Robust White test with bootstrap and effect sizes

Extends performWhiteTest() with optional bootstrap resampling, confidence intervals, effect size reporting, and power analysis.

Usage

performWhiteTestRobust(
  model,
  data,
  method = c("standard", "reduced"),
  bootstrap = FALSE,
  B = 1000,
  ci_level = 0.95,
  parallel = FALSE
)

Arguments

model

A fitted stats::lm object representing the mean specification to be diagnosed.

data

A base::data.frame (or object coercible to one) containing the variables referenced by model. It must include the observations used to fit model and will be checked for missing values.

method

Character string selecting the auxiliary specification. "standard" retains squares and cross-products, while "reduced" excludes cross-products for high-dimensional designs.

bootstrap

Logical, compute bootstrap diagnostics for the statistic and p-value?

B

Number of bootstrap replications when bootstrap = TRUE.

ci_level

Confidence level for reported intervals.

parallel

Logical, allow parallel bootstrap evaluation when the parallel package is available.

Value

An object of class htest augmented with a robust_details list containing bootstrap, effect size, and power information.

Details

Provides enriched inference around the White test by combining bootstrap resampling (Efron & Tibshirani, 1993) with asymptotic approximations. The robust_details element summarises interval estimates, effect sizes, and power calculations to aid decision-making.

References

White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity. Econometrica, 48(4), 817–838.

Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall.

See also

performWhiteTest() for the base statistic and performWhiteTestBootstrap() for a lighter-weight resampling option.

Examples

data(mtcars)
mod <- lm(mpg ~ wt + qsec, data = mtcars)
performWhiteTestRobust(mod, mtcars, bootstrap = TRUE, B = 200)
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 11.8225 df = 5 p = 0.0373
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 11.8225 df = 5 p = 0.0373
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1767 df = 5 p = 0.5243
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.0809 df = 5 p = 0.2984
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2523 df = 5 p = 0.9398
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.6931 df = 5 p = 0.2445
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.918 df = 5 p = 0.8604
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.0071 df = 5 p = 0.5484
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1534 df = 5 p = 0.8275
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.7347 df = 5 p = 0.1715
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.2991 df = 5 p = 0.1993
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.4501 df = 5 p = 0.1893
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.0159 df = 5 p = 0.4139
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.8661 df = 5 p = 0.2308
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1635 df = 5 p = 0.5261
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.579 df = 5 p = 0.3494
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.0149 df = 5 p = 0.3048
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4289 df = 5 p = 0.4895
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3186 df = 5 p = 0.8035
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7419 df = 5 p = 0.5871
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0753 df = 5 p = 0.8386
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.477 df = 5 p = 0.2625
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5982 df = 5 p = 0.7616
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8615 df = 5 p = 0.7213
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.622 df = 5 p = 0.2503
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.3353 df = 5 p = 0.1387
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.606 df = 5 p = 0.2516
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.477 df = 5 p = 0.2625
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.0425 df = 5 p = 0.1539
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.897 df = 5 p = 0.1132
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.763 df = 5 p = 0.2389
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.8873 df = 5 p = 0.2292
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7811 df = 5 p = 0.5813
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.7303 df = 5 p = 0.3334
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.2091 df = 5 p = 0.944
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.0323 df = 5 p = 0.3031
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4925 df = 5 p = 0.7776
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.7703 df = 5 p = 0.1694
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.9617 df = 5 p = 0.31
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.0401 df = 5 p = 0.411
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.5743 df = 5 p = 0.1273
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.511 df = 5 p = 0.1853
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 16.2019 df = 5 p = 0.0063
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2043 df = 5 p = 0.6685
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9169 df = 5 p = 0.5614
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7393 df = 5 p = 0.7401
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0214 df = 5 p = 0.8462
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3601 df = 5 p = 0.7974
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5255 df = 5 p = 0.9101
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.6327 df = 5 p = 0.1246
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1921 df = 5 p = 0.5221
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.4546 df = 5 p = 0.2645
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.2667 df = 5 p = 0.2016
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.3407 df = 5 p = 0.0962
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4697 df = 5 p = 0.628
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9137 df = 5 p = 0.5619
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9094 df = 5 p = 0.5625
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3211 df = 5 p = 0.9327
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6198 df = 5 p = 0.6053
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 17.0505 df = 5 p = 0.0044
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.776 df = 5 p = 0.5821
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5799 df = 5 p = 0.7644
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8022 df = 5 p = 0.5782
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5673 df = 5 p = 0.9052
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 11.1876 df = 5 p = 0.0478
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8519 df = 5 p = 0.7228
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4725 df = 5 p = 0.4836
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.3638 df = 5 p = 0.3731
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.4215 df = 5 p = 0.2673
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8322 df = 5 p = 0.7258
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.9934 df = 5 p = 0.9631
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.4471 df = 5 p = 0.0925
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.4629 df = 5 p = 0.1325
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.3381 df = 5 p = 0.1386
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.1474 df = 5 p = 0.3982
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6314 df = 5 p = 0.6036
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.8092 df = 5 p = 0.1671
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 13.954 df = 5 p = 0.0159
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9582 df = 5 p = 0.421
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.8924 df = 5 p = 0.8638
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.8284 df = 5 p = 0.3233
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.9002 df = 5 p = 0.2282
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.1329 df = 5 p = 0.3999
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.0347 df = 5 p = 0.5444
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4592 df = 5 p = 0.7826
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.7969 df = 5 p = 0.2362
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9575 df = 5 p = 0.5555
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2667 df = 5 p = 0.8112
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6153 df = 5 p = 0.8994
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3384 df = 5 p = 0.648
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9048 df = 5 p = 0.4276
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.7732 df = 5 p = 0.2381
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7647 df = 5 p = 0.7362
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.6891 df = 5 p = 0.3377
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.3546 df = 5 p = 0.7982
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.1588 df = 5 p = 0.2911
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.17 df = 5 p = 0.1025
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.2025 df = 5 p = 0.8205
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.9139 df = 5 p = 0.2271
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.4368 df = 5 p = 0.3649
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.6308 df = 5 p = 0.4626
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5989 df = 5 p = 0.7615
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.4488 df = 5 p = 0.3636
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1464 df = 5 p = 0.5285
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.9026 df = 5 p = 0.228
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9435 df = 5 p = 0.5576
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6266 df = 5 p = 0.6043
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.4537 df = 5 p = 0.2645
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2265 df = 5 p = 0.5173
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3779 df = 5 p = 0.6419
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.1516 df = 5 p = 0.2917
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.5335 df = 5 p = 0.3543
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7997 df = 5 p = 0.8761
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.5202 df = 5 p = 0.6203
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.3786 df = 5 p = 0.0949
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.2734 df = 5 p = 0.2805
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.8067 df = 5 p = 0.117
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.5553 df = 5 p = 0.9066
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4 df = 5 p = 0.7915
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2586 df = 5 p = 0.5128
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.8074 df = 5 p = 0.0809
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6529 df = 5 p = 0.7533
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.7948 df = 5 p = 0.3267
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.7307 df = 5 p = 0.1203
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 9.1046 df = 5 p = 0.105
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.6269 df = 5 p = 0.6043
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.3419 df = 5 p = 0.2744
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.0381 df = 5 p = 0.3025
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8921 df = 5 p = 0.5651
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3237 df = 5 p = 0.6502
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.5892 df = 5 p = 0.468
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8667 df = 5 p = 0.7205
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.6435 df = 5 p = 0.7547
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.3285 df = 5 p = 0.5032
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4567 df = 5 p = 0.4857
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.5274 df = 5 p = 0.355
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1142 df = 5 p = 0.5331
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4519 df = 5 p = 0.6307
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.2859 df = 5 p = 0.2794
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.5797 df = 5 p = 0.6114
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9347 df = 5 p = 0.5589
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 14.1721 df = 5 p = 0.0146
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 0.611 df = 5 p = 0.9875
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.4925 df = 5 p = 0.4809
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.9229 df = 5 p = 0.7119
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9737 df = 5 p = 0.5532
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.7254 df = 5 p = 0.8857
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.6509 df = 5 p = 0.895
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8507 df = 5 p = 0.723
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.5346 df = 5 p = 0.7713
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2405 df = 5 p = 0.663
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.9087 df = 5 p = 0.2275
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.3416 df = 5 p = 0.1384
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.2685 df = 5 p = 0.5114
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.257 df = 5 p = 0.282
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.1846 df = 5 p = 0.2073
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8569 df = 5 p = 0.5702
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.2654 df = 5 p = 0.6591
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.9463 df = 5 p = 0.8565
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.698 df = 5 p = 0.3367
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8464 df = 5 p = 0.7237
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1464 df = 5 p = 0.5285
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.4331 df = 5 p = 0.2663
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.0383 df = 5 p = 0.1541
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7375 df = 5 p = 0.5878
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.8084 df = 5 p = 0.4397
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.3841 df = 5 p = 0.1363
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.1429 df = 5 p = 0.829
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.0997 df = 5 p = 0.8352
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.3836 df = 5 p = 0.3709
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.9156 df = 5 p = 0.227
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.3496 df = 5 p = 0.2737
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.4359 df = 5 p = 0.6331
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.2655 df = 5 p = 0.1422
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.5519 df = 5 p = 0.2562
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 8.2113 df = 5 p = 0.145
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.6273 df = 5 p = 0.463
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.562 df = 5 p = 0.4716
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.3304 df = 5 p = 0.3769
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.1483 df = 5 p = 0.5283
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.4834 df = 5 p = 0.779
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.0321 df = 5 p = 0.5448
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 1.3426 df = 5 p = 0.9305
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.7542 df = 5 p = 0.4466
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3901 df = 5 p = 0.6401
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 6.244 df = 5 p = 0.2832
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.2579 df = 5 p = 0.2022
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.9805 df = 5 p = 0.5522
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.7762 df = 5 p = 0.7344
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 5.3421 df = 5 p = 0.3756
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.1147 df = 5 p = 0.2123
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.0189 df = 5 p = 0.5467
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.7546 df = 5 p = 0.4466
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 7.4592 df = 5 p = 0.1887
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.631 df = 5 p = 0.4626
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7232 df = 5 p = 0.5899
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 2.8306 df = 5 p = 0.7261
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 12.3172 df = 5 p = 0.0307
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.7105 df = 5 p = 0.5918
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 4.9682 df = 5 p = 0.4198
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.8066 df = 5 p = 0.5776
#> [INFO] Running White test
#> [INFO] White test completed: statistic = 3.3544 df = 5 p = 0.6455
#> 
#> 	White's test for heteroscedasticity (robust)
#> 
#> data:  model
#> X-squared = 11.822, df = 5, p-value = 0.0373
#> alternative hypothesis: heteroscedasticity present
#>