advanced statistical methods for observational studiesrogosateaching.com/somgen290/20_lect05.pdfby...
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![Page 1: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/1.jpg)
L E C T U R E 0 5
Advanced Statistical Methods for Observational Studies
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unobserved confounding
Design of Observational Studies: chapter 3.4-3.8
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unobserved confounding
There are more things in heaven and earth, Horatio,Than are dreamt of in your philosophy.-Hamlet (1.5.167-8)
Design of Observational Studies: chapter 3.4-3.8
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naïve model
Model
Assumptions
Implementation
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naïve model: “natural” experiments
What if we design our study such that 𝑍𝑙 + 𝑍𝑘 = 1?
Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 … , 𝑍𝑙 + 𝑍𝑘 = 1
=Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 …
Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 … +Pr 𝑍𝑘 = 0, 𝑍𝑙 = 1 …
=𝜋𝑘1+0 1−𝜋𝑘
1−1 +(1−0)
Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 … +Pr 𝑍𝑘 = 0, 𝑍𝑙 = 1 …
=𝜋𝑘1+0 1−𝜋𝑘
1−1 +(1−0)
𝜋𝑘1+0 1−𝜋𝑘
1−1 +(1−0) +𝜋𝑘0+1 1−𝜋𝑘
1−0 +(1−1) =1
2
IF we can do this then we get to use the same tools developed for RCTs!
lecture 04
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naïve model: assumption one
Strongly Ignorable Treatment Assignment: Those that look alike (in our data set) are alike
𝜋𝑖 = Pr 𝑍𝑖 = 1 𝑟𝑇𝑖 , 𝑟𝐶𝑖 , 𝒙𝑖 , 𝑢𝑖 = Pr 𝑍𝑖 = 1 𝒙𝑖and
0 < 𝜋𝑖 < 1 for all i = 1, 2, …, n
If two subjects have the same propensity score then their values of xmay be different.
By SITA, if these two subjects have the same e(x) then the differences in their x are not predictive of treatment assignment (i.e., 𝒙 ⊥ 𝑍|𝑒(𝒙)).
Therefore the mismatches in x will be due to chance and will tend to balance. (more details)
∥
lecture 04
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naïve model: assumption two
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naïve model: assumption two
No Interference Between Units:
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naïve model: assumption two
No Interference Between Units: the observation on one unit should be unaffected by particular assignment of treatments to other units.
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naïve model: assumption two
No Interference Between Units: the observation on one unit should be unaffected by particular assignment of treatments to other units.
Can be written as:𝑅𝑖(𝑍𝑖 = 𝑧𝑖) = 𝑅𝑖(𝒁
∗)
where 𝑍𝑖 = 𝑧𝑖 indicates the treatment level for the ith unit and 𝒁∗ is a particular randomization from the set of all randomizations that have 𝑍𝑖 = 𝑧𝑖.
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naïve model: assumption two
No Interference Between Units: the observation on one unit should be unaffected by particular assignment of treatments to other units.
Can be written as:𝑅𝑖(𝑍𝑖 = 𝑧𝑖) = 𝑅𝑖(𝒁
∗)
where 𝑍𝑖 = 𝑧𝑖 indicates the treatment level for the ith unit and 𝒁∗ is a particular randomization from the set of all randomizations that have 𝑍𝑖 = 𝑧𝑖.
Not true for most educational interventions and infectious disease applications.
![Page 12: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/12.jpg)
naïve model: assumption two
No Interference Between Units: the observation on one unit should be unaffected by particular assignment of treatments to other units.
Can be written as:𝑅𝑖(𝑍𝑖 = 𝑧𝑖) = 𝑅𝑖(𝒁
∗)
where 𝑍𝑖 = 𝑧𝑖 indicates the treatment level for the ith unit and 𝒁∗ is a particular randomization from the set of all randomizations that have 𝑍𝑖 = 𝑧𝑖.
Not true for most educational interventions and infectious disease applications.
More details here and here.
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naïve model: assumption two
No Interference Between Units (part of SUTVA): the observation on one unit should be unaffected by particular assignment of treatments to other units.
Can be written as:𝑅𝑖(𝑍𝑖 = 𝑧𝑖) = 𝑅𝑖(𝒁
∗)
where 𝑍𝑖 = 𝑧𝑖 indicates the treatment level for the ith unit and 𝒁∗ is a particular randomization from the set of all randomizations that have 𝑍𝑖 = 𝑧𝑖.
Not true for most educational interventions and infectious disease applications.
More details here and here.
![Page 14: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/14.jpg)
naïve model: implementation
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naïve model: implementation
Collect a bunch of covariates that are related to treatment level and to the outcome.
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naïve model: implementation
Collect a bunch of covariates that are related to treatment level and to the outcome.
Exact match if you can.
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naïve model: implementation
Collect a bunch of covariates that are related to treatment level and to the outcome.
Exact match if you can.
You probably can’t exact match so estimate propensity scores and match on a hybrid of pscores and Mahalanobisdistance.
![Page 18: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/18.jpg)
naïve model: implementation
Collect a bunch of covariates that are related to treatment level and to the outcome.
Exact match if you can.
You probably can’t exact match so estimate propensity scores and match on a hybrid of pscores and Mahalanobisdistance.
Play around with the matching until you achieve acceptable comparison groups.
![Page 19: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/19.jpg)
naïve model: implementation
Collect a bunch of covariates that are related to treatment level and to the outcome.
Exact match if you can.
You probably can’t exact match so estimate propensity scores and match on a hybrid of pscores and Mahalanobisdistance.
Play around with the matching until you achieve acceptable comparison groups.
Die a little bit inside when you read your critics’ reviews because they point out all of the confounding that could exist.
![Page 20: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/20.jpg)
naïve model: implementation
Collect a bunch of covariates that are related to treatment level and to the outcome.
Exact match if you can.
You probably can’t exact match so estimate propensity scores and match on a hybrid of pscores and Mahalanobisdistance.
Play around with the matching until you achieve acceptable comparison groups.
Die a little bit inside when you read your critics’ reviews because they point out all of the confounding that could exist. Reevaluate life choices.
![Page 21: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/21.jpg)
sensitivity analysis
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sensitivity analysis
Sensitivity models are a means for moving past the “you didn’t do X which could lead to bias” argument.
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sensitivity analysis
Sensitivity models are a means for moving past the “you didn’t do X which could lead to bias” argument.
A useful sensitivity model addresses one assumption at a time, quantifying and making understandable the impact of departures from the assumption being assessed.
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sensitivity analysis
Sensitivity models are a means for moving past the “you didn’t do X which could lead to bias” argument.
A useful sensitivity model addresses one assumption at a time, quantifying and making understandable the impact of departures from the assumption being assessed.
We’re going to discuss the Γ sensitivity model which addresses the ignorable treatment assignment (SITA)
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sensitivity analysis
Sensitivity models are a means for moving past the “you didn’t do X which could lead to bias” argument.
A useful sensitivity model addresses one assumption at a time, quantifying and making understandable the impact of departures from the assumption being assessed.
We’re going to discuss the Γ sensitivity model which addresses the ignorable treatment assignment (SITA), not interference (SUTVA).
![Page 26: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/26.jpg)
sensitivity analysis
A word of warning:
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sensitivity analysis
A word of warning: many people find the Γ sensitivity model confusing.
![Page 28: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/28.jpg)
sensitivity analysis
A word of warning: many people find the Γ sensitivity model confusing. This lecture will only give you a sense of what’s going on with this model;
it isn’t intended to be sufficient to fully understand Γ sensitivity.
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sensitivity analysis
A word of warning: many people find the Γ sensitivity model confusing. This lecture will only give you a sense of what’s going on with this model;
it isn’t intended to be sufficient to fully understand Γ sensitivity.
Read section 3.4-3.8.
![Page 30: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/30.jpg)
sensitivity analysis
A word of warning: many people find the Γ sensitivity model confusing. This lecture will only give you a sense of what’s going on with this model;
it isn’t intended to be sufficient to fully understand Γ sensitivity.
Read section 3.4-3.8.
If you are so inclined then this might be a very nice place to produce your own framework for sensitivity.
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model: sensitivity analysis
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model: sensitivity analysis
Start with two observational units who have probability of treatment 𝜋𝑖 and 𝜋𝑗 (which may not be the same values).
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model: sensitivity analysis
Start with two observational units who have probability of treatment 𝜋𝑖 and 𝜋𝑗 (which may not be the same values).
Recall we defined this as 𝜋𝑖 = Pr 𝑍𝑖 = 1 𝑟𝑇𝑖 , 𝑟𝐶𝑖 , 𝒙𝑖 , 𝑢𝑖 .
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model: sensitivity analysis
Start with two observational units who have probability of treatment 𝜋𝑖 and 𝜋𝑗 (which may not be the same values).
Recall we defined this as 𝜋𝑖 = Pr 𝑍𝑖 = 1 𝑟𝑇𝑖 , 𝑟𝐶𝑖 , 𝒙𝑖 , 𝑢𝑖 .
We can talk about the odds of i receiving treatment:𝜋𝑖
1 − 𝜋𝑖
![Page 35: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/35.jpg)
model: sensitivity analysis
Start with two observational units who have probability of treatment 𝜋𝑖 and 𝜋𝑗 (which may not be the same values).
Recall we defined this as 𝜋𝑖 = Pr 𝑍𝑖 = 1 𝑟𝑇𝑖 , 𝑟𝐶𝑖 , 𝒙𝑖 , 𝑢𝑖 .
We can talk about the odds of i receiving treatment:𝜋𝑖
1 − 𝜋𝑖
And we can put the odds into a ratio:𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)
![Page 36: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/36.jpg)
model: sensitivity analysis
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model: sensitivity analysis
Our sensitivity model asserts that we can bound the odds ratio like so:
![Page 38: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/38.jpg)
model: sensitivity analysis
Our sensitivity model asserts that we can bound the odds ratio like so:
1
Γ≤𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)≤ Γ
![Page 39: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/39.jpg)
model: sensitivity analysis
Our sensitivity model asserts that we can bound the odds ratio like so:
1
Γ≤𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)≤ Γ
whenever 𝒙𝑖 = 𝒙𝑗.
![Page 40: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/40.jpg)
model: sensitivity analysis
Our sensitivity model asserts that we can bound the odds ratio like so:
1
Γ≤𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)≤ Γ
whenever 𝒙𝑖 = 𝒙𝑗.
We are making a particular statement about how “far off” the actual treatment probabilities are from the pscore(which only depends on the observed covariates).
![Page 41: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/41.jpg)
model: sensitivity analysis
Our sensitivity model asserts that we can bound the odds ratio like so:
1
Γ≤𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)≤ Γ
whenever 𝒙𝑖 = 𝒙𝑗.
We are making a particular statement about how “far off” the actual treatment probabilities are from the pscore(which only depends on the observed covariates).
If Γ = 1 then this forces 𝜋𝑖 = 𝜋𝑗.
![Page 42: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/42.jpg)
model: sensitivity analysis
Our sensitivity model asserts that we can bound the odds ratio like so:
1
Γ≤𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)≤ Γ
whenever 𝒙𝑖 = 𝒙𝑗.
We are making a particular statement about how “far off” the actual treatment probabilities are from the pscore(which only depends on the observed covariates).
If Γ = 1 then this forces 𝜋𝑖 = 𝜋𝑗.
If Γ = 2 then 𝜋𝑖 can depart from 𝜋𝑗
![Page 43: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/43.jpg)
model: sensitivity analysis
Our sensitivity model asserts that we can bound the odds ratio like so:
1
Γ≤𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)≤ Γ
whenever 𝒙𝑖 = 𝒙𝑗.
We are making a particular statement about how “far off” the actual treatment probabilities are from the pscore(which only depends on the observed covariates).
If Γ = 1 then this forces 𝜋𝑖 = 𝜋𝑗.
If Γ = 2 then 𝜋𝑖 can depart from 𝜋𝑗 For example: if 𝜋𝑖 = 1/2 and 𝜋𝑗 = 2/3 then
𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)=
0.5/(1 − 0.5)
0. ഥ6 /(1 − 0. ഥ6)= 2
![Page 44: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/44.jpg)
model: sensitivity analysis
Our sensitivity model asserts that we can bound the odds ratio like so:
1
Γ≤𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)≤ Γ
whenever 𝒙𝑖 = 𝒙𝑗.
We are making a particular statement about how “far off” the actual treatment probabilities are from the pscore(which only depends on the observed covariates).
If Γ = 1 then this forces 𝜋𝑖 = 𝜋𝑗.
If Γ = 2 then 𝜋𝑖 can depart from 𝜋𝑗 For example: if 𝜋𝑖 = 1/2 and 𝜋𝑗 = 2/3 then
𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)=
0.5/(1 − 0.5)
0. ഥ6 /(1 − 0. ഥ6)= 2
![Page 45: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/45.jpg)
model: sensitivity analysis
Our sensitivity model asserts that we can bound the odds ratio like so:
1
Γ≤𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)≤ Γ
whenever 𝒙𝑖 = 𝒙𝑗.
We are making a particular statement about how “far off” the actual treatment probabilities are from the pscore(which only depends on the observed covariates).
If Γ = 1 then this forces 𝜋𝑖 = 𝜋𝑗.
If Γ = 2 then 𝜋𝑖 can depart from 𝜋𝑗 For example: if 𝜋𝑖 = 1/2 and 𝜋𝑗 = 2/3 then
𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)=
0.5/(1 − 0.5)
0. ഥ6 /(1 − 0. ഥ6)= 2
![Page 46: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/46.jpg)
model: sensitivity analysis
Our sensitivity model asserts that we can bound the odds ratio like so:
1
Γ≤𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)≤ Γ
whenever 𝒙𝑖 = 𝒙𝑗.
We are making a particular statement about how “far off” the actual treatment probabilities are from the pscore(which only depends on the observed covariates).
If Γ = 1 then this forces 𝜋𝑖 = 𝜋𝑗.
If Γ = 2 then 𝜋𝑖 can depart from 𝜋𝑗 For example: if 𝜋𝑖 = 1/2 and 𝜋𝑗 = 2/3 then
𝜋𝑖/(1 − 𝜋𝑖)
𝜋𝑗/(1 − 𝜋𝑗)=
0.5/(1 − 0.5)
0. ഥ6 /(1 − 0. ഥ6)= 2
![Page 47: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/47.jpg)
model: sensitivity analysis
With this model in place we can think about “worst case” scenarios regarding violations of SITA.
![Page 48: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/48.jpg)
model: sensitivity analysis
With this model in place we can think about “worst case” scenarios regarding violations of SITA.
If someone is willing to give you a particular framework for how the violation must occur…
![Page 49: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/49.jpg)
model: sensitivity analysis
With this model in place we can think about “worst case” scenarios regarding violations of SITA.
If someone is willing to give you a particular framework for how the violation must occur (to the exclusion of all other possible ways it can fail)…
![Page 50: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/50.jpg)
model: sensitivity analysis
With this model in place we can think about “worst case” scenarios regarding violations of SITA.
If someone is willing to give you a particular framework for how the violation must occur (to the exclusion of all other possible ways it can fail) then use that parametric model.
![Page 51: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/51.jpg)
model: sensitivity analysis
With this model in place we can think about “worst case” scenarios regarding violations of SITA.
If someone is willing to give you a particular framework for how the violation must occur (to the exclusion of all other possible ways it can fail) then use that parametric model.
The Γ sensitivity model is non-parametric and we look at the extreme values that might occur when Γ > 1.
![Page 52: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/52.jpg)
model: sensitivity analysis
With this model in place we can think about “worst case” scenarios regarding violations of SITA.
If someone is willing to give you a particular framework for how the violation must occur (to the exclusion of all other possible ways it can fail) then use that parametric model.
The Γ sensitivity model is non-parametric and we look at the extreme values that might occur when Γ > 1. We’ll get ranges of p-values and estimates
![Page 53: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/53.jpg)
model: sensitivity analysis
With this model in place we can think about “worst case” scenarios regarding violations of SITA.
If someone is willing to give you a particular framework for how the violation must occur (to the exclusion of all other possible ways it can fail) then use that parametric model.
The Γ sensitivity model is non-parametric and we look at the extreme values that might occur when Γ > 1. We’ll get ranges of p-values and estimates
Every study is sensitive to sufficiently large violations of the SITA assumption.
![Page 54: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/54.jpg)
model: sensitivity analysis
With this model in place we can think about “worst case” scenarios regarding violations of SITA.
If someone is willing to give you a particular framework for how the violation must occur (to the exclusion of all other possible ways it can fail) then use that parametric model.
The Γ sensitivity model is non-parametric and we look at the extreme values that might occur when Γ > 1. We’ll get ranges of p-values and estimates
Every study is sensitive to sufficiently large violations of the SITA assumption. Just let Γ → ∞.
![Page 55: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/55.jpg)
model: sensitivity analysis
With this model in place we can think about “worst case” scenarios regarding violations of SITA.
If someone is willing to give you a particular framework for how the violation must occur (to the exclusion of all other possible ways it can fail) then use that parametric model.
The Γ sensitivity model is non-parametric and we look at the extreme values that might occur when Γ > 1. We’ll get ranges of p-values and estimates
Every study is sensitive to sufficiently large violations of the SITA assumption. Just let Γ → ∞.
If we’re going to make progress then the question becomes what level of Γ is sufficiently large to proceed.
![Page 56: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/56.jpg)
naïve model: “natural” experiments
What if we design our study such that 𝑍𝑙 + 𝑍𝑘 = 1?
Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 … , 𝑍𝑙 + 𝑍𝑘 = 1
=Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 …
Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 … +Pr 𝑍𝑘 = 0, 𝑍𝑙 = 1 …
=𝜋𝑘1+0 1−𝜋𝑘
1−1 +(1−0)
Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 … +Pr 𝑍𝑘 = 0, 𝑍𝑙 = 1 …
=𝜋𝑘1+0 1−𝜋𝑘
1−1 +(1−0)
𝜋𝑘1+0 1−𝜋𝑘
1−1 +(1−0) +𝜋𝑘0+1 1−𝜋𝑘
1−0 +(1−1) =1
2
IF we can do this then we get to use the same tools developed for RCTs!
lecture 04
![Page 57: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/57.jpg)
model: sensitivity analysis
If we design our study such that 𝑍𝑙 + 𝑍𝑘 = 1:
Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 … , 𝑍𝑙 + 𝑍𝑘 = 1 =𝜋𝑖
𝜋𝑖+𝜋𝑗
![Page 58: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/58.jpg)
model: sensitivity analysis
If we design our study such that 𝑍𝑙 + 𝑍𝑘 = 1:
Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 … , 𝑍𝑙 + 𝑍𝑘 = 1 =𝜋𝑖
𝜋𝑖+𝜋𝑗
Combining this with the sensitivity model, and doing some vaguely enjoyable algebra, we get:
![Page 59: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/59.jpg)
model: sensitivity analysis
If we design our study such that 𝑍𝑙 + 𝑍𝑘 = 1:
Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 … , 𝑍𝑙 + 𝑍𝑘 = 1 =𝜋𝑖
𝜋𝑖+𝜋𝑗
Combining this with the sensitivity model, and doing some vaguely enjoyable algebra, we get:
1
1 + Γ≤
𝜋𝑖𝜋𝑖 + 𝜋𝑗
≤Γ
1 + Γ
![Page 60: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/60.jpg)
model: sensitivity analysis
If we design our study such that 𝑍𝑙 + 𝑍𝑘 = 1:
Pr 𝑍𝑘 = 1, 𝑍𝑙 = 0 … , 𝑍𝑙 + 𝑍𝑘 = 1 =𝜋𝑖
𝜋𝑖+𝜋𝑗
Combining this with the sensitivity model, and doing some vaguely enjoyable algebra, we get:
1
1 + Γ≤
𝜋𝑖𝜋𝑖 + 𝜋𝑗
≤Γ
1 + Γ
We get ½ if Γ = 1.
![Page 61: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/61.jpg)
model: sensitivity analysis
The randomization tests we have can be reworked under the understanding that we can vary the odds ratio within
1
1 + Γ≤
𝜋𝑖𝜋𝑖 + 𝜋𝑗
≤Γ
1 + Γ
![Page 62: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/62.jpg)
model: sensitivity analysis
The randomization tests we have can be reworked under the understanding that we can vary the odds ratio within
1
1 + Γ≤
𝜋𝑖𝜋𝑖 + 𝜋𝑗
≤Γ
1 + Γ
Setting 𝜋𝑖
𝜋𝑖+𝜋𝑗=
Γ
1+Γwill get you one extreme.
![Page 63: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/63.jpg)
model: sensitivity analysis
The randomization tests we have can be reworked under the understanding that we can vary the odds ratio within
1
1 + Γ≤
𝜋𝑖𝜋𝑖 + 𝜋𝑗
≤Γ
1 + Γ
Setting 𝜋𝑖
𝜋𝑖+𝜋𝑗=
Γ
1+Γwill get you one extreme.
Setting1
1+Γ=
𝜋𝑖
𝜋𝑖+𝜋𝑗will get you the other.
![Page 64: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/64.jpg)
model: sensitivity analysis
The randomization tests we have can be reworked under the understanding that we can vary the odds ratio within
1
1 + Γ≤
𝜋𝑖𝜋𝑖 + 𝜋𝑗
≤Γ
1 + Γ
Setting 𝜋𝑖
𝜋𝑖+𝜋𝑗=
Γ
1+Γwill get you one extreme.
Setting1
1+Γ=
𝜋𝑖
𝜋𝑖+𝜋𝑗will get you the other.
For notational purposes, let’s say that the usual Wilcoxon signed rank test (when Γ = 1) is written as 𝑇.
![Page 65: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/65.jpg)
( S E E L E C T U R E 0 4 F O R M O R E D E T A I L )
Wilcoxon rank-sign test
![Page 66: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/66.jpg)
the Wilcoxon signed rank
So… what’s a “good” test statistic to use?
![Page 67: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/67.jpg)
the Wilcoxon signed rank
So… what’s a “good” test statistic to use?
It depends…
![Page 68: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/68.jpg)
the Wilcoxon signed rank
So… what’s a “good” test statistic to use?
It depends…
A reasonable one is the Wilcoxon signed rank statistic
![Page 69: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/69.jpg)
the Wilcoxon signed rank
So… what’s a “good” test statistic to use?
It depends…
A reasonable one is the Wilcoxon signed rank statistic
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude, call these ranks 𝑞𝑖3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
![Page 70: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/70.jpg)
the Wilcoxon signed rank
So… what’s a “good” test statistic to use?
It depends…
A reasonable one is the Wilcoxon signed rank statistic
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude, call these ranks 𝑞𝑖3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
![Page 71: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/71.jpg)
the Wilcoxon signed rank
So… what’s a “good” test statistic to use?
It depends…
A reasonable one is the Wilcoxon signed rank statistic
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude, call these ranks 𝑞𝑖3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
![Page 72: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/72.jpg)
the Wilcoxon signed rank
So… what’s a “good” test statistic to use?
It depends…
A reasonable one is the Wilcoxon signed rank statistic
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude, call these ranks 𝑞𝑖3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
![Page 73: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/73.jpg)
the Wilcoxon signed rank
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude from smallest to largest, call these ranks 𝑞𝑖
3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
pair r_Ci1 r_Ti2 y_i
1 153 159 -6
2 163 148 15
3 159 150 9
4 142 159 -17
5 155 137 18
6 140 149 -9
7 148 154 -6
8 145 140 5
9 148 134 14
10 159 133 26
![Page 74: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/74.jpg)
the Wilcoxon signed rank
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude from smallest to largest, call these ranks 𝑞𝑖
3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
pair r_Ci1 r_Ti2 y_i abs(y_i) q_i
1 153 159 -6
2 163 148 15
3 159 150 9
4 142 159 -17
5 155 137 18
6 140 149 -9
7 148 154 -6
8 145 140 5
9 148 134 14
10 159 133 26
![Page 75: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/75.jpg)
the Wilcoxon signed rank
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude from smallest to largest, call these ranks 𝑞𝑖
3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
pair r_Ci1 r_Ti2 y_i abs(y_i) q_i
1 153 159 -6 6
2 163 148 15 15
3 159 150 9 9
4 142 159 -17 17
5 155 137 18 18
6 140 149 -9 9
7 148 154 -6 6
8 145 140 5 5
9 148 134 14 14
10 159 133 26 26
![Page 76: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/76.jpg)
the Wilcoxon signed rank
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude from smallest to largest, call these ranks 𝑞𝑖
3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
pair r_Ci1 r_Ti2 y_i abs(y_i) q_i
1 153 159 -6 6
2 163 148 15 15
3 159 150 9 9
4 142 159 -17 17
5 155 137 18 18
6 140 149 -9 9
7 148 154 -6 6
8 145 140 5 5 1
9 148 134 14 14
10 159 133 26 26
![Page 77: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/77.jpg)
the Wilcoxon signed rank
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude from smallest to largest, call these ranks 𝑞𝑖
3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
pair r_Ci1 r_Ti2 y_i abs(y_i) q_i
1 153 159 -6 6 2.5
2 163 148 15 15
3 159 150 9 9
4 142 159 -17 17
5 155 137 18 18
6 140 149 -9 9
7 148 154 -6 6 2.5
8 145 140 5 5 1
9 148 134 14 14
10 159 133 26 26
![Page 78: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/78.jpg)
the Wilcoxon signed rank
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude from smallest to largest, call these ranks 𝑞𝑖
3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
pair r_Ci1 r_Ti2 y_i abs(y_i) q_i
1 153 159 -6 6 2.5
2 163 148 15 15 7
3 159 150 9 9 4.5
4 142 159 -17 17 8
5 155 137 18 18 9
6 140 149 -9 9 4.5
7 148 154 -6 6 2.5
8 145 140 5 5 1
9 148 134 14 14 6
10 159 133 26 26 10
![Page 79: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/79.jpg)
the Wilcoxon signed rank
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude from smallest to largest, call these ranks 𝑞𝑖
3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
4. Average ties
pair r_Ci1 r_Ti2 y_i abs(y_i) q_i
1 153 159 -6 6 2.5
2 163 148 15 15 7
3 159 150 9 9 4.5
4 142 159 -17 17 8
5 155 137 18 18 9
6 140 149 -9 9 4.5
7 148 154 -6 6 2.5
8 145 140 5 5 1
9 148 134 14 14 6
10 159 133 26 26 10
![Page 80: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/80.jpg)
the Wilcoxon signed rank
How it works:
1. Take the absolute value of the difference between pairs, |𝑌𝑖|
2. Assign ranks to these |𝑌𝑖| based on magnitude from smallest to largest, call these ranks 𝑞𝑖
3. The Wilcoxon signed rank statistic is the sum of the 𝑞𝑖 where 𝑌𝑖 > 0
4. Average ties
5. You can remove zeros
pair r_Ci1 r_Ti2 y_i abs(y_i) q_i
1 153 159 -6 6 2.5
2 163 148 15 15 7
3 159 150 9 9 4.5
4 142 159 -17 17 8
5 155 137 18 18 9
6 140 149 -9 9 4.5
7 148 154 -6 6 2.5
8 145 140 5 5 1
9 148 134 14 14 6
10 159 133 26 26 10
![Page 81: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/81.jpg)
model: sensitivity analysis
The randomization tests we have can be reworked under the understanding that we can vary the odds ratio within
1
1 + Γ≤
𝜋𝑖𝜋𝑖 + 𝜋𝑗
≤Γ
1 + Γ
Setting 𝜋𝑖
𝜋𝑖+𝜋𝑗=
Γ
1+Γwill get you one extreme.
Setting1
1+Γ=
𝜋𝑖
𝜋𝑖+𝜋𝑗will get you the other.
For notational purposes, let’s say that the usual Wilcoxon signed rank test (when Γ = 1) is written as 𝑇.
Then we’ll write the test statistic under our sensitivity model as ത𝑇.
![Page 82: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/82.jpg)
( E N D I N T E R L U D E )
Sensitivity analysis
![Page 83: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/83.jpg)
model: sensitivity analysis
![Page 84: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/84.jpg)
model: sensitivity analysis
We can calculate the exact distribution of ത𝑇 under either extreme, but for large matched sets it’ll be easier (and not far off) to use an approximation.
![Page 85: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/85.jpg)
model: sensitivity analysis
We can calculate the exact distribution of ത𝑇 under either extreme, but for large matched sets it’ll be easier (and not far off) to use an approximation.
The ത𝑇 has known expected value and variance
![Page 86: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/86.jpg)
model: sensitivity analysis
We can calculate the exact distribution of ത𝑇 under either extreme, but for large matched sets it’ll be easier (and not far off) to use an approximation.
The ത𝑇 has known expected value and variance
𝐸 ത𝑇 =Γ
1 + Γ
I(I + 1)
2
![Page 87: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/87.jpg)
model: sensitivity analysis
We can calculate the exact distribution of ത𝑇 under either extreme, but for large matched sets it’ll be easier (and not far off) to use an approximation.
The ത𝑇 has known expected value and variance
𝐸 ത𝑇 =Γ
1 + Γ
I(I + 1)
2
𝑣𝑎𝑟(ത𝑇) =Γ
1 + Γ 2
I(I + 1)(2I + 1)
6
![Page 88: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/88.jpg)
model: sensitivity analysis
We can calculate the exact distribution of ത𝑇 under either extreme, but for large matched sets it’ll be easier (and not far off) to use an approximation.
The ത𝑇 has known expected value and variance
𝐸 ത𝑇 =Γ
1 + Γ
I(I + 1)
2
𝑣𝑎𝑟(ത𝑇) =Γ
1 + Γ 2
I(I + 1)(2I + 1)
6
where I is the number of matched pairs.
![Page 89: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/89.jpg)
model: sensitivity analysis
We can calculate the exact distribution of ത𝑇 under either extreme, but for large matched sets it’ll be easier (and not far off) to use an approximation.
The ത𝑇 has known expected value and variance
𝐸 ത𝑇 =Γ
1 + Γ
I(I + 1)
2
𝑣𝑎𝑟(ത𝑇) =Γ
1 + Γ 2
I(I + 1)(2I + 1)
6
where I is the number of matched pairs.
![Page 90: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/90.jpg)
model: sensitivity analysis
We can calculate the exact distribution of ത𝑇 under either extreme, but for large matched sets it’ll be easier (and not far off) to use an approximation.
The ത𝑇 has known expected value and variance
𝐸 ത𝑇 =1
1 + Γ
I(I + 1)
2
𝑣𝑎𝑟(ത𝑇) =Γ
1 + Γ 2
I(I + 1)(2I + 1)
6
where I is the number of matched pairs.
![Page 91: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/91.jpg)
model: sensitivity analysis
The standardized deviate of 𝑇 (the Wilcoxon signed rank statistic) can be approximated using:
𝑇 − 𝐸[ത𝑇]
𝑣𝑎𝑟(ത𝑇)~𝑁(0,1)
![Page 92: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/92.jpg)
example: sensitivity analysis
obs b_weight gest_age dose death
1 2412 36 1 0
2 2205 29 1 1
3 2569 36 1 0
4 2443 34 1 0
5 2569 36 0 0
6 2436 35 0 0
7 2461 34 0 0
8 2759 32 0 0
9 2324 27 0 1
10 2667 34 0 0
… … … … …
500 2349 33 1 0
![Page 93: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/93.jpg)
example: sensitivity analysis
obs b_weight gest_age dose death
1 2412 36 1 0
2 2205 29 1 1
3 2569 36 1 0
4 2443 34 1 0
5 2569 36 0 0
6 2436 35 0 0
7 2461 34 0 0
8 2759 32 0 0
9 2324 27 0 1
10 2667 34 0 0
… … … … …
500 2349 33 1 0
Similar to data set from lecture 03, but different number of observations...
![Page 94: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/94.jpg)
example: sensitivity analysis
obs b_weight gest_age dose death
1 2412 36 1 0
2 2205 29 1 1
3 2569 36 1 0
4 2443 34 1 0
5 2569 36 0 0
6 2436 35 0 0
7 2461 34 0 0
8 2759 32 0 0
9 2324 27 0 1
10 2667 34 0 0
… … … … …
500 2349 33 1 0
Similar to data set from lecture 03, but different number of observations and outcome of interest.
![Page 95: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/95.jpg)
example: sensitivity analysis
Similar to data set from lecture 03, but different number of observations and outcome of interest.
obs b_weight gest_age dose hearing
1 2412 36 1 0.12
2 2205 29 1 0.24
3 2569 36 1 0.02
4 2443 34 1 -0.16
5 2569 36 0 0.58
6 2436 35 0 -0.22
7 2461 34 0 -0.07
8 2759 32 0 -0.55
9 2324 27 0 -0.36
10 2667 34 0 0.28
… … … … …
500 2349 33 1 -0.55
![Page 96: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/96.jpg)
example: sensitivity analysis
Outcome of interest: Hearing is some standardized metric with population mean=0 and sd=1.
obs b_weight gest_age dose hearing
1 2412 36 1 0.12
2 2205 29 1 0.24
3 2569 36 1 0.02
4 2443 34 1 -0.16
5 2569 36 0 0.58
6 2436 35 0 -0.22
7 2461 34 0 -0.07
8 2759 32 0 -0.55
9 2324 27 0 -0.36
10 2667 34 0 0.28
… … … … …
500 2349 33 1 -0.55
![Page 97: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/97.jpg)
example: sensitivity analysis
![Page 98: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/98.jpg)
example: sensitivity analysis
Create 250 pair matches.
![Page 99: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/99.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using T, the usual Wilcoxon signed rank statistic:
![Page 100: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/100.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using T, the usual Wilcoxon signed rank statistic:
We know that E[T]=15,688 and sd(T)=812
![Page 101: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/101.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using T, the usual Wilcoxon signed rank statistic:
We know that E[T]=15,688 and sd(T)=812
Get T=13,250
![Page 102: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/102.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using T, the usual Wilcoxon signed rank statistic:
We know that E[T]=15,688 and sd(T)=812
Get T=13,250
Using the approximation:
𝑇 − 𝐸[𝑇]
𝑣𝑎𝑟(𝑇)~𝑁(0,1)
![Page 103: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/103.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using T, the usual Wilcoxon signed rank statistic:
We know that E[T]=15,688 and sd(T)=812
Get T=13,250
Using the approximation:
𝑇 − 𝐸[𝑇]
𝑣𝑎𝑟(𝑇)~𝑁(0,1)
13,250−15,688
812=
![Page 104: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/104.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using T, the usual Wilcoxon signed rank statistic:
We know that E[T]=15,688 and sd(T)=812
Get T=13,250
Using the approximation:
𝑇 − 𝐸[𝑇]
𝑣𝑎𝑟(𝑇)~𝑁(0,1)
13,250−15,688
812= −3.00, which has a small p-value.
![Page 105: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/105.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using T, the usual Wilcoxon signed rank statistic:
We know that E[T]=15,688 and sd(T)=812
Get T=13,250
Using the approximation:
𝑇 − 𝐸[𝑇]
𝑣𝑎𝑟(𝑇)~𝑁(0,1)
13,250−15,688
812= −3.00, which has a small p-value, under the
naïve model.
![Page 106: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/106.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using T, the usual Wilcoxon signed rank statistic:
We know that E[T]=15,688 and sd(T)=812
Get T=13,250
Using the approximation:
𝑇 − 𝐸[𝑇]
𝑣𝑎𝑟(𝑇)~𝑁(0,1)
13,250−15,688
812= −3.00, which has a small p-value, under the
naïve model.
𝐸 ത𝑇 =Γ
1 + Γ
I(I + 1)
2
![Page 107: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/107.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using T, the usual Wilcoxon signed rank statistic:
We know that E[T]=15,688 and sd(T)=812
Get T=13,250
Using the approximation:
𝑇 − 𝐸[𝑇]
𝑣𝑎𝑟(𝑇)~𝑁(0,1)
13,250−15,688
812= −3.00, which has a small p-value, under the
naïve model.
𝐸 ത𝑇 =Γ
1 + Γ
I(I + 1)
2
upper bound
![Page 108: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/108.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using T, the usual Wilcoxon signed rank statistic:
We know that E[T]=15,688 and sd(T)=812
Get T=13,250
Using the approximation:
𝑇 − 𝐸[𝑇]
𝑣𝑎𝑟(𝑇)~𝑁(0,1)
13,250−15,688
812= −3.00, which has a small p-value, under the
naïve model.
𝐸 ത𝑇 =Γ
1 + Γ
I(I + 1)
2
Set Γ=1.11
upper bound
![Page 109: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/109.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using ത𝑇, the usual Wilcoxon signed rank statistic:
We know that E[ത𝑇]=16,540 and sd(ത𝑇)=810
Get T=13,250
Using the approximation:
𝑇 − 𝐸[ത𝑇]
𝑣𝑎𝑟(ത𝑇)~𝑁(0,1)
13,250−16,540
810= −4.00, which has a small p-value.
𝐸 ത𝑇 =Γ
1 + Γ
I(I + 1)
2
Set Γ=1.11
upper bound
![Page 110: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/110.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using ത𝑇, the usual Wilcoxon signed rank statistic:
We know that E[ത𝑇]=16,540 and sd(ത𝑇)=810
Get T=13,250
Using the approximation:
𝑇 − 𝐸[ത𝑇]
𝑣𝑎𝑟(ത𝑇)~𝑁(0,1)
13,250−16,540
810= −4.00, which has a small p-value.
𝐸 ത𝑇 =Γ
1 + Γ
I(I + 1)
2
Set Γ=1.11
upper bound
![Page 111: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/111.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using ത𝑇, the usual Wilcoxon signed rank statistic:
We know that E[ത𝑇]=16,540 and sd(ത𝑇)=810
Get T=13,250
Using the approximation:
𝑇 − 𝐸[ത𝑇]
𝑣𝑎𝑟(ത𝑇)~𝑁(0,1)
13,250−16,540
810= −4.00, which has a small p-value.
𝐸 ത𝑇 =Γ
1 + Γ
I(I + 1)
2
Set Γ=1.11
upper bound
![Page 112: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/112.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using ത𝑇, the usual Wilcoxon signed rank statistic:
We know that E[ത𝑇]=16,540 and sd(ത𝑇)=810
Get T=13,250
Using the approximation:
𝑇 − 𝐸[ത𝑇]
𝑣𝑎𝑟(ത𝑇)~𝑁(0,1)
13,250−16,540
810= −4.00, which has a small p-value.
𝐸 ത𝑇 =Γ
1 + Γ
I(I + 1)
2
Set Γ=1.11
upper bound
![Page 113: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/113.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using ത𝑇, the usual Wilcoxon signed rank statistic:
We know that E[ത𝑇]=14,835 and sd(ത𝑇)=810
Get T=13,250
Using the approximation:
𝑇 − 𝐸[ത𝑇]
𝑣𝑎𝑟(ത𝑇)~𝑁(0,1)
13,250−14,835
810= −1.95, which has a p-value close to 0.05.
𝐸 ത𝑇 =𝟏
1 + Γ
I(I + 1)
2
Set Γ=1.11
lower bound
![Page 114: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/114.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using ത𝑇, the usual Wilcoxon signed rank statistic:
We know that E[ത𝑇]=14,835 and sd(ത𝑇)=810
Get T=13,250
Using the approximation:
𝑇 − 𝐸[ത𝑇]
𝑣𝑎𝑟(ത𝑇)~𝑁(0,1)
13,250−14,835
810= −1.95, which has a p-value close to 0.05.
𝐸 ത𝑇 =𝟏
1 + Γ
I(I + 1)
2
Set Γ=1.11
lower bound
![Page 115: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/115.jpg)
example: sensitivity analysis
Create 250 pair matches.
Using ത𝑇, the usual Wilcoxon signed rank statistic:
We know that E[ത𝑇]=14,835 and sd(ത𝑇)=810
Get T=13,250
Using the approximation:
𝑇 − 𝐸[ത𝑇]
𝑣𝑎𝑟(ത𝑇)~𝑁(0,1)
13,250−14,835
810= −1.95, which has a p-value close to 0.05.
Interpretation: If there was a small amount of bias Γ = 1.12then this would nullify our qualitative claims.
𝐸 ത𝑇 =1
1 + Γ
I(I + 1)
2
Set Γ=1.11
lower bound
![Page 116: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/116.jpg)
implementation: sensitivity analysis
![Page 117: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/117.jpg)
implementation: sensitivity analysis
In practice, software will do this for you and you will interpret.
![Page 118: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/118.jpg)
implementation: sensitivity analysis
In practice, software will do this for you and you will interpret.
The key to keep in mind is that there are two different way things could go wrong:
![Page 119: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/119.jpg)
implementation: sensitivity analysis
In practice, software will do this for you and you will interpret.
The key to keep in mind is that there are two different way things could go wrong: (i) units could be sorted into treatment or (ii) into control.
![Page 120: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/120.jpg)
implementation: sensitivity analysis
In practice, software will do this for you and you will interpret.
The key to keep in mind is that there are two different way things could go wrong: (i) units could be sorted into treatment or (ii) into control.
This gives rise to three different distributions:
![Page 121: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/121.jpg)
implementation: sensitivity analysis
In practice, software will do this for you and you will interpret.
The key to keep in mind is that there are two different way things could go wrong: (i) units could be sorted into treatment or (ii) into control.
This gives rise to three different distributions:
Naïve model: T~N(I I+1
4,I(I+1)(2I+1)
24)
![Page 122: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/122.jpg)
implementation: sensitivity analysis
In practice, software will do this for you and you will interpret.
The key to keep in mind is that there are two different way things could go wrong: (i) units could be sorted into treatment or (ii) into control.
This gives rise to three different distributions:
Naïve model: T~N(I I+1
4,I(I+1)(2I+1)
24)
Biased toward one way: T~N(Γ
1+Γ
I I+1
2,
Γ
1+Γ 2
I(I+1)(2I+1)
6)
![Page 123: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/123.jpg)
implementation: sensitivity analysis
In practice, software will do this for you and you will interpret.
The key to keep in mind is that there are two different way things could go wrong: (i) units could be sorted into treatment or (ii) into control.
This gives rise to three different distributions:
Naïve model: T~N(I I+1
4,I(I+1)(2I+1)
24)
Biased toward one way: T~N(Γ
1+Γ
I I+1
2,
Γ
1+Γ 2
I(I+1)(2I+1)
6)
Biased other way: T~N(1
1+Γ
I I+1
2,
Γ
1+Γ 2
I(I+1)(2I+1)
6)
![Page 124: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/124.jpg)
implementation: sensitivity analysis
Use the new distributions to test your statistic to see where its critical values are.
![Page 125: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/125.jpg)
implementation: sensitivity analysis
Use the new distributions to test your statistic to see where its critical values are.
This will lead you to provide wider intervals for everything
![Page 126: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/126.jpg)
implementation: sensitivity analysis
Use the new distributions to test your statistic to see where its critical values are.
This will lead you to provide wider intervals for everything: If you had a point estimate of (to pick a random number): 5 then, for a
particular Γ, you may end up with a “point estimate” of (4, 6).
![Page 127: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/127.jpg)
implementation: sensitivity analysis
Use the new distributions to test your statistic to see where its critical values are.
This will lead you to provide wider intervals for everything: If you had a point estimate of (to pick a random number): 5 then, for a
particular Γ, you may end up with a “point estimate” of (4, 6). This new interval is not due to randomness in assignment, it is due to the difference in treatment assignment probabilities.
![Page 128: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/128.jpg)
implementation: sensitivity analysis
Use the new distributions to test your statistic to see where its critical values are.
This will lead you to provide wider intervals for everything: If you had a point estimate of (to pick a random number): 5 then, for a
particular Γ, you may end up with a “point estimate” of (4, 6). This new interval is not due to randomness in assignment, it is due to the difference in treatment assignment probabilities.
If you had a p-value of 0.012 , for a particular Γ, you may end up with a p-value interval of (0.032, 0.0001).
![Page 129: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/129.jpg)
implementation: sensitivity analysis
Use the new distributions to test your statistic to see where its critical values are.
This will lead you to provide wider intervals for everything: If you had a point estimate of (to pick a random number): 5 then, for a
particular Γ, you may end up with a “point estimate” of (4, 6). This new interval is not due to randomness in assignment, it is due to the difference in treatment assignment probabilities.
If you had a p-value of 0.012 , for a particular Γ, you may end up with a p-value interval of (0.032, 0.0001). Generally (though not always), you’ll report the largest p-value in the interval.
![Page 130: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/130.jpg)
implementation: sensitivity analysis
![Page 131: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/131.jpg)
implementation: sensitivity analysis
In practice, it’s common to just report the value of Γ which nullifies your qualitative conclusions (i.e., goes from significant to insignificant)…
![Page 132: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/132.jpg)
implementation: sensitivity analysis
In practice, it’s common to just report the value of Γ which nullifies your qualitative conclusions (i.e., goes from significant to insignificant), and to help the reader in interpreting the meaning of Γ.
![Page 133: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/133.jpg)
implementation: sensitivity analysis
In practice, it’s common to just report the value of Γ which nullifies your qualitative conclusions (i.e., goes from significant to insignificant), and to help the reader in interpreting the meaning of Γ.
For example, Γ = 2 means that within a given pair…
![Page 134: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/134.jpg)
implementation: sensitivity analysis
In practice, it’s common to just report the value of Γ which nullifies your qualitative conclusions (i.e., goes from significant to insignificant), and to help the reader in interpreting the meaning of Γ.
For example, Γ = 2 means that within a given pair – even though the two matched individuals looked identical in the data set
![Page 135: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/135.jpg)
implementation: sensitivity analysis
In practice, it’s common to just report the value of Γ which nullifies your qualitative conclusions (i.e., goes from significant to insignificant), and to help the reader in interpreting the meaning of Γ.
For example, Γ = 2 means that within a given pair – even though the two matched individuals looked identical in the data set – the actual odds of assignment was up to twice as likely for one member in the pair than the other.
![Page 136: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/136.jpg)
implementation: sensitivity analysis
In practice, it’s common to just report the value of Γ which nullifies your qualitative conclusions (i.e., goes from significant to insignificant), and to help the reader in interpreting the meaning of Γ.
For example, Γ = 2 means that within a given pair – even though the two matched individuals looked identical in the data set – the actual odds of assignment was up to twice as likely for one member in the pair than the other. Likely this difference is due to the unobserved covariates.
![Page 137: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/137.jpg)
implementation: sensitivity analysis
In practice, it’s common to just report the value of Γ which nullifies your qualitative conclusions (i.e., goes from significant to insignificant), and to help the reader in interpreting the meaning of Γ.
For example, Γ = 2 means that within a given pair – even though the two matched individuals looked identical in the data set – the actual odds of assignment was up to twice as likely for one member in the pair than the other. Likely this difference is due to the unobserved covariates.
The question then becomes: Is what’s left lingering out there, outside of your data set, enough to cause that level of confounding?
![Page 138: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/138.jpg)
sensitivity analysis: understanding gamma
![Page 139: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/139.jpg)
sensitivity analysis: understanding gamma
When you report Gamma, you report the maximal Gamma that still returns a qualitatively similar conclusion (e.g., if you found the treatment positive and significant then you report the value of Gamma that just barely makes the test not-significant).
![Page 140: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/140.jpg)
sensitivity analysis: understanding gamma
When you report Gamma, you report the maximal Gamma that still returns a qualitatively similar conclusion (e.g., if you found the treatment positive and significant then you report the value of Gamma that just barely makes the test not-significant).
Note: This says nothing about the case at hand.
![Page 141: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/141.jpg)
sensitivity analysis: understanding gamma
When you report Gamma, you report the maximal Gamma that still returns a qualitatively similar conclusion (e.g., if you found the treatment positive and significant then you report the value of Gamma that just barely makes the test not-significant).
Note: This says nothing about the case at hand. You are looking at the strength of the argument (i.e., how many in treated need to be switched to control before null).
![Page 142: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/142.jpg)
sensitivity analysis: understanding gamma
When you report Gamma, you report the maximal Gamma that still returns a qualitatively similar conclusion (e.g., if you found the treatment positive and significant then you report the value of Gamma that just barely makes the test not-significant).
Note: This says nothing about the case at hand. You are looking at the strength of the argument (i.e., how many in treated need to be switched to control before null). But we haven’t measured the level of unobserved (it’s still unobserved!!).
![Page 143: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/143.jpg)
sensitivity analysis: understanding gamma
When you report Gamma, you report the maximal Gamma that still returns a qualitatively similar conclusion (e.g., if you found the treatment positive and significant then you report the value of Gamma that just barely makes the test not-significant).
Note: This says nothing about the case at hand. You are looking at the strength of the argument (i.e., how many in treated need to be switched to control before null). But we haven’t measured the level of unobserved (it’s still unobserved!!).
This is like saying a building can withstand a magnitude 8 earthquake.
![Page 144: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/144.jpg)
sensitivity analysis: understanding gamma
When you report Gamma, you report the maximal Gamma that still returns a qualitatively similar conclusion (e.g., if you found the treatment positive and significant then you report the value of Gamma that just barely makes the test not-significant).
Note: This says nothing about the case at hand. You are looking at the strength of the argument (i.e., how many in treated need to be switched to control before null). But we haven’t measured the level of unobserved (it’s still unobserved!!).
This is like saying a building can withstand a magnitude 8 earthquake. There is no statement in there about what magnitude earthquake the building will experience.
![Page 145: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/145.jpg)
sensitivity analysis: understanding gamma
![Page 146: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/146.jpg)
sensitivity analysis: understanding gamma
Explaining Gamma sensitivity is hard because the concept of confounding is tough.
![Page 147: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/147.jpg)
sensitivity analysis: understanding gamma
Explaining Gamma sensitivity is hard because the concept of confounding is tough.
Confounding (usually) arises from sorting into treatment/control…
![Page 148: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/148.jpg)
sensitivity analysis: understanding gamma
Explaining Gamma sensitivity is hard because the concept of confounding is tough.
Confounding (usually) arises from sorting into treatment/control on variables important in determining the outcome.
![Page 149: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/149.jpg)
sensitivity analysis: understanding gamma
Explaining Gamma sensitivity is hard because the concept of confounding is tough.
Confounding (usually) arises from sorting into treatment/control (propensity) on variables important in determining the outcome.
![Page 150: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/150.jpg)
sensitivity analysis: understanding gamma
Explaining Gamma sensitivity is hard because the concept of confounding is tough.
Confounding (usually) arises from sorting into treatment/control (propensity) on variables important in determining the outcome (prognostic).
![Page 151: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/151.jpg)
sensitivity analysis: understanding gamma
Explaining Gamma sensitivity is hard because the concept of confounding is tough.
Confounding (usually) arises from sorting into treatment/control (propensity - Fisher) on variables important in determining the outcome (prognostic).
![Page 152: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/152.jpg)
sensitivity analysis: understanding gamma
Explaining Gamma sensitivity is hard because the concept of confounding is tough.
Confounding (usually) arises from sorting into treatment/control (propensity - Fisher) on variables important in determining the outcome (prognostic –Mill).
![Page 153: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/153.jpg)
sensitivity analysis: understanding gamma
Explaining Gamma sensitivity is hard because the concept of confounding is tough.
Confounding (usually) arises from sorting into treatment/control (propensity - Fisher) on variables important in determining the outcome (prognostic – Mill).
Gamma sensitivity is kind of weird because it only focusses on the propensity, and then assumes the worst case for prognostic.
![Page 154: Advanced Statistical Methods for Observational Studiesrogosateaching.com/somgen290/20_lect05.pdfBy SITA, if these two subjects have the same e(x) then the differences in their x are](https://reader034.vdocuments.site/reader034/viewer/2022050111/5f484f9f796098406229b4c6/html5/thumbnails/154.jpg)
an example
Design of Observational Studies: chapter 5ish
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matching on more than one metric
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matching on more than one metric
Intuition: matching on just propensity scores is like uniform randomization, whereas a Mahalanobis & pscoresis more like a matched pairs randomization.
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example
House, MD:
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example
House, MD:
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example
Example: House examines the patient and wants to treat for sarcoidosis.
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example
Example: House examines the patient and wants to treat for sarcoidosis. He is always considering treating for sarcoidosis…
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example
Example: House examines the patient and wants to treat for sarcoidosis. He is always considering treating for sarcoidosis… but in a way that is unrelated to how sick the patient is.
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example
Example: House examines the patient and wants to treat for sarcoidosis. He is always considering treating for sarcoidosis… but in a way that is unrelated to how sick the patient is.
To make this example easier to follow, let’s consider two data generating functions
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example
Example: House examines the patient and wants to treat for sarcoidosis. He is always considering treating for sarcoidosis… but in a way that is unrelated to how sick the patient is.
To make this example easier to follow, let’s consider two data generating functions 1. Treatment:
2. Outcome:
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example
Example: House examines the patient and wants to treat for sarcoidosis. He is always considering treating for sarcoidosis… but in a way that is unrelated to how sick the patient is.
To make this example easier to follow, let’s consider two data generating functions 1. Treatment:
2. Outcome:
0 1
0 0.5 0.1
1 0.5 0.1A
B
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example
Example: House examines the patient and wants to treat for sarcoidosis. He is always considering treating for sarcoidosis… but in a way that is unrelated to how sick the patient is.
To make this example easier to follow, let’s consider two data generating functions 1. Treatment:
2. Outcome:
0 1
0 0.5 0.1
1 0.5 0.1A
B
0 1
0 0.1 0.1
1 0.5 0.5A
B
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propensity score vs. prognostic score
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propensity score vs. prognostic score
This departure arises when the variables predictive of treatment differs from the prognostically relevant variables
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propensity score vs. prognostic score
This departure arises when the variables predictive of treatment differs from the prognostically relevant variables
This insight led to an interesting paper: Bhattacharya & Vogt “Do Instrumental Variables Belong in Propensity Scores?”
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propensity score vs. prognostic score
This departure arises when the variables predictive of treatment differs from the prognostically relevant variables
This insight led to an interesting paper: Bhattacharya & Vogt “Do Instrumental Variables Belong in Propensity Scores?”
Prognostic score is one way to address this: Ben Hansen “The prognostic analog of the propensity score”
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Pilot Matching
There are more ways to use information that is commonly considered.
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Pilot Matching
There are more ways to use information that is commonly considered.
Dylan Greaves
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Pilot Matching
There are more ways to use information that is commonly considered.
Dylan Greaves
Rocky Aikens
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Pilot Matching
There are more ways to use information that is commonly considered.
What if, instead of using many controls to match to a treated, we use some of the controls to understand outcome variation.
Dylan Greaves
Rocky Aikens
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Pilot Matching
Dylan Greaves
There are more ways to use information that is commonly considered.
What if, instead of using many controls to match to a treated, we use some of the controls to understand outcome variation.
Problematic because we are looking at the outcome for some observations, but maybe we can “burn” those and still benefit.
Rocky Aikens
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T Y
C
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T Y
C
treatment effect
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T Y
C
treatment effectp
rop
ensi
ty
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T Y
C
treatment effectp
rop
ensi
ty
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treated
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treated
control
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Pilot Matching
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Pilot Matching
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Pilot Matching
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Pilot Matching
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person id x1 x2 … x_p treated propensity prognostic
1 109 119 71 0 0.32 10
2 106 48 91 1 0.54 18
3 106 81 35 1 0.32 16
4 102 70 79 0 0.32 6
5 65 68 38 0 0.32 13
6 110 118 74 0 0.54 13
7 73 101 69 1 0.54 7
8 50 90 81 0 0.54 18
9 94 71 54 0 0.54 16
10 38 120 42 1 0.32 18
11 54 116 56 0 0.32 18
12 56 106 50 0 0.54 5
13 112 88 75 1 0.32 5
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person id x1 x2 … x_p treated propensity prognostic
1 109 119 71 0 0.32 10
2 106 48 91 1 0.54 18
3 106 81 35 1 0.32 16
4 102 70 79 0 0.32 6
5 65 68 38 0 0.32 13
6 110 118 74 0 0.54 13
7 73 101 69 1 0.54 7
8 50 90 81 0 0.54 18
9 94 71 54 0 0.54 16
10 38 120 42 1 0.32 18
11 54 116 56 0 0.32 18
12 56 106 50 0 0.54 5
13 112 88 75 1 0.32 5
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person id x1 x2 … x_p treated propensity prognostic
1 109 119 71 0 0.32 10
3 106 81 35 1 0.32 16
4 102 70 79 0 0.32 6
5 65 68 38 0 0.32 13
10 38 120 42 1 0.32 18
11 54 116 56 0 0.32 18
13 112 88 75 1 0.32 5
2 106 48 91 1 0.54 18
6 110 118 74 0 0.54 13
7 73 101 69 1 0.54 7
8 50 90 81 0 0.54 18
9 94 71 54 0 0.54 16
12 56 106 50 0 0.54 5
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person id x1 x2 … x_p treated propensity prognostic
1 109 119 71 0 0.32 10
3 106 81 35 1 0.32 16
4 102 70 79 0 0.32 6
5 65 68 38 0 0.32 13
10 38 120 42 1 0.32 18
11 54 116 56 0 0.32 18
13 112 88 75 1 0.32 5
2 106 48 91 1 0.54 18
6 110 118 74 0 0.54 13
7 73 101 69 1 0.54 7
8 50 90 81 0 0.54 18
9 94 71 54 0 0.54 16
12 56 106 50 0 0.54 5
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person id x1 x2 … x_p treated propensity prognostic
13 112 88 75 1 0.32 5
12 56 106 50 0 0.54 5
4 102 70 79 0 0.32 6
7 73 101 69 1 0.54 7
1 109 119 71 0 0.32 10
5 65 68 38 0 0.32 13
6 110 118 74 0 0.54 13
3 106 81 35 1 0.32 16
9 94 71 54 0 0.54 16
10 38 120 42 1 0.32 18
11 54 116 56 0 0.32 18
2 106 48 91 1 0.54 18
8 50 90 81 0 0.54 18
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takeaway
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takeaway
This toy example highlights that the propensity score focuses on treatment, which may be unrelated to outcomes.
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takeaway
This toy example highlights that the propensity score focuses on treatment, which may be unrelated to outcomes.
This is OK – the theory of inference is predicated on randomization, not identical units going into the groups (Fisher)
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takeaway
This toy example highlights that the propensity score focuses on treatment, which may be unrelated to outcomes.
This is OK – the theory of inference is predicated on randomization, not identical units going into the groups (Fisher)
But it is better to start with similar groups (Mill)
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takeaway
This toy example highlights that the propensity score focuses on treatment, which may be unrelated to outcomes.
This is OK – the theory of inference is predicated on randomization, not identical units going into the groups (Fisher)
But it is better to start with similar groups (Mill)
Pilot designs: arxiv
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takeaway
This toy example highlights that the propensity score focuses on treatment, which may be unrelated to outcomes.
This is OK – the theory of inference is predicated on randomization, not identical units going into the groups (Fisher)
But it is better to start with similar groups (Mill)
Pilot designs: arxiv
Stratified pilot matching: arxiv
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a second outcome
Design of Observational Studies: chapter 5.2.3 and 5.2.4Rosenbaum, “The Role of Known Effects in Observational Studies”
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the structure of the argument: two outcomes
If your theory is well developed then you might be able to locate multiple outcomes that will support your understanding of the mechanism of the intervention.
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the structure of the argument: two outcomes
If your theory is well developed then you might be able to locate multiple outcomes that will support your understanding of the mechanism of the intervention.
Two ways this can happen: The second outcome can be compatible (show violation)
The confirmation of a “null effect” can help rebuff claims of unobserved biases
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the structure of the argument: two outcomes
If your theory is well developed then you might be able to locate multiple outcomes that will support your understanding of the mechanism of the intervention.*
Two ways this can happen: The second outcome can be compatible (show violation)
The confirmation of a “null effect” can help rebuff claims of unobserved biases
*Keep this idea separate from “intermediate effects,” not because there’s a deep fundamental difference in these concepts but rather conflating them will tend to confuse discussions.
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coherence
(Rough) Definition:
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coherence
(Rough) Definition: A claim is made that an intervention must have a certain form (i.e., there’s a detailed hypothesis).
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coherence
(Rough) Definition: A claim is made that an intervention must have a certain form (i.e., there’s a detailed hypothesis). In this situation, coherence means a pattern of observed associations compatible with this anticipated form, and incoherence means a pattern of observed associations incompatible with this form.
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coherence
(Rough) Definition: A claim is made that an intervention must have a certain form (i.e., there’s a detailed hypothesis). In this situation, coherence means a pattern of observed associations compatible with this anticipated form, and incoherence means a pattern of observed associations incompatible with this form.
Claims of coherence or incoherence are arguable to the extent that the anticipated form of treatment effect is arguable.
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coherence
(Rough) Definition: A claim is made that an intervention must have a certain form (i.e., there’s a detailed hypothesis). In this situation, coherence means a pattern of observed associations compatible with this anticipated form, and incoherence means a pattern of observed associations incompatible with this form.
Claims of coherence or incoherence are arguable to the extent that the anticipated form of treatment effect is arguable.
If you want to see the technical details of how to build a statistical argument around this then check out Observational Studies, section 17.2 (coherent signed rank statistic).
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X1
X2
X3
Xp
…coherence
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X1
X2
X3
Xp
…coherence
I
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X1
X2
X3
Xp
…
P1
P2
P4
P3
coherence
I
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X1
X2
X3
Xp
…
P1
P2
P4
P3
coherence
Y1 = rape
I
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Y2
X1
X2
X3
Xp
…
P1
P2
P4
P3
coherence
Y1 = rape
= unplanned
pregnancyI
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Y2
X1
X2
X3
Xp
…
P1
P2
P4
P3
coherence
Y1
Y3
= rape
= unplanned
pregnancy
= STI
I
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Y2
X1
X2
X3
Xp
…
P1
P2
P4
P3
coherence
Y1
Y3
= rape
= unplanned
pregnancy
= STI
Y4 = test scores
I
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Y2
X1
X2
X3
Xp
…
P1
P2
P4
P3
coherence
Y1
Y3
= rape
= unplanned
pregnancy
= STI
Y4 = test scores
I
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the structure of the argument: null effect outcomes
Basic idea:
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the structure of the argument: null effect outcomes
Basic idea: Suppose that a treatment is known to not change a particular outcome.
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the structure of the argument: null effect outcomes
Basic idea: Suppose that a treatment is known to not change a particular outcome. Then if we see differences between the treatment and control groups on this particular outcome…
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the structure of the argument: null effect outcomes
Basic idea: Suppose that a treatment is known to not change a particular outcome. Then if we see differences between the treatment and control groups on this particular outcome, this must mean that there are differences between the treatment and control group on unmeasured covariates and thus there is hidden bias.
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example: methylmercury fish
Example:
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example: methylmercury fish
Example: Skerfving (1974) studied whether eating fish contaminated with methylmercury causes chromosome damage.
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example: methylmercury fish
Example: Skerfving (1974) studied whether eating fish contaminated with methylmercury causes chromosome damage. The outcomes of interest was the percentage of cells exhibiting chromosome damage. Pairs were matched for age and sex.
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example: methylmercury fish
Example: Skerfving (1974) studied whether eating fish contaminated with methylmercury causes chromosome damage. The outcomes of interest was the percentage of cells exhibiting chromosome damage. Pairs were matched for age and sex.
control.cu.cells <- c(2.7,.5,0,0,5,0,0,1.3,0,1.8,0,0,1,1.8,0,3.1)
exposed.cu.cells <- c(.7,1.7,0,4.6,0,9.5,5,2,2,2,1,3,2,3.5,0,4);
library(exactRankTests)
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example: methylmercury fish
Example: Skerfving (1974) studied whether eating fish contaminated with methylmercury causes chromosome damage. The outcomes of interest was the percentage of cells exhibiting chromosome damage. Pairs were matched for age and sex.
control.cu.cells <- c(2.7,.5,0,0,5,0,0,1.3,0,1.8,0,0,1,1.8,0,3.1)
exposed.cu.cells <- c(.7,1.7,0,4.6,0,9.5,5,2,2,2,1,3,2,3.5,0,4);
library(exactRankTests)
wilcox.exact(exposed.cu.cells,control.cu.cells,paired=TRUE)
Exact Wilcoxon signed rank test
data: exposed.cu.cells and control.cu.cells
V = 84, p-value = 0.04712
alternative hypothesis: true mu is not equal to 0
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example: methylmercury fish
In the absence of hidden bias, there’s evidence that eating large quantities of fish containing methylmercury causes chromosome damage.
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example: methylmercury fish
In the absence of hidden bias, there’s evidence that eating large quantities of fish containing methylmercury causes chromosome damage.
Going further, Skerfving described other health conditions of these subjects including other diseases such as…
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example: methylmercury fish
In the absence of hidden bias, there’s evidence that eating large quantities of fish containing methylmercury causes chromosome damage.
Going further, Skerfving described other health conditions of these subjects including other diseases such as (i) hypertension, (ii) asthma, (iii) drugs taken regularly, (iv) diagnostic X-rays over the previous three years, (v) and viral diseases such as influenza.
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example: methylmercury fish
In the absence of hidden bias, there’s evidence that eating large quantities of fish containing methylmercury causes chromosome damage.
Going further, Skerfving described other health conditions of these subjects including other diseases such as (i) hypertension, (ii) asthma, (iii) drugs taken regularly, (iv) diagnostic X-rays over the previous three years, (v) and viral diseases such as influenza.
These can be considered outcomes since they describe the period when the exposed subjects were consuming contaminated fish.
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example: methylmercury fish
In the absence of hidden bias, there’s evidence that eating large quantities of fish containing methylmercury causes chromosome damage.
Going further, Skerfving described other health conditions of these subjects including other diseases such as (i) hypertension, (ii) asthma, (iii) drugs taken regularly, (iv) diagnostic X-rays over the previous three years, (v) and viral diseases such as influenza.
These can be considered outcomes since they describe the period when the exposed subjects were consuming contaminated fish.
However, it is difficult to imagine that eating fish contaminated with methylmercury causes influenza or asthma, or prompts X-rays of the hip or lumbar spine.
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example: methylmercury fish
The data
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example: methylmercury fish
The data control.other.health.conditions <- c(rep(0,8),2,rep(0,3),2,1,4,1)
exposed.other.health.conditions <- c(0,0,2,0,2,0,0,1,1,2,0,9,0,0,1,0)
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example: methylmercury fish
The data control.other.health.conditions <- c(rep(0,8),2,rep(0,3),2,1,4,1)
exposed.other.health.conditions <- c(0,0,2,0,2,0,0,1,1,2,0,9,0,0,1,0)
> wilcox.exact(control.other.health.conditions,exposed.other.health.conditions)
Exact Wilcoxon rank sum test
data: control.other.health.conditions and exposed.other.health.conditions
W = 112.5, p-value = 0.5257
alternative hypothesis: true mu is not equal to 0
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example: methylmercury fish
The data control.other.health.conditions <- c(rep(0,8),2,rep(0,3),2,1,4,1)
exposed.other.health.conditions <- c(0,0,2,0,2,0,0,1,1,2,0,9,0,0,1,0)
> wilcox.exact(control.other.health.conditions,exposed.other.health.conditions)
Exact Wilcoxon rank sum test
data: control.other.health.conditions and exposed.other.health.conditions
W = 112.5, p-value = 0.5257
alternative hypothesis: true mu is not equal to 0
There is no evidence of hidden bias.
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example: methylmercury fish
The data control.other.health.conditions <- c(rep(0,8),2,rep(0,3),2,1,4,1)
exposed.other.health.conditions <- c(0,0,2,0,2,0,0,1,1,2,0,9,0,0,1,0)
> wilcox.exact(control.other.health.conditions,exposed.other.health.conditions)
Exact Wilcoxon rank sum test
data: control.other.health.conditions and exposed.other.health.conditions
W = 112.5, p-value = 0.5257
alternative hypothesis: true mu is not equal to 0
There is no evidence of hidden bias.
But absence of evidence is not evidence of absence.
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example: methylmercury fish
Questions: (1) When does such a test have a reasonable prospect of detecting hidden bias?
(2) If no evidence of hidden bias is found, does this imply reduced sensitivity to bias in the comparisons involving the outcomes of primary interest?
(3) If evidence of bias is found, what can be said about its magnitude and its impact on the primary comparisons?
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null effect outcomes
Power of the test of hidden bias:
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null effect outcomes
Power of the test of hidden bias: Let y denote the outcome for which there is a known effect of zero. For a particular unobserved covariate u, what unaffected outcome y would be useful in detecting hidden bias from u?
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null effect outcomes
Power of the test of hidden bias: Let y denote the outcome for which there is a known effect of zero. For a particular unobserved covariate u, what unaffected outcome y would be useful in detecting hidden bias from u?
Precise statement of results in: “The Role of Known Effects in Observational Studies”
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null effect outcomes
Power of the test of hidden bias: Let y denote the outcome for which there is a known effect of zero. For a particular unobserved covariate u, what unaffected outcome y would be useful in detecting hidden bias from u?
Precise statement of results in: “The Role of Known Effects in Observational Studies”
Basic result:
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null effect outcomes
Power of the test of hidden bias: Let y denote the outcome for which there is a known effect of zero. For a particular unobserved covariate u, what unaffected outcome y would be useful in detecting hidden bias from u?
Precise statement of results in: “The Role of Known Effects in Observational Studies”
Basic result: The power of the test of whether y is affected by the treatment increases with the strength of the relationship between y and u. If one is concerned about a particular unobserved covariate u, one should search for an unaffected outcome y that is strongly related to u.
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takeaway
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takeaway
Having a detailed understanding of how your intervention functions, what the causal pathway includes and excludes, will give you more data sources that may validate or refute your hypothesis.
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takeaway
Having a detailed understanding of how your intervention functions, what the causal pathway includes and excludes, will give you more data sources that may validate or refute your hypothesis.
Coherence is trying to flesh out your hypothesis.
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takeaway
Having a detailed understanding of how your intervention functions, what the causal pathway includes and excludes, will give you more data sources that may validate or refute your hypothesis.
Coherence is trying to flesh out your hypothesis.
Known null effects may help to address unobserved confounding
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T W O P R O B L E M S T W O C O N T R O L S
a second control group
Design of Observational Studies: chapter 5.2.2Rosenbaum, “The Role of a Second Control Group in an Observational Study”
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structure of argument
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structure of argument
In an RCT the control and treatment groups are created from a pool of study participants. The assignment to C or T is due to a researcher-directed mechanism (e.g., flipping a coin, or matched pairs). Importantly: all participants can receive C or T.
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structure of argument
In an RCT the control and treatment groups are created from a pool of study participants. The assignment to C or T is due to a researcher-directed mechanism (e.g., flipping a coin, or matched pairs). Importantly: all participants can receive C or T.
In an observational study there are possibly many different reasons for people to have not received the treatment.
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structure of argument
In an RCT the control and treatment groups are created from a pool of study participants. The assignment to C or T is due to a researcher-directed mechanism (e.g., flipping a coin, or matched pairs). Importantly: all participants can receive C or T.
In an observational study there are possibly many different reasons for people to have not received the treatment.
In some situations there are discernable subgroups within the non-treatment group, each subgroup being identifiable by the reason for the subgroup not receiving the treatment.
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structure of argument
In an RCT the control and treatment groups are created from a pool of study participants. The assignment to C or T is due to a researcher-directed mechanism (e.g., flipping a coin, or matched pairs). Importantly: all participants can receive C or T.
In an observational study there are possibly many different reasons for people to have not received the treatment.
In some situations there are discernable subgroups within the non-treatment group, each subgroup being identifiable by the reason for the subgroup not receiving the treatment.
In some subset of these situations these subgroups will be open to critiques of bias when compared to the treatment group, but at least two of the subgroups will differ in the nature of their bias.
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structure of argument
In an RCT the control and treatment groups are created from a pool of study participants. The assignment to C or T is due to a researcher-directed mechanism (e.g., flipping a coin, or matched pairs). Importantly: all participants can receive C or T.
In an observational study there are possibly many different reasons for people to have not received the treatment.
In some situations there are discernable subgroups within the non-treatment group, each subgroup being identifiable by the reason for the subgroup not receiving the treatment.
In some subset of these situations these subgroups will be open to critiques of bias when compared to the treatment group, but at least two of the subgroups will differ in the nature of their bias.
The contrast of these two control groups with the treatment group may strengthen your analysis.
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second control group: army toxicity
Example:
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second control group: army toxicity
Example: The army is interested in the long term effects of exposure to a list of specific chemical agents that were suspected of being toxic. Relatively few soldiers were exposed to these chemicals.
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second control group: army toxicity
Example: The army is interested in the long term effects of exposure to a list of specific chemical agents that were suspected of being toxic. Relatively few soldiers were exposed to these chemicals.
At first pass, one might think to compare these exposed (“treated”) service members to service members who were not exposed at all.
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second control group: army toxicity
Example: The army is interested in the long term effects of exposure to a list of specific chemical agents that were suspected of being toxic. Relatively few soldiers were exposed to these chemicals.
At first pass, one might think to compare these exposed (“treated”) service members to service members who were not exposed at all.
Complicating that comparison, though, is that the army sorted people into jobs which exposed them or to jobs which did not.
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second control group: army toxicity
Example: The army is interested in the long term effects of exposure to a list of specific chemical agents that were suspected of being toxic. Relatively few soldiers were exposed to these chemicals.
At first pass, one might think to compare these exposed (“treated”) service members to service members who were not exposed at all.
Complicating that comparison, though, is that the army sorted people into jobs which exposed them or to jobs which did not.
The army used medical examinations – which were not well documented – to sort some individuals out of high-exposure jobs.
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second control group: army toxicity
Example: The army is interested in the long term effects of exposure to a list of specific chemical agents that were suspected of being toxic. Relatively few soldiers were exposed to these chemicals.
At first pass, one might think to compare these exposed (“treated”) service members to service members who were not exposed at all.
Complicating that comparison, though, is that the army sorted people into jobs which exposed them or to jobs which did not.
The army used medical examinations – which were not well documented – to sort some individuals out of high-exposure jobs. This leaves the comparison between exposed and strictly unexposed potentially biased due to baseline conditions.
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second control group: army toxicity
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second control group: army toxicity
A second control group was constructed using service members who were in jobs which exposed them to chemical agents…
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second control group: army toxicity
A second control group was constructed using service members who were in jobs which exposed them to chemical agents, but not the specific list of chemical agents under consideration.
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second control group: army toxicity
A second control group was constructed using service members who were in jobs which exposed them to chemical agents, but not the specific list of chemical agents under consideration. These other chemical agents were thought to have little or no longer term effects.
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second control group: army toxicity
A second control group was constructed using service members who were in jobs which exposed them to chemical agents, but not the specific list of chemical agents under consideration. These other chemical agents were thought to have little or no longer term effects. Thus this group is thought to have received an “ineffective dose” of the exposure.
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second control group: army toxicity
A second control group was constructed using service members who were in jobs which exposed them to chemical agents, but not the specific list of chemical agents under consideration. These other chemical agents were thought to have little or no longer term effects. Thus this group is thought to have received an “ineffective dose” of the exposure.
Each of these control groups is problematic…
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second control group: army toxicity
A second control group was constructed using service members who were in jobs which exposed them to chemical agents, but not the specific list of chemical agents under consideration. These other chemical agents were thought to have little or no longer term effects. Thus this group is thought to have received an “ineffective dose” of the exposure.
Each of these control groups is problematic: the first group is open to critiques of baseline differences in medical conditions…
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second control group: army toxicity
A second control group was constructed using service members who were in jobs which exposed them to chemical agents, but not the specific list of chemical agents under consideration. These other chemical agents were thought to have little or no longer term effects. Thus this group is thought to have received an “ineffective dose” of the exposure.
Each of these control groups is problematic: the first group is open to critiques of baseline differences in medical conditions; the second group has individuals who were potentially exposed to actively toxic chemical agents.
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second control group: army toxicity
A second control group was constructed using service members who were in jobs which exposed them to chemical agents, but not the specific list of chemical agents under consideration. These other chemical agents were thought to have little or no longer term effects. Thus this group is thought to have received an “ineffective dose” of the exposure.
Each of these control groups is problematic: the first group is open to critiques of baseline differences in medical conditions; the second group has individuals who were potentially exposed to actively toxic chemical agents.
But the first control group is unlikely to suffer from the bias encounter in the second control group, and vice versa.
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second control group: army toxicity
The hope is that the two control groups will not differ from each other in a meaningful way.
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second control group: army toxicity
The hope is that the two control groups will not differ from each other in a meaningful way.
A rejection of a test of equivalency between the control groups is a strong warning sign of potential bias.
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second control group: army toxicity
The hope is that the two control groups will not differ from each other in a meaningful way.
A rejection of a test of equivalency between the control groups is a strong warning sign of potential bias.
A non-rejection may arise for several reasons. A false-negative would be problematic.
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second control group: army toxicity
The hope is that the two control groups will not differ from each other in a meaningful way.
A rejection of a test of equivalency between the control groups is a strong warning sign of potential bias.
A non-rejection may arise for several reasons. A false-negative would be problematic.
The hope is that the control reservoir (i.e., the ratio of controls to treated observations) is large enough that we can reach adequate levels of statistical power for our tests.
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second control group: army toxicity
The hope is that the two control groups will not differ from each other in a meaningful way.
A rejection of a test of equivalency between the control groups is a strong warning sign of potential bias.
A non-rejection may arise for several reasons. A false-negative would be problematic.
The hope is that the control reservoir (i.e., the ratio of controls to treated observations) is large enough that we can reach adequate levels of statistical power for our tests.
Precise statements of how this argument works statistically, as well as a couple more examples from the literature, can be found in “The Role of a Second Control Group in an Observational Study”
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structure of argument
In an RCT the control and treatment groups are created from a pool of study participants. The assignment to C or T is due to a researcher-directed mechanism (e.g., flipping a coin, or matched pairs). All participants can receive either T or C.
In an observational study there are possibly many different reasons for people to have not received the treatment.
In some situations there are discernable subgroups within the non-treatment group, each subgroup being identifiable by the reason for the subgroup not receiving the treatment.
In some subset of these situations these subgroups will be open to critiques of bias when compared to the treatment group, but at least two of the subgroups will differ in the nature of their bias.
The contrast of these two control groups with the treatment group may strengthen your analysis.
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fin.