Background

In social science (and educational) research, we often wish to understand how robust inferences about effects are to unobserved (or controlled for) covariates, possible problems with measurement, and other sources of bias. 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 package provides functions based on developments in sensitivity analysis by Frank and colleagues, which previously have been implemented in Stata and through an Excel spreadsheet, in R through the konfound package. In particular, we provide functions for both values from analyses carried out outside of R as well as from models (lm(), glm(), and lme4::lmer() fit in R) for:

  • Quantifying the bias necessary to alter an inference from the framework of Rubin’s (1974) causal model
  • The robustness of causal inference in terms of the impact threshold of a confounding variable

Installation konfound

Presently, konfound is available only on GitHub. You can install konfound from GitHub with:

install.packages("devtools")
devtools::install_github("jrosen48/konfound")

You can then load konfound with the library() function:

library(konfound)
#> Loading konfound
#> Sensitivity analysis as described in Frank, Maroulis, Duong, and Kelcey (2013) and in Frank (2000).
#> For more information visit https://jmichaelrosenberg.shinyapps.io/shinykonfound/.

Use of pkonfound() for values from an already-conducted analysis

pkonfound() is used when we have values from an already-conducted analysis (like a regression analysis), such as one in an already-published study or from an analysis carried out using other software.

In the case of a regression analysis, values from the analysis would simply be used as the inputs to the pkonfound() function. For example, in the use below, we simply enter the values for the estimated effect (an unstandardardized beta coefficient) (2), its standard error (.4), the sample size (100), and the number of covariates (3):

pkonfound(2, .4, 100, 3)
#> Replacement of Cases Approach:
#> To invalidate an inference, 60.3% of the estimate would have to be due to bias. This is based on a threshold of 0.794 for statistical significance (alpha = 0.05).
#> To invalidate an inference, 60 observations would have to be replaced with cases for which the effect is 0.
#> 
#> Correlation-based Approach:
#> An omitted variable would have to be correlated at 0.568 with the outcome and at 0.568 with the predictor of interest (conditioning on observed covariates) to invalidate an inference based on a threshold of 0.201 for statistical significance (alpha = 0.05).
#> Correspondingly the impact of an omitted variable (as defined in Frank 2000) must be 0.568 X 0.568 = 0.323 to invalidate an inference.
#> For other forms of output, change `to_return` to table, raw_output, thres_plot, or corr_plot.
#> For models fit in R, consider use of konfound().

For this set of values, around 60% would need to be false due to a source of bias for the inference to be invalidated (based on statistical significance and a p-value (or alpha) of .05), possible a very robust effect. An omitted, confounding variable (sometimes referred to as a covariate) would need to have an impact (defined as the product of the confounding variable’s correlation with both the predictor of interest and the outcome) of 0.323, presenting a different interpretation of how robust this (hypothetical) effect is to a variable which is important but not included in the analysis.

Here is another example, but one in which the unstandardized beta coefficient is smaller than its standard error:

pkonfound(.4, 2, 100, 3)
#> Replacement of Cases Approach:
#> To sustain an inference, 89.924% of the estimate would have to be due to bias. This is based on a threshold of 3.97 for statistical significance (alpha = 0.05).
#> To sustain an inference, 90 of the cases with 0 effect would have to be replaced with cases at the threshold of inference.
#> 
#> Correlation-based Approach:
#> An omitted variable would have to be correlated at 0.387 with the outcome and at 0.387 with the predictor of interest (conditioning on observed covariates) to sustain an inference based on a threshold of 3.97 for statistical significance (alpha = 0.05).
#> Correspondingly the impact of an omitted variable (as defined in Frank 2000) must be 0.387 X 0.387 = 0.15 to sustain an inference.
#> For other forms of output, change `to_return` to table, raw_output, thres_plot, or corr_plot.
#> For models fit in R, consider use of konfound().

Note that this use of pkonfound() is equivalent to naming the arguments, i.e. for a different set of values:

pkonfound(est_eff = -2.2,
          std_err = .65, 
          n_obs = 200,
          n_covariates = 3)
#> Replacement of Cases Approach:
#> To invalidate an inference, 41.732% of the estimate would have to be due to bias. This is based on a threshold of -1.282 for statistical significance (alpha = 0.05).
#> To invalidate an inference, 83 observations would have to be replaced with cases for which the effect is 0.
#> 
#> Correlation-based Approach:
#> An omitted variable would have to be correlated at 0.334 with the outcome and at 0.334 with the predictor of interest (conditioning on observed covariates) to invalidate an inference based on a threshold of -0.14 for statistical significance (alpha = 0.05).
#> Correspondingly the impact of an omitted variable (as defined in Frank 2000) must be 0.334 X 0.334 = 0.112 to invalidate an inference.
#> For other forms of output, change `to_return` to table, raw_output, thres_plot, or corr_plot.
#> For models fit in R, consider use of konfound().

We notice that the output includes a message that says we can view other forms of output by changing the to_return argument. Here are the two plots - for the bias necessary to alter an inference (thresh_plot) and for the robustness of an inference in terms of the impact of a confounding variable (corr_plot) that can be returned:

pkonfound(.4, 2, 100, 3, to_return = "thresh_plot")

pkonfound(.4, 2, 100, 3, to_return = "corr_plot")

You can also specify multiple forms of output at once.

model_output <- pkonfound(2, .4, 200, 3, to_return = c("raw_output", "thresh_plot", "corr_plot"))
#> Replacement of Cases Approach:
#> To invalidate an inference, 60.557% of the estimate would have to be due to bias. This is based on a threshold of 0.789 for statistical significance (alpha = 0.05).
#> To invalidate an inference, 121 observations would have to be replaced with cases for which the effect is 0.
#> 
#> Correlation-based Approach:
#> An omitted variable would have to be correlated at 0.479 with the outcome and at 0.479 with the predictor of interest (conditioning on observed covariates) to invalidate an inference based on a threshold of 0.14 for statistical significance (alpha = 0.05).
#> Correspondingly the impact of an omitted variable (as defined in Frank 2000) must be 0.479 X 0.479 = 0.229 to invalidate an inference.
#> Print output created by default. Created 3 other forms of output. Use list indexing or run summary() on the output to see how to access.
summary(model_output)
#> Created 3 forms of output. To access type: 
#> 
#> model_output$raw_output
#> model_output$thresh_plot
#> model_output$corr_plot

When we type the name of the object, we see that we created three types of output that we can access as follows:

model_output$raw_output
#> # A tibble: 1 x 8
#>   action    inference  percent_bias_to_change… replace_null_ca… unstd_beta
#>   <chr>     <chr>                        <dbl>            <dbl>      <dbl>
#> 1 to_inval… reject_nu…                    60.6             121.         2.
#> # ... with 3 more variables: beta_threshhold <dbl>,
#> #   omitted_variable_corr <dbl>, itcv <dbl>
model_output$thresh_plot

model_output$corr_plot

Finally, you can return the raw output, for use in other analyses.

pkonfound(.4, 2, 100, 3, to_return = "raw_output")
#> # A tibble: 1 x 8
#>   action   inference     percent_bias_to_chan… replace_null_ca… unstd_beta
#>   <chr>    <chr>                         <dbl>            <dbl>      <dbl>
#> 1 to_sust… fail_to_reje…                  89.9              90.      0.400
#> # ... with 3 more variables: beta_threshhold <dbl>,
#> #   omitted_variable_corr <dbl>, itcv <dbl>

Use of konfound() for models fit in R

Where pkonfound() can be used with values from already-conducted analyses, konfound() can be used with models (lm(), glm(), and lme4::lmer()) fit in R.

For linear models fit with lm()

m1 <- lm(mpg ~ wt + hp + qsec, data = mtcars)
m1
#> 
#> Call:
#> lm(formula = mpg ~ wt + hp + qsec, data = mtcars)
#> 
#> Coefficients:
#> (Intercept)           wt           hp         qsec  
#>    27.61053     -4.35880     -0.01782      0.51083

konfound(m1, hp)
#> Note that this output is calculated based on the correlation-based approach used in mkonfound()
#> Replacement of Cases Approach:
#> To sustain an inference, 41.327% of the estimate would have to be due to bias. This is based on a threshold of -0.031 for statistical significance (alpha = 0.05).
#> To sustain an inference, 13 of the cases with 0 effect would have to be replaced with cases at the threshold of inference.
#> 
#> Correlation-based Approach:
#> An omitted variable would have to be correlated at 0.322 with the outcome and at 0.322 with the predictor of interest (conditioning on observed covariates) to sustain an inference based on a threshold of -0.031 for statistical significance (alpha = 0.05).
#> Correspondingly the impact of an omitted variable (as defined in Frank 2000) must be 0.322 X 0.322 = 0.104 to sustain an inference.
#> For more detailed output, consider setting `to_return` to table
#> To consider other predictors of interest, consider setting `test_all` to TRUE.

Like with pkonfound(), we can also output multiple forms of output at once with konfound():

konfound_output <- konfound(m1, hp, to_return = c("raw_output", "thresh_plot", "corr_plot"))
#> Note that this output is calculated based on the correlation-based approach used in mkonfound()
#> Replacement of Cases Approach:
#> To sustain an inference, 41.327% of the estimate would have to be due to bias. This is based on a threshold of -0.031 for statistical significance (alpha = 0.05).
#> To sustain an inference, 13 of the cases with 0 effect would have to be replaced with cases at the threshold of inference.
#> 
#> Correlation-based Approach:
#> An omitted variable would have to be correlated at 0.322 with the outcome and at 0.322 with the predictor of interest (conditioning on observed covariates) to sustain an inference based on a threshold of -0.031 for statistical significance (alpha = 0.05).
#> Correspondingly the impact of an omitted variable (as defined in Frank 2000) must be 0.322 X 0.322 = 0.104 to sustain an inference.
#> Print output created by default. Created 3 other forms of output. Use list indexing or run summary() on the output to see how to access.
summary(konfound_output)
#> Created 3 forms of output. To access type: 
#> 
#> konfound_output$raw_output
#> konfound_output$thresh_plot
#> konfound_output$corr_plot

Again, we can type each of those, i.e.:

konfound_output$raw_output
#> # A tibble: 1 x 8
#>   action   inference     percent_bias_to_chan… replace_null_ca… unstd_beta
#>   <chr>    <chr>                         <dbl>            <dbl>      <dbl>
#> 1 to_sust… fail_to_reje…                  41.3              13.    -0.0180
#> # ... with 3 more variables: beta_threshhold <dbl>,
#> #   omitted_variable_corr <dbl>, itcv <dbl>
konfound_output$thresh_plot

We can also test all of the variables as predictors of interest:

konfound(m1, wt, test_all = TRUE)
#>   var_name         t df        action           inference
#> 1       wt -5.788845 29 to_invalidate         reject_null
#> 2       hp -1.200000 29    to_sustain fail_to_reject_null
#> 3     qsec  1.164009 29    to_sustain fail_to_reject_null
#>   pct_bias_to_change_inference   itcv r_con
#> 1                       51.508  0.585 0.765
#> 2                       38.740 -0.102 0.319
#> 3                       40.494 -0.106 0.326

Whereas this cannot be carried out with pkonfound(), with konfound() you can also return a table with some key output from the correlation-based approach.

konfound(m1, wt, to_return = "table")
#> Note that this output is calculated based on the correlation-based approach used in mkonfound()
#> Dependent variable is mpg
#>          term estimate std.error statistic p.value  itcv impact
#> 1 (Intercept)   27.611     8.420     3.279   0.003    NA     NA
#> 2          wt   -4.359     0.753    -5.791   0.000 0.243     NA
#> 3          hp   -0.018     0.015    -1.190   0.244    NA  0.511
#> 4        qsec    0.511     0.439     1.163   0.255    NA  0.073

If the impact threshhold is greater than the impacts of the Zs (the other covariates) then an omitted variable would have to have a greater impact than any of the observed covariates to change the inference. Note that in fields in which there is a lot known about covariates given the outcome of interest, then the omitted ones are likely less important than those that are known an included (i.e., we have a good sense of the factors that matter in terms of educational achievement).

For generalized linear models fit with glm()

Effects for these models are interpreted on the basis of average partial (or marginal) effects (calculated using the margins package).

# if forcats is not installed, this install it first using install.packages("forcats") for this to run
if (requireNamespace("forcats")) {
    d <- forcats::gss_cat
    
    d$married <- ifelse(d$marital == "Married", 1, 0)
    
    m2 <- glm(married ~ age, data = d, family = binomial(link = "logit"))
    konfound(m2, age)
}
#> Replacement of Cases Approach:
#> To sustain an inference, 80.978% of the estimate would have to be due to bias. This is based on a threshold of 0.013 for statistical significance (alpha = 0.05).
#> To sustain an inference, 17334 of the cases with 0 effect would have to be replaced with cases at the threshold of inference.
#> 
#> Correlation-based Approach:
#> An omitted variable would have to be correlated at 5.535 with the outcome and at 5.535 with the predictor of interest (conditioning on observed covariates) to invalidate an inference based on a threshold of 1.003 for statistical significance (alpha = 0.05).
#> Correspondingly the impact of an omitted variable (as defined in Frank 2000) must be 5.535 X 5.535 = 30.636 to invalidate an inference.
#> NULL

As with models fit with lm() (and use of pkonfound()), multiple forms of output can be specified with the to_return argument to konfound(), i.e. konfound(m2, age, to_return = c("raw_output", "corr_plot", "thresh_plot")).

For mixed effects (or multi-level) models fit with the lmer() function from the lme4 package

konfound also works with models fit with the lmer() function from the package lme4, for mixed-effects or multi-level models. One challenge with carrying out sensitivity analysis for fixed effects in mixed effects models is calculating the correct denominator degrees of freedom for the t-test associated with the coefficients. This is not unique to sensitivity analysis, as, for example, lmer() does not report degrees of freedom (or p-values) for fixed effects predictors (see this information in the lme4 FAQ here). While it may be possible to determine the correct degrees of freedom for some models (i.e., models with relatively simple random effects structures), it is difficult to generalize this approach, and so in this package the Kenward-Roger approximation for the denominator degrees of freedom as implemented in the pbkrtest package (described in Halekoh and Højsgaard, 2014).

Here is an example of the use of konfound() with a model fit with lmer():

if (requireNamespace("lme4")) {
    library(lme4)
    m3 <- fm1 <- lmer(Reaction ~ Days + (1 | Subject), sleepstudy)
    konfound(m3, Days)
}
#> Loading required package: Matrix
#> Replacement of Cases Approach:
#> To invalidate an inference, 84.83% of the estimate would have to be due to bias. This is based on a threshold of 1.588 for statistical significance (alpha = 0.05).
#> To invalidate an inference, 137 observations would have to be replaced with cases for which the effect is 0.
#> 
#> Correlation-based Approach:
#> An omitted variable would have to be correlated at 0.817 with the outcome and at 0.817 with the predictor of interest (conditioning on observed covariates) to invalidate an inference based on a threshold of 0.155 for statistical significance (alpha = 0.05).
#> Correspondingly the impact of an omitted variable (as defined in Frank 2000) must be 0.817 X 0.817 = 0.667 to invalidate an inference.
#> Note that the Kenward-Roger approximation is used to estimate degrees of freedom for the predictor(s) of interest.
#> NULL

Use of mkonfound() for meta-analyses that include sensitivity analysis

We can also use konfound to carry out sensitivity analysis as part of meta-analyses. For example, here, d represents output from a number (30 in this case) of past studies, read in a CSV file from a website:

d <- read.csv("https://msu.edu/~kenfrank/example%20dataset%20for%20mkonfound.csv")
head(d)
#>           t  df
#> 1  7.076763 178
#> 2  4.127893 193
#> 3  1.893137  47
#> 4 -4.166395 138
#> 5 -1.187599  97
#> 6  3.585478  87
mkonfound(d, t, df)
#> # A tibble: 30 x 7
#>         t    df action        inference  pct_bias_to_chan…     itcv  r_con
#>     <dbl> <int> <chr>         <chr>                  <dbl>    <dbl>  <dbl>
#>  1  7.08    178 to_invalidate reject_nu…             68.8   0.378   0.614 
#>  2  4.13    193 to_invalidate reject_nu…             50.6   0.168   0.410 
#>  3  1.89     47 to_sustain    fail_to_r…              5.47 -0.0120  0.110 
#>  4 -4.17    138 to_invalidate reject_nu…             50.3   0.202   0.449 
#>  5 -1.19     97 to_sustain    fail_to_r…             39.4  -0.0650  0.255 
#>  6  3.59     87 to_invalidate reject_nu…             41.9   0.190   0.436 
#>  7  0.282   117 to_sustain    fail_to_r…             85.5  -0.131   0.361 
#>  8  2.55     75 to_invalidate reject_nu…             20.6   0.0750  0.274 
#>  9 -4.44    137 to_invalidate reject_nu…             53.0   0.225   0.475 
#> 10 -2.05    195 to_invalidate reject_nu…              3.51  0.00600 0.0770
#> # ... with 20 more rows

We can also return a plot summarizing the percent bias needed to sustan or invalidate an inference across all of the past studies:

mkonfound(d, t, df, return_plot = T)
#> `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Other information

How to learn more about sensitivity analysis

To learn more about sensitivity analysis, please visit:

Feedback, issues, and feature requests

konfound is actively under development as of January, 2018. We welcome feedback and requests for improvement. We prefer for issues to be filed via GitHub (link to the issues page for konfound here) though we also welcome questions or feedback via email.

Code of Conduct

Please note that this project is released with a Contributor Code of Conduct available at http://contributor-covenant.org/version/1/0/0/

References