    y_time = y_time / 365.25 #change time-scale to years
   )
 ```
 
@@ -216,3 +181,5 @@ res_rb_coxph <- cmest(
 
 summary(res_rb_coxph)
 ```
+
+## References

sessions/group_work_day1.qmd

@@ -14,6 +14,7 @@ in the dataset.
 
 ```{r setup}
 library(here)
+
 load(here::here("data/frmghamdata.RData"))
 ```
 

sessions/motivation-for-mediation-analysis.qmd

@@ -4,7 +4,7 @@
 
 ```{r setup}
 #| include: false
-# Creating The causal diagram for a mediation model
+
 library(DiagrammeR)
 ```
 
@@ -23,7 +23,7 @@ uncovering the underlying pathways, and mechanisms.
 
 ```{r}
 #| echo: false
-library(DiagrammeR)
+
 grViz("
 digraph {
   graph []
@@ -46,8 +46,6 @@ digraph {
 
 ### Confirmation and refutation of theory
 
-<!-- DBI: this section is a little unclear. There are 2 mediators in the example above. -->
-
 Let's assume we estimate a familial risk of diabetes and that risk
 estimates follow a normal distribution at the population level. Then we
 select the extremes of the distribution (2.5%) and assess the relative
@@ -57,19 +55,17 @@ individuals with a strong/weak familial risk of type 2 diabetes.
 ### To refine interventions
 
 Let's consider the evidence from the landmark prevention trials. The
-Diabetes Prevention Programme (DPP, USA), The Diabetes Prevention Study
-(DPS, Finland), and The Da Quin study (DPS, China). These studies
-randomized high risk individuals to either metformin or a lifestyle
-intervention consisting of a physical activity or physical activity +
-diet. The main finding was a \~58% lower risk of incidence type 2
-diabetes in the lifestyle intervention group compared to the metformin
-group
+Diabetes Prevention Programme @knowler_reduction_2002, The Diabetes
+Prevention Study @tuomilehto_prevention_2001, and The Da Quin study
+@pan_effects_1997. These studies randomized high risk individuals to
+either metformin or a lifestyle intervention consisting of a physical
+activity or physical activity + diet. The main finding was a \~58% lower
+risk of incidence type 2 diabetes in the lifestyle intervention group
+compared to the metformin group
 
 ```{r}
 #| echo: false
 
-# Creating The causal diagram for a mediation model
-library(DiagrammeR)
 grViz("
 digraph {
   graph []
@@ -98,8 +94,7 @@ effectiveness of these prevention efforts
 
 ```{r}
 #| echo: false
-# Creating The causal diagram for a mediation model
-library(DiagrammeR)
+
 grViz("
 digraph {
   graph []