Inverse Probability Weighting Method, Multiple Treatments with An Ordinal Variable

[Amy Zhang]

I am currently working on a model with an ordinal outcome (i.e., self-rated health: 1=very unhealthy, 2=unhealthy, 3=fair, 4=healthy, 5=very healthy). My treatment variable is a binary variable (good economic condition=1, others=0). This is how I used the IPW method:

***Generate ATT weight
.logit econ_good urban female age edu i.occupation
.predict ps, p
.gen _att_wt=econ_good+(1-econ_good)*(ps/1-ps)

***Run the model with the weight
ologit selfrate econ_good urban female age edu i.occupation [pw=_att_wt]

Now I really want to try a three-level treatment--economic condition (1=difficult economic condition, 2=fairt economic condition, 3=good economic condition). I don't know how I am going to use the IPW method with (1) A three-level treatment and (2) an ordinal outcome. I also want to keep using the doubly robust method (i.e., having the control variables when I run the Ordinal Logistic Regression model).

Thank you!

[Jeff Wooldridge]

Amy: When I get to my desktop computer I can make some suggestions.

[Jeff Wooldridge]

Amy:

Here's a link to a co-authored paper of mine, where we show how to do the weighting with more than two groups. Now you have different ATTs, and the weights that we propose identify those. See equations (43) and (50) in particular. In your case, the estimated J probability is the hth treatment unit. So you estimate the parameters for each (g,h) combination by using the ratios of the probabilities for the two treatment groups. Then, do the averaging as in (50).

Sloczynski_Wooldridge_2018

I should have mentioned that you can use either ordered logit or multinomial logit to estimate the treatment probabilities.

[Stephen Jenkins]

thanks for the link, Jeff, but it appears that one needs to have MSU credentials in order to log in and get the paper.

[Jeff Wooldridge]

Sorry Stephen; that was dumb. Here's the working paper version, which is almost the published version. For some reason, the key equations are (44) and (51) in the WP version ....

Sloczynski_Wooldridge_WP

[Amy Zhang]

Thank you so much for sharing the paper, Prof. Wooldridge!

[Amy Zhang]

edit it later

[Jeff Wooldridge]

Amy: When I get a bit more time I can suggest some Stata code.

[Amy Zhang]

Amy: When I get a bit more time I can suggest some Stata code.

Thank you so much for your help, Jeff and Stephen! I really appreciate it. I think the paper is a bit general when it comes to this specific case and I found another paper that is similar to my models- https://onlinelibrary.wiley.com/doi/....1002/jae.2386.

It seems that I need to mean-center all the covariates (including the categorical variables) except for the treatment variable at the second stage of the model. Following the steps of this paper, here are my Stata codes:

***Stage 1, Generate ATE weight .

ologit econ urban female age i.edu occupation [pw=sampleweight]
predcit m1 m2 m3

***ATE weight
gen p1=1/m1
gen p2=1/m2
gen p3=1/m3

gen pweight=p1 if econ==1
replace pweight=p2 if econ==2
replace pweight=p3 if econ==3



***Mean Centering all the covariates!
egen female_mean=mean(female)
gen female_center=female-female_mean

egen age_mean=mean(age)
gen age_center=age-age_mean

egen edu2_mean=mean(edu2)
gen edu2_center=edu2-edu2_mean
egen edu3_mean=mean(edu3)
gen edu3_center=edu3-edu3_mean

...

***Stage 2, Run the model with the weight
ologit selfrate urban_center female_center age_center edu2_center edu3_center edu4_center occupation_center i.econ [pw=sampleweight*pweight]

Paper: Uysal, S. D. (2015) Doubly Robust Estimation of Causal Effects with Multivalued Treatments: An Application to the Returns to Schooling. J. Appl. Econ., 30: 763– 786. doi: 10.1002/jae.2386.

I am not sure if I got the right model. Could you please help me with this when you have time? Thank you!

[Jeff Wooldridge]

Ah, you even threw in the twist of needing sampling weights.

What you have shown seems correct to me. I'd probably just do three separate estimations but it seems correct to me. Now, your standard errors do not account for the first-stage estimation, but they are conservative -- based on my 2007 Journal of Econometrics paper -- so you may be fine with that.

I guess my one question is: How will you summarize the outcome variable because it is also ordered? I guess I would look at the differences in the probabilities of being at different levels of selfrate by treatment status. Derya's paper used linear regression for the outcome variable but you can use any method combined with IPW.

It looks like you have this figured out.

JW

[Amy Zhang]

Ah, you even threw in the twist of needing sampling weights.

What you have shown seems correct to me. I'd probably just do three separate estimations but it seems correct to me. Now, your standard errors do not account for the first-stage estimation, but they are conservative -- based on my 2007 Journal of Econometrics paper -- so you may be fine with that.

I guess my one question is: How will you summarize the outcome variable because it is also ordered? I guess I would look at the differences in the probabilities of being at different levels of selfrate by treatment status. Derya's paper used linear regression for the outcome variable but you can use any method combined with IPW.

It looks like you have this figured out.

JW

Thank you, Jeff! This is super helpful!! It's great to know that I will get conservative results and I can generalize the method to ordered logistic regression!

I know this is A LOT to ask. I have another question about the next step. I am considering the mediation analysis using the KHB method. For example, I want to examine whether economic conditions influence self-rated health through social support. In other words, those with better economic conditions might have a larger social network that brings them more social supports, contributing to their better self-rated health. Of course, the causal direction is unclear, and it is just my assumption.

Stata codes:

gen pw_all=sampleweight*pweight

khb ologit selfrate i.econ || support [pw=pw_all], concomitant (urban_center female_center age_center edu2_center edu3_center edu4_center occupation_center) summary d

My questions are: (1) Can I combine IPW with the KHB method at all? (2) If I can combine the two methods, should I mean center my assumed mediator (in this case, social support) as well? (3) If my treatment variable is binary (e.g., having good economic condition VS others), can I combine IPW with the KHB method? Thank you so much!

Here are two papers about the KHB method:

Comparing Regression Coefficients Between Same-sample Nested Models Using Logit and Probit: A New Method https://journals.sagepub.com/doi/ful...81175012444861

Total, Direct, and Indirect Effects in Logit and Probit Models https://journals.sagepub.com/doi/10....49124113494572