potential outcomes
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Potential Outcomes Brady Neal causalcourse.com What are potential - PowerPoint PPT Presentation

Potential Outcomes Brady Neal causalcourse.com What are potential outcomes? The fundamental problem of causal inference Getting around the fundamental problem of causal inference A complete example with estimation 2 / 41 Brady Neal What


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T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual do( T = 0) 5 / 41 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) 5 / 41 Brady Neal

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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) 5 / 41 Brady Neal

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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) 5 / 41 Brady Neal

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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) 5 / 41 Brady Neal

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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) 5 / 41 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 : 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) 5 / 41 Brady Neal

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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) 5 / 41 Brady Neal

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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) 5 / 41 Brady Neal

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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 5 / 41 Brady Neal

  11. What are potential outcomes? The fundamental problem of causal inference Getting around the fundamental problem of causal inference A complete example with estimation 6 / 41 Brady Neal

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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? 7 / 41 Brady Neal

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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? 7 / 41 Brady Neal

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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? 7 / 41 Brady Neal

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observed outcome Y : used in subscript to denote a i specific unit/individual Y (1) Y (0) Y (1) − Y (0) i T Y : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) 1 0 0 ? 0 ? 2 1 1 1 ? ? 3 1 0 0 ? ? 4 0 0 ? 0 ? 5 0 1 ? 1 ? 6 1 1 1 ? ? 8 / 41 Brady Neal

  16. Question: What is the fundamental problem of causal inference?

  17. What are potential outcomes? The fundamental problem of causal inference Getting around the fundamental problem of causal inference A complete example with estimation 10 / 41 Brady Neal

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sha1_base64="bg29vTMGqhBy203y1BQJ/yjmYZM=">AEz3icbVNtaxNBEN7W+BbfWv3ol9VSaOEaEylUwYAgiogfFGxVmlD29ibJkn1jd84YjxO/+hv8quBf8t84d7mqid3j4JmZ152Zjb1WkXsdn+trZ9rnb9w8dLl9pWr167f2Ni8eRdHiQcSqdeJeKCFpZOESFGt75AMKkGt6m0yeV/e0HCFE5+wbnHoZGjK0aKSmQVCcbN96fqJ3eLt/jFejunmxsdTvd+vD/Qa8BW6w5r042z/0cZE7mBixKLWI87nU9DgsRUEkNZXuQR/BCTsUYjglaYSAOi7rykm+TJuMjF+i3yGvtvx6FMDHOTUpMI3ASV2V8izbcY6jB8NCWZ8jWLlINMo1R8erNvBMBZCo5wSEDIpq5XIigpBIzWovpUlNuSw7N0WRxpLzbf6MSrdKAieNBvLc5m/U9BNvyMsFLy69pEIil3+8IiAqO47ceVRGfWp6I0UehebjIPwkdoj8Io9Vs/x8z4uIlHyiIqUMWMeuYmqVBhHmRZwIDzHJQLpQjzwmIgQ3i4kUWja4YwBFMlKYeBdVxaIiqM4qEIUr2pzO3kuB8DERObrEuqxqYURh6er9HpeGC5vxCiQRjKC5DSp/YiEcK8Tca6JWoDWykdIeBbELOFGWVyw2cqwnv06odUIxy4eqdsvjHlRdSBVqpU1dlLQRqme/Tfu5T4pHSelGapGWifYz9omxCpWoxciaeP2ivk9zq0zECWQNF/Q/Of9mue8x4UKrse1rGBGO2nlyatfze0IjcYyG0OdoLlbmDVCMXhaFoNqR9O0eFqWy7YjERprMEUlrNiVzcATw0PwfHCn+ngtrPDsKdE6JNIZdHrbvdWX/D84ut/p7Xcevt7fevyoeWX2G12l+2wHjtgj9lz9odMsly9o19Zz9ar1uz1ufWlwV1fa3xucWTuvrb3Urk9U=</latexit> Average treatment effect (ATE) Y i (1) − Y i (0) Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? : observed treatment T 2 1 1 1 ? ? : observed outcome Y 3 1 0 0 ? ? : used in subscript to denote a i specific unit/individual 4 0 0 ? 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 ? 1 ? Y i (0) 6 1 1 1 ? ? 11 / 41 Brady Neal

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sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> Average treatment effect (ATE) E [ Y i (1) − Y i (0)] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? : observed treatment T 2 1 1 1 ? ? : observed outcome Y 3 1 0 0 ? ? : used in subscript to denote a i specific unit/individual 4 0 0 ? 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 ? 1 ? Y i (0) 6 1 1 1 ? ? 11 / 41 Brady Neal

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(ATE) E [ Y (1) − Y (0)] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? : observed treatment T 2 1 1 1 ? ? : observed outcome Y 3 1 0 0 ? ? : used in subscript to denote a i specific unit/individual 4 0 0 ? 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 ? 1 ? Y i (0) 6 1 1 1 ? ? 11 / 41 Brady Neal

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E [ Y (0)] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? : observed treatment T 2 1 1 1 ? ? : observed outcome Y 3 1 0 0 ? ? : used in subscript to denote a i specific unit/individual 4 0 0 ? 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 ? 1 ? Y i (0) 6 1 1 1 ? ? 11 / 41 Brady Neal

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[ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? : observed treatment T 2 1 1 1 ? ? : observed outcome Y 3 1 0 0 ? ? : used in subscript to denote a i specific unit/individual 4 0 0 ? 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 ? 1 ? Y i (0) 6 1 1 1 ? ? 11 / 41 Brady Neal

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treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? 11 / 41 Brady Neal

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sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? 2 / 3 11 / 41 Brady Neal

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treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? 2 / − 1 / 3 3 11 / 41 Brady Neal

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sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? / 2 / − 1 / / − 1 / 3 − 3 3 3 = 3 11 / 41 Brady Neal

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(ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? / 2 / − 1 / / − 1 / 3 − 3 3 3 = 3 11 / 41 Brady Neal

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sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? / 2 / − 1 / / − 1 / 3 − 3 3 3 = 3 11 / 41 Brady Neal

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(ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] associational difference Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? / 2 / − 1 / / − 1 / 3 − 3 3 3 = 3 11 / 41 Brady Neal

  30. Association is not causation Sleeping with shoes on is strongly correlated with waking up with a headache 12 / 41 Brady Neal

  31. Association is not causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 12 / 41 Brady Neal

  32. Association is not causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 1. Confounding 12 / 41 Brady Neal

  33. <latexit sha1_base64="uGwb7/40FtxyqwMkvjDesmjJa3g=">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</latexit> Association is not causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 1. Confounding X T Y 12 / 41 Brady Neal

  34. <latexit sha1_base64="uGwb7/40FtxyqwMkvjDesmjJa3g=">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</latexit> Association is not causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 1. Confounding 2. Shoe-sleepers differ from non-shoe- sleepers in a key way X T Y 12 / 41 Brady Neal

  35. <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> <latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">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</latexit> Why? Because the groups are not comparable E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) 13 / 41 Brady Neal

  36. <latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">AE+XicbVNLaxRBEJ4k62t9RT16aQ2BCJN1NwRiwAUhREQ8RMhLskvo6andbZfdNcY12F+i3gLXv0NXvXsv7FmdvLYmB4Gvqr6tnViVMyYLv9d25+oXHj5q3bd5p3791/8HDx0eP9YDMvYE9YZf1hwgMoaWAPJSo4dB64ThQcJOt0n7wGXyQ1uzixEFf86GRAyk4kup4cbO3fRpfOiz1ZBdsEuxVkPS1TtktS58x6oWr3jxeX2q12dj/oFODpag+O8ePFk57qRWZBoNC8RCOm2H/Zx7lEJB0exlARwXYz6EI4KGawj9vOqxYMukSdnAevoNskp72SPnOoSJToipOY7CVupvM52lOHgVT+XxmUIRkwTDTLF0LJyYCyVHgSqCQEuvKRamRhxzwXSWJszaRJdzMrWjpEnoWBsmb2l0o0UwEijgDyX2a4cf2U1ebgadMzKiRyce4VAFGaYWDWodTyaz0bwbPAFRt67kahReT3WSiH5Sarjgek5CMZKXHKnYZU8nEcz/Jw4g7CHEKwvpqOULMvbcnIRZciRq3NCPBxJjZ4MsWVQE1VkGonB5k9FZ/cARvsQ8Qxsbm5YjDMgNtd7tMKEZNykrQRxASypIjOPKj0gIL1sBJ4qoOSglXYCYpZ6fxExLI3Wm2YlMcVRuYGuDYhRTV2elwXNXlgvpaXOXKUx4Glkrkv7uU6JB1KpaWmClon2MXTzog6VyOmNQ1rH6+ZVP3VXKQ8jSGsuqEs5L7KsOYwZV3JougoGhIOyjpya1f1t0ZVYTZm1pknQvRs4qYW8t13kvXJHkyTfLopZ2z73tdXrvBSu2KVJwRHDgXes96z8WCVc4ZkzorFIpGvo9LY7V1/y/2B/rdVZb21+XF9687p+5bejp9HzaCXqRBvRm+hdtBPtRSL6Fv2Kfkd/Gnje+O08WNKnZ+rfZ5EM6fx8x/+yKCY</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> Why? Because the groups are not comparable E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober sober drunk drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 13 / 41 Brady Neal

  37. <latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> Why? Because the groups are not comparable E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober sober drunk drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 13 / 41 Brady Neal

  38. <latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> Why? Because the groups are not comparable E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober sober drunk drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 13 / 41 Brady Neal

  39. <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">AE7nicbVRLaxRBEJ4kq8b1lejRS2sIRJisuyEQBRcCISLiIUJekA2hp6d2t9l+0V1jshnmR3jxoIhX/45H/401s5OYTdLDQHXV91V9XV0ziVMyYLv9d2Z2rnHn7r35+80HDx89frKw+HQ/2MwL2BNWX+Y8ABKGthDiQoOnQeuEwUHyWirjB98AR+kNbs4dnCs+cDIvhQcyXWwsu6rP3qZGp3WpXi90OrWxFNVr52Rx7k8vtSLTYFAoHsJRp+3wOcepVBQNHtZAMfFiA/giEzDNYTjvNJbsGXypKxvPb0GWeW9ysi5DmGsE0JqjsNwPVY6b4sdZdh/c5xL4zIEIyaF+pliaFl5eJZKDwLVmAwuvCStTAy5wKpRc2pMokupvfWjpAnoWBsmb0n6UYKYORMK3vrE/5plGlp2w3lVhmu3J0zmrwNHPSnSkXEri4ZAVAlGYQmHUotTyvmyh4FrhiA8/dMLQI/DELZVfdeNXxgKRyKAOV9FjlLnMqmXjux3kYcgchTkFYX01EiLn39jTEgitR2y0NyO+xNjZIEsUiSCdZSJKlzcZrdVPHOEs5hna2Ni07HVAbuj03Q4TmnGTstKIA2hJgsQorngEQnjdCjhWBM1BKekCxCz1/DRmWhqpM81OZYrDclRbG5SjmFCdlQYvqSwX0tPsXVClMeCpZa5Lg7xOhftSqYk0QVNHgxu6eVGnSuRkNCt83Xz6jz1qVIehpDWFBXav6vsuYwZlzJgekq6JMdlHVEalb3t0VXYjV1po6Qfdu4LTe5L3tIu+Vw5wk+XZRTMf2ua+jXufl5lpcmhQcIRx4x3ovyodVm2s4cwE0Fgl0C5x+Ap3rn/xNY3+t1Vlvf28vrT5rv4dzEfPo5fRStSJNqLN6EO0E+1FIhpFX6Pv0Y+Ga3xr/Gz8mkBnZ2rOs2hqNX7/A4LxoMk=</latexit> <latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">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</latexit> What would comparable groups look like? E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober sober drunk drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 14 / 41 Brady Neal

  40. <latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> What would comparable groups look like? E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk drunk sober sober sober sober sober sober drunk drunk drunk drunk drunk drunk sober sober sober sober drunk drunk 14 / 41 Brady Neal

  41. <latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">AE7nicbVRLaxRBEJ4kq8b1lejRS2sIRJisuyEQBRcCISLiIUJekA2hp6d2t9l+0V1jshnmR3jxoIhX/45H/401s5OYTdLDQHXV91V9XV0ziVMyYLv9d2Z2rnHn7r35+80HDx89frKw+HQ/2MwL2BNWX+Y8ABKGthDiQoOnQeuEwUHyWirjB98AR+kNbs4dnCs+cDIvhQcyXWwsu6rPqZGp3WpXi90OrWxFNVr52Rx7k8vtSLTYFAoHsJRp+3wOcepVBQNHtZAMfFiA/giEzDNYTjvNJbsGXypKxvPb0GWeW9ysi5DmGsE0JqjsNwPVY6b4sdZdh/c5xL4zIEIyaF+pliaFl5eJZKDwLVmAwuvCStTAy5wKpRc2pMokupvfWjpAnoWBsmb0n6UYKYORMK3vrE/5plGlp2w3lVhmu3J0zmrwNHPSnSkXEri4ZAVAlGYQmHUotTyvmyh4FrhiA8/dMLQI/DELZVfdeNXxgKRyKAOV9FjlLnMqmXjux3kYcgchTkFYX01EiLn39jTEgitR2y0NyO+xNjZIEsUiSCdZSJKlzcZrdVPHOEs5hna2Ni07HVAbuj03Q4TmnGTstKIA2hJgsQorngEQnjdCjhWBM1BKekCxCz1/DRmWhqpM81OZYpDGtV2a4NyFBOqs9LgJZXlQnqavQuqNAY8tcx1aZDXqXBfKjWRJmjqaHBDNy/qVImcjAakdb5uXp2nPlXKwxDSGgvqSs3/VdYcxowrOTBdBX2yg7KOSM3q/rboSqymylpTJ+jeDZzWm7y3XeS9cpiTJN8uiunYPvd1Ou83FyLS5OCI4QD71jvRfmwanMNZy6AxiKBboHT6Bz/ZO/aeyvtTrbef15c239W/g/noefQyWok60Ua0GX2IdqK9SESj6Gv0PfrRcI1vjZ+NXxPo7EzNeRZNrcbvf4d2oMo=</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> What would comparable groups look like? E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk drunk sober sober sober sober sober sober drunk drunk drunk drunk drunk drunk sober sober sober sober drunk drunk 14 / 41 Brady Neal

  42. Question: Why is association not causation?

  43. What assumptions would make the ATE equal to the associational difference?

  44. <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T 17 / 41 Brady Neal

  45. <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] (ignorability) 17 / 41 Brady Neal

  46. <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] 17 / 41 Brady Neal

  47. <latexit 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<latexit 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Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? 2 1 1 1 ? ? 3 1 0 0 ? ? 4 0 0 ? 0 ? 5 0 1 ? 1 ? 6 1 1 1 ? ? 17 / 41 Brady Neal

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<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? 6 1 1 1 ? 17 / 41 Brady Neal

  49. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? 6 1 1 1 ? 2 / 3 17 / 41 Brady Neal

  50. <latexit 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<latexit 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Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? 6 1 1 1 ? 2 / − 1 / 3 3 17 / 41 Brady Neal

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Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? 6 1 1 1 ? / 2 / / − 1 / − 1 / 3 = 3 3 − 3 3 17 / 41 Brady Neal

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Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? 6 1 1 1 ? / 2 / / − 1 / − 1 / 3 = 3 3 − 3 3 17 / 41 Brady Neal

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Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? X 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? T Y 6 1 1 1 ? / 2 / / − 1 / − 1 / 3 = 3 3 − 3 3 17 / 41 Brady Neal

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Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? X 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? T Y 6 1 1 1 ? / 2 / / − 1 / − 1 / 3 = 3 3 − 3 3 17 / 41 Brady Neal

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Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? X 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? T Y 6 1 1 1 ? / 2 / / − 1 / − 1 / 3 = 3 3 − 3 3 17 / 41 Brady Neal

  56. <latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">AFh3icfVTbjhNHEB0uDmCSsBCJl7x0MCvlYTD2asVFiUitFEU5YFIuwtox1r19NTYLfdN3TUspmfyWvyQ/mbVI8Hdr1EacujupyqOlVd6tIpGXAy+efK1WvXB1/duHlrePvrb769s3P3nGwjRdwJKy/k3JAyhp4AglKnjPHBdKnhdrl4m/+t34IO05hDXDuaL4yspeBIptOd+8XByVtWaFmxQzZjkzl91qeT053RZDzpDvtSmPbCKOvPq9O7198WlRWNBoNC8RBOphOH8g9SqGgHRZNAMfFi/ghETDNYR57Bpo2S5ZKlZbT3+DrLNejIhch7DWJSE1x2W47EvG/KdNFg/m0dpXINgxKZQ3SiGlqVpsEp6EKjWJHDhJXFlYsk9F0gzG26VKXW7rVu7Ql6GlrFd9gtRN1IAI4uCbX7va8q3jUqWNH8qscsO5eoD68HbkZvpbJmQwO3nqACI0iwCsw6lh/6IQreBK7YwnO3DGMC/9aENFW3fuR4QGK5lIFKeuxyp5xKlp7dQxL7iDkFQjruxUJOfenoVcCV6eawBeV5LzJ0NMqGIBPFMiShdHDI6j37nCO9z3qDNja3SrANyQ93Ppkxoxk3FkpAH0JIiVXexREI4fE4FoRNIJS0gXIWeX5Wc60NFI3mp3JCpdpYcdPKUe7CXVWGvwcyqKQnbvU6g0BjyNzM1okfepcC2V2lATtHW0uGEW2z5VKTerAVWfbxa7fvquKh6WUPVYUBdqnlfZc5gzruTCzBTUJAdlHQUNu/t7SVdiNVXWmiZB927grFdicdDGIi1zWcaDt32HXPfe72OSbnkl6YCRwgH3rHih/RjnXIJZz4BjUC/T98kxXoY5BSe4s0m46E4qWEM6N58jDnkViWEF90TOajvb6JkDFwivu4sNkfNgWegXe7OkmJr2lYHqMpefni+F473xdH/8/I/90Yuf+mfpZvZ9iD7MZtmT7MX2a/Zq+woE9nH7M/sr+zvwa3B48GTwbMN9OqVPua7bOsMfv4XHAzX0g=</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> Another perspective: exchangeability T = 0 T = 1 E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 18 / 41 Brady Neal

  57. <latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> Another perspective: exchangeability T = 0 T = 1 Group A Group B Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 18 / 41 Brady Neal

  58. <latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> Another perspective: exchangeability T = 0 T = 1 Group B Group A Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 18 / 41 Brady Neal

  59. <latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> Another perspective: exchangeability T = 0 T = 1 Group B Group A Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 18 / 41 Brady Neal

  60. <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> <latexit 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<latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="gTr4smGTrhcKjcTg6MmgOZTfxMo=">AFpnicfVTbihtHEB1fFNvKxWv70S+dyAsbGCuaZY1tsMBg1oQgN7sdkRm56eGqlR3+iu8Voe5kvyNXlNvsB/k+rR2LvaDWmhofrUqarT1UXTsmAk8mna9dv3Bx8dev2neHX3z73d2te/ePgq29gENhlfVvCx5ASQOHKFHBW+eB60LBcbF8Ff3H78EHac0BrhzMNJ8bWUnBkaDTrSf5/sm7nexHlmtZsgM2ZdmMPlfQyQV0xk63RpPxpFvsqpH1xijp15vTezf5aUVtQaDQvEQTrKJw1nDPUqhoB3mdQDHxZLP4YRMwzWEWdOdr2XbhJSsp7+BlmHXoxouA5hpQtiao6LcNkXwf/yndRYPZs10rgawYh1oapWDC2LzWKl9CBQrcjgwkvSysSCey6QWjrcKFPodnNv7RJ5EVrGtlrkm6kAEaIgk19HyrKt8mKSLweKrHNDuTyI+vJm5Hr7mxASOT2S1QARGnmgVmHUsuPfRMFrwNXbO65W4QxkX+pQ+yqWz12PCpXMhAJT12uWNOJQvP/aoJC+4gpCUI67sJCin3p6FVHAlenusAXlaSUydDTKySATpjIkoXTNktB7/yhE+pLxGmxpbxl4H5IZOP82Y0IybkUjDaAlCRLtIsjEsJP4ArRdQGlJIuQMpKz89SpqWRutbsTJa4iGM7fko52nWos9Lgl1DWCOlp9j6HSmPAU8vclAZ5jwpXUqm1NEFTR4Mbpk3bpyrkejSg7PNm+48/alKHhZQ9lxQF2qeV9l1mDKu5NxMFVRkB2UdBQ27+3tFV2I1VdaOkH3buCs3zT5ftvkcZiLotlv203fEfe91+smbi75pSnBEcOBdyz/Pv5Yt7nEM5+JxiKR/p+zgr0MUipvUXqTSfCcUVDCOfgOfOgVxEVlBd9Iy0W5/CFBN7hV3zaMIPmpzvQRvdnXdxH1LwfQYZefnqvG0e42xs/31v9PJF/yzdTh4mPyQ7SZY8TV4mPydvksNEJH8mfyV/J/8Mdga/DQ4Hx2vq9Wt9zINkYw3+Bc92+Bf</latexit> Another perspective: exchangeability T = 0 T = 1 Group B Group A Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 Before switch After switch E [ Y (1) | T = 1] = E [ Y (1) | T = 0] 18 / 41 Brady Neal

  61. <latexit 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<latexit 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<latexit 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<latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="gTr4smGTrhcKjcTg6MmgOZTfxMo=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> Another perspective: exchangeability T = 0 T = 1 Group B Group A Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 Before switch After switch E [ Y (1) | T = 1] = E [ Y (1) | T = 0] 0] = E [ Y (1)] 18 / 41 Brady Neal

  62. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="WFjqyGlwAO5sBmjhLTYUe7gk=">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</latexit> <latexit sha1_base64="WFjqyGlwAO5sBmjhLTYUe7gk=">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</latexit> Another perspective: exchangeability T = 0 T = 1 Group B Group A Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 Before switch After switch E [ Y (1) | T = 1] = E [ Y (1) | T = 0] 0] = E [ Y (1)] E [ Y (0) | T = 0] = E [ Y (0) | T = 1] 1] = E [ Y (0)] 18 / 41 Brady Neal

  63. <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Aside: identifiability E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] 19 / 41 Brady Neal

  64. <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">AF83icfVRfbxtFED+XGoqBksIDSLwsuEFuhfFKkgYalSFYQD0VK0lY5K9rbm7NX3n/anSN1T/fC1+AN8coH4hvwMZg9X5tcWrGWrdmZ3/zmt7PjLZySAefzf0a3rk9fve9O+9Pvjwo7sf79375CzY2gs4FVZ/6zgAZQ0cIoSFTxzHrguFDwtNo9j/Olv4IO05gS3Dpar4yspOBIrou93/Pj8+cPsm+W7IB15pzMrxesd7Ncy5KdsAXLriGuvPMly5GvGrky1vNCKonbluX5hNHqad7CMSC42JvOZ/NusTeNrDemSb+eXNy7/Twvrag1GBSKh3CezR0uG+5RCgXtJK8DOC42fAXnZBquISybrlkt2ydPySr6WuQd7rGQ3XIWx1QUjNcR1uxqLzbHzGqvlo0rkYwYleoqhVDy2LnWSk9CFRbMrjwkrQyseaeC6T7mQzKFLod7q3dIC9Cy9g+5GkGymAkUfBUN+LiviGqOiJd0l9tmJ3LxkPXiYuevOwIUEbl9nBUCUZhWYdSi1fNk3UfA6cMVWnrt1mBH45zrErtgeMBSeVaBirpseOnEoWnvtE9bcQUhLEDQ7ncSUe28vQyq4Er0904A8rSmzgYZUSCdEYiomu6STv4hSO8SHmNjW2jL0OyA2dfpExoRk3JYtGkBLEiQ2aZdHIRvZwG3iqANKCVdgJSVnl+mTEsjda3ZpSxHWd19pA42l2qs9Lg61TWCOlp9l6lSmPAU8vcgb5iApXUqmdNEFTR4MbFk3bUxVyNxpQ9nyLpjtPf6qShzWUPRbUtZpXVQ4dpowr+hsuFRkB2UdJU26+3tMV2I1VdaOkH3buCy3zT5cdvkcZiLojlu2HsjPs+6nUTNzfi0pTgCOHAO5Z/GT+s29zAmVdAY5FA/w/fsQL9GCRqb5F604lwXNEQwpXzCnSq4gKS6iuR6bZ9LA/BKgm94q75n503m9zvQFvDnXdxH1LyfQYZTefnjeNs8NZdjT7/tej6aMf+mfpTvJF8lXyIMmSh8mj5KfkSXKaiOTf0d3RZ6Px/X4j/Gf47920FujPufTZLDGf/8HwkH2Ow=</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Aside: identifiability E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Causal quantities 19 / 41 Brady Neal

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  66. <latexit 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<latexit 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Aside: identifiability E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Causal quantities (accessible, since we Statistical quantities have ) P ( x, t, y ) A causal quantity (e.g. ) is identifiable if we can E [ Y ( t )] compute it from a purely statistical quantity (e.g. ) E [ Y | t ] 19 / 41 Brady Neal

  67. <latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> Randomized control trial (RCT) T 20 / 41 Brady Neal

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  69. <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> Randomized control trial (RCT) T Slept with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober drunk sober drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 20 / 41 Brady Neal

  70. <latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> Randomized control trial (RCT) T Slept with shoes on Slept without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober drunk sober drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 20 / 41 Brady Neal

  71. <latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">AE7nicbVRLaxRBEJ4kq8b1lejRS2sIRJisuyEQBRcCISLiIUJekA2hp6d2t9l+0V1jshnmR3jxoIhX/45H/401s5OYTdLDQHXV91V9XV0ziVMyYLv9d2Z2rnHn7r35+80HDx89frKw+HQ/2MwL2BNWX+Y8ABKGthDiQoOnQeuEwUHyWirjB98AR+kNbs4dnCs+cDIvhQcyXWwsu6rPqZGp3WpXi90OrWxFNVr52Rx7k8vtSLTYFAoHsJRp+3wOcepVBQNHtZAMfFiA/giEzDNYTjvNJbsGXypKxvPb0GWeW9ysi5DmGsE0JqjsNwPVY6b4sdZdh/c5xL4zIEIyaF+pliaFl5eJZKDwLVmAwuvCStTAy5wKpRc2pMokupvfWjpAnoWBsmb0n6UYKYORMK3vrE/5plGlp2w3lVhmu3J0zmrwNHPSnSkXEri4ZAVAlGYQmHUotTyvmyh4FrhiA8/dMLQI/DELZVfdeNXxgKRyKAOV9FjlLnMqmXjux3kYcgchTkFYX01EiLn39jTEgitR2y0NyO+xNjZIEsUiSCdZSJKlzcZrdVPHOEs5hna2Ni07HVAbuj03Q4TmnGTstKIA2hJgsQorngEQnjdCjhWBM1BKekCxCz1/DRmWhqpM81OZYpDGtV2a4NyFBOqs9LgJZXlQnqavQuqNAY8tcx1aZDXqXBfKjWRJmjqaHBDNy/qVImcjAakdb5uXp2nPlXKwxDSGgvqSs3/VdYcxowrOTBdBX2yg7KOSM3q/rboSqymylpTJ+jeDZzWm7y3XeS9cpiTJN8uiunYPvd1Ou83FyLS5OCI4QD71jvRfmwanMNZy6AxiKBboHT6Bz/ZO/aeyvtTrbef15c239W/g/noefQyWok60Ua0GX2IdqK9SESj6Gv0PfrRcI1vjZ+NXxPo7EzNeRZNrcbvf4d2oMo=</latexit> Randomized control trial (RCT) T Slept with shoes on Slept without shoes on ( T = 1) ( T = 0) drunk drunk sober sober sober sober sober sober drunk drunk drunk drunk drunk drunk sober sober sober sober drunk drunk 20 / 41 Brady Neal

  72. Graphical interpretation of RCT 21 / 41 Brady Neal

  73. <latexit sha1_base64="uGwb7/40FtxyqwMkvjDesmjJa3g=">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</latexit> Graphical interpretation of RCT X T Y 21 / 41 Brady Neal

  74. <latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> <latexit sha1_base64="uGwb7/40FtxyqwMkvjDesmjJa3g=">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</latexit> Graphical interpretation of RCT T X T Y 21 / 41 Brady Neal

  75. <latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> <latexit sha1_base64="wt2lCE4iu6jp6hQfvYVw6vDRF60=">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</latexit> Graphical interpretation of RCT T X T Y 21 / 41 Brady Neal

  76. Question: What important property does an RCT give us?

  77. <latexit sha1_base64="vdhnOVYCs5M1CLTr9/rvuY12bRc=">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</latexit> Conditional exchangeability Exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T 23 / 41 Brady Neal

  78. <latexit sha1_base64="vdhnOVYCs5M1CLTr9/rvuY12bRc=">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</latexit> <latexit sha1_base64="DuN46LKiETzH5PvG/Zwp4JQMaEA=">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</latexit> Conditional exchangeability Exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T T Y 23 / 41 Brady Neal

  79. <latexit sha1_base64="DuN46LKiETzH5PvG/Zwp4JQMaEA=">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</latexit> <latexit sha1_base64="nx+h6pLGXoiKdS8RQWDjcXaAg8c=">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</latexit> <latexit sha1_base64="vdhnOVYCs5M1CLTr9/rvuY12bRc=">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</latexit> Conditional exchangeability Exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T T Y X T Y 23 / 41 Brady Neal

  80. <latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">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</latexit> <latexit 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<latexit 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<latexit 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Conditional exchangeability Exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T T Y Conditional exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T | X T Y 23 / 41 Brady Neal

  81. <latexit 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<latexit 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<latexit sha1_base64="vdhnOVYCs5M1CLTr9/rvuY12bRc=">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</latexit> <latexit sha1_base64="cNh0Io/0qaIFOZ30xwVEGc7NFQg=">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</latexit> Conditional exchangeability Exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T T Y Conditional exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T | X T Y 23 / 41 Brady Neal

  82. <latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">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</latexit> Identification of conditional average treatment effect Conditional exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T | X 24 / 41 Brady Neal

  83. <latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">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</latexit> <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> Identification of conditional average treatment effect Conditional exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T | X E [ Y (1) − Y (0) | X ] = E [ Y (1) | X ] − E [ Y (0) | X ] 24 / 41 Brady Neal

  84. <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">AFjXicfVTbhs3EN1c1CbqJU72DywVQzYwEaVDBdJgSgIEKQtij6kgO3Y8BoGlzsrEeIN5GwdhdiXfE1f27/p3S42sSWU5SChLmcmTkzHLF0SgacTP65cfPW7cEn965O/zs8y+vLd1/6ujYBsv4FBYZf1xyQMoaeAQJSo4dh64LhW8Lpcvkv/1H+CDtOYAVw7ONJ8bWUvBkUznWw92Tnamuzk72Zns7rJCmgocO2CFlhU7Pt8aTcaT7rCPhWkvjL+vDq/f/ukqKxoNBgUiodwOp04PIvcoxQK2mHRBHBcLPkcTk0XEM4i10bLdsmS8Vq6+lrkHXWqxGR6xBWuiSk5rgI13J+F+0wbrJ2dRGtcgGLEuVDeKoWVpJqySHgSqFQlceElcmVhwzwXS5IYbZUrdburWLpGXoWVsm/1E1I0UwMiYJPfm5rybaKSJd0CldhmB3L5lvXgzcj1dDZMSOD2Q1QARGnmgVmHUsu3/RAFbwJXbO65W4QxgX9tQpqWz1yPCxXMhAJT12uVNOJUvP/SqGBXcQ8gqE9d2ihJx7by9CLrgSvTzWgDyvJebOBplQRIJ4pkSULg4ZnUe/cYQ3OW/Q5sZWadYBuaHuZ1MmNOmYknIA2hJhMQy7+IhPD9OBKETSCUtIFyFnl+UXOtDRSN5pdyAoXbEY7+ZhytOtQZ6XBD6EsCulp96HSmPA08jcjBZ5nwrXUqk1NUFbR4sbZrHtU5VyvRpQ9flmseun76riYQFVjwV1peZlT2HOeNKzs1MQU1yUNZR0LC7vxd0JVZTZa1pEnTvBi56JRYv21ikZS7L+LJtN31H3Pder2NSrvm7PzIhHjHim/Th3XKNZx5DzQWCfT/8HVWoB+DlNpbpNl0JBxXtIRwabxEHvQsEsMK6que0XS01zcBKhZecRcfJuPDtBL8GZPNzHpLQXTYzS9/vR8LBztjaf74x9/3x89f9o/S3eyb7Lvsp1smj3Onme/ZK+yw0xk7I/s7+yvwf3Bj8Mng6eraE3b/QxX2cbZ/Dzv3Kc2YA=</latexit> Identification of conditional average treatment effect Conditional exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T | X E [ Y (1) − Y (0) | X ] = E [ Y (1) | X ] − E [ Y (0) | X ] | | | − − = E [ Y (1) | T = 1 , X ] − E [ Y (0) | T = 0 , X ] 24 / 41 Brady Neal

  85. <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">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</latexit> Identification of conditional average treatment effect Conditional exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T | X E [ Y (1) − Y (0) | X ] = E [ Y (1) | X ] − E [ Y (0) | X ] | | | − − = E [ Y (1) | T = 1 , X ] − E [ Y (0) | T = 0 , X ] | | − = E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ] 24 / 41 Brady Neal

  86. <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> Identification of conditional average treatment effect Conditional exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T | X E [ Y (1) − Y (0) | X ] = E [ Y (1) | X ] − E [ Y (0) | X ] | | | − − = E [ Y (1) | T = 1 , X ] − E [ Y (0) | T = 0 , X ] | | − = E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ] What about the ATE? E [ Y (1) − Y (0)] 24 / 41 Brady Neal

  87. <latexit sha1_base64="QBQ/SlmBKmhrRqHJ9awVHyCWcFA=">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</latexit> The Adjustment Formula (identification of ATE) E [ Y (1) − Y (0)] = E X E [ Y (1) − Y (0) | X ] = E X [ E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ]] 25 / 41 Brady Neal

  88. <latexit sha1_base64="nx+h6pLGXoiKdS8RQWDjcXaAg8c=">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</latexit> <latexit sha1_base64="QBQ/SlmBKmhrRqHJ9awVHyCWcFA=">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</latexit> The Adjustment Formula (identification of ATE) E [ Y (1) − Y (0)] = E X E [ Y (1) − Y (0) | X ] = E X [ E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ]] X T Y 25 / 41 Brady Neal

  89. <latexit sha1_base64="nx+h6pLGXoiKdS8RQWDjcXaAg8c=">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</latexit> <latexit sha1_base64="QBQ/SlmBKmhrRqHJ9awVHyCWcFA=">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</latexit> The Adjustment Formula (identification of ATE) E [ Y (1) − Y (0)] = E X E [ Y (1) − Y (0) | X ] = E X [ E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ]] X T Y 25 / 41 Brady Neal

  90. <latexit sha1_base64="cNh0Io/0qaIFOZ30xwVEGc7NFQg=">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</latexit> <latexit sha1_base64="QBQ/SlmBKmhrRqHJ9awVHyCWcFA=">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</latexit> The Adjustment Formula (identification of ATE) E [ Y (1) − Y (0)] = E X E [ Y (1) − Y (0) | X ] = E X [ E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ]] X T Y 25 / 41 Brady Neal

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