# Introduction to natstrat

The `natstrat` package helps to create natural strata that balance many covariates.

Strata are groups of people or units that share certain traits.

Natural strata in particular have a fixed ratio of controls to treated units in every stratum. The fixed ratio allows us to combine strata, either when plotting covariate or outcome distributions or when calculating treatment effects, without needing to weight the individuals in different strata. This allows for simple plots, analyses, and interpretations.

To achieve a fixed ratio of controls to treated units within each stratum, a subset of controls will be selected to form the refined control group. While selecting this subset of controls, many covariates can be balanced. This assures that the control and treated groups are as similar as possible in terms of measured covariates, meaning that if there is no unmeasured confounding, the observed difference in the outcomes of the two groups is due to the treatment received by one group.

There are roughly 6 steps to an analysis using natural strata:

1. Create strata definitions

2. Select a ratio of controls to treated units

3. Specify which covariates should be balanced and how using `generate_constraints()`

4. Select the control units that achieve the fixed ratio and optimal covariate balance using `optimize_controls()`

5. Check whether the balance achieved was satisfactory using `check_balance()`. If not, change some aspects from steps 2-3 and repeat steps 4-5

6. Conduct outcome analysis and visualizations

We will demonstrate each of these steps using an extended example, and then go into Further details about some of the functions and arguments.

Note that natural strata can also be used with any number of treatment or control groups and any number of simultaneous comparisons between subsets of treatment and control groups, as shown in Multiple comparisons. In addition, natural strata can deal with deviations from proportionality, as demonstrated in Deviating from proportionality.

## Example: Smoking and Homocysteine Levels

In this vignette, we demonstrate the use of the `natstrat` package with a motivating example of the effect of smoking on homocysteine levels using data from NHANES 2005-2006 that is provided in the package.

    library(natstrat)
    #> Loading required package: caret
    #> Loading required package: ggplot2
    #> Loading required package: lattice
    data('nh0506')

### Creating strata

Our first step is to choose factors for our strata. We choose to stratify by age and sex, cutting age into three categories:

    age_cat <- cut(nh0506$age,
                   breaks = c(19, 39, 50, 85),
                   labels = c('< 40 years', '40 - 50 years', '> 50 years'))
    strata <- age_cat : nh0506$sex

We can now see how many treated and control units are in each of our strata. Our treated units are composed of daily smokers, whereas control units have never smoked. We look at the ratio of controls to treated units available in each stratum:

### Selecting a ratio of controls to treated

Natural strata are defined by having a fixed ratio of controls to treated units within each stratum, so we now select the fixed ratio of controls we will strive for. As the minimum ratio of controls to treated units available is 2.3, we set a fixed ratio of 1 for our natural strata. Note that we can select a noninteger ratio, such as 1.5 controls for every treated unit. However, if we select too large a ratio compared to the minimum available ratio, it will become difficult to balance covariates in this small dataset.

If we had multiple control or treated groups, as in `nh0506_3groups`, we could also use natural strata, specifying a ratio for each group relative to the primary treated group.

### Generating covariate balance constraints

Our data contains 6 covariates, sex, age, race, education, poverty ratio, and bmi, that we’d like to be balanced between the control and treated groups. `Sex` is already balanced through the definition of the strata. While the strata force balance of our coarse age groups, we still seek to balance `age` more finely. We thus seek to balance `age` and our additional four covariates both across and within strata.

To do this, we generate balance constraints using the function `generate_constraints()` in the package. We must specify `balance_formulas`, `z`, and `data`.

- `balance_formulas`: Here we specify a list of formulas that demonstrate the ways in which we want to balance. These formulas are extensions to `R` formulas, which follow many of the same conventions but have added functionality. Specifically, we include multiple terms on the left hand side, specifying the covariates we’d like to balance. On the right hand side, we explicitly specify `1` which indicates to balance these covariates across the whole population and is included implicitly unless `- 1` or `+ 0` are specified. We also include on the right hand side our `strata` variable, indicating that we’d like to balance the covariates within each level of `strata` as well. To learn more about how to specify balance formulas, skip to the Balance formula syntax section below.
- `z`: Here we specify the 0/1 vector indicating control/treated status.
- `data`: Here we specify the `data.frame` that contains some or all of the covariates included in our `balance_formulas`.

There are many additional arguments which can be adjusted to control the desired covariate balance. These can be found in the Additional `generate_constraints()` arguments section below. Here, we leave all additional arguments as the default.

    constraints <- generate_constraints(
      balance_formulas = list(age + race + education + povertyr + bmi ~ 1 + strata),
      z = nh0506$z,
      data = nh0506)

    names(constraints)
    #> [1] "X"           "importances"
    dim(constraints$X)
    #> [1] 2928  105
    length(constraints$importances)
    #> [1] 105

We see that the return of `generate_constraints()` is a list with two elements.

The first element, `X`, has a row for each unit and a column for each constraint. We see there are 105 constraints generated, which is much more than the 5 covariates entered on the left hand side of `balance_formulas`. This is because those 5 covariates imply 15 variables to balance: age, 5 levels of race, 5 levels of education, poverty ratio, missing poverty ratio, bmi, and missing bmi. Each of these 15 variables is balanced across the population and within each of the 6 strata. This leads to \(15 \cdot (1 + 6) = 105\) balance constraints.

The second element, `importances`, contains an entry for each of the 105 constraints, stating how important it is to balance that constraint. This plays a bigger role as you do more fine tuning as illustrated in the Balance formula syntax and Additional `generate_constraints()` arguments sections.

### Optimizing the control group

We now run the linear program and randomized rounding that will choose the best controls to balance our covariates. We input 5 arguments:

- `z`: Treatment vector
- `X`: Constraint matrix
- `importances`: Named vector of the importance of balancing each of the constraints in `X`
- `st`: Strata vector
- `ratio`: Desired ratio of controls

Additional arguments can be found in in the Additional `optimize_controls()` arguments section. One notable argument is `solver`, which is set by default to ‘Rglpk’. The other solver implemented is ‘gurobi’, and is highly recommended for use on larger data sets if available to you.

    results <- optimize_controls(z = nh0506$z,
                                 X = constraints$X,
                                 importances = constraints$importances,
                                 st = strata,
                                 ratio = 1)

    names(results)
    #>  [1] "objective"                "objective_wo_importances"
    #>  [3] "eps"                      "eps_star"
    #>  [5] "importances"              "weight_star"
    #>  [7] "selected"                 "selected_star"
    #>  [9] "pr"                       "pr_star"
    #> [11] "rrdetails"                "lpdetails"

We see the return of `optimize_controls()` is a list. The most useful element is `selected`, which is a boolean vector stating whether each unit should be included in the analysis or not. This includes both treated and control units. The rest of the elements in the list are useful in analyzing the performance of the linear program and randomized rounding solutions, and are described in the help page for the function.

This package also supports optimizing for multiple comparisons simultaneously; see the Additional `optimize_controls()` arguments section to learn more.

### Checking balance achieved

We now see how well we have balanced our covariates of interest, specified in the `cov_data` dataframe below. The `check_balance` function below will generate standardized differences for all variables in the specified `X` dataframe, whether or not they were included in the original optimization. The `selected` argument simply takes the `selected` element of the results of the optimization in the last step, and the `plot` argument is a boolean stating whether to generate ggplot objects containing love plots demonstrating balance.

    cov_data <- nh0506[, c('sex', 'age', 'race', 'education', 'povertyr', 'bmi')]

    stand_diffs <- check_balance(z = nh0506$z,
                                 X = cov_data,
                                 st = strata,
                                 selected = results$selected,
                                 plot = TRUE)
    #> Excellent balance achieved across strata (<.1 standardized difference in all covariates).
    #>  The mean standardized difference across strata for the covariates is 0.016,
    #>    the median is 0.001,
    #>    and the maximum is 0.079.
    #> Taking a weighted average of standardized differences within strata,
    #>  we have the mean covariate imbalance is 0.025,
    #>    the median is 0.013,
    #>    and the maximum is 0.092.
    #>  Within strata, we see imbalances (>=.2 standardized difference in some covariates in some strata).
    #>  The mean standardized difference within strata for the covariates is 0.025,
    #>    the median is 0,
    #>    and the maximum is 0.298.

We see below that there is excellent balance across strata as all standardized differences in means after refining the control group fall below the first dashed line at 0.1. We also see that when we take the weighted average of the standardized differences in means within strata, where the average is weighted by stratum size, these values also fall below 0.1.

    stand_diffs$plot_pair

We also take a look within each of the 6 strata, and find only two cases of slight imbalance where the standardized differences in means fall outside the second dashed line at 0.2: the ‘Mexican American’ category for males less than 40 years old, and the ‘9-11th grade’ category for females 40-50 years old.

    stand_diffs$plot_strata
    #> $`< 40 years:Male`

    #>
    #> $`< 40 years:Female`

    #>
    #> $`40 - 50 years:Male`

    #>
    #> $`40 - 50 years:Female`

    #>
    #> $`> 50 years:Male`

    #>
    #> $`> 50 years:Female`

As the imbalances are not too severe, we will accept this balance. However, there are many ways we could fine-tune the balance by shifting the arguments specified in the Further details sections.

### Outcome analysis

Once you have balanced treated and control groups formed by taking the units with `TRUE` values in `results$selected`, you can use these to analyze your outcomes. Recall that since the individuals in these groups are not weighted and there is covariate balance both within and across strata, it is very easy to make plots and conduct analyses.

For example, we can make boxplots of the log of the homocysteine levels in the two groups both within each stratum and across the strata:

    nh0506$group <- factor(nh0506$z, levels = c(0, 1),
                              labels = c("Never smoker", "Daily smoker"))
    nh0506$strata <- strata

    ggplot(nh0506[results$selected, ], aes(x = group, y = log(homocysteine))) +
      geom_boxplot() +
      facet_wrap(~ strata)

    ggplot(nh0506[results$selected, ], aes(x = group, y = log(homocysteine))) +
      geom_boxplot()

We can also calculate the average treatment effect on the treated (ATT) in terms of log of homocysteine, both within the strata and across:

    att <- mean(log(nh0506$homocysteine[results$selected & nh0506$z == 1])) -
      mean(log(nh0506$homocysteine[results$selected & nh0506$z == 0]))
    cat(paste0("\nThe ATT across the strata is ", round(att, 3), " on the log scale,
        which converted back to a regular scale is ", round(exp(att), 3), " umol/L."))
    #>
    #> The ATT across the strata is 0.13 on the log scale,
    #>     which converted back to a regular scale is 1.139 umol/L.

    for (st in levels(nh0506$strata)) {
      att <- mean(log(nh0506$homocysteine[results$selected & nh0506$z == 1 & nh0506$strata == st])) -
        mean(log(nh0506$homocysteine[results$selected & nh0506$z == 0 & nh0506$strata == st]))
      cat(paste0("\nThe ATT for stratum ", st, " is ", round(att, 3), " on the log scale,
          which converted back to a regular scale is ", round(exp(att), 3), " umol/L."))
    }
    #>
    #> The ATT for stratum < 40 years:Male is 0.081 on the log scale,
    #>       which converted back to a regular scale is 1.084 umol/L.
    #> The ATT for stratum < 40 years:Female is 0.246 on the log scale,
    #>       which converted back to a regular scale is 1.279 umol/L.
    #> The ATT for stratum 40 - 50 years:Male is 0.04 on the log scale,
    #>       which converted back to a regular scale is 1.041 umol/L.
    #> The ATT for stratum 40 - 50 years:Female is 0.213 on the log scale,
    #>       which converted back to a regular scale is 1.237 umol/L.
    #> The ATT for stratum > 50 years:Male is 0.081 on the log scale,
    #>       which converted back to a regular scale is 1.085 umol/L.
    #> The ATT for stratum > 50 years:Female is 0.17 on the log scale,
    #>       which converted back to a regular scale is 1.185 umol/L.

We see a similar effect of smoking increasing homocysteine levels across all the strata, but the effect is greater for females of all ages than for males.

## Further details

### Deviating from proportionality

In some cases, the minimum feasible ratio is defined by a small stratum, and you might want to choose a larger ratio. In this case, you should specify the number of controls you want to select from each stratum in the `qs` argument of `optimize_controls()` instead of specifying the ratio.

One way to do this would be to set `qs` to be the maximum of the number of controls available and the desired amount of the ratio times the number of treated units in the stratum.

An alternative way would be to make up for the deficit in one or more strata by selecting additional controls from other similar strata such that the overall ratio is maintained. An example of this is below, although it is not recommended for this stratification as the stratum with the smallest ratio available is not particularly small relative to the others and thus forcing all controls from that stratum to be included will limit the optimal balance you can achieve.

First, we will specify what it means for strata to be close to one another, by defining the three age groups to be 0, 1, or 2 distance units apart from one another and defining the two sexes to be 1 distance unit apart from one another. We combine the two distance matrices for these factors using the `create_dist_matrix()` function in this package which will add the distances in the two dimensions to create one distance matrix.

    age_dist <- matrix(data = c(0, 1, 2, 1, 0, 1, 2, 1, 0),
                       nrow = 3,
                       byrow = TRUE,
                       dimnames = list(levels(age_cat), levels(age_cat)))
    age_dist
    #>               < 40 years 40 - 50 years > 50 years
    #> < 40 years             0             1          2
    #> 40 - 50 years          1             0          1
    #> > 50 years             2             1          0

    sex_dist <- matrix(data = c(0, 1, 1, 0),
                       nrow = 2,
                       dimnames = list(levels(nh0506$sex), levels(nh0506$sex)))
    sex_dist
    #>        Male Female
    #> Male      0      1
    #> Female    1      0

    strata_dist <- create_dist_matrix(age_dist, sex_dist)
    strata_dist
    #>                      < 40 years:Male 40 - 50 years:Male > 50 years:Male
    #> < 40 years:Male                    0                  1               2
    #> 40 - 50 years:Male                 1                  0               1
    #> > 50 years:Male                    2                  1               0
    #> < 40 years:Female                  1                  2               3
    #> 40 - 50 years:Female               2                  1               2
    #> > 50 years:Female                  3                  2               1
    #>                      < 40 years:Female 40 - 50 years:Female > 50 years:Female
    #> < 40 years:Male                      1                    2                 3
    #> 40 - 50 years:Male                   2                    1                 2
    #> > 50 years:Male                      3                    2                 1
    #> < 40 years:Female                    0                    1                 2
    #> 40 - 50 years:Female                 1                    0                 1
    #> > 50 years:Female                    2                    1                 0

Next, we specify that we’d like a ratio of 2.5 overall, meaning in an ideal world, we’d like the following amount of control units selected for each stratum:

    #>      < 40 years:Male    < 40 years:Female   40 - 50 years:Male
    #>                  320                  220                  202
    #> 40 - 50 years:Female      > 50 years:Male    > 50 years:Female
    #>                  182                  308                  220

However, only 185 controls are available in the `40 - 50 years:Male` stratum, so 17 controls must be taken from elsewhere.

We also specify the amount of extra units we will tolerate from any one stratum. To specify this allowable deviation from proportionality, we specify one or both of the following:

- `max_ratio`: We set this to 2.6, meaning the ratio of controls to treated units chosen from a single stratum cannot exceed 2.6. By default, this is set to 1.1 times the `ratio`, which in this case would be 2.75. To have no maximum ratio, this can be set to `Inf`.
- `max_extra_s`: If we prefer setting maximum extra units in absolute amounts, we can set this to either a number of extra units per stratum, or a named vector stating the maximum extra units for each stratum. The maximum of the extra units allowed by `max_ratio` and by `max_extra_s` will be taken. We set this to 0, such that our maximum extra units are solely controlled by `max_ratio`. By default, this is set to 5. This can be set to `Inf` to have no maximum amount.

It is not recommended to set either of the above arguments to `Inf` as it will place as much of the burden as possible on the closest stratum due to the optimization algorithm and potentially cause large deviations from proportionality in certain strata.

We set the `strata_dist` argument to specify that certain strata are closer to one another. If we leave this `NULL`, all strata will be assumed to be equidistant.

    qs <- generate_qs(z = nh0506$z,
                      st = strata,
                      ratio = 2.5,
                      max_ratio = 2.6,
                      max_extra_s = 0,
                      strata_dist = strata_dist)
    qs[1, ]
    #>      < 40 years:Male    < 40 years:Female   40 - 50 years:Male
    #>                  320                  220                  185
    #> 40 - 50 years:Female      > 50 years:Male    > 50 years:Female
    #>                  190                  317                  220

We see that for the control group (row 1 of `qs`), 8 extra units will be drawn from the `40 - 50 years:Female` stratum, yielding a ratio of 2.60, and 9 extra units will be drawn from the `> 50 years:Male` stratum, yielding a ratio of 2.58. These are two of the three strata that are 1 distance unit away from our deficient stratum.

Once we have these `qs` generated, we can plug these into `optimize_controls()` and set the `ratio` argument to `NULL`. Everything else holds the same.

### Multiple comparisons

This package supports optimizing covariate balance for multiple separate comparisons simultaneously. This is useful when there are multiple treatment or control groups of vastly varying sizes, such that you might want to have a main analysis with all groups, and supplement this with another analysis of a subset of groups with larger sample sizes.

In the NHANES data, splitting into three exposures yields the following:

    data('nh0506_3groups')
    table(nh0506_3groups$z)
    #>
    #> never smoker some smoking daily smoker
    #>         2347         1529          581

In this case, we might want a primary analysis of 581 people from each exposure, and then a larger analysis of 1529 people from both the never smokers and occasional smokers. This larger analysis would yield greater power to detect differences between these two groups, where the treatment effect would be calculated for the occasional smoker population, as opposed to the daily smoker population as in the primary analysis.

To set this up, we first repeat the stratifying and constraint generating process from before. In this case, we only stratify on age and instead balance for sex later:

    strata2 <- cut(nh0506_3groups$age,
                   breaks = c(19, 39, 50, 85),
                   labels = c('< 40 years', '40 - 50 years', '> 50 years'))

    constraints2 <- generate_constraints(
      balance_formulas = list(age + race + education + povertyr + bmi + sex ~ 1 + strata2),
      z = nh0506_3groups$z,
      data = nh0506_3groups,
      treated = 'daily smoker')

    names(constraints2)
    #> [1] "X"           "importances"
    dim(constraints2$X)
    #> [1] 4457   68
    length(constraints2$importances)
    #> [1] 68

Note that although we have 68 constraints, these will be enforced for each pair of groups within each comparison, such that we will have 4 times this many constraints– 3 for the main comparison and 1 for the supplemental comparison.

The optimization step is where we need to specify information about each comparison. We will provide the supplemental comparison information in the `_star` arguments.

For the second comparison, although we want to use all occasional smokers and an equivalent amount of never smokers, we will already have selected some of each for the main comparison, which we will reuse in the second comparison. Note that we do not have to reuse these people, but we would need to be careful about specifying the treatment effect of interest if we did not use all the occasional smokers.

We thus want to select the difference between the number of people we want from each group for the second analysis and for the main analysis, which we will split into strata and specify in the `q_star_s` argument. We also add arguments for `treated` and `treated_star`, which will specify the primary treated group in each comparison. Finally, we add `correct_sizes = FALSE`, meaning that sample sizes will only be correct in expectation, which is needed for multiple comparisons.

    q_star_s <- matrix(c(rep(table(nh0506_3groups$z, strata2)['some smoking', ] -
                               table(nh0506_3groups$z, strata2)['daily smoker', ], 2),
                         rep(0, 3)), byrow = TRUE, nrow = 3,
                       dimnames = list(levels(nh0506_3groups$z), levels(strata2)))
    q_star_s
    #>              < 40 years 40 - 50 years > 50 years
    #> never smoker        228           116        604
    #> some smoking        228           116        604
    #> daily smoker          0             0          0

    results <- optimize_controls(z = nh0506_3groups$z,
                                 X = constraints2$X,
                                 importances = constraints2$importances,
                                 st = strata2,
                                 ratio = 1,
                                 treated = 'daily smoker',
                                 treated_star = 'some smoking',
                                 q_star_s = q_star_s,
                                 correct_sizes = FALSE)

We can then analyze our balance results and estimate our treatment effects in the same way as before.

### Balance formula syntax

The `balance_formulas` argument of `generate_constraints()` allows for us to specify what we want to balance, within what sections of the population, and how to prioritize it.

For example, to balance age and race across the population and within the levels of `strata`, we specify `age + race ~ 1 + strata` within the list of `balance_formulas`.

We now look at variations on this formula, which show how to change which groups to balance in and how to incorporate coefficients to indicate how to prioritize different aspects of balance:

- `age + race ~ strata`: Equivalent
- `age + race ~ strata - 1` or `age + race ~ strata + 0`: Only balance within the levels of `strata`
- `age + race ~ 1`: Only balance across the population
- `2 * age + race ~ 1 + strata`: Twice as great penalty for imbalanced `age`
- `age + race ~ 2 + strata`: Twice as great penalty for imbalanced covariates across the population
- `age + race ~ 1 + 2 * strata`: Twice as great penalty for imbalanced covariates within `strata`
- `age + race ~ 0 + age_cat + sex`: Balance within the levels of `age_cat` and `sex` individually instead of within their interaction that forms `strata`

If we want to balance all, or almost all, covariates in a data frame with the same penalty coefficient, we can use the `.` and `-` options as usual. `. - sex ~ strata` would mean to balance all the variables in our specified data frame except for `sex` across the population and within the strata.

We combine any amount of such balance formulas in a list. For example, another way to specify `age + race ~ 1 + strata` would be to specify two entries in the list: `age ~ 1 + strata, race ~ 1 + strata`. Yet another way would be to enter `age, race` in the `balance_formulas` list and also set `default_rhs = '1 + strata'`. This would apply the right hand side `1 + strata` to any element of the `balance_formulas` list that does not have a right hand side specified.

### Additional `generate_constraints()` arguments

- `weight_by_size`: The default value of 0 signifies that imbalance in each group is viewed in absolute terms, not relative to the group size. A value of 1 signifies that constraints for groups will be weighted such that imbalance is viewed relative to group size instead of in absolute terms. Values between 0 and 1 fall in between these two extremes.
- `denom_variance`: The default value of `treated` signifies that covariates will be standardized by dividing by the treated variance across the population. To instead divide by the average of the control and treated variance, set `denom_variance = 'pooled'`.
- `autogen_missing`: The default value of 50 means that missingness indicators will automatically be generated for any covariate with missing values, and these missingness indicators will be balanced according the the same right hand side as the original covariate. `50` indicates that the penalty for having imbalance in the missingness indicator will be amplified 50 times. This argument can be set to a different number greater than 0, with larger numbers indicating more pressure to balance the missingness indicator, or can be set to `FALSE`, which means that missingness indicators will not be generated and is only recommended if you have already accounted for missing data in some way.

### Additional `optimize_controls()` arguments

- `q_s`: This is a named vector specifying how many controls to take from each stratum. This should only be used if the ratio of controls to treated units in each stratum is not constant. This could occur if a certain ratio is not feasible in a stratum and thus you’d like to select fewer controls from that stratum or compensate by selecting controls from other specified strata. The function `generate_qs()`, described in the Deviating from proportionality section above, is useful for specifying this argument
- `weight_star`: This is a scalar or vector specifying how much to prioritize balancing each supplemental comparison relative to the main comparison.
- `integer`: Whether to solve the integer solution as opposed to using randomized rounding. Note that using the integer solver is not recommended for large datasets as the optimization may take an excessively long time or never finish. It is instead suggested to leave `integer = FALSE` (default), which will conduct randomized rounding of the linear program as described in the manuscript `Brumberg et al. (2021+)`
- `solver`: The solver we’d like to use (‘Rglpk’ or ‘gurobi’). Note that we use the ‘Rglpk’ solver here as we have a small dataset and the software is publicly available. However, we recommend using the ‘gurobi’ software for larger datasets if available
- `seed`: The seed we’d like to set when conducting randomized rounding of the linear program solution if `integer = FALSE`
- `runs`: The number of times we’d like to conduct randomized rounding if `integer = FALSE` before taking the best solution
- `time_limit`: Time limit for the optimization
- `threads`: A scalar indicating how many threads the solver should use
- `correct_sizes`: If set to `TRUE`, the sampling algorithm will not strictly enforce the sample size constraints. This must be set to `TRUE` when using multiple comparisons
- `low_memory`: If set to `TRUE`, the result will not contain the epsilon matrices, which show the imbalance that each covariate contributes to the objective function