Drawing Inferences How to use statistical models to get at scientific estimands? Need to incorporate causal thinking into (1) How we draw the statistical models (2) How we process the results
Drawing Inferences (1) How we draw the statistical models Generative model + multiple estimands => multiple estimators (2) How we process the results Only very simple causal estimates show up in summary tables
Categories & Curves Linear models can do extra-linear things Categories (indicator & index variables) Splines and other additive structures We require these tools to build causal estimators
Categories How to cope with causes that are not continuous? Categories: discrete, unordered types Want to stratify by category: Fit a separate line for each
The causes aren’t in the data S H S H women men 30 40 50 60 0.00 0.04 0.08 weight (kg) Density 140 150 160 170 180 0.00 0.04 0.08 height (cm) Density women men S W S W
The causes aren’t in the data S H S H women men 30 40 50 60 0.00 0.04 0.08 weight (kg) Density 140 150 160 170 180 0.00 0.04 0.08 height (cm) Density women men S W S W
W H S U V W # S=1 female; S=2 male sim_HW N H W data.frame(S,H,W) } S dat head(dat) S H W 1 1 149.2428 80.07163 2 2 164.8629 94.79647 3 2 158.1800 90.59944 4 1 148.2153 70.68285 5 2 155.5794 91.43798 6 2 154.0733 92.18237
Think scientifically first Different causal questions may need different statistical models Q: Causal effect of H on W? Q: Causal effect of S on W? Q: Direct causal effect of S on W? W H S
Think scientifically first Different causal questions need different statistical models Q: Causal effect of H on W? Q: Causal effect of S on W? Q: Direct causal effect of S on W? W H S
Think scientifically first Different causal questions need different statistical models Q: Causal effect of H on W? Q: Causal effect of S on W? Q: Direct causal effect of S on W? W H S
Think scientifically first Different causal questions need different statistical models Q: Causal effect of H on W? Q: Causal effect of S on W? Q: Direct causal effect of S on W? W H S
From estimand to estimate Q: Causal effect of S on W? W H S W H S Q: Direct causal effect of S on W? Stratify by S: different estimate for each value S can take
Drawing Categorical Owls Several ways to code categorical variables (1) indicator (0/1) variables (2) index variables: 1,2,3,4,… We will use index variables: Extend to many categories with no change in code Better for specifying priors Extend effortlessly to multi-level models
Using Index Variables 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 Wi ⇠ Normal(µi, ) µi = ↵S[i] + S[i] (Hi ¯ H) sex of i-th person AAACDHicbZDLSsNAFIYn9VbrrerSzWARXEhJSlE3QtGNywr2Ak0oJ9NJO3QyCTMToYQ+gBtfxY0LRdz6AO58GydtFtr6w8DHf85hzvn9mDOlbfvbKqysrq1vFDdLW9s7u3vl/YO2ihJJaItEPJJdHxTlTNCWZprTbiwphD6nHX98k9U7D1QqFol7PYmpF8JQsIAR0MbqlytuCHrkBy7weAT4Cvfm1HfOcqh5psuu2jPhZXByqKBczX75yx1EJAmp0ISDUj3HjrWXgtSMcDotuYmiMZAxDGnPoICQKi+dHTPFJ8YZ4CCS5gmNZ+7viRRCpSahbzqz1dViLTP/q/USHVx6KRNxoqkg84+ChGMd4SwZPGCSEs0nBoBIZnbFZAQSiDb5lUwIzuLJy9CuVZ3zav2uXmlc53EU0RE6RqfIQReogW5RE7UQQY/oGb2iN+vJerHerY95a8HKZw7RH1mfP1Wnmog= ↵ = [↵1, ↵2] S = 1 indicates female S = 2 indicates male Two intercepts, one for each value in S
Testing 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 Wi ⇠ Normal(µi, ) µi = ↵S[i] + S[i] (Hi ¯ H) AAACCnicbVDLSgMxFM3UV62vUZduokWpIGVGirosunFZwWkLnaFk0rQNTWaG5I5Yhq7d+CtuXCji1i9w59+YPhZavRByOOdc7r0nTATX4DhfVm5hcWl5Jb9aWFvf2Nyyt3fqOk4VZR6NRayaIdFM8Ih5wEGwZqIYkaFgjXBwNdYbd0xpHke3MExYIEkv4l1OCRiqbe/7mvckwUfml9gHdg+ZZwyxkqOSc+I6x2276JSdSeG/wJ2BIppVrW1/+p2YppJFQAXRuuU6CQQZUcCpYKOCn2qWEDogPdYyMCKS6SCbnDLCh4bpYDPevAjwhP3ZkRGp9VCGxikJ9PW8Nib/01opdC+CjEdJCiyi00HdVGCI8TgX3OGKURBDAwhV3OyKaZ8oQsGkVzAhuPMn/wX107J7Vq7cVIrVy1kcebSHDlAJuegcVdE1qiEPUfSAntALerUerWfrzXqfWnPWrGcX/Srr4xt9+5l9 ⇠ Uniform(0, 10) AAACC3icbVA9SwNBEN3zM8avqKXNEhUiSLgTUcugjZUoGCPkQpjbbMya3btjd04MR3ob/4qNoCJ24h+w84do7Sax0OiDgcd7M8zMC2IpDLruuzMyOjY+MZmZyk7PzM7N5xYWT02UaMbLLJKRPgvAcClCXkaBkp/FmoMKJK8E7f2eX7nk2ogoPMFOzGsKzkPRFAzQSvVc3gcZt6B+QX0jFPWRX2F6GGkFslvYdjc8d72eW3GLbh/0L/G+yUpp9eP+5XL686iee/MbEUsUD5FJMKbquTHWUtAomOTdrJ8YHgNrwzmvWhqC4qaW9n/p0jWrNGgz0rZCpH3150QKypiOCmynAmyZYa8n/udVE2zu1lIRxgnykA0WNRNJMaK9YGhDaM5QdiwBpoW9lbIWaGBo48vaELzhl/+S082it13cOrZp7JEBMmSZ5EmBeGSHlMgBOSJlwsg1uSUP5NG5ce6cJ+d50DrifM8skV9wXr8A+UieIw== ↵j ⇠ Normal(60, 10) # female sample S simF # male sample S simM # effect of sex (male-female) mean(simM$W - simF$W) What is the total causal effect of sex? Causal effect of sex is difference made intervening: [1] 21.13852
Always Be Contrasting Need to compute contrast, the difference between the categories It is never legitimate to compare overlap in distributions Must compute contrast distribution 20 30 40 50 60 70 0.00 0.04 0.08 posterior predicted weight (kg) Density
-3 -2 -1 0 1 2 3 0.0 0.5 1.0 1.5 2.0 value Density -1.5 -0.5 0.0 0.5 1.0 1.5 -0.5 0.5 1.0 1.5 2.0 2.5 parameter 1 parameter 2 parameter 1 parameter 2 difference It is never legitimate to compare overlap in parameters or lines This means no comparing confidence intervals or p-values either!
From estimand to estimate Q: Causal effect of S on W? W H S W H S Q: Direct causal effect of S on W? 3 4 5 6 7 8 9 10 0.0 0.3 0.6 posterior mean weight contrast (kg) Density -20 -10 0 10 20 30 0.00 0.02 0.04 posterior weight contrast (kg) Density
140 145 150 155 160 165 170 70 80 90 100 H W S dat direct indirect Indirect: H influences W differently (slopes) Direct: W differs net any slope differences
Centering variables 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 Wi ⇠ Normal(µi, ) µi = ↵ + (Hi ¯ H) “centered” height Centering H makes it so that alpha is the average weight of a person with average height average height Centering makes it easier to define scientific priors for alpha
140 145 150 155 160 165 170 70 80 90 100 H W Centering H makes it so that alpha is the average weight of a person with average height average W average H 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 Wi ⇠ Normal(µi, ) µi = ↵ + (Hi ¯ H)
140 145 150 155 160 165 170 70 80 90 100 H W Centering H makes it so that alpha is the average weight of a person with average height average W average H 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 Wi ⇠ Normal(µi, ) µi = ↵ + (Hi ¯ H)
Index Variables & Lines 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 Wi ⇠ Normal(µi, ) µi = ↵S[i] + S[i] (Hi ¯ H) sex of i-th person AAACCXicbVDLSsNAFJ3UV62vqEs3g0VwISUpRd0IRTcuK9gHJCFMppN26GQSZiZCCd268VfcuFDErX/gzr9xkmahrQeGOZxzL/feEySMSmVZ30ZlZXVtfaO6Wdva3tndM/cPejJOBSZdHLNYDAIkCaOcdBVVjAwSQVAUMNIPJje5338gQtKY36tpQrwIjTgNKUZKS74J3QipcRC6AVEIXkGnIL59Nv+bnm/WrYZVAC4TuyR1UKLjm1/uMMZpRLjCDEnp2FaivAwJRTEjs5qbSpIgPEEj4mjKUUSklxWXzOCJVoYwjIV+XMFC/d2RoUjKaRToynxvuejl4n+ek6rw0ssoT1JFOJ4PClMGVQzzWOCQCoIVm2qCsKB6V4jHSCCsdHg1HYK9ePIy6TUb9nmjddeqt6/LOKrgCByDU2CDC9AGt6ADugCDR/AMXsGb8WS8GO/Gx7y0YpQ9h+APjM8f3ZGZLA== = [ 1, 2] AAACDHicbZDLSsNAFIYn9VbrrerSzWARXEhJSlE3QtGNywr2Ak0oJ9NJO3QyCTMToYQ+gBtfxY0LRdz6AO58GydtFtr6w8DHf85hzvn9mDOlbfvbKqysrq1vFDdLW9s7u3vl/YO2ihJJaItEPJJdHxTlTNCWZprTbiwphD6nHX98k9U7D1QqFol7PYmpF8JQsIAR0MbqlytuCHrkBy7weAT4Cvfm1HfOcqh5psuu2jPhZXByqKBczX75yx1EJAmp0ISDUj3HjrWXgtSMcDotuYmiMZAxDGnPoICQKi+dHTPFJ8YZ4CCS5gmNZ+7viRRCpSahbzqz1dViLTP/q/USHVx6KRNxoqkg84+ChGMd4SwZPGCSEs0nBoBIZnbFZAQSiDb5lUwIzuLJy9CuVZ3zav2uXmlc53EU0RE6RqfIQReogW5RE7UQQY/oGb2iN+vJerHerY95a8HKZw7RH1mfP1Wnmog= ↵ = [↵1, ↵2] S = 1 indicates female S = 2 indicates male Two intercepts and two slopes, one for each value in S
Contrasts at each height (1) Compute posterior predictive for women (2) Compute posterior predictive for men (3) Subtract (2) from (1) (4) Plot distribution at each height (on right) 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg)
From estimand to estimate Q: Causal effect of S on W? W H S W H S Q: Direct causal effect of S on W? 3 4 5 6 7 8 9 10 0.0 0.3 0.6 posterior mean weight contrast (kg) Density -20 -10 0 10 20 30 0.00 0.02 0.04 posterior weight contrast (kg) Density 130 140 150 160 170 180 190 -6 -4 -2 0 2 4 6 8 height (cm) weight contrast (F–M)
Categorical variables Easy to use with index coding Use samples to compute relevant contrasts Always summarize (mean, interval) as the last step Want mean difference and not difference of means 140 150 160 170 180 30 35 40 45 50 55 60 height (cm) weight (kg)
Curves from lines H –> W obviously not linear Linear models can easily fit curves But this is not mechanistic library(rethinking) data(Howell1) 60 80 100 140 180 10 20 30 40 50 60 height (cm) weight (kg)
Polynomial linear models Strategy: polynomial functions Problems: strange symmetries, explosive uncertainty at edges No local smoothing; only global DO NOT USE μi = α + β1 xi + β2 x2 i
Splines Basis-Splines: Wiggly function built from many local functions Basis function: A local function Local functions make splines better than polynomials, but equally geocentric Richard McElreath McElreath with Examples in R and Stan SECOND EDITION ical Rethinking o advanced multilevel mod- d Gaussian process models h (DAG) approach to causal w edition also contains new ered categorical predictors, ampling, instrumental vari- rely new chapter that goes n-specific scientific models mples umptions are reflected in in optional sections analyze causal graphs bsite and on GitHub nd is a Director at the Max Germany. He has published analysis of social behavior, Models of Social Evolution.
Height as a function of age Obviously not linear Want function approximately right Fit spline as example But biological model would do a lot better 0 20 40 60 80 60 80 100 140 180 age (years) height (cm)
Height as a function of age W H S A age How can age cause anything? No definable intervention? Association between age and height due to accumulated growth
Curves and Splines Can build very non-linear functions from linear pieces Splines are powerful geocentric devices Adding scientific information helps e.g. Average weight only increases with height e.g. Height increases with age, then levels off (or declines) Ideally statistical model has some form as scientific model
Full Luxury Bayes We used two models for two estimands But alternative and equivalent approach is to use one model of entire causal system Then use joint posterior to compute each estimand W H S
Full Luxury Bayes m_SHW_full alist( # weight W ~ dnorm(mu,sigma), mu a[S] ~ dnorm(60,10), b[S] ~ dunif(0,1), sigma ~ dunif(0,10), # height H ~ dnorm(nu,tau), nu h[S] ~ dnorm(160,10), tau ~ dunif(0,10) ), data=dat ) W H S
Full Luxury Bayes m_SHW_full alist( # weight W ~ dnorm(mu,sigma), mu a[S] ~ dnorm(60,10), b[S] ~ dunif(0,1), sigma ~ dunif(0,10), # height H ~ dnorm(nu,tau), nu h[S] ~ dnorm(160,10), tau ~ dunif(0,10) ), data=dat ) W H S H S W H S
Full Luxury Bayes m_SHW_full alist( # weight W ~ dnorm(mu,sigma), mu a[S] ~ dnorm(60,10), b[S] ~ dunif(0,1), sigma ~ dunif(0,10), # height H ~ dnorm(nu,tau), nu h[S] ~ dnorm(160,10), tau ~ dunif(0,10) ), data=dat ) Causal effect is consequence of intervention Now simulate each intervention
Simulating Interventions post Hbar n with( post , { # simulate W for S=1 H_S1 W_S1 b[,1]*(H_S1-Hbar) , sigma) # simulate W for S=2 H_S2 W_S2 b[,2]*(H_S2-Hbar) , sigma) # compute contrast W_do_S <}) Total causal effect of S on W: Consequence of changing S at birth 3 4 5 6 7 8 9 10 0.0 0.3 0.6 posterior mean weight contrast (kg) Density -20 -10 0 10 20 30 0.00 0.02 0.04 posterior weight contrast (kg) Density “p(W|do(S))”
Simulating Interventions post Hbar n with( post , { # simulate W for S=1 H_S1 W_S1 b[,1]*(H_S1-Hbar) , sigma) # simulate W for S=2 H_S2 W_S2 b[,2]*(H_S2-Hbar) , sigma) # compute contrast W_do_S <}) Total causal effect of S on W: Consequence of changing S at birth 3 4 5 6 7 8 9 10 0.0 0.3 0.6 posterior mean weight contrast (kg) Density -20 -10 0 10 20 30 0.00 0.02 0.04 posterior weight contrast (kg) Density “p(W|do(S))” # automated way HWsim data=list(S=c(1,2)), vars=c("H","W")) W_do_S_auto
Inference With Linear Models With more than two variables, scientific (causal) model and statistical model not always same (1) State each estimand (2) Design unique statistical model for each (3) Compute each estimand Or (1) State each estimand (2) Compute joint posterior for causal system (3) Simulate each estimand as an intervention ONE STAT MODEL FOR EACH ESTIMAND ONE SIMULATION FOR EACH ESTIMAND
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