a brief introduction to causal inference
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A Brief Introduction to Causal Inference Brady Neal - PowerPoint PPT Presentation

Mar 31, 2024 •1.72k likes •2.8k views

A Brief Introduction to Causal Inference Brady Neal causalcourse.com What is causal inference? Inferring the effects of any treatment/policy/intervention/etc. 2 / 28 Brady Neal What is causal inference? Inferring the effects of any

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  2. <latexit sha1_base64="upoWhkgWh/gDf1sbvpAmAOKwsw4=">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</latexit> <latexit sha1_base64="hTyqvd9qpY0L5hIjmF/eBUI1tr0=">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</latexit> <latexit sha1_base64="YuoRGm8uMvFe8newXYncLQ+OLCs=">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</latexit> Simpson’s paradox: mortality rate table Condition Mild Severe Total Treatment 15% 30% 16% 1400 + 100 A 0.15 0.30 = 0.16 1500 1500 (210/1400) (30/100) (240/1500) 10% 20% 19% 50 + 500 B 0.10 0.20 = 0.19 (5/50) (100/500) (105/550) 550 550 E [ Y | T, C = 0] E [ Y | T, C = 1] E [ Y | T ] Motivating example: Simpson’s paradox 6 / 28 Brady Neal

  3. <latexit sha1_base64="upoWhkgWh/gDf1sbvpAmAOKwsw4=">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</latexit> <latexit sha1_base64="hTyqvd9qpY0L5hIjmF/eBUI1tr0=">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</latexit> <latexit sha1_base64="YuoRGm8uMvFe8newXYncLQ+OLCs=">AEo3icbVNLb9NAEHbACW8WjhyWagqcXBDUlUqHCJVqioh4FBQXygJ1Xo9SVbZl3bHhNRE/A2u8K/4N4wdlzZp17L07cw379nEKRmw2fy7tLxSu3P3ur9+oOHjx4/WVt/ehJs5gUcC6usP0t4ACUNHKNEBWfOA9eJgtNktF/oT7+BD9KaI5w46Gk+MLIvBUcSfe0edL78OIrZPmuzZu98baPZaJaH3QStCmxE1Tk8X1/52U2tyDQYFIqH0Gk1HfZy7lEKBdN6NwvguBjxAXQIGq4h9PIy7SnbJEnK+tbTb5CV0usWOdchTHRCTM1xGBZ1hfA2XSfD/pteLo3LEIyYBepniqFlRQ9YKj0IVBMCXHhJuTIx5J4LpE7V58Ikelqvb7IjObpglWw+j1ktcyIk8pVAERpBoFZh1Li6pkwbPAFRt47oahQeT3WSh64CZbjgcEhkMZKTH0nfhU8nEcz/Jw5A7CHEKwvpyjCHm3txiAVXosINDcjvsTY2SALFiVBeRaOyF1eZ3S2PnKE7zHP0MbGpkVnAnIjgLVbTGjGTcoKEAfQkhISo7i0IxLC60bAiSJqDkpJFyBmqefjmGlpM40G8sUh8ViNXbJx3Rm6qw0+N+U5UJ62pRLU2kMeGqZa9Pa7VDgvlRqlpqgHaE1C+18WrlK5GyQkFb+2nlZT1VysMQ0oL6lrMqyjbDmPGlRyYtoI+4aCsI6N6Ob9GonVFlr6gTN3cC4uTdg2neLVYvSfKD6XRed8J9pfU6Ly4LemlScMRw4B3rvig+Vl4WeOaSaCwS6RY6PdnW4gO9CU62G62dxtPOxt729XjXY2eRy+jV1Er2o32onfRYXQcichHv6Lf0Z/aZu1D7XPtaEZdXqpsnkVzp9b7B027g9Q=</latexit> Simpson’s paradox: mortality rate table Condition Mild Severe Total Treatment 15% 30% 16% 1400 + 100 A 0.15 0.30 = 0.16 1500 1500 (210/1400) (30/100) (240/1500) 10% 20% 19% 50 + 500 B 0.10 0.20 = 0.19 (5/50) (100/500) (105/550) 550 550 E [ Y | T, C = 0] E [ Y | T, C = 1] E [ Y | T ] Motivating example: Simpson’s paradox 6 / 28 Brady Neal

  4. <latexit sha1_base64="upoWhkgWh/gDf1sbvpAmAOKwsw4=">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</latexit> <latexit sha1_base64="hTyqvd9qpY0L5hIjmF/eBUI1tr0=">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</latexit> <latexit sha1_base64="YuoRGm8uMvFe8newXYncLQ+OLCs=">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</latexit> Simpson’s paradox: mortality rate table Condition Mild Severe Total Treatment 15% 30% 16% 1400 + 100 A 0.15 0.30 = 0.16 1500 1500 (210/1400) (30/100) (240/1500) 10% 20% 19% 50 + 500 B 0.10 0.20 = 0.19 (5/50) (100/500) (105/550) 550 550 E [ Y | T, C = 0] E [ Y | T, C = 1] E [ Y | T ] Which treatment should you choose? Motivating example: Simpson’s paradox 6 / 28 Brady Neal

  5. Simpson’s paradox: scenario 1 (treatment B) Condition Mild Severe Total 15% 30% 16% t A n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B (5/50) (100/500) (105/550) Motivating example: Simpson’s paradox 7 / 28 Brady Neal

  6. <latexit sha1_base64="6nzlYT7Z3LXF3cG2BdYfPCoT2Q=">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</latexit> Simpson’s paradox: scenario 1 (treatment B) Condition Mild Severe Total 15% 30% 16% t A n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B (5/50) (100/500) (105/550) Causal Graph C T Y Motivating example: Simpson’s paradox 7 / 28 Brady Neal

  7. <latexit sha1_base64="6nzlYT7Z3LXF3cG2BdYfPCoT2Q=">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</latexit> Simpson’s paradox: scenario 1 (treatment B) Condition Mild Mild Severe Total 15% 30% 16% t A n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B (5/50) (100/500) (105/550) Causal Graph C Treatment A T Y Motivating example: Simpson’s paradox 7 / 28 Brady Neal

  8. <latexit sha1_base64="6nzlYT7Z3LXF3cG2BdYfPCoT2Q=">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</latexit> Simpson’s paradox: scenario 1 (treatment B) Condition Mild Severe Mild Severe Total 15% 30% 16% t A n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B (5/50) (100/500) (105/550) Causal Graph C Treatment A Treatment B T Y Motivating example: Simpson’s paradox 7 / 28 Brady Neal

  9. Simpson’s paradox: scenario 2 (treatment A) Condition Mild Severe Total 15% 30% 16% t A n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B (5/50) (100/500) (105/550) Motivating example: Simpson’s paradox 8 / 28 Brady Neal

  10. <latexit sha1_base64="WDTVIbXn/a2k0zCUek3TNQgock=">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</latexit> Simpson’s paradox: scenario 2 (treatment A) Condition Mild Severe Total 15% 30% 16% t A n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B (5/50) (100/500) (105/550) T C Causal Graph Y Motivating example: Simpson’s paradox 8 / 28 Brady Neal

  11. <latexit sha1_base64="WDTVIbXn/a2k0zCUek3TNQgock=">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</latexit> Simpson’s paradox: scenario 2 (treatment A) Condition Mild Severe Total Treatment B 15% 30% 16% t A n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B (5/50) (100/500) (105/550) T C Severe Causal Graph Y Motivating example: Simpson’s paradox 8 / 28 Brady Neal

  12. <latexit sha1_base64="WDTVIbXn/a2k0zCUek3TNQgock=">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</latexit> Simpson’s paradox: scenario 2 (treatment A) Condition Mild Severe Total Treatment A Treatment B 15% 30% 16% t A n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B (5/50) (100/500) (105/550) T C Mild Severe Causal Graph Y Motivating example: Simpson’s paradox 8 / 28 Brady Neal

  13. Motivating example: Simpson’s paradox Correlation does not imply causation Then, what does imply causation? Causation in observational studies Correlation does not imply causation 9 / 28 Brady Neal

  14. Correlation does not imply causation Correlation does not imply causation 10 / 28 Brady Neal

  15. Correlation does not imply causation Sleeping with shoes on is strongly correlated with waking up with a headache Correlation does not imply causation 11 / 28 Brady Neal

  16. Correlation does not imply causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before Correlation does not imply causation 11 / 28 Brady Neal

  17. Correlation does not imply causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 1. Shoe-sleepers differ from non-shoe- sleepers in a key way Correlation does not imply causation 11 / 28 Brady Neal

  18. Correlation does not imply causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 1. Shoe-sleepers differ from non-shoe- sleepers in a key way 2. Confounding Correlation does not imply causation 11 / 28 Brady Neal

  19. Correlation does not imply causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 1. Shoe-sleepers differ from non-shoe- sleepers in a key way 2. Confounding Correlation does not imply causation 11 / 28 Brady Neal

  20. Correlation does not imply causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 1. Shoe-sleepers differ from non-shoe- sleepers in a key way 2. Confounding Causal association Correlation does not imply causation 11 / 28 Brady Neal

  21. Correlation does not imply causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 1. Shoe-sleepers differ from non-shoe- sleepers in a key way 2. Confounding Total association (e.g. correlation): mixture of causal and confounding association Causal association Correlation does not imply causation 11 / 28 Brady Neal

  22. “Correlation = Causation” is a cognitive bias 1 Causal association 1 The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018) Correlation does not imply causation 12 / 28 Brady Neal

  23. “Correlation = Causation” is a cognitive bias 1 confounder Causal association 1 The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018) Correlation does not imply causation 12 / 28 Brady Neal

  24. “Correlation = Causation” is a cognitive bias 1 Availability heuristic (another cognitive bias) gives us confounder Causal association 1 The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018) Correlation does not imply causation 12 / 28 Brady Neal

  25. “Correlation = Causation” is a cognitive bias 1 Availability heuristic (another cognitive bias) gives us Why? confounder Causal association 1 The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018) Correlation does not imply causation 12 / 28 Brady Neal

  26. “Correlation = Causation” is a cognitive bias 1 Availability heuristic (another cognitive bias) gives us Motivated reasoning (another cognitive bias) Why? confounder Causal association 1 The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018) Correlation does not imply causation 12 / 28 Brady Neal

  27. “Correlation = Causation” is a cognitive bias 1 Availability heuristic (another cognitive bias) gives us Motivated reasoning (another cognitive bias) Why? explains confounder correlation = causation Causal association 1 The illusion of causality: A cognitive bias underlying pseudoscience (Blanco & Matute, 2018) Correlation does not imply causation 12 / 28 Brady Neal

  28. Nicolas Cage drives people to drown themselves ��������������������������������������������������� ����������������� ������������������������������ ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ������������� ������� ����������������������� ������������� ������� ������������� ������������� ������� ������������ ������� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ������������� ����������������������� �������������� https://www.tylervigen.com/spurious-correlations Correlation does not imply causation 13 / 28 Brady Neal

  29. Correlation does not imply causation Correlation does not imply causation 14 / 28 Brady Neal

  30. Then, what does imply causation?

  31. Motivating example: Simpson’s paradox Correlation does not imply causation Then, what does imply causation? Causation in observational studies Then, what does imply causation? 16 / 28 Brady Neal

  32. Potential outcomes: intuition Inferring the effect of treatment/policy on some outcome Then, what does imply causation? 17 / 28 Brady Neal

  33. Potential outcomes: intuition Inferring the effect of treatment/policy on some outcome Take pill Then, what does imply causation? 17 / 28 Brady Neal

  34. Potential outcomes: intuition Inferring the effect of treatment/policy on some outcome Take pill Don’t take pill Then, what does imply causation? 17 / 28 Brady Neal

  35. Potential outcomes: intuition Inferring the effect of treatment/policy on some outcome Take pill causal effect Don’t take pill Then, what does imply causation? 17 / 28 Brady Neal

  36. Potential outcomes: intuition Inferring the effect of treatment/policy on some outcome Take pill no causal effect Don’t take pill Then, what does imply causation? 17 / 28 Brady Neal

  37. <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> Potential outcomes: notation : observed treatment T do( T = 1) : observed outcome Y do( T = 0) Then, what does imply causation? 18 / 28 Brady Neal

  38. <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="ankFpkUc9naoITIsD9NjYr0YunY=">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</latexit> Potential outcomes: notation Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual do( T = 0) Then, what does imply causation? 18 / 28 Brady Neal

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=1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual Y i | do( T =0) do( T = 0) Then, what does imply causation? 18 / 28 Brady Neal

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sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> Potential outcomes: notation ) , Y i (1) Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) Y i | do( T =0) do( T = 0) Then, what does imply causation? 18 / 28 Brady Neal

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sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> Potential outcomes: notation ) , Y i (1) Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) ) , Y i (0) Y i | do( T =0) do( T = 0) Then, what does imply causation? 18 / 28 Brady Neal

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sha1_base64="wHBl5SOG0FiXEwPvpVB6ZgvKM3g=">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</latexit> Potential outcomes: notation ) , Y i (1) Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) ) , Y i (0) Y i | do( T =0) do( T = 0) Causal effect Y i (1) − Y i (0) Then, what does imply causation? 18 / 28 Brady Neal

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sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> Fundamental problem of causal inference Y i (1) = 1 : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) Y i (0) = 0 do( T = 0) Causal effect Y i (1) − Y i (0) = 1 Then, what does imply causation? 19 / 28 Brady Neal

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sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> Fundamental problem of causal inference Y i (1) = 1 : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i Counterfactual specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) Y i (0) = 0 do( T = 0) Causal effect Factual Y i (1) − Y i (0) = 1 Then, what does imply causation? 19 / 28 Brady Neal

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sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">AEp3icbVNtSxtBED5r2tr0TduP/bKtCApnmohgCw0IpRSqAWNghHZ25skS/aN3TltPI7+kX5t/1P/TecuZzXRPQ6enXnmfTZxSgZst/8uPFhsPHz0eOlJ8+mz5y9eLq+86gWbeQFHwirTxIeQEkDRyhRwYnzwHWi4DgZ75X64wvwQVpziBMHZ5oPjRxIwZFE58vLfYQfmKe2WD9kXdbeOF9ebfa1WF3QacGq1F9Ds5XFn/2UysyDQaF4iGcdtoOz3LuUQoFRbOfBXBcjPkQTgkariGc5VXqBVsjScoG1tNvkFXS2xY51yFMdEJMzXEU5nWl8D7daYaD2e5NC5DMGIaJAphpaVfWCp9CBQTQhw4SXlysSIey6QutWcCZPotlcY4dyfMVq2Wwe01pmREjkG6sAiNIMA7MOpZXdcmCZ4ErNvTcjUKLyF+yUPbATYdDwgMRzJQSI+V79KnkonfpKHEXcQ4hSE9dUoQ8y9t5chFlyJGrc0I8HEmNngyxZlATlWToid3mT0dn8ymkFYp6hjY1Ny84E5EYA63aY0IyblJUgDqAlJSTGcWVHJIT3rYATRdQclJIuQMxSzy9jpqWROtPsUqY4KhertUM+iqmps9Lgf1OWC+lpU65NpTHgqWuS2u3TYEHUqlpaoJ2hNYsdPOidpXI6SAhrf186qeuqUhxGkNRfUrZg3UbYcxowrOTRdBQPCQVlHRs1qfns0EqspstbUCZq7gcv6kvf3i7xfrl6S5PtFMavrcV9rvc7Ly5xemhQcMRx4x/pvy49VlzmeuSYai0S6h05PtjP/QO+C3lars936+H17dfdT/XiXojfRu2g96kQ70W70OTqIjiIRXUS/ot/Rn8ZG41uj1ziZUh8s1Davo5nT4P8Ayh6Fow=</latexit> Fundamental problem of causal inference Y i (1) = 1 : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i Factual specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) Y i (0) = 0 do( T = 0) Causal effect Counterfactual Y i (1) − Y i (0) = 1 Then, what does imply causation? 19 / 28 Brady Neal

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treatment effect (ITE): Y i (1) − Y i (0) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) : population-level potential outcome Y ( t ) Then, what does imply causation? 20 / 28 Brady Neal

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<latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">AElnicbVNbaxNBFN7aqDXeWn0RfBktBR+2MSmB6kOkUIoiPrTQGzSlzM6eJEPmxsxZY7os/gBf9cf5bzy72dom7SwL35znfuZxCkZsN3+u3RvuXH/wcOVR83HT54+e769uI42MwLOBJWX+a8ABKGjhCiQpOnQeuEwUnyXi31J98Bx+kNYc4dXCu+dDIgRQcSXQgL1bX2612dht0KnBelSf/Yu15Z/91IpMg0GheAhnbD85x7lEJB0exnARwXYz6EM4KGawjneZVpwTZIkrKB9fQbZJX0pkXOdQhTnRBTcxyFRV0pvEt3luHgw3kujcsQjJgFGmSKoWVl2SyVHgSqKQEuvKRcmRhxzwVSc5pzYRJdNJsb7FCOL1ktm89jVsucCIl8bRUAUZphYNah1PKyLlnwLHDFhp67UWgR+WsWyh646abjAYHhSAYK6bHyXfpUMvHcT/Mw4g5CnIKwvpciLn3dhJiwZWocUsD8ngMXY2yJFSVCepSNylzcZnc1vHOFHzDO0sbFp2ZmA3AhgvQ4TmnGTshLEAbSkhMQ4ruyIhPC+FXCqiJqDUtIFiFnq+SRmWhqpM80mMsUR69EGbZOPYmbqrDT435TlQnralCtTaQx4apnr0dp1KfBAKjVLTdCO0JqFXl7UrhI5GySktb9eXtVTV5XyMIK05oK6EfM6ypbDmHElh6anYEA4KOvIqFnNb5dGYjVF1po6QXM3MKkveX+vyPvl6iVJvlcU87pj7mut13l5WdBLk4IjhgPvWP9N+bHqsAzV0RjkUh30OnJdhYf6G1wvNXqdFsfD7rO9368a5Er6O30buoE21HO9GXaD86ikQE0a/od/Sn8arxqbHX+Dyj3luqbV5Gc6ex/w80lH/</latexit> Average treatment effect (ATE) : observed treatment T Individual treatment effect (ITE): Y i (1) − Y i (0) : observed outcome Y : used in subscript to denote a i specific unit/individual Average treatment effect (ATE): : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) E [ Y i (1) � Y i (0)] = E [ Y (1)] � E [ Y (0)] : population-level potential outcome Y ( t ) Then, what does imply causation? 20 / 28 Brady Neal

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Average treatment effect (ATE) : observed treatment T Individual treatment effect (ITE): Y i (1) − Y i (0) : observed outcome Y : used in subscript to denote a i specific unit/individual Average treatment effect (ATE): : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) E [ Y i (1) � Y i (0)] = E [ Y (1)] � E [ Y (0)] � � : population-level potential outcome Y ( t ) 6 = E [ Y | T = 1] � E [ Y | T = 0] Then, what does imply causation? 20 / 28 Brady Neal

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sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> Average treatment effect (ATE) : observed treatment T Individual treatment effect (ITE): Y i (1) − Y i (0) : observed outcome Y : used in subscript to denote a i specific unit/individual Average treatment effect (ATE): : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) E [ Y i (1) � Y i (0)] = E [ Y (1)] � E [ Y (0)] � � : population-level potential outcome Y ( t ) 6 = E [ Y | T = 1] � E [ Y | T = 0] C Recall: correlation does not imply causation T Y Causal association Then, what does imply causation? 20 / 28 Brady Neal

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sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> Average treatment effect (ATE) : observed treatment T Individual treatment effect (ITE): Y i (1) − Y i (0) : observed outcome Y : used in subscript to denote a i specific unit/individual Average treatment effect (ATE): : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) E [ Y i (1) � Y i (0)] = E [ Y (1)] � E [ Y (0)] � � : population-level potential outcome Y ( t ) 6 = E [ Y | T = 1] � E [ Y | T = 0] C Recall: correlation does not imply causation T Y Causal association Then, what does imply causation? 20 / 28 Brady Neal

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sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> Average treatment effect (ATE) : observed treatment T Individual treatment effect (ITE): Y i (1) − Y i (0) : observed outcome Y : used in subscript to denote a i specific unit/individual Average treatment effect (ATE): : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) E [ Y i (1) � Y i (0)] = E [ Y (1)] � E [ Y (0)] � � : population-level potential outcome Y ( t ) 6 = E [ Y | T = 1] � E [ Y | T = 0] C Recall: correlation does not imply causation T Y Causal association Then, what does imply causation? 20 / 28 Brady Neal

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<latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> Randomized control trials (RCTs) ATE when there is confounding: : observed treatment T : observed outcome Y E [ Y (1)] � E [ Y (0)] 6 = E [ Y | T = 1] � E [ Y | T = 0] : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) C : population-level potential outcome Y ( t ) T Y Causal association Then, what does imply causation? 21 / 28 Brady Neal

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sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> Randomized control trials (RCTs) ATE when there is confounding: : observed treatment T : observed outcome Y E [ Y (1)] � E [ Y (0)] 6 = E [ Y | T = 1] � E [ Y | T = 0] : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) C C : population-level potential outcome Y ( t ) T Y T Y Causal association Causal association Then, what does imply causation? 21 / 28 Brady Neal

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Randomized control trials (RCTs) ATE when there is confounding: : observed treatment T : observed outcome Y E [ Y (1)] � E [ Y (0)] 6 = E [ Y | T = 1] � E [ Y | T = 0] : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) C C : population-level potential outcome Y ( t ) RCTs: experimenter randomizes T Y T Y subjects into treatment group Causal association Causal association or control group 1. T cannot have any causal parents Then, what does imply causation? 21 / 28 Brady Neal

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Randomized control trials (RCTs) ATE when there is confounding: : observed treatment T : observed outcome Y E [ Y (1)] � E [ Y (0)] 6 = E [ Y | T = 1] � E [ Y | T = 0] : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) C C : population-level potential outcome Y ( t ) RCTs: experimenter randomizes T Y T Y subjects into treatment group Causal association Causal association or control group 1. T cannot have any causal parents 2. Groups are comparable Then, what does imply causation? 21 / 28 Brady Neal

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<latexit sha1_base64="ndfgSdpCNxkzlSHeFg0kf2fPuEo=">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</latexit> <latexit sha1_base64="uVLJKjW1XV4vsecLsVOJSqsUw0U=">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</latexit> Randomized control trials (RCTs) ATE when there is confounding: : observed treatment T : observed outcome Y E [ Y (1)] � E [ Y (0)] 6 = E [ Y | T = 1] � E [ Y | T = 0] : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) C C : population-level potential outcome Y ( t ) RCTs: experimenter randomizes T Y T Y subjects into treatment group Causal association Causal association or control group 1. T cannot have any causal ATE when there is no confounding (e.g. RCTs): parents E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] 2. Groups are comparable Then, what does imply causation? 21 / 28 Brady Neal

  57. Motivating example: Simpson’s paradox Correlation does not imply causation Then, what does imply causation? Causation in observational studies Causation in observational studies 22 / 28 Brady Neal

  58. <latexit sha1_base64="6nzlYT7Z3LXF3cG2BdYfPCoT2Q=">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</latexit> <latexit sha1_base64="2pM8GJC8It2zwHmHDEMBJXZBQYg=">AFiXicbVNtaxtHED4lVZsqfXHaj/kyrWOw4aJKxpCkQRAQhlJKSUF2HCRh9vZG0uJ9Y3cuinKo+Zj+xf6N/oLOnc6J5HiPg9mZ96fzbxWkXq9f1t37n7R/vKre1937n/z7Xf7z34Ty6Ikg8k067cJGJiFpZPCNFGi98QGEyja+yq2Flf/UGQ1TOjmjlcWrE3KqZkoJYdbn3yTDubIlqat3XkqAq47wOcA/nQ5xlqeWBbHkQThFA6HRyAIDntp7wjKR8NH6+e3gEZHM5QuyVonBEM3AyGU4aPboe/gPar7Ywr+u8E1Bp/n8uiAvaFXgqyr3J9v60fb+gnafKe9y739XrdXH/hc6DfCftKcl5cP7r6f5E4WBi1JLWIc93uepqUIpKTmeU2KiF7IKzHMYtWGIzTsl7OGg5Yk8PMBf4tQa3d9iFiXFlMkYarj7etFXK2zjgmZPp6WyviC0cpNoVmgB9WmIVcBJekVC0IGxbWCXIgJDEfOjtpMrPudA5gxFOCRrdbx6aXHVU10k9eEYmUnUdwnpR75qWpSi0DAPwi9il8G/F7GagV89iISAi1U5JSB6thVTK2yIMKqjAvhMaY5ShdqsZUhOCWMZVCy0buGiSRzhSl3kVobgIrMKxOHKmhWP/2CSvU1FQS6tWMcDYd5ZiTDogzQgbA6VkEY0iguSV2ntV5Pzl26klWZoiVorHzGFPIhlCkZQoDS5Uz5wbMoCcY71x9U5Z+ugKpVSBmXLtqzFwCPzA6bdCSeKa03pUnmCNMsDsp1EypTm0Vi3sQblHU/TVe5iAvMGyzqrZyfshx7SkFoNbeD6jWmELXz7NSp9zfklTjDmY3hSfDeLS6bSzk5XZeTinpZVp6u17u2cxEazBldblhVzZHzwiPwcPkp+qD+nIDZ6+B1hGDboHzk+3fKCfC+fH3f5J9lfx/svTprHey95mPycHCb95EnyIvkteZmcJbJ10fq79aH1T/t+u9+2v51A73Tanx+THZOe/g/0UHIeA=</latexit> Observational studies Ideal C T Y Causal association Observational studies C T Y Causal association Causation in observational studies 23 / 28 Brady Neal

  59. <latexit sha1_base64="2pM8GJC8It2zwHmHDEMBJXZBQYg=">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</latexit> <latexit sha1_base64="6nzlYT7Z3LXF3cG2BdYfPCoT2Q=">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</latexit> Observational studies Ideal Can’t always randomize treatment C T Y Causal association Observational studies C T Y Causal association Causation in observational studies 23 / 28 Brady Neal

  60. <latexit sha1_base64="6nzlYT7Z3LXF3cG2BdYfPCoT2Q=">AFonicbVNtbxNHED4T3AYXWlI+8mUgiZRIh7EjJFqBJSQrUoUQAtUJRD4r2tsb26vsW3f3MOZ04mfwtf1Z/TfMnS8QB+/pNmZ96fTa0UPvR6/7dubN1s/Tz9q3OL7fv/Prb3Z3fT73JHcTbqRx71PmUQqNJ0Eie+tQ6ZSie/Si2Flf/cBnRdGj8LS4kSxmRZTwVkg1flOq5OkOBO6COLikxU85A7LDtDZh9cmQ1/LiSZx7AMLOIGD4SGwAe9uHcIxd5wr3y2ATQ6hHGK0ixA4jTAwExhOCH4aDP87Bvcidn8Cv6swjcFHWezy4IsC/O6EiRdle3ZRv3Zun50VZ+gztbaPr+72+v26gM/Cv1G2I2a8+Z8Z+tzkhmeK9SBS+b9uN+zYVIwFwSXNMck92gZv2AzHJOomUI/KeqtlbBPmgymxtGvA9Taqx4FU94vVUpIRdX767ZKuck2zsP0j0khtM0Dar5KNM0lBAMVBSATDnmQSxIYd4JqBT5njvFAROmspUlV2ensw4imBI1uvY5VL2uqaqTfvTyGIPTMg7FBKPGpaZmz3DMJM8fs3HcJ/DL31Qzs8pFlPiCEufCU0oU6dhVTitQxtyz8nFn0cYbcuJrFPmbOmYWPOZO8kbsKA4unIsTWeFGhqAiqswpE4YqaFY9eEfk+xiwPJq7YSAMhPmqOMOgDV8B0BpUQe1SCuIXce1Xk/Zx14elJGiBUgrMYbMsUMSmihcgULkRHnBsSgpxSjXLlaI3T45goF46YcukqtEZHI7MDot0TSjwVUq5K48QRopkfFGUTKhWrRWLWxBsUdT9NVxnzc8waLMorOb9nObIhBibFTA+qVxqDl8aSU6fe35BWYhRlVomQXvXuGguRXJcFklFvTQtjsty3XbKXGN1qgu1+xCZ2gJYdFZSB5UH9SXazh9CdQmEGgDnJ5s/oD/VE4Per2n3T/fHu0+J583i3o/vRw+g6kdPoxfRX9Gb6CTirX9aX1r/tv5r7Vft+2/15Bb7Qan3vR2mknXwGKxs53</latexit> <latexit sha1_base64="2pM8GJC8It2zwHmHDEMBJXZBQYg=">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</latexit> Observational studies Ideal Can’t always randomize treatment C • Ethical reasons (e.g. unethical to randomize people to smoke for T Y measuring effect on lung cancer) Causal association Observational studies C T Y Causal association Causation in observational studies 23 / 28 Brady Neal

  61. <latexit sha1_base64="2pM8GJC8It2zwHmHDEMBJXZBQYg=">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</latexit> <latexit sha1_base64="6nzlYT7Z3LXF3cG2BdYfPCoT2Q=">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</latexit> Observational studies Ideal Can’t always randomize treatment C • Ethical reasons (e.g. unethical to randomize people to smoke for T Y measuring effect on lung cancer) Causal association • Infeasibility (e.g. can’t randomize Observational countries into communist/capitalist studies systems to measure effect on GDP) C T Y Causal association Causation in observational studies 23 / 28 Brady Neal

  62. <latexit sha1_base64="6nzlYT7Z3LXF3cG2BdYfPCoT2Q=">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</latexit> <latexit sha1_base64="2pM8GJC8It2zwHmHDEMBJXZBQYg=">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</latexit> Observational studies Ideal Can’t always randomize treatment C • Ethical reasons (e.g. unethical to randomize people to smoke for T Y measuring effect on lung cancer) Causal association • Infeasibility (e.g. can’t randomize Observational countries into communist/capitalist studies systems to measure effect on GDP) C • Impossibility (e.g. can’t change a living person’s DNA at birth for measuring T Y effect on breast cancer) Causal association Causation in observational studies 23 / 28 Brady Neal

  63. How do we measure causal effects in observational studies?

  64. <latexit sha1_base64="RGeNzbGahBlVlAS3Mmu4WXVfuH4=">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</latexit> Solution: adjust/control for confounders C T M Y Causal association Causation in observational studies 25 / 28 Brady Neal

  65. <latexit sha1_base64="29OrfUGpP2tF9xLa5KD4vhmSkPs=">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</latexit> <latexit sha1_base64="RGeNzbGahBlVlAS3Mmu4WXVfuH4=">AF2HicbVNbaxtHF4lVZuqt7h97Mu0jsGjSIZQxqCwCAMpVQlBdmJKwkzO3skDZ4bM2ejKsvSvpW+9k/01/S1pf+mZ1brxnI0y8KZ73znMme+yZySAXu9f1v37r/Xfv+DBx92Pvr4k08/e7j3+UWwhRdwLqy/lXGAyhp4BwlKnjlPHCdKXiZXQ+j/+Vr8EFaM8a1g5nmCyPnUnAk6GqvdTrNYCFNifL6jZMCw9Vh9E6YD/YHEJtTw2Zk4AcYcYOh0eMIzvspb0jVj4aPqe7yCNjtgkA2VXbGDnbDgj5mg3c0xMBXOsiaNIHO8mXhLRy8XyFvMyMpt2z/LFTbuO47LuEwiLFZ7vxC+38fENPtrGR7f5UzD51rCuHu73ur16sXeNfmPsJ816cbV3/5dpbkWhwaBQPIRJv+dwVnKPUia/rQI4Li45guYkGm4hjAr67u2AEhOZtbT79BVqO3I0quQ1jrjJiaug93fRHc5ZsUOP9mVkrjCgQjNoXmhWJoWRQOy6UHgWpNBhdeUq9MLnAklena0yma46nQM2pimxBtvuY3OWLSiO9G1UAERpFoFZh1LN82RBS8CV2zhuVuGLpG/K0KcgVs/djwgMFzKQCU91rljTiUz/26DEvuIKQ5COtr7YeUe29XIRVcicbuakCeziWmzgYZWdQE9RkTUbqyVsXj70mOP6e8QJtGfdJASKFGABv0mdCMm5xFIw2gJTUkrtM6rpbxk27AtSJqCUpJFyBluerlGlpC40W8mcNDcgBT2lHNUm1Flp8P9QVgrpSk3odIY8DQyNyDZnVDhuVRq05ogjZDMwqCsmlSZ3Fwk5E2+QVmfpzlVzsMS8oYL6lbNt1WOHaMK7kwg/huUxaUdRTUqe9vSFdiNVXWmiZB925g1WzK6VlVTqP0sqw8q6pt3wX3jdfrMm7u+KXJwRHDgXds+lX8WL25wzM3RGORSDvo9GT7dx/ou8bFcbd/0n324/H+6UnzeB8kXyZfJ4dJP3manCbfJi+S80S0/mz91fq79U/7p/av7d/av2+o91pNzBfJ1mr/8R9CF969</latexit> Solution: adjust/control for confounders Adjust/control for the right variables . W C T M Y Causal association Causation in observational studies 25 / 28 Brady Neal

  66. <latexit sha1_base64="29OrfUGpP2tF9xLa5KD4vhmSkPs=">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</latexit> <latexit sha1_base64="RGeNzbGahBlVlAS3Mmu4WXVfuH4=">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</latexit> <latexit sha1_base64="wd+vx2usHaSjfhLEk/lCQTkN61U=">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</latexit> <latexit sha1_base64="SvIFhPpjgdYoT5MviGQldEUfN8=">AEw3icbVNbaxNBFN62UWu8VX30ZbQIFbYxKQUVDAgSEPGhQm+ShDI7e5IMmRszZ43pdvE/+Gt81X/hv/HsZmtNdJbdPZfv3M8kTsmA7favtfWNxrXrNzZvNm/dvnP3tb9B8fBZl7AkbDK+tOEB1DSwBFKVHDqPHCdKDhJpm9L/cln8EFac4hzB0PNx0aOpOBIorOteNDrf7oYIHzBPLXFziHrMnwWsxP6z4b0KfXsgmFM7NnWdrvVrg7l+jUxHZUn4Oz+xtfB6kVmQaDQvEQ+p2w2HOPUqhoGgOsgCOiykfQ59IwzWEYV6VbCnJEnZyHp6DbJK+rdFznUIc50QUnOchFVdKfyfrp/h6OUwl8ZlCEYsAo0yxdCyskcslR4EqjkRXHhJuTIx4Z4LpE42l8Ikumg2n7JDOT1ntWw5j0UtSyIk8JVAERpxoFZh1L87pkwbPAFRt7iahReD3WSh74Oa7jgcEhMZKTHynfpU8nEcz/Pw4Q7CHEKwvpqzCHm3tZiAVXoqZbGpDHI4mxs0GWKEqC8iwdkbu8yejsfuC0GTHP0MbGpmVnAnIjgHU7TGjGTcpKIg6gJSUkpnFlRyCE562Ac0XQHJSLkDMUs9nMdPSJ1pNpMpTmjF2q0X5KNYmDorDf4xZbmQnjbl0lQaA5a5rq0dvsUeCSVWqQmaEdozUI3L2pXiVwMEtLaXzev6qmrSnmYQFpjQf0V8yrKnqPF50qOTVfBiOigrCOjZjW/tzQSqymy1tQJmruBWc3kg16RD8rVS5K8VxTLumPua63Xecms6KVJwRHCgXds8Lh8WMWs4Mwl0Fgk0H/gdGU7qxf0X+J4r9XZb736uL/95nV9eTejR9GTaCfqRC+iN9G76CA6ikT0Lfoe/Yh+NnqNacM3cAFdX6tHkZLp1H8BnTZjuY=</latexit> <latexit sha1_base64="29OrfUGpP2tF9xLa5KD4vhmSkPs=">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</latexit> Solution: adjust/control for confounders Adjust/control for the right variables . W If is a sufficient adjustment set, we have W E [ Y ( t ) | W = w ] , E [ Y | do( T = t ) , W = w ] = E [ Y | t, w ] C T M Y Causal association Causation in observational studies 25 / 28 Brady Neal

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right variables . W If is a sufficient adjustment set, we have W E [ Y ( t ) | W = w ] , E [ Y | do( T = t ) , W = w ] = E [ Y | t, w ] C T M Y Causal association Causation in observational studies 25 / 28 Brady Neal

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sha1_base64="29OrfUGpP2tF9xLa5KD4vhmSkPs=">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</latexit> Solution: adjust/control for confounders Adjust/control for the right variables . W If is a sufficient adjustment set, we have W E [ Y ( t ) | W = w ] , E [ Y | do( T = t ) , W = w ] = E [ Y | t, w ] E [ Y ( t )] , C E [ Y | do( T = t )] = E W E [ Y | t, W ] T M Y Causal association Causation in observational studies 25 / 28 Brady Neal

  69. <latexit sha1_base64="Zf7+Iu3daDl81Gj/9ZAxRda9yw=">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</latexit> <latexit sha1_base64="RGeNzbGahBlVlAS3Mmu4WXVfuH4=">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</latexit> Solution: backdoor adjustment E [ Y | do( T = t )] = E W E [ Y | t, W ] C T M Y Causal association Causation in observational studies 26 / 28 Brady Neal

  70. <latexit sha1_base64="Zf7+Iu3daDl81Gj/9ZAxRda9yw=">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</latexit> <latexit sha1_base64="8lMEwfbHG0mova6DtPTZwP8voxw=">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</latexit> Solution: backdoor adjustment E [ Y | do( T = t )] = E W E [ Y | t, W ] C T M Y Causal association Causation in observational studies 26 / 28 Brady Neal

  71. <latexit sha1_base64="Zf7+Iu3daDl81Gj/9ZAxRda9yw=">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</latexit> <latexit sha1_base64="8lMEwfbHG0mova6DtPTZwP8voxw=">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</latexit> <latexit sha1_base64="29OrfUGpP2tF9xLa5KD4vhmSkPs=">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</latexit> Solution: backdoor adjustment E [ Y | do( T = t )] = E W E [ Y | t, W ] Shaded nodes are examples of sufficient adjustment sets W C T M Y Causal association Causation in observational studies 26 / 28 Brady Neal

  72. <latexit sha1_base64="ox3/eARyg5DmkB3v3xif3G1icUY=">AG03icdVRtaxtHEL4orZqb3H7MV+2dQw2XFSdbEiKEQSEoZSqpCBbMZIwe3sjafG+sbsXRTmOQsjX/on+mn5tv/XfdO50sn2KvMYwN8/zMzOzig2gjvf6fz3oPHwk0+bnz36vPXFl19/c3jvW8vnE4tg3OmhbavY+pAcAXnsBr40FKmMBo/i6X+CjN2Ad12roVwamks4Vn3FGPbqu9hqnkxjmXGWeX78znPnUQt4ieA7IbzoBV9oThebYephSg5H3SNCPTnshJ0jkj0dXWf5qe7eNERGcg9JImHnS0zMy6k5LSXSP5PhGYvl8Udc79b0byS35P5u6qBG7RfMwW7mEJk3RQ8K4nA38RKJt6WzMuCWfXwLJlvemioX6ybB+gs23N6D3K8hUQbZLgFHG+AyzrQv0fQv4c/3PgHdf/gLn8CKqnNydXj/U67Ux7ysRFVxn5QnVdXew/mCSapRKUZ4I6N46xk8zaj1nAgdvkjowlF3TOYzRVFSCm2blmOfkAD0JmWmL/8qT0ntXkVHp3ErGyJRYvdvGCucubJz62YtpxpVJPSi2TjRLBfGaFDtDEm6BebFCgzLsVbCFtRS5nGzWrU0scxbrQMyxC6RylevY32Xmqto6a3KgfdczR3RxnPJ31VXZjR1VJC5pWbh2kj+JXVFD8zqmaHOA/EL7jCl9WXsIqbgsaV2lbkFNeDCBJi25dq7kFqrly5kVLDKbkvwNJxHxrteMHCIrDOIhCGy8qpePYrDv3bkKZeh8UWYENwDxQD0osIk4SqhBRG6EByLIhdh6WuXJYf286vBFIzEIbByFJLF2GRHLFZSrJkic4cz2coOcYI19LjebK30hJxrjFSdlIuVJgsWmh2N3golnXIh1aQxnBMfM9bK8ChXz9UNCUsXrZeV9qlsl1C0gqbg7uS8zdI1PiRU8LnqFb8OIXFCGxS1yvfr45NoiZmlxE7guytYVh/Z5CzPJsXoxXF2lud17ILaCrUyKz62cK4SMgwYA2ZfF/8kfJji6c2RKU9knbQcWj7QX92LjotqOT9k+/d/dfnlTL+yh4EvwQHAZR8Dx4GfwcvArOA9b4q/F345/Gv83zZtZ83/ywpjYeVJrvgtp/vk/D04lLg=</latexit> <latexit sha1_base64="Zf7+Iu3daDl81Gj/9ZAxRda9yw=">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</latexit> <latexit sha1_base64="8lMEwfbHG0mova6DtPTZwP8voxw=">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</latexit> <latexit sha1_base64="29OrfUGpP2tF9xLa5KD4vhmSkPs=">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</latexit> Solution: backdoor adjustment E [ Y | do( T = t )] = E W E [ Y | t, W ] Shaded nodes are examples of sufficient adjustment sets W W 2 C W 1 W 3 C T M Y Causal association T M Y Causal association Causation in observational studies 26 / 28 Brady Neal

  73. <latexit sha1_base64="Z7rlzBpTi+dPmukn+wKc7khHqvg=">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</latexit> <latexit sha1_base64="Zf7+Iu3daDl81Gj/9ZAxRda9yw=">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</latexit> <latexit sha1_base64="8lMEwfbHG0mova6DtPTZwP8voxw=">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</latexit> <latexit sha1_base64="29OrfUGpP2tF9xLa5KD4vhmSkPs=">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</latexit> Solution: backdoor adjustment E [ Y | do( T = t )] = E W E [ Y | t, W ] Shaded nodes are examples of sufficient adjustment sets W W 2 C W 1 W 3 C T M Y Causal association T M Y Causal association Causation in observational studies 26 / 28 Brady Neal

  74. <latexit sha1_base64="51Yegy1qsnh/tCWQIBxk6KQHY=">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</latexit> <latexit sha1_base64="Zf7+Iu3daDl81Gj/9ZAxRda9yw=">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</latexit> <latexit sha1_base64="8lMEwfbHG0mova6DtPTZwP8voxw=">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</latexit> <latexit sha1_base64="29OrfUGpP2tF9xLa5KD4vhmSkPs=">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</latexit> Solution: backdoor adjustment E [ Y | do( T = t )] = E W E [ Y | t, W ] Shaded nodes are examples of sufficient adjustment sets W W 2 C W 1 W 3 C T M Y Causal association T M Y Causal association Causation in observational studies 26 / 28 Brady Neal

  75. <latexit sha1_base64="RTOfT2kT5YRxU5DAqe+VA6VyPZ0=">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</latexit> <latexit sha1_base64="Zf7+Iu3daDl81Gj/9ZAxRda9yw=">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</latexit> <latexit sha1_base64="8lMEwfbHG0mova6DtPTZwP8voxw=">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</latexit> <latexit sha1_base64="29OrfUGpP2tF9xLa5KD4vhmSkPs=">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</latexit> Solution: backdoor adjustment E [ Y | do( T = t )] = E W E [ Y | t, W ] Shaded nodes are examples of sufficient adjustment sets W W 2 C W 1 W 3 C T M Y Causal association T M Y Causal association Causation in observational studies 26 / 28 Brady Neal

  76. <latexit 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Solution: backdoor adjustment E [ Y | do( T = t )] = E W E [ Y | t, W ] Shaded nodes are examples of sufficient adjustment sets W W 2 Z 1 Z 3 C W 1 W 3 Z 2 C T M Y Causal association T M Y T M Y Causal association Causal association Causation in observational studies 26 / 28 Brady Neal

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<latexit sha1_base64="vIo63CkJuJpAQihvFztsK1i5sk=">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</latexit> Application to the COVID-27 example Causal Graph X E [ Y | do( T = t )] = E C E [ Y | t, C ] = E [ Y | t, c ] P ( c ) C c Condition T Y Mild Severe Total Causal 15% 30% 16% t A 19.4% n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B 12.9% (5/50) (100/500) (105/550) E [ Y | do( t )] E [ Y | t, C = 0] E [ Y | t, C = 1] E [ Y | t ] Causation in observational studies 27 / 28 Brady Neal

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<latexit sha1_base64="vIo63CkJuJpAQihvFztsK1i5sk=">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</latexit> Application to the COVID-27 example Causal Graph X E [ Y | do( T = t )] = E C E [ Y | t, C ] = E [ Y | t, c ] P ( c ) C c Condition T Y Mild Severe Total Causal 15% 30% 16% t A 19.4% n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B 12.9% (5/50) (100/500) (105/550) E [ Y | do( t )] E [ Y | t, C = 0] E [ Y | t, C = 1] E [ Y | t ] Causation in observational studies 27 / 28 Brady Neal

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<latexit sha1_base64="vIo63CkJuJpAQihvFztsK1i5sk=">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</latexit> Application to the COVID-27 example Causal Graph X E [ Y | do( T = t )] = E C E [ Y | t, C ] = E [ Y | t, c ] P ( c ) C c Condition T Y Mild Severe Total Causal 15% 30% 16% t A 19.4% n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B 12.9% (5/50) (100/500) (105/550) E [ Y | do( t )] E [ Y | t, C = 0] E [ Y | t, C = 1] E [ Y | t ] Causation in observational studies 27 / 28 Brady Neal

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<latexit sha1_base64="vIo63CkJuJpAQihvFztsK1i5sk=">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</latexit> Application to the COVID-27 example Causal Graph X E [ Y | do( T = t )] = E C E [ Y | t, C ] = E [ Y | t, c ] P ( c ) C c Condition T Y Mild Severe Total Causal 15% 30% 16% t A 19.4% n e (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% r T B 12.9% (5/50) (100/500) (105/550) E [ Y | do( t )] E [ Y | t, C = 0] E [ Y | t, C = 1] E [ Y | t ] Causation in observational studies 27 / 28 Brady Neal

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Application to the COVID-27 example Causal Graph X E [ Y | do( T = t )] = E C E [ Y | t, C ] = E [ Y | t, c ] P ( c ) C c Condition T Y Mild Severe Total Causal 15% 30% 16% 1450 + 600 t A 19.4% 0.15 0.30 ≈ 0.194 n e 2050 2050 (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% 1450 + 600 r T B 12.9% 0.10 0.20 ≈ 0.129 (5/50) (100/500) (105/550) 2050 2050 E [ Y | do( t )] E [ Y | t, C = 0] E [ Y | t, C = 1] E [ Y | t ] Causation in observational studies 27 / 28 Brady Neal

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Application to the COVID-27 example Causal Graph X E [ Y | do( T = t )] = E C E [ Y | t, C ] = E [ Y | t, c ] P ( c ) C c Condition T Y Mild Severe Total Causal 15% 30% 16% 1450 + 600 t A 19.4% 0.15 0.30 ≈ 0.194 n e 2050 2050 (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% 1450 + 600 r T B 12.9% 0.10 0.20 ≈ 0.129 (5/50) (100/500) (105/550) 2050 2050 E [ Y | do( t )] E [ Y | t, C = 0] E [ Y | t, C = 1] E [ Y | t ] Causation in observational studies 27 / 28 Brady Neal

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Application to the COVID-27 example Causal Graph X E [ Y | do( T = t )] = E C E [ Y | t, C ] = E [ Y | t, c ] P ( c ) C c Condition T Y Mild Severe Total Causal 15% 30% 16% 1450 + 600 t A 19.4% 0.15 0.30 ≈ 0.194 n e 2050 2050 (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% 1450 + 600 r T B 12.9% 0.10 0.20 ≈ 0.129 (5/50) (100/500) (105/550) 2050 2050 E [ Y | do( t )] E [ Y | t, C = 0] E [ Y | t, C = 1] E [ Y | t ] Causation in observational studies 27 / 28 Brady Neal

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Application to the COVID-27 example Causal Graph X E [ Y | do( T = t )] = E C E [ Y | t, C ] = E [ Y | t, c ] P ( c ) C c Condition T Y Mild Severe Total Causal 15% 30% 16% 1450 + 600 t A 19.4% 0.15 0.30 ≈ 0.194 n e 2050 2050 (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% 1450 + 600 r T B 12.9% 0.10 0.20 ≈ 0.129 (5/50) (100/500) (105/550) 2050 2050 E [ Y | do( t )] E [ Y | t, C = 0] E [ Y | t, C = 1] E [ Y | t ] Causation in observational studies 27 / 28 Brady Neal

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Application to the COVID-27 example Causal Graph X E [ Y | do( T = t )] = E C E [ Y | t, C ] = E [ Y | t, c ] P ( c ) C c Condition T Y Mild Severe Total Causal 15% 30% 16% 1450 + 600 t A 19.4% 0.15 0.30 ≈ 0.194 n e 2050 2050 (210/1400) (30/100) (240/1500) m t a e 10% 20% 19% 1450 + 600 r T B 12.9% 0.10 0.20 ≈ 0.129 (5/50) (100/500) (105/550) 2050 2050 E [ Y | do( t )] E [ Y | t, C = 0] E [ Y | t, C = 1] E [ Y | t ] Causation in observational studies 27 / 28 Brady Neal

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