# Fast Romano-Wolf Multiple Testing Corrections for fixest 🐺 (2023)

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For the final chapter of my dissertation, I had examined the effects of ordinal class rank on the academic achievement of Danish primary school students (following a popular identification strategy introduced in a paper by Murphy and Weinhard). Because of the richness of the Danish register data, I had a large number of potential outcome variables at my disposal, and as a result, I was able to examine literally all the outcomes that the previous literature had covered in individual studies – the effect of rank on achievement, personality, risky behaviour, mental health, parental investment, etc. – in one paper. Figure 1: The Effect of Ordinal Class Rank on quite a few outcome variables

But with (too) many outcome variables comes a risk: inflated type 1 error rates, or an increased risk of ‘false positives’. So I was wondering: were all the significant effects I estimated – shown in the figure above – “real”, or was I simply being fooled by randomness?

A common way to control the risk of false positive when testing multiple hypothesis is to use methods that control the ‘family-wise’ error rate, i.e.the risk of at least one false positive in a family of S hypotheses.

Among such methods, Romano and Wolf’s correction is particularly popular, because it is “uniformly the most powerful”. Without going into too much detail, I’ll just say that if you have a choice between a number of methods that control the family-wise error rate at a desired level $$\alpha$$, you might want to choose the “most powerful” one, i.e.the one that has the highest success rate of finding a true effect.

Now, there is actually an amazing Stata package for the Romano-Wolf method called rwolf.

But this is an R blog, and I’m an R guy … In addition, my regression involved several million rows and some high-dimensional fixed effects, and rwolf quickly showed its limitations. It just wasn’t fast enough!

While playing around with the rwolf package, I finally did my due diligence on the method it implements, and after a little background reading, I realized that both the Romano and Wolf method – as well as its main rival, the method proposed by Westfall and Young – are based on the bootstrap!

But wait! Had I not just spent a lot of time porting a super-fast bootstrap algorithm from R to Stata? Could I not use Roodman et al’s “fast and wild” cluster bootstrap algorithm for bootstrap-based multiple hypothesis correction?

Of course it was inevitable: I ended up writing an R package. Today I am happy to present the first MVP version of the wildwrwolf package!

## The wildrwolf package

You can simply install the package from github or r-universe by typing

# install.packages("devtools")devtools::install_github("s3alfisc/wildrwolf")# from r-universe (windows & mac, compiled R > 4.0 required)install.packages('wildrwolf', repos ='https://s3alfisc.r-universe.dev')

The Romano Wolf correction in wildrwolf is implemented as a post-estimation commands for multiple estimation objects from the fabulous fixest package.

To demonstrate how wildrwolf's main function, rwolf, works, let’s first simulate some data:

library(wildrwolf)library(fixest)set.seed(1412)library(wildrwolf)library(fixest)set.seed(1412)N <- 5000X1 <- rnorm(N)X2 <- rnorm(N)rho <- 0.5sigma <- matrix(rho, 4, 4); diag(sigma) <- 1u <- MASS::mvrnorm(n = N, mu = rep(0, 4), Sigma = sigma)Y1 <- 1 + 1 * X1 + X2 Y2 <- 1 + 0.01 * X1 + X2Y3 <- 1 + 0.4 * X1 + X2Y4 <- 1 + -0.02 * X1 + X2for(x in 1:4){ var_char <- paste0("Y", x) assign(var_char, get(var_char) + u[,x])}group_id <- sample(1:100, N, TRUE)data <- data.frame(Y1 = Y1, Y2 = Y2, Y3 = Y3, Y4 = Y4, X1 = X1, X2 = X2, group_id = group_id, splitvar = sample(1:2, N, TRUE))

We now estimate a regression via the fixest package:

fit <- feols(c(Y1, Y2, Y3, Y4) ~ csw(X1,X2), data = data, cluster = ~group_id, ssc = ssc(cluster.adj = TRUE))# clean workspace except for res & datarm(list= ls()[!(ls() %in% c('fit','data'))])

For all eight estimated regressions, we want to apply the Romano-Wolf correction to the parameter of interest, X1. We simply need to provide the regression object of type fixest_multi (it is also possible to simply provide a list of objects of type fixest), the parameter of interest, the number of bootstrap draws, and possibly a seed (for replicability). That’s it! If the regressions use clustered standard errors, rwolf will pick this up and run a wild cluster bootstrap, otherwise just a robust wild bootstrap.

pracma::tic()res_rwolf <- wildrwolf::rwolf( models = fit, param = "X1", B = 9999, seed = 23)## | | | 0% | |========= | 12% | |================== | 25% | |========================== | 38% | |=================================== | 50% | |============================================ | 62% | |==================================================== | 75% | |============================================================= | 88% | |======================================================================| 100%pracma::toc()## elapsed time is 3.980000 seconds

For $$N=5000$$ observations with $$G=100$$ clusters, $$S=8$$ hypothesis and $$B=9999$$ bootstrap draws, the calculation of Romano-Wolf corrected p-values takes less than 5 seconds. If you ask me, that is pretty fast! =) 🚀

We can now compare the results of rwolf with the uncorrected p-values and another popular multiple testing adjustment, Holm’s method.

pvals <- lapply(fit, function(x) pvalue(x)["X1"]) |> unlist() df <- data.frame( "uncorrected" = pvals, "Holm" = p.adjust(pvals, method = "holm"), "rwolf" = res_rwolf\$pval)rownames(df) <- NULLround(df, 3)## uncorrected Holm rwolf## 1 0.000 0.000 0.000## 2 0.000 0.000 0.000## 3 0.140 0.420 0.367## 4 0.033 0.132 0.128## 5 0.000 0.000 0.000## 6 0.000 0.000 0.000## 7 0.398 0.420 0.394## 8 0.152 0.420 0.367

Both Holm’s method and rwolf produce similar corrected p-values, which - in general - are larger than the uncorrected p-values.

## But does it actually work? Monte Carlo Experiments

We test $$S=6$$ hypotheses and generate data as

$Y_{i,s,g} = \beta_{0} + \beta_{1,s} D_{i} + u_{i,g} + \epsilon_{i,s}$where $$D_i = 1(U_i > 0.5)$$ and $$U_i$$ is drawn from a uniformdistribution, $$u_{i,g}$$ is a cluster level shock with intra-clustercorrelation $$0.5$$, and the idiosyncratic error term is drawn from amultivariate random normal distribution with mean $$0_S$$ and covariancematrix

S <- 6rho <- 0.5Sigma <- matrix(1, 6, 6)diag(Sigma) <- rhoSigma

This experiment imposes a data generating process as in equation (9) inClarke, Romano and Wolf, with anadditional error term $$u_g$$ for $$G=20$$ clusters and intra-clustercorrelation 0.5 and $$N=1000$$ observations.

You can run the simulations via the run_fwer_sim() function attachedin the package.

# note that this will take some timeres <- run_fwer_sim( seed = 232123, n_sims = 500, B = 499, N = 1000, s = 6, rho = 0.5 #correlation between hypotheses, not intra-cluster!)# > res# reject_5 reject_10 rho# fit_pvalue 0.274 0.460 0.5# fit_pvalue_holm 0.046 0.112 0.5# fit_padjust_rw 0.052 0.110 0.5

Both Holm’s method and wildrwolf control the family wise error rates, at both the 5 and 10% significance level. Very cool!

## The method by Westfall and Young

A package for Westfall and Young’s (1993) method is currently in development. I am not yet fully convinced that it works as intented - in simulations, I regularly find that wildwyoung overrejects.

## Literature

• Clarke, Damian, Joseph P. Romano, and Michael Wolf. “The Romano–Wolf multiple-hypothesis correction in Stata.” The Stata Journal 20.4 (2020): 812-843.

• Murphy, Richard, and Felix Weinhardt. “Top of the class: The importance of ordinal rank.” The Review of Economic Studies 87.6 (2020): 2777-2826.

• Romano, Joseph P., and Michael Wolf. “Stepwise multiple testing as formalized data snooping.” Econometrica 73.4 (2005): 1237-1282.

• Roodman, David, et al.“Fast and wild: Bootstrap inference in Stata using boottest.” The Stata Journal 19.1 (2019): 4-60.

• Westfall, Peter H., and S. Stanley Young. Resampling-based multiple testing: Examples and methods for p-value adjustment. Vol. 279. John Wiley & Sons, 1993.

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