class: center, middle, inverse, title-slide # Effect modification ## What If: Chapter 4 ### Elena Dudukina ### 2020-11-26 --- <style>.shareagain-bar { --shareagain-foreground: rgb(255, 255, 255); --shareagain-background: rgba(0, 0, 0, 0.5); --shareagain-facebook: none; --shareagain-linkedin: none; --shareagain-pinterest: none; --shareagain-pocket: none; --shareagain-reddit: none; }</style> # Definition of the effect modification * The causal effect changes with the characteristics of the population under study ```r greek_gods_po <- tibble::tribble( ~greek, ~V, ~Y_a0, ~Y_a1, "Rheia", 1, 0, 1, "Demeter", 1, 0, 0, "Hestia", 1, 0, 0, "Hera", 1, 0, 0, "Artemis", 1, 1, 1, "Leto", 1, 0, 1, "Athena", 1, 1, 1, "Aphrodite", 1, 0, 1, "Persephone", 1, 1, 1, "Hebe", 1, 1, 0, "Kronos", 0, 1, 0, "Hades", 0, 0, 0, "Poseidon", 0, 1, 0, "Zeus", 0, 0, 1, "Apollo", 0, 1, 0, "Ares", 0, 1, 1, "Hephaestus", 0, 0, 1, "Cyclope", 0, 0, 1, "Hermes", 0, 1, 0, "Dionysus", 0, 1, 0 ) ``` --- # Definition of the effect modification ```r # had we known counterfactual outcomes greek_gods_po %>% count(Y_a0) ``` ``` ## # A tibble: 2 x 2 ## Y_a0 n ## <dbl> <int> ## 1 0 10 ## 2 1 10 ``` ```r greek_gods_po %>% count(Y_a1) ``` ``` ## # A tibble: 2 x 2 ## Y_a1 n ## <dbl> <int> ## 1 0 10 ## 2 1 10 ``` - `\(Pr[Y^{a=1}=1] - Pr[Y^{a=0}=1]\)` = 0 - `\(\frac{Pr[Y^{a=1}=1]}{Pr[Y^{a=0}=1]}\)` = 1 --- # What is the average causal effect of A on Y in women and in men? * `\(Pr[Y^{a=1}=1|V=1]\)` - `\(Pr[Y^{a=0}=1|V=1]\)` * `\(Pr[Y^{a=1}=1|V=0]\)` - `\(Pr[Y^{a=0}=1|V=0]\)` -- * `\(\frac{Pr[Y^{a=1}=1|V=1]}{Pr[Y^{a=0}=1|V=1]}\)` * `\(\frac{Pr[Y^{a=1}=1|V=0]}{Pr[Y^{a=0}=1|V=0]}\)` -- ``` ## # A tibble: 2 x 6 ## # Groups: V [2] ## V label risk_Y_a1 risk_Y_a0 RR RD ## <dbl> <chr> <dbl> <dbl> <dbl> <dbl> ## 1 0 men 0.4 0.6 0.667 -0.200 ## 2 1 women 0.6 0.4 1.50 0.200 ``` --- # Results - Heart transplant A decreases the risk of death Y in men, but increases the risk of death in women - Perfectly "neutralizing" stratum-specific results - V is a modifier of the effect of A on Y when the average causal effect of A on Y varies across levels of V --- # Effect measure modification - For binary outcome: - Additive effect modification: `\(Pr[Y^{a=1} - Y^{a=0}|V=1] \neq Pr[Y^{a=1} - Y^{a=0}|V=0]\)` - Multiplicative effect modification: `\(\frac{Pr[Y^{a=1}|V=1]}{Pr[Y^{a=0}|V=1]}\)` `\(\neq\)` `\(\frac{Pr[Y^{a=1}|V=0]}{Pr[Y^{a=0}|V=0]}\)` - Effect *measure* modification - Example: - `\(Pr[Y^{a=1}=1|V=1]\)` = 0.9 - `\(Pr[Y^{a=0}=1|V=1]\)` = 0.8 - `\(Pr[Y^{a=1}=1|V=0]\)` = 0.2 - `\(Pr[Y^{a=0}=1|V=0]\)` = 0.1 - Additive scale: `\(0.9-0.8 = 0.2-0.1 = 0.1\)` - Multiplicative scale: `\(\frac{0.9}{0.8}\)` `\(\neq\)` `\(\frac{0.2}{0.1}\)` --> `\(1.13 \neq 2.00\)` --- # Stratification to identify effect modification - Marginally randomized experiment: V-stratum specific effects to evaluate effect measure modification --- # Conditionally randomized "experiment" ``` ## # A tibble: 2 x 4 ## # Groups: L [2] ## L A n pr ## <dbl> <dbl> <int> <dbl> ## 1 0 1 4 0.5 ## 2 1 1 9 0.75 ``` --- # Causal effect among romans ```r pr_a <- glm(data = romans, formula = A ~ 1, family = binomial(link = "logit")) pr_a_l <- glm(data = romans, formula = A ~ L, family = binomial(link = "logit")) romans %<>% mutate( p_a = predict(object = pr_a, type = "response"), p_a_l = predict(object = pr_a_l, type = "response"), # IPTW sw_iptw = if_else(A==1, p_a/p_a_l, (1-p_a)/(1-p_a_l)) # stabilized weights ) ``` --- ```r romans %>% distinct(A, L, sw_iptw) ``` ``` ## # A tibble: 4 x 3 ## L A sw_iptw ## <dbl> <dbl> <dbl> ## 1 0 0 0.7 ## 2 0 1 1.30 ## 3 1 0 1.40 ## 4 1 1 0.867 ``` ```r romans %>% group_by(A) %>% count(Y) %>% mutate(risk = n/sum(n)) %>% filter(Y == 1) ``` ``` ## # A tibble: 2 x 4 ## # Groups: A [2] ## A Y n risk ## <dbl> <dbl> <int> <dbl> ## 1 0 1 2 0.286 ## 2 1 1 8 0.615 ``` ```r romans %>% group_by(A) %>% count(Y, wt = sw_iptw) %>% mutate(risk = n/sum(n)) %>% filter(Y == 1) ``` ``` ## # A tibble: 2 x 4 ## # Groups: A [2] ## A Y n risk ## <dbl> <dbl> <dbl> <dbl> ## 1 0 1 2.10 0.300 ## 2 1 1 7.8 0.6 ``` --- ```r glm(data = romans, formula = Y ~ A, weights = sw_iptw, family = binomial(link = "log")) %>% broom::tidy(., exponentiate = T) %>% filter(term == "A") %>% select(1, 2) ``` ``` ## # A tibble: 1 x 2 ## term estimate ## <chr> <dbl> ## 1 A 2. ``` ```r glm(data = romans, formula = Y ~ A, weights = sw_iptw, family = binomial(link = "identity")) %>% broom::tidy() %>% filter(term == "A") %>% select(1, 2) ``` ``` ## # A tibble: 1 x 2 ## term estimate ## <chr> <dbl> ## 1 A 0.3 ``` --- # Causal effect among greeks ```r greeks ``` ``` ## # A tibble: 20 x 5 ## name L A Y V ## <chr> <dbl> <dbl> <dbl> <dbl> ## 1 Rheia 0 0 0 1 ## 2 Kronos 0 0 1 1 ## 3 Demeter 0 0 0 1 ## 4 Hades 0 0 0 1 ## 5 Hestia 0 1 0 1 ## 6 Poseidon 0 1 0 1 ## 7 Hera 0 1 0 1 ## 8 Zeus 0 1 1 1 ## 9 Artemis 1 0 1 1 ## 10 Apollo 1 0 1 1 ## 11 Leto 1 0 0 1 ## 12 Ares 1 1 1 1 ## 13 Athena 1 1 1 1 ## 14 Hephaestus 1 1 1 1 ## 15 Aphrodite 1 1 1 1 ## 16 Cyclope 1 1 1 1 ## 17 Persephone 1 1 1 1 ## 18 Hermes 1 1 0 1 ## 19 Hebe 1 1 0 1 ## 20 Dionysus 1 1 0 1 ``` --- ```r pr_a_gr <- glm(data = greeks, formula = A~1, family = binomial("logit")) pr_a_l_gr <- glm(data = greeks, formula = A~L, family = binomial("logit")) greeks %<>% mutate( p_a = predict(object = pr_a_gr, type = "response"), p_a_l = predict(object = pr_a_l_gr, type = "response"), # IPTW sw_iptw = if_else(A==1, p_a/p_a_l, (1-p_a)/(1-p_a_l)) # stabilized weights ) ``` --- ```r glm(data = greeks, formula = Y ~ A, family = binomial("log"), weights = sw_iptw) %>% broom::tidy(exponentiate = T) %>% filter(term == "A") %>% select(term, estimate) ``` ``` ## # A tibble: 1 x 2 ## term estimate ## <chr> <dbl> ## 1 A 1.00 ``` ```r glm(data = greeks, formula = Y ~ A, family = binomial("identity"), weights = sw_iptw) %>% broom::tidy(exponentiate = F) %>% filter(term == "A") %>% select(term, estimate) ``` ``` ## # A tibble: 1 x 2 ## term estimate ## <chr> <dbl> ## 1 A 2.96e-13 ``` ```r round(2.96e-13, 2) ``` ``` ## [1] 0 ``` --- ```r # combined greeks and romans greeks_romans <- bind_rows(greeks, romans) pr_a_greeks_romans <- glm(data = greeks_romans, formula = A~1, family = binomial("logit")) pr_a_l_greeks_romans <- glm(data = greeks_romans, formula = A~L, family = binomial("logit")) greeks_romans %<>% mutate( p_a = predict(object = pr_a_greeks_romans, type = "response"), p_a_l = predict(object = pr_a_l_greeks_romans, type = "response"), # IPTW sw_iptw = if_else(A==1, p_a/p_a_l, (1-p_a)/(1-p_a_l)) # stabilized weights ) ``` --- ```r glm(data = greeks_romans, formula = Y ~ A, family = binomial("log"), weights = sw_iptw) %>% broom::tidy(exponentiate = T) %>% filter(term == "A") %>% select(term, estimate) ``` ``` ## # A tibble: 1 x 2 ## term estimate ## <chr> <dbl> ## 1 A 1.37 ``` ```r glm(data = greeks_romans, formula = Y ~ A, family = binomial("identity"), weights = sw_iptw) %>% broom::tidy(exponentiate = F) %>% filter(term == "A") %>% select(term, estimate) ``` ``` ## # A tibble: 1 x 2 ## term estimate ## <chr> <dbl> ## 1 A 0.15 ``` --- # Results - No effect of A on Y (mortality) among greeks (V=1): RR = 1.0 and RD=0.0 - A increases the risk of Y (mortality) among romans (V=0): RR = 2.00 and RD=0.3 - An average causal effect was: RR=1.375 and RD=0.15 - Additive and multiplicative effect modification --- # Treatment effect in the treated - If `\(Pr[Y=1|A=1] \neq Pr[Y^{a=0}=1|A=1]\)` - "there is a causal effect in the treated if the observed risk among the treated individuals does not equal the counterfactual risk had the treated individuals been untreated" - Causal risk ratio in the treated (SMR): - SMR = `\(\frac{Pr[Y=1|A=1]}{Pr[Y^{a=0}=1|A=1]}\)` - Numerator: Probability of the outcome in those, who actually received treatment - Denominator: Probability of the outcome (potential outcome) in the treated had they been untreated - Average effect in the treated will differ from the average effect in the whole population when if the distribution of individual causal effects is unequal between treated and untreated - "Treatment modifies treatment effect" --- # Computing the conditional risks For `\(Pr[Y=1|A=1] \neq Pr[Y^{a=0}=1|A=1]\)` - Stratification by the stratum level `\(V=v\)` - Standardization by the confounding variable(s) L - Standardization formula: `\(Pr(Y^a=1)=\Sigma_lPr(Y=1|A=a, L=l, V=v)*Pr(L=l)\)` - Same result as IPTW --- # Surrogate vs causal effect modifier - Effect modification does not imply causality - It may be a proxy for a causal factor or a proxy for a factor that invokes an association, which is not causal (eg, a proxy of a collider) - **Effect heterogeneity** = effect measure modification - **Effect heterogeneity** term is agnostic about the modifier role being potentially causal or not --- # Why care? - Populations with different prevalence of effect modifiers (V) will show different average causal effects of exposures/interventions - Extrapolate the effect of the treatment computed in one population to a different population --> transportability - External validity: from the sample of Z population extrapolate to the whole Z population - Transportability: from the population Z extrapolate to the population W - Conditional effects within the strata of the effect modifier may be easier to transport - Known and unknown effect measure modifiers (outcome predictors) - Expert knowledge --- # Additive vs multiplicative effect measure modification * Additive scale to identify the target groups * Report (absolute) risks in each stratum of the effect modifier --- # Effect measure modification (EMM) vs interaction teaser <blockquote class="twitter-tweet"><p lang="en" dir="ltr">Trying to clear up confusion on effect modification vs. interaction. After diving into the literature, my Qs &gt;&gt; As. Is the difference purely a conceptual one? Do you just focus on stat. vs bio. interaction? Do you only talk about EMM? The lit is all over the place! <a href="https://twitter.com/hashtag/epitwitter?src=hash&amp;ref_src=twsrc%5Etfw">#epitwitter</a></p>&mdash; Jess Rohmann is on staycation (@JLRohmann) <a href="https://twitter.com/JLRohmann/status/1133032390291611648?ref_src=twsrc%5Etfw">May 27, 2019</a></blockquote> <script async src="https://platform.twitter.com/widgets.js" charset="utf-8"></script> --- <blockquote class="twitter-tweet"><p lang="en" dir="ltr">(2) In contrast, if I claim that smoking interacts with alcohol, what this means is that if I randomize into four groups, and compare alcohol use vs no use in those who were assigned to smoking, and the same among those assigned to no smoking, the results would be different</p>&mdash; Anders Huitfeldt (@AndersHuitfeldt) <a href="https://twitter.com/AndersHuitfeldt/status/1133073700885016576?ref_src=twsrc%5Etfw">May 27, 2019</a></blockquote> <script async src="https://platform.twitter.com/widgets.js" charset="utf-8"></script> --- <blockquote class="twitter-tweet"><p lang="en" dir="ltr">(3) Effect modification often occurs in the absence of interaction. For example, if you look at people with tobacco stained fingers vs those who don&#39;t, the effect of alcohol may differ by groups. But if you randomly assign them to stain their fingers, it probably won&#39;t interact</p>&mdash; Anders Huitfeldt (@AndersHuitfeldt) <a href="https://twitter.com/AndersHuitfeldt/status/1133074508594778113?ref_src=twsrc%5Etfw">May 27, 2019</a></blockquote> <script async src="https://platform.twitter.com/widgets.js" charset="utf-8"></script> --- <blockquote class="twitter-tweet"><p lang="en" dir="ltr">(5) It is much more difficult to imagine something that interacts, but is not an effect modifier. I guess it could happen if multiple interactions balance out perfectly, but it seems very unlikely.</p>&mdash; Anders Huitfeldt (@AndersHuitfeldt) <a href="https://twitter.com/AndersHuitfeldt/status/1133075071176773635?ref_src=twsrc%5Etfw">May 27, 2019</a></blockquote> <script async src="https://platform.twitter.com/widgets.js" charset="utf-8"></script> --- # Stratification as adjustment - Stratify by modifier variable V before adjusting for L - Stratify to assess effect heterogeneity - Adjust for L to reach conditional exchangeability - If L is the only factor needed to reach cond. exchangeability, L-specific effect estimates have a causal interpretation - **Conditional** effect measures as opposed to **marginal** effect estimates we gain by using IPW or standardization - Stratification cannot be used to compute marginal average causal effects, only to compute effects in non-overlapping subsets of the population - Different tools for different aims --- # Matching as a form of adjustment - Matching of treated to untreated based on a number of characteristics (confounders, L) ```r library(MatchIt) # matched on PS here matched_greeks <- matchit(A ~ L, method = "nearest", data = greeks) ``` ```r # Sample sizes: # Control Treated # All 7 12 # Matched 7 7 # Unmatched 0 5 # Discarded 0 0 ``` --- # 1:1 matching - In the matched sub-population of 7 pairs, treated and untreated are marginally exchangeable - Based on the group for which the matches are chosen, the computed effect may be 1) effect in the treated 2) effect in the untreated - Initial population `\(\neq\)` matched population --- # Estimating different types of effects - IPW and standardization * marginal and conditional effects - Stratification/restriction, matching * conditional effects in population subsets - All require cond. exchangeability and positivity - In the absence of causal effect modifiers may give the same numerical results --- # Example ```r table_4.3 ``` ``` ## # A tibble: 20 x 4 ## name L A Z ## <chr> <dbl> <dbl> <dbl> ## 1 Rheia 0 0 0 ## 2 Kronos 0 0 1 ## 3 Demeter 0 0 0 ## 4 Hades 0 0 0 ## 5 Hestia 0 1 0 ## 6 Poseidon 0 1 0 ## 7 Hera 0 1 1 ## 8 Zeus 0 1 1 ## 9 Artemis 1 0 1 ## 10 Apollo 1 0 1 ## 11 Leto 1 0 0 ## 12 Ares 1 1 1 ## 13 Athena 1 1 1 ## 14 Hephaestus 1 1 1 ## 15 Aphrodite 1 1 0 ## 16 Cyclope 1 1 0 ## 17 Persephone 1 1 0 ## 18 Hermes 1 1 0 ## 19 Hebe 1 1 0 ## 20 Dionysus 1 1 0 ``` - outcome Z: blood pressure - cond. exchangeability holds: `\(Z^a \perp\perp\ A|L\)` --- # IPTW: average causal effect in the entire population ```r pr_a_table_4.3 <- glm(data = table_4.3, formula = A~1, family = binomial("logit")) pr_a_l_table_4.3 <- glm(data = table_4.3, formula = A~L, family = binomial("logit")) table_4.3 %<>% mutate( p_a = predict(object = pr_a_table_4.3, type = "response"), p_a_l = predict(object = pr_a_l_table_4.3, type = "response"), # IPTW sw_iptw = if_else(A==1, p_a/p_a_l, (1-p_a)/(1-p_a_l)) # stabilized weights ) glm(data = table_4.3, formula = Z ~ A, family = binomial("log"), weights = sw_iptw) %>% broom::tidy(exponentiate = T) %>% filter(term == "A") %>% select(term, estimate) ``` ``` ## # A tibble: 1 x 2 ## term estimate ## <chr> <dbl> ## 1 A 0.800 ``` --- # Stratification: sub-groups specific effects ```r table_4.3 %>% group_by(L, A) %>% count(Z) %>% mutate( risk = n / sum(n) ) %>% filter(Z == 1) %>% ungroup() %>% group_by(L) %>% select(-n, -Z) %>% mutate( RR = lead(risk) / risk ) ``` ``` ## # A tibble: 4 x 4 ## # Groups: L [2] ## L A risk RR ## <dbl> <dbl> <dbl> <dbl> ## 1 0 0 0.25 2 ## 2 0 1 0.5 NA ## 3 1 0 0.667 0.5 ## 4 1 1 0.333 NA ``` --- # 1:1 Matching: treatment effect in the untreated or treated ```r matched_table_4.3 <- matchit(A ~ L, method = "nearest", data = table_4.3) # Sample sizes: # Control Treated # All 7 13 # Matched 7 7 # Unmatched 0 6 # Discarded 0 0 matched_table_4.3_data <- match.data(matched_table_4.3) glm(data = matched_table_4.3_data, formula = Z ~ A, family = binomial("log")) %>% broom::tidy(exponentiate = T) %>% filter(term == "A") %>% select(term, estimate) ``` ``` ## # A tibble: 1 x 2 ## term estimate ## <chr> <dbl> ## 1 A 1. ``` --- # Effect modification - Treatment doubles the risk among individuals in noncritical condition (L = 0, causal risk ratio 2.0) and halves the risk among individuals in critical condition (L = 1, causal risk ratio 0.5) - Average causal effect is 0.8 because L=1 group is larger ``` ## # A tibble: 2 x 3 ## L n pr ## <dbl> <int> <dbl> ## 1 0 8 0.4 ## 2 1 12 0.6 ``` - Causal effect in the untreated (had they been treated) is null because most of them (57%) are in non-critical condition (while in the whole population only 40%) ``` ## # A tibble: 4 x 4 ## # Groups: A [2] ## A L n pr ## <dbl> <dbl> <int> <dbl> ## 1 0 0 4 0.571 ## 2 0 1 3 0.429 ## 3 1 0 4 0.308 ## 4 1 1 9 0.692 ``` --- # Collapsibility - No multiplicative effect measure modification by V: - The causal **marginal** RR in the whole population is equal to the **conditional** causal RRs in every stratum of V - `\(\frac{Pr[Y^{a=1}=1]}{Pr[Y^{a=0}=1]}\)` = `\(\frac{Pr[Y^{a=1}=1|V=v]}{Pr[Y^{a=0}=1|V=v]}\)` - Causal RR is a **weighted average** of the stratum specific RRs <img src="index_files/figure-html/unnamed-chunk-23-1.png" width="504" /> - Collapsible over levels of V --- # Non-collapsibility - Population effect measure `\(\neq\)` weighted average of the stratum-specific measures - E.g., OR --- ```r table_4.4 ``` ``` ## # A tibble: 20 x 5 ## name V A Y sex ## <chr> <fct> <dbl> <dbl> <chr> ## 1 Rheia 1 0 0 female ## 2 Demeter 1 0 0 female ## 3 Hestia 1 0 0 female ## 4 Hera 1 0 0 female ## 5 Artemis 1 0 1 female ## 6 Leto 1 1 0 female ## 7 Athena 1 1 1 female ## 8 Aphrodite 1 1 1 female ## 9 Persephone 1 1 0 female ## 10 Hebe 1 1 1 female ## 11 Kronos 0 0 0 male ## 12 Hades 0 0 0 male ## 13 Poseidon 0 0 1 male ## 14 Zeus 0 0 1 male ## 15 Apollo 0 0 0 male ## 16 Ares 0 1 1 male ## 17 Hephaestus 0 1 1 male ## 18 Cyclope 0 1 1 male ## 19 Hermes 0 1 0 male ## 20 Dionysus 0 1 1 male ``` --- # RR - A is assigned at random - Unconditional exchangeability holds `\(Y^a \perp\perp\ A\)` * `\(\frac{Pr[Y^{a=1}=1]}{Pr[Y^{a=0}=1]}\)` = * under exchangeability: `\(\frac{Pr[Y^{a=1}=1|A=1]}{Pr[Y^{a=0}=1|A=0]}\)` = * under consistency: `\(\frac{Pr[Y=1|A=1]}{Pr[Y=1|A=0]}\)` ```r glm(Y~A, data = table_4.4, family=binomial(link = "log")) %>% broom::tidy(exponentiate = T) %>% filter(term == "A") %>% select(1, 2) ``` ``` ## # A tibble: 1 x 2 ## term estimate ## <chr> <dbl> ## 1 A 2.33 ``` - Marginal causal RR=2.3 --- # OR * `\(\frac{{Pr[Y^{a=1}=1]}/{Pr[Y^{a=1}=0]}}{{Pr[Y^{a=0}=1]}/{Pr[Y^{a=0}=0]}}\)` = * under exchangeability: `\(\frac{{Pr[Y^{a=1}=1|A=1]}/{Pr[Y^{a=1}=0|A=1]}}{{Pr[Y^{a=0}=1|A=0]}/{Pr[Y^{a=0}=0|A=0]}}\)` = * under consistency: `\(\frac{{Pr[Y=1|A=1]}/{Pr[Y=0|A=1]}}{{Pr[Y=1|A=0]}/{Pr[Y=0|A=0]}}\)` ```r glm(Y~A, data = table_4.4, family=binomial(link = "logit")) %>% broom::tidy(exponentiate = T) %>% filter(term == "A") %>% select(1, 2) ``` ``` ## # A tibble: 1 x 2 ## term estimate ## <chr> <dbl> ## 1 A 5.44 ``` - Marginal causal OR=5.4 --- # Conditional RR - `\(\frac{Pr[Y=1|A=1|V=v]}{Pr[Y=1|A=0|V=v]}\)` ``` ## # A tibble: 1 x 3 ## V term estimate ## <chr> <chr> <dbl> ## 1 men A 2.00 ``` ``` ## # A tibble: 1 x 3 ## V term estimate ## <chr> <chr> <dbl> ## 1 women A 3.00 ``` --- # Conditional OR - `\(\frac{{Pr[Y=1|A=1, V=v]}/{Pr[Y=0|A=1, V=v]}}{{Pr[Y=1|A=0, V=v]}/{Pr[Y=0|A=0, V=v]}}\)` ``` ## # A tibble: 1 x 3 ## V term estimate ## <chr> <chr> <dbl> ## 1 men A 6.00 ``` ``` ## # A tibble: 1 x 3 ## V term estimate ## <chr> <chr> <dbl> ## 1 women A 6.00 ``` - Average RR of 2.3 is between sex stratum-specific estimates of RR=2 for men and RR=3 for women - Average OR of 5.4 is smaller than sex stratum-specific estimates of OR=6 for men and OR=6 for women - OR is not collapsible in this example - Change in the conditional OR vs marginal OR due to non-collapsibility; exchangeability holds --- # Take home messages - Effect **measure** modification = effect **heterogeneity** - There's no **the** causal effect - Effect measure modifiers **may or may not** be proxies of the causal factors; no causal claims are made about effect measure modifiers - Do not confuse non-collapsibility with the lack of exchangeability - **IPW and standardization** are appropriate approaches for computing **marginal** causal effects in the **entire population**; stratification and matching are not