# Introduction to propertee

#### propertee Authors

#### 2025-10-28

## Main Features

The propertee package, Prognostic Regression Offsets with Propagation of ERrors (for Treatment Effect Estimation), facilitates direct adjustment or standardization for experiments and observational studies, with the option of using a separately fitted prognostic regression model for covariance adjustment. Propertee calls for a self-standing specification of a study’s design and treatment allocations, using these to create weights needed to align treatment and control samples for estimation of treatment effects averaged across a common standard population, with standard errors that appropriately reflect the study design. For covariance adjustment it enables offsetting the outcome against predictions from a dedicated covariance model, with standard error calculations propagating error as appropriate from the covariance model.

The main workflow consists of two main steps and one optional step:

1. Encode the study’s design, including the unit of assignment, treatment status of each unit of assignment, and block membership information as appropriate, in a `StudySpecification` object. This is accomplished with the `obs_spec()`, `rct_spec()` or `rd_spec()` functions.
2. Optionally, fit a covariate adjustment model. This is done with any of a number of modeling functions external to `propertee`, for example `lm()` or `glm()` from the `stats` package, or `lmrob` from the `robustbase` package.
3. Use `lm()` to contrast the separately reweighted treatment and control groups, incorporating optional covariance adjustments as an `offset` to `lm()`. The `StudySpecification` is unobtrusively retained, as is error propagation from the optional covariance adjustment, for use in subsequent standard error calculations. This is done using the `lmitt()` function, which either takes and enriches a fitted `lm` or, in the more fully supported usage, internally calls `lm()` to itself create a directly adjusted contrast of treatment conditions.

## Example Data

The example dataset comes from the state of Tennessee’s Student-Teacher Achievement Ratio (STAR) experiment. Students were randomly assigned to three possible classroom conditions: small (13 to 17 students per teacher), regular class (22 to 25 students per teacher), and regular-with-aide class (22 to 25 students with a full-time teacher’s aide).

    data("STARplus")
    table(STARplus$cond_at_entry)
    #>
    #>         regular    regular+aide           small nonexperimental
    #>            4328            4249            3024            1780

For simplicity for this first example, we will examine a single binary treatment - “small” classrooms versus “regular” and “regular+aide” classrooms.

    STARplus$cond_small <- STARplus$cond_at_entry == "small"
    table(STARplus$cond_small)
    #>
    #> FALSE  TRUE
    #> 10357  3024

After this basic example, we will see how propertee makes it easy to handle non-binary treatment variables by introducing `dichotomy`s.

The outcome of interest is a reading score at the end of kindergarten.

    summary(STARplus$g1treadss)
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's
    #>   404.0   478.0   516.0   521.5   558.0   651.0    5805

We first take the random assignment scheme to have been either complete or Bernoulli randomization of students, separately by school. (Later we’ll change this to the more realistic assumption of complete or Bernoulli randomization, separately by school and year of entry into the study.) Accordingly, experimental blocks are given by the `school_at_entry` variable.

    length(unique(STARplus$school_at_entry))
    #> [1] 81
    head(table(STARplus$school_at_entry))
    #>
    #> 112038 123056 128068 128076 128079 130085
    #>    129     85     54     96    135    135

We need a unique identifier for the unit by which treatment was allocated. STAR treatments were assigned to students, as opposed to classrooms or other aggregates of students; accordingly we need a unique identifier per student. The `stdntid` variable fills this role.

    length(unique(STARplus$stdntid))
    #> [1] 13381
    head(STARplus$stdntid)
    #> [1] 10000 10001 10002 10003 10004 10005

## A Basic Example

### Defining the `StudySpecification`

The three `_spec` functions (`rct_spec()`, `obs_spec()`, and `rd_spec()`) operate similarly. The first argument is the most important, and encodes all the specification information through the use of a formula. The left-hand side of the formula identifies the treatment variable. The right-hand side of the formula consists of the following potential pieces of information:

1. `unit_of_assignment()`: This identifies the variable(s) which indicate the units of assignment. This is required for all specifications. The alias `uoa()` can be used in its place.
2. `block()`: The identifies the variable(s) which contain block information. Optional.
3. `forcing()`: In regression discontinuity specifications (`rd_spec()`), this identifies the variable(s) which contain forcing information.

To define a `StudySpecification` in our example:

    spec <- obs_spec(cond_small ~ unit_of_assignment(stdntid) + block(school_at_entry),
                      data = STARplus, na.fail = FALSE)
    summary(spec)
    #> Observational Study
    #>
    #>  Structure          Variables
    #>  ---------          ---------
    #>  Treatment          cond_small
    #>  Unit of Assignment stdntid
    #>  Block              school_at_entry
    #>
    #> Number of units per Treatment group:
    #>  Txt Grp Num Units
    #>    FALSE      8577
    #>     TRUE      3024

Should additional variables be needed to identify the unit of assignment, block, or forcing, they can be included. Had the variable identifying the years in which participants entered the study been available, for example, we might have identified the experimental blocks as combinations of school and year of study entry, rather than as the school alone. To do this we would pass `block(year_at_entry, school_at_entry)`, rather than `block(school_at_entry)`, in the above.

### Estimating the treatment effect

The main function for estimating treatment effects is the `lmitt()` function. It takes in three main required arguments:

1. A formula specifying the outcome and the desired treatment effect.
2. The data set containing the outcome information.
3. A `StudySpecification`.

Note that the data set does not need to be the same data set which generated the `StudySpecification`; it does however need to include the same variables to identify the units of assignment. (If the variable names differ, the `by=` argument can be used to link them, though we recommend renaming to reduce the likelihood of issues.)

For example, you may have one dataset containing school-level information, and a separate dataset containing student-level information. Assume school is the unit of assignment. While you could of course merge those two data-sets, propertee can instead use the school-level data to define the `StudySpecification`, and the student-level data to estimate the treatment effect.

The formula entering `lmitt()` can take on one of two forms:

    y ~ 1

will estimate the main treatment effect on outcome variable `y`, and

    y ~ x

will estimate subgroup specific treatment effects for each level of `x` for the outcome `y`, if `x` is categorical. For continuous `x`, a main effect and a treatment-`x` interaction effect is estimated.

Therefore, to estimate the treatment effect in our example, we can run:

    te <- lmitt(g1treadss ~ 1, data = STARplus, specification = spec)
    #> The StudySpecification object contains block-level information, but it is not used in this model. Block information is used when weights are defined via `ate()` or `ett()` or if the `absorb=TRUE` argument is passed.
    summary(te)
    #>
    #> Call:
    #> lmitt(g1treadss ~ 1, data = STARplus, specification = spec)
    #>
    #>  Treatment Effects :
    #>                       Estimate Std. Error t value Pr(>|t|)
    #> cond_small.TRUE        12.6024     1.5938   7.907 2.84e-15 ***
    #> g1treadss:(Intercept) 517.4628     0.7898 655.150  < 2e-16 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #> Std. Error calculated via type "HC0"

The data includes ethnicity; we can estimate subgroup effects by ethnicity:

    te_s <- lmitt(g1treadss ~ race, data = STARplus, specification = spec)
    #> The StudySpecification object contains block-level information, but it is not used in this model. Block information is used when weights are defined via `ate()` or `ett()` or if the `absorb=TRUE` argument is passed.
    summary(te_s)
    #> Warning: The following subgroups do not have sufficient degrees of freedom for
    #> standard error estimates and will be returned as NA: racemssng
    #>
    #> Call:
    #> lmitt(g1treadss ~ race, data = STARplus, specification = spec)
    #>
    #> Treatment Effects: (1 not defined because of singularities)
    #>                           Estimate Std. Error t value Pr(>|t|)
    #> cond_small.TRUE_racewhite   9.6229     1.9779   4.865 1.16e-06 ***
    #> cond_small.TRUE_raceblack  16.4164     2.3296   7.047 1.92e-12 ***
    #> cond_small.TRUE_raceasian -19.3846    29.7818  -0.651    0.515
    #> cond_small.TRUE_racehispa  43.1667    50.6159   0.853    0.394
    #> cond_small.TRUE_racenatam  12.5000    37.2522   0.336    0.737
    #> cond_small.TRUE_raceother   8.0000    30.4991   0.262    0.793
    #> cond_small.TRUE_racemssng       NA         NA      NA       NA
    #> g1treadss:racewhite       530.4149     0.9873 537.252  < 2e-16 ***
    #> g1treadss:raceblack       492.0263     1.0665 461.335  < 2e-16 ***
    #> g1treadss:raceasian       552.3846    16.8166  32.848  < 2e-16 ***
    #> g1treadss:racehispa       526.5000    25.1480  20.936  < 2e-16 ***
    #> g1treadss:racenatam       490.0000    26.1639  18.728  < 2e-16 ***
    #> g1treadss:raceother       547.0000    19.0245  28.752  < 2e-16 ***
    #> g1treadss:racemssng       480.0000         NA      NA       NA
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #> Std. Error calculated via type "HC0"

### Including specification weights

Study specification weights can be easily included in this estimation. propertee supports average treatment effect (ATE) and effect of the treatment on the treated (ETT) weights.

To include one of the weights, simply include the `weights = "ate"` or `weights = "ett"` argument to `lmitt()`:

    lmitt(g1treadss ~ 1, data = STARplus, specification = spec, weights = "ate")
    #>       cond_small.TRUE g1treadss:(Intercept)
    #>              11.35214             517.91270
    lmitt(g1treadss ~ 1, data = STARplus, specification = spec, weights = "ett")
    #>       cond_small.TRUE g1treadss:(Intercept)
    #>              10.88851             519.17669

Internally, these call the `ate()` or `ett()` functions which can be used directly.

    head(ate(spec, data = STARplus))
    #> [1] 1.448276 1.237013 1.433071 1.413793 1.675676 1.361345

When included inside `lmitt()`, you do not need to specify any additional arguments to `ate()` or `ett()`, enabling easy functions of weights. For example if you had some other weight variable, say `wgt`, you could include `weights = wgt*ate()` in the `lmitt()` call.

### Covariance Adjustment models

By itself, `lmitt()` does not allow for other covariates; e.g. something like `lmitt(y ~ 1 + control_var,...` will fail. To adjust for covariates, a separate covariate model should be fit. Any model which supports a `predict()` function should work.

    camod <- lm(g1treadss ~ gender * dob + race, data = STARplus)

The `cov_adj()` function can be used to process the covariance adjustment model and produce the required values; and its output can be passed as an `offset=`.

    lmitt(g1treadss ~ 1, data = STARplus, specification = spec,
          weights = "ate", offset = cov_adj(camod))
    #>       cond_small.TRUE g1treadss:(Intercept)
    #>              11.03055             517.91270

Similarly to the weight functions, `cov_adj()` attempts to locate the correct arguments (in this case, mainly the `data=` argument) to use in the model command; while `cov_adj()` does fall back to using the data which is in the covariance model, its safer to use the `newdata=` argument if calling `cov_adj()` outside of the model.

    head(cov_adj(camod, newdata = STARplus))
    #> [1] 519.6613 528.8581 495.1975 531.1456 500.2958 490.3070

Also, similarly to weights, `cov_adj()` can be used in normal modeling commands as well.

    lm(g1treadss ~ cond_small, data = STARplus, weights = ate(spec),
       offset = cov_adj(camod))
    #>
    #> Call:
    #> lm(formula = g1treadss ~ cond_small, data = STARplus, weights = ate(spec),
    #>     offset = cov_adj(camod))
    #>
    #> Coefficients:
    #>    (Intercept)  cond_smallTRUE
    #>         -3.266          11.031

### Absorbing Blocks

If fixed effects for blocks are desired, which can be absorbed away to avoid estimating, the `absorb=TRUE` argument can be passed.

    lmitt(g1treadss ~ 1, data = STARplus, specification = spec, absorb = TRUE)
    #>       cond_small.TRUE g1treadss:(Intercept)
    #>               11.0594              518.8194