The goal of konfound is to carry out sensitivity analysis to help analysts to quantify how robust inferences are to potential sources of bias. This R package provides tools to carry out sensitivity analysis as described in Frank, Maroulis, Duong, and Kelcey (2013) based on Rubin’s (1974) causal model as well as in Frank (2000) based on the impact threshold for a confounding variable.

Installation

You can install the CRAN version of konfound with:

install.packages("konfound")

You can install the development version from GitHub with:

install.packages("devtools")
devtools::install_github("konfound-project/konfound")

Use of konfound

pkonfound() for published studies

pkonfound(), for published studies, calculates (1) how much bias there must be in an estimate to invalidate/sustain an inference, and interprets in terms of how much data would need to be replaced to nullify an inference (Robustness of an Inference to Replacement, RIR); (2) the impact of an omitted variable necessary to invalidate/sustain an inference for a regression coefficient (Impact Threshold of a Confounding Variable, ITCV). RIR is reported by default. ITCV can be generated by specifying index = "IT".

library(konfound)
#> Sensitivity analysis as described in Frank, 
#> Maroulis, Duong, and Kelcey (2013) and in 
#> Frank (2000).
#> For more information visit http://konfound-it.com.

pkonfound(est_eff = 2, 
          std_err = .4, 
          n_obs = 100, 
          n_covariates = 3)
#> Robustness of Inference to Replacement (RIR):
#> RIR = 60
#> 
#> To invalidate the inference of an effect using the threshold of 0.794 for
#> statistical significance (with null hypothesis = 0 and alpha = 0.05), 60.295%
#> of the (2) estimate would have to be due to bias. This implies that to
#> invalidate the inference one would expect to have to replace 60 (60.295%)
#> observations with data points for which the effect is 0 (RIR = 60).
#> 
#> See Frank et al. (2013) for a description of the method.
#> 
#> Citation: Frank, K.A., Maroulis, S., Duong, M., and Kelcey, B. (2013).
#> What would it take to change an inference?
#> Using Rubin's causal model to interpret the robustness of causal inferences.
#> Education, Evaluation and Policy Analysis, 35 437-460.
#> 
#> Accuracy of results increases with the number of decimals reported.
#> For other forms of output, run
#>           ?pkonfound and inspect the to_return argument
#> For models fit in R, consider use of konfound().
pkonfound(est_eff = 2, 
          std_err = .4, 
          n_obs = 100, 
          n_covariates = 3, 
          index = "IT")
#> Impact Threshold for a Confounding Variable (ITCV):
#> 
#> The minimum impact of an omitted variable to invalidate an inference for
#> a null hypothesis of an effect of nu (0) is based on a correlation of 0.566
#> with the outcome and 0.566 with the predictor of interest (conditioning
#> on all observed covariates in the model; signs are interchangeable). This is
#> based on a threshold effect of 0.2 for statistical significance (alpha = 0.05).
#> 
#> Correspondingly the impact of an omitted variable (as defined in Frank 2000) must be 
#> 0.566 X 0.566 = 0.321 to invalidate an inference for a null hypothesis of an effect of nu (0).
#> 
#> For calculation of unconditional ITCV using pkonfound(), additionally include
#> the R2, sdx, and sdy as input, and request raw output.
#> 
#> See Frank (2000) for a description of the method.
#> 
#> Citation:
#> Frank, K. (2000). Impact of a confounding variable on the inference of a
#> regression coefficient. Sociological Methods and Research, 29 (2), 147-194
#> 
#> Accuracy of results increases with the number of decimals reported.
#> 
#> The ITCV analysis was originally derived for OLS standard errors. If the
#> standard errors reported in the table were not based on OLS, some caution
#> should be used to interpret the ITCV.
#> For other forms of output, run
#>           ?pkonfound and inspect the to_return argument
#> For models fit in R, consider use of konfound().

konfound() for models fit in R

konfound() calculates the robustness of inferences for models fit in R. For example, below are the coefficients for a linear model fit with lm() using the built-in dataset mtcars:

m1 <- lm(mpg ~ wt + disp, data = mtcars)
m1
#> 
#> Call:
#> lm(formula = mpg ~ wt + disp, data = mtcars)
#> 
#> Coefficients:
#> (Intercept)           wt         disp  
#>    34.96055     -3.35083     -0.01772
summary(m1)
#> 
#> Call:
#> lm(formula = mpg ~ wt + disp, data = mtcars)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -3.4087 -2.3243 -0.7683  1.7721  6.3484 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept) 34.96055    2.16454  16.151 4.91e-16 ***
#> wt          -3.35082    1.16413  -2.878  0.00743 ** 
#> disp        -0.01773    0.00919  -1.929  0.06362 .  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 2.917 on 29 degrees of freedom
#> Multiple R-squared:  0.7809, Adjusted R-squared:  0.7658 
#> F-statistic: 51.69 on 2 and 29 DF,  p-value: 2.744e-10

Sensitivity analysis for the effect for wt on mpg can be carried out as follows, specifying the fitted model object:

konfound(m1, wt)
#> Robustness of Inference to Replacement (RIR):
#> RIR = 9
#> 
#> To invalidate the inference of an effect using the threshold of -2.381 for
#> statistical significance (with null hypothesis = 0 and alpha = 0.05), 28.946%
#> of the (-3.351) estimate would have to be due to bias. This implies that to
#> invalidate the inference one would expect to have to replace 9 (28.946%)
#> observations with data points for which the effect is 0 (RIR = 9).
#> 
#> See Frank et al. (2013) for a description of the method.
#> 
#> Citation: Frank, K.A., Maroulis, S., Duong, M., and Kelcey, B. (2013).
#> What would it take to change an inference?
#> Using Rubin's causal model to interpret the robustness of causal inferences.
#> Education, Evaluation and Policy Analysis, 35 437-460.
#> 
#> Accuracy of results increases with the number of decimals reported.
#> NULL

Similar to pkonfound, ITCV can be generated by specifying index = "IT".

konfound(m1, wt, index = "IT")
#> Impact Threshold for a Confounding Variable (ITCV):
#> 
#> The minimum (in absolute value) impact of an omitted variable to invalidate
#> an inference for a null hypothesis of an effect of nu (0) is based on
#> a correlation of -0.425 with the outcome and 0.425 with the predictor of
#> interest (conditioning on all observed covariates in the model; signs are
#> interchangeable). This is based on a threshold effect of -0.355 for statistical
#> significance (alpha = 0.05).
#> 
#> Correspondingly the impact of an omitted variable (as defined in Frank 2000) must be
#> -0.425 X 0.425 = -0.18 to invalidate an inference for a null hypothesis of an effect of nu (0).
#> 
#> See Frank (2000) for a description of the method.
#> 
#> Citation:
#> Frank, K. (2000). Impact of a confounding variable on the inference of a
#> regression coefficient. Sociological Methods and Research, 29 (2), 147-194
#> 
#> Accuracy of results increases with the number of decimals reported.
#> 
#> The ITCV analysis was originally derived for OLS standard errors. If the
#> standard errors reported in the table were not based on OLS, some caution
#> should be used to interpret the ITCV.
#> NULL

mkonfound for meta-analyses including sensitivity analysis

mkonfound() supports sensitivity that can be compared or synthesized across multiple analyses. We can use an existing (and built-in) dataset, such as mkonfound_ex.

mkonfound_ex
#> # A tibble: 30 × 2
#>         t    df
#>     <dbl> <dbl>
#>  1  7.08    178
#>  2  4.13    193
#>  3  1.89     47
#>  4 -4.17    138
#>  5 -1.19     97
#>  6  3.59     87
#>  7  0.282   117
#>  8  2.55     75
#>  9 -4.44    137
#> 10 -2.05    195
#> # ℹ 20 more rows
mkonfound(mkonfound_ex, t, df)
#> # A tibble: 30 × 7
#>         t    df action        inference      pct_bias_to_change_i…¹   itcv r_con
#>     <dbl> <dbl> <chr>         <chr>                           <dbl>  <dbl> <dbl>
#>  1  7.08    178 to_invalidate reject_null                     68.8   0.378 0.614
#>  2  4.13    193 to_invalidate reject_null                     50.6   0.168 0.41 
#>  3  1.89     47 to_sustain    fail_to_rejec…                   5.47 -0.012 0.11 
#>  4 -4.17    138 to_invalidate reject_null                     50.3   0.202 0.449
#>  5 -1.19     97 to_sustain    fail_to_rejec…                  39.4  -0.065 0.255
#>  6  3.59     87 to_invalidate reject_null                     41.9   0.19  0.436
#>  7  0.282   117 to_sustain    fail_to_rejec…                  85.5  -0.131 0.361
#>  8  2.55     75 to_invalidate reject_null                     20.6   0.075 0.274
#>  9 -4.44    137 to_invalidate reject_null                     53.0   0.225 0.475
#> 10 -2.05    195 to_invalidate reject_null                      3.51  0.006 0.077
#> # ℹ 20 more rows
#> # ℹ abbreviated name: ¹pct_bias_to_change_inference

Other information

How to learn more about sensitivity analysis

To learn more about sensitivity analysis, please visit:

The KonFound-It website, with latest news, links to tools and support
The Introduction to konfound vignette, with detailed information about each of the functions (pkonfound(), konfound(), and mkounfound())
The Konfound-It! interactive web application, with links to PowerPoints and key publications

Issues, feature requests, and contributing

We prefer for issues to be filed via GitHub (link to the issues page for konfound here) though we also welcome questions or feedback requests via email (see the DESCRIPTION file).

Contributing guidelines are here.

Code of Conduct

Please note that the konfound project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.