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BACKGROUND: Before performing a t-test to compare means between two samples it is common to check whether the variances can be assumed to be the same. Usually this is done with either an $F$ or a Levene's test. However, there is (apparently) a strong dependency on normalcy in the underlying data. Bootstrapping is suggested as a better alternative in a Coursera course.

QUESTION: I am considering a simple dataset put together by just subsetting the miles per gallon in the mtcars dataset and comparing cars with four and eight cylinders:

(four_cyl    <- mtcars$mpg[mtcars$cyl==4])
[1] 22.8 24.4 22.8 32.4 30.4 33.9 21.5 27.3 26.0 30.4 21.4
(eight_cyl   <- mtcars$mpg[mtcars$cyl==8])
[1] 18.7 14.3 16.4 17.3 15.2 10.4 10.4 14.7 15.5 15.2 13.3 19.2 15.8 15.0

If I run an $F$ test, there doesn't seem to be any reason to conclude that the variances are different:

var.test(four_cyl, eight_cyl)   
F test to compare two variances

    data:  four_cyl and eight_cyl
    F = 3.1033, num df = 10, denom df = 13, p-value = 0.05925
    alternative hypothesis: true ratio of variances is not equal to 1
    95 percent confidence interval:
      0.9549589 11.1197133
    sample estimates:
    ratio of variances 
              3.103299 

I also ran a Levene's test but I had problems with the size of the samples being unequal: library(reshape2); library(car); sample <- as.data.frame(cbind(four_cyl, eight_cyl)); dataset <- melt(sample); leveneTest(value ~ variable, dataset) yielding a $p$ value of 0.1061.

Now if I try to put together some bootstrapping attempt, the visual plotting doesn't seem to be so reassuring:

dat <- replicate(1e4, sample(four_cyl, length(four_cyl), replace = T))
sd_4 <- apply(dat, 2, sd)
hist(sd_4, freq = F, col="moccasin", ylim=c(0,1), border = F)

dat <- replicate(1e4, sample(eight_cyl, length(eight_cyl), replace = T))
sd_8 <- apply(dat, 2, sd)
hist(sd_8, add=T, freq = F, col="navajowhite4", border = F)

enter image description here

However, I don't know how to test the difference shown in the bootstrap simulation numerically. The temptation is to run a t-test: t.test(sd_4, sd_8), which yields a p-value = < 2.2e-16. But this doesn't make too much sense...

Perhaps its possible to just run the ratios of the different observed $10,000$ bootstrap samples, and calculate a Wald interval on them:

boot_ratios <- sd_4/sd_8
mean(boot_ratios) + c(-1, 1) * qnorm(0.975) * sd(boot_ratios)
[1] 0.7533392 2.9471311

? - this is not a typo...

This would include $1$ and hence be concordant with the failure to reject $H_o$ with the $F$ test.

ADDENDUM: Following @Scortchi comment I calculated the quantiles as:

quantile(boot_ratios, c(0.025, 0.975))
    2.5%    97.5% 
1.075067 3.243562 

and they don't include $1$.

Is this procedure overall sound? How can interpret the results?

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  • $\begingroup$ I'd imagine they mean using the bootstrap to estimate the distribution of the test statistic under the null. Something along the lines of resampling from the combined distribution of centred observations. (BTW I don't think the F test for equal variances is usually called the "Fisher test", & Levene's test is supposed to be quite robust.) $\endgroup$ – Scortchi Oct 22 '15 at 13:21
  • $\begingroup$ @Scortchi: I found a sentence that may contain the key on a slide: "Alternatively, for moderate sample sizes, consider creating a bootstrap confidence interval for the ratio of the two variables." Does this help? $\endgroup$ – Antoni Parellada Oct 22 '15 at 13:53
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    $\begingroup$ That's an alternative. You could also directly find the 0.025 & 0.975 percentiles of the bootstrap ratios. $\endgroup$ – Scortchi Oct 22 '15 at 16:29
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    $\begingroup$ The approach of "test for equality of variance then if you don't reject, use a t-test that assumes equality of variance otherwise use one that doesn't assume equality of variance" is in general not as good as the much simpler approach "if you're not in a position to assume the variances are equal, just don't assume the variances are equal" (i.e. if you were going to use say a Welch test if you rejected the equality of variance test, just use the Welch test without testing variance first). There's a number of posts on site that advise against such "preliminary" testing, with references. $\endgroup$ – Glen_b Oct 22 '15 at 22:22
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    $\begingroup$ I didn't look closely (I expect your implementation is okay). However, as a general principle, I would tend to have concerns about the properties of the bootstrap as such small sample sizes (I'd want to see that it would give me the sort of properties I seek; so if one is using CIs as the basis of a test, do the CI's at such sample sizes have close to the desired coverage level under some plausible assumptions? I'm not especially fussy about significance levels being off a bit, but if I get effective rejection rates under the null that are quite far from what I think I would get, it's a worry) $\endgroup$ – Glen_b Oct 22 '15 at 22:38
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A test proper could be devised by assuming that the two groups differ only in location $\mu$ & scale $\sigma$, so that their distribution functions are $$F_1(x_1) = F((x_1-\mu_1)/\sigma_1)$$ & $$F_2(x_2) = F((x_2-\mu_2)/\sigma_2)$$; then under the null hypothesis of a common scale $\sigma=\sigma_1=\sigma_2$, the centred variable $Y= X_i-\mu_i$ has the distribution $$F_\mathrm{c}(y) = F(y/\sigma)$$, which can be approximated by the empirical distribution function $$\hat F_\mathrm{c}(y) = \frac{\sum_{i=1}^{2}\sum_{j=1}^{n_i} I(x_{ij} - \bar x_i \leq y)}{n_1+n_2}$$ where $I$ is the indicator function.

So the bootstrap procedure would resample from the pooled differences between each observation & the mean of its group, & compare the distribution of the test statistic, say the usual F statistic, whose distribution shouldn't depend overmuch on the unknown characteristics of $F_c(\cdot)$, with the value in fact observed.

(This is a rather off-the-cuff answer—it illustrates a basic bootstrap test, but I daresay there are better methods to use in practice.)

Here's some code to test it:

# sample sizes
n1 <- 12
n2 <- 20
# location parameters
mu1 <- 3
mu2 <- 6
# scale parameters (alt. hyp sigma1 > sigma2)
sigma1 <- 1
sigma2 <- 1
# distribution function, e.g. Student's t with 3 degrees of freedom
rdist <- function(n) rt(n, df=3)

# no. simulations to perform
no.sims <- 1000
# no. bootstrap samples to take in each simulation
no.boots <- 1000

# initialize vector of p-values - for normal F test
p.normFtest <- numeric(no.sims)
# initialize vector of p-values - for bootstrap test
p.bsFtest <- numeric(no.sims)

# simulate!
for (j in 1:no.sims){
  # simulate samples
  rdist(n1)*sigma1 + mu1 -> x1
  rdist(n2)*sigma2 + mu2 -> x2
  # calculate observed test statistic
  var(x1)/var(x2) -> F.obs
  # calculate its p-value
  1-pf(F.obs,n1-1,n2-1) -> p.normFtest[j]
  # initialize vector of test statistics
  F.boot <- numeric(no.boots)
  # define bootstrap population
  c(x1-mean(x1),x2-mean(x2)) -> boot.pop
  # bootstrap!
  for (i in 1:no.boots){
    # 1st sample
    x1.boot <- sample(boot.pop, n1, replace=T)
    # 2nd sample
    x2.boot <- sample(boot.pop, n2, replace=T)
    # calculate bootstrap test statistic
    var(x1.boot)/var(x2.boot) -> F.boot[i]
  }
# estimate bootstrap p-value
sum(F.boot >= F.obs)/no.boots -> p.bsFtest[j]
}

# examine distributions of p-values (should be uniform under null)
plot(ecdf(p.normFtest), col="black", do.points=F, verticals=T, xlab="calculated p-value", ylab="simulated distribution function", main="")
plot(ecdf(p.bsFtest), col="red", add=T, do.points=F, verticals=T)
abline(a=0,b=1, col="grey", lty="dashed")
legend("bottomright", legend=c("normal", "bootstrapped"), lty=1, col=c("black","red")) 

EDFs of simulated p-values

So in this particular case this bootstrap F test maintains size rather better than the F test based on an incorrect assumption of normality. It'd be interesting to compare it with more robust tests like Levene's, & to examine its power vs the normal F test when the normality assumption is correct. And using a double bootstrap might help to keep the test's size closer to the nominal level.

† With the commonly used Brown–Forsythe modification.

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