Explaining or predicting?

What we want vs what we can see

• Everyone has two potential outcomes; we only ever see one. So we aim for an average: the average treatment effect, ATE = E[Y(1) − Y(0)]
• The data give E[Y | A = 1] − E[Y | A = 0]: 7.9% − 13.0%. The two match only if the groups are exchangeable — alike except for obesity
• Here they are not: people without obesity are older (54.6 vs 50.5 years), more often men and more often smokers (24.7% vs 18.0%)

Positivity, consistency and a tricky exposure

• Positivity — is it violated here? No. It needs every kind of person to appear in both groups, and they do: given age, sex, race/ethnicity, smoking and activity, each person’s predicted chance of obesity lies between 35% and 97% — never 0% or 100%
• But it is thin in places: among women aged 18–39, 671 have obesity and only 53 do not, so what we conclude about young women rests on 53 people
• Consistency: “not obese” must mean one thing. It doesn’t — diet, exercise, surgery, a drug, smoking or cancer can each make someone not obese, with different effects on death. What If uses exactly this example, down to asking whether we mean “a massive liposuction”

A DAG, and the two kinds of path in it

• Arrow = “directly causes”; no arrow = a claim of no direct effect
• Causal paths (green) follow the arrows forward: obesity → death; obesity → diabetes → death. Back-door paths (red) start with an arrow into obesity: obesity ← age → death — they carry confounding

Three building blocks

A collider (two arrows collide into it) blocks its path on its own. Adjusting for it — or for something it causes — opens the path.

Selecting on FLI ≥ 60: a collider you can see

FLI rises with BMI and with triglycerides (TG). Keep only FLI ≥ 60, and people with a low BMI stay in only if their TG is high: ρ = +0.28 among everyone, −0.30 among the selected.

Selection can make obesity look protective

People without obesity reach FLI ≥ 60 with higher TG (174 vs 126 mg/dL) and GGT (36 vs 25 U/L). If these raise the risk of death, obesity looks protective — index-event bias. The birth-weight paradox (left) is the classic case.

A working DAG for obesity and death

Reading the moves with the DAG

Four more roles a variable can play

Can’t draw the whole DAG? Rules of thumb

Table 1 describes; a DAG chooses

• Every p-value is <0.001 — with 5,911 people, even small gaps are “significant”. The SMD (standardised mean difference) measures the gap’s size: at EpiMethods’ cut-off of 0.2, sex (0.52) and age (0.26) are imbalanced; smoking (0.16) and diabetes (0.13) are not
• A gap says a variable differs between the groups — not whether it is a confounder, a mediator or neither. That takes the DAG

Change-in-estimate: “if it moves, keep it”

The rule: add a variable; keep it if the exposure’s estimate changes by more than 10%.

• Age and sex: +34% → “confounders” ✓ right
• The recorded conditions: −15% → “confounders” ✗ probably partly mediators
• Smoking and sedentary time: +4% → “not confounders” ✗ smoking is a confounder

p-values, stepwise and machine learning

• Confounding is not a significance test: a strong confounder can be “not significant” in a small sample
• Stepwise selection keeps whatever predicts death best — here, diabetes, kidney disease, heart attack — and its confidence intervals ignore the search: too narrow
• Purposeful selection is more careful, but still blind to causal structure: it keeps colliders, because colliders move the estimate
• Machine learning (e.g., LASSO) is excellent at prediction; in causal work, use it only for the prediction parts of a method (propensity scores, later in the course)

Collapsibility: ORs move without confounding

Made-up numbers. Half of each age group is exposed, so age is not a confounder. OR = odds ratio · RR = risk ratio · RD = risk difference.

• RD and RR are collapsible: the overall value is an average of the subgroups’
• OR is not: 4.00 in both age groups, 3.45 overall. HRs behave like ORs
• Adjusting for age moves the OR away from 1 (3.45 → 4.00) — change-in-estimate would wrongly call age a confounder

Conditional vs marginal — both are right

• Conditional: among people with the same covariate values — what a regression coefficient gives. Here, OR = 4.00
• Marginal: in the whole population — everyone exposed vs everyone unexposed, what a randomised trial gives. Here, OR = 3.45
• The gap is not bias. It needs a real effect, and it always pushes the conditional OR or HR away from 1

Getting a marginal effect: g-computation

1. Fit an outcome model with the exposure and the covariates
2. Set everyone to obese; predict each person’s risk and average
3. Set everyone to not obese; predict and average again — then compare

# fit = glm(died ~ obese + age + female + ..., family = binomial, data = d)
d1 <- transform(d, obese = 1);  d0 <- transform(d, obese = 0)
p1 <- mean(predict(fit, newdata = d1, type = "response"))
p0 <- mean(predict(fit, newdata = d0, type = "response"))
c(RD = p1 - p0, RR = p1 / p0, OR = (p1 / (1 - p1)) / (p0 / (1 - p0)))

One logistic model gives the marginal RD, RR and OR together. For survival, the same idea gives a standardised survival curve: everyone obese vs no one obese.

Table 2 fallacy: one model, many questions

One Cox model, as a paper’s Table 2 would print it:

Four questions, four adjustment sets (1 of 2)

Same DAG, different question — different roles. Waist fat is a mediator in Q1 but the exposure in Q2; obesity is the exposure in Q1 but a confounder in Q2.

Four questions, four adjustment sets (2 of 2)

Age needs no adjustment — nothing causes it. Diabetes needs a variable nobody measured (genetic risk), so no model in our data answers Q4.

The one model vs each question’s own model

• Obesity looks “protective” (0.80) in the one model because it holds waist fat and diabetes fixed — both on obesity’s path. Its own model says 0.95
• Even where the numbers are close, each row means something different

Key takeaways

1. An adjusted estimate answers a causal question only under assumptions — draw them as a DAG
2. Adjust for confounders; not for mediators (for the total effect), colliders or instruments. Selecting who is in the study counts as adjusting
3. Choose confounders from knowledge, not p-values or change-in-estimate
4. ORs and HRs move on adjustment even without confounding; RDs and RRs don’t
5. One exposure per model — the other rows are not effects
6. Obesity’s apparent protection (0.67) mostly disappears with pre-exposure variables (0.95). Adjustment can’t fix what’s unmeasured (serious illness) or the selection (FLI ≥ 60)

Where to read more

References (1 of 2)

References (2 of 2)

Appendix: our working DAG as dagitty code

Paste into the Model code box at dagitty.net. With Ill, Before and After marked latent (never measured), dagitty reports that no adjustment set exists. Remove the latent tags from Ill and Before, and it returns {Before, Ill, L}.

dag {
  Obese [exposure]  Death [outcome]  Ill [latent]  Before [latent]  After [latent]
  L -> Obese  L -> Death  L -> Before
  Ill -> Obese  Ill -> Death
  Before -> Obese  Before -> Death  Before -> After
  Obese -> After  After -> Death  Obese -> Death
  Before -> Blend  After -> Blend
}

L = age, sex, race/ethnicity, income, smoking, physical activity. Ill = serious illness that causes weight loss. Blend = the recorded “has diabetes (etc.)” variable.