Linear regression with the Lin (2013) covariate adjustment

This function is a wrapper for lm_robust that is useful for estimating treatment effects with pre-treatment covariate data. This implements the method described by Lin (2013).

Usage

lm_lin(
  formula,
  covariates,
  data,
  weights,
  subset,
  clusters,
  se_type = NULL,
  ci = TRUE,
  alpha = 0.05,
  return_vcov = TRUE,
  try_cholesky = FALSE
)

Arguments

formula: an object of class formula, as in lm, such as Y ~ Z with only one variable on the right-hand side, the treatment
covariates: a right-sided formula with pre-treatment covariates on the right hand side, such as ~ x1 + x2 + x3.
data: A data.frame
weights: the bare (unquoted) names of the weights variable in the supplied data.
subset: An optional bare (unquoted) expression specifying a subset of observations to be used.
clusters: An optional bare (unquoted) name of the variable that corresponds to the clusters in the data.
se_type: The sort of standard error sought. If clusters is not specified the options are "HC0", "HC1" (or "stata", the equivalent), "HC2" (default), "HC3", or "classical". If clusters is specified the options are "CR0", "CR2" (default), or "stata" are permissible.
ci: logical. Whether to compute and return p-values and confidence intervals, TRUE by default.
alpha: The significance level, 0.05 by default.
return_vcov: logical. Whether to return the variance-covariance matrix for later usage, TRUE by default.
try_cholesky: logical. Whether to try using a Cholesky decomposition to solve least squares instead of a QR decomposition, FALSE by default. Using a Cholesky decomposition may result in speed gains, but should only be used if users are sure their model is full-rank (i.e., there is no perfect multi-collinearity)

Value

An object of class "lm_robust".

The post-estimation commands functions summary and tidy

return results in a data.frame. To get useful data out of the return, you can use these data frames, you can use the resulting list directly, or you can use the generic accessor functions coef, vcov, confint, and predict. Marginal effects and uncertainty about them can be gotten by passing this object to margins from the margins.

Users who want to print the results in TeX of HTML can use the extract function and the texreg package.

An object of class "lm_robust" is a list containing at least the following components:

coefficients: the estimated coefficients
std.error: the estimated standard errors
statistic: the t-statistic
df: the estimated degrees of freedom
p.value: the p-values from a two-sided t-test using coefficients, std.error, and df
conf.low: the lower bound of the 1 - alpha percent confidence interval
conf.high: the upper bound of the 1 - alpha percent confidence interval
term: a character vector of coefficient names
alpha: the significance level specified by the user
se_type: the standard error type specified by the user
res_var: the residual variance
N: the number of observations used
k: the number of columns in the design matrix (includes linearly dependent columns!)
rank: the rank of the fitted model
vcov: the fitted variance covariance matrix
r.squared: The $R^2$, $$R^2 = 1 - Sum(e[i]^2) / Sum((y[i] - y^*)^2),$$ where $y^*$ is the mean of $y[i]$ if there is an intercept and zero otherwise, and $e[i]$ is the ith residual.
adj.r.squared: The $R^2$ but penalized for having more parameters, rank
weighted: whether or not weights were applied
call: the original function call
fitted.values: the matrix of predicted means

We also return terms and contrasts, used by predict, and scaled_center (the means of each of the covariates used for centering them).

Details

This function is simply a wrapper for lm_robust and implements the Lin estimator (see the reference below). This method pre-processes the data by taking the covariates specified in the `covariates` argument, centering them by subtracting from each covariate its mean, and interacting them with the treatment. If the treatment has multiple values, a series of dummies for each value is created and each of those is interacted with the demeaned covariates. More details can be found in the Getting Started vignette and the mathematical notes.

References

Freedman, David A. 2008. "On Regression Adjustments in Experiments with Several Treatments." The Annals of Applied Statistics. JSTOR, 176-96. doi:10.1214/07-AOAS143 .

Lin, Winston. 2013. "Agnostic Notes on Regression Adjustments to Experimental Data: Reexamining Freedman's Critique." The Annals of Applied Statistics 7 (1). Institute of Mathematical Statistics: 295-318. doi:10.1214/12-AOAS583 .

Examples

library(fabricatr)
library(randomizr)
dat <- fabricate(
  N = 40,
  x = rnorm(N, mean = 2.3),
  x2 = rpois(N, lambda = 2),
  x3 = runif(N),
  y0 = rnorm(N) + x,
  y1 = rnorm(N) + x + 0.35
)

dat$z <- complete_ra(N = nrow(dat))
dat$y <- ifelse(dat$z == 1, dat$y1, dat$y0)

# Same specification as lm_robust() with one additional argument
lmlin_out <- lm_lin(y ~ z, covariates = ~ x, data = dat)
tidy(lmlin_out)
#>          term   estimate std.error  statistic      p.value   conf.low conf.high
#> 1 (Intercept)  2.0074002 0.2012088  9.9767032 6.611620e-12  1.5993299 2.4154705
#> 2           z  0.7263528 0.3037565  2.3912337 2.214176e-02  0.1103061 1.3423996
#> 3         x_c  1.0673003 0.2721532  3.9216889 3.786550e-04  0.5153480 1.6192527
#> 4       z:x_c -0.2815330 0.3235071 -0.8702529 3.899283e-01 -0.9376358 0.3745698
#>   df outcome
#> 1 36       y
#> 2 36       y
#> 3 36       y
#> 4 36       y

# Works with multiple pre-treatment covariates
lm_lin(y ~ z, covariates = ~ x + x2, data = dat)
#>               Estimate Std. Error    t value     Pr(>|t|)    CI Lower  CI Upper
#> (Intercept)  2.0331904  0.2035346  9.9894110 1.198631e-11  1.61955838 2.4468224
#> z            0.6874490  0.3118326  2.2045450 3.435353e-02  0.05372891 1.3211692
#> x_c          1.1154732  0.2539604  4.3923109 1.038122e-04  0.59936348 1.6315829
#> x2_c         0.2456971  0.2805576  0.8757456 3.873136e-01 -0.32446459 0.8158588
#> z:x_c       -0.3294245  0.3067964 -1.0737561 2.904938e-01 -0.95290983 0.2940608
#> z:x2_c      -0.1146326  0.3618812 -0.3167687 7.533560e-01 -0.85006364 0.6207984
#>             DF
#> (Intercept) 34
#> z           34
#> x_c         34
#> x2_c        34
#> z:x_c       34
#> z:x2_c      34

# Also centers data AFTER evaluating any functions in formula
lmlin_out2 <- lm_lin(y ~ z, covariates = ~ x + log(x3), data = dat)
lmlin_out2$scaled_center["log(x3)"]
#>    log(x3) 
#> -0.7815509 
mean(log(dat$x3))
#> [1] -0.7815509

# Works easily with clusters
dat$clusterID <- rep(1:20, each = 2)
dat$z_clust <- cluster_ra(clusters = dat$clusterID)

lm_lin(y ~ z_clust, covariates = ~ x, data = dat, clusters = clusterID)
#>              Estimate Std. Error   t value     Pr(>|t|)     CI Lower  CI Upper
#> (Intercept) 2.1869918  0.2599259 8.4139039 6.338923e-05  1.573233032 2.8007505
#> z_clust     0.1547201  0.3441864 0.4495242 6.598847e-01 -0.582978576 0.8924188
#> x_c         0.6678927  0.2632028 2.5375595 5.052778e-02 -0.002227515 1.3380129
#> z_clust:x_c 0.5497024  0.3559260 1.5444288 1.576286e-01 -0.258197236 1.3576021
#>                    DF
#> (Intercept)  7.049192
#> z_clust     14.103467
#> x_c          5.165385
#> z_clust:x_c  8.804119

# Works with multi-valued treatments
dat$z_multi <- sample(1:3, size = nrow(dat), replace = TRUE)
lm_lin(y ~ z_multi, covariates = ~ x, data = dat)
#>                  Estimate Std. Error      t value     Pr(>|t|)   CI Lower
#> (Intercept)   2.087111175  0.2397513  8.705318218 3.583451e-10  1.5998780
#> z_multi2      0.632076161  0.3949708  1.600311246 1.187819e-01 -0.1706010
#> z_multi3      0.225302890  0.4595353  0.490284219 6.270820e-01 -0.7085851
#> x_c           0.971855697  0.3265662  2.975983428 5.346867e-03  0.3081933
#> z_multi2:x_c -0.164498171  0.4184305 -0.393131454 6.966767e-01 -1.0148512
#> z_multi3:x_c  0.007310125  0.8138866  0.008981748 9.928862e-01 -1.6467064
#>               CI Upper DF
#> (Intercept)  2.5743444 34
#> z_multi2     1.4347533 34
#> z_multi3     1.1591909 34
#> x_c          1.6355181 34
#> z_multi2:x_c 0.6858548 34
#> z_multi3:x_c 1.6613267 34

# Stratified estimator with blocks
dat$blockID <- rep(1:5, each = 8)
dat$z_block <- block_ra(blocks = dat$blockID)

lm_lin(y ~ z_block, ~ factor(blockID), data = dat)
#>                                Estimate Std. Error    t value     Pr(>|t|)
#> (Intercept)                   2.0704132  0.3451408  5.9987493 1.399160e-06
#> z_block                       0.5931938  0.4501499  1.3177696 1.975558e-01
#> (factor(blockID)2)_c          1.1858143  1.1686199  1.0147134 3.183567e-01
#> (factor(blockID)3)_c          1.4533957  1.1962207  1.2149896 2.338464e-01
#> (factor(blockID)4)_c          0.3103012  1.2793724  0.2425417 8.100110e-01
#> (factor(blockID)5)_c          0.6828532  1.1941371  0.5718382 5.716919e-01
#> z_block:(factor(blockID)2)_c -1.5162764  1.4447190 -1.0495303 3.023158e-01
#> z_block:(factor(blockID)3)_c -1.2130004  1.4862041 -0.8161735 4.208346e-01
#> z_block:(factor(blockID)4)_c -0.4459309  1.5448894 -0.2886491 7.748351e-01
#> z_block:(factor(blockID)5)_c -1.9553208  1.3328098 -1.4670666 1.527638e-01
#>                                CI Lower CI Upper DF
#> (Intercept)                   1.3655416 2.775285 30
#> z_block                      -0.3261349 1.512523 30
#> (factor(blockID)2)_c         -1.2008259 3.572455 30
#> (factor(blockID)3)_c         -0.9896129 3.896404 30
#> (factor(blockID)4)_c         -2.3025258 2.923128 30
#> (factor(blockID)5)_c         -1.7559001 3.121606 30
#> z_block:(factor(blockID)2)_c -4.4667862 1.434233 30
#> z_block:(factor(blockID)3)_c -4.2482341 1.822233 30
#> z_block:(factor(blockID)4)_c -3.6010159 2.709154 30
#> z_block:(factor(blockID)5)_c -4.6772816 0.766640 30

if (FALSE) {
  # Can also use 'margins' package if you have it installed to get
  # marginal effects
  library(margins)
  lmlout <- lm_lin(y ~ z_block, ~ x, data = dat)
  summary(margins(lmlout))

  # Can output results using 'texreg'
  library(texreg)
  texregobj <- extract(lmlout)
}