pvaluefunctions

P-value functions

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Accompanying paper

We published an accompanying paper to illustrate the use of p-value functions:

Infanger D, Schmidt-Trucksäss A. (2019): P value functions: An underused method to present research results and to promote quantitative reasoning. Statistics in Medicine, 1-9. doi: 10.1002/sim.8293.

Recreation of the graphics in the paper

The code and instructions to reproduce all graphics in our paper can be found in the following GitHub repository: https://github.com/DInfanger/pvalue_functions

Overview

This is the repository for the R-package pvaluefunctions. The package contains R functions to create graphics of p-value functions, confidence distributions, confidence densities, or the Surprisal value (S-value) (Greenland 2019).

Installation

You can install the package directly from CRAN by typing install.packages("pvaluefunctions"). After installation, load it in R using library(pvaluefunctions).

Dependencies

The function depends on the following R packages, which need to be installed beforehand:

Use the command install.packages(c("ggplot2", "scales", "zipfR")) in R to install those packages.

Important information!

The newest version of ggplot2 (3.1.1) has a bug in sec_axis that will lead to the secondary y-axis being labelled wrongly.

It is therefore recommended that you install the developmental version of ggplot2 until the bug has been fixed. You can install the developmental version using the following command (after installing the devtools package): devtools::install_github("tidyverse/ggplot2")

To see what version of ggplot2 has been used to create the plots on this page, see the Session info.

This warning will be deleted upon the release of a new version of ggplot2 that fixes the bug.

Usage

There is only one function needed to create the plots: conf_dist(). The function has the following arguments:

Required arguments for different estimate types

Returned values

The main function conf_dist() returns five objects in a list:

Examples

Two-sample t-test with unequal variances (Welch-Test)


#-----------------------------------------------------------------------------
# Installing package from GitHub and load package
#-----------------------------------------------------------------------------

devtools::install_github("DInfanger/pvaluefunctions")

library(pvaluefunctions)

#-----------------------------------------------------------------------------
# T-Test
#-----------------------------------------------------------------------------

with(sleep, mean(extra[group == 1])) - with(sleep, mean(extra[group == 2]))
#> [1] -1.58
t.test(extra ~ group, data = sleep, var.equal = FALSE)
#> 
#>  Welch Two Sample t-test
#> 
#> data:  extra by group
#> t = -1.8608, df = 17.776, p-value = 0.07939
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -3.3654832  0.2054832
#> sample estimates:
#> mean in group 1 mean in group 2 
#>            0.75            2.33

#-----------------------------------------------------------------------------
# Create p-value function
#-----------------------------------------------------------------------------

res <- conf_dist(
  estimate = c(-1.58)
  , df = c(17.77647)
  , tstat = c(-1.860813)
  , type = "ttest"
  , plot_type = "p_val"
  , n_values = 1e4L
  # , est_names = c("")
  , conf_level = c(0.95, 0.90, 0.80)
  , null_values = c(0)
  , trans = "identity"
  , alternative = "two_sided"
  , log_yaxis = FALSE
  , cut_logyaxis = 0.05
  , xlab = "Mean difference (group 1 - group 2)"
  , together = FALSE
  , plot_p_limit = 1 - 0.999
  , plot_counternull = TRUE
)

Single coefficient from a linear regression model

P-value function

Because it’s difficult to see very small p-values in the graph, you can set the option log_yaxis = TRUE so that p-values (i.e. the y-axes) below the value set in cut_logyaxis will be plotted on a logarithmic scale. This will make it much easier to see small p-values but has the disadvantage of creating a “kink” in the p-value function which is a pure artifact and puts an undue emphasis on the specified cutoff.

#-----------------------------------------------------------------------------
# Model
#-----------------------------------------------------------------------------

mod <- lm(Infant.Mortality~Agriculture + Fertility + Examination, data = swiss)

summary(mod)
#> 
#> Call:
#> lm(formula = Infant.Mortality ~ Agriculture + Fertility + Examination, 
#>     data = swiss)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -8.5375 -1.4021 -0.0066  1.7381  5.9150 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)   
#> (Intercept) 11.01896    4.47291   2.463  0.01784 * 
#> Agriculture -0.02143    0.02394  -0.895  0.37569   
#> Fertility    0.13115    0.04145   3.164  0.00285 **
#> Examination  0.04913    0.08351   0.588  0.55942   
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 2.645 on 43 degrees of freedom
#> Multiple R-squared:  0.2291, Adjusted R-squared:  0.1753 
#> F-statistic:  4.26 on 3 and 43 DF,  p-value: 0.01014

#-----------------------------------------------------------------------------
# Create p-value function
#-----------------------------------------------------------------------------

res <- conf_dist(
  estimate = c(-0.02143)
  , df = c(43)
  , stderr = (0.02394)
  , type = "linreg"
  , plot_type = "p_val"
  , n_values = 1e4L
  # , est_names = c("")
  , conf_level = c(0.95, 0.90, 0.80)
  , null_values = c(0)
  , trans = "identity"
  , alternative = "two_sided"
  , log_yaxis = TRUE
  , cut_logyaxis = 0.05
  , xlab = "Coefficient Agriculture"
  , xlim = c(-0.12, 0.065)
  , together = FALSE
  , plot_p_limit = 1 - 0.999
  , plot_counternull = FALSE
)

Confidence distribution

res <- conf_dist(
  estimate = c(-0.02143)
  , df = c(43)
  , stderr = (0.02394)
  , type = "linreg"
  , plot_type = "cdf"
  , n_values = 1e4L
  # , est_names = c("")
  , conf_level = c(0.95, 0.90, 0.80)
  , null_values = c(0)
  , trans = "identity"
  , alternative = "two_sided"
  # , log_yaxis = TRUE
  # , cut_logyaxis = 0.05
  , xlab = "Coefficient Agriculture"
  , xlim = c(-0.12, 0.065)
  , together = FALSE
  # , plot_p_limit = 1 - 0.999
  , plot_counternull = FALSE
)

The point where the confidence distribution is (0.5) is the median unbiased estimator (see Xie & Singh (2013) for a review and proofs).

Multiple coefficients from a linear regression model

P-value functions

res <- conf_dist(
  estimate = c(0.13115, 0.04913)
  , df = c(43, 43)
  , stderr = c(0.04145, 0.08351)
  , type = "linreg"
  , plot_type = "p_val"
  , n_values = 1e4L
  , est_names = c("Fertility", "Examination")
  , conf_level = c(0.95, 0.90, 0.80)
  , null_values = c(0)
  , trans = "identity"
  , alternative = "two_sided"
  , log_yaxis = FALSE
  , cut_logyaxis = 0.05
  , xlab = "Regression coefficients"
  , together = TRUE
  , plot_p_limit = 1 - 0.999
  , plot_counternull = FALSE
)

Surprisal values

res <- conf_dist(
  estimate = c(0.13115, 0.04913)
  , df = c(43, 43)
  , stderr = c(0.04145, 0.08351)
  , type = "linreg"
  , plot_type = "s_val"
  , n_values = 1e4L
  , est_names = c("Fertility", "Examination")
  , conf_level = c(0.95, 0.90, 0.80)
  , null_values = c(0)
  , trans = "identity"
  , alternative = "two_sided"
  # , log_yaxis = TRUE
  # , cut_logyaxis = 0.05
  , xlab = "Regression coefficients"
  , together = TRUE
  , plot_p_limit = 1 - 0.999
  , plot_counternull = TRUE
)

Pearson correlation coefficient (one-sided)

#-----------------------------------------------------------------------------
# Calculate Pearson's correlation coefficient
#-----------------------------------------------------------------------------

cor.test(swiss$Fertility, swiss$Agriculture, alternative = "two.sided", method = "pearson")
#> 
#>  Pearson's product-moment correlation
#> 
#> data:  swiss$Fertility and swiss$Agriculture
#> t = 2.5316, df = 45, p-value = 0.01492
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#>  0.07334947 0.58130587
#> sample estimates:
#>       cor 
#> 0.3530792

#-----------------------------------------------------------------------------
# Create p-value function
#-----------------------------------------------------------------------------

res <- conf_dist(
  estimate = c(0.3530792)
  , n = 47
  , type = "pearson"
  , plot_type = "p_val"
  , n_values = 1e4L
  # , est_names = c("")
  , conf_level = c(0.95, 0.90, 0.80)
  , null_values = c(0)
  , trans = "identity"
  , alternative = "one_sided"
  , log_yaxis = FALSE
  , cut_logyaxis = 0.05
  , xlab = "Pearson correlation"
  , together = TRUE
  , plot_p_limit = 1 - 0.999
  , plot_counternull = FALSE
)

Odds ratio from logistic regression

#-----------------------------------------------------------------------------
# Calculate logistic regression model using a dataset from UCLA
#-----------------------------------------------------------------------------

dat_tmp <- read.csv("https://stats.idre.ucla.edu/stat/data/binary.csv")

dat_tmp$rank <- factor(dat_tmp$rank)
logistic_mod <- glm(admit ~ gre + gpa + rank, data = dat_tmp, family = "binomial")

summary(logistic_mod)
#> 
#> Call:
#> glm(formula = admit ~ gre + gpa + rank, family = "binomial", 
#>     data = dat_tmp)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -1.6268  -0.8662  -0.6388   1.1490   2.0790  
#> 
#> Coefficients:
#>              Estimate Std. Error z value Pr(>|z|)    
#> (Intercept) -3.989979   1.139951  -3.500 0.000465 ***
#> gre          0.002264   0.001094   2.070 0.038465 *  
#> gpa          0.804038   0.331819   2.423 0.015388 *  
#> rank2       -0.675443   0.316490  -2.134 0.032829 *  
#> rank3       -1.340204   0.345306  -3.881 0.000104 ***
#> rank4       -1.551464   0.417832  -3.713 0.000205 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 499.98  on 399  degrees of freedom
#> Residual deviance: 458.52  on 394  degrees of freedom
#> AIC: 470.52
#> 
#> Number of Fisher Scoring iterations: 4

rm(dat_tmp)

#-----------------------------------------------------------------------------
# Create p-value function
#-----------------------------------------------------------------------------

res <- conf_dist(
  estimate = c(0.804037549)
  , stderr = c(0.331819298)
  , type = "logreg"
  , plot_type = "p_val"
  , n_values = 1e4L
  , est_names = c("GPA")
  , conf_level = c(0.95, 0.90, 0.80)
  , null_values = c(log(1)) # null value on the log-odds scale
  , trans = "exp"
  , alternative = "two_sided"
  , log_yaxis = FALSE
  , cut_logyaxis = 0.05
  , xlab = "Odds Ratio (GPA)"
  , xlim = log(c(0.7, 5.2)) # axis limits on the log-odds scale
  , together = FALSE
  , plot_p_limit = 1 - 0.999
  , plot_counternull = TRUE
)

Proportion

The p-value function (and thus the confidence intervals) are based on Wilson’s score interval and not the normal approximation. This means that the p-value function will never be outside the interval [0, 1].

res <- conf_dist(
  estimate = c(0.44)
  , n = c(50)
  , type = "prop"
  , plot_type = "p_val"
  , n_values = 1e4L
  # , est_names = c("")
  , conf_level = c(0.95, 0.90, 0.80)
  , null_values = c(0.5)
  , trans = "identity"
  , alternative = "two_sided"
  , log_yaxis = FALSE
  , cut_logyaxis = 0.05
  , xlab = "Proportion"
  # , xlim = log(c(0.95, 1.2))
  , together = FALSE
  , plot_p_limit = 1 - 0.999
  , plot_counternull = FALSE
)

Difference between two independent proportions: Wilson’s score by Newcombe with continuity correction

res <- conf_dist(
  estimate = c(68/100, 98/150)
  , n = c(100, 150)
  , type = "propdiff"
  , plot_type = "p_val"
  , n_values = 1e4L
  , conf_level = c(0.95, 0.90, 0.80)
  , null_values = c(0)
  , trans = "identity"
  , alternative = "two_sided"
  , log_yaxis = FALSE
  , cut_logyaxis = 0.05
  , xlab = "Difference between proportions"
  , together = FALSE
  , plot_p_limit = 1 - 0.9999
  , plot_counternull = FALSE
)

Difference between two independent proportions: Agresti-Caffo adjusted Wald interval

The standard Wald interval can be modified in a simple manner to drastically improve its coverage probabilities. Simply add 1 to the number of successes and add 2 to the sample size for both proportions. Then proceed to calculate the Wald interval with these modified data. The point estimate for the difference between proportions is still calculated using the unmodified data. The function conf_dist does not have a dedicaded type for this kind of estimator but as the Wald interval is based on the normal distribution, we can use type = general_z to create the p-value function.


# First proportion

x1 <- 8
n1 <- 40

# Second proportion

x2 <- 11
n2 <- 30

# Apply the correction 

p1hat <- (x1 + 1)/(n1 + 2)
p2hat <- (x2 + 1)/(n2 + 2)

# The estimator (unmodified)

est0 <- (x1/n1) - (x2/n2)

# The modified estimator and its standard error using the correction

est <- p1hat - p2hat
se <- sqrt(((p1hat*(1 - p1hat))/(n1 + 2)) + ((p2hat*(1 - p2hat))/(n2 + 2)))

res <- conf_dist(
  estimate = c(est)
  , stderr = c(se)
  , type = "general_z"
  , plot_type = "p_val"
  , n_values = 1e4L
  # , est_names = c("Estimate")
  , log_yaxis = FALSE
  , cut_logyaxis = 0.05
  , conf_level = c(0.95, 0.99)
  , null_values = c(0, 0.3)
  , trans = "identity"
  , alternative = "two_sided"
  , xlab = "Difference of proportions"
  # , xlim = c(-0.75, 0.5)
  , together = FALSE
  , plot_p_limit = 1 - 0.9999
  , plot_counternull = FALSE
)

Confidence density of a variance estimate from a normal distribution

The confidence density of a variance estimate is skewed. This means that the mean, mode and median of the confidence density will not be identical, in general.


# Simulate some data from a normal distribution

set.seed(142857)
var_est <- var(x <- rnorm(20, 100, 15))

res <- conf_dist(
  estimate = var_est
  , n = length(x)
  , type = "var"
  , plot_type = "pdf"
  , n_values = 1e4L
  , est_names = c("Variance")
  , log_yaxis = FALSE
  , cut_logyaxis = 0.05
  , conf_level = c(0.95)
  # , null_values = c(15^2, 18^2)
  , trans = "identity"
  , alternative = "two_sided"
  , xlab = "Variance"
  , xlim = c(100, 900)
  , together = TRUE
  , plot_p_limit = 1 - 0.999
  , plot_counternull = TRUE
)
# Add vertical lines at the point estimates (mode, median, mean)

res$plot + ggplot2::geom_vline(xintercept = as.numeric(res$point_est[1, 1:3]), linetype = 2, size = 1)

References

Bender R, Berg G, Zeeb H. (2005): Tutorial: using confidence curves in medical research. Biom J. 47(2): 237-47.

Fraser D. A. S. (2019): The p-value function and statistical inference. Am Stat, 73:sup1, 135-147.

Greenland S (2019): Valid P-Values Behave Exactly as They Should: Some Misleading Criticisms of P-Values and Their Resolution with S-Values. Am Stat, 73sup1, 106-114.

Infanger D, Schmidt-Trucksäss A. (2019): P value functions: An underused method to present research results and to promote quantitative reasoning. Stat Med, 1-9. doi: 10.1002/sim.8293.

Poole C. (1987a): Beyond the confidence interval. Am J Public Health. 77(2): 195-9.

Poole C. (1987b) Confidence intervals exclude nothing. Am J Public Health. 77(4): 492-3.

Rosenthal R, Rubin DB. (1994): The counternull value of an effect size: A new statistic. Psychol Sci. 5(6): 329-34.

Schweder T, Hjort NL. (2016): Confidence, likelihood, probability: statistical inference with confidence distributions. New York, NY: Cambridge University Press.

Xie M, Singh K, Strawderman WE. (2011): Confidence Distributions and a Unifying Framework for Meta-Analysis. J Am Stat Assoc 106(493): 320-33. doi: 10.1198/jasa.2011.tm09803.

Xie Mg, Singh K. (2013): Confidence distribution, the frequentist distribution estimator of a parameter: A review. Internat Statist Rev. 81(1): 3-39.

Contact

Denis Infanger

Session info

#> R version 3.6.1 (2019-07-05)
#> Platform: x86_64-w64-mingw32/x64 (64-bit)
#> Running under: Windows 10 x64 (build 17134)
#> 
#> Matrix products: default
#> 
#> locale:
#> [1] LC_COLLATE=German_Switzerland.1252  LC_CTYPE=German_Switzerland.1252   
#> [3] LC_MONETARY=German_Switzerland.1252 LC_NUMERIC=C                       
#> [5] LC_TIME=German_Switzerland.1252    
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] pvaluefunctions_1.3.0
#> 
#> loaded via a namespace (and not attached):
#>  [1] Rcpp_1.0.1         RColorBrewer_1.1-2 pillar_1.4.2      
#>  [4] compiler_3.6.1     prettyunits_1.0.2  remotes_2.1.0     
#>  [7] tools_3.6.1        testthat_2.1.1     digest_0.6.20     
#> [10] pkgbuild_1.0.3     pkgload_1.0.2      tibble_2.1.3      
#> [13] gtable_0.3.0       evaluate_0.14      memoise_1.1.0     
#> [16] pkgconfig_2.0.2    rlang_0.4.0        cli_1.1.0         
#> [19] curl_3.3           yaml_2.2.0         xfun_0.8          
#> [22] dplyr_0.8.3        withr_2.1.2        stringr_1.4.0     
#> [25] knitr_1.23         desc_1.2.0         fs_1.3.1          
#> [28] devtools_2.1.0     tidyselect_0.2.5   rprojroot_1.3-2   
#> [31] grid_3.6.1         glue_1.3.1         R6_2.4.0          
#> [34] processx_3.4.0     rmarkdown_1.13     sessioninfo_1.1.1 
#> [37] zipfR_0.6-10       purrr_0.3.2        callr_3.3.0       
#> [40] ggplot2_3.2.0.9000 magrittr_1.5       scales_1.0.0      
#> [43] backports_1.1.4    ps_1.3.0           htmltools_0.3.6   
#> [46] usethis_1.5.1      assertthat_0.2.1   colorspace_1.4-1  
#> [49] labeling_0.3       stringi_1.4.3      lazyeval_0.2.2    
#> [52] munsell_0.5.0      crayon_1.3.4

License

License: GPL v3