Diferenças entre Coeficientes Padronizados vs Brutos
Em tabelas de regressão temos geralmente temos duas opções de como reportar os coeficientes:
Para explicar melhor esses conceitos, simularemos alguns dados:
N <- 1000
x <- rnorm(N, 1, 0.1)
error <- rnorm(N, 0, 1)
y <- rep(10, N) + (100 * x) + error
df <- data.frame(x, y)
Observem os dados com o pacote skimr (Waring et al., 2021) usando a função skim():
| Name | df |
| Number of rows | 1000 |
| Number of columns | 2 |
| _______________________ | |
| Column type frequency: | |
| numeric | 2 |
| ________________________ | |
| Group variables | None |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| x | 0 | 1 | 1 | 0.1 | 0.72 | 0.94 | 1 | 1.1 | 1.3 | ▁▅▇▃▁ |
| y | 0 | 1 | 110 | 10.1 | 79.76 | 103.59 | 110 | 116.8 | 142.3 | ▁▅▇▃▁ |
Prestem atenção:
Agora vamos executar uma regressão1 e mostrar coeficientes tanto como brutos, assim como padronizados:
Call:
lm(formula = y ~ x, data = df)
Residuals:
Min 1Q Median 3Q Max
-3.028 -0.691 0.004 0.709 3.291
Coefficients:
Estimate Standardized Std. Error t value
(Intercept) 9.161 0.000 0.323 28.4
x 100.880 0.995 0.321 314.2
Pr(>|t|)
(Intercept) <0.0000000000000002 ***
x <0.0000000000000002 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1 on 998 degrees of freedom
Multiple R-squared: 0.99, Adjusted R-squared: 0.99
F-statistic: 9.87e+04 on 1 and 998 DF, p-value: <0.0000000000000002Ambas colunas o output de summary() mostram a mesma coisa:
Estimate: a cada 1 unidade que \(x\) aumenta, \(y\) aumenta 100.88Standardized: a cada 1 desvio padrão de \(x\) de incremento (dp = 0.1), há um aumento de 0.99 desvio padrão de \(y\) (10). Um total de 100.88. \(\big( \frac{0.955 * \operatorname{sd}_y}{\operatorname{sd}_x}\big)\)R version 4.1.0 (2021-05-18)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods
[7] base
other attached packages:
[1] lm.beta_1.5-1 loo_2.4.1 DescTools_0.99.41
[4] gapminder_0.3.0 ggExtra_0.9 dplyr_1.0.6
[7] rstan_2.21.2 StanHeaders_2.21.0-7 MASS_7.3-54
[10] ggforce_0.3.3 gganimate_1.0.7 plotly_4.9.3
[13] carData_3.0-4 DiagrammeR_1.0.6.1 brms_2.15.0
[16] rstanarm_2.21.1 Rcpp_1.0.6 skimr_2.1.3
[19] readr_1.4.0 readxl_1.3.1 tibble_3.1.2
[22] ggplot2_3.3.3 patchwork_1.1.1 cowplot_1.1.1
loaded via a namespace (and not attached):
[1] utf8_1.2.1 tidyselect_1.1.1 lme4_1.1-27
[4] htmlwidgets_1.5.3 grid_4.1.0 munsell_0.5.0
[7] codetools_0.2-18 ragg_1.1.2 distill_1.2
[10] DT_0.18 gifski_1.4.3-1 miniUI_0.1.1.1
[13] withr_2.4.2 Brobdingnag_1.2-6 colorspace_2.0-1
[16] highr_0.9 knitr_1.33 rstudioapi_0.13
[19] stats4_4.1.0 bayesplot_1.8.0 labeling_0.4.2
[22] repr_1.1.3 mnormt_2.0.2 polyclip_1.10-0
[25] farver_2.1.0 bridgesampling_1.1-2 rprojroot_2.0.2
[28] coda_0.19-4 vctrs_0.3.8 generics_0.1.0
[31] xfun_0.23 R6_2.5.0 markdown_1.1
[34] isoband_0.2.4 gamm4_0.2-6 projpred_2.0.2
[37] assertthat_0.2.1 promises_1.2.0.1 scales_1.1.1
[40] rootSolve_1.8.2.1 gtable_0.3.0 downlit_0.2.1
[43] processx_3.5.2 lmom_2.8 rlang_0.4.11
[46] systemfonts_1.0.2 splines_4.1.0 lazyeval_0.2.2
[49] checkmate_2.0.0 inline_0.3.19 yaml_2.2.1
[52] reshape2_1.4.4 abind_1.4-5 threejs_0.3.3
[55] crosstalk_1.1.1 backports_1.2.1 httpuv_1.6.1
[58] rsconnect_0.8.18 tools_4.1.0 bookdown_0.22
[61] ellipsis_0.3.2 jquerylib_0.1.4 RColorBrewer_1.1-2
[64] proxy_0.4-25 ggridges_0.5.3 plyr_1.8.6
[67] base64enc_0.1-3 visNetwork_2.0.9 progress_1.2.2
[70] purrr_0.3.4 ps_1.6.0 prettyunits_1.1.1
[73] zoo_1.8-9 here_1.0.1 magrittr_2.0.1
[76] data.table_1.14.0 magick_2.7.2 colourpicker_1.1.0
[79] tmvnsim_1.0-2 mvtnorm_1.1-1 matrixStats_0.59.0
[82] hms_1.1.0 shinyjs_2.0.0 mime_0.10
[85] evaluate_0.14 xtable_1.8-4 shinystan_2.5.0
[88] gridExtra_2.3 rstantools_2.1.1 compiler_4.1.0
[91] V8_3.4.2 crayon_1.4.1 minqa_1.2.4
[94] htmltools_0.5.1.1 mgcv_1.8-35 later_1.2.0
[97] tidyr_1.1.3 expm_0.999-6 Exact_2.1
[100] RcppParallel_5.1.4 lubridate_1.7.10 DBI_1.1.1
[103] tweenr_1.0.2 boot_1.3-28 Matrix_1.3-3
[106] cli_2.5.0 parallel_4.1.0 igraph_1.2.6
[109] pkgconfig_2.0.3 xml2_1.3.2 dygraphs_1.1.1.6
[112] bslib_0.2.5.1 stringr_1.4.0 callr_3.7.0
[115] digest_0.6.27 rmarkdown_2.8 cellranger_1.1.0
[118] gld_2.6.2 curl_4.3.1 shiny_1.6.0
[121] gtools_3.8.2 nloptr_1.2.2.2 lifecycle_1.0.0
[124] nlme_3.1-152 jsonlite_1.7.2 viridisLite_0.4.0
[127] fansi_0.5.0 pillar_1.6.1 lattice_0.20-44
[130] fastmap_1.1.0 httr_1.4.2 pkgbuild_1.2.0
[133] survival_3.2-11 glue_1.4.2 xts_0.12.1
[136] png_0.1-7 shinythemes_1.2.0 class_7.3-19
[139] stringi_1.6.2 sass_0.4.0 textshaping_0.3.4
[142] e1071_1.7-7 estou rodando uma regressão simples frequentista pois o foco não é a robustez da inferência mas apenas uma mera exemplificação de como interpretar coeficientes.↩︎
If you see mistakes or want to suggest changes, please create an issue on the source repository.
Text and figures are licensed under Creative Commons Attribution CC BY-SA 4.0. Source code is available at https://github.com/storopoli/Estatistica-Bayesiana, unless otherwise noted. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in their caption: "Figure from ...".
For attribution, please cite this work as
Storopoli (2021, Aug. 1). Estatística Bayesiana com R e Stan: Coeficientes de uma Regressão. Retrieved from https://storopoli.github.io/Estatistica-Bayesiana/aux-Regressao_Coeficientes.html
BibTeX citation
@misc{storopoli2021regressaocoeficientesR,
author = {Storopoli, Jose},
title = {Estatística Bayesiana com R e Stan: Coeficientes de uma Regressão},
url = {https://storopoli.github.io/Estatistica-Bayesiana/aux-Regressao_Coeficientes.html},
year = {2021}
}