betaDelta: Example Using the BetaDelta Function

In this example, a multiple regression model is fitted using program quality ratings (QUALITY) as the regressand/outcome variable and number of published articles attributed to the program faculty members (NARTIC), percent of faculty members holding research grants (PCTGRT), and percentage of program graduates who received support (PCTSUPP) as regressor/predictor variables using a data set from 1982 ratings of 46 doctoral programs in psychology in the USA (National Research Council, 1982). Confidence intervals for the standardized regression coefficients are generated using the BetaDelta() function from the betaDelta package following Yuan & Chan (2011) and Jones & Waller (2015).

library(betaDelta)

df <- betaDelta::nas1982

Fit the regression model using the `lm()` function.

object <- lm(QUALITY ~ NARTIC + PCTGRT + PCTSUPP, data = df)

Estimate the standardized regression slopes and the corresponding sampling covariance matrix.

Multivariate Normal-Theory Approach

BetaDelta(object, type = "mvn", alpha = 0.05)
#> Call:
#> BetaDelta(object = object, type = "mvn", alpha = 0.05)
#> 
#> Standardized regression slopes with MVN standard errors:
#>            est     se      t df     p   2.5%  97.5%
#> NARTIC  0.4951 0.0759 6.5272 42 0.000 0.3421 0.6482
#> PCTGRT  0.3915 0.0770 5.0824 42 0.000 0.2360 0.5469
#> PCTSUPP 0.2632 0.0747 3.5224 42 0.001 0.1124 0.4141

Asymptotic Distribution-Free Approach

BetaDelta(object, type = "adf", alpha = 0.05)
#> Call:
#> BetaDelta(object = object, type = "adf", alpha = 0.05)
#> 
#> Standardized regression slopes with ADF standard errors:
#>            est     se      t df      p   2.5%  97.5%
#> NARTIC  0.4951 0.0674 7.3490 42 0.0000 0.3592 0.6311
#> PCTGRT  0.3915 0.0710 5.5164 42 0.0000 0.2483 0.5347
#> PCTSUPP 0.2632 0.0769 3.4231 42 0.0014 0.1081 0.4184

Methods

mvn <- BetaDelta(object, type = "mvn")
adf <- BetaDelta(object, type = "adf")

summary

Summary of the results of BetaDelta().

summary(mvn)
#> Call:
#> BetaDelta(object = object, type = "mvn")
#> 
#> Standardized regression slopes with MVN standard errors:
#>            est     se      t df     p   0.05%   0.5%   2.5%  97.5%  99.5%
#> NARTIC  0.4951 0.0759 6.5272 42 0.000  0.2268 0.2905 0.3421 0.6482 0.6998
#> PCTGRT  0.3915 0.0770 5.0824 42 0.000  0.1190 0.1837 0.2360 0.5469 0.5993
#> PCTSUPP 0.2632 0.0747 3.5224 42 0.001 -0.0011 0.0616 0.1124 0.4141 0.4649
#>         99.95%
#> NARTIC  0.7635
#> PCTGRT  0.6640
#> PCTSUPP 0.5276
summary(adf)
#> Call:
#> BetaDelta(object = object, type = "adf")
#> 
#> Standardized regression slopes with ADF standard errors:
#>            est     se      t df      p   0.05%   0.5%   2.5%  97.5%  99.5%
#> NARTIC  0.4951 0.0674 7.3490 42 0.0000  0.2568 0.3134 0.3592 0.6311 0.6769
#> PCTGRT  0.3915 0.0710 5.5164 42 0.0000  0.1404 0.2000 0.2483 0.5347 0.5830
#> PCTSUPP 0.2632 0.0769 3.4231 42 0.0014 -0.0088 0.0558 0.1081 0.4184 0.4707
#>         99.95%
#> NARTIC  0.7335
#> PCTGRT  0.6426
#> PCTSUPP 0.5353

coef

Calculate the standardized regression slopes.

coef(mvn)
#>    NARTIC    PCTGRT   PCTSUPP 
#> 0.4951451 0.3914887 0.2632477
coef(adf)
#>    NARTIC    PCTGRT   PCTSUPP 
#> 0.4951451 0.3914887 0.2632477

vcov

Calculate the sampling covariance matrix of the standardized regression slopes.

vcov(mvn)
#>               NARTIC       PCTGRT      PCTSUPP
#> NARTIC   0.005754524 -0.003360334 -0.002166127
#> PCTGRT  -0.003360334  0.005933462 -0.001769723
#> PCTSUPP -0.002166127 -0.001769723  0.005585256
vcov(adf)
#>               NARTIC       PCTGRT      PCTSUPP
#> NARTIC   0.004539472 -0.002552698 -0.001742698
#> PCTGRT  -0.002552698  0.005036538 -0.001906216
#> PCTSUPP -0.001742698 -0.001906216  0.005914088

confint

Generate confidence intervals for standardized regression slopes.

confint(mvn, level = 0.95)
#>             2.5 %    97.5 %
#> NARTIC  0.3420563 0.6482339
#> PCTGRT  0.2360380 0.5469395
#> PCTSUPP 0.1124272 0.4140682
confint(adf, level = 0.95)
#>             2.5 %    97.5 %
#> NARTIC  0.3591757 0.6311146
#> PCTGRT  0.2482683 0.5347091
#> PCTSUPP 0.1080509 0.4184444

References

Jones, J. A., & Waller, N. G. (2015). The normal-theory and asymptotic distribution-free (ADF) covariance matrix of standardized regression coefficients: Theoretical extensions and finite sample behavior. Psychometrika, 80(2), 365–378. https://doi.org/10.1007/s11336-013-9380-y

National Research Council. (1982). An assessment of research-doctorate programs in the United States: Social and behavioral sciences. National Academies Press. https://doi.org/10.17226/9781

Pesigan, I. J. A., Sun, R. W., & Cheung, S. F. (2023). betaDelta and betaSandwich: Confidence intervals for standardized regression coefficients in R. Multivariate Behavioral Research, 58(6), 1183–1186. https://doi.org/10.1080/00273171.2023.2201277

Yuan, K.-H., & Chan, W. (2011). Biases and standard errors of standardized regression coefficients. Psychometrika, 76(4), 670–690. https://doi.org/10.1007/s11336-011-9224-6

Ivan Jacob Agaloos Pesigan

Fit the regression model using the lm() function.