ICTIR2016tutorial

Topic Set Size Design and
Power Analysis in Practice
Tetsuya Sakai
@tetsuyasakai
tetsuyasakai@acm.org
Waseda University
ICTIR 2016 Tutorial: September 13, 2016, Delaware.

This half-day tutorial will teach you
• How to determine the number of topics when building
a new test collection (prerequisite: you already have
some pilot data from which you can construct a topic-
by-run score matrix). You will kind of know how it
works.
• How to check whether a reported experiment is
overpowered/underpowered and decide on a better
sample size for a future experiment.

Before attending the tutorial, please
download on your laptop
- Sample topic-by-run matrix:
https://waseda.box.com/20topics3runs
- Excel topic set size design tools:
http://www.f.waseda.jp/tetsuya/CIKM2014/samplesizeTTEST.xlsx
http://www.f.waseda.jp/tetsuya/CIKM2014/samplesizeANOVA.xlsx
http://www.f.waseda.jp/tetsuya/FIT2014/samplesizeCI.xlsx
[OPTIONAL]
- (Install R first and then) R scripts for power analysis:
https://waseda.box.com/SIGIR2016PACK

TUTORIAL OUTLINE
1. Significance testing basics and limitations
1.1 Preliminaries
1.2 How the t-test works
1.3 T-test with Excel and R (hands-on)
1.4 How ANOVA works
1.5 ANOVA with Excel and R (hands-on)
1.6 What's wrong with significance tests?
1.7 Significance tests in the IR literature, or lack thereof
2. Using the Excel topic set size design tools
2.1 Topic set sizes in IR
2.2 Topic set size design
<30min coffee break>
2.3 With paired t-tests (hands-on)
2.4 With one-way ANOVA (hands-on)
2.5 With confidence intervals (hands-on)
2.6 Estimating the variance (hands-on)
2.7 How much pilot data do we need?
3. Using the R power analysis scripts
3.1 Power analysis
3.2 With paired t-tests (hands-on)
3.3 With unpaired t-tests (hands-on)
3.4 With one-way ANOVA (hands-on)
3.5 With two-way ANOVA without replication (hands-on)
3.6 With two-way ANOVA (hands-on)
3.7 Overpowered and underpowered experiments in IR
4. Summary, a few additional remarks, and Q&A
30min
70min
20min
50min
10min
Appendix

1.1 Preliminaries (1)
• 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 with parameters)
- paired and unpaired t-tests (comparing m=2 means)
- ANOVA (comparing m (>2) means)
one-way, two-way, two-way without replication
Are the two population
means equal?
Are the m population
means equal?
scores
EXAMPLE
n topics
m systems
Sample mean for a system

• 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

Reject H0
if p-value <= α
test statistic t0 t(φ; α)
Accept 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

Accept H0 Reject H0
H0 is true
Correct conclusion
(1-α)
Type I error
α
H0 is false
Type II error
β
Correct conclusion
(1-β)
Statistical power:
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

For a continuous random variable x and its probability density function
f(x), the expectation of a function g(x) (including g(x)=x) is given by:
How likely x will take a
particular value
Population mean
Population variance
Population standard
deviation
The central position of x as
it is observed an infinite
number of times
How x varies from
the population
mean

A normal distribution with
population parameters
is denoted by .
Properties of a normal distribution
:
Probability density function
of a normal distribution
μ = 100
σ = 20

If x obeys , then
obeys .
Standardisation
Population
mean: 0
Population
standard
deviation: 1
Standard normal distribution

For random variables x, y, a function that satisfies the following is called
a joint probability density function:
Whereas, marginal probability density functions are defined as:
If the following holds for any (x,y), x and y are said to be independent.

If are independent and obey
then obeys
Reproductive property:
Adding normally distributed variables
still gives you a normal distribution
Population mean Population variance

If are independent and obey
then obeys
obeys and therefore
obeys .
Corollary: If we let ai = 1/n, μi = μ, σi = σ ...
Sample mean

Sample mean
Sum of squares
Sample variance
Sample standard deviation
If are independent
and obey , then
holds.
Sample variance V is an unbiased
estimator of the population variance
cf. 2.5 (3):
s is NOT an unbiased estimator of the population standard deviation

If are independent and
then:
• Law of large numbers
As n approaches infinity, approaches .
• Central Limit Theorem
Provided that n is large, the distribution of
can be approximated by .
It’s a good thing to observe
lots of data to estimate the
population mean.
If you have lots of observations, then the sample mean can be regarded as normally
distributed even if we don’t know much about individual random variables {xi}
Not necessarily normal

If are independent and obey then
the probability distribution that the following random variable obeys is
called a chi-square distribution with φ = k degrees of freedom:
The pdf of the above distribution is given by:
Gamma
function
Denoted by

If obeys then .
If are independent and obey then:
(a) obeys .
(b) and are independent.
(c) obeys .
Corollary from previous slide since
(xi – μ)/σ obeys
[Nagata03] p.57
[Nagata03] p.58

If and they are independent,
called a t distribution with φ degrees of freedom, denoted by t(φ).
IMPORTANT PROPERTY:
If and are independent, then:
obeys
Sample mean and
sample variance
as defined in 1.1 (11)

If and they are independent,
called an F distribution with degrees of freedom,
denoted by .
IMPORTANT PROPERTY:
If
and they are all independent, then:

1.2 How the t-test works (1) paired t-test
What does this
sample tell us about
the populations?

Comparing Systems X and Y with n topics with
(say) Mean nDCG over n topics
ASSUMPTIONS:
are independent and obey
are independent and obey .
Under these assumptions:
In Slide 1.1 (9), let a1 = 1, a2 = -1.

⇒
⇒
⇒
is an unbiased estimator of :
t distribution with n-1 degrees of
freedom, which is basically like the
standard normal distribution
(See also 1.1 (15))
1.1 (10)
1.1 (7)
We don’t know the
population variance so use a
sample variance instead.
1.1 (11)

Since under our assumptions,
if we further assume , then
.
Hypotheses:
Same population means: X and Y are equally effective
Two-sided test
0
test statistic t0

Hypotheses:
test statistic t0critical t value t(n-1; α)
α/2 α/2
Under , .
0
So if ,
something highly unlikely has happened.
We assumed but that must have been
wrong! Reject !
is probably true,
with 100(1-α)% confidence.
α: significance criterion

α/2 α/2
0
Using Excel to do a t-test:
- Reject if = TINV(α, n-1) = T.INV.2T(α, n-1).
- P-value = TDIST(|t0|, n-1, 2) = T.DIST.2T(|t0|, n-1).
Blue areas under the curve:
probability of observing the
data at hand or something
more extreme, if H0 is true

1.2 How the t-test works (7) confidence intervals
From 1.2 (3),
⇒
critical t value t(n-1; α)
α/2 α/2
0
t obeys t(n-1)

1.2 How the t-test works (8) confidence intervals
From 1.2 (3),
⇒
⇒
where .
So 95% CI for the difference in means is given by:
Margin of
Eerror
Different samples yield different CIs. 95% of the CIs will capture the true difference in means.

1.2 How the t-test works (9) unpaired t-test

Comparing Systems X and Y, based on a sample of size n1 for X and
another sample of size n2 for Y.
ASSUMPTIONS: the above observations are all independent and
and furthermore
Homoscedasticity (equal variance)
but the t-test is quite robust to the
assumption violation [Sakai16SIGIRshort]
cf. 1.2 (15)

Under the assumptions, it is known that
where
Pooled variance

Hypotheses:
Since under our assumptions,
if we further assume , then
Same population means: X and Y are equally effective
Two-sided test
0
test statistic t0

Hypotheses:
Under , .
α/2 α/2
0
α: significance level
So if ,
something highly unlikely has happened.
We assumed but that must have been
wrong! Reject !
is probably true,
with 100(1-α)% confidence.

α/2 α/2
0
Using Excel to do a t-test:
- Reject if = TINV(α, φ) = T.INV.2T(α, φ).
- P-value = TDIST(|t0|, φ, 2) = T.DIST.2T(|t0|, φ).
Blue areas under the curve:
probability of observing the
data at hand or something
more extreme, if H0 is true

• Unpaired (i.e., two-sample) t-tests:
- Student’s t-test: equal variance assumption
- Welch’s t-test: no equal variance assumption, but involves
approximations – use this if (1) two sample sizes are very different
AND (2) two sample variances are very different [Sakai16SIGIRshort].
The Welch t-statistic and the degrees of freedom:

Difference measured in standard deviation units
Paired data [Sakai14SIGIRForm] :
Unpaired data:
WARNING: Different books define “Cohen’s d” differently. [Okubo12]
1.2 How the t-test works (15) effect sizes
effect size
Pooled variance
effect size
cf. Hedges’ g, Glass’s Δ

1.3 T-test with Excel and R (hands-on) (1)
- Sample topic-by-run matrix:
https://waseda.box.com/20topics3runs
The easiest way to obtain the p-values:
Paired t-test:
= TTEST(A1:A20,B1:B20,2,1) = 0.2058
Unpaired, Student’s t-test:
= TTEST(A1:A20,B1:B20,2,2) = 0.5300
Unpaired, Welch’s t-test:
= TTEST(A1:A20,B1:B20,2,3) = 0.5302
0.4695 0.3732 0.3575
0.2813 0.3783 0.2435
0.3914 0.3868 0.3167
0.6884 0.5896 0.6024
0.6121 0.4725 0.4766
0.3266 0.233 0.2429
0.5605 0.4328 0.4066
0.5916 0.5073 0.4707
0.4385 0.3889 0.3384
0.5821 0.5551 0.4597
0.2871 0.3274 0.2769
0.5186 0.5066 0.4066
0.5188 0.5198 0.3859
0.5019 0.4981 0.4568
0.4702 0.3878 0.3437
0.329 0.4387 0.2649
0.4758 0.4946 0.4045
0.3028 0.34 0.3253
0.3752 0.4895 0.3205
0.2796 0.2335 0.224
Runs A, B, C
20 topics
two-sided
But this makes you treat the t-test as a black box.
To obtain the test statistic, degrees of freedom etc., let’s do it “by hand”...

A B C D
=A1-B1
= AVERAGE(D1:D20)
= 0.022375
= DEVSQ(D1:D20)/(20-1)
= 0.005834
Paired t-test
= 1.3101
P-value = T.DIST.2T(|t0|, 19) = 0.2058.
0.4695 0.3732 0.3575 0.0963
0.2813 0.3783 0.2435 -0.097
0.3914 0.3868 0.3167 0.0046
0.6884 0.5896 0.6024 0.0988
0.6121 0.4725 0.4766 0.1396
0.3266 0.233 0.2429 0.0936
0.5605 0.4328 0.4066 0.1277
0.5916 0.5073 0.4707 0.0843
0.4385 0.3889 0.3384 0.0496
0.5821 0.5551 0.4597 0.027
0.2871 0.3274 0.2769 -0.0403
0.5186 0.5066 0.4066 0.012
0.5188 0.5198 0.3859 -0.001
0.5019 0.4981 0.4568 0.0038
0.4702 0.3878 0.3437 0.0824
0.329 0.4387 0.2649 -0.1097
0.4758 0.4946 0.4045 -0.0188
0.3028 0.34 0.3253 -0.0372
0.3752 0.4895 0.3205 -0.1143
0.2796 0.2335 0.224 0.0461

A B C
=A1-B1
= AVERAGE(A1:A20)-AVERAGE(B1:B20)
= 0.022375
= DEVSQ(A1:A20) = 0.291139
Unpaired, Student’s t-test
= 0.012463
P-value = T.DIST.2T(|t0|, 38) = 0.5300.
0.4695 0.3732 0.3575
0.2813 0.3783 0.2435
0.3914 0.3868 0.3167
0.6884 0.5896 0.6024
0.6121 0.4725 0.4766
0.3266 0.233 0.2429
0.5605 0.4328 0.4066
0.5916 0.5073 0.4707
0.4385 0.3889 0.3384
0.5821 0.5551 0.4597
0.2871 0.3274 0.2769
0.5186 0.5066 0.4066
0.5188 0.5198 0.3859
0.5019 0.4981 0.4568
0.4702 0.3878 0.3437
0.329 0.4387 0.2649
0.4758 0.4946 0.4045
0.3028 0.34 0.3253
0.3752 0.4895 0.3205
0.2796 0.2335 0.224
= DEVSQ(B1:B20) = 0.182445
= 0.6338

A B C
=A1-B1 Unpaired, Welch’s t-test
= 0.015323
P-value = T.DIST.2T(|t0|, φ*) = 0.5302.
0.4695 0.3732 0.3575
0.2813 0.3783 0.2435
0.3914 0.3868 0.3167
0.6884 0.5896 0.6024
0.6121 0.4725 0.4766
0.3266 0.233 0.2429
0.5605 0.4328 0.4066
0.5916 0.5073 0.4707
0.4385 0.3889 0.3384
0.5821 0.5551 0.4597
0.2871 0.3274 0.2769
0.5186 0.5066 0.4066
0.5188 0.5198 0.3859
0.5019 0.4981 0.4568
0.4702 0.3878 0.3437
0.329 0.4387 0.2649
0.4758 0.4946 0.4045
0.3028 0.34 0.3253
0.3752 0.4895 0.3205
0.2796 0.2335 0.224
= 0.6338
= 0.009602
= 36.0985

Compare with the Excel results.

Also try:
R uses Welch as the default!

1.5 How ANOVA works (1)
System Per-topic
performances
1 x11, x12, … , x1n
2 x21, x22, … , x1n
3 x31, x32, … , x3n
Topic→
↓System
1 2 … n
1 x11 x12 … x1n
2 y21 y22 … y2n
3 z31 z32 … z3n
One-way ANOVA with
equal number of replicates
Two-way ANOVA without replication
(If xi corresponds to yi and zi,
this should be preferred over one-way ANOVA)
ANOVA can ask: “Are ALL systems equally effective?” when there are m (>2) systems.
In this tutorial, let’s first consider the following two simplest types of ANOVA.
Generalises the
unpaired t-test
Generalises the
paired t-test

1.5 How ANOVA works (2) one-way ANOVA
System Per-topic
performances
1 x11, x12, … , x1n
2 x21, x22, … , x1n
3 x31, x32, … , x3n
i=1, … , m
j=1, … , n
: score of i-th system for topic j
ASSUMPTIONS: are independent and
, or, equivalently,
and .
Let and .
Then it is easy to show that .
Homoscedasticity
(equal variance)
assumption
Population grand mean i-th system effect

Hypotheses:
: At least one of the system effects is non-zero.
Let
.
Note that
ALL population means
are equal
Diff between
score and
grand mean
Diff between
system mean and
grand mean
Diff between
score and
system mean
Sample grand mean Sample system mean

Similarly, ST = SA + SE holds, where System Per-topic
performances
1 x11, x12, … , x1n
2 x21, x22, … , x1n
3 x31, x32, … , x3n
Total variation
Between-system
variation
Within-system
variation

ST = SA + SE
Under the i.i.d. and normality assumptions on ,
(a)
⇒
(b) .
So, under H0 (ai = 0),
φE =m(n-1)
φA =m-1
1.1 (14)(c)
φT =mn-1
= φA + φE
Degrees of freedom:
how accurate is the sum of
squares?
1.1 (14)(c)
1.1 (10)

ST = SA + SE φT = φA + φE
[Under H0]
⇒ Under H0,
Is the between-system variation large compared to the within system variation?
φE = m(n-1)
φA = m-1
1.1 (16)

m=3,n=10 m=5, n=10 m=20, n=10
Hypotheses:
Test statistic:
Reject H0 if
F0 >= F(φA,φE;α).
φE = m(n-1)
φA = m-1
Critical F value
F(φA,φE;α)
F0
α
0
SE from
1.5 (4)

Sum of
squares
Degrees of
freedom
Mean
squares
F0
Between
System
SA φA = m-1 VA = SA/φA =
SA/(m-1)
VA/VE =
m(n-1)SA
(m-1)SE
Within
System
SE φE = m(n-1) VE = SE/φE =
SE/m(n-1)
Total ST φT = mn-1
- Reject H0 if F0 >= F(φA,φE;α) = F.INV.RT(φA,φE,α)
- P-value = F.DIST.RT(F0,φA,φE)
If n varies across the m systems,
let φE = (total #observations) – m.

Population effect size
Simplest estimator of the above from a sample
(more accurate)
How much of the total variance can be accounted
for by the between-system variance?
Effect sizes for one-way ANOVA [Okubo12]
More accurate
estimator in
[Okubo12, Sakai14SIGIRforum]

1.5 How ANOVA works (10) two-way ANOVA w/o replication
Topic→
↓System
1 2 … n
1 x11 x12 … x1n
2 y21 y22 … y2n
3 z31 z32 … z3n
ASSUMPTIONS: are independent and
homoscedasticity
System and topic effects are additive
and linearly related to xij
Sample grand mean
Sample system mean Sample topic mean

Hypothesis for the system effects
: at least one differs
Hypothesis for the topic effects
Note that
Diff between
score and
grand mean
Diff between
system mean and
grand mean
Diff between
topic mean and
grand mean
The rest
Green part for one-way ANOVA in 1.5 (3)

Similarly, ST = SA + SB + SE holds, where
Total variation
Between-system
variation
Residual
Between-topic
variation Within-system variance
for one-way ANOVA
in 1.5 (4)

ST = SA + SB + SE φT = φA + φB + φE
Hypotheses for the system effects
Under H0,
Hypotheses for the topic effects
Under H0,
i
φE = (m-1)(n-1)
φA = m-1
φB = n-1

m=3,n=10 m=5, n=10 m=20, n=10
Hypotheses (for system effects):
Test statistic:
Reject H0 if
F0 >= F(φA,φE;α).
φE = (m-1)(n-1)
φA = m-1
Critical F value
F(φA,φE;α)
F0
α
0
For topic effects,
use SB and φB
instead of
SA and φA.SE from
1.5 (12)

Sum of
squares
Degrees of
freedom
Mean squares F0
Between
system
SA φA =m-1 VA = SA/φA =
SA/(m-1)
VA/VE =
(n-1)SA/SE
Between
topic
SB φB = n-1 VB = SB/φB =
SB/(n-1)
VB/VE =
(m-1)SB/SE
SE φE = (m-1)(n-1) VE = SE/φE =
SE/(n-1)(m-1)
Total ST φT = mn-1
For system effects: - Reject H0 if F0 >= F(φA,φE;α) = F.INV.RT(φA,φE,α)
- P-value = F.DIST.RT(F0,φA,φE)

ST = SA + SB + SAxB + SE
1.5 How ANOVA works (16) two-way ANOVA
φT = φA + φB + φAxB + φE
B→
↓A
1 2 … n
1 x111,
:
x11r
x121
:
x12r
… x1n1
:
x1nr
2 x211,
:
x21r
: … :
: : : … :
m xm11
:
xm1r
xm21
:
xm2r
… xmn1
:
xmnr
Not discussed in detail in this tutorial as this design is rare in system-based evaluation
• Two factors A and B
• Each cell contains r observations
(total #observations = N = mnr)
• Interaction between A and B considered
A levels
score
B level 1
B level 2
score seems
high if A level
is high AND B
level is high!
No interaction

Sum of
squares
Degrees of
freedom
Mean squares F0
A SA φA =m-1 VA = SA/φA VA/VE
B SB φB = n-1 VB = SB/φB VB/VE
AxB SAB φAxB = (m-1)(n-1) VAxB = SAxB/φAxB VAxB/VE
SE φE = mn(r-1) VE = SE/φE
Total ST φT = mnr-1
P-value = F.DIST.RT( F0, φA, φE )
P-value = F.DIST.RT( F0, φB, φE )
P-value = F.DIST.RT( F0, φAxB, φE )
1.5 How ANOVA works (17) two-way ANOVA
ST = SA + SB + SAxB + SE φT = φA + φB + φAxB + φE
Definitions of
SAxB and SE for two-way ANOVA
can be found in text books.

Population effect sizes
Simplest estimators of the above from a sample
Variances we’re not
interested in removed
from denominator
(more accurate)
Effect sizes for two-way ANOVA w and w/o replication [Okubo12]
How much of the total variance does the
between-system variance account for?
1.5 How ANOVA works (18)
without replication: ST = SA + SB + SE
with replication: ST = SA + SB + SAB + SE
More accurate
estimators in
[Okubo12, Sakai14SIGIRforum]

1.6 ANOVA with Excel and R (1)
one-way ANOVA
• = DEVSQ(A1:C20) = 0.726229
• = DEVSQ(A1:A20)
+ DEVSQ(B1:B20)
+ DEVSQ(C1:C20) = 0.650834
= ST – SE = 0.075395
0.4695 0.3732 0.3575
0.2813 0.3783 0.2435
0.3914 0.3868 0.3167
0.6884 0.5896 0.6024
0.6121 0.4725 0.4766
0.3266 0.233 0.2429
0.5605 0.4328 0.4066
0.5916 0.5073 0.4707
0.4385 0.3889 0.3384
0.5821 0.5551 0.4597
0.2871 0.3274 0.2769
0.5186 0.5066 0.4066
0.5188 0.5198 0.3859
0.5019 0.4981 0.4568
0.4702 0.3878 0.3437
0.329 0.4387 0.2649
0.4758 0.4946 0.4045
0.3028 0.34 0.3253
0.3752 0.4895 0.3205
0.2796 0.2335 0.224
A
20 topics
B C

one-way ANOVA
0.4695 0.3732 0.3575
0.2813 0.3783 0.2435
0.3914 0.3868 0.3167
0.6884 0.5896 0.6024
0.6121 0.4725 0.4766
0.3266 0.233 0.2429
0.5605 0.4328 0.4066
0.5916 0.5073 0.4707
0.4385 0.3889 0.3384
0.5821 0.5551 0.4597
0.2871 0.3274 0.2769
0.5186 0.5066 0.4066
0.5188 0.5198 0.3859
0.5019 0.4981 0.4568
0.4702 0.3878 0.3437
0.329 0.4387 0.2649
0.4758 0.4946 0.4045
0.3028 0.34 0.3253
0.3752 0.4895 0.3205
0.2796 0.2335 0.224
A B C
Sum of
squares
Degrees of
freedom
Mean
squares
F0
Between
System
SA
= 0.075395
φA = m-1
= 2
VA = SA/φA
= 0.037697
VA/VE
= 3.3015
Within
System
SE
= 0.650834
φE = m(n-1)
= 57
VE = SE/φE
= 0.011418
Total ST
= 0.726229
P-value = F.DIST.RT( F0, φA, φE ) = 0.0440

one-way ANOVA
0.4695 0.3732 0.3575
0.2813 0.3783 0.2435
0.3914 0.3868 0.3167
0.6884 0.5896 0.6024
0.6121 0.4725 0.4766
0.3266 0.233 0.2429
0.5605 0.4328 0.4066
0.5916 0.5073 0.4707
0.4385 0.3889 0.3384
0.5821 0.5551 0.4597
0.2871 0.3274 0.2769
0.5186 0.5066 0.4066
0.5188 0.5198 0.3859
0.5019 0.4981 0.4568
0.4702 0.3878 0.3437
0.329 0.4387 0.2649
0.4758 0.4946 0.4045
0.3028 0.34 0.3253
0.3752 0.4895 0.3205
0.2796 0.2335 0.224
A B C
Data that we used for the t-test

one-way ANOVA
0.4695 0.3732 0.3575
0.2813 0.3783 0.2435
0.3914 0.3868 0.3167
0.6884 0.5896 0.6024
0.6121 0.4725 0.4766
0.3266 0.233 0.2429
0.5605 0.4328 0.4066
0.5916 0.5073 0.4707
0.4385 0.3889 0.3384
0.5821 0.5551 0.4597
0.2871 0.3274 0.2769
0.5186 0.5066 0.4066
0.5188 0.5198 0.3859
0.5019 0.4981 0.4568
0.4702 0.3878 0.3437
0.329 0.4387 0.2649
0.4758 0.4946 0.4045
0.3028 0.34 0.3253
0.3752 0.4895 0.3205
0.2796 0.2335 0.224
A B C

two-way ANOVA w/o replication
• = DEVSQ(A1:C20) = 0.726229
= 20*((0.4501-0.4146)^2
+(0.4277-0.4146)^2
+(0.3662-0.4146)^2 = 0.075395
= 0.579826
= ST – SA – SB = 0.726229
0.4695 0.3732 0.3575
0.2813 0.3783 0.2435
0.3914 0.3868 0.3167
0.6884 0.5896 0.6024
0.6121 0.4725 0.4766
0.3266 0.233 0.2429
0.5605 0.4328 0.4066
0.5916 0.5073 0.4707
0.4385 0.3889 0.3384
0.5821 0.5551 0.4597
0.2871 0.3274 0.2769
0.5186 0.5066 0.4066
0.5188 0.5198 0.3859
0.5019 0.4981 0.4568
0.4702 0.3878 0.3437
0.329 0.4387 0.2649
0.4758 0.4946 0.4045
0.3028 0.34 0.3253
0.3752 0.4895 0.3205
0.2796 0.2335 0.224
A B C
cf. 1.6 (1)

0.4695 0.3732 0.3575
0.2813 0.3783 0.2435
0.3914 0.3868 0.3167
0.6884 0.5896 0.6024
0.6121 0.4725 0.4766
0.3266 0.233 0.2429
0.5605 0.4328 0.4066
0.5916 0.5073 0.4707
0.4385 0.3889 0.3384
0.5821 0.5551 0.4597
0.2871 0.3274 0.2769
0.5186 0.5066 0.4066
0.5188 0.5198 0.3859
0.5019 0.4981 0.4568
0.4702 0.3878 0.3437
0.329 0.4387 0.2649
0.4758 0.4946 0.4045
0.3028 0.34 0.3253
0.3752 0.4895 0.3205
0.2796 0.2335 0.224
A B C
Sum of
squares
Degrees of
freedom
Mean squares F0
Between
system
SA
= 0.075395
φA =m-1
= 2
VA = SA/φA
= 0.037697
VA/VE
= 20.1737
Between
topic
SB
= 0.579826
φB = n-1
= 19
VB = SB/φB
= 0.030517
VB/VE
= 16.3312
SE
= 0.071008
φE = (m-1)(n-1)
= 38
VE = SE/φE
= 0.001869
Total ST
= 0.726229
P-value (system) = F.DIST.RT( F0, φA, φE ) = 1.070E-06
P-value (topic) = F.DIST.RT( F0, φB, φE ) = 8.173E-13

0.4695 0.3732 0.3575
0.2813 0.3783 0.2435
0.3914 0.3868 0.3167
0.6884 0.5896 0.6024
0.6121 0.4725 0.4766
0.3266 0.233 0.2429
0.5605 0.4328 0.4066
0.5916 0.5073 0.4707
0.4385 0.3889 0.3384
0.5821 0.5551 0.4597
0.2871 0.3274 0.2769
0.5186 0.5066 0.4066
0.5188 0.5198 0.3859
0.5019 0.4981 0.4568
0.4702 0.3878 0.3437
0.329 0.4387 0.2649
0.4758 0.4946 0.4045
0.3028 0.34 0.3253
0.3752 0.4895 0.3205
0.2796 0.2335 0.224
A B C

1.7 What's wrong with significance tests? (1)
[Johnson99]
• Deming (1975) commented that the reason students have problems
understanding hypothesis tests is that they may be trying to think.
• Carver (1978) recommended that statistical significance testing
should be eliminated; it is not only useless, it is also harmful because
it is interpreted to mean something else.
• Cohen (1994:997) noted that statistical testing of the null hypothesis
"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!"

• We want to know P(H|D), but classical significance testing only gives
us something like P(D|H). (Alternative: Bayesian statistics etc.)
• Reporting α (e.g. 0.05) instead of the actual p-values leads to
dichotomous thinking (“Signifcant or not”?)
• Even if p-values are reported, p-values reflect not only the effect size
(magnitude of the actual difference) but also the sample size:
p-value = f( sample_size, effect_size )
large effect size ⇒ small p-value
large sample size ⇒ small p-value
H: Hypothesis, D: Data
Anything can be made statistically significant by using lots of data
1.2 (15)

[Sakai14SIGIRForum]
So what should we do?
Whenever using a classical significance test, report not only p-values,
but also effect sizes and confidence intervals.
Difference between
two systems
measured in standard
deviation units

[Sakai14SIGIRForum]
So what should we do?
Whenever using a classical significance test, report not only p-values,
but also effect sizes and confidence intervals.
Difference between
two systems
measured in standard
deviation units
Actually, if you want p-values for every system pair, you can
apply randomised Tukey HSD
[Carterette12,Sakai14PROMISE] WITHOUT doing ANOVA.
More accurate
estimators of
and
cf. 1.5 (18)

Randomised Tukey HSD test for m>=2 systems
http://research.nii.ac.jp/ntcir/tools/discpower-en.html
• Input: a topic-by-run score matrix.
• Can be used to compute
p-values for 2 or more systems.
• Unlike classical tests, it does not
rely on assumptions such as normality.
• It is a kind of multiple comparison
procedure (free from the familywise
error rate problem).

(1) [Sakai16SIGIR]

(2) [Sakai16SIGIR]

(3) [Sakai16SIGIR]

(4) [Sakai16SIGIR]

(5) [Sakai16SIGIR]

2.1 Topic set sizes in IR (1) [Sakai16IRJ]
According to Sparck Jones and Van Rijsbergen [SparckJones75],
fewer than 75 topics “are of no real value”;
250 topics “are minimally acceptable”;
more than 1000 topics “are needed for some purposes”
because “real collections are large”; “statistically significant results are
desirable” and “scaling up must be studied.”

In 1979, in a report that considered the number of relevance
assessments required from a statistical viewpoint, Gilbert and Sparck
Jones remarked [Gilbert79]:
“Since there is some doubt about the feasibility of getting 1000
requests, or the convenience of such a large set for future experiments,
we consider 500 requests.”

2.1 Topic set sizes in IR (3)
The default topic set size at TREC: 50.
Exceptions include the million query track that created 1800+ topics
[Carterette08] but creating a “reusable” test collection was not the
objective of the track. Round Documents Topics
TREC-1 disks 1 + 2 51-100
TREC-2 disks 1 + 2 101-150
TREC-3 disks 1 + 2 151-200
TREC-4 disks 2 + 3 201-250
TREC-5 disks 2 + 4 251-300
TREC-6 disks 4 + 5 301-350
TREC-7 disks 4 + 5 351-400
TREC-8 disks 4 + 5 401-450
Early TREC ad hoc tasks and topics
[Voorhees05, p.24]

In 2009, Voorhees conducted an experiment where she randomly
split 100 TREC topics in half to count discrepancies in statistically
significant results, and concluded that
“Fifty-topic sets are clearly too small to have confidence 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.” [Voorhees09]
TREC-7 + 8 topics
with TREC 2004
robust track systems
100 topics
random split 50
topics
50
topics
Paired t-test says
System A > B!
Paired t-test says
System A < B!
conflict
But if randomised Tukey HSD (i.e. a multiple comparison procedure) is used for filtering system pairs,
discrepancies across test collections almost never occur [Sakai16ICTIR].

At CIKM 2008, [Webber08] pointed out that the topic set size should be
determined based on the required statistical power.
Accept H0 Reject H0
H0 is true
Correct conclusion
(1-α)
Type I error
α
H0 is false
Type II error
β
Correct conclusion
(1-β)
Statistical power:
differences

The approach of [Webber08]:
• Incremental test collection building – adding topics with relevance
assessments one by one until the desired power is achieved;
• Considered the t-test without addressing the familywise error rate
problem;
• Estimated the variance of score deltas using non-standard methods;
We want a more straightforward answer to “How many topics should I create?”
In addition to the t-test, we can consider one-way ANOVA and confidence intervals as the basis.
Residual variances from ANOVA are unbiased estimators of the within-system variances.

2.2 Topic set size design (1) [Sakai16IRJ]
• Provides answers to the following question:
“I’m building a new test collection. How many topics should I create?”
• A prerequisite: a small topic-by-run score matrix based on pilot data,
for estimating within-system variances.
• Three approaches (with easy-to-use Excel tools), based on:
(1) paired t-test power
(2) one-way ANOVA power
(3) confidence interval width upperbound.

Test collection designs should evolve based on past data
topic-by-run
score matrix with
pilot data
About 25 topics
with runs from
a few teams
probably sufficient
[Sakai16EVIA]
n1 topics
m runs
Estimate n1 based on the
within-system variance
estimate
TREC 201X TREC 201(X+1)
n2 topics
n0 topics
Estimate n2 based on the
within-system variance
estimate
A more accurate estimate

Method Input required
Paired t-test α (Type I error probability), β (Type II error probability),
minDt (minimum detectable difference: whenever the diff between two systems is this
much or larger, we want to guarantee (1-β)% power),
: variance estimate for the score delta.
one-way ANOVA α (Type I error probability), β (Type II error probability), m (number of systems),
minD (minimum detectable range: whenever the diff between the best and worst
systems is this much or larger, we want to guarantee (1-β)% power),
: estimate of the within-system variance under the homoscedasticity assumption.
Confidence intervals α (Type I error probability),
δ (CI width upperbound: you want the CI for the diff between any system pair to be this
much or smaller),

ANOVA-based results for
m=10 can be used instead
of CI-based results
ANOVA-based results for
m=2 can be used instead of
t-test-based results
In practice, you can deduce t-test-based and CI-based results from ANOVA-based results
Caveat: the ANOVA-based tool can only
handle (α, β)=(0.05, 0.20), (0.01, 0.20),
(0.05, 0.10), (0.01, 0.10).

2.3 Paired t-tests (1)
Example situation: You plan to compare a system pair with the paired t-test
with α=5%. You plan to use nDCG as a primary evaluation measure, and
want to guarantee 80% power whenever the diff between two systems
>= minDt.
You know from pilot data that the variance of the nDCG delta is around .
What is the required number of topics n?
Paired t-test α (Type I error probability), β (Type II error probability),
minDt (minimum detectable difference: whenever the diff between two systems is this
much or larger, we want to guarantee (1-β)% power),

Notations (some slightly different from Part 1)
t: a random variable that obeys t(φ) where φ=n-1;
: two-sided critical t value for sig. criterion α
= T.INV.2T(α, φ)
α/2 α/2
0

Under our assumptions, holds.
In a t-test, we let
and consider . Due to the t-test procedure,
regardless of what t0 obeys, the probability of rejecting H0 is
.

Regardless of what t0 obeys, the probability of rejecting H0 is
... (a)
If H0 is true, then t0 obeys t(n-1) and (a) is exactly α
(that’s how was defined).
Alternatively, if H1 is true, the distribution that t0 obeys is known as
a noncentral t distribution with φ degrees of freedom,
and (a) is exactly the power, (1-β). Rejecting the
incorrect
hypothesis H0
Rejecting the
correct
hypothesis H0

Accept H0 Reject H0
H0 is true
Correct conclusion
(1-α)
Type I error
α
H0 is false
Type II error
β
Correct conclusion
(1-β)
t0 obeys a (central) t distribution
t0 obeys a noncentral t distribution
=
≠
(a)

If H1 is true, the distribution that t0 obeys is known as a noncentral t
distribution with φ degrees of freedom, and (a) is exactly the power,
(1-β).
The noncentral t distribution in fact has another parameter called
the noncentrality parameter λt :
≠
population
effect size
Population variance of the score differences: See 1.2 (2)

If H1 is true, the distribution that t0 obeys is known as a noncentral t
distribution with φ degrees of freedom and a noncentrality parameter
λt, and (a) is exactly the power, (1-β).
We want to compute (a) , but the computation involving the noncentral
t distribution is too complex...
... (a)
Power =

Fortunately, a good approximation is available [Nagata03] .
t’: a random variable that obeys a noncentral t distribution with φ, λt ;
u: a random variable that obeys a standard normal distribution;
... (a)
Power =
Appendix
Theorem A’

... (a)
Power =
where .
... (a’)
Theorem A’

Power = 1-β
Now we know how to compute power given (α, Δt, n).
But we want to compute n given (α, β, Δt).
... (a’)

But we want to compute n given (α, β, Δt). Starting again with:
Power =
Appendix
Theorem A

But we want to compute n given (α, β, Δt). Starting again with:
Power =
Theorem A
If λt > 0
λt < 0 will lead to the same final
result
Ignore

Power
⇒
one-sided z value for probability 1-β
Let
⇒
cf. This is rougher than Theorem A’

When λt > 0 or λt < 0
(i.e. H1 is true)
Similarly, when λt = 0
(i.e. H0 is true),
two-sided t value
one-sided z value
≠ 0

Appendix
Theorem A’’
Appendix
Theorem B

Let and recall that . Substituting these to the above gives
≠ 0
when H1 is true

Given (α, β, minΔt), the minimal sample size n can be approximated as
by letting Δt = minΔt .
But this involved a lot of approximations, so we need to go back to (a’)
and check that n actually achieves 100(1-β)% power:
minimum detectable effect size
... (a’)

EXAMPLE: α=0.05, β=0.20, detectable effect size regardless of
evaluation measure minΔt = 0.50 (i.e. half a std deviation of the diff)
→
= 33.3
(z α/2 = z 0.025 = NORM.S.INV(1-0.025)=1.960, z 1-β = z 0.80 = -0.842)
So if we let n=33, the achieved power according to (a’)
= 0.795 ... doesn’t quite achieve 80%!

EXAMPLE: α=0.05, β=0.20, detectable effect size regardless of
evaluation measure minΔt = 0.50 (i.e. half a std deviation of the diff)
If we let n=34, the achieved power according to (a’)
= 0.808 ... so n=34 is what we need!

Don’t worry,
http://www.f.waseda.jp/tetsuya/CIKM2014/samplesizeTTEST.xlsx
will do this for you! Use the “From effect size” sheet and fill out the
orange cells.
n=34 is what you
want!

2.3 Paired t-tests (21) [Sakai16IRJ]
Topic set sizes for typical requirements based on effect sizes

In practice, you might want to specify a minimum detectable diff
(minDt) in (say) nDCG instead of minΔt for guaranteeing 100(1-β)%
power.
Given minD and , so n can be obtained as before.
A conservative estimate for the delta variance would be
where is a within-system variance estimate obtained under a
homoscedasticity assumption. See 2.6

EXAMPLE: For nDCG, α=0.05, β=0.20, minDt =0.1 (i.e., one-tenth of
nDCG’s score range), = 0.50 (from some pilot data)
→ Use the “From the absolute diff” sheet:
n=395 is what you
want!

one-way ANOVA α (Type I error probability), β (Type II error probability), m (number of systems),
minD (minimum detectable range: whenever the diff between the best and worst
systems is this much or larger, we want to guarantee (1-β)% power),
: estimate of the within-system variance under the homoscedasticity assumption.
Example situation: You plan to compare m systems with one-way ANOVA with
α=5%. You plan to use nDCG as a primary evaluation measure, and want to
guarantee 80% power whenever the diff between the best and the worst systems
>= minD.
You know from pilot data that the within-system variance for nDCG is around .
2.4 One-way ANOVA (1) m systems
best
worst
minD <= D

2.4 One-way ANOVA (2)
Notations (some slightly different from Part 1)
F: random variable that obeys an F distribution with (φA, φE) degrees of
freedom;
: critical F value for sig. criterion α
= F.INT.RT(α, φA, φE)
α
0
φA = m-1
φE = m(n-1)

Due to the one-way ANOVA procedure, regardless of what F0 obeys, the
probability of rejecting H0 is:
If H0 is true, then F0 obeys F(φA, φE) and (c) is exactly α
(that’s how is defined).
Alternatively, if H1 is true, the distribution that F0 obeys is known as a
noncentral F distribution with (φA, φE) degrees of freedom,
and (c) is exactly the power, (1-β).
... (c)

Accept H0 Reject H0
H0 is true
Correct conclusion
(1-α)
Type I error
α
H0 is false
Type II error
β
Correct conclusion
(1-β)
F0 obeys a (central) F distribution
F0 obeys a noncentral F
distribution
(c)

If H1 is true, the distribution that F0 obeys is known as a noncentral F
distribution with (φA, φE) degrees of freedom, and (c) is exactly the
power, (1-β).
The noncentral F distribution in fact has another parameter called
the noncentrality parameter λ :
Measures the total system effects in
variance units
Within-system variance
under homoscedasticity

If H1 is true, the distribution that F0 obeys is known as a noncentral F
distribution with (φA, φE) degrees of freedom and a noncentrality
parameter λ, and (c) is exactly the power, (1-β).
... (c)
Appendix
Theorem C ... (c’)
Denoted F’(φA, φE, λ)

Let us ensure that when Δ≠0 (i.e., H1 is true), we guarantee 100(1-β)%
power whenever the difference between best and worst systems is
minD or larger (minimum detectable range).
m systems
best
worst
minD <= D
H1: At least
one system is
different
≠ 0

Let us ensure that when Δ≠0 (i.e., H1 is true), we guarantee 100(1-β)%
power whenever the difference D between best and worst systems is
minD or larger (minimum detectable range).
Define .
Then
holds.
Appendix
Theorem D
minD does not uniquely determine Δ,
but minΔ can be used as the worst-case Δ.

The worst-case sample size:
The λ is the noncentrality parameter for
F’(φA, φE, λ), which can be approximated by
, for which these linear
approximations are available α β
0.01 0.10
0.01 0.20
0.05 0.10
0.05 0.20
Appdendix Theorem E
λ for noncentral chi-square
distributions [Nagata03]

Given (α, β, minD, m, ), the minimal sample size n can be
approximated as
.
But this involved a lot of approximations, so we need to go back to (c’)
and check that n actually achieves 100(1-β)% power:
... (c’)Power

EXAMPLE: α=0.05, β=0.20, minD=0.5, m=3, =0.5^2.
→
So let n=19 ⇒
Hence from (c’) we get power =
= 0.791 ... doesn’t quite achieve 80%!

EXAMPLE: α=0.05, β=0.20, minD=0.5, m=3, =0.5^2.
→
Try n=20 ⇒
From (c’) we get power =
= 0.813 ... so n=20 is what we need!

Don’t worry,
http://www.f.waseda.jp/tetsuya/CIKM2014/samplesizeANOVA.xlsx
will do this for you! Use the appropriate sheet for a given (α, β) and fill
out the orange cells.
:
n=20 is what you
want!

2.5 Confidence Intervals (1)
Confidence intervals α (Type I error probability),
δ (CI width upperbound: you want the CI for the diff between any system pair to be this
much or smaller),
Example situation: You plan to compare a system pair by means of 95% CI
for the difference in nDCG. You want to guarantee that the CI width for any
system pair is δ or smaller. You know from pilot data that the variance of
the nDCG delta is around .

2.5 Confidence Intervals (2) cf. 1.2 (8)
The 100(1-α)% CI for a difference in means (paired data) is given by
where
.
Let’s consider a sample size n which guarantees that the CI width
(=2*MOE) for any difference will be no larger than δ.
But since MOE contains a random variable V, let’s consider the above
requirement using an expectation:
.

Now, it is known that
so we want to find the smallest n that
satisfies:
.
sample standard deviation
population standard deviation
gamma function:
(see Theorem A)
cf. 1.1 (11)

We want to find the smallest n that satisfies:
To obtain an initial n, instead of ,
consider where the variance is known.
Thus, let and start with .
Increment n’ until (d) is satisfied.
... (d)

EXAMPLE: α=0.05, δ=0.5, = 0.5 (from some pilot data)
= 30.7
Try n=31 → LHS=0.257 > 0.25
n=32 → LHS=0.253 > 0.25
n=33 → LHS=0.249 < 0.25
... (d)
=0.25
LHS
n=33 is what you
want!

Don’t worry,
http://www.f.waseda.jp/tetsuya/FIT2014/samplesizeCI.xlsx
will do this for you! Just fill out the orange cells.
n=33 is what you
want!

2.6 Estimating the variance (1)
We need for topic set size design based on one-way ANOVA
and for that based on the paired t-test or CI.
From a pilot topic-by-run score matrix, obtain:
Then, if possible, pool multiple estimates to enhance accuracy:
Pooled estimate
By-product of one-way
ANOVA
(use two-way w/o
replilcation for tighter
estimates)

• = DEVSQ(A1:A20)
+ DEVSQ(B1:B20)
+ DEVSQ(C1:C20) = 0.650834
φE = m(n-1) = 3(20-1)= 57
= VE = SE / φE = 0.011
0.4695 0.3732 0.3575
0.2813 0.3783 0.2435
0.3914 0.3868 0.3167
0.6884 0.5896 0.6024
0.6121 0.4725 0.4766
0.3266 0.233 0.2429
0.5605 0.4328 0.4066
0.5916 0.5073 0.4707
0.4385 0.3889 0.3384
0.5821 0.5551 0.4597
0.2871 0.3274 0.2769
0.5186 0.5066 0.4066
0.5188 0.5198 0.3859
0.5019 0.4981 0.4568
0.4702 0.3878 0.3437
0.329 0.4387 0.2649
0.4758 0.4946 0.4045
0.3028 0.34 0.3253
0.3752 0.4895 0.3205
0.2796 0.2335 0.224
A
20 topics
B C
2.6 Estimating the variance (2)
cf. 1.6 (1)
cf. 1.6 (2)
If there is no other topic-by-run matrix available, use this as .

2.7 How much pilot data do we need? (1)
[Sakai16EVIA]
100
topics
44 runs from 16 teams
Pilot data
Variance
estimates
(best estimates
available)
Official
NTCIR-12 STC
qrels based on
16 teams
(union of
contributions
from 16 teams)
Can we obtain a reliable even from a few teams and a small number of topics?

[Sakai16EVIA] Can we obtain a reliable even from a few teams and a small number of topics?
100
topics
Runs from 15 teams
Pilot data
New variance
estimates
Try
leave-1-out
10 times
Leaving out k teams
k=1
(k=1,...,15)

100
topics
Runs from 1 team
Pilot data
New variance
estimates
Leaving out k teams
k=15
(k=1,...,15)
Try
leave-15-out
10 times

100
topics
44 runs from 16 teams
Variance
estimates
(best estimates
available)
50
25
Variance
estimates
Variance
estimates
Removing topics
100 → 90 → 75 → 50 → 25 → 10
Official NTCIR-12
STC qrels

100
topics
Runs from 15 teams
Variance
estimates
(best estimates
available)
50
25
Variance
estimates
Variance
estimates
Removing topics
100 → 90 → 75 → 50 → 25 → 10
Leave-k-out qrels
k=1
(k=1,...,15)

Starting with n’=100 topics Starting with n’=10 topics
[Sakai16EVIA] About 25 topics with a few teams seems sufficient,
provided that a reasonably stable measure is used.

3.1 Power analysis (1) [Ellis10, pp.56-57]
1. Effect size describes the degree to which the phenomenon is
present in the population;
2. Sample size determines the amount of sampling error inherent in a
result;
3. Significance criterion α defines the risk of committing a Type I error;
4. power (1-β) refers to the chosen or implied Type II error rate.
“The four power parameters are related, meaning that the value of any
parameter can be determined from the other three.”
We had a quick look at how the computations can be done in Part 2.

3.1 Power analysis (2) [Toyoda09]
If a paper reports
- The parametric significance test type (paired/unpaired t-test, one-way
ANOVA, two-way ANOVA w and w/o replication)
- either p-value or test statistic (t-value or F-value)
- actual sample size
we can easily compute the sample effect size.
Then, using the library pwr of R, we can compute
- the achieved power of the experiment
- future sample size for achieving given (α, β).
cf. 1.7 (2)
https://cran.r-project.org/web/packages/pwr/pwr.pdf
power=(1-β)

3.1 Power analysis (3) [Sakai16SIGIR]
My R power analysis scripts, adapted from [Toyoda09] with Professor
Toyoda’s kind permission, are available at
https://waseda.box.com/SIGIR2016PACK
- Works with paired/unpaired t-test, one-way ANOVA, two-way ANOVA
w and w/o replication.
- SIGIR2016PACK also contains an Excel file from [Sakai16SIGIR]
(manual analysis of 1055 papers from SIGIR+TOIS 2006-2015).

3.2 With paired t-tests (1)
future.sample.pairedt arguments:
- t statistic (t)
- sample size (n)
- two-sided/one-sided (default: two-sided)
- α (default: 0.05)
- desired power (1-β) (default:0.80)
OUTPUT:
- effect size
- achieved power
- future sample size n’
1.2 (15)
Calls power.t.test

3.2 With paired t-tests (2)
A paper from SIGIR 2012 reports
“t(27)=0.953 with (two-sided) paired t-test”
⇒ t = 0.953, n = 28 (φ = n-1 = 27)
Line 270 in the raw Excel file from [Sakai16SIGIR]
very low power (15.1%)
For this kind of effect, we need a much larger
sample if we want 80% power

3.3 With unpaired t-tests (1)
future.sample.unpairedt arguments:
- t statistic (t)
- sample sizes (n1, n2)
- two-sided/one-sided (default: two-sided)
- desired power (1-β) (default: 0.80)
OUTPUT:
- effect size
- achieved power
- future sample size n’ per group
1.2 (15)
Calls pwr.t2n.test

3.3 With unpaired t-tests (2)
A paper from SIGIR 2007 reports:
“t(188403) = 2.81, n1 = 150610, n2 = 37795 with (two-sided) two-
sample t-test”
φ = n1 + n2 -2 = 188403
Appropriate level of power
n1 = n2 = 60066 would be the typical setting for 80% power

3.4 With one-way ANOVA (1)
future.sample.1wayanova arguments:
- F statistic (F, i.e. FA)
- #groups (systems) compared (m)
- #observations (topics) per group (n)
OUTPUT:
- effect size
- achieved power
- future sample size per group n’
φA = m-1, φE = m(n-1)
Calls pwr.anova.test
1.5 (9)
Compares
between-system
variation against
within-system

3.4 With one-way ANOVA (2) φA = m-1, φE = m(n-1)
“m=3 groups, n=12 subjects per group,
F(2, 33)=1.284 with (one-way) ANOVA”
(φA = m-1 = 2, φE = m(n-1) = 3*(12-1) = 33)
Very low power (27.9%)
For this kind of effect, we need more subjects
if we want 80% power

future.sample.2waynorep arguments:
same as future.sample.1wayanova.
OUTPUT:
- effect size
- achieved power
- future sample size per group n’
3.5 With two-way ANOVA without replication (1)
φA = m-1, φE = (m-1)(n-1)
A little different from 1.5 (18)
Calls pwr.f2.test, which requires the above squared effect size
p stands for partial:
effect of B has been removed

3.5 With two-way ANOVA without replication (2)
“m=4 groups,
F(3, 48)=0.63 with a repeated-measures ANOVA”
⇒ m = φA +1 = 4, φE = (m-1)(n-1) = 48, n = 17 per group
Same procedure as two-way
ANOVA w/o replication
(second factor e.g. topics
regarded as repeated
observations)
Very low power (18.3%)
For this kind of effect, we need more subjects
if we want 80% power

3.6 With two-way ANOVA (1)
future.sample.2wayanova2 arguments:
- F statistics (FA, FB, FAB)
- #groups compared (m)
- #cells per group (n)
- #total observations (N=mnr)
OUTPUT:
- effect size
- achieved power
- Total sample size N’
φA = m-1, φB = n-1, φAB = (m-1)(n-1)
φE = mn(r-1)
And similarly for B and ABCalls pwr.anova.test
p stands for partial:
effects of B and AB have
been removed
Version 2

3.6 With two-way ANOVA (2)
“m=2, n=2, two-way ANOVA,
A: F(1, 960)=24.00,
B: F(1, 960)=24.89,
AxB: F(1, 960)=10.03”
φA = m-1 = 1, φB = n-1 = 1,
φAxB = (m-1)(n-1)=1,
φE = mn(r-1) = 960
⇒ r= 960/4+1 = 241,
N = mnr = 964
Very high power
Smaller sample sizes
suffice
φE/(φA+1) + 1
= 960/(1+1) + 1
= 481
[Cohen88, p.365]

3.7 Overpowered and underpowered experiments in IR (1)
[Sakai16SIGIR]
SSR = sample size ratio = actual size/recommended size for future
SSR is extremely large ⇔ extremely overpowered
SSR is extremely small ⇔ extremely underpowered
133 SIGIR+TOIS papers from the past decade (2006-2015) were
examined using the R power analysis tools.
(106 with t-tests; 27 with ANOVAs)

[Sakai16SIGIR]

A paper on personalisation from a search engine company (paired t-test)
t=16.00, n=5,352,460, effect size=0.007, achieved power=1
recommended future sample size=164,107
Effect size very small (though this may translate into substantial profit for a
company)
[Sakai16SIGIR]

User experiments, paired t-test
t=0.95, n=28,
effect size=0.180,
achieved power=0.152
future sample size=244
(similar results for other t-test
results in the same paper)
[Sakai16SIGIR]

[Sakai16SIGIR]

Experiments with a commercial social media
application data (one-way ANOVA)
F=243.42, m=3,
sample size per group=2551,
effect size fhat=2.252, achieved power=1,
recommended future sample size per group=52
[Sakai16SIGIR]

User experiments, two-way
ANOVA w/o replication
F=0.63, m=4,
sample size per group=17,
effect size fhat^2 = 0.039,
achieved power=0.183,
recommended future sample
size per group=75
(similar results for other
ANOVA results in the same
paper)
[Sakai16SIGIR]

Now you know
• How to determine the number of topics when building
a new test collection using a topic-by-run matrix from
pilot data and a simple Excel tool. And you kind of
know how it works!
• How to check whether a reported experiment is
overpowered/underpowered and decide on a better
sample size for a future experiment using simple R
scripts.

What now?
• Be aware of the limitations of classical significance testing. But while
we are still using classical tests, report effect sizes, p-values etc. for
collective wisdom [Sakai14SIGIRforum,Sakai16SIGIR]. And use topic
set size design and power analysis! Some guidance is better than
none!
• My personal wish is that the classical significance tests will soon be
replaced by Bayesian tests, so we can discuss P(H|D) instead of
P(D|H) for various H’s, not just “equality of means” etc.
Using score standardisation can give you smaller topic set sizes in topic set size design.
Please have a look at [Sakai16ICTIR].

Thank you for staying with me until the end!
Questions?

Acknowledgements
This tutorial is rather heavily based
on what I learnt from Professor
Yasushi Nagata’s and Professor
Hideki Toyoda’s books (written in
Japanese).
I thank Professor Nagata (Waseda
University) for his valuable advice
and Professor Toyoda (Waseda
University) for letting me modify
his R code and distribute it.
If there are any errors in this
tutorial, I am solely responsible.

References
[Carterette08] Carterette, B., Pavlu, V., Kanoulas, E., Aslam, J. A., and Allan, J.: Evaluation over
Thousands of Queries, ACM SIGIR 2008.
[Carterette12] Carterette, B.: Multiple Testing in Statistical Analysis of Systems-Based Information
Retrieval Experiments, ACM TOIS 30(1), 2012.
[Cohen88] Cohen. J.: Statistical Power Analysis for the Behavioral Sciences (Second Edition),
Psychology Press, 1988.
[Ellis10] Ellis, P. D.: The Essential Guide to Effect Sizes, Cambridge, 2010.
[Gilbert79] Gilbert, H. and Sparck Jones, K. S.:, Statistical Bases of Relevance assessment for the
`IDEAL’ Information Retrieval Test Collection, Computer Laboratory, University of Cambridge, 1979.
[Johnson99] Johnson, D. H.: The Insignificance of Statistical Significance Testing, Journal of Wildlife
Management, 63(3), 1999.
[Nagata03] Nagata, Y.: How to Design the Sample Size (In Japanese), Asakura Shoten, 2003.
[Okubo12] Okubo, G. and Okada, K.: Psychological Statistics to Tell Your Story: Effect Size,
Confidence Interval, and Power (in Japanese), Keisho Shobo, 2012.

References
[Sakai14SIGIRforum] Sakai, T.: Statistical Reform in Information Retrieval?, SIGIR
Forum, 48(1), 2016.
http://sigir.org/files/forum/2014J/2014J_sigirforum_Article_TetsuyaSakai.pdf
[Sakai16EVIA] Sakai, T. and Shang, L.: On Estimating Variances for Topic Set Size
Design, EVIA 2016.
[Sakai16ICTIR] Sakai, T.: A Simple and Effective Approach to Score Standardisation,
ACM ICTIR 2016.
[Sakai16IRJ] Sakai, T.: Topic Set Size Design, Information Retrieval Journal, 19(3),
2016. [OPEN ACCESS] http://link.springer.com/content/pdf/10.1007%2Fs10791-
015-9273-z.pdf
[Sakai16SIGIR] Sakai, T.: Statistical Significance, Power, and Sample Sizes: A
Systematic Review of SIGIR and TOIS, 2006-2015, ACM SIGIR 2016.
[Sakai16SIGIRshort] Sakai, T.: Two Sample T-tests for IR Evaluation: Student or
Welch?, ACM SIGIR 2016.

References
[SparckJones75] Sparck Jones, K.S. and Van Rijsbergen, C.J.: Report on the
Need for and Provision on an `Ideal’ Information Retrieval Test Collection,
Computer Laboratory, University of Cambridge, 1975.
[Toyoda09] Tokoda, H.: Introduction to Statistical Power Analysis: A Tutorial
with R (in Japanese). Tokyo Tosyo, 2009.
[Voorhees05] Voorhees, E. M. and Harman, D. K.: TREC: Experiment and
Evaluation in Information Retrieval, The MIT Press, 2005.
[Voorhees09] Voorhees, E. M.: Topic Set Size Redux, ACM SIGIR 2009.
[Webber08] Webber, W., Moffat, A., and Zobel, J.: Statistical Power in
Retrieval Experimentation, ACM CIKM 2008.

Appendix (everything adapted from [Nagata03])
• Definition: noncentral t distribution
• Definition: noncentral chi-square distribution
• Definition: noncentral F distribution
• Theorem A: normal approximation of a noncentral t distribution
• Theorem A’: corollary of A
• Theorem A’’: corollary of A (approximating a z value using a t value)
• Theorem B: approximating a t value using a z value
• Theorem C: normal approximation of a noncentral F distribution
• Theorem D: inequality for system effects
• Theorem E: approximating a noncentral F distribution with a noncentral chi-
square distribution

Definition: noncentral t distribution
Let ,
where the two random variables are independent.
The probability distribution of the following random variable is called a
noncentral t distribution with φ degrees of freedom and a
noncentrality parameter λ:
When λ=0,
it is reduced to the central t distribution with φ degrees of freedom, t(φ).
Denoted by t’(φ, λ)

Let where the random variables are independent.
noncentral chi-square distribution with φ=k degrees of freedom and a
noncentrality parameter λ:
where .
Definition: noncentral chi-square distribution
When λ=0,
it is reduced to the central chi-square distribution
with φ degrees of freedom, .
Denoted by

Let ,
where the two random variables are independent.
noncentral F distribution with (φ1, φ2) degrees of freedom and a
noncentrality parameter λ.
Definition: noncentral F distribution
noncentral chi-square distribution central chi-square distribution
When λ=0,
it is reduced to the central F distribution with (φ1, φ2) degrees of freedom, F(φ1, φ2).
Denoted by

Theorem A: normal approximation of a
noncentral t distribution
Let , .
Then:
where:
.
Gamma function
noncentral t distribution
Brief derivation given in
[Sakai16IRJ Appendix 1]

Theorem A’: corollary of A
Let , .
Then:
PROOF: Let
,
in Theorem A.

Theorem A’’: corollary of A (approximating a z
value using a t value)
one-sided z value
two-sided t value
PROOF: In Theorem A, when λ=0,
then t=t’ obeys a (central) t distribution.
Also let
.
1
1.5
2
2.5
3
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
z f(t)
φ
2P=α=0.05
Verified with Excel

Theorem B: approximating a t value using a z
value
This is a special case of Johnson and Welch’s theorem on the noncentral t statistic. [Nagata03]
Two-sided t value one-sided z value
P = α = 0.05
1.5
2
2.5
3
3.5
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
t f(z)
Verified with Excel

Theorem C: normal approximation of a
noncentral F distribution
Let ,
Then:
where
.
noncentral F distribution

Theorem D: inequality for system effects
For ,
Let
.
Then
.
The equality holds when = D/2, = -D/2
and ai = 0 for all others.
Proof in [Sakai16IRJ footnote 19]

Theorem E: approximating a noncentral F distribution
with a noncentral chi-square distribution
Let ,
Then:
Letting φE ≒ ∞
F value for probability P
chi-square value for probability P

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