Outline 1 Causal inference Confounding and lurking variables Imbalance and lack of overlap Fundamental problem of causal inference Blocking Randomization Statistical adjustment Causal inference versus prediction In prediction we make comparisons between outcomes across different combinations of values of input variables In causal inference we ask what would happen to an outcome y as a result of a treatment or intervention Predictive inference relates to comparisons between units Causal inference addresses comparisons of different treatments when applied to the same unit Example of prediction not causal inference The second to fourth digit 2D 4D ratio length of index finger ring finger was used to predict financial traders performance profit and loss P L It was stated to serve as a surrogate for prenatal androgen effects Coates et al 2009 PNAS Counterfactual outcomes Let T represent a treatment variable For a categorical treatment 1 if unit i receives the treatment Ti 1li is treated 0 if unit i receives the control yi1 outcome of the ith unit if the treatment is given yi0 outcome of the ith unit if the control is given One of these is observed the other is counterfactual what would have been observed if the other treatment had been given For a continuous treatment Ti the numerical value of the treatment assigned to unit i Confounding A variable or covariate is a confounder if it predicts both treatment and outcome confounder X lurking Treatment T response Y We can estimate a causal effect in regression if 1 the regression model includes all confounders and 2 the regression model is correct A confounder left out of a regression model is called a lurking variable Causal inference and estimation of treatment effects can be misleading when confounders are omitted from a model Ex Y FEV T smoking and X age Algebraic explanation Suppose a true model has a treatment T and a confounder x for outcome y yi 0 1 Ti 2 xi ei If x is related to the treatment we could write xi 0 1 Ti i 1 Treatment T confounder X lurking 1 2 response Y Algebra cont for misleading estimation If we ignore the confounder x we would fit the model yi 0 1 Ti ei The correct model then becomes yi 0 1 Ti 2 xi ei 0 1 Ti 2 0 1 Ti i ei 0 2 0 1 2 1 Ti 2 i ei 1 Treatment T confounder X lurking 1 2 1 1 2 response Y Algebra cont From before yi 0 2 0 1 2 1 Ti 2 i ei true treatment effect 1 estimated effect if x is omitted 1 1 2 1 Estimation without x is correct only if 2 1 0 which happens if 2 0 i e x does not predict y or if 1 0 i e x does not predict T Causal inference can mislead when there are lurking variables FEV example age as a confounder fit2 lm fev age smoke data fev fit1 lm fev smoke data fev fitage lm age smoke data fev coef fit2 true beta0 beta2 beta1 Intercept age smoketrue 0 3673730 0 2306046 0 2089949 coef fitage gamma0 gamma1 Intercept smoketrue 9 534805 3 988272 Beneficial estimated smoking effect when age is ignored lurking coef fit1 Intercept 2 5661426 wrong beta0star beta1star smoketrue 0 7107189 check that beta0star beta0 beta2 gamma0 0 3673730 0 2306046 9 534805 1 2 566143 check that wrong beta1star beta1 beta2 gamma1 0 2089949 0 2306046 3 988272 1 0 710719 Factors that affect causal inference Imbalance and lack of complete overlap can make causal inference difficult for comparing between two treatments Imbalance when treatment groups differ with respect to an important covariate Lack of complete overlap when some combination of treatment level and covariate level is lacking no observations Imbalance Baby food example 842 bottle fed 274 breastfed Not same numbers but no issue Imagine these proportions of girls boys per food type food boy girl bottle fed 0 80 0 20 breast fed 0 35 0 65 There would be a serious difference between groups bottle breast fed in an important covariate gender Gender would be a confounding covariate because it can help predict both the treatment food and the outcome disease We prefer balanced experiments when all treatment groups are similar with respect to important covariates Randomization is the only way go get balanced groups with respect to all covariates even those we don t suspect Imbalance in the FEV example To explain fev sex seems to matter especially among older individuals plot fev age col sex data fev legend topleft pch 1 col 1 2 levels dat sex 5 female male 4 2 3 fev 1 5 10 age 15 Imbalance in the FEV example Slight gender imbalance among age categories for ages 9 0 0 0 2 female 0 4 sex 0 6 male 0 8 1 0 par mar c 3 3 1 2 mgp c 1 4 4 0 tck 02 plot sex age data subset fev age 9 3 5 6 7 age 8 with subset fev age 9 table age sex sex age female male 3 1 1 4 6 3 5 14 14 6 15 22 7 29 25 8 46 39 Imbalance For imbalanced samples simple comparisons of sample means between groups are not good estimates of treatment effects After the experiment a model adjustment is one way to better estimate a treatment effect where we add the covariate to the model Before the experiment matching is another strategy to overcome avoid imbalance Ex If gender is thought to affect respiratory disease risk match each breast fed baby with a bottle fed baby of the same gender Ex blocking later Lack of complete overlap in the FEV data Lack of smoking children below age 9 false smoke true plot smoke age data fev with fev table age smoke 2 6 8 10 age 12 14 smoke age false true 3 2 0 4 9 0 5 28 0 6 37 0 7 54 0 8 85 0 9 93 1 10 76 5 11 81 9 12 50 7 13 30 13 Lack of complete overlap Lack of complete overlap is when there are no observations at some combination s of treatment levels covariate levels For lack of complete overlap there is no data available for some comparisons This requires extrapolation using a model to make comparisons This is a more serious problem than imbalance Fundamental problem of causal inference The fundamental problem of causal inference is that at most one of yi0 and yi1 can be observed Causal inference is predictive inference in a potential outcome framework Estimation of causal effects requires some combination of close substitutes for potential outcomes randomization or statistical adjustment Close substitutes Several ways to attempt to use close substitutes the same unit can be measured for all treatments but are the effects the same Ex treatment 1 in the first week tmt 2 in week 2 etc But at least randomize the order a unit can be subdivided into groups and subjected to different treatments but do the parts behave identically Ex plots farms etc a pre …