Sharp Bounds on Functionals of the Joint Distribution in the Analysis of Treatment Effects

2017 ◽  
Author(s):  
Thomas Russell
Keyword(s):  
Joint Distribution ◽  
Treatment Effects ◽  
Sharp Bounds

Combining planned and discovered comparisons in observational studies

Biostatistics ◽  
2018 ◽  
Vol 21 (3) ◽  
pp. 384-399 ◽  
Author(s):  
Paul R Rosenbaum
Keyword(s):  
Cognitive Decline ◽  
Observational Studies ◽  
Multiple Testing ◽  
Joint Distribution ◽  
Treatment Effects ◽  
A Priori ◽  
Principal Component ◽  
R Package ◽  
Test Statistic

Summary In observational studies of treatment effects, it is common to have several outcomes, perhaps of uncertain quality and relevance, each purporting to measure the effect of the treatment. A single planned combination of several outcomes may increase both power and insensitivity to unmeasured bias when the plan is wisely chosen, but it may miss opportunities in other cases. A method is proposed that uses one planned combination with only a mild correction for multiple testing and exhaustive consideration of all possible combinations fully correcting for multiple testing. The method works with the joint distribution of $\kappa^{T}\left( \mathbf{T}-\boldsymbol{\mu}\right) /\sqrt {\boldsymbol{\kappa}^{T}\boldsymbol{\Sigma\boldsymbol{\kappa}}}$ and $max_{\boldsymbol{\lambda}\neq\mathbf{0}}$$\,\lambda^{T}\left( \mathbf{T} -\boldsymbol{\mu}\right) /$$\sqrt{\boldsymbol{\lambda}^{T}\boldsymbol{\Sigma \lambda}}$ where $\kappa$ is chosen a priori and the test statistic $\mathbf{T}$ is asymptotically $N_{L}\left( \boldsymbol{\mu},\boldsymbol{\Sigma}\right) $. The correction for multiple testing has a smaller effect on the power of $\kappa^{T}\left( \mathbf{T}-\boldsymbol{\mu }\right) /\sqrt{\boldsymbol{\kappa}^{T}\boldsymbol{\Sigma\boldsymbol{\kappa} }}$ than does switching to a two-tailed test, even though the opposite tail does receive consideration when $\lambda=-\kappa$. In the application, there are three measures of cognitive decline, and the a priori comparison $\kappa$ is their first principal component, computed without reference to treatment assignments. The method is implemented in an R package sensitivitymult.

Sharp Bounds on the Largest of some Linear Combinations of Random Variables with Given Marginal Distributions

1991 ◽  
Vol 5 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Isaac Meilijson
Keyword(s):  
Network Analysis ◽  
Design Of Experiments ◽  
Joint Distribution ◽  
Linear Combinations ◽  
Worst Case ◽  
Marginal Distributions ◽  
Sharp Bounds ◽  
Analysis And Design ◽  
Joint Distributions ◽  
Increasing Functions

Let X be a random vector and A a matrix. Let M be the maximal coordinate of the vector AX. For given marginal distributions of the coordinates of X, we present sharp bounds on the expectations of convex increasing functions of M. We derive joint distributions of X that achieve some of these bounds, and under these “worst case” distributions we study the joint distribution of M and the index of the largest coordinate of AX. Some possible applications are PERT network analysis and design of experiments.

scholarly journals Partial identification of the distribution of treatment effects with an application to the Knowledge is Power Program (KIPP)

2021 ◽  
Vol 12 (1) ◽  
pp. 143-171 ◽  
Author(s):  
Brigham R. Frandsen ◽  
Lars J. Lefgren
Keyword(s):  
Student Achievement ◽  
Charter School ◽  
Joint Distribution ◽  
Treatment Effects ◽  
Math Achievement ◽  
Partial Identification ◽  
Traditional Classroom ◽  
Potential Outcomes ◽  
Knowledge Is Power Program

We bound the distribution of treatment effects under plausible and testable assumptions on the joint distribution of potential outcomes, namely that potential outcomes are mutually stochastically increasing. We show how to test the empirical restrictions implied by those assumptions. The resulting bounds substantially sharpen bounds based on classical inequalities. We apply our method to estimate bounds on the distribution of effects of attending a Knowledge is Power Program (KIPP) charter school on student achievement, and find that a substantial majority of students' math achievement benefited from attendance, especially those who would have fared poorly in a traditional classroom.

SHARP BOUNDS ON THE DISTRIBUTION OF TREATMENT EFFECTS AND THEIR STATISTICAL INFERENCE

2009 ◽  
Vol 26 (3) ◽  
pp. 931-951 ◽  
Author(s):  
Yanqin Fan ◽  
Sang Soo Park
Keyword(s):  
Sample Size ◽  
Statistical Inference ◽  
Confidence Intervals ◽  
Simulation Study ◽  
Treatment Effect ◽  
Treatment Effects ◽  
Asymptotic Distributions ◽  
Finite Sample ◽  
Sharp Bounds ◽  
Nonparametric Estimators

In this paper, we propose nonparametric estimators of sharp bounds on the distribution of treatment effects of a binary treatment and establish their asymptotic distributions. We note the possible failure of the standard bootstrap with the same sample size and apply the fewer-than-nbootstrap to making inferences on these bounds. The finite sample performances of the confidence intervals for the bounds based on normal critical values, the standard bootstrap, and the fewer-than-nbootstrap are investigated via a simulation study. Finally we establish sharp bounds on the treatment effect distribution when covariates are available.

A Phonomotor Approach to Apraxia of Speech Treatment

2020 ◽  
Vol 29 (4) ◽  
pp. 2109-2130
Author(s):  
Lauren Bislick
Keyword(s):  
Phase I ◽  
Treatment Effects ◽  
Single Case ◽  
Motor Planning ◽  
The Other ◽  
Multimodal Approach ◽  
Apraxia Of Speech ◽  
Duration Of Treatment ◽  
Treatment Gains ◽  
Future Work

Purpose This study continued Phase I investigation of a modified Phonomotor Treatment (PMT) Program on motor planning in two individuals with apraxia of speech (AOS) and aphasia and, with support from prior work, refined Phase I methodology for treatment intensity and duration, a measure of communicative participation, and the use of effect size benchmarks specific to AOS. Method A single-case experimental design with multiple baselines across behaviors and participants was used to examine acquisition, generalization, and maintenance of treatment effects 8–10 weeks posttreatment. Treatment was distributed 3 days a week, and duration of treatment was specific to each participant (criterion based). Experimental stimuli consisted of target sounds or clusters embedded nonwords and real words, specific to each participants' deficit. Results Findings show improved repetition accuracy for targets in trained nonwords, generalization to targets in untrained nonwords and real words, and maintenance of treatment effects at 10 weeks posttreatment for one participant and more variable outcomes for the other participant. Conclusions Results indicate that a modified version of PMT can promote generalization and maintenance of treatment gains for trained speech targets via a multimodal approach emphasizing repeated exposure and practice. While these results are promising, the frequent co-occurrence of AOS and aphasia warrants a treatment that addresses both motor planning and linguistic deficits. Thus, the application of traditional PMT with participant-specific modifications for AOS embedded into the treatment program may be a more effective approach. Future work will continue to examine and maximize improvements in motor planning, while also treating anomia in aphasia.

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