xBalance: Standardized Differences for Stratified Comparisons In RItools: Randomization Inference Tools

Description

Given covariates, a treatment variable, and a stratifying factor, calculates standardized mean differences along each covariate, with and without the stratification and tests for conditional independence of the treatment variable and the covariates within strata.

Usage

 ```1 2 3 4 5``` ```xBalance(fmla, strata = list(unstrat = NULL), data, report = c("std.diffs", "z.scores", "adj.means", "adj.mean.diffs", "adj.mean.diffs.null.sd", "chisquare.test", "p.values", "all")[1:2], stratum.weights = harmonic, na.rm = FALSE, covariate.scaling = NULL, normalize.weights = TRUE, impfn = median, post.alignment.transform = NULL) ```

Arguments

 `fmla` A formula containing an indicator of treatment assignment on the left hand side and covariates at right. `strata` A list of right-hand-side-only formulas containing the factor(s) identifying the strata, with `NULL` entries interpreted as no stratification; or a factor with length equal to the number of rows in data; or a data frame of such factors. See below for examples. `data` A data frame in which `fmla` and `strata` are to be evaluated. `report` Character vector listing measures to report for each stratification; a subset of ```c("adj.means", "adj.mean.diffs", "adj.mean.diffs.null.sd", "chisquare.test", "std.diffs", "z.scores", "p.values", "all")```. P-values reported are two-sided for the null-hypothesis of no effect. The option "all" requests all measures. `stratum.weights` Weights to be applied when aggregating across strata specified by `strata`, defaulting to weights proportional to the harmonic mean of treatment and control group sizes within strata. This can be either a function used to calculate the weights or the weights themselves; if `strata` is a data frame, then it can be such a function, a list of such functions, or a data frame of stratum weighting schemes corresponding to the different stratifying factors of `strata`. See details. `na.rm` Whether to remove rows with NAs on any variables mentioned on the RHS of `fmla` (i.e. listwise deletion). Defaults to `FALSE`, wherein rows aren't deleted but for each variable with `NA`s a missing-data indicator variable is added to the variables on which balance is calculated and medians are imputed for the variable with missing data (in RItools versions 0.1-9 and before the default imputation was the mean, in RItools versions 0.1-11 and henceforth the default is the median). See the example below. `covariate.scaling` A scale factor to apply to covariates in calculating `std.diffs`. If `NULL`, `xBalance` pools standard deviations of each variable in the treatment and control group (defining these groups according to whether the LHS of `formula` is greater than or equal to 0). Also, see details. `normalize.weights` If `TRUE`, then stratum weights are normalized so as to sum to 1. Defaults to `TRUE`. `impfn` A function to impute missing values when `na.rm=FALSE`. Currently `median`. To impute means use `mean.default`. `post.alignment.transform` Optional transformation applied to covariates just after their stratum means are subtracted off.

Details

In the unstratified case, the standardized difference of covariate means is the mean in the treatment group minus the mean in the control group, divided by the S.D. (standard deviation) in the same variable estimated by pooling treatment and control group S.D.s on the same variable. In the stratified case, the denominator of the standardized difference remains the same but the numerator is a weighted average of within-stratum differences in means on the covariate. By default, each stratum is weighted in proportion to the harmonic mean 1/[(1/a + 1/b)/2]=2*a*b/(a+b) of the number of treated units (a) and control units (b) in the stratum; this weighting is optimal under certain modeling assumptions (discussed in Kalton 1968, Hansen and Bowers 2008). This weighting can be modified using the `stratum.weights` argument; see below.

When the treatment variable, the variable specified by the left-hand side of `fmla`, is not binary, `xBalance` calculates the covariates' regressions on the treatment variable, in the stratified case pooling these regressions across strata using weights that default to the stratum-wise sum of squared deviations of the treatment variable from its stratum mean. (Applied to binary treatment variables, this recipe gives the same result as the one given above.) In the numerator of the standardized difference, we get a “pooled S.D.” from separating units into two groups, one in which the treatment variable is 0 or less and another in which it is positive. If `report` includes "adj.means", covariate means for the former of these groups are reported, along with the sums of these means and the covariates' regressions on either the treatment variable, in the unstratified (“pre”) case, or the treatment variable and the strata, in the stratified (“post”) case.

`stratum.weights` can be either a function or a numeric vector of weights. If it is a numeric vector, it should be non-negative and it should have stratum names as its names. (i.e., its names should be equal to the levels of the factor specified by `strata`.) If it is a function, it should accept one argument, a data frame containing the variables in `data` and additionally `Tx.grp` and `stratum.code`, and return a vector of non-negative weights with stratum codes as names; for an example, do `getFromNamespace("harmonic", "RItools")`.

If `covariate.scaling` is not `NULL`, no scaling is applied. This behavior is likely to change in future versions. (If you want no scaling, set `covariate.scaling=1`, as this is likely to retain this meaning in the future.)

`adj.mean.diffs.null.sd` returns the standard deviation of the Normal approximated randomization distribution of the strata-adjusted difference of means under the strict null of no effect.

Value

An object of class `c("xbal", "list")`. There are `plot`, `print`, and `xtable` methods for class `"xbal"`; the `print` method is demonstrated in the examples.

Note

Evidence pertaining to the hypothesis that a treatment variable is not associated with differences in covariate values is assessed by comparing the differences of means (or regression coefficients), without standardization, to their distributions under hypothetical shuffles of the treatment variable, a permutation or randomization distribution. For the unstratified comparison, this reference distribution consists of differences (more generally, regression coefficients) when the treatment variable is permuted without regard to strata. For the stratified comparison, the reference distribution is determined by randomly permuting the treatment variable within strata, then re-calculating the treatment-control differences (regressions of each covariate on the permuted treatment variable). Significance assessments are based on the large-sample Normal approximation to these reference distributions.

Author(s)

Ben Hansen and Jake Bowers and Mark Fredrickson

References

Hansen, B.B. and Bowers, J. (2008), “Covariate Balance in Simple, Stratified and Clustered Comparative Studies,” Statistical Science 23.

Kalton, G. (1968), “Standardization: A technique to control for extraneous variables,” Applied Statistics 17, 118–136.

Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57``` ```data(nuclearplants) ##No strata, default output xBalance(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, data=nuclearplants) ##No strata, all output xBalance(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, data=nuclearplants, report=c("all")) ##Stratified, all output xBalance(pr~.-cost-pt, strata=factor(nuclearplants\$pt), data=nuclearplants, report=c("adj.means", "adj.mean.diffs", "adj.mean.diffs.null.sd", "chisquare.test", "std.diffs", "z.scores", "p.values")) ##Comparing unstratified to stratified, just adjusted means and #omnibus test xBalance(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, strata=list(unstrat=NULL, pt=~pt), data=nuclearplants, report=c("adj.means", "chisquare.test")) ##Comparing unstratified to stratified, just adjusted means and #omnibus test xBalance(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, strata=data.frame(unstrat=factor('none'), pt=factor(nuclearplants\$pt)), data=nuclearplants, report=c("adj.means", "chisquare.test")) ##Missing data handling. testdata<-nuclearplants testdata\$date[testdata\$date<68]<-NA ##na.rm=FALSE by default xBalance(pr ~ date, data = testdata, report="all") xBalance(pr ~ date, data = testdata, na.rm = TRUE,report="all") ##To match versions of RItools 0.1-9 and older, impute means #rather than medians. ##Not run, impfn option is not implemented in the most recent version ## Not run: xBalance(pr ~ date, data = testdata, na.rm = FALSE, report="all", impfn=mean.default) ## End(Not run) ##Comparing unstratified to stratified, just one-by-one wilcoxon #rank sum tests and omnibus test of multivariate differences on #rank scale. xBalance(pr~ date + t1 + t2 + cap + ne + ct + bw + cum.n, strata=data.frame(unstrat=factor('none'), pt=factor(nuclearplants\$pt)), data=nuclearplants, report=c("adj.means", "chisquare.test"), post.alignment.transform=rank) ```

RItools documentation built on June 20, 2018, 1:04 a.m.