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fixing spelling error
Signed-off-by: Nathaniel <[email protected]>
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docs/source/notebooks/inv_prop_latent.ipynb

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"However, this joint approach introduces a feedback loop: the outcome $Y$ can influence the estimation of the treatment mechanism $P(T | X)$. This violates the original logic of design-based inference, where treatment assignment should be modeled independently of the observed outcomes. This phenomenon is often subtle but can lead to biased treatment effect estimates.\n",
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"Across several examples, we have shown that fitting a full joint model distorts the treatment effect estimate relative to a two-step (modular) approach.\n",
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"In other cases, joint and modular approaches yield nearly identical estimates — usually when the treatment mechanism is well-identified from covariates alone. With these observations in scope, we recommend that practioners generally follow a two-step or modular approach. Either two-stage inverse propensity score weighting or regression adjustment with the propensity score as an additional covariate. Both methods are available now in `CausalPy`. \n",
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"In other cases, joint and modular approaches yield nearly identical estimates — usually when the treatment mechanism is well-identified from covariates alone. With these observations in scope, we recommend that practitioners generally follow a two-step or modular approach. Either two-stage inverse propensity score weighting or regression adjustment with the propensity score as an additional covariate. Both methods are available now in `CausalPy`. \n",
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"Framed this way we can see that joint model violates the temporal precedence of the treatment assignment and outcome process. The 2-stage Bayesian procedures ensure that the causal ordering encoded in the actual data generating process is respected in the estimation process. The confounding adjustment achieved with propensity score must occur without access to information about the outcome. A well-specified propensity score model can substantially improve causal estimates (as we've seen), especially when the outcome model is weak or mis-specified. Propensity scores do not only serve to reduce dimensionality; they formalize the treatment mechanism and encode information that the outcome model might fail to recover. This explains their continued prominence in modern causal inference.\n",
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