estimation
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Estimation Brady Neal causalcourse.com <latexit - PowerPoint PPT Presentation

Estimation Brady Neal causalcourse.com <latexit


  1. <latexit 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<latexit 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Preliminaries Conditional average treatment effects (CATEs): τ ( x ) , E [ Y (1) − Y (0) | X = x ] Always assuming unconfoundedness and positivity τ , E [ Y (1) − Y (0)] = E W [ E [ Y | T = 1 , W ] − E [ Y | T = 0 , W ]] Given is a sufficient adjustment set <latexit 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W Brady Neal 3 / 35

  2. <latexit 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<latexit sha1_base64="nkm/C9Jn2NA5cqfAGWnaB02y2ao=">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</latexit> Preliminaries Conditional average treatment effects (CATEs): τ ( x ) , E [ Y (1) − Y (0) | X = x ] Always assuming unconfoundedness and positivity τ , E [ Y (1) − Y (0)] = E W [ E [ Y | T = 1 , W ] − E [ Y | T = 0 , W ]] Given is a sufficient adjustment set <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W τ ( x ) , E [ Y (1) − Y (0) | X = x ] = E W [ E [ Y | T = 1 , X = x, W ] − E [ Y | T = 0 , X = x, W ]] Given is a sufficient adjustment set <latexit sha1_base64="HJoRYiEmPGuhM+q3HZJwgKHYVJI=">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</latexit> W ∪ X Brady Neal 3 / 35

  3. Conditional Outcome Modeling Increasing Data Efficiency Propensity Scores and IPW Other Methods Brady Neal 4 / 35

  4. Conditional Outcome Modeling Increasing Data Efficiency Propensity Scores and IPW Other Methods Brady Neal Conditional Outcome Modeling 5 / 35

  5. Conditional outcome modeling (COM) <latexit sha1_base64="6c1Bz+GSJyXbD1v3ABdCutoCeLI=">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</latexit> τ = E W [ E [ Y | T = 1 , W ] − E [ Y | T = 0 , W ]] Brady Neal Conditional Outcome Modeling 6 / 35

  6. Conditional outcome modeling (COM) <latexit sha1_base64="6c1Bz+GSJyXbD1v3ABdCutoCeLI=">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</latexit> τ = E W [ E [ Y | T = 1 , W ] − E [ Y | T = 0 , W ]] model model Brady Neal Conditional Outcome Modeling 6 / 35

  7. Conditional outcome modeling (COM) <latexit sha1_base64="6c1Bz+GSJyXbD1v3ABdCutoCeLI=">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</latexit> τ = E W [ E [ Y | T = 1 , W ] − E [ Y | T = 0 , W ]] model model Brady Neal Conditional Outcome Modeling 6 / 35

  8. Conditional outcome modeling (COM) <latexit sha1_base64="6c1Bz+GSJyXbD1v3ABdCutoCeLI=">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</latexit> τ = E W [ E [ Y | T = 1 , W ] − E [ Y | T = 0 , W ]] model model <latexit sha1_base64="sDoQAtycoL6a2NeSyENAcLTzMto=">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</latexit> τ = E W [ µ (1 , W ) − µ (0 , W )] Brady Neal Conditional Outcome Modeling 6 / 35

  9. <latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">ATBHicrVhZb9tGEF6nR1z3ctTHvrBRCjgArUhuCudFQA7QVE0gAP4aqNAoChKIsQrPKw6BF/7a/pW9LH9He2fKTrz7VIkdSuIBZGr2ZlvZufaXfcCx47iZvPfnTsfPjRx3d3P9n79LPv/hy/17tMvKT0LQuTN/xw+ueEVmO7VkXsR071nUQWobc6yr3viE569urDCyfe8vg2s164x9OyBbRoxkbr7YWdkxGknNpJMa2udQWiYaStLvUzTOlHidm2t41iD+ECTfG6SHbR0rWP63sBPvD4h3xhO136oHWoluZClk5oD0fxw+5+vdlo4k+bH7TUoC7U35l/b/cv0RF94QtTJMIVlvBETGNHGCKizyvREk0REO21SIkW0sjGvCUysUeyCXFZxGEQdUzPIf16page/WbMCNImaXHoG5KkJr5VPH0aD0CVb9avlXiX6UiBzTbe0runMF2ixmJE1HVyOemcrymCx8grXYZGcACq/SrKxoQG+HfsdkPz9vidOiUZ+kQhqZRHOIKimsI6S39CuvfAQ/G+CzaMQ2LV9Nj2xfHgme9+kzJiyDxhEsZVs18Vx53YNmC7Yyj4OILUf8lVYo7VuFNZiuwUZU5SqY95woY/GWRlXkVTrLubOcK1bI2QJdzB+DwyPuiCg+8tom/9nEUc1EkzBZh4FIDrGWAPnRUMg/Yj7P1YDieQh7IkRNQzbZ0BOouinszu106N0DdkjyKc2NoId9oZM9FrBD5GbuR3cIf2a4JcJG80ZegN1x/HUaU2cXTrh+jRjT7GkJ6Q/c4ukdSnRNPU5FD9Bv0Vx1+EVzmudZH1YKLM4gi5Pxb5NXYP94tKTqezZnKJDhws7RsiWMdEKfRKJ9T2iVUSoH0ehpvR24LUAKDq0s38mGLtYE0c0ge4JjfvQw9IadbKGOFZ2ZBWt7BsbvWdeq4b849qVfW9WK8t5qFRNZWUbmprisVoxR4DtLnvNVL1OdtwIq8tmrOrB/qJrcI5W7WOpIj7VWPWRjSNIVXHZi4vXuWgtR6gTHbFk3w+Jow2pgaJHqM1Aador1d+JqhIf8TDxdlVOyHpnbZOZmZTorvKUQ3yznH4Zk3hGkgpzrwXnl+FeIzJZ/3+h59UlBXy12isqyIY3S6cxqeRsjC/6RI464xAvg6wBr+Wb61Uozm2K/HzxvDpEzNlZIm6L3kRuLV34u6tQL6pRVX9ypJkS4uwR0PjBlPMB4TLvGFo8onCFp9P5bG0MTomSKY8NMJfXYWGDjc7XB2dHZ7kBUVz3BwVa/Wynm5KG8kfbVsf5rvVlJXy3LlBvkpq36QSs6zVyMeLAEm4pS8/XSPnYEyRd2vzBvZ52GsTlRehsvBqjaSLHiNX5auqfLHWvhdMFQ+YZlf3sGDN+h/GU5A2/qxkI238mYhd/tOPi3kJ1t6tpB0t/RvIfl2jaSFvcGe0ljm5dqaYq6ztchvpucSWU6nj3yDu/FfdVxNLWPcLTlrtdGbetqD8t3sfz0GZX6mw3txck0xSliojphfp4hCa5th+ol7QHs3jE/EdnQ9a9Hxe6ZqbovI5PVH1VsY9IuRj6nen6Ivb4wbqNH1YOXNXdTwjDfk3U7eTgtRZ8hlSNvoXYwsc70/d5ad1SRPKyPikif42zWIRfUsxlsd38XI8h7t4wRZnOPfB3L5jrIZXpEzy2+U6+65y29fAb0H9HTQESLFe4qsc9AZLHGhbohy9z2Zu+ty/h5RVXDupvdrM/q+LO3pyFuPwlxFDzyFBxDWtLKiMvuYQFkYnXaiHBnKW5mjcpNS+LN2haV8spVeG3VrSzVlfa6+/XW7P9m5geXR43W943my8f1p0/U/212xdfivjgPx2Lp1SZ+RXU/wj/tu5u7Nb+632e+2P2p+S9c6OkvlKVP5qf/8Ptlm8jQ=</latexit> Conditional outcome modeling (COM) <latexit sha1_base64="6c1Bz+GSJyXbD1v3ABdCutoCeLI=">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</latexit> τ = E W [ E [ Y | T = 1 , W ] − E [ Y | T = 0 , W ]] model model <latexit sha1_base64="sDoQAtycoL6a2NeSyENAcLTzMto=">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</latexit> τ = E W [ µ (1 , W ) − µ (0 , W )] Model-assisted estimator: τ = 1 X ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) n i Brady Neal Conditional Outcome Modeling 6 / 35

  10. <latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit> COM estimation of CATEs τ = 1 ATE COM Estimator: X ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) n i Brady Neal Conditional Outcome Modeling 7 / 35

  11. <latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit> <latexit sha1_base64="KWSzAeHNia9cnBsm5+I3e5sEX8M=">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</latexit> COM estimation of CATEs τ = 1 ATE COM Estimator: X ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) n i CATE Estimand: τ ( x ) , E [ Y (1) − Y (0) | X = x ] = E W [ E [ Y | T = 1 , X = x, W ] − E [ Y | T = 0 , X = x, W ]] Brady Neal Conditional Outcome Modeling 7 / 35

  12. <latexit 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<latexit 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COM estimation of CATEs τ = 1 ATE COM Estimator: X ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) n i CATE Estimand: τ ( x ) , E [ Y (1) − Y (0) | X = x ] = E W [ E [ Y | T = 1 , X = x, W ] − E [ Y | T = 0 , X = x, W ]] µ ( t, w, x ) , E [ Y | T = t, W = w, X = x ] Brady Neal Conditional Outcome Modeling 7 / 35

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sha1_base64="NUn2yjqlAs8cjD5+3hs3RYaw7yo=">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</latexit> <latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit> COM estimation of CATEs τ = 1 ATE COM Estimator: X ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) n i CATE Estimand: τ ( x ) , E [ Y (1) − Y (0) | X = x ] = E W [ E [ Y | T = 1 , X = x, W ] − E [ Y | T = 0 , X = x, W ]] µ ( t, w, x ) , E [ Y | T = t, W = w, X = x ] CATE COM τ ( x ) = 1 X ˆ (ˆ µ (1 , w i , x ) − ˆ µ (0 , w i , x )) Estimator: n x i : x i = x Brady Neal Conditional Outcome Modeling 7 / 35

  14. τ i = ˆ <latexit sha1_base64="9Zf9BmLnXQJnoqhOEu7stCDJKjM=">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</latexit> ˆ τ ( x i ) = ˆ µ (1 , w i , x i ) − ˆ µ (0 , w i , x i ) Question: What could go wrong with this estimator?

  15. COM estimation’s many faces Brady Neal Conditional Outcome Modeling 9 / 35

  16. COM estimation’s many faces • G-computation estimators Brady Neal Conditional Outcome Modeling 9 / 35

  17. COM estimation’s many faces • G-computation estimators Brady Neal Conditional Outcome Modeling 9 / 35

  18. COM estimation’s many faces • G-computation estimators • Parametric G-formula Brady Neal Conditional Outcome Modeling 9 / 35

  19. COM estimation’s many faces • G-computation estimators • Parametric G-formula Brady Neal Conditional Outcome Modeling 9 / 35

  20. COM estimation’s many faces • G-computation estimators • Parametric G-formula • Standardization Brady Neal Conditional Outcome Modeling 9 / 35

  21. COM estimation’s many faces • G-computation estimators • Parametric G-formula • Standardization • S-learner where “S” is for “Single” Brady Neal Conditional Outcome Modeling 9 / 35

  22. <latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit> Problem with COM estimation in high dimensions τ = 1 X ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) n i Brady Neal Conditional Outcome Modeling 10 / 35

  23. <latexit 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<latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit> Problem with COM estimation in high dimensions τ = 1 T X <latexit 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ˆ ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) µ n i Y <latexit 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W Brady Neal Conditional Outcome Modeling 10 / 35

  24. <latexit 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<latexit 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Problem with COM estimation in high dimensions τ = 1 T X <latexit 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ˆ ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) µ n i Y <latexit 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W Brady Neal Conditional Outcome Modeling 10 / 35

  25. <latexit 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<latexit 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Problem with COM estimation in high dimensions τ = 1 T X <latexit 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ˆ ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) µ n i Y Problem: estimate can <latexit 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W be biased toward zero (Künzel et al., 2019) Brady Neal Conditional Outcome Modeling 10 / 35

  26. How can we ensure that the model doesn’t ignore T?

  27. <latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit> Grouped COM (GCOM) estimation τ = 1 COM: X ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) n i Brady Neal Conditional Outcome Modeling 12 / 35

  28. <latexit sha1_base64="ls0sKV6pN4O/vMX7/T0OvDurI=">ATAnicrVhLb9tGEF6nTeq6L0c9sJGKeAsiq5KZyLgAB2gqJoAfwq40CgaIoiRBf4cOqQ/DWX9Nb0Wv7Nwr0v/TQmW+XIqkXpSAWRK5mZ76Zndfu/bVhi1Wv/u3Pngw7v3Ptr9eO+Tz/7/Iv9+7XL0IsDw7wPNsLrvt6aNqWa15EVmSb135g6k7fNq/6kxOev7oxg9Dy3Po1jdfO/rItYaWoUdE6u373bEeJd1Ij1Oto3WHgW4k7TRxU03rhrHTs7SubQ6jA03yOXHax90Dc8derE7IOAb3e5Zj7RDrcDRWsbRDazROHrU26+3mi38aYuDthrUhfo78+7v/i26YiA8YhYOMIUrohobAtdhPR5JdqiJXyivRYJ0QIaWZg3RSr2SDYmLpM4dKJO6DmiX68U1aXfjBlC2iAtNn0DktTEN4pnQOMhqPLN+rUC7yodCbDZxlt69xWmQ9RIjIlaJZdxbirHa4rIwidYi0V2+qDwKo3Siob0tul3RPbz85Y4TRoNSCqgkUE0m6iSwjoCeku/8srH8LMOPpNGbNPq1fTJ9tWR4HmPhPC0mkcwlK2VRPldaDZhK/PYiNhqxF9phdK+dVjD2RosRFWugnPiTIRb2lURl6ns5g7q7kihZwu0cX8EThc4g6J4iGvLfKfRzlTDQIk3XoiOQIa/GRH02F/CPms1z1KZ6HsCdE1DRkwU9vqb3O7MTpvefWAHJ/Q3Bh62BcNscEdoDczLzYAHdAv6b4ZcBGY47eRN1xPBu0Js6uBuF6NGPNsKQnpD8zi6R1CdE09TkUP0G/SXFvwCuc1w2S9WChzOIQulwV+w51DfaLQ0+msmczSgM6HNgxRrZMiJbrk0is71taRYj6sRVqQm8bXvOB0oB29s8UYwdr4ojG0D2l8QB6WFqjTtYUx8qOtKSVfWOh9yxq1ZB/XLuy781rZTkXlaqprOxAU0s8VivmCLDdRa8ZqtfJjhtidemcVX3Yn3cNztGyfSyVx6cqwGycQypMi57cfk6l63lCHXSQCzZ9yPi6EBqOghatNXmvYK9XeiqsRDPAy8HZUTst5Z23RuJiG6ozxlE5+sc95xeOYNYerIqS68V5xfh/hMyWe9vk+fBNT1cpeorLJsQKNkNrNe3sLIhH/kiCMu8Xz42sdavp59tcLMptjvB89dQOSMjRTSpugD5MbylZ+LOvWCOmV2Z8caYEOHv4NH43xIuMw7gRaXKFzhyWw+rYzBKVFS5bEh5rI6zG2w0PkG4OyqzJMcsqKiOQ6u6vVaOS+X5Y2kr5cdzPK9LCvp62WZcoPctFQ/CIF1XSEXIQ4s4RSy9LxCysOeIOnS5p83sM/FXhurvAiUhVcVkg56jFyVp6ryRaV9EbpgoHzCMr+8gwdv0P9SnIC29WMuG23lzVzu9p18mstPt/RsLuls6d9c8m2FpIm9wZrRWOZlZU0x1kl8pvZuURWUQPnmH9+KB6jia2kc42nLX6C2G2oPy3ax7PQZFvqbBe35yTBKWKqOmF2njiEJrm2H6iWXtAezeMT8R2dD9r0fF7qmpui8jk9VvVWxD0i5GPqd6foi9vj+uo0fVg6c5d1PCMN2TdVt5Oc21ZnyFVI2+hdjixzfbBwlp3XJE8rY+KSJ/jbCsS8epbjrY/vcmR5j/ZwgszP8e8DuXhH2Qwvz5nVN8qe+7q25dP7yE9bXSEUPGeIutsdAZTXKgbotx9Txbupy/R1QVnLvJg9qcvjL+/O1phNtPTBw5jzwFR5CWtCLiqnuYD5lInTZC3Fnym1mzdNOSePO2hYW8chReR3UrU3Wlvd5+vT3/v5nFweVRs/19s/Xycf3pE/V/m13xlXgDshPx+IpVeYZ+dUQ/4j/du7u3Kv9Vvu9kftT8l6Z0fJfClKf7W/gfRqbyf</latexit> <latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">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</latexit> Grouped COM (GCOM) estimation τ = 1 COM: X ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) n i τ = 1 GCOM: X ˆ (ˆ µ 1 ( w i ) − ˆ µ 0 ( w i )) n i Brady Neal Conditional Outcome Modeling 12 / 35

  29. <latexit sha1_base64="2blVoQ+k5PyAuiF7BR/rXP61udw=">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</latexit> <latexit sha1_base64="WX2frOmrP2Yjr1CIxDypCtqJ0GQ=">ATBHicrVhZb9tGEF6nR1z3ctTHvrBRCjgArUhuCudFQA7QVE0gAP4aqNAoChKIsQrPKw6BF/7a/pW9LH9He2fKTrz7VIkdSuIBZGr2ZlvZufaXfcCx47iZvPfnTsfPjRx3d3P9n79LPv/hy/17tMvKT0LQuTN/xw+ueEVmO7VkXsR071nUQWobc6yr3viE569urDCyfe8vg2s164x9OyBbRoxkbr7YWdkxGknNpJMa2udQWiYaStLvUzTOlHidm2t41iD+ECTfG6SHbR0rWP63sBPvD4h3xhO136oHWoluZClk5oD0fxw+5+vdlo4k+bH7TUoC7U35l/b/cv0RF94QtTJMIVlvBETGNHGCKizyvREk0REO21SIkW0sjGvCUysUeyCXFZxGEQdUzPIf16page/WbMCNImaXHoG5KkJr5VPH0aD0CVb9avlXiX6UiBzTbe0runMF2ixmJE1HVyOemcrymCx8grXYZGcACq/SrKxoQG+HfsdkPz9vidOiUZ+kQhqZRHOIKimsI6S39CuvfAQ/G+CzaMQ2LV9Nj2xfHgme9+kzJiyDxhEsZVs18Vx53YNmC7Yyj4OILUf8lVYo7VuFNZiuwUZU5SqY95woY/GWRlXkVTrLubOcK1bI2QJdzB+DwyPuiCg+8tom/9nEUc1EkzBZh4FIDrGWAPnRUMg/Yj7P1YDieQh7IkRNQzbZ0BOouinszu106N0DdkjyKc2NoId9oZM9FrBD5GbuR3cIf2a4JcJG80ZegN1x/HUaU2cXTrh+jRjT7GkJ6Q/c4ukdSnRNPU5FD9Bv0Vx1+EVzmudZH1YKLM4gi5Pxb5NXYP94tKTqezZnKJDhws7RsiWMdEKfRKJ9T2iVUSoH0ehpvR24LUAKDq0s38mGLtYE0c0ge4JjfvQw9IadbKGOFZ2ZBWt7BsbvWdeq4b849qVfW9WK8t5qFRNZWUbmprisVoxR4DtLnvNVL1OdtwIq8tmrOrB/qJrcI5W7WOpIj7VWPWRjSNIVXHZi4vXuWgtR6gTHbFk3w+Jow2pgaJHqM1Aador1d+JqhIf8TDxdlVOyHpnbZOZmZTorvKUQ3yznH4Zk3hGkgpzrwXnl+FeIzJZ/3+h59UlBXy12isqyIY3S6cxqeRsjC/6RI464xAvg6wBr+Wb61Uozm2K/HzxvDpEzNlZIm6L3kRuLV34u6tQL6pRVX9ypJkS4uwR0PjBlPMB4TLvGFo8onCFp9P5bG0MTomSKY8NMJfXYWGDjc7XB2dHZ7kBUVz3BwVa/Wynm5KG8kfbVsf5rvVlJXy3LlBvkpq36QSs6zVyMeLAEm4pS8/XSPnYEyRd2vzBvZ52GsTlRehsvBqjaSLHiNX5auqfLHWvhdMFQ+YZlf3sGDN+h/GU5A2/qxkI238mYhd/tOPi3kJ1t6tpB0t/RvIfl2jaSFvcGe0ljm5dqaYq6ztchvpucSWU6nj3yDu/FfdVxNLWPcLTlrtdGbetqD8t3sfz0GZX6mw3txck0xSliojphfp4hCa5th+ol7QHs3jE/EdnQ9a9Hxe6ZqbovI5PVH1VsY9IuRj6nen6Ivb4wbqNH1YOXNXdTwjDfk3U7eTgtRZ8hlSNvoXYwsc70/d5ad1SRPKyPikif42zWIRfUsxlsd38XI8h7t4wRZnOPfB3L5jrIZXpEzy2+U6+65y29fAb0H9HTQESLFe4qsc9AZLHGhbohy9z2Zu+ty/h5RVXDupvdrM/q+LO3pyFuPwlxFDzyFBxDWtLKiMvuYQFkYnXaiHBnKW5mjcpNS+LN2haV8spVeG3VrSzVlfa6+/XW7P9m5geXR43W943my8f1p0/U/212xdfivjgPx2Lp1SZ+RXU/wj/tu5u7Nb+632e+2P2p+S9c6OkvlKVP5qf/8Ptlm8jQ=</latexit> <latexit 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Grouped COM (GCOM) estimation τ = 1 COM: X ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) T n Y i W τ = 1 GCOM: X ˆ (ˆ µ 1 ( w i ) − ˆ µ 0 ( w i )) n i Brady Neal Conditional Outcome Modeling 12 / 35

  30. <latexit 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Grouped COM (GCOM) estimation τ = 1 COM: X ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) T n Y i W τ = 1 GCOM: X ˆ (ˆ µ 1 ( w i ) − ˆ µ 0 ( w i )) n i T = 1 network T = 0 network W Y W Y Brady Neal Conditional Outcome Modeling 12 / 35

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  32. <latexit sha1_base64="8jMDGluqeHMSJvwUG4K/zbsc1Ow=">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</latexit> <latexit sha1_base64="8jMDGluqeHMSJvwUG4K/zbsc1Ow=">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</latexit> Trained with treatment group data Trained with control group data T = 1 network T = 0 network W Y W Y Problem: networks have higher variance than they would if they were trained with all the data (not efficient) Brady Neal Conditional Outcome Modeling 13 / 35

  33. Question: Write down the general form of a COM estimator and a GCOM estimator.

  34. Conditional Outcome Modeling Increasing Data Efficiency Propensity Scores and IPW Other Methods Brady Neal Increasing Data Efficiency 15 / 35

  35. TARNet Brady Neal Increasing Data Efficiency 16 / 35

  36. <latexit sha1_base64="LSZwUq8CzOLCijicp5leilkvu/E=">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</latexit> TARNet COM T Y W Brady Neal Increasing Data Efficiency 16 / 35

  37. <latexit sha1_base64="LSZwUq8CzOLCijicp5leilkvu/E=">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</latexit> TARNet COM T Y W Too biased! Brady Neal Increasing Data Efficiency 16 / 35

  38. <latexit sha1_base64="txXJgFDQnPuQf6d7IXvEg2lQk=">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</latexit> <latexit sha1_base64="LSZwUq8CzOLCijicp5leilkvu/E=">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</latexit> TARNet GCOM COM T = 1 network W Y T Y W T = 0 network Too biased! W Y Brady Neal Increasing Data Efficiency 16 / 35

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  40. <latexit 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TARNet GCOM TARNet (Shalit et al., 2017) COM T = 1 network Y T = 1 W Y T Y W W T = 0 T = 0 network Too much variance! Y W Y Brady Neal Increasing Data Efficiency 16 / 35

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TARNet GCOM TARNet (Shalit et al., 2017) COM T = 1 network Y T = 1 W Y T Y W W T = 0 T = 0 network Just right! Y W Y Brady Neal Increasing Data Efficiency 16 / 35

  42. <latexit 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<latexit 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<latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">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</latexit> TARNet GCOM TARNet (Shalit et al., 2017) COM T = 1 network Y T = 1 W Y T Y W W T = 0 T = 0 network Y Used in a COM W Y estimator: τ = 1 X ˆ (ˆ µ (1 , w i ) − ˆ µ (0 , w i )) n i Brady Neal Increasing Data Efficiency 16 / 35

  43. <latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">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</latexit> TARNet inefficiency Y T = 1 W T = 0 Y Brady Neal Increasing Data Efficiency 17 / 35

  44. <latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">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</latexit> TARNet inefficiency Y T = 1 W T = 0 Y Uses all the data Brady Neal Increasing Data Efficiency 17 / 35

  45. <latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">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</latexit> TARNet inefficiency Only uses treated group data Y T = 1 W T = 0 Y Brady Neal Increasing Data Efficiency 17 / 35

  46. <latexit sha1_base64="m4is/x1qKs6D+47izt0tLlAKSRI=">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</latexit> TARNet inefficiency Only uses treated group data Y T = 1 W T = 0 Y Only uses control group data Brady Neal Increasing Data Efficiency 17 / 35

  47. X-Learner (Künzel et al., 2019) Brady Neal Increasing Data Efficiency 18 / 35

  48. X-Learner (Künzel et al., 2019) 1. Estimate and µ 1 ( x ) <latexit sha1_base64="osuKu1R2XKoXiNSjtTsB5nyHo=">ASrXicrVjbtGEN2kN9e9OepjXtgoBVJAViQ3hfNiICdoCgawAF8SRslBkVREiHeQlJWHUIP/Zq+th/QD+nfdObsUCR1oxTEgsjV7MyZ2bntruh68RJq/XfrdsfzJp5/tfL7xZdf3N3p3aRyMI8s+twI3iF52zdh2Hd8+T5zEtV+GkW16Xde+7I6Oef7y2o5iJ/DPkpvQfu2ZA9/pO5aZEOlq725naCZpxtPr9oPOlZwbUaOmdjxten+cLVXbzVb+DMWB20Z1JX8nQZ3dv5VHdVTgbLUWHnKVr5KaOwqU8X0eaXaqVCor1WKdEiGjmYt9VU7ZLsmLhs4jCJOqLngH69EqpPvxkzhrRFWlz6RiRpqO+Fp0fjPqj6zfqNAu8qHSmw2cYbencF0yNqoZErZLODeV4zUlZOFjrMUhO0NQeJVWaUV9erv0OyH7+XlDnDaNeiQV0cgimktUTWEdEb21X3nlQ/jZBJ9NI7Zp9Wq6ZPvqSPB8QJ8RYZk0jmEp2qoZ+J1H5pt2Mo8LiK2GvEPWqG2bx1Wf7YGB1HVq2DeM6KM1DsalZHX6SzmzmquRJCnS3QxfwIOn7hjogTIa4f85xBHORMtwmQdJiI5wFpC5EdTkH/BfJarIcVzH/bEiJqBbHKgJ5S6ye3O7HTp3QV2RPIpzQ2h3RIHtsYEfIzcyLDXBH9GuCXxZstOboTdQdx7NBa+LsahBuQDPODEt7Qvszs0hblxLNkM+hX6bYp7A17hvG6QbALdRbH0OVL7I+oa7BfPHoylT2bURrQ4cGOIbJlRLRcn0ZifQ9pFTHqxXUlN4uvBYCpQHt7J8Jxh7WxBEdQ/eExj3oYWmDOlTHYod05JW9o2D3rOo1UD+ce3qvjevleV8VKohWXkETS31SFbMEWC7i16zpNfpjhtjdM5q7qwP+8anKNl+1gqj085Vj1k4xBSZVz24vJ1LlvLAeqkgViy7wfEcQSpvtBj1GYomnYL9XcsVRIgHhbenuSErnfWNpmbSYnuiadc4tN1zjsOz7wlTBM51YH3ivPrEJ+KfNbru/RJQV0vd4HKstGNEpnM+vlHYxs+EePOIaL4SvQ6zlu9nXKMxsiv1h8PwFRM7YRJA2Re8hN5av/EzVqRfUKavK/uRIMyXC2SOk8f0Z53CZd4RtPhE4QpPZ/PTyhicEGUqHutjLqvD3AYHna8Hzo5knubQFZXMcXBVr9fKebksbzR9vWxvlu9lWU1fL8uUa+SmI/0gBtbLCrkEcWAJr5ClZxVSAfYETdc2/7aBfT72rHkRSQWXlZIeugxelWBVOXzSvsSdMFIfMIyv7+HB6/R/6Y4AW3rx1w2cqbudzNe/k0l59s6dlc0tvSv7nkuwpJG3uDM6OxzIvKmKu0rkt7Nzia6iBp5d8g7vxT3pOIbsIxtvesdobYbsodlu1h2+owL/c2B9vxkmuIUMZFOmJ0n9qFJr+1nqXntEfz+Fj9SOeDNj2flbrmpqh8Th9LvRVxDwj5kPrdCfri9rihnKb3S2fuso6npCH7TuV2knO7coZchbSN3uXIOtd7C2fZeU36tDIkLn2Cv6lAzKtnOd76+C5H1vfoACfI/Bz/IZCLd5TN8PKcWX2jrLrnr59hfTu09NFR4iF9wRZ56Iz2Opcboh69z1euOty/h5QVXDupvdqc/rK+PO3pwFuP2PiyHn0KTiBtKYVEVfdw0LIJHLaiHFnyW9mzdJNS+PN2xYX8soTvCPpVrZ0pd2rvXp7/n8zi4OLg2b7p2brxaP6k8fyf5sdVfdUw/IT4fqCVXmKfnVUn+qv9Tf6p/aw9p5rVN7o1lv3xKZb1Xprzb4HyFZnJ8=</latexit> ˆ µ 0 ( x ) <latexit sha1_base64="kf8cwsMZn/C6e41SHA8x7p9rqo=">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</latexit> ˆ Brady Neal Increasing Data Efficiency 18 / 35

  49. X-Learner (Künzel et al., 2019) 1. Estimate and µ 1 ( x ) <latexit sha1_base64="osuKu1R2XKoXiNSjtTsB5nyHo=">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</latexit> ˆ µ 0 ( x ) <latexit sha1_base64="kf8cwsMZn/C6e41SHA8x7p9rqo=">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</latexit> ˆ Assume X is a sufficient adjustment set and is all observed covariates Brady Neal Increasing Data Efficiency 18 / 35

  50. X-Learner (Künzel et al., 2019) 1. Estimate and µ 1 ( x ) <latexit sha1_base64="osuKu1R2XKoXiNSjtTsB5nyHo=">ASrXicrVjbtGEN2kN9e9OepjXtgoBVJAViQ3hfNiICdoCgawAF8SRslBkVREiHeQlJWHUIP/Zq+th/QD+nfdObsUCR1oxTEgsjV7MyZ2bntruh68RJq/XfrdsfzJp5/tfL7xZdf3N3p3aRyMI8s+twI3iF52zdh2Hd8+T5zEtV+GkW16Xde+7I6Oef7y2o5iJ/DPkpvQfu2ZA9/pO5aZEOlq725naCZpxtPr9oPOlZwbUaOmdjxten+cLVXbzVb+DMWB20Z1JX8nQZ3dv5VHdVTgbLUWHnKVr5KaOwqU8X0eaXaqVCor1WKdEiGjmYt9VU7ZLsmLhs4jCJOqLngH69EqpPvxkzhrRFWlz6RiRpqO+Fp0fjPqj6zfqNAu8qHSmw2cYbencF0yNqoZErZLODeV4zUlZOFjrMUhO0NQeJVWaUV9erv0OyH7+XlDnDaNeiQV0cgimktUTWEdEb21X3nlQ/jZBJ9NI7Zp9Wq6ZPvqSPB8QJ8RYZk0jmEp2qoZ+J1H5pt2Mo8LiK2GvEPWqG2bx1Wf7YGB1HVq2DeM6KM1DsalZHX6SzmzmquRJCnS3QxfwIOn7hjogTIa4f85xBHORMtwmQdJiI5wFpC5EdTkH/BfJarIcVzH/bEiJqBbHKgJ5S6ye3O7HTp3QV2RPIpzQ2h3RIHtsYEfIzcyLDXBH9GuCXxZstOboTdQdx7NBa+LsahBuQDPODEt7Qvszs0hblxLNkM+hX6bYp7A17hvG6QbALdRbH0OVL7I+oa7BfPHoylT2bURrQ4cGOIbJlRLRcn0ZifQ9pFTHqxXUlN4uvBYCpQHt7J8Jxh7WxBEdQ/eExj3oYWmDOlTHYod05JW9o2D3rOo1UD+ce3qvjevleV8VKohWXkETS31SFbMEWC7i16zpNfpjhtjdM5q7qwP+8anKNl+1gqj085Vj1k4xBSZVz24vJ1LlvLAeqkgViy7wfEcQSpvtBj1GYomnYL9XcsVRIgHhbenuSErnfWNpmbSYnuiadc4tN1zjsOz7wlTBM51YH3ivPrEJ+KfNbru/RJQV0vd4HKstGNEpnM+vlHYxs+EePOIaL4SvQ6zlu9nXKMxsiv1h8PwFRM7YRJA2Re8hN5av/EzVqRfUKavK/uRIMyXC2SOk8f0Z53CZd4RtPhE4QpPZ/PTyhicEGUqHutjLqvD3AYHna8Hzo5knubQFZXMcXBVr9fKebksbzR9vWxvlu9lWU1fL8uUa+SmI/0gBtbLCrkEcWAJr5ClZxVSAfYETdc2/7aBfT72rHkRSQWXlZIeugxelWBVOXzSvsSdMFIfMIyv7+HB6/R/6Y4AW3rx1w2cqbudzNe/k0l59s6dlc0tvSv7nkuwpJG3uDM6OxzIvKmKu0rkt7Nzia6iBp5d8g7vxT3pOIbsIxtvesdobYbsodlu1h2+owL/c2B9vxkmuIUMZFOmJ0n9qFJr+1nqXntEfz+Fj9SOeDNj2flbrmpqh8Th9LvRVxDwj5kPrdCfri9rihnKb3S2fuso6npCH7TuV2knO7coZchbSN3uXIOtd7C2fZeU36tDIkLn2Cv6lAzKtnOd76+C5H1vfoACfI/Bz/IZCLd5TN8PKcWX2jrLrnr59hfTu09NFR4iF9wRZ56Iz2Opcboh69z1euOty/h5QVXDupvdqc/rK+PO3pwFuP2PiyHn0KTiBtKYVEVfdw0LIJHLaiHFnyW9mzdJNS+PN2xYX8soTvCPpVrZ0pd2rvXp7/n8zi4OLg2b7p2brxaP6k8fyf5sdVfdUw/IT4fqCVXmKfnVUn+qv9Tf6p/aw9p5rVN7o1lv3xKZb1Xprzb4HyFZnJ8=</latexit> ˆ µ 0 ( x ) <latexit sha1_base64="kf8cwsMZn/C6e41SHA8x7p9rqo=">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</latexit> ˆ Assume X is a sufficient adjustment set and is all observed covariates 2a. Impute ITEs Brady Neal Increasing Data Efficiency 18 / 35

  51. X-Learner (Künzel et al., 2019) 1. Estimate and µ 1 ( x ) <latexit sha1_base64="osuKu1R2XKoXiNSjtTsB5nyHo=">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</latexit> ˆ µ 0 ( x ) <latexit sha1_base64="kf8cwsMZn/C6e41SHA8x7p9rqo=">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</latexit> ˆ Assume X is a sufficient adjustment set and is all observed covariates 2a. Impute ITEs Treatment group: τ 1 ,i = Y i (1) − ˆ <latexit sha1_base64="PFp+vEFb+AGb+R5v0HjXiL4voUE=">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</latexit> ˆ µ 0 ( x i ) Brady Neal Increasing Data Efficiency 18 / 35

  52. X-Learner (Künzel et al., 2019) 1. Estimate and µ 1 ( x ) <latexit sha1_base64="osuKu1R2XKoXiNSjtTsB5nyHo=">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</latexit> ˆ µ 0 ( x ) <latexit sha1_base64="kf8cwsMZn/C6e41SHA8x7p9rqo=">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</latexit> ˆ Assume X is a sufficient adjustment set and is all observed covariates 2a. Impute ITEs Treatment group: Control group: τ 1 ,i = Y i (1) − ˆ <latexit sha1_base64="PFp+vEFb+AGb+R5v0HjXiL4voUE=">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</latexit> ˆ µ 0 ( x i ) τ 0 ,i = ˆ <latexit sha1_base64="e2TR8WsXFJDPNOPL2kEvLBGOwo=">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</latexit> ˆ µ 1 ( x i ) − Y i (0) Brady Neal Increasing Data Efficiency 18 / 35

  53. X-Learner (Künzel et al., 2019) 1. Estimate and µ 1 ( x ) <latexit sha1_base64="osuKu1R2XKoXiNSjtTsB5nyHo=">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</latexit> ˆ µ 0 ( x ) <latexit sha1_base64="kf8cwsMZn/C6e41SHA8x7p9rqo=">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</latexit> ˆ Assume X is a sufficient adjustment set and is all observed covariates 2a. Impute ITEs Treatment group: Control group: τ 1 ,i = Y i (1) − ˆ <latexit sha1_base64="PFp+vEFb+AGb+R5v0HjXiL4voUE=">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</latexit> ˆ µ 0 ( x i ) τ 0 ,i = ˆ <latexit sha1_base64="e2TR8WsXFJDPNOPL2kEvLBGOwo=">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</latexit> ˆ µ 1 ( x i ) − Y i (0) 2b. Fit a model to predict from in treatment group τ 1 ( x ) <latexit sha1_base64="g+Zn8NhynI+z7xkN0fwgB9i4C6U=">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</latexit> ˆ <latexit sha1_base64="jVUZKdmo4+tDg+lFEkTEJMR+RCs=">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</latexit> ˆ τ 1 ,i <latexit sha1_base64="+NXEOUHMx5ONP5URL5/25IiHVCg=">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</latexit> x i Brady Neal Increasing Data Efficiency 18 / 35

  54. X-Learner (Künzel et al., 2019) 1. Estimate and µ 1 ( x ) <latexit sha1_base64="osuKu1R2XKoXiNSjtTsB5nyHo=">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</latexit> ˆ µ 0 ( x ) <latexit sha1_base64="kf8cwsMZn/C6e41SHA8x7p9rqo=">ASrXicrVjbtGEN2kN9e9OepjXtQoBVJAVmQ3hfNiICdoCgawAF8SRslBkVREiFSZEjKqkPoV/T1/YD+iH9m545uxRJ3R3EgsjV7MyZ2bntrtuh58ZJs/nfrdsfzJp59tfb79xZdf3Nzp3KeRyMIts5swMviF62rdjx3KFzlriJ57wMI8fy25z0R4cyfzFlRPFbjA8Ta5D57Vv9YZu17WtBKTLnbutvpWkLX80uWw+aNnBlRW5VuLEV5b3w+VOrdlo8q86P9gzg5oyfyfBna1/VUt1VKBsNVK+ctRQJRh7ylIxPq/UnmqELTXKgUtwsjlvKMmahuyI3A54LBAHeDZw69XhjrEb8GMKW1Di4dvBMmq+t7wdDukqrfor9a4F2mIyW2HiNd9tg+qAmqg/qOrmMc1M5WVMCx9zLS7sDEmRVdqlFXx9vA7gf3yvAang1EHUhFGNmgeqJoiOiK8tV9l5X362SKfg5HYtHw1bdi+PBIyH+AzAJaFcUxLxdaqema8PqRmh7YKj8eILUf8AyvU9q3C6k7X4DKqehXCewrKQL3DqIy8Smcxd5ZzJQZ5skCX8CfkGI7BiVgXrvwnwuOcibawBQdFiPZ41pC5kfDIP/C+SxXQ8Rzl/bEjFqV2eRST2jqJrc7s9PDu03sCPIp5vrUI76owx6H2BFzM/NindwRfo35y6aN9gy9wbqTeNaxJsmuOnADzLhTLO0J7c/MIm1dClrVfHbVr9TvIO51ekXyug7ZgBbqLI6pa2hif4iuIX7x8RSqeDaj1KnDpx19ZsAtFyfRhJ9D7GKmPXjGdQUb49eC4lSp3bxz5hjn2uSiI6oe4xh3pEuopO1lAHxo5JSav4xmXvmdaZf5J7eq+N6tV5Ias1KrJykNqaqpHZsUSAbG76DXb9DrdcWOubjJjVZv251DcrRsn0jl8SnHqsNs7FOqjCteXLzORWvZ53UGUvxfQ8ch5TqGnrM2gyNpu1C/R2ZKgkYD5tv3+SErnfRNp6ZSUH3jac8Ok6lx1HZt4C02JOtei94vwqxKdGPuv1bXxSUlfLnbOyrIRul0ZrW8y5FD/+iRFzjhfR1yLV8N/1WCzObYn8YvOEcomRsYpA2Re8wNxav/FTV0AtqyKqyPyXSQol49gxvj/lvA9c4R1QyxAUqfB0Oj9ZG4NjUCbGY13OZXWY2+Cy83XI2TKZpzl0RSUzHFLVq7VKXi7KG01fLduZ5ntZVtNXywrlirnpmn4QE+vlGrmEcRAJv5Clp2ukAu4Jmq5t/m0D+4bca0cmLyJj4cUaSZ89Rq8qMFX5fK19CbtgZHwiMr+/hwev2P8mPAHd1I+5bHIjb+Zy1+/l01x+fEP5pL+Df2bS75bI+lwb3CnNJF5sbamhOtkLfLb6blEV1Gdza8I3tx3ScqtlHJNp61ztkbdfNHpbtYtnpMy70N5fa85NpylPE2HTC7DyxS016bT+jlp5j5bxkfoR54M9PJ+VuamqHJOH5l6K+LuA/kA/e6YfHmuKE5Te+WztxlHU+hIftOzO0k5/bMGXIZ0k30LkbWud6ZO8vOatKnlT649An+eg1iXj2L8VbHdzGyvkcHPEHm5/gPgVy8o2yGl+fM8hvlunvu8tXiHcXT48dITa8x8w6j53BUWfmhqh36O5u67k7z6qQnI3vVeZ0VfGn709Xj7GYEj59Gn4ITSmlZEXHYPCymTmNGzDtLfjNrlG5aGm/WtriQV7BOzTdyjFdaftyp7Y3+7+Z+cH5fmPvp0bzxaPak8fm/zZb6q6px7ATwfqCSrzBH61Z/qL/W3+qfysHJWaVXeaNbt4zMt6r0V+n9Dw8cnJ4=</latexit> ˆ Assume X is a sufficient adjustment set and is all observed covariates 2a. Impute ITEs Treatment group: Control group: τ 1 ,i = Y i (1) − ˆ <latexit sha1_base64="PFp+vEFb+AGb+R5v0HjXiL4voUE=">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</latexit> ˆ µ 0 ( x i ) τ 0 ,i = ˆ <latexit sha1_base64="e2TR8WsXFJDPNOPL2kEvLBGOwo=">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</latexit> ˆ µ 1 ( x i ) − Y i (0) 2b. Fit a model to predict from in treatment group τ 1 ( x ) <latexit sha1_base64="g+Zn8NhynI+z7xkN0fwgB9i4C6U=">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</latexit> ˆ <latexit sha1_base64="jVUZKdmo4+tDg+lFEkTEJMR+RCs=">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</latexit> ˆ τ 1 ,i <latexit sha1_base64="+NXEOUHMx5ONP5URL5/25IiHVCg=">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</latexit> x i Fit a model to predict from in control group τ 0 ( x ) <latexit sha1_base64="+B6aB/oNTxH5htFMqtizYEt8Og=">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</latexit> ˆ <latexit sha1_base64="znIcbhd9m39l256M9OyeAVNyrEs=">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</latexit> ˆ τ 0 ,i <latexit sha1_base64="+NXEOUHMx5ONP5URL5/25IiHVCg=">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</latexit> x i Brady Neal Increasing Data Efficiency 18 / 35

  55. X-Learner (Künzel et al., 2019) 1. Estimate and µ 1 ( x ) <latexit sha1_base64="osuKu1R2XKoXiNSjtTsB5nyHo=">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</latexit> ˆ µ 0 ( x ) <latexit sha1_base64="kf8cwsMZn/C6e41SHA8x7p9rqo=">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</latexit> ˆ Assume X is a sufficient adjustment set and is all observed covariates 2a. Impute ITEs Treatment group: Control group: τ 1 ,i = Y i (1) − ˆ <latexit sha1_base64="PFp+vEFb+AGb+R5v0HjXiL4voUE=">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</latexit> ˆ µ 0 ( x i ) τ 0 ,i = ˆ <latexit sha1_base64="e2TR8WsXFJDPNOPL2kEvLBGOwo=">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</latexit> ˆ µ 1 ( x i ) − Y i (0) 2b. Fit a model to predict from in treatment group τ 1 ( x ) <latexit sha1_base64="g+Zn8NhynI+z7xkN0fwgB9i4C6U=">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</latexit> ˆ <latexit sha1_base64="jVUZKdmo4+tDg+lFEkTEJMR+RCs=">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</latexit> ˆ τ 1 ,i <latexit sha1_base64="+NXEOUHMx5ONP5URL5/25IiHVCg=">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</latexit> x i Fit a model to predict from in control group τ 0 ( x ) <latexit sha1_base64="+B6aB/oNTxH5htFMqtizYEt8Og=">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</latexit> ˆ <latexit sha1_base64="znIcbhd9m39l256M9OyeAVNyrEs=">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</latexit> ˆ τ 0 ,i <latexit sha1_base64="+NXEOUHMx5ONP5URL5/25IiHVCg=">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</latexit> x i 3. τ ( x ) = g ( x ) ˆ <latexit sha1_base64="EMunY3MlgWATYCP6VALasYQMI4g=">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</latexit> ˆ τ 0 ( x ) + (1 − g ( x )) ˆ τ 1 ( x ) where is some weighing function between 0 and 1. Example: propensity score <latexit sha1_base64="yWtQkPwj1Y5gecocBX7yqmiIGQI=">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</latexit> g ( x ) Brady Neal Increasing Data Efficiency 18 / 35

  56. Question: What would motivate someone to consider a more complex type of estimation than COM/GCOM?

  57. Conditional Outcome Modeling Increasing Data Efficiency Propensity Scores and IPW Other Methods Brady Neal Propensity Scores and IPW 20 / 35

  58. Propensity scores Brady Neal Propensity Scores and IPW 21 / 35

  59. <latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit> Propensity scores , P ( T = 1 | W ) Brady Neal Propensity Scores and IPW 21 / 35

  60. <latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit> <latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit> Propensity scores e ( W ) , , P ( T = 1 | W ) Brady Neal Propensity Scores and IPW 21 / 35

  61. <latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit> <latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit> Propensity scores e ( W ) , , P ( T = 1 | W ) Given positivity, unconfoundedness given implies unconfoundedness <latexit 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W given the propensity score . <latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit> e ( W ) Brady Neal Propensity Scores and IPW 21 / 35

  62. <latexit 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<latexit sha1_base64="oFsTp51xydUEyPGVLoEpiS4jiV8=">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</latexit> Propensity scores e ( W ) , , P ( T = 1 | W ) Given positivity, unconfoundedness given implies unconfoundedness <latexit 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W given the propensity score . <latexit 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e ( W ) Even if is high-dimensional, is only 1-dimensional! <latexit 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e ( W ) <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W Brady Neal Propensity Scores and IPW 21 / 35

  63. Propensity Score Theorem Given positivity, unconfoundedness given implies unconfoundedness <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W given the propensity score . <latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit> e ( W ) Brady Neal Propensity Scores and IPW 22 / 35

  64. Propensity Score Theorem Given positivity, unconfoundedness given implies unconfoundedness <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W given the propensity score . Equivalently, <latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit> e ( W ) <latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit> ( Y (1) , Y (0)) ⊥ ⊥ T | W = ⇒ ( Y (1) , Y (0)) ⊥ ⊥ T | e ( W ) Brady Neal Propensity Scores and IPW 22 / 35

  65. <latexit sha1_base64="+nLdlJQ0Pu/fpBeP9g62yMnaeY=">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</latexit> Propensity Score Theorem Given positivity, unconfoundedness given implies unconfoundedness <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W given the propensity score . Equivalently, <latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit> e ( W ) <latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit> ( Y (1) , Y (0)) ⊥ ⊥ T | W = ⇒ ( Y (1) , Y (0)) ⊥ ⊥ T | e ( W ) W Graphical Proof: T Y Brady Neal Propensity Scores and IPW 22 / 35

  66. <latexit sha1_base64="+nLdlJQ0Pu/fpBeP9g62yMnaeY=">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</latexit> Propensity Score Theorem Given positivity, unconfoundedness given implies unconfoundedness <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W given the propensity score . Equivalently, <latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit> e ( W ) <latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit> ( Y (1) , Y (0)) ⊥ ⊥ T | W = ⇒ ( Y (1) , Y (0)) ⊥ ⊥ T | e ( W ) W Graphical Proof: <latexit sha1_base64="RknOQawkM/4W5EV5APGqYeOoRPY=">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</latexit> P ( T | W ) T Y Brady Neal Propensity Scores and IPW 22 / 35

  67. <latexit sha1_base64="+nLdlJQ0Pu/fpBeP9g62yMnaeY=">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</latexit> Propensity Score Theorem Given positivity, unconfoundedness given implies unconfoundedness <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W given the propensity score . Equivalently, <latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">ASlHicrVjbtGEN2kN9e9OSrQl76wUQqkgKxKbgrnoQYC2A6KoikcwLc2DgKJoixCvIWkrDqCfqGv7c/0Q/o3PXN2KZK6K4gEkcvZmTOzc+Ou2pHnJmj8d+du+9/8GH219vP3Jp59/sXOvcp5Eg5i2zmzQy+ML9utxPHcwDlL3dRzLqPYafltz7lo9w9l/uLGiRM3DE7T28h56beuA7fr2q1USM7Di+9e7VQb9QY/1uygaQZVZT4n4b2tf9WV6qhQ2WqgfOWoQKUYe6qlEnxfqKZqAi0l2oEWoyRy3lHjdU2ZAfgcsDRArWP6zWeXhqgGfBTChtQ4uHXwxJS31reDoYd0nVd9FvFXgX6RgRW2y8xb1tMH1QU9UDdZVcxrmunKwphYWPuRYXdkakyCrt0oq6uHt4TmG/XG/B6WDUgVSMkQ2aB6qmiI4Yd+1XWXmPfm6Rz8FIbFq8mjZsXxwJmQ/x7QOrhXFCS8VWSz01Xg+o2aGtwuMxYosR/8QKtX3LsLqTNbiMql6F8J6C0ldvMCojL9NZzJ3FXKlBHs/RJfwpOQJwJ6CEzGsX/nPBUc5EG5io8VIXnMtEfOjbpB/4XyWqxHiuUt7EkbNYja51BOZusntzuz0cG8TO4b8CHM96hFf1GCPQ+yYuZl5sUbuGE9DPtm0Z6i1l3Es8a1iTZVQNuiBl3gqU9of2ZWaStG4Fme+u+pX6HcS9Rq9IXtcgG9JCncUJdQUm9gfoGuIXH1ehimczSo06fNrRY7b0Qcv1aSTR9z1WkbB+PIM6wt2j1yKi1Khd/DPk2OeaJKID6h5i3KEekbQyepq39gxLmkV37jsPbNaLeaf1K7ue9NaRS5gpVomKw+oqaEemRVLBMTuotds0+t0x024uvGUVW3an3cNydGyfSKVx6cqw6zsUepMq54cf46561lj3VSYyzF9fgOKBU19AT1mZkNG0X6u/QVEnIeNi8+yYndL2LtuHUzAh03jKA5+uc3njyMxrYLaYU1f0XnF+GeKxkc96fRvfEanL5c5ZWXZGKPRZGa5vMuRQ/okURc40X0dcS1fDP5WYWZdbHfDV4wgygZmxqkdE7zI35Kz9VfSCKrKq7E+JtFBi7j0ijB9MOB8AV3j71BKAIhU+msyPV8bgCJSx8ViXc1kd5ja47Hwdcl6ZzNMcuqLSKQ6p6uVaJS/n5Y2mL5ftTPK9LKvpy2WFcsPcdE0/SIh1uUIuZRxEwi9k6ekKqZDvBE3XNv+hn0B37UDkxexsfBihaTPHqNXFZqfLbSvpRdMDY+EZk/3sKDN+x/Y+6ANvVjLptu5M1c7vatfJrLDzf0bC7pb+jfXPLNCkmH7wZ3QhOZ5ytrSrhOViK/nuxLdBXVeG3DO/Iu7piOY5n3iERbv/UOWNs18w7L3mLZ7jMp9DeX2vOd6Yi7iKHphNl+Ypea9Np+Ri09wztaxofqB+wPmrg+LXNdVFlnz4w9VbE3QPyPvrdEfvi5riR2U3vlvbcZR3H0JD9xuZ0knN7Zg+5CGkTvfORda53Zvay05r0bqUHLr2Dv12BmFfPfLzl8Z2PrM/RIXeQ+T7+XSAXzyjr4eU5s/hEueqcu/j0FeHexdVjR0gM7xGzmNncNSZOSHqt+/hzFlX8ncPVSG5O7pfmdJXxp8+PV3z9DMAR86jd8EpTWtiLjoHBZRJjW7jYRnlvxkVi+dtDTetG1JIa98g3dgupVjutL2q51qc/q/mdnB+V69+WO98fxR9clj87/Nlvpa3VcP4ad9QSVeQK/yv8Gf6m/1T+Vryo/VQ4rx5r17h0j86UqfSq/Q9/qpIV</latexit> e ( W ) <latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit> ( Y (1) , Y (0)) ⊥ ⊥ T | W = ⇒ ( Y (1) , Y (0)) ⊥ ⊥ T | e ( W ) W Graphical Proof: <latexit sha1_base64="HKOiwODvc4MQlTpe9cVjLda9EJo=">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</latexit> P ( T = 1 | W ) T Y Brady Neal Propensity Scores and IPW 22 / 35

  68. <latexit sha1_base64="+nLdlJQ0Pu/fpBeP9g62yMnaeY=">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</latexit> Propensity Score Theorem Given positivity, unconfoundedness given implies unconfoundedness <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W given the propensity score . Equivalently, <latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit> e ( W ) <latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit> ( Y (1) , Y (0)) ⊥ ⊥ T | W = ⇒ ( Y (1) , Y (0)) ⊥ ⊥ T | e ( W ) W Graphical Proof: <latexit sha1_base64="K1oMagBWgBWi4J9/qCI4hSIC3MQ=">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</latexit> e ( W ) T Y Brady Neal Propensity Scores and IPW 22 / 35

  69. <latexit sha1_base64="UopTeG8ASlwcISOBmlfrReRpz+I=">AUyHicrVjbtGEN2oN9dtmj1W1/YyAEcQFZlN0UKBAICOA6KoCkcwHSWkZAkSuJEG8lV1EdQi/9jL62P9TH/k1nzi5FUlcriASLy9mZM5edmd1N/a9VLVa/92ofTxJ59+tvX59hdf3vzq1u2dO+dpNEoc+dKJ/Ch53bVT6XuhfKk85cvXcSLtoOvLV93hMc+/eiuT1IvCM3UVy8vA7odez3NsRaQ3O7Wbna7se2GmvOG72HPUKJGTbct87lm/RK5Mp+dkF4vUmUreWntd5wo7EWj0CX8+5atrP1Wo3XfyvbKE3uTR8vEY5K6Eo/GlvtqGeVpS4ZJU6rKlZAKfJZBTJUVcRYA01nVyBEI+VEgST5xOsPlNU+cgKLMabCgDJsOVApUiduvxyp2FaD2RBJYtF+P5pjIv5wpl5tkV4uenz3KWwLOTtyNCtrPyb2/VWs4WPNT84NIO6MJ/TaGfrX9ERroiEI0YiEFKEQtHYF7ZI6XshDkVLxES7FBnREhp5mJdiIrZJdkRckjhsog7pt09vF4Ya0jtjpB2SItPfwlJWuKe4XFp3ANVP1m/VeJdpiMDNt4Rc+uwQyIqsSAqOvkcs7ryrFPiz8Eb54ZGcMCnvpVDzq0dOnd0X28+8VcUoauSV0Mghmk9UTWEdCT1XNnzAeJsg0/SiG1a7k2XbF+Ejwf0XdIWDaNU1jKtlriqYl6CM0StjKPjxVbjvgHeajtW4XVm/rgYVW1F8x7RpSheEejKvIqneXcWc6lDPJkgS7mV+AIiTslSoS89ih+HnFUM9EhTNZhYyX78CVGfjQN8jPM57ka03oewJ4Uq2YhmzoiU3dFHbndvr07AI7IfmM5gbQw7FokD0S2AlyM49iA9wJvY3x5sBGZ4beRN3xejbIJ86uBuFGNONsXQkdDxzi7R1GdEs8z0QP0O/pHVvICqc1w2SjWChzuIUukKz9m3qGhyXgH6ZypHNKQ3oCGDHANkyJFqhTyOxvu/IixT14xvUjJ4+ohYDpQHtHJ8xgF84hUdQfeYxi70sLRFnawpHho7JhWtHBsPvWdeq4X849rVfW9WK8uFqFTLZGUbmlrigfGYV4DtLkfNMb1Od9wU3k1mrOrC/qJrcI5W7WOpYn2qa+UiGweQquJyFBf7uciXI9RJA2vJse8TRxtSPUNPUZux0bRdqr9jUyUR1sPBMzA5oeudtY1nZjKiByZSPvHpOucdh2d+J0wbOdVB9MrzqxBPjHze67v0zUBdLXeOyqrKJjTKpjOr5T2MJOKjR7ziGi9GrGP48u30zyrNXBf7w+CFc4icscogXRfdRW4s9vxM1KkX1CmrqvHklWZKgrNHTO9Kece4TLvEFpConCFZ9P5ydo1eEKUiYlYD3N5HRY2eOh8Ljg7JvM0h64oNcPBVb1aK+florzR9NWy7jTfq7KavlqWKW+Rm57pBymwXq+RU1gHlghKWXq2RirCnqDp2uZfr2FfiL12ZPIiMRa+WiMZoMdoryJTlc/X2qfQBRMTE5b57T0i+Bb9b4IT0KZxLGTVRtEs5K7eK6aF/HjDyBaSwYbxLSTfrZGU2Bu8KY1lXqytKeY6Xcul82ndniBpN8nPL7raGvjtUhR5z3ZNZ7LMfsNZoXfHNnpAw+x1+W6Xn1LTUh/0oL04wWY4bYxNx8zPHQfQpGPwE9Xc9rLeXwsvqdzxCH9Pq101+ui8nl+ZOqyjHtEyA+pLz5B/9wcNzan7oPK2byq4Q05H8Tc4spuH1z1lyGtInexci6Jty5M+sJn2qGRCXPulfrUEsqmwx3ur1XYys79sRTprFef9DIJfvMtfDK3Jm+c1z3X14+S0tpmePfn10jtTwPkHW+egUrw0N0m9Sx/P3Yk5f4+oKjh3s7t3ZvRV8WdvWX3ckbEUfDo07KCtKaVEZfd12LIKHMqSXG3KW5wzcqNTOPN2paW8ioweG3TraTpStvbtcPZ/+HMz84P2oe/tBsvTiqP35g/r+zJb4Rd8U+xemheEyVeUpxdWq9lft79o/u892493x7pVmrd0wMl+Lymf3z/8BZTo/zg=</latexit> Propensity Score Theorem Given positivity, unconfoundedness given implies unconfoundedness <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W given the propensity score . Equivalently, <latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit> e ( W ) <latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit> ( Y (1) , Y (0)) ⊥ ⊥ T | W = ⇒ ( Y (1) , Y (0)) ⊥ ⊥ T | e ( W ) W Graphical Proof: e ( W ) T Y Brady Neal Propensity Scores and IPW 22 / 35

  70. <latexit sha1_base64="UPh9SvP40VYuMCNm0FqUqLNhHZA=">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</latexit> Propensity Score Theorem Given positivity, unconfoundedness given implies unconfoundedness <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W given the propensity score . Equivalently, <latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit> e ( W ) <latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit> ( Y (1) , Y (0)) ⊥ ⊥ T | W = ⇒ ( Y (1) , Y (0)) ⊥ ⊥ T | e ( W ) W Graphical Proof: e ( W ) T Y Brady Neal Propensity Scores and IPW 22 / 35

  71. <latexit sha1_base64="ZwD937NUd3UMuUBHyuHRqe+s4wk=">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</latexit> Propensity Score Theorem Given positivity, unconfoundedness given implies unconfoundedness <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W given the propensity score . Equivalently, <latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">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</latexit> e ( W ) <latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit> ( Y (1) , Y (0)) ⊥ ⊥ T | W = ⇒ ( Y (1) , Y (0)) ⊥ ⊥ T | e ( W ) W Graphical Proof: e ( W ) T Y Brady Neal Propensity Scores and IPW 22 / 35

  72. <latexit sha1_base64="ZwD937NUd3UMuUBHyuHRqe+s4wk=">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</latexit> Propensity Score Theorem Given positivity, unconfoundedness given implies unconfoundedness <latexit sha1_base64="y6nSigj2UkfW7tvC9oZQnFjROI=">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</latexit> W given the propensity score . Equivalently, <latexit sha1_base64="Xx4JzALq6sUvKk6mTfd9apHdEfA=">ASlHicrVjbtGEN2kN9e9OSrQl76wUQqkgKxKbgrnoQYC2A6KoikcwLc2DgKJoixCvIWkrDqCfqGv7c/0Q/o3PXN2KZK6K4gEkcvZmTOzc+Ou2pHnJmj8d+du+9/8GH219vP3Jp59/sXOvcp5Eg5i2zmzQy+ML9utxPHcwDlL3dRzLqPYafltz7lo9w9l/uLGiRM3DE7T28h56beuA7fr2q1USM7Di+9e7VQb9QY/1uygaQZVZT4n4b2tf9WV6qhQ2WqgfOWoQKUYe6qlEnxfqKZqAi0l2oEWoyRy3lHjdU2ZAfgcsDRArWP6zWeXhqgGfBTChtQ4uHXwxJS31reDoYd0nVd9FvFXgX6RgRW2y8xb1tMH1QU9UDdZVcxrmunKwphYWPuRYXdkakyCrt0oq6uHt4TmG/XG/B6WDUgVSMkQ2aB6qmiI4Yd+1XWXmPfm6Rz8FIbFq8mjZsXxwJmQ/x7QOrhXFCS8VWSz01Xg+o2aGtwuMxYosR/8QKtX3LsLqTNbiMql6F8J6C0ldvMCojL9NZzJ3FXKlBHs/RJfwpOQJwJ6CEzGsX/nPBUc5EG5io8VIXnMtEfOjbpB/4XyWqxHiuUt7EkbNYja51BOZusntzuz0cG8TO4b8CHM96hFf1GCPQ+yYuZl5sUbuGE9DPtm0Z6i1l3Es8a1iTZVQNuiBl3gqU9of2ZWaStG4Fme+u+pX6HcS9Rq9IXtcgG9JCncUJdQUm9gfoGuIXH1ehimczSo06fNrRY7b0Qcv1aSTR9z1WkbB+PIM6wt2j1yKi1Khd/DPk2OeaJKID6h5i3KEekbQyepq39gxLmkV37jsPbNaLeaf1K7ue9NaRS5gpVomKw+oqaEemRVLBMTuotds0+t0x024uvGUVW3an3cNydGyfSKVx6cqw6zsUepMq54cf46561lj3VSYyzF9fgOKBU19AT1mZkNG0X6u/QVEnIeNi8+yYndL2LtuHUzAh03jKA5+uc3njyMxrYLaYU1f0XnF+GeKxkc96fRvfEanL5c5ZWXZGKPRZGa5vMuRQ/okURc40X0dcS1fDP5WYWZdbHfDV4wgygZmxqkdE7zI35Kz9VfSCKrKq7E+JtFBi7j0ijB9MOB8AV3j71BKAIhU+msyPV8bgCJSx8ViXc1kd5ja47Hwdcl6ZzNMcuqLSKQ6p6uVaJS/n5Y2mL5ftTPK9LKvpy2WFcsPcdE0/SIh1uUIuZRxEwi9k6ekKqZDvBE3XNv+hn0B37UDkxexsfBihaTPHqNXFZqfLbSvpRdMDY+EZk/3sKDN+x/Y+6ANvVjLptu5M1c7vatfJrLDzf0bC7pb+jfXPLNCkmH7wZ3QhOZ5ytrSrhOViK/nuxLdBXVeG3DO/Iu7piOY5n3iERbv/UOWNs18w7L3mLZ7jMp9DeX2vOd6Yi7iKHphNl+Ypea9Np+Ri09wztaxofqB+wPmrg+LXNdVFlnz4w9VbE3QPyPvrdEfvi5riR2U3vlvbcZR3H0JD9xuZ0knN7Zg+5CGkTvfORda53Zvay05r0bqUHLr2Dv12BmFfPfLzl8Z2PrM/RIXeQ+T7+XSAXzyjr4eU5s/hEueqcu/j0FeHexdVjR0gM7xGzmNncNSZOSHqt+/hzFlX8ncPVSG5O7pfmdJXxp8+PV3z9DMAR86jd8EpTWtiLjoHBZRJjW7jYRnlvxkVi+dtDTetG1JIa98g3dgupVjutL2q51qc/q/mdnB+V69+WO98fxR9clj87/Nlvpa3VcP4ad9QSVeQK/yv8Gf6m/1T+Vryo/VQ4rx5r17h0j86UqfSq/Q9/qpIV</latexit> e ( W ) <latexit sha1_base64="GnarY3MBDQ50l4QKbX7xledsOgM=">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</latexit> ( Y (1) , Y (0)) ⊥ ⊥ T | W = ⇒ ( Y (1) , Y (0)) ⊥ ⊥ T | e ( W ) W Graphical Proof: See non-graphical proof in Appendix A.2 of the course book e ( W ) T Y Brady Neal Propensity Scores and IPW 22 / 35

  73. Implications for the Positivity-Unconfoundedness Tradeoff Brady Neal Propensity Scores and IPW 23 / 35

  74. Implications for the Positivity-Unconfoundedness Tradeoff Recall that overlap decreases with the dimensionality of the adjustment set Brady Neal Propensity Scores and IPW 23 / 35

  75. Implications for the Positivity-Unconfoundedness Tradeoff Recall that overlap decreases with the dimensionality of the adjustment set <latexit sha1_base64="VPvP5OxvTxlPGVcROXV6+AtJi24=">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</latexit> W Brady Neal Propensity Scores and IPW 23 / 35

  76. Implications for the Positivity-Unconfoundedness Tradeoff Recall that overlap decreases with the dimensionality of the adjustment set The propensity score magically reduces the dimensionality of the adjustment set done to 1! <latexit sha1_base64="VPvP5OxvTxlPGVcROXV6+AtJi24=">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</latexit> W <latexit sha1_base64="yHnJqHzwl4G8pVn8qDn+vJ3Kk=">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</latexit> e ( W ) Brady Neal Propensity Scores and IPW 23 / 35

  77. <latexit sha1_base64="jLupkGKl6KgsQ3x2L+BioPxjiM=">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</latexit> Implications for the Positivity-Unconfoundedness Tradeoff Recall that overlap decreases with the dimensionality of the adjustment set The propensity score magically reduces the dimensionality of the adjustment set done to 1! <latexit sha1_base64="VPvP5OxvTxlPGVcROXV6+AtJi24=">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</latexit> W <latexit sha1_base64="yHnJqHzwl4G8pVn8qDn+vJ3Kk=">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</latexit> e ( W ) Unfortunately, we don’t have access to it. The best we can do is model it, shifting the high-dimensionality problem to the modeling of e ( W ) , P ( T = 1 | W ) Brady Neal Propensity Scores and IPW 23 / 35

  78. Questions: 1. What is the intuition behind why we can condition on e(W) instead of W? 2. What is attractive about conditioning on e(W) as opposed to W? 3. Why does this not solve positivity issues when W is high-dimensional?

  79. Pseudo-populations Brady Neal Propensity Scores and IPW 25 / 35

  80. <latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">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</latexit> Pseudo-populations Regular population W T Y Brady Neal Propensity Scores and IPW 25 / 35

  81. <latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">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</latexit> Pseudo-populations Regular population W T Y <latexit sha1_base64="nXPx5E5upvT2MNCbVd0JFQJD9c=">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</latexit> P ( T | W ) 6 = P ( T ) Brady Neal Propensity Scores and IPW 25 / 35

  82. <latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">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</latexit> Pseudo-populations Reweighted population Regular population (Pseudo-population) W T Y or <latexit sha1_base64="nXPx5E5upvT2MNCbVd0JFQJD9c=">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</latexit> P ( T | W ) 6 = P ( T ) <latexit 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P ( T | W ) = P ( T ) <latexit sha1_base64="NYa6JfkFwEVZRzboxfl8ts8u2dQ=">AS3nicrVhbxtFJ6Wwi31DzystRBKpJj7FCUvliqlLRCiEqplBvUVbDX63jlvbG7jktv/KAhHjlZ/AK/4JH/g3f+WbW6/U9VWN5d3zmnO+cObeZSTvy3CSt1f67c/et95972t97c/+PCjz/ZuVc6S8JBbDunduiF8UW7lTieGzinqZt6zkUOy2/7Tn7f6hzJ9fO3HihsFJehM5L/3WVeB2XbuVgnS5s9uMHjRTSKS+E6RW03c7VtMOg24CDqQ+9JqWPXLnXKtWuOfNT+om0FZmb/j8N7Wv6qpOipUthoXzkqUCnGnmqpBJ8Xq5qKgLtpRqBFmPkct5RY7UN2QG4HC0QO3jeYVfLw1wG/BTChtQ4uHbwxJS31heDoYd0nVb9FvTfEu0zEith4g3fbYPqgpqoH6jq5jHNTOVlTCgsfcS0u7IxIkVXahRV18fbwO4X98rwBp4NRB1IxRjZoHqiaIjpivLVfZeU9+rlFPgcjsWn5atqwfXkZD7Epw+sFsYJLRVbLfXUeD2gZoe2Co/HiC1H/AUr1PatwupO1uAyqnoVwnsCSl+9wqiIvErndO4s50oN8niBLuFPyRGAOwElZF678J8LjmIm2sAUHS1G8opriZgfVYP8HezXI0Qz3akzBqFrPJpZ7I1E1ud2anh3eb2DHkR5jrUY/4ogJ7HGLHzM3MixVyx/g15C+bNtoz9CrTuJZwZokuyrADTHjTrC0J7Q/M4u0dSPQLPZU9Tv4O4V+gVyesKZENaqLM4oa7AxL6BriF+8fEUqng2o1Sow6cdPWZLH7Rcn0YSfV9hFQnrxzOoI7w9ei0iSoXaxT9Djn2uSI6oO4hxh3qEWkLnayqDowd4JW8Y3L3jOv1WL+Se3qvjerVeQCVqplsrJBTX10KxYIiB2T3vNr1Od9yEqxvPWNWm/XnXkBwt2idSeXyKseowG3uUKuKFxevc9Fa9lknFcZSfH8FjgaluoaesDYjo2l7qv4OTZWEjIfNt29yQte7aBvOzIxA942nPDpOpcdR2Z+BmaLOdWk96bnVyE+MfJZr2/jMyJ1tdwZK6soG2M0msyslnc5cugfPZKIa7yIvo64ls8nX2tqZlPsN4MXzCFKxqYGaVP0DnNj8cpPVBm9oIysKvpTIi2UmGePCOPdCecucIW3Ty0BKFLho8n8eG0MjkAZG491OZfVYW6Dy87XIWfTZJ7m0BWVznBIVa/WKnm5KG80fbVsZ5LvRVlNXy0rlGvmpmv6QUKsizVyKeMgEv5Ulp6skQq5J2i6tvmHDewLuNcOTF7ExsLzNZI+e4xeVWiq8tla+1J2wdj4RGR+fA0PXrP/jXkCuq0fc9n0Vt7M5W5ey6e5/PCWns0l/Vv6N5d8tUbS4d7gTmgi83xtTQnX8VounU/r9gQHu0l2ftHVuGzDS/Knt0xncky+41khd4dG+wBFbPXZbtdkpNpvqgS+35CXbE08bQdMzs3LFHTdoH36LmnmEvl/Gh+hrniDqeTwvdVNUOc8PTF1O4+4D+QB98Yj98/a4kTl17xXO5kUdT6Ah+47NLSbn9sxZcxnSbfQuRtY10Zk785q0qeaHrj0Sf9mDWJeZYvxVsd3MbK+b4c8aebn/TeBPH2X2Qwvz5nlN8919+Hlt7QI7y6eHjtHYniPmHUeO4ijTs1NUu/Sh3N3YsnfVSF5O7ofmlGXxF/9pZ1xVvSABw5jz4tp5TWtGnEZfe1iDKpOZUkvNvkN7hq4Uam8WZtS6byjd4DdOtHNOVti93yvXZ/+HMD872q/VvqrXnD8uPH5n/72ypz9R9QB+OlCPUZnH8KutflN/qb/VP6WfSr+Wfi/9oVnv3jEyn6rCX+nP/wGU6hL</latexit> P ( T | W ) = 1 Brady Neal Propensity Scores and IPW 25 / 35

  83. <latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">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</latexit> <latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">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</latexit> Pseudo-populations Reweighted population Regular population (Pseudo-population) W W T Y T Y or <latexit 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P ( T | W ) 6 = P ( T ) <latexit 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P ( T | W ) = P ( T ) <latexit 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P ( T | W ) = 1 Brady Neal Propensity Scores and IPW 25 / 35

  84. <latexit 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<latexit 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<latexit sha1_base64="7bYfiGO9BLg1PVktDdKv2r6qy5s=">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</latexit> Pseudo-populations Reweighted population Regular population (Pseudo-population) W W T Y T Y or <latexit 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P ( T | W ) 6 = P ( T ) <latexit 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P ( T | W ) = P ( T ) <latexit sha1_base64="NYa6JfkFwEVZRzboxfl8ts8u2dQ=">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</latexit> P ( T | W ) = 1 1 Reweighting intuition: P ( T | W ) · P ( T | W ) = 1 Brady Neal Propensity Scores and IPW 25 / 35

  85. Inverse probability weighting (IPW) Brady Neal Propensity Scores and IPW 26 / 35

  86. <latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">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</latexit> Inverse probability weighting (IPW) W T Y Brady Neal Propensity Scores and IPW 26 / 35

  87. <latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">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</latexit> <latexit sha1_base64="x9MwVMOEAp+s2GhuQc6v5IjE2Eg=">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</latexit> Inverse probability weighting (IPW) t ) Y W T Y Brady Neal Propensity Scores and IPW 26 / 35

  88. <latexit 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<latexit 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<latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">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</latexit> Inverse probability weighting (IPW)  t ) Y W E P ( t | W ) T Y Brady Neal Propensity Scores and IPW 26 / 35

  89. <latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">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</latexit> <latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">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</latexit> Inverse probability weighting (IPW)  ( T = t ) Y � W E [ Y ( t )] = E P ( t | W ) T Y Brady Neal Propensity Scores and IPW 26 / 35

  90. <latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">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</latexit> <latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">AUcHicrVjpbtGEN64l+sesRP/KNACZWIHcApZld0WCVAICOokKNqkSAfaS3DoMiVRIhXyZVm9Bj9UH6s6/RJ+jMt0uR1GkHkWBxOTvzbEzs7tux76Xqkbj31sr73/wYcfrX689smn31+e3jznEaDRJHjmRHyVv2nYqfS+UR8pTvnwTJ9IO2r48afcPeP7kQiapF4WH6jKWZ4HdDb2O59iKSOcbt/5utWXCzPl9a9iz1GDRI7WLPN5YP0WuTIdv7dCej1Nla3kmbXTcqKwEw1Cl/AfWraydhq1xkMr2y5PbI9+nCeuyFAVyFA9tE7b0o+Gli87ympGHauMcMaIY94FeNFAOVEgx2iJ1+3NgTOsOVjJ4Wdut+Twg1Zsq96kq5J4KvaXTFrAn9s3zV2KxEzelgzdygqdr2816g18rOnBnhlsCfN5FW2s/iNawhWRcMRABEKUCga+8IWKX1PxZ5oiJhoZyIjWkIjD/NSjMQayQ6ISxKHTdQ+/Xbp7dRQ3pnzBTSDmnx6S8hSUs8MDwujTug6ifrt0q83RkwGYbL+nZNpgBUZXoEXWZXM5XTn2SZGFj+GLR3bGoLCXTsWjDj19eldkP/9eEqekUtSCY0covlE1RTWkdBTx5U97yHONvgkjdim+d60yfb5K8HzEX37hGXTOIWlbKslnpuoh9AsYSvz+Fix+Yh/kYfavkVYnbEPHlZVe8G8h0TpiysaVZEX6SznznwuZBHM3QxvwJHSNwpUSLktUfx84ijmokOYbIOGyvZhS8x8qNukH/BfJ6rMa3nLuxJsWoWsmDntjUTWF3bqdPzawE5LPaK4HPRyLGtkjgZ0gN/Mo1sCd0NsQbw5sdCbodQdr2eNfOLsqhFuRDPeGEtHQsczt0hblxHNMt9d8QL6Ja17DVHhvK6RbAQLdRan0BWatW9S1+C4BPTLVI5sTqlBRwA7esiWPtEKfRqJ9X1LXqSoH9+gZvT0EbUYKDVo5/gMQ7gE6/oALqHNHah6Ut6mR18cjYMapo5dh46D3TWi3kH9eu7nuTWlkuRKVaJiub0NQ3xuPeQXY7nLUHNPrdMdN4d1owqo27C+6Budo1T6WKtanulYusrEHqSouR3G2n7N82Ued1LCWHPsucTQh1TH0FLUZG01rpfo7MFUSYT0cPAOTE7reWdtwYiYjemAi5ROfrnPecXjmT8K0kVMtRK8vwjxmZHPe32bvhmoi+WOUVlV2YRG2XhmsbyHkUR89IhXOPFiHUMX+6N/6zSzHWx3w1eOIXIGasM0nXRXeTGbM8PxRb1gi3Kqmo8eaWZkuDsEdN4e8y5TbjM24eWkChc4dl4frR0DZ4SZWQi1sFcXoeFDR46nwvOlsk8zaErSk1wcFUv1sp5OStvNH2xrDvO96qspi+WZcoFctMz/SAF1pslcgrwBJBKUsPl0hF2BM0Xdv8+zXsC7HXDkxeJMbCkyWSAXqM9ioyVflyqX0KXTAxMWGZP94ighfofyOcgG4ax0JW3SiahdzlW8W0kB/eMLKFZHD+BaSV0skJfYGb0xjmdLa4q5Xi3l0vm0bE+QtJvk5xdbTX8timKvGe7pjNZr/hrNC7YxM9oGb2uny3y0+pakPetBenGAznDaGpmPm545daNIx+Jlq7iXt5Tw+EN/ROWKPfp9Xut1Ufk8PzB1WcbdJ+RH1Befon/eHDc2p+7dytm8quMZacj/RuYWU3D75qw5D+kmemcj65pwp868k5r0qaZHXPqkf7kEsaiy2XiL13c2sr5vRzhpFuf9d4FcvstcD6/Imfk3z2X34fm3tJieHfr10TlSw/sUWejg0hxZG6Sepc+mLoTc/7uU1Vw7mb370zoq+JP3rK6uCUNiKPg0adlBWlNKyPOu6/FkFHmVJLiblPc4OqVG5nGm7QtLeVYPCapltJ05XWzte39ib/hzM9ON6v7/1Qb7ze3ry2Px/Z1V8Ke6LHYrTI/GEKvMVxdVZ+Wrlp5VfV17c/W/zi82vN+9p1pVbRuauqHw2v/kfEPEkg=</latexit> Inverse probability weighting (IPW)  ( T = t ) Y � W E [ Y ( t )] = E P ( t | W ) See proof in Appendix A.3 of the course book T Y Brady Neal Propensity Scores and IPW 26 / 35

  91. <latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">AUcHicrVjpbtGEN64l+sesRP/KNACZWIHcApZld0WCVAICOokKNqkSAfaS3DoMiVRIhXyZVm9Bj9UH6s6/RJ+jMt0uR1GkHkWBxOTvzbEzs7tux76Xqkbj31sr73/wYcfrX689smn31+e3jznEaDRJHjmRHyVv2nYqfS+UR8pTvnwTJ9IO2r48afcPeP7kQiapF4WH6jKWZ4HdDb2O59iKSOcbt/5utWXCzPl9a9iz1GDRI7WLPN5YP0WuTIdv7dCej1Nla3kmbXTcqKwEw1Cl/AfWraydhq1xkMr2y5PbI9+nCeuyFAVyFA9tE7b0o+Gli87ympGHauMcMaIY94FeNFAOVEgx2iJ1+3NgTOsOVjJ4Wdut+Twg1Zsq96kq5J4KvaXTFrAn9s3zV2KxEzelgzdygqdr2816g18rOnBnhlsCfN5FW2s/iNawhWRcMRABEKUCga+8IWKX1PxZ5oiJhoZyIjWkIjD/NSjMQayQ6ISxKHTdQ+/Xbp7dRQ3pnzBTSDmnx6S8hSUs8MDwujTug6ifrt0q83RkwGYbL+nZNpgBUZXoEXWZXM5XTn2SZGFj+GLR3bGoLCXTsWjDj19eldkP/9eEqekUtSCY0covlE1RTWkdBTx5U97yHONvgkjdim+d60yfb5K8HzEX37hGXTOIWlbKslnpuoh9AsYSvz+Fix+Yh/kYfavkVYnbEPHlZVe8G8h0TpiysaVZEX6SznznwuZBHM3QxvwJHSNwpUSLktUfx84ijmokOYbIOGyvZhS8x8qNukH/BfJ6rMa3nLuxJsWoWsmDntjUTWF3bqdPzawE5LPaK4HPRyLGtkjgZ0gN/Mo1sCd0NsQbw5sdCbodQdr2eNfOLsqhFuRDPeGEtHQsczt0hblxHNMt9d8QL6Ja17DVHhvK6RbAQLdRan0BWatW9S1+C4BPTLVI5sTqlBRwA7esiWPtEKfRqJ9X1LXqSoH9+gZvT0EbUYKDVo5/gMQ7gE6/oALqHNHah6Ut6mR18cjYMapo5dh46D3TWi3kH9eu7nuTWlkuRKVaJiub0NQ3xuPeQXY7nLUHNPrdMdN4d1owqo27C+6Budo1T6WKtanulYusrEHqSouR3G2n7N82Ued1LCWHPsucTQh1TH0FLUZG01rpfo7MFUSYT0cPAOTE7reWdtwYiYjemAi5ROfrnPecXjmT8K0kVMtRK8vwjxmZHPe32bvhmoi+WOUVlV2YRG2XhmsbyHkUR89IhXOPFiHUMX+6N/6zSzHWx3w1eOIXIGasM0nXRXeTGbM8PxRb1gi3Kqmo8eaWZkuDsEdN4e8y5TbjM24eWkChc4dl4frR0DZ4SZWQi1sFcXoeFDR46nwvOlsk8zaErSk1wcFUv1sp5OStvNH2xrDvO96qspi+WZcoFctMz/SAF1pslcgrwBJBKUsPl0hF2BM0Xdv8+zXsC7HXDkxeJMbCkyWSAXqM9ioyVflyqX0KXTAxMWGZP94ighfofyOcgG4ax0JW3SiahdzlW8W0kB/eMLKFZHD+BaSV0skJfYGb0xjmdLa4q5Xi3l0vm0bE+QtJvk5xdbTX8timKvGe7pjNZr/hrNC7YxM9oGb2uny3y0+pakPetBenGAznDaGpmPm545daNIx+Jlq7iXt5Tw+EN/ROWKPfp9Xut1Ufk8PzB1WcbdJ+RH1Befon/eHDc2p+7dytm8quMZacj/RuYWU3D75qw5D+kmemcj65pwp868k5r0qaZHXPqkf7kEsaiy2XiL13c2sr5vRzhpFuf9d4FcvstcD6/Imfk3z2X34fm3tJieHfr10TlSw/sUWejg0hxZG6Sepc+mLoTc/7uU1Vw7mb370zoq+JP3rK6uCUNiKPg0adlBWlNKyPOu6/FkFHmVJLiblPc4OqVG5nGm7QtLeVYPCapltJ05XWzte39ib/hzM9ON6v7/1Qb7ze3ry2Px/Z1V8Ke6LHYrTI/GEKvMVxdVZ+Wrlp5VfV17c/W/zi82vN+9p1pVbRuauqHw2v/kfEPEkg=</latexit> <latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">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</latexit> <latexit sha1_base64="drjK7+xGMxArDmUZ1XIxVBgk+bo=">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</latexit> Inverse probability weighting (IPW)  ( T = t ) Y � W E [ Y ( t )] = E P ( t | W ) See proof in Appendix A.3 of the course book T Y  ( T = 1) Y  ( T = 0) Y � � τ , E [ Y (1) − Y (0)] = E − E e ( W ) 1 − e ( W ) Brady Neal Propensity Scores and IPW 26 / 35

  92. <latexit 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<latexit 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<latexit sha1_base64="CtPqyah4M0xn8axhaP4crWih7V8=">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</latexit> <latexit sha1_base64="KdwhJ0Jld01DCjPFpk6t3N087eM=">ATn3icrVhZb9tGEF6lV+IesaO35oWJUiABZFVyUzgvAgLYTouiKRzAV2oaBkVSEiFeJSmrDqH/sj+hv6Jzny7FEldlINYELmanflmdq7dS90nThpt/+t3fvs8y+/Or+g62v/n2u4fbO4/O4mAcmfapGbhBdNEzYt1fPs0cRLXvgj2/B6rn3eGx3w/PmNHcVO4J8kt6F95RkD3+k7pEQ6Xr7H/3oUg/GiRl49nM9IdHEs/3kxnBfXGldT/SdNfuJ5ea3o8M9Ud3yqwMUdJRsuwpqkelvE03XMsTcDvx+MfYtsejHV9MgZDJOr6+1Gu9XGn7Y46KhBQ6i/42DnQU3owhKBMVYeMIWvkho7ApDxPS5FB3RFiHRrkRKtIhGDuZtMRVbJDsmLps4DKO6DmgX5eK6tNvxowhbZIWl74RSWriB8Vj0bgPqnyzfq3Au0pHCmy28ZbePYXpETURQ6JWyWcm8v16ONVrDqhNbzCah1aSQgK+8EsrblPb5d+J7RCft4Sp0ji6QiGplEc4kqKawjorf0PtmiEgY4LNpxFavs3ud1Twf0GdEWAaNY1jKtmrijYqLD802bGUeFzFdjfg3rVDatw6rP1uDg7jLVTDvCVFG4gONysjrdBazazVXopCnS3QxfwIOn7hjogTIfIf85xBHOVdNwmQdBiI5wFpCZFBLIf+G+SybQ4rnLuyJETUN+eZAT6gqK7c7s9Oldw/YEcmnNDeEHvZFk+yxgR0hezMvNsEd0a8Jfpmw0Zyjt1CZHM8mrYmzq0m4Ac04MyzpCenPzCJpXUo0TX12xe/Qb1Pcm/AK53WTZANYKLM4hi5fxb5LfYX94tGTqezZjNKEDg92DJEtI6Ll+iQS6/uRVhGjflyFmtLbhdCoDShnf0zwdjDmjiY+ie0NiCHpbWqNe1xL6yY1rSyr5x0J0WtWrIP65d2RntbKcj0rVFZ2oaktXqoVcwTY7qLXTNUNZU+OsbrpnFU92J93Dc7Rsn0slcenHCsL2TiEVBmXvbh8ncvWsoc6aSKW7PsBcXQh1Vf0GLUZKk1bhfo7UFUSIB4m3p7KCVnvrG0yN5MS3VOecolP1jnvSTzF2EayCkd3ivOr0M8UvLZbsAdPwV1vdwZKqsG9Eonc2sl3cwsuEfOeKIS7wQvg6xliezr1aY2RT70+D5C4icsYlC2hTdQm4sX/mJaFAvaFBWlf3JkWZKhNJSONnM85nhMu8I2jxicIVns7mp5UxOCTKVHmsj7msDnMbHQ+C5y6yjzJISsqmePgql6vlfNyWd5I+npZa5bvZVlJXy/LlBvkpqP6Qysiwq5BHFgCa+QpScVUgH2BEmXNr/fwD4fe+1Y5UWkLDyvkPTQY+SqAlWVbyvtS9AFI+UTlvnzIzx4g/43xQnorn7MZM7eTOXu/0on+bykzt6Npf07ujfXPJDhaSNvcGZ0VjmXWVNMdxJZfMp6o9wabdJDu/yGpr4tkjL/KebanOpKn9hrNC7o5d9IDmRv15cd/hSu6oOs73yuyMGxe6qAPs/Pyb4qwyUf02O7Xswk6p6Veq2Ld0EuDxgfiJTiEder4p9eZNUfk2MFZVXcTdI+R96qH6L53xw3VmX23dLIv6zgiDdl3qu5AOberTqrkO6idzmyrChr4cQ8r0meiYbEJe8JtxWIeY0ux1sf3+XI8j4f4Jya3xY+BXLxJrQZXp4zq+tVbfp1Xe8kN59eroO7HiPUTWueg/tjhV91C5x8s3Kg5f/eoKjh306eP5vSV8efvaAPcscbEkfPIs3YCaUkrIq67YWQSdSZJsbNKL/tUr3OYk3b1tcyCtP4XVL7JVT9u63m505v9HtDg42t1fm61371svH6l/n90XzwWT8Vz8tO+eE2VeUx+NcV/tZ3a97XH9Sf1X+p/1I8l672akqmL0l/9/f8Zut1</latexit> Inverse probability weighting (IPW)  ( T = t ) Y � W E [ Y ( t )] = E P ( t | W ) See proof in Appendix A.3 of the course book T Y  ( T = 1) Y  ( T = 0) Y � � τ , E [ Y (1) − Y (0)] = E − E e ( W ) 1 − e ( W ) τ = 1 e ( w i ) − 1 y i y i X X ˆ ˆ 1 − ˆ e ( w i ) n 1 n 0 i : t i =1 i : t i =0 Brady Neal Propensity Scores and IPW 26 / 35

  93. Questions: 1. What happens if the estimated propensity score for some unit is 1 or 0? 2. What happens if the estimated propensity score is near 1 or 0?

  94. IPW CATE estimation Brady Neal Propensity Scores and IPW 28 / 35

  95. IPW CATE estimation Not quite as natural with IPW as with COM, so beyond scope of course Brady Neal Propensity Scores and IPW 28 / 35

  96. <latexit sha1_base64="WNmK867A4qokDcAGsHv/v/sp128=">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</latexit> IPW CATE estimation Not quite as natural with IPW as with COM, so beyond scope of course Simple extension: ✓ ( t i = 1) y i τ ( x ) = 1 ( t i = 0) y i ◆ X ˆ − e ( w i ) ˆ 1 − ˆ e ( w i ) n x i : x i = x Brady Neal Propensity Scores and IPW 28 / 35

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