Sample size calculation

Allergol

Immunopathol

(Madr).

2014;42(5):485---492

Allergologia

et

immunopathologia

Sociedad

Espa

˜nola

de

Inmunolog´ıa

Cl´ınica,

Alergolog´ıa

y

Asma

Pedi

´atrica

www.elsevier.es/ai

SERIES:

BASIC

STATISTICS

FOR

BUSY

CLINICIANS

(XI)

Sample

size

calculation

MM

Rodríguez

del

Águila

a,∗

,

AR

González-Ramírez

b

a

UGC

Medicina

Preventiva,

Vigilancia

y

Promoción

de

la

Salud.

Hospital

Virgen

de

las

Nieves,

Granada,

Spain

b

Fundación

Pública

Andaluza

para

la

Investigación

Biosanitaria

de

Andalucía

Oriental.

Hospital

Clínico

San

Cecilio,

Granada,

Spain

Received

29

January

2013;

accepted

25

March

2013

Available

online

23

November

2013

Series’

Editor:

V.

Pérez-Fernández.

Abstract

When

designing

any

research

project,

deﬁnition

is

required

of

the

sample

size

needed

in

order

to

carry

out

the

study.

This

sample

size

is

an

estimate

of

the

number

of

patients

required

in

accordance

with

the

pursued

study

objective.

In

this

context,

it

is

more

efﬁcient

in

terms

of

both

cost

and

time

to

use

samples

than

to

work

with

the

entire

population.

The

present

article

describes

the

way

to

establish

sample

size

in

the

kinds

of

studies

most

frequently

found

in

health

research,

and

how

to

calculate

it

using

the

epicalc

package

included

in

the

shareware

R

program.

A

description

is

provided

of

the

formulae

used

to

calculate

sample

sizes

for

the

estimation

of

a

mean

and

percentage

(referring

to

both

ﬁnite

and

inﬁnite

popu-

lations)

and

for

the

comparison

of

two

proportions

and

two

means.

Likewise,

examples

of

the

application

of

the

mentioned

statistical

package

are

provided.

©

2013

SEICAP.

Published

by

Elsevier

España,

S.L.U.

All

rights

reserved.

Introduction

The

design

phase

of

any

epidemiological

study

includes

the

determination

of

the

sample

size

needed

to

carry

out

the

study.

1 --- 4

In

order

to

conﬁrm

the

established

study

hypothe-

ses,

there

must

be

a

coherent

relationship

among

the

amount

or

number

of

observations

made

and

their

possible

repetitions,

their

representativeness

and

the

quality

of

the

evidence

---

in

addition

to

a

solid

and

rigorous

experimental

design.

4

This

number

of

observations

or

samples

is

called

the

sample

size

(SS),

5

referred

to

as

the

letter

n.

∗

Corresponding

author.

E-mail

address:

mmar[email protected]

(M.

Rodríguez

del

Águila).

The

SS

calculation

involves

the

application

of

a

series

of

mathematical

formulae

that

have

been

designed

to

secure

precision

in

estimating

the

population

parameters

or

to

obtain

signiﬁcant

results

in

those

studies

that

compare

several

treatment

regimens

or

groups.

It

is

important

to

establish

the

SS

before

the

study

is

carried

out,

since

in

this

way

we

can

be

sure

of

recruiting

an

adequate

number

of

patients.

If

this

is

not

done,

we

run

the

risk

of

conducting

an

unnecessary

number

of

tests,

with

the

associated

waste

of

time

and

money,

or

of

collecting

an

insufﬁcient

body

of

data

---

thereby

generating

imprecision

and

very

probably

leading

to

failure

to

detect

signiﬁcant

differences,

when

in

fact

such

differences

might

indeed

exist.

6,7

It

is

common

for

the

number

of

observations

to

be

deﬁned

by

the

investigator,

according

to

the

existing

economic

and

human

resources,

and

on

the

time

available

for

carrying

out

the

study.

8

0301-0546/$

–

see

front

matter

©

2013

SEICAP.

Published

by

Elsevier

España,

S.L.U.

All

rights

reserved.

http://dx.doi.org/10.1016/j.aller.2013.03.008

486

MM

Rodríguez

del

Águila,

AR

González-Ramírez

As

has

been

mentioned,

the

SS

of

a

study

is

deter-

mined

using

mathematical

formulae

designed

to

the

effect.

Accordingly,

we

will

need

prior

information

that

can

be

obtained

from

historical

studies,

the

literature,

or

from

a

pilot

study.

In

this

context,

pilot

studies

are

small

studies

carried

out

under

the

same

conditions

as

the

global

or

larger

study,

but

involving

a

limited

sample

size

of

10

or

20

subjects

---

thereby

allowing

us

to

correct

possible

errors

in

imple-

menting

the

project.

The

preliminary

results

afforded

by

such

pilot

studies

produce

information

for

establishing

the

deﬁnitive

sample

size.

There

are

studies

in

which

it

proves

difﬁcult

to

recruit

the

necessary

number

of

patients;

in

most

such

cases

the

study

involves

a

rare

disease

for

which

the

number

of

cases

is

limited,

such

as

for

example

idiopathic

solar

urticaria.

Even

in

these

cases,

however,

it

is

advisable

to

determine

the

SS,

attempting

to

carry

out

the

study

on

a

multicentre

basis

in

which

each

participating

centre

con-

tributes

a

certain

number

of

cases.

When

a

study

of

this

type

is

not

possible,

the

considerations

referring

to

SS

are

made

according

to

the

maximum

number

of

patients

that

can

be

recruited

in

the

course

of

the

study

---

but

this

implies

the

important

inconvenience

of

a

decrease

in

precision.

It

is

not

unusual

to

ﬁnd

studies

that

establish

two

pri-

mary

objectives

and/or

several

secondary

objectives.

In

theory,

each

primary

objective

should

be

associated

to

its

own

SS.

Choosing

a

smaller

SS

results

in

diminished

statisti-

cal

power.

Theoretically,

we

should

choose

the

largest

SS

of

the

primary

objectives,

since

failure

to

do

so

would

cause

the

primary

objectives

without

a

sufﬁcient

SS

to

automat-

ically

become

secondary

objectives

or

simply

exploratory

objectives.

9

The

SS

is

partially

dependent

on

the

size

of

the

popula-

tion

of

origin.

In

order

to

establish

the

necessary

number

of

patients

in

a

study,

we

generally

start

by

assuming

that

populations

of

unknown

or

inﬁnite

size

must

be

sampled.

In

some

studies

we

will

need

to

sample

populations

of

ﬁnite

size

(or

N),

particularly

in

descriptive

surveys

where

this

size

must

be

incorporated

into

the

calculations.

In

fact,

the

SS

in

the

formulae

that

include

N

in

the

calculation

tends

to

con-

verge

with

the

size

in

which

this

parameter

is

not

included.

Most

authors

consider

a

population

to

be

ﬁnite

if

N

is

less

than

100,000

subjects.

Factors

inﬂuencing

the

calculation

of

sample

size

In

calculating

the

sample

size,

we

must

ﬁrst

take

a

number

of

factors

into

account,

since

they

condition

the

different

formulae

used

to

establish

SS.

10

These

factors

are

the

fol-

lowing:

(1)

The

type

of

study

involved:

descriptive,

observational

or

experimental.

In

descriptive

studies

with

ﬁnite

popu-

lations,

we

also

need

to

know

the

population

size,

N.

(2)

The

˛

(type

I

error)

and

ˇ

(type

II

error)

errors

we

are

willing

to

accept.

In

case

of

doubt,

we

adopt

˛

=

0.05

and

ˇ

=

0.10

or

0.20

as

standard

values,

with

the

following

exceptions:

(a)

In

descriptive

studies

we

only

require

the

˛

error

or

conﬁdence

level

in

the

estimation,

together

with

the

precision

(magnitude

or

width

of

the

conﬁdence

interval).

(b)

In

experimental

or

observational

studies

we

require

both

˛

and

ˇ

error.

(3)

The

response

variables

to

be

observed

and

their

level

of

measurement

(i.e.,

whether

they

are

quantitative

or

qualitative:

means

or

proportions

(%)).

(4)

The

minimum

difference

to

be

detected

between

the

treatment

groups

or

between

the

null

hypothesis

and

the

alternative

hypothesis.

This

will

depend

on

the

study

involved.

The

smaller

the

difference

we

wish

to

detect,

the

larger

the

number

of

subjects

we

must

include

in

the

study.

This

difference

should

be

not

only

clinically

signiﬁcant

but

also

realistic.

In

descriptive

studies,

the

difference

is

reﬂected

by

the

amplitude

of

the

conﬁ-

dence

interval

calculated

in

the

estimation.

(5)

When

the

variables

analysed

in

the

study

are

of

a

quan-

titative

nature,

their

variability

must

be

considered,

measured

in

terms

of

variance

or

standard

deviation

(SD).

If

there

is

little

variability,

the

required

number

of

subjects

is

much

smaller

than

when

the

variability

referred

to

the

analysed

characteristic

is

large.

The

vari-

ability

can

be

obtained

from

the

literature

sources

or

from

pilot

studies.

(6)

Skewness

(laterality)

of

the

hypothesis

test:

i.e.,

whether

it

is

a

one-

or

two-tailed

test.

Studies

involv-

ing

one-tailed

testing

generally

require

a

smaller

sample

size

than

those

with

two-tailed

testing,

though

the

former

should

only

be

contemplated

when

the

direction

of

the

test

is

evident.

(7)

Losses

referred

to

patient

localisation

or

follow-up.

These

losses

should

be

added

to

the

sample

calculation

made.

(8)

The

different

groups

to

be

compared

and

the

compar-

isons

to

be

made

between

them.

When

several

groups

are

contrasted,

the

formulae

used

to

determine

SS

must

document

information

on

the

number

of

groups

con-

sidered

in

the

study.

Failure

to

do

so

can

result

in

the

propagation

of

˛

error,

which

would

exceed

the

initially

deﬁned

level

of

5%.

11

Example:

Suppose

we

wish

to

examine

the

effectiveness

of

four

different

treatments

(A,

B,

C,

and

D)

in

patients

with

atopic

dermatitis,

evaluating

the

number

of

successes

or

failures

with

each

of

them.

If

we

perform

two-by-two

comparisons,

a

total

of

six

comparisons

would

have

to

be

made

(A---B,

A---C,

A---D,

B---C,

B --- D

and

C---D).

The

probability

of

obtaining

a

correct

decision

referred

to

H

0

in

one

such

test

would

be

(1

−

˛)

=

95%,

and

the

probability

of

obtaining

a

correct

decision

in

all

the

above

tests

therefore

would

be

0.95

6

=

0.74.

The

probability

of

a

wrong

decision

in

any

of

them

would

be

1

−

0.74

=

0.26,

which

is

far

higher

than

the

generally

established

value

of

˛

=

0.05.

The

formulae

referred

to

SS

are

little

affected

by

the

magnitude

of

N

referred

to

the

population,

since

the

larger

the

latter,

the

more

stable

the

SS

value

tends

to

become.

The

result

of

applying

a

formula

for

calculating

SS

gener-

ally

yields

a

non-whole

number.

In

this

sense,

SS

is

taken

to

be

the

rounded

whole

number

or

integer

(e.g.,

for

n

=

120.34

we

take

121).

Sample

size

calculation

487

Calculation

of

sample

size

The

different

formulae

available

for

calculating

sample

size

are

described

below,

distinguishing

between:

-

Descriptive

studies

-

Experimental

or

observational

studies

Descriptive

studies

Descriptive

studies

are

those

designed

to

estimate

popula-

tion

parameters,

generally

proportions

or

percentages

and

means.

Among

these

studies,

a

distinction

is

established

between

those

with

ﬁnite

populations

and

those

with

inﬁnite

populations.

12

Finite

populations

Estimation

of

a

proportion.

The

estimation

of

a

pro-

portion,

percentage

or

prevalence

is

obtained

from

the

following

formula:

n

=

t

2

˛

∗

p

∗

q

∗

N

(N

−

1)

∗

e

2

+

t

2

˛

∗

p

∗

q

where

n

=

sample

size

to

be

calculated;

N

=

size

of

the

population

from

which

the

sample

is

drawn;

p

=

expected

percentage

of

the

response

variable;

q

=

1

−

p

(inverse

of

the

above);

e

=

accepted

margin

of

error

(usually

between

5

and

10%).

(This

error

is

one-half

of

the

width

of

the

conﬁ-

dence

interval

calculated

for

the

parameter,

or

equivalently

2*e

is

the

width

or

amplitude

of

the

interval.

It

is

expressed

as

a

percentage);

t

˛

=

value

of

the

normal

curve

associated

to

the

conﬁdence

level.

For

a

conﬁdence

of

95%,

this

value

is

1.96;

for

a

conﬁdence

of

90%,

the

value

is

1.64,

and

for

a

conﬁdence

of

99%

it

is

found

to

be

2.57

(for

two-tailed

testing).

In

calculating

n,

it

is

important

for

all

the

ﬁgures

to

receive

the

same

format,

i.e.,

all

as

fractions

or

all

as

per-

centages.

Estimation

of

a

mean

(normal

variable).

The

formula

used

to

calculate

sample

size

in

the

estimation

of

a

mean

is

sim-

ilar

to

that

given

above:

n

=

t

2

˛

∗

s

2

∗

N

(N

−

1)

∗

e

2

+

t

2

˛

∗

s

2

where

n

=

sample

size

to

be

calculated;

N

=

size

of

the

pop-

ulation

from

which

the

sample

is

drawn;

s

2

=

variance

of

the

variable

for

which

we

want

to

estimate

the

mean;

e

=

margin

of

error.

It

is

expressed

in

the

same

units

as

the

variable

for

which

the

mean

is

to

be

estimated.

The

interpretation

is

the

same

as

before;

accordingly,

the

amplitude

of

the

conﬁ-

dence

interval

will

depend

on

the

measure

we

are

estimating

(for

variables

with

a

very

broad

range

of

values,

the

margin

is

larger

than

in

the

case

of

variables

with

a

smaller

range

of

values.

Example:

For

cholesterol

we

take

a

margin

of

error

of

5

or

10

units,

while

in

the

case

of

bilirubin

we

would

take

0.5---0.7);

t˛

=

value

of

the

normal

curve

associated

to

the

conﬁdence

level.

Examples:

In

a

population

of

2000

males

of

the

same

age,

race

and

height,

and

with

very

similar

habits,

simple

random

sampling

has

been

decided

to

determine

the

following

characteristics:

•

Prevalence

of

seasonal

allergy;

•

Mean

inspiratory

reserve

volume

(IRV).

In

reference

to

the

ﬁrst

parameter

we

accept

an

error

of

5%,

while

for

the

second

we

accept

5

ml.

Based

on

the

data

obtained

from

a

the

proportion

of

individuals

who

are

allergic

is

in

the

order

of

20%,

and

the

variance

of

the

mean

IRV

is

350

ml.

We

wish

to

determine

the

sample

size

that

would

be

needed

in

order

to

make

these

estimations,

with

a

95%

conﬁdence

level:

-

p

=

80%,

q

=

20%,

s

2

=

350;

-

t˛

=

1.96

(conﬁdence

95%);

-

e

=

5%

(for

the

ﬁrst

case);

e

=

5

(for

the

second

case);

-

N

=

2000.

By

substituting

in

both

formulae,

we

have:

n

=

t

2

˛

∗

p

∗

q

∗

N

(N

−

1)

∗

e

2

+

t

2

˛

∗

p

∗

q

=

1.96

2

∗

0.8

∗

0.2

∗

2000

(2000

−

1)

∗

0.05

2

+

1.96

2

∗

0.8

∗

0.2

=

219.04

∼

=

220

n

=

t

2

˛

∗

s

2

∗

N

(N

−

1)

∗

e

2

+

t

2

˛

∗

s

2

=

1.96

2

∗

350

∗

2000

(2000

−

1)

∗

5

2

+

1.96

2

∗

350

=

52.40

∼

=

53

Thus,

in

order

to

estimate

the

percentage

of

allergic

sub-

jects

we

require

220

people;

while

for

estimating

the

mean

IRV

we

need

53.

If

only

53

individuals

are

taken,

we

would

be

able

to

estimate

the

mean

IRV

but

would

be

unable

to

ensure

estimation

of

the

prevalence

of

allergic

subjects

with

an

error

of

5%

(the

error

would

be

greater).

Therefore,

in

order

to

cover

both

objectives,

we

would

have

to

use

the

larger

sample,

i.e.,

220

cases.

Inﬁnite

populations

In

the

case

of

inﬁnite

populations,

the

size

of

the

popu-

lation

exerts

no

inﬂuence,

and

the

formulae

referring

to

sample

size

can

be

simpliﬁed,

giving

rise

to

the

following

expressions:

Estimation

of

a

proportion.

n

=

t

2

˛

∗

p

∗

q

e

2

Estimation

of

a

mean

(normal

variable).

n

=

t

2

˛

∗

s

2

e

2

where

n

=

sample

size

to

be

calculated;

p

=

percentage

or

presence

of

the

study

characteristic;

q

=

1

−

p;

s

2

=

variance

of

the

variable

for

which

we

want

to

estimate

the

mean;

488

MM

Rodríguez

del

Águila,

AR

González-Ramírez

Table

1

Sample

size

values

for

estimating

a

percentage

with

an

error

of

5%

and

a

conﬁdence

level

of

95%,

in

inﬁnite

populations.

p

q

p*q

n

(rounded)

0.5

0.25

384.16

385

0.4

0.6

0.24

368.79

369

0.3

0.7

0.21

322.69

323

0.9

0.1

0.09

138.29

139

e

=

accepted

margin

of

error;

t˛

=

1.96

(95%

conﬁdence

level).

Examples:

We

wish

to

estimate

the

parameters,

but

in

this

case

in

inﬁnite

populations

or

populations

of

more

than

100,000

individuals.

Estimation

of

a

proportion.

n

=

t

2

˛

∗

p

∗

q

e

2

=

1.96

2

∗

0.8

∗

0.2

0.05

2

=

245.86

∼

=

246

Estimation

of

a

mean

(normal

variable).

n

=

t

2

˛

∗

s

2

e

2

=

1.96

2

∗

350

5

2

=

53.78

∼

=

54

As

can

be

seen,

the

sample

sizes

are

somewhat

larger

than

those

calculated

above

for

ﬁnite

populations.

Both

sizes

(those

of

ﬁnite

and

inﬁnite

populations)

become

more

simi-

lar

as

the

population

subjected

to

sampling

becomes

larger.

When

we

have

no

a

priori

values

for

the

proportions

to

be

estimated,

we

can

use

p-

and

q-values

of

50%.

This

is

the

least

favourable

situation,

in

the

sense

that

it

yields

a

larger

sample

size,

as

can

be

seen

in

Table

1.

It

is

always

advisable

to

use

prior

pilot

studies

capable

of

giving

us

an

idea

of

the

proportion

we

wish

to

estimate,

in

order

not

to

draw

more

sample

than

is

actually

needed.

In

the

case

of

estimating

means,

when

the

variance

is

not

known

but

the

mean

has

been

established,

we

can

take

a

standard

deviation

(square

root

of

the

variance)

equiva-

lent

to

at

least

half

of

the

mean,

for

example:

to

estimate

a

mean

of

14

referred

to

haemoglobin,

we

can

take

a

devia-

tion

(variance)

of

7

(49),

whereby

the

sample

size

for

an

error

of

two

units

and

a

95%

conﬁdence

level

would

be

48

subjects.

Experimental

studies

In

the

case

of

experimental

or

observational

studies,

cal-

culation

is

made

of

sample

sizes

when

comparing

two

proportions

and

two

means.

12

The

size

of

the

population

N

does

not

intervene

in

these

calculations.

Comparison

of

two

proportions

When

considering

the

comparison

of

two

proportions,

for

example

the

percentage

improvement

after

the

administration

of

two

different

antihistamines

in

patients

with

allergic

rhinitis,

the

formula

for

calculating

SS

would

be:

n

=



t

˛

∗



2

∗

p

∗

q

+

t

ˇ

∗

√

p

1

∗

q

1

+

p

2

+

q

2



2

(p

1

−

p

2

)

where

n

=

number

of

patients

required

in

each

of

the

groups;

p

1

=

proportion

in

the

usual

treatment

group;

p

2

=

proportion

in

the

new

treatment

group;

q

1

,

q

2

=

inverse

of

the

above

(q

1

=

1

−

p

1

,

q

2

=

1

−

p

2

);

p

=

mean

value

of

the

two

propor-

tions

(p

1

+

p

2

)/2;

q

=

1

−

p;

t˛

=

value

of

the

normal

curve

associated

to

type

I

error

(˛);

the

value

of

˛

is

usually

between

5

and

10%;

tˇ

=

value

of

the

normal

curve

associ-

ated

to

type

II

error

(ˇ);

the

value

of

ˇ

is

between

10

and

20%.

Comparison

of

two

means

The

formula

for

calculating

SS

when

comparing

two

means

would

be

as

follows:

n

=

2

∗

s

2

∗

(t

˛

+

t

ˇ

)

2

(x

1

−

x

2

)

2

where

n

=

number

of

patients

required

in

each

of

the

two

groups;

x

1

,

x

2

=

estimated

means

of

the

groups

to

be

com-

pared;

s

2

=

mean

estimated

variance

of

the

two

groups.

If

a

pilot

study

has

been

made,

this

variance

can

be

estimated

as

a

weighted

average

of

the

variances

of

both

groups:

s

2

=

(n

1

−

1)

∗

S

1

2

+

(n

2

−

1)

∗

S

2

n

1

+

n

2

−

2

with

s

1

2

,

s

2

,

n

1

,

and

n

2

as

the

variances

and

initial

sample

sizes

in

each

group,

respectively;

t˛

=

value

of

the

normal

curve

associated

to

type

I

error

(˛);

the

value

of

˛

is

usually

between

5

and

10%;

tˇ

=

value

of

the

normal

curve

associ-

ated

to

type

II

error

(ˇ);

the

value

of

ˇ

is

between

10

and

20%.

Losses

in

sample

calculation

The

losses

in

a

study

are

those

subjects

for

which

it

has

not

been

possible

to

obtain

information,

on

the

grounds

that

they

were

not

available

at

the

time

of

the

investigation.

12

Losses

are

due

to

different

causes:

•

Patients

who

abandon

the

study

(for

example,

in

a

clinical

trial);

•

Individuals

who

do

not

wish

to

form

part

of

the

study

(fail

to

answer

a

questionnaire,

etc.);

•

Individuals

failing

to

report

for

the

visit,

revision,

etc.;

•

Subjects

that

cannot

be

located

at

the

time

of

the

study;

•

Adverse

effects

with

some

of

the

treatments;

•

Loss

of

information

due

to

other

reasons

(measurement

error,

imprecision

of

the

measuring

devices,

etc.).

In

sum,

the

causes

of

loss

can

be

grouped

into

three

categories:

Sample

size

calculation

489

•

Withdrawal

or

dropout;

•

No

participation

of

the

subject;

•

No

location

of

the

subject.

All

these

causes

imply

that

the

initially

calculated

sample

size

becomes

smaller,

thereby

reducing

the

power

of

the

study

(for

example,

more

often

declaring

that

there

are

no

differences

between

treatments

when

in

fact

there

are

such

differences).

In

order

to

avoid

this

restriction

in

sample

size,

once

the

latter

has

been

calculated,

we

increase

it

by

the

expected

or

foreseeable

percentage

of

losses,

based

on

the

following

formula:

n

′

=

n

1

−

d

n

′

:

deﬁnitive

sample

size;

n:

initial

sample

size;

d:

expected

proportion

of

losses

expressed

as

a

fraction.

The

percentage

of

losses

is

estimated

from

pilot

studies.

Example:

in

the

study

referred

to

calculation

of

the

sample

size

for

estimating

the

percentage

of

individuals

with

seasonal

allergy

in

ﬁnite

populations,

a

2%

loss

rate

is

estimated

in

relation

to

the

localisation

of

individuals

par-

ticipating

in

the

investigation.

On

incrementing

the

initial

size

by

this

percentage

of

losses,

we

have:

n

′

=

n

1

−

d

=

220

1

−

0.02

=

224.48

∼

=

225

With

a

total

of

225

subjects

we

ensure

that

in

the

event

of

a

2%

loss

rate

or

lower,

the

precision

will

be

as

initially

established

(i.e.,

a

margin

of

error

of

5%)

---

the

precision

increasing

(i.e.,

error

decreasing)

as

the

number

of

cases

increases

(fewer

losses)

and

decreasing

(increased

error)

as

the

losses

increase.

Calculation

of

sample

size

with

R

The

calculations

of

sample

sizes

with

the

R

program

are

made

using

software

packages

developed

for

speciﬁc

pur-

poses,

such

as

epicalc,

13

samplesize

and

pwr.

In

the

present

article

we

use

epicalc,

which

must

be

downloaded

and

installed

as

a

ﬁrst

step.

The

R

packages

can

be

accessed

from

http://cran.r-project.org/web/packages/,

and

the

epicalc

application

is

downloaded

in

compressed

format

suited

for

the

corresponding

operating

system.

Once

the

package

has

been

installed,

it

must

be

loaded

every

time

we

wish

to

use

the

program.

The

installation

is

carried

out

from

the

R

menu,

following

the

instruction

Packages→Install

package(s)

from

local

zip

ﬁles,

and

select-

ing

the

previously

downloaded

ﬁle

in

the

corresponding

folder.

For

loading

the

package

we

follow

the

instruction:

>

library(epicalc)

or

use

the

option

Packages->Load

packages,

from

the

R

menu.

Estimation

of

a

proportion

We

aim

to

calculate

the

SS

for

the

same

study

proposed

in

the

sections

above

(estimation

of

the

prevalence

of

seasonal

allergy).

The

instruction

used

for

both

ﬁnite

and

inﬁnite

popula-

tions

is:

n.for.survey

(p,

delta

=

‘‘auto’’,

popsize

=

NULL ,

deff

=

1,

alpha

=

0.05)

where

p

=

the

proportion

to

be

estimated;

delta

=

accepted

error

(half

of

the

conﬁdence

interval,

with

a

default

value

of

5%);

popsize

=

size

of

the

population.

In

inﬁnite

popula-

tions

this

is

left

in

blank

(default

value);

deff

=

design

of

the

effect,

number

of

patients

required

for

each

sampled

sub-

ject

(default

value

1);

alpha

=

level

of

signiﬁcation

(default

value

5%).

Example:

Thus,

219

subjects

are

needed

to

estimate

the

prevalence

of

seasonal

allergy

in

this

population

of

2000

individuals.

For

inﬁnite

populations,

the

corresponding

instruction

would

be:

In

this

case,

246

individuals

are

required.

490

MM

Rodríguez

del

Águila,

AR

González-Ramírez

Estimation

of

a

mean

Epicalc

does

not

provide

instructions

for

calculating

the

SS

when

we

wish

to

estimate

a

mean

in

either

ﬁnite

or

inﬁnite

populations.

Instead,

we

create

a

function

that

serves

for

this

purpose

by

simply

entering

the

formula

for

the

ﬁnite

case

(in

the

inﬁnite

case

we

enter

a

population

size

of

over

100,000).

The

function

will

comprise

the

parameters

needed

to

apply

the

formula:

a

t-value

associated

to

the

conﬁdence

level

(generally

1.96),

variance,

the

size

of

the

population,

and

the

accepted

error.

On

applying

the

function

to

the

above

described

example

of

the

estimation

of

the

mean

inspiratory

reserve

volume,

we

obtain

the

following:

We

thus

require

53

subjects

in

the

case

of

a

ﬁnite

popu-

lation,

and

54

in

the

case

of

an

inﬁnite

population.

Comparison

of

two

means

The

instruction

calculating

the

SS

for

comparing

two

means

is:

n.for.2means

(mu1,

mu2,

sd1,

sd2,

ratio

=

1,

alpha

=

0.05,

power

=

0.8)

where

mu1:

mean

of

the

ﬁrst

group;

mu2:

mean

of

the

sec-

ond

group;

sd1:

standard

deviation

of

the

ﬁrst

group;

sd2:

standard

deviation

of

the

second

group;

ratio:

ratio

of

cases

between

the

ﬁrst

and

second

group.

The

default

value

is

1

(i.e.,

the

same

sample

size

in

both

groups);

alpha:

level

of

signiﬁcance.

The

default

value

is

5%;

power:

power

of

the

test

(1-beta).

The

default

value

is

80%.

Example:

We

wish

to

compare

two

egg

white-free

diets,

with

the

purpose

of

lowering

the

speciﬁc

IgE-KUi/l

(immunoglobulin

E)

levels

in

individuals

with

egg

white

allergy.

ies

have

shown

that

the

mean

speciﬁc

IgE

level

with

diet

1

is

0.342,

with

a

standard

deviation

of

0.051,

while

in

the

case

of

diet

2

the

mean

speciﬁc

IgE

level

is

0.391,

with

a

standard

deviation

of

0.091.

We

need

to

determine

the

sample

size

needed

to

evaluate

differences

between

the

mean

speciﬁc

IgE

values

after

application

of

the

two

diets

in

two

differ-

ent

groups

of

patients

with

egg

white

allergy,

considering

an

alpha

error

of

5%

and

a

beta

error

of

20%.

Substituting

the

values

of

the

means,

deviations,

alpha

and

power

of

the

test,

we

obtain:

We

therefore

need

a

total

of

74

individuals

with

egg

white

allergy

(37

per

group)

in

order

to

detect

differences

in

spe-

ciﬁc

IgE

between

the

two

diets.

Comparison

of

two

proportions

The

comparison

of

two

proportions

is

performed

in

R

with

the

instruction:

n.for.2p

(p1,

p2,

alpha

=

0.05,

power

=

0.8,

ratio

=

1)

where

in

a

way

equivalent

to

the

section

above,

the

param-

eters

to

be

entered

are:

p1:

proportion

of

the

ﬁrst

group;

p2:

proportion

of

the

second

group;

alpha:

level

of

signiﬁ-

cance.

The

default

value

is

5%;

power:

power

of

the

test.

The

default

value

is

80%;

ratio:

ratio

of

cases

between

the

two

groups

(by

default

we

have

the

same

number

of

cases

in

both

groups).

Sample

size

calculation

491

Example:

We

wish

to

determine

the

percentage

of

dropouts

follow-

ing

the

administration

of

a

treatment

in

two

different

groups

of

individuals.

It

is

known

that

the

dropout

rate

in

group

1

can

be

23%,

versus

about

15%

in

group

2.

We

need

to

deter-

mine

the

sample

size

required

to

evaluate

whether

there

are

differences

in

the

percentages

of

dropouts

between

the

two

groups,

knowing

the

frequency

for

group

2

to

be

half

that

of

group

1,

with

an

alpha

error

of

5%

and

a

beta

error

of

80%.

Substituting

in

the

formula,

we

have:

We

thus

need

613

subjects

in

group

1,

and

306

in

group

2.

It

is

estimated

that

there

will

be

a

loss

rate

of

approxi-

mately

5%

in

each

group,

thus

the

above

sample

size

would

have

to

be

expanded

by

this

percentage:

We

therefore

would

have

to

select

646

subjects

in

the

ﬁrst

group,

and

323

in

the

second.

The

above

operation

can

also

be

deﬁned

from

the

function:

Final

considerations

In

this

article

we

have

addressed

the

calculation

of

sample

size

using

the

formulae

deﬁned

to

the

effect,

applied

by

means

of

the

epicalc

program

to

the

type

of

studies

most

commonly

found

in

health

research.

The

instructions

of

this

package

refer

to

two-tailed

hypotheses.

In

the

case

of

a

one-tailed

hypothesis

test,

the

formulae

change,

and

these

instructions

are

therefore

not

applicable.

The

epicalc

package

comprises

other

instructions

referred

to

the

calculation

of

sample

size

for

cases

in

which

bioequivalence

evaluations

and

non-inferiority

tests

are

contemplated,

and

which

are

found

in

the

documenta-

tion

that

comes

with

the

package.

There

are

also

other

software

packages

14

for

the

calcu-

lation

of

sample

size,

such

as

pwr

(calculation

of

power)

or

MBESS

(for

social

and

behavioural

sciences),

which

are

more

complicated

to

use.

In

any

case,

we

can

always

calculate

any

sample

size

in

a

study

by

simply

applying

the

formula

directly

from

the

instructions

line,

or

by

deﬁning

a

speciﬁc

function

as

we

have

seen

in

some

cases.

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