Inverse Probability Tilting

This page explains the details of estimating weights using inverse probability tilting by setting method = "ipt" in the call to weightit() or weightitMSM(). This method can be used with binary and multi-category treatments.

In general, this method relies on estimating propensity scores using a modification of the usual generalized linear model score equations to enforce balance and then converting those propensity scores into weights using a formula that depends on the desired estimand. This method relies on code written for WeightIt using rootSolve::multiroot().

Binary Treatments

For binary treatments, this method estimates the weights using formulas described by Graham, Pinto, and Egel (2012). The following estimands are allowed: ATE, ATT, and ATC. When the ATE is requested, the optimization is run twice, once for each treatment group.

Multi-Category Treatments

For multi-category treatments, this method estimates the weights using modifications of the formulas described by Graham, Pinto, and Egel (2012). The following estimands are allowed: ATE and ATT. When the ATE is requested, estimation is performed once for each treatment group. When the ATT is requested, estimation is performed once for each non-focal (i.e., control) group.

Continuous Treatments

Inverse probability tilting is not compatible with continuous treatments.

Longitudinal Treatments

For longitudinal treatments, the weights are the product of the weights estimated at each time point. This method is not guaranteed to yield exact balance at each time point. NOTE: the use of inverse probability tilting with longitudinal treatments has not been validated!

Sampling Weights

Sampling weights are supported through s.weights in all scenarios.

Missing Data

In the presence of missing data, the following value(s) for missing are allowed:

"ind" (default): First, for each variable with missingness, a new missingness indicator variable is created which takes the value 1 if the original covariate is NA and 0 otherwise. The missingness indicators are added to the model formula as main effects. The missing values in the covariates are then replaced with the covariate medians (this value is arbitrary and does not affect estimation). The weight estimation then proceeds with this new formula and set of covariates. The covariates output in the resulting weightit object will be the original covariates with the NAs.

M-estimation

M-estimation is supported for all scenarios. See glm_weightit() and vignette("estimating-effects") for details.

Details

Inverse probability tilting (IPT) involves specifying estimating equations that fit the parameters of two or more generalized linear models with a modification that ensures exact balance on the covariate means. These estimating equations are solved, and the estimated parameters are used in the (generalized) propensity score, which is used to compute the weights. Conceptually and mathematically, IPT is very similar to entropy balancing and just-identified CBPS. For the ATT and ATC, entropy balancing, just-identified CBPS, and IPT will yield identical results. For the ATE or when link is specified as something other than "logit", the three methods differ.

Treatment effect estimates for binary treatments are consistent if the true propensity score is a logistic regression or the outcome model is linear in the covariates and their interaction with treatments. For entropy balancing, this is only true for the ATT, and for just-identified CBPS, this is only true if there is no effect modification by covariates. In this way, IPT provides additional theoretical guarantees over the other two methods, though potentially with some cost in precision.

Additional Arguments

moments: integer; the highest power of each covariate to be balanced. For example, if moments = 3, each covariate, its square, and its cube will be balanced. Can also be a named vector with a value for each covariate (e.g., moments = c(x1 = 2, x2 = 4)). Values greater than 1 for categorical covariates are ignored. Default is 1 to balance covariate means.
int: logical; whether first-order interactions of the covariates are to be balanced. Default is FALSE.
quantile: a named list of quantiles (values between 0 and 1) for each continuous covariate, which are used to create additional variables that when balanced ensure balance on the corresponding quantile of the variable. For example, setting quantile = list(x1 = c(.25, .5. , .75)) ensures the 25th, 50th, and 75th percentiles of x1 in each treatment group will be balanced in the weighted sample. Can also be a single number (e.g., .5) or a vector (e.g., c(.25, .5, .75)) to request the same quantile(s) for all continuous covariates.
link: the link used to determine the inverse link for computing the (generalized) propensity scores. Default is "logit", which is used in the original description of the method by Graham, Pinto, and Egel (2012), but "probit", "cauchit", "cloglog", "loglog", "log", and "clog" are also allowed. Note that negative weights are possible with these last two and they should be used with caution. An object of class "link-glm" can also be supplied. The argument is passed to quasibinomial().

The stabilize argument is ignored.

Additional Outputs

obj: When include.obj = TRUE, the output of the call to optim(), which contains the coefficient estimates and convergence information. For ATE fits or with multi-category treatments, a list of rootSolve::multiroot() outputs, one for each weighted group.

References

`estimand = "ATE"`

Graham, B. S., De Xavier Pinto, C. C., & Egel, D. (2012). Inverse Probability Tilting for Moment Condition Models with Missing Data. The Review of Economic Studies, 79(3), 1053–1079. doi:10.1093/restud/rdr047

`estimand = "ATT"`

Sant'Anna, P. H. C., & Zhao, J. (2020). Doubly robust difference-in-differences estimators. Journal of Econometrics, 219(1), 101–122. doi:10.1016/j.jeconom.2020.06.003

Examples

data("lalonde", package = "cobalt")

#Balancing covariates between treatment groups (binary)
(W1 <- weightit(treat ~ age + educ + married +
                  nodegree + re74, data = lalonde,
                method = "ipt", estimand = "ATT"))
#> A weightit object
#>  - method: "ipt" (inverse probability tilting)
#>  - number of obs.: 614
#>  - sampling weights: none
#>  - treatment: 2-category
#>  - estimand: ATT (focal: 1)
#>  - covariates: age, educ, married, nodegree, re74

summary(W1)
#>                   Summary of weights
#> 
#> - Weight ranges:
#> 
#>           Min                                 Max
#> treated 1.                 ||               1.   
#> control 0.017 |---------------------------| 2.263
#> 
#> - Units with the 5 most extreme weights by group:
#>                                       
#>              1     2     3     4     5
#>  treated     1     1     1     1     1
#>            410   404   224   111    84
#>  control 1.464 1.485 1.576 1.743 2.263
#> 
#> - Weight statistics:
#> 
#>         Coef of Var   MAD Entropy # Zeros
#> treated       0.000 0.000   0.000       0
#> control       0.839 0.707   0.341       0
#> 
#> - Effective Sample Sizes:
#> 
#>            Control Treated
#> Unweighted  429.       185
#> Weighted    252.12     185

cobalt::bal.tab(W1)
#> Balance Measures
#>                Type Diff.Adj
#> prop.score Distance   0.0164
#> age         Contin.  -0.0000
#> educ        Contin.   0.0000
#> married      Binary  -0.0000
#> nodegree     Binary  -0.0000
#> re74        Contin.  -0.0000
#> 
#> Effective sample sizes
#>            Control Treated
#> Unadjusted  429.       185
#> Adjusted    252.12     185

#Balancing covariates with respect to race (multi-category)
(W2 <- weightit(race ~ age + educ + married +
                  nodegree + re74, data = lalonde,
                method = "ipt", estimand = "ATE"))
#> A weightit object
#>  - method: "ipt" (inverse probability tilting)
#>  - number of obs.: 614
#>  - sampling weights: none
#>  - treatment: 3-category (black, hispan, white)
#>  - estimand: ATE
#>  - covariates: age, educ, married, nodegree, re74

summary(W2)
#>                   Summary of weights
#> 
#> - Weight ranges:
#> 
#>          Min                                  Max
#> black  1.57   |------------|               15.787
#> hispan 1.711  |--------------------------| 29.07 
#> white  1.102 |--|                           4.693
#> 
#> - Units with the 5 most extreme weights by group:
#>                                          
#>            203    166    163   153    152
#>   black  6.567   6.77  7.096 9.976 15.787
#>             67     43     39    36     28
#>  hispan 17.436 21.673 23.033 24.23  29.07
#>            291    285    258   205      6
#>   white  3.841  3.912  3.934 4.177  4.693
#> 
#> - Weight statistics:
#> 
#>        Coef of Var   MAD Entropy # Zeros
#> black        0.618 0.398   0.133       0
#> hispan       0.618 0.442   0.164       0
#> white        0.389 0.316   0.070       0
#> 
#> - Effective Sample Sizes:
#> 
#>             black hispan  white
#> Unweighted 243.     72.  299.  
#> Weighted   176.11   52.3 259.76

cobalt::bal.tab(W2)
#> Balance summary across all treatment pairs
#>             Type Max.Diff.Adj
#> age      Contin.            0
#> educ     Contin.            0
#> married   Binary            0
#> nodegree  Binary            0
#> re74     Contin.            0
#> 
#> Effective sample sizes
#>             black hispan  white
#> Unadjusted 243.     72.  299.  
#> Adjusted   176.11   52.3 259.76