Conducting Statistical Experiments Hands-On

Conducting Laboratory
Experiments Properly
with Statistical Tools:
An Easy Hands-on Tutorial
Tetsuya Sakai (Waseda University)
tetsuyasakai@acm.org
8th July@SIGIR 2018, Ann Arbor Michigan, USA.

Tutorial materials:
If you want a hands-on experience at the tutorial,
please do the following BEFORE ATTENDING:
- Download
http://waseda.box.com/SIGIR2018tutorial
- Install R on your laptop. Then install the tidyr (for
reformatting data) and pwr libraries (for power
calculation)
If you have reshape2 instead of tidyr and
know how to use melt that’s okay too

Tutorial Outline
• Part I (9:00-10:00)
- Preliminaries
- Paired and two-sample t-tests, confidence intervals
- One-way ANOVA and two-way ANOVA without replication
- Familywise error rate
[Coffee Break (10:00-10:30)]
• Part II (10:30-12:00)
- Tukey’s HSD test, simultaneous confidence intervals
- Randomisation test and randomised Tukey HSD test
- What’s wrong with statistical significance tests?
- Effect sizes, statistical power
- Topic set size design and power analysis
- Summary: how to report your results
IF YOU WANT MORE, PLEASE READ:
Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes,
and statistical power, to appear, Springer, 2018.

Which search engine is better?
(paired data)
0.4 0.4
0.8 0.6
0.7 0.5
Some evaluation
measure score
Sample size n = 3

Which search engine is better?
(unpaired data)
0.4
0.8
0.7
1.0
0.8
0.1
0.5
n1 = 3
n2 = 4

Statistical significance testing
[Robertson81, p.23]
“More particularly, having performed a comparison
of two systems on specific samples of documents
and requests, we may be interested in the statistical
significance of the difference, that is in whether the
difference we observe could be simply an accidental
property of the sample or can be assumed to
represent a genuine characteristic of the
populations.”
“Document collection as a sample” [Robertson12] not discussed in this tutorial.

Statistically significant result may not be
practically significant (and vice versa)
“It must nevertheless be admitted that the basis for
applying significance tests to retrieval results is not well
established, and it should also be noted that
statistically significant performance differences may be
too small to be of much operational interest.”
[SparckJones81, Chapter 12, p.243]
Karen Sparck Jones
1935-2007
Roger Needham
1935-2003

What do samples tell us about
population means?
Are they the same?

Parametric tests for comparing means
• In IR experiments, we often compare sample means to
guess if the population means are different.
• We often employ parametric tests (assume specific
population distributions, e.g., normal)
- paired and two-sample t-tests
(Are the m(=2) population means equal?)
- ANOVA (Are the m(>2) population means equal?)
- Tukey HSD test for m(m-1)/2 system pairs
scores
EXAMPLE
(paired data) n topics
m systems
Sample mean for a system

Null hypothesis, test statistic, p-value
• H0: tentative assumption that all
population means are equal
• test statistic: what you compute
from observed data – under H0,
this should obey a known
distribution (e.g. t-distribution)
• p-value: probability of observing
what you have observed (or
something more extreme)
assuming H0 is true
Null hypothesis
test statistic t0

Type I error and statistical power
Reject H0
if p-value <= α
test statistic t0
tinv(φ; α)
Can’t reject H0 Reject H0
H0 is true
systems are equivalent
Correct conclusion
(1-α)
Type I error
α
H0 is false
systems are different
Type II error
β
Correct conclusion
(1-β)
α/2 α/2
Statistical power: ability to
detect real differences

Type II error
Can’t reject H0
if p-value > α
test statistic t0
tinv(φ; α)
H0 is true
Correct conclusion
(1-α)
Type I error
α
H0 is false
Type II error
β
Correct conclusion
(1-β)
α/2 α/2

Cohen’s five-eighty convention
H0 is true
Correct conclusion
(1-α)
Type I error
α
H0 is false
Type II error
β
Correct conclusion
(1-β)
Statistical power:
ability to detect
real differencesCohen’s five-eighty convention:
α=5%, 1-β=80% (β=20%)
Type I errors 4 times as serious as Type II errors
The ratio may be set depending on specific situations

Population means and variances
x: a random variable
f(x): probability density function (pdf) of x
The expectation of any function g(x) is:
Population mean: E(x) i.e. expectation of g(x)=x
Population variance:
How does x move around the
population mean?

Law of large numbers
ANY distribution is okay!
If want a good estimate of the population mean,
just take a large sample and compute the sample mean.

Normal distribution
For given μ and σ (>0), the pdf of a normal
distribution is given by:
where
The distribution is denoted by .

Upper 100P% z-value
For any u ～
that satisfies
zinv(0.05)
5%

Properties of the normal distribution
[Sakai18book]

Central Limit Theorem [Sakai18book]
ANY distribution is okay!

Does CLT really hold? Test it with
uniform distributions (1)
n = 1

n = 2

n = 4

n = 8
Already looking quite like a
normal distribution
Variance getting
smaller

Sample variance
V is an unbiased estimator of the population variance
(just as is an unbiased estimator of the population
mean : )

Chi-square distribution
The distribution is denoted by
φ=5

t distribution
The distribution is denoted by t(φ).
φ=5

Two-sided 100P% t-value
For any t ～ t(φ)
that satisfies
2.5% 2.5%
tinv(5; 0.05)
qt returns one-sided
t-values

Basis of the paired t-test [Sakai18book]

t-distribution is like a standard
normal distribution with uncertainty
Compare this with:
population variance
sample variance

Basis of the two-sample t-test
[Sakai18book]
Pooled variance
homoscedasticity
(equal variance)

F distribution
The distribution is denoted by
φ1 =10 φ2 =20

Upper 100P% F-value
For any F ～ F(φ1 , φ2 )
that satisfies 5%
Finv(10, 20; 0.05)

Basis of ANOVA
See “Basis of the two-sample t-test

Historically, IR people were
hesitant about the t-test (1)
“since this normality is generally hard to prove for
statistics derived from a request-document
correlation process, the sign test probabilities may
provide a better indicator of system performance
than the t-test” [Salton+68] (p.15)

Historically, IR people were
hesitant about the t-test (2)
“Parametric tests are inappropriate because we do
not know the form of the underlying distribution”
[VanRijsbergen79] (p.136)
“Since the form of the population distributions
underlying the observed performance values is not
known, only weak tests can be applied; for example,
the sign test.” [SparckJones+97]

But actually the t-test is robust to
minor assumption violations
This assumes normality BUT
CLT says ANYTHING can be viewed as normally distributed once averaged over a large sample.
The robustness of the t-test can also be demonstrated using the randomisation test.

Paired t-test (1)
x1j : nDCG of System 1 for the j-th topic
x2j: nDCG of System 2 for the j-th topic
Assume that the scores are independent and that
Then for per-topic differences

Paired t-test (2)
Therefore (from Corollary 5)
where
Sample mean Sample variance

Paired t-test (3)
Two-sided test:
H0 : μ1 = μ2 H1 : μ1 ≠ μ2
Under H0 the following should hold:
Two-sided vs one-sided tests: See [Sakai18book] Ch.1

Paired t-test (4)
Under H0 , should hold.
So reject H0 iff
The difference is statistically significant
at the significance criterion α

http://www.f.waseda.jp/tetsuya/20topics3runs.mat.csv
n=20 topics
m=3 systems

Lazy paired t-test with Excel
p-value
two-sided test
paired t-test

Loading 20topics3runs.mat.csv to R
sample means

Paired t-test with R
Compare with the Excel case
Two-sided test

Two-sample t-test (1)
x1j : nDCG of System 1 for the j-th topic (n1 topics)
x2j: nDCG of System 2 for the j-th topic (n2 topics)
Assume that the scores are independent and that
Homoscedasticity (equal variance) assumption.
But the t-test is actually quite robust to the assumption
violation. For a discussion on Student’s and Welch’s
t-tests, see [Sakai16SIGIRshort, Sakai18book]

(From Corollary 6)
Pooled variance
Sample means

H0 : μ1 = μ2 H1 : μ1 ≠ μ2
Under H0 the following should hold:
So reject H0 iff

Lazy two-sample t-test with Excel
p-value
two-sided test
Student’s t-test
Two sets of nDCG scores
treated as unpaired data
⇒ a much larger p-value
In practice, if the scores are
paired, use a paired test.

Two-sample (Student’s) t-test with R
Two-sided test
Compare with the Excel case

Confidence intervals for the difference in
population means - paired data (1)
From the paired t-test,
therefore

Confidence intervals for the difference in
population means - paired data (2)
hence
where
100(1-α)% CI:
Margin
of error
Difference in population means

What does a 95% CI mean?
(a constant)
:
0.05
If you compute
100 different CIs
from 100
different
samples
(i.e. topic sets)
…
About 95 of
the CIs will
capture the
true difference

CIs for the difference in population means
- unpaired data
From the two-sample t-test,
we obtain the following in a similar manner:
where

Computing 95% CIs in practice
(paired data)
We had it already!
To compute
with Excel, use T.INV.2T(α, n-1)
with R: qt(α/2, n-1, lower.tail=FALSE)

Computing 95% CIs in practice
(unpaired data)
We had it already!
To compute
with Excel, use T.INV.2T(α, n1 + n2 - 2)
with R: qt(α/2, n1+n2-2, lower.tail=FALSE)

Analysis of Variance
• A typical question ANOVA addresses:
Given observed scores for m systems,
are the m population means all equal or not?
• ANOVA does NOT tell you which system means are
different from others.
• If you are interested in the difference between
every system pair (i.e. obtaining m(m-1)/2 p-values),
conduct an appropriate multiple comparison
procedure, e.g. Tukey HSD test. No need to do
ANOVA before Tukey HSD.

One-way ANOVA, equal group sizes (1)
• Data format:
• Basic assumption:
or
• Question: Are the m population means equal?
unpaired
data, but
equal
group sizes
(e.g. #topics)
homoscedasticity
Generalises the two-sample t-test, and can handle unequal
group sizes as well. See [Sakai18book]
population mean for System i

Let
Null hypothesis:
⇔
μ2 = μ3 = 0.2
μ = 0.3
μ1 = 0.5
a1 = 0.2
a2 = -0.1 a3 = -0.1
population grand mean i-th system effect
All population means are equal (to μ)
m=3

Let
Clearly,
sample grand mean System i’s sample mean
Diff between
an individual score
and the grand mean
can be broken down
into…
Diff between
the system mean
and the grand mean
and…
Diff between
the individual score
and the grand mean

Interestingly, this also holds:
Between-system
sum of squares
Within-system sum of squares
Total sum of squares (variations)

From a property of the chi-square distribution,
hence

As for SA , since
Corollary 1 gives us
Hence, under H0 ,
From a property of the chi-square distribution,
under H0.

Hence, by definition, under H0 ,
Under H0 :

Under H0 ,
so reject H0 iff
Conclusion: probably
not all population
means are equal

One-way ANOVA with R (1)
Here, just as an exercise, treat the matrix as if it’s unpaired
data (i.e., sample sizes equal but no common topic set)
The sample means (mean nDGG scores) suggest
System1 > System2 > System3.
But is the system effect statistically significant?

mat is a 20x3 topic-by-run matrix:
Let’s convert the format for convenience…

A 60x2 data.frame Gather all
columns of mat

• φA = m-1 = 3-1 = 2
• φE1 = m(n-1) = 3(20-1) = 57
The system effect
is statistically significant
at α = 0.05
p-value
The three systems are probably not all equally effective,
but we don’t know where the difference lies.

Two-way ANOVA without replication (1)
• Data format:
• Basic assumption:
i-th system effect
j-th topic effect
A common
topic set
for all m
systems
(paired data)

Clearly,
sample grand mean System i’s sample mean
Topic j’s sample mean

Similarly:
Between-topic
sum of squaresFrom one-way ANOVA

It can be shown that under H0,
so reject H0 iff
The system effect is
statistically significant at α
All population system
means are equal

If also interested in the topic effect, under H0
so reject H0 iff
The topic effect is
statistically significant at α
All population topic
means are equal

Two-way ANOVA without
replication with R (1)
Just inserting a column for topic IDs

Just converting the data format
Gather all
columns of mat
except Topic
A 60x3 data.frame

• φA = 3-1 = 2
• φB = 20-1 = 19
• φE1 = (3-1)*(20-1)= 38
The system effect
is statistically highly significant
(so is the topic effect)
The three systems are probably not all equally effective,
but we don’t know where the difference lies.

Interested in the differences for all system pairs.
So just repeat t-tests m(m-1)/2 times? (1)
The following is the same as repeating t.test with
paired=TRUE for every system pair...
Compare with the
Paired t-test with R slide
... but is NOT the right thing to do.

Interested in the differences for all system pairs.
So just repeat t-tests m(m-1)/2 times? (2)
The following is the same as repeating t.test with
var.equal=TRUE for every system pair...
Compare with the
Two-sample (Student’s) t-test with R slide
This means using Vp rather
than VE1 from one-way
ANOVA
... but is NOT the right thing to do.

Don’t repeat a regular t-test m(m-1)/2
times!
Why? Suppose a restaurant has a wine cellar. It is
known that one in every twenty bottles is sour.
Pick a bottle; the probability
that it is sour is
1/20 = 0.05
(Assume that we have an
infinite number of bottles)
VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN
VIN VIN VIN VIN VIN VIN VIN VIN VIN VIN

A customer orders one bottle
A bottle of red
please
The probability that
sour wine is served
to him is 0.05
VIN

A customer orders two bottles
Two bottles
please
sour wine is served
to him is
1-P(both bottles are good)
= 1- 0.95^2
= 0.0975
VIN VIN

A customer orders three bottles
Three bottles
please
sour wine is served
to him is
1-P(all bottles are good)
= 1- 0.95^3
= 0.1426
VIN VIN VIN

Comparisonwise vs Familywise error rate
(restaurant owner)
• The restaurant is worried not about the probability
of each bottle being sour, but about the probability
of accidentally serving sour wine to the customer
who orders k bottles. The latter probability should
be no larger than (say) 5%.
YOU SERVED ME SOUR WINE
I’M GONNA TWEET ABOUT IT

Comparisonwise vs Familywise error rate
(researcher)
• We should be worried not about the
comparisonwise Type I error rate, but about the
familywise error rate – the probability of making at
least one Type I error among the k=m(m-1)/2 tests.
• Just repeating a t-test k times gives us a familywise
error rate of 1-(1-α)^k if the tests are independent.
e.g. α=0.05, k=10 ⇒ familywise error rate = 40%!

Multiple comparison procedures
[Carterette12][Nagata+97][Sakai18book]
• Make sure that the familywise error rate is no more
than α.
• Stepwise methods: outcome of one hypothesis test
determines what to do next
• Single step methods: test all hypotheses at the
same time – we discuss these only.
- Bonferroni correction (considered obsolete)
- Tukey’s Honestly Significant Difference (HSD) test
- others (e.g. those available in pairwise.t.test)

Bonferroni correction
[Crawley15 ](pp.17-18)
“The old fashioned approach was to use
Bonferroni’s correction: in looking up a
value for Student’s t, you divide your α value
by the number of comparisons you have
done. […] Bonferroni’s correction is very
harsh and will often throw out the baby
with the bathwater. […] The modern
approach is […] to use the wonderfully
named Tukey’s honestly significant
differences”
Or, equivalently,
multiply each p-value
by k

• Instead of conducting a t-test k = m(m-1)/2 times,
consider the maximum difference (best system –
worst system) among the k differences.
• The distribution that the max difference obeys is
called a studentised range distribution. Its upper
100P% value is denoted by
• We compare the k differences against the above
distribution. By construction, if the maximum is not
statistically significant, the other differences are not
statistically significant either. Hence the familywise
error rate can be controlled.
How Tukey HSD works
qtukey(P, m, φ, lower.tail=FALSE) in R

Tukey HSD with equal group sizes (1)
Data structure: same as one-way ANOVA with equal
group sizes
Tukey HSD can handle unequal group sizes as well. See [Sakai18book]
sample mean
for System i
unpaired data

Null hypothesis :
the population means for
systems i and i’ are equal
Test statistic:
Reject iff
Tukey HSD with equal group sizes (2)

R: Tukey HSD with equal group sizes
The data.frame we made for
one-way ANOVA
Only the diff between Systems 1
and 3 statistically significant at
α=0.05

Tukey HSD with paired observations (1)
Data structure: same as two-way ANOVA without
replication
sample mean
for System i
sample mean
for Topic j
paired data

Null hypothesis :
the population means for
systems i and i’ are equal
Test statistic:
Reject iff
Tukey HSD with paired observations (2)

R: Tukey HSD with paired observations
The data.frame we made for two-way
ANOVA without replication
The difference between Systems 1 and 3
and that between Systems 2 and 3 are
statistically highly significant

If you want a 95%CI for the diff between
every system pair…
• If you use the t-test-based MOE for every system
pair, this approach has the same problem as
repeating t-tests multiple times.
• Use a Tukey-based MOE instead to construct a
simultaneous CI – to capture all true means at the
same time, not individually.

Computing the MOE for simultaneous CIs
From Tukey HSD with equal group sizes (unpaired data)
From Tukey HSD with paired observations
Apply the above MOE to each of the k differences

R: Simultaneous 95%CI,
equal group sizes (unpaired data)
MOE = 0.081314
Upper limit = diff + MOELower limit = diff - MOE

R: Simultaneous 95%CI,
paired observations
MOE = 0.033342
(CIs are narrower than the unpaired case)
Upper limit = diff + MOELower limit = diff - MOE

Computer-based tests
• Unlike classical significance tests, do not require
assumptions about the underlying distribution
• Bootstrap test [Sakai06SIGIR][Savoy97] – assumes
the observed data are a random sample from the
population. Samples with replacement from the
observed data.
• Randomisation test [Smucker+07] – no random
sampling assumption. Permutes the observed data.

Randomisation test for paired data (1)
Suppose we have an nDCG matrix for
two systems with n topics.
Are these systems equally effective?

two systems with n topics.
Let’s assume there is a single hidden
system. For each topic, it generates
two nDCG scores. They are randomly
assigned to the two systems.

If H0 is right, these alternative matrices
(obtained by randomly permuting each
row of U) could also have occurred

There are 2 possible matrices, but
let’s just consider B of them
(e.g. 10000 trials)
n

How likely is the observed difference (or something
even more extreme) under H0? → p-value

Randomisation test for paired data -
pseudocode
The exact p-value changes slightly depending on B.

Random-test in Discpower [Sakai14PROMISE]
http://research.nii.ac.jp/ntcir/tools/discpower-en.html
Contains a tool for
conducting a
randomisation test or
randomised Tukey HSD
test

http://www.f.waseda.jp/tetsuya/20topics2runs.scorematrix
Mean difference and
p-value (compare
with paired t-test)
Paired randomisation test, B=5000 trials
A 20x2 matrix,
white-space-separated

Randomised Tukey HSD test for paired data (1)
more than two systems with n topics.

more than two systems with n topics.
Which system pairs are really
different?
Let’s assume there is a single hidden
system. For each topic, it generates m
nDCG scores. They are randomly
assigned to the m systems.

If H0 is right, these alternative matrices
(obtained by randomly permuting each
row of U) could also have occurred

There are (m!) possible matrices, but
let’s just consider B of them
(e.g. 10000 trials)
n

How likely are the observed differences
given the null distribution of the maximum differences?
→ Tukey HSD p-value

Randomised Tukey HSD – pseudocode
(adapted from [Carterette12])
The exact p-value changes slightly depending on B.

http://www.f.waseda.jp/tetsuya/20topics3runs.scorematrix
Randomised Tukey HSD test, B=5000 trials
Compare the p-values with
those of the Tukey HSD test
with R (paired data)
A 20x3 matrix,
white-space-separated

[Bakan66]
“The test of significance does not provide the
information concerning psychological
phenomena characteristically attributed to it;
and a great deal of mischief has been
associated with its use.”

[Deming75]
“Little advancement in the teaching of
statistics is possible, and little hope for
statistical methods to be useful in the frightful
problems that face man today, until the
literature and classroom be rid of terms so
deadening to scientific enquiry as null
hypothesis, population (in place of frame),
true value, level of significance for comparison
of treatments, representative sample.”

[Loftus91]
“Despite the stranglehold that hypothesis
testing has on experimental psychology, I find
it difficult to imagine a less insightful means of
transiting from data to conclusions.”

[Cohen94] (1)
“And we, as teachers, consultants, authors,
and otherwise perpetrators of quantitative
methods, are responsible for the ritualization
of null hypothesis significance testing (NHST; I
resisted the temptation to call it statistical
hypothesis inference testing) to the point of
meaninglessness and beyond. I argue herein
that NHST has not only failed to support the
advances of psychology as a science but also
has seriously impeded it.”

[Cohen94] (2)
“What’s wrong with NHST? Well, among many other
things, it does not tell us what we want to know, and
we so much want to know what we want to know
that, out of desperation, we nevertheless believe
that it does! What we want to know is “Given these
data, what is the probability that H0 is true?” But as
most of us know, what it tells us is “Given that H0 is
true, what is the probability of these (or more
extreme) data?””
p-value = Pr(D+| H)
not Pr(H|D)!
See also
[Carterette15]
[Sakai17SIGIR]

[Rothman98]
“When writing for Epidemiology, you can also
enhance your prospects if you omit tests of statistical
significance. Despite a wide spread belief that many
journals require significance tests for publication, […]
every worthwhile journal will accept papers that
omit them entirely. In Epidemiology, we do not
publish them at all. Not only do we eschew
publishing claims of the presence or absence of
statistical significance, we discourage the use of this
type of thinking in the data analysis [….]”

[Ziliak+08]
“Statistical significance is surely not the only
error in modern science, although it has been,
as we will show, an exceptionally damaging
one.”
“Most important is to minimise Error of the
Third Kind, “the error of undue inattention,”
which is caused by trying to solve a scientific
problem using statistical significance or
insignificance only.”

[Harlow+16]
“The main opposition to NHST then and now
is the tendency for researchers to narrowly
focus on making a dichotomous decision to
retain or reject a null hypothesis, which is
usually not very informative to current or
future research[…] Although there is still not a
universally agreed upon set of practices
regarding statistical inference, there does
seem to be more consistency in agreeing on
the need to move away from an exclusive
focus on NHST […]”

Statistical significance: Problem 1
Many misinterpret and/or misuse significance tests.
American Statistical Association statement (March
2016):
• P-values do not measure the probability that the
studied hypothesis is true, or the probability that
the data were produced by random chance alone.
• A p-value, or statistical significance, does not
measure the size of an effect or the importance of a
result.
p-value = Pr(D+|H)
not Pr(H|D)!

Dichotomous thinking: statistically significant or not?
(More important questions:
- How much is the difference?
- What does a difference of that magnitude mean to us?)
p=0.049 ⇒ statistically significant at α=0.05
p=0.051 ⇒ NOT statistically significant at α=0.05
Reporting the exact p-value is more informative than
saying “p<0.05” BUT THIS IS STILL NOT SUFFICIENT.

P-value = f(sample_size, effect_size)
- A large effect size ⇒ a small p-value
- A large sample size ⇒ a small p-value
For example, consider:
From the paired
t-test
Magnitude of the difference
A large effect size (standardised mean difference)
⇒ a large t-value ⇒ a small p-value
A large sample size (topic set size)
⇒ a large t-value ⇒ a small p-value
Anything can be made
statistically significant
by making n large enough!

Effect size definition [Cohen88]
“it is convenient to use the phrase “effect size”
to mean “the degree to which the
phenomenon is present in the population,” or
“the degree to which the null hypothesis is
false.” Whatever the manner of
representation of a phenomenon in a
particular research in the present treatment,
the null hypothesis always means that the
effect size is zero.”

Effect size definition [Olejnik+03]
“An effect-size measure is a standardized index
and estimates a parameter that is
independent of sample size and quantifies the
magnitude of the difference between
populations or the relationship between
explanatory and response variable.”

Effect size definition [Kelley+12]
“Effect size is defined as a quantitative
reflection of the magnitude of some
phenomenon that is used for the
purpose of addressing a question of
interest.”

Various effect sizes
• Effect sizes for t-tests (and Tukey HSD):
standardized mean differences (covered in
this lecture)
• Effect sizes for ANOVAs: contribution to the
overall variance: see
[Olejnik+03][Sakai18book] etc. for details
• Other forms of effect sizes: see [Cohen88]
etc.

Paired t-test effect size
Standardised mean difference (diff measured in standard deviation units)
Note that
sample size effect size
Reporting dpaired along with the p-value is more informative than just
reporting the p-value. But dpaired uses the standard deviation of the
differences – works only with the paired t-test.

Hedge’s g (1)
Standardised mean difference for the two-sample
case:
Hedge’s g estimates the above:
Common standard
deviation
Pooled variance

Hedge’s g (2)
From Student’s t-test,
so, since ,
note that
Reporting Hedge’s g along with the p-value is more informative than
just reporting the p-value. See [Sakai18book] for bias correction.
See [Sakai18book] for Cohen’s d

Glass’s Δ (my favourite)
• No homoscedasticity assumption!
• Works for both paired and unpaired data!
• A bias-corrected Δ:
“How much is the difference, when measured in standard deviation units
computed from the control group (i.e., baseline) data that we are familiar with?”

Effect sizes for (randomised)
Tukey HSD
If homoscedasticity is assumed (as in classical Tukey
HSD):
If not, and if there is a common baseline, use Glass’ Δ
by using the standard deviation of that baseline.
Bottom line: report effect sizes as standardised mean differences,
along with p-values.
See “One-way ANOVA,
equal group sizes (8)”
See “Two-way ANOVA
without replication (4)”

Statistical power: ability to detect
real differences
Given α and an effect size that you are interested in
(e.g. standardized mean difference >=0.2),
increasing the sample size n improves statistical power (1-β).
- An overpowered experiment: n larger than necessary
- An underpowered experiment: n smaller than necessary
(cannot detect real differences – a waste of research effort!)
H0 is true
Correct conclusion
(1-α)
Type I error
α
H0 is false
Type II error
β
Correct conclusion
(1-β)

http://sigir.org/files/museum/pub-14/pub_14.pdf
[SparckJones+75]

On TREC topic set sizes [Voorhees09]
“Fifty-topic sets are clearly too small to have
confidence intervals in a conclusion when
using a measure as unstable as P(10). Even for
stable measures, researchers should remain
skeptical of conclusions demonstrated on only
a single test collection.”
TREC 2007 Million Query track [Allan+08] had “sparsely-judged”
1,800 topics, but this was an exception…

Deciding on the number of topics to
create based on statistical
requirements
• Desired statistical power [Webber+08][Sakai16IRJ]
• A cap on the confidence interval width for the mean
difference [Sakai16IRJ]
• Sakai’s Excel tools based on [Nagata03]:
samplesizeTTEST2.xlsx (paired t-test power)
samplesize2SAMPLET.xlsx (two-sample t-test power)
samplesizeANOVA2.xlsx (one-way ANOVA power)
samplesizeCI2.xlsx (paired data CI width)
samplesize2SAMPLECI (two-sample CI width)

Recommendations on topic set
size design tools
• If you’re interested in the statistical power of the
paired t-test, two-sample t-test, or one-way ANOVA,
use samplesizeANOVA2.
• If you’re interested in the CI width of the mean
difference for paired or two-sample data, use
samplesize2SAMPLECI.
• … unless you have an accurate estimate of the
population variance of the score differences
which the paired-data tools
require.
See “Paired t-test (1)”
samplesizeTTEST2
samplesizeCI2

samplesizeANOVA2 input
α: Type I Error probability
β: Type II Error probability, i.e., you want 100(1-β)%
power (see below)
m: number of systems to be compared in one-way
ANOVA
minD: minimum detectable range, i.e.,
whenever the true difference D between
the best and the worst systems is minD
or larger, you want to guarantee 100(1-β)% power
: variance estimate for a particular evaluation
measure (under the homoscedasticity assumption)
μbest
μworst
D
m system means

samplesizeANOVA2:
“alpha=.05, beta=.20” sheet (1)
Enter values in the orange cells (α=5%, β=20%):
m=10, minD=0.1, =0.1
To ensure 80% power (at α=5%) for one-way ANOVA
with any m=10 systems with a minimum detectable
range of 0.1 in terms of a measure whose variance is
0.1, we need n=312 topics. μbest
μworst
D>= 0.1
m system means

samplesizeANOVA2:
Enter values in the orange cells (α=5%, β=20%):
m=2, minD=0.1, =0.1
To ensure 80% power (at α=5%) for one-way ANOVA
with any m=2 systems with a minimum detectable
difference of 0.1 in terms of a measure whose
variance is 0.1, we need n=154 topics. μbest
μworst
D>= 0.1 Two system means

samplesizeANOVA2:
Since one-way ANOVA with m=2 systems is strictly
equivalent to the two-sample t-test [Sakai18book],
To ensure 80% power (at α=5%) for the two-sample
t-test with a minimum detectable difference of 0.1 in
terms of a measure whose variance is 0.1, we need
n=154 topics. μbest
μworst
D>= 0.1 Two system meansThis n can also be regarded as a pessimistic estimate
for the paired data case.

samplesizeANOVA2: how does it work? (1)
[Nagata03][Sakai16IRJ][Sakai18book]
Remember what we do in one-way ANOVA:
we reject H0 (all m means are equal)
iff See “One-way ANOVA,
equal group sizes (8)”

So, whether H0 is true or not, the probability of
rejecting H0 is:
If H0 is true, then F0 ～ F(φA, φE1)
and the above = α
Probability of incorrectly
concluding that the system
means are different

So, whether H0 is true or not, the probability of
rejecting H0 is:
If H0 is false, then F0 ～ F’(φA, φE1, λ)
and the above = 1-β
statistical power: probability of
correctly concluding that the two
system means are different
A noncentral F
distribution with a
noncentrality
parameter λ
Accumulates squared
system effects

If H0 is false, the power can be approximated as:
where
Let’s call it
Formula P

Let’s ensure 100(1-β)% power
whenever D >= minD (e.g., 0.1 in
mean nDCG). To do this, we define:
so that Δ >= minΔ holds.
μbest
μworst
D
m system means
minΔ is the worst-case Δ for topic set sizes

• λ=nΔ so the worst-case topic set size n can be
estimated very roughly as:
where λ can be approximated using a linear function
of φA for given (α, β).
• Having thus obtained an n, check with Formula P to
see if the desired power is really achieved. If not,
n++. If the power is too high, n--, etc.
• The excel tool does just that.

samplesize2SAMPLECI input
α: Type I Error probability for
100(1-α)% CIs
δ: cap on the CI width for the
difference between two systems
(two-sample data). That is, you
want the width of any CI to be δ
or smaller.
: variance estimate for a
particular evaluation measure
(under the homoscedasticity
assumption)
The n returned by samplesize2SAMPLECI can also be
regarded as a pessimistic estimate for the paired data case.
(a constant)
:
Width of this CI

Enter values in the orange cells: α=5%, δ=0.1, =0.1
To ensure the CI width of any between-system difference
to be 0.1 or smaller, we need n=309 topics.
samplesize2SAMPLECI:
“approximation” sheet

samplesize2SAMPLECI: how does it work? (1)
[Nagata03][Sakai16IRJ][Sakai18book]
From “CIs for the difference in population means –
unpaird data” we have:
and the CI width is twice the MOE.
We want 2MOE <= δ, but since Vp is a random
variable, we use E(Vp) instead:

Consider a balanced design, n = n1 = n2 . Then the
above can be approximated as:
Let’s call it Inequality C

To obtain an initial estimate of n that satisfies the
above, consider the CI for an ideal case where σ is
known:
cf.
Inequality C
Remember, the t-
distribution is like a
standard normal
distribution with
uncertainty

Hence
• This gives us an optimistic estimate of n,
so check with Inequality C. If the condition is not
satisfied, n++, etc.
• The excel tool does just that.

Estimating the common variance
If you have a topic-by-system score matrix or two
from some pilot data,
an unbiased estimator can be
obtained as:
Pooled estimate
Estimate from one matrix See “Two-way ANOVA without replication (4)”
A score matrix from
test collection C

Some real estimates based on TREC
data (using VE1 rather than VE2)
See “One-way ANOVA, equal group sizes (8)”
Some measures are less stable ⇒ require larger topic set sizes under the same requirement

Some topic set size design results
[Sakai18book]
The paired t-test tool does not return tight estimates due to my crude estimate
(covariance not considered)

So, to build a test collection…
1. Build a small data set first (or borrow one from a past
task similar to your own).
2. Decide on a primary evaluation measure, and create
a small topic-by-system score matrix with the small
data set.
3. Compute as VE1 or VE2 and use a topic set size
design tool to decide on n.
4. You can advertise your test collection as follows:
“We created 70 topics, which, according to topic set size
design with = 0.044, is more than sufficient for
achieving 80% power with a (paired) t-test whenever the
true difference in Mean nDCG@10 is 0.10 or larger.”
See previous two slides

Power analysis with R scripts
[Sakai16SIGIR] (adapted from [Toyoda09])
• Given an adequately reported significance test
result in a paper,
- compute the effect size and the achieved power in
that experiment.
- propose a new sample size to achieve a desired
power.
Relies on the pwr library of R

The five R power analysis scripts
[Sakai16SIGIR]
• future.sample.paired (for paired t-tests)
• future.sample.unpairedt (for two-sample t-tests)
• future.sample.1wayanova (for one-way ANOVAs)
• future.sample.2waynorep (for two-way ANOVAs
without replication)
• future.sample.2wayanova2 (for two-way ANOVAs)

future.sample.pairedt
Basically just enter t0 and the actual sample size
OUTPUT:
- Effect size dpaired
- Achieved power of the experiment
- future sample size for achieving 80% power

future.sample.pairedt: an actual
example from a SIGIR paper
A highly underpowered experiment.
In future, use 244 topics, not 28, to achieve 80%
power for this small effect (dpaired = 0.18).
Only 15% power!
Underpowered experiments can be a waste of research effort:
there’s a high chance that you will miss a true difference!
about 85%

future.sample.2waynorep
Basically just enter F0, the number of systems m
and the actual sample size
OUTPUT:
- A partial effect size [Sakai18book]
- Achieved power of the experiment
- future sample size for achieving 80% power

future.sample.2waynorep: an actual
example from a SIGIR paper
A highly underpowered experiment.
In future, use 75 topics, not 17, to achieve 80%
power for this small effect.
Only 18% power!
Underpowered experiments can be a waste of research effort:
there’s a high chance that you will miss a true difference!
about 82%

Underpowered/Overpowered
experiments: t-tests

Underpowered/Overpowered
experiments: ANOVAs

Reporting a paired t-test result (1)
Clearly state the sample
size within the table!
Clearly indicate what the
numbers are! Which
evaluation measure?
Are they mean values?

Reporting a paired t-test result (2)
“We conducted a paired t-test for the difference
between our proposed system and the baseline in
terms of mean nDCG over n=20 topics. The
difference is not statistically significant (t(19) =
1.3101, p=0.2058, 95%CI[-0.0134, 0.0581]), with the
effect size dpaired = t0/√n = 0.2929.”
Provide as much information as
possible for future researchers.
There are several types of effect
sizes. Clarify which one you are
talking about.

Reporting a two-sample t-test result (1)
Clearly indicate what the
numbers are! Which
evaluation measure?
Are they mean values?
Clearly state the sample
sizes within the table!

Reporting a two-sample t-test result (2)
“We conducted a Student’s t-test for the difference
between the our proposed system (sample size n1 =
20) and the baseline (sample size n2 = 20) in terms of
mean nDCG. The difference is not statistically
significant (t(38) = 0.6338, p=0.53,
95%CI[-0.0491, 0.0938]), with Hedge’s g = 0.2004.”
Note that
If you used Welch’s t-test [Sakai16SIGIRshort], you consciously avoided the
homoscedasticity assumption, so I recommend using Glass’ Δ, not Hedge’s d.

Reporting a (randomised) Tukey
HSD result (1)

Reporting a (randomised) Tukey
HSD result (2)
“We conducted a randomised Tukey HSD test with
B=5,000 trials to compare every system pair […] It
can be observed that System 3 statistically
significantly underperforms Systems 1 (p≒0.0000)
and 2 (p=0.0024). Moreover, we computed an effect
size ESE2 [Sakai18book] for each system pair. It can be
observed that Systems 1 and 3 are almost two
standard deviations apart from each other.”
Also visualise each system’s performance using graphs with error bars,
or boxplots! State clearly what the error bars represent (e.g., 95% CIs)!

Summary
• To design a test collection, use some pilot data to
estimate the variance of a particular evaluation
measure for sample size considerations.
• To design an experiment, use a pilot or existing
study for sample size considerations to ensure
sufficient statistical power. Underpowered
experiments can be a waste of research effort.
• When reporting on statistical significance test
results, report the sample sizes, test statistics,
degrees of freedom, p-values (not the stars
*/**/***), and effect sizes!
Despite what R outputs, it is wrong to decide on α AFTER looking at the results
(e.g. “this difference is statistically significant at α=1% while that is significant at α=5%”)

Thank you for staying
until the end!

References: A-B
[Allan+08] Allan, J., Carterette, B., Aslam, J.A., Pavlu,
V., Dachev, B., and Kanoulas, E.: Million Query Track
2007 Overview, Proceedings of TREC 2007, 2008.
[Bakan66] Bakan, D.: The Test of Significance in
Psychological Research, Psychological Bulletin, 66(6),
pp.423-437, 1966.

References: C-D
[Carterette12] Carterette, B.: Multiple Testing in Statistical
Analysis of Systems-based Information Retrieval Experiments,
ACM TOIS 30(1), 2012.
[Carterette15] Carterette, B.: Bayesian Inference for
Information Retrieval Evaluation, ACM ICTIR 2015, pp.31-40,
2015.
[Cohen88] Cohen, J.: Statistical Power Analysis for the
Behavioral Sciences (Second Edition), Psychology Press, 1988.
[Cohen94] Cohen, J.: The Earth is Round (p<.05), American
Psychologist, 49(12), pp.997-1003, 1994.
[Crawley15] Crawly, M.J.: Statistics: An Introduction Using R
(Second Edition), Wiley, 2015.
[Deming75] Deming, W.E., On Probability as a Basic for Action,
American Statistician, 29(4), pp.146-152, 1975.

References: G-H
[Good05] Good, P.: Permutation, Parametric, and
Bootstrap Tests of Hypothesis (Third Edition),
Springer, 2005.
[Harlow+16] Harlow, L.L., Mulaik, S.A., and Steiger,
J.H.: What If There Were No Significance Tests?
(Classic Edition), Routledge, 2016.

References: K-L
[Kelley+12] Kelly, K. and Preacher, K.J.: On Effect Size,
Psychological Methods, 17(2), pp.137-152, 2012.
[Loftus91] On the Tyranny of Hypothesis Testing in
the Social Sciences, Contemporary Psychology, 36(2),
pp.102-105, 1991.

References: N-O
[Nagata+97] Nagata, Y. and Yoshida, M.: Introduction
to Multiple Comparison Procedures (in Japanese),
Scientist press, 1997.
[Nagata03] Nagata, Y.: How to Design the Sample Size
(in Japanese), Asakura Shoten, 2003.
[Olejnik+03] Olejnik, S. and Algina, J.: Generalized Eta
and Omega Squared Statistics: Measures of Effect
Size for Some Common Research Designs,
Psychological Research, 8(4), pp.434-447, 2003.

References: R
[Robertson81] Robertson, S.E.: The Methodology of
Information Retrieval Experiment, In Sparck Jones, K.
(ed.): Information Retrieval Experiment, Chapter 1,
Butterworths, 1981.
[Robertson12] Robertson, S.E., Kanoulas, E.: On Per-
topic Variance in IR Evaluation, ACM SIGIR 2012,
pp.891-900, 2012.
[Rothman98] Rothman, K.J.: Writing for Epidemiology,
Epidemiology, 9(3), pp.333-337, 1998.

References: S
[Sakai06SIGIR] Sakai, T.: Evaluating Evaluation Metrics based on the Bootstrap, ACM SIGIR 2006,
pp.525-532, 2006.
[Sakai14PROMISE] Sakai, T.: Metrics, Statistics, Tests, PROMISE Winter School 2013: Bridging
between Information Retrieval and Databases (LNCS 8173), Springer, 2014.
[Sakai16IRJ] Topic Set Size Design, Information Retrieval, 19(3), pp.256-283, 2016.
[Sakai16SIGIR] Sakai, T.: Statistical Significance, Power, and Sample Sizes: A Systematic Review of
SIGIR and TOIS, 2006-2015, Proceedings of ACM SIGIR 2016, pp.5-14, 2016.
[Sakai16SIGIRshort] Sakai, T.: Two Sample T-tests for IR Evaluation: Student or Welch?, Proceedings
of ACM SIGIR 2016, pp.1045-1048, 2016.
[Sakai17SIGIR] Sakai, T.: The Probability That Your Hypothesis Is Correct, Credible Intervals, and
Effect Sizes for IR Evaluation, Proceedings of ACM SIGIR 2017, pp.25-34, 2017.
[Sakai18book] Sakai, T.: Laboratory Experiments in Information Retrieval: Sample Sizes, Effect Sizes,
and statistical power, to appear, Springer, 2018.
[Salton+68] Salton, G. and Lesk, M.E.: Computer Evaluation of indexing and text processing, Journal
of the ACM, 15, pp.8-36, 1968.

References: S (continued)
[Savoy97] Savoy, J.: Statistical Inference in Retrieval Effectiveness
Evaluation, Information Processing and Management, 33(4),
pp.495-512, 1997.
[Smucker+07] Smucker, M.D., Allan, J. and Carterette, B.: A
Comparison of Statistical Significance Tests for Information Retrieval
Evaluation, CIKM 2007, pp.623-632, 2007.
[SparckJones+75] Sparck Jones, K. and van Rijsbergen, C.J.: Report
on the Need for and Provision of an ‘Ideal’ Information Retrieval
Test Collection, Computer Laboratory, University of Cambridge,
British Library Research and Development Report No.5266, 1975.
[SparckJones81] Sparck Jones, K. (ed.): Information Retrieval
Experiment, Butterworths, 1981.
[SparckJones+97] Sparck Jones, K. and Willet, P.: Readings in
Information Retrieval, Morgan Kaufmann, 1997.

References: T
[Toyoda09] Toyoda, H.: Introduction to Statistical
Power Analysis: A Tutorial with R (in Japanese), Toyo
Tosyo, 2009.

References: V-W
[VanRijsbergen79] Van Rijsbergen, C.J., Information
Retrieval, Chapter 7, Butterworths, 1979.
[Voorhees09] Voorhees, E.M.: Topic Set Size Redux,
Proceedings of ACM SIGIR 2009, pp.806-807, 2009.
[Webber+08] Webber, W., Moffat, A., and Zobel, J.:
Statistical Power in Retrieval Experimentation,
Proceedings of CIKM 2008, pp.571-580, 2008.

References: Z
[Ziliak+08] The Cult of Statistical Significance: How
the Standard Error Costs Us Jobs, Justice, and Lives,
The University of Michigan Press, 2008.

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