# Effect size and bootstrapping in paired t-test

I have multiple paired $t$-tests, such as one giving results:

$t_{14} = 2.7,\ p = .017$

Although people seem to do effect sizes in different ways in repeated samples, I have taken the mean difference divided by the standard deviation of the differences (I'll call this $d$, though maybe I should call it something else?) and get $0.70$. I also have a very strong correlation between the samples, not sure if that is problematic.

I would like to put confidence limits around my effect size estimate. To do so, I randomly resample from the difference scores, compute $d$ in the same way and repeat 1000 times. My question is whether this is a good approach, rather than, say, just giving confidence limits around the unstandardised difference or resampling from the original samples. My bootstrap gives me a mean $d$ of $0.79$ with confidence limits of $[0.4, 1.4]$. I've tried this on other random data too. Why am I getting a consistently higher $d$ from bootstrapping, and why are the intervals asymmetric? Is this because of skew in the (difference) scores, and does this make this approach more or less robust?

Edit: here is an example of the data involved. 15 people were measured two times.

Mean A = 1742; SD = 435
Mean B = 1820; SD = 426
Mean difference = 78, SD of differences = 111, $d$ = 0.70

    A    B
1999 2040
1501 1601
1552 1623
2385 2386
2488 2671
1257 1218
1806 1719
1348 1405
2048 2079
1810 2017
1308 1356
2310 2324
1247 1616
1839 1878
1235 1370

• Just to say that I have found useful material on these pages (though I haven't got a specific answer to the case of bootstrapping CIs for an effect size) stats.stackexchange.com/questions/71525/… , stats.stackexchange.com/questions/73818/… – splint Nov 30 '15 at 15:45
• I'm not quite sure I'm following this. Can you give a simple example / some example data? Is this a multiple comparisons issue? – gung - Reinstate Monica Dec 3 '15 at 14:59
• @gung Thanks for looking. The t quoted is a simple example though I can fish out some data if you want. The issue is not about multiple comparisons. It is about (1) how to calculate effect sizes in a paired t-test; (2) whether it makes sense to bootstrap a confidence interval around this; and (3) why this interval might be asymmetric. – splint Dec 4 '15 at 14:16
• What are the "repeated samples" that supposedly lead people to "do effect sizes in different ways"? For people here to get a sense of why the mean of your bootsamples is different & the CI is asymmetric, you will probably need to paste your data & your code. – gung - Reinstate Monica Dec 4 '15 at 16:51
• I have added some data. Is there a better way to do tables on here? For background on the different ways to calculate effect sizes in repeated measures, see the links in my first comment (essentially, some prefer to use the pooled SD as a denominator rather than the SD of the difference scores). – splint Dec 4 '15 at 17:47

I will attempt to answer but I am not totally sure on my own knowledge on the subject.

Bootstrap, as far as I know is always done on the original data. In your case the original data is pairs of data. So to do a bootstrap, you would have to random sample (with replacement) on the pairs of the original data. That is equivalent to do the bootstrap on the difference scores and performing the effect size calculation as you described on the samples.

I get a different result from you (in R)

a=read.table(header=F,text="
1999 2040
1501 1601
1552 1623
2385 2386
2488 2671
1257 1218
1806 1719
1348 1405
2048 2079
1810 2017
1308 1356
2310 2324
1247 1616
1839 1878
1235 1370
")
d=a$V2-a$V1
mean(d)/sd(d)
[1] 0.7006464
aux=function(x,i) mean(x[i])/sd(x[i])
bb=boot::boot(d,aux,R=1000)
mean(bb$t) [1] 0.7530415 boot::boot.ci(bb) BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS Based on 1000 bootstrap replicates CALL : boot::boot.ci(boot.out = bb) Intervals : Level Normal Basic 95% ( 0.1840, 1.0846 ) ( 0.1454, 1.0570 ) Level Percentile BCa 95% ( 0.3443, 1.2559 ) ( 0.1634, 1.0722 ) Calculations and Intervals on Original Scale Some BCa intervals may be unstable  (code corrected as per the comments) Indeed the direct calculation of the effect size (mean(d)/sd(d)) is not similar to the bootstrap calculation (mean(bb$t)). I dont know how to explain it

The only confidence interval that matches yours in the percentile (I dont really know which interval to choose on theoretical grounds - I use the BCa - I think it was suggested somewhere)

The second way to calculate a CI on effect size is to use analytical formulas. This question on CV discussed the formulas How can i calculate the 95% confidence interval of an effect size if I have the mean difference score, CI of that difference score

Using the MBESS package I get the following CI

MBESS::ci.sm(Mean = mean(d), SD=sd(d),N=length(d))
[1] "The 0.95 confidence limits for the standardized mean are given as:"
$Lower.Conf.Limit.Standardized.Mean [1] 0.1231584$Standardized.Mean
[1] 0.7006464

$Upper.Conf.Limit.Standardized.Mean [1] 1.258396  As for your suggestion on computing the confidence interval for the difference score and using it to compute a confidence interval on the effect size, I have never heard of it, and I would suggest not using it. • +1 to @amoeba, I think you want to use mean(bb$t). Nice answer; +1 as soon as you fix that issue. – usεr11852 says Reinstate Monic Dec 6 '15 at 8:16
• Very helpful answer. I do have questions though. mean(bb$t) returns 0.76 which, as in my example, is considerably greater than the sample value. All of the intervals are also asymmetric whereas my understanding was that this should not be the case for analytically computed CIs. – splint Dec 6 '15 at 12:07 • There is a missing close bracket in the function/mean call, apparently edits of 1 character are not allowed! – splint Dec 6 '15 at 12:10 • thanks folks. mean(bb$t) and the ")" in the mean corrected. Indeed the values for the bootstrap mean and the full data effect size are not close. I dont know how to explain. – Jacques Wainer Dec 7 '15 at 13:44
• @splint and Jacques: what happens here has to do with bias and bias correcting in bootstrap, see e.g. on wikipedia. I am not a specialist, but rougly what happens is that the difference between your empirical value 0.7 and your bootstrapped value 0.75 indicates a bias. You can correct this bias, by subtracting this difference from 0.7 and arrive to the bias-corrected estimate of d as 0.65. The intuition is that if your bootstrapped samples were on average 0.05 higher [ctd.] – amoeba says Reinstate Monica Dec 7 '15 at 14:00