This function performs sensitivity analysis to omitted variables as discussed in Cinelli and Hazlett (2020). It returns an object of class sensemakr with several pre-computed sensitivity statistics for reporting. After running sensemakr you may directly use the plot, print and summary methods in the returned object.

sensemakr(...)

# S3 method for lm
sensemakr(
  model,
  treatment,
  benchmark_covariates = NULL,
  kd = 1,
  ky = kd,
  q = 1,
  alpha = 0.05,
  r2dz.x = NULL,
  r2yz.dx = r2dz.x,
  bound_label = "Manual Bound",
  reduce = TRUE,
  ...
)

# S3 method for formula
sensemakr(
  formula,
  data,
  treatment,
  benchmark_covariates = NULL,
  kd = 1,
  ky = kd,
  q = 1,
  alpha = 0.05,
  r2dz.x = NULL,
  r2yz.dx = r2dz.x,
  bound_label = "",
  reduce = TRUE,
  ...
)

# S3 method for numeric
sensemakr(
  estimate,
  se,
  dof,
  treatment = "D",
  q = 1,
  alpha = 0.05,
  r2dz.x = NULL,
  r2yz.dx = r2dz.x,
  bound_label = "manual_bound",
  r2dxj.x = NULL,
  r2yxj.dx = r2dxj.x,
  benchmark_covariates = "manual_benchmark",
  kd = 1,
  ky = kd,
  reduce = TRUE,
  ...
)

Arguments

...

arguments passed to other methods. First argument should either be (i) an lm model with the outcome regression (argument model); (ii) a formula describing the model along with the data.frame containing the variables of the model (arguments formula and data); or, (iii) the numerical estimated value of the coefficient, along with the numeric values of standard errors and degrees of freedom (arguments estimate, se and df).

model

An lm object with the outcome regression.

treatment

A character vector with the name of the treatment variable of the model.

benchmark_covariates

The user has two options: (i) character vector of the names of covariates that will be used to bound the plausible strength of the unobserved confounders. Each variable will be considered separately; (ii) a named list with character vector names of covariates that will be used, as a group, to bound the plausible strength of the unobserved confounders. The names of the list will be used for the benchmark labels. Note: for factor variables with more than two levels, you need to provide the name of each level as encoded in the lm model (the columns of model.matrix).

kd

numeric vector. Parameterizes how many times stronger the confounder is related to the treatment in comparison to the observed benchmark covariate. Default value is 1 (confounder is as strong as benchmark covariate).

ky

numeric vector. Parameterizes how many times stronger the confounder is related to the outcome in comparison to the observed benchmark covariate. Default value is the same as kd.

q

percent change of the effect estimate that would be deemed problematic. Default is 1, which means a reduction of 100% of the current effect estimate (bring estimate to zero). It has to be greater than zero.

alpha

significance level.

r2dz.x

Hypothetical partial R2 of unobserved confounder Z with treatment D, given covariates X.

r2yz.dx

Hypothetical partial R2 of unobserved confounder Z with outcome Y, given covariates X and treatment D.

bound_label

label to bounds provided manually in r2dz.x and r2yz.dx.

reduce

Should the bias adjustment reduce or increase the absolute value of the estimated coefficient? Default is TRUE.

formula

an object of the class formula: a symbolic description of the model to be fitted.

data

data needed only when you pass a formula as first parameter. An object of the class data.frame containing the variables used in the analysis.

estimate

Coefficient estimate.

se

Standard error of the coefficient estimate.

dof

Residual degrees of freedom of the regression.

r2dxj.x

partial R2 of covariate Xj with the treatment D (after partialling out the effect of the remaining covariates X, excluding Xj).

r2yxj.dx

partial R2 of covariate Xj with the outcome Y (after partialling out the effect of the remaining covariates X, excluding Xj).

Value

An object of class sensemakr, containing:

info

A data.frame with the general information of the analysis, including the formula used, the name of the treatment variable, parameter values such as q, alpha, and whether the bias is assumed to reduce the current estimate.

sensitivity_stats

A data.frame with the sensitivity statistics for the treatment variable, as computed by the function sensitivity_stats.

bounds

A data.frame with bounds on the strength of confounding according to some benchmark covariates, as computed by the function ovb_bounds.

References

Cinelli, C. and Hazlett, C. (2020), "Making Sense of Sensitivity: Extending Omitted Variable Bias." Journal of the Royal Statistical Society, Series B (Statistical Methodology).

See also

The function sensemakr is a convenience function. You may use the other sensitivity functions of the package directly, such as the functions for sensitivity plots (ovb_contour_plot, ovb_extreme_plot) the functions for computing bias-adjusted estimates and t-values (adjusted_estimate, adjusted_t), the functions for computing the robustness value and partial R2 (robustness_value, partial_r2), or the functions for bounding the strength of unobserved confounders (ovb_bounds), among others.

Examples

# loads dataset data("darfur") # runs regression model model <- lm(peacefactor ~ directlyharmed + age + farmer_dar + herder_dar + pastvoted + hhsize_darfur + female + village, data = darfur) # runs sensemakr for sensitivity analysis sensitivity <- sensemakr(model, treatment = "directlyharmed", benchmark_covariates = "female", kd = 1:3) # short description of results sensitivity
#> Sensitivity Analysis to Unobserved Confounding #> #> Model Formula: peacefactor ~ directlyharmed + age + farmer_dar + herder_dar + #> pastvoted + hhsize_darfur + female + village #> #> Null hypothesis: q = 1 and reduce = TRUE #> #> Unadjusted Estimates of ' directlyharmed ': #> Coef. estimate: 0.09732 #> Standard Error: 0.02326 #> t-value: 4.18445 #> #> Sensitivity Statistics: #> Partial R2 of treatment with outcome: 0.02187 #> Robustness Value, q = 1 : 0.13878 #> Robustness Value, q = 1 alpha = 0.05 : 0.07626 #> #> For more information, check summary.
# long description of results summary(sensitivity)
#> Sensitivity Analysis to Unobserved Confounding #> #> Model Formula: peacefactor ~ directlyharmed + age + farmer_dar + herder_dar + #> pastvoted + hhsize_darfur + female + village #> #> Null hypothesis: q = 1 and reduce = TRUE #> -- This means we are considering biases that reduce the absolute value of the current estimate. #> -- The null hypothesis deemed problematic is H0:tau = 0 #> #> Unadjusted Estimates of 'directlyharmed': #> Coef. estimate: 0.0973 #> Standard Error: 0.0233 #> t-value (H0:tau = 0): 4.1844 #> #> Sensitivity Statistics: #> Partial R2 of treatment with outcome: 0.0219 #> Robustness Value, q = 1: 0.1388 #> Robustness Value, q = 1, alpha = 0.05: 0.0763 #> #> Verbal interpretation of sensitivity statistics: #> #> -- Partial R2 of the treatment with the outcome: an extreme confounder (orthogonal to the covariates) that explains 100% of the residual variance of the outcome, would need to explain at least 2.19% of the residual variance of the treatment to fully account for the observed estimated effect. #> #> -- Robustness Value, q = 1: unobserved confounders (orthogonal to the covariates) that explain more than 13.88% of the residual variance of both the treatment and the outcome are strong enough to bring the point estimate to 0 (a bias of 100% of the original estimate). Conversely, unobserved confounders that do not explain more than 13.88% of the residual variance of both the treatment and the outcome are not strong enough to bring the point estimate to 0. #> #> -- Robustness Value, q = 1, alpha = 0.05: unobserved confounders (orthogonal to the covariates) that explain more than 7.63% of the residual variance of both the treatment and the outcome are strong enough to bring the estimate to a range where it is no longer 'statistically different' from 0 (a bias of 100% of the original estimate), at the significance level of alpha = 0.05. Conversely, unobserved confounders that do not explain more than 7.63% of the residual variance of both the treatment and the outcome are not strong enough to bring the estimate to a range where it is no longer 'statistically different' from 0, at the significance level of alpha = 0.05. #> #> Bounds on omitted variable bias: #> #> --The table below shows the maximum strength of unobserved confounders with association with the treatment and the outcome bounded by a multiple of the observed explanatory power of the chosen benchmark covariate(s). #> #> Bound Label R2dz.x R2yz.dx Treatment Adjusted Estimate Adjusted Se #> 1x female 0.0092 0.1246 directlyharmed 0.0752 0.0219 #> 2x female 0.0183 0.2493 directlyharmed 0.0529 0.0204 #> 3x female 0.0275 0.3741 directlyharmed 0.0304 0.0187 #> Adjusted T Adjusted Lower CI Adjusted Upper CI #> 3.4389 0.0323 0.1182 #> 2.6002 0.0130 0.0929 #> 1.6281 -0.0063 0.0670
# plot bias contour of point estimate plot(sensitivity)
# plot bias contour of t-value plot(sensitivity, sensitivity.of = "t-value")
# plot extreme scenario plot(sensitivity, type = "extreme")
# latex code for sensitivity table ovb_minimal_reporting(sensitivity)
#> \begin{table}[!h] #> \centering #> \begin{tabular}{lrrrrrr} #> \multicolumn{7}{c}{Outcome: \textit{peacefactor}} \\ #> \hline \hline #> Treatment: & Est. & S.E. & t-value & $R^2_{Y \sim D |{\bf X}}$ & $RV_{q = 1}$ & $RV_{q = 1, \alpha = 0.05}$ \\ #> \hline #> \textit{directlyharmed} & 0.097 & 0.023 & 4.184 & 2.2\% & 13.9\% & 7.6\% \\ #> \hline #> df = 783 & & \multicolumn{5}{r}{ \small \textit{Bound (1x female)}: $R^2_{Y\sim Z| {\bf X}, D}$ = 12.5\%, $R^2_{D\sim Z| {\bf X} }$ = 0.9\%} \\ #> \end{tabular} #> \end{table}