Summarising Regression Models Using the Bootstrap

Summaries for regression models, including "lm", "glm", "glm.nb", "nls", "rlm", "polr", and "merMod" ("lmer", "glmer") objects, using the bootstrap for p-values and confidence intervals.

Usage

boot_summary(
  model,
  type = "perc",
  method = NULL,
  conf.level = 0.95,
  R = 999,
  coef = "raw",
  pval_precision = NULL,
  adjust.method = "none",
  ...
)

Arguments

model: An object fitted using e.g. "lm", "glm", "glm.nb", "nls", "rlm", "polr", lmer", or "glmer".
type: A vector of character strings representing the type of interval to base the test on. The value should be one of "norm", "basic", "bca", and "perc" (the default). "bca" is not supported for "lmer" and "glmer" models.
method: The method used for bootstrapping. For "lm" and "nls" objects use either "residual" (for resampling of scaled and centred residuals, the default) or "case" (for case resampling). For "glm" objects, use "case" (the default). For "merMod" objects (mixed models) use either "parametric" (the default) or "semiparametric".
conf.level: The confidence level for the confidence intervals. The default is 0.95.
R: The number of bootstrap replicates. The default is 999.
coef: A string specifying whether to use exponentiated coefficients in the summary table. Either "exp" (for exponentiated coefficients, i.e. odds ratios in the case of a logistic regression model) or "raw" (for coefficients on their original scale). The default is "raw", which is recommended for linear models.
pval_precision: The desired precision for the p-value. The default is 1/R.
adjust.method: Adjustment of p-values for multiple comparisons using p.adjust. The default is "none", in which case the p-values aren't adjusted. The other options are "holm", "hochberg", "hommel", "bonferroni", "BH", "BY", and "fdr"; see ?p.adjust for details on these methods.
...: Additional arguments passed to Boot or bootMer, such as parallel for parallel computations. See ?car::Boot and ?lme4::bootMer for details.

Value

A data frame containing coefficient estimates, bootstrap confidence intervals, and bootstrap p-values.

Details

p-values can be computed by inverting the corresponding confidence intervals, as described in Section 14.2 of Thulin (2024) and Section 3.12 in Hall (1992). This function computes p-values for coefficients of regression models in this way. The approach relies on the fact that:

the p-value of the two-sided test for the parameter theta is the smallest alpha such that theta is not contained in the corresponding 1-alpha confidence interval,
for a test of the parameter theta with significance level alpha, the set of values of theta that aren't rejected by the two-sided test (when used as the null hypothesis) is a 1-alpha confidence interval for theta.

The function can be used with "lm", "glm", "glm.nb", "nls", "rlm", and "merMod" ("lmer", "glmer") objects. In addition, it should work for any regression model such that: residuals(object, type="pearson") returns Pearson residuals; fitted(object) returns fitted values; hatvalues(object) returns the leverages, or perhaps the value 1 which will effectively ignore setting the hatvalues. In addition, the data argument should contain no missing values among the columns actually used in fitting the model.

References

hall92boot.pval thulin21boot.pval

Examples

# Bootstrap summary of a linear model for mtcars:
model <- lm(mpg ~ hp + vs, data = mtcars)
boot_summary(model, R = 99)
#>                Estimate Lower.bound Upper.bound p.value
#> (Intercept) 26.96300111 22.46858043 33.04643462   <0.01
#> hp          -0.05453412 -0.08515078 -0.03300863   <0.01
#> vs           2.57622314 -1.62537273  6.29452416    0.37
# (Values for R greater than 99 are recommended for most applications.)

# Adjust p-values for multiplicity using Holm's method:
boot_summary(model, R = 99, adjust.method = "holm")
#>                Estimate Lower.bound Upper.bound p.value Adjusted p-value
#> (Intercept) 26.96300111 20.64731661 33.01217023   <0.01             0.03
#> hp          -0.05453412 -0.08505702 -0.02204617   <0.01             0.03
#> vs           2.57622314 -1.10406035  7.69178446    0.23             0.23