Sample Size Determination€¦ · • Calculate the sample size for the detectable difference of...

Assoc Prof Dr Sarimah binti Abdullah

Unit of Biostatistics & Research Methodology

Universiti Sains Malaysia

Sample Size Determination

Why do we need to calculate

the sample size?

Sample Size Calculation

A) for Estimation

i. a mean

ii. a proportion

B) for Hypothesis Testing

i. compare means

ii. compare proportions

iii. correlation

C) Diagnostic test

D) Validation

SINGLE MEAN FORMULA

SINGLE PROPORTION FORMULA


for Estimation

1. to estimate the duration of exercise among

adult in Kg X.

2. to estimate the prevalence of obesity

in Kg X.

Estimating a mean

• A study is planned to estimate the duration

of exercise among adult in Kampong X.

• The result should be reported as "mean

duration of exercise (DEx.) and its 95% CI".

e.g. mean DEx. 16.5 mins/day (95% CI: 15.5, 17.5)

The value of a study can be judged by the width

of Confidence Interval.

Wide CI means .. a poor study.

If we plan for 95% confidence (5% error), so Z = 1.96;And SD (σ ) is estimated as 4.3 (Duration of Exercise) (either by previous study or a pilot study; if previous study, state

the reference)

To estimate duration of exercise

If we plan for 95% confidence (5% error), so Z = 1.96;And SD (σ ) is estimated as 4.3 (Duration of Exercise) (either by previous study or a pilot study; if previous study, state

the reference)

Impossible to check for

normality assumption

Now, it is the researcher decision to select which sample size

will be appropriate for the study.

How to report? (in Methodology)

• Sample size was determined as follows.

• The following formula (Daniel, 1999) is used to

calculate the sample size.

Z = 1.96 for 95% confidence

σ = SD of DEx. = 4.3 (Brian, 2002??)

∆ = Precision = 1 min/day

• We need 72 people in order to estimate the mean DEx. with

the precision of 1 min/day.

• We decided to take 90 people (additional 20%) for

anticipated non-response cases.


for Estimation

1. to estimate the duration of exercise among

adult in Kg X.

2. to estimate the prevalence of obesity

in Kg X.

Estimating a Proportion

• A study is planned to estimate the prevalence of

obesity in Kampong X.

• The result should be reported as Prevalence

(Proportion) of obesity and its 95% CI".

In our example data,

we get 37% (95% CI: 27%, 47%).

If we plan for 95% confidence (5% error), Z = 1.96,

and P is estimated as 40% (Prevalence of Obesity) (Literature or Pilot study).

(1) Generally, smaller precision is better.

(2) However, commonly, researchers are limited with

the availability of resources .

(3) It may depend on previous studies:

- Previous studies have reported with a certain width of CIs

in their studies. Somehow, if we want to repeat the study,

we should come out with a better width of CI (added value).

Setting the level of confidence is conventional at 95% (Z = 1.96).

P or SD is estimated by the literature or pilot study.

The remaining question is "HOW TO DECIDE THE PRECISION?".

If we plan for 95% confidence (5% error), Z = 1.96,

and P is estimated as 40% (Prevalence of Obesity) (Literature or Pilot study).

Now, it is the

researcher

decision to

select which

sample size will

be appropriate

for the study.

Relationship between P & Sample Size

0

100

200

300

400

500

600

700

800

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

P

Sa

mp

le S

ize


for Hypothesis Testing


- For Hypothesis Testing

3. to compare duration of exercise between male

and female

4. to compare prevalence of obesity between male

and female

Important Concepts

1. Type I (α error)

2. Type II (β error) / Power of the Study (1-β)

3. Detectable Difference (Detectable

Alternative)

Test

DecisionCorrectType I

Type IICorrect

FalseTrue

Null Hypothesis (Ho)

Reject Ho

Do not reject Ho

Power of the study

"Power to reject the false null

hypothesis"

0.05

0.2

80%

Important Concepts

Detectable Difference

1. What is Detectable Difference?

2. How to decide the Detectable

Difference?

Important Concepts

What is Detectable Difference (Detectable Alternative)?

• The “minimum size of the difference between groups”

that the study could detect !!!

Important Concepts

60.0 Kg 60.1 Kg≈

60.0 Kg 60.5 Kg<


Important Concepts


• The “minimum size of the difference between groups”

that the study could detect !!!

• "The study could detect" means ...

➢ Let's say, you are comparing means of 2 groups, and in

reality, the 2 means are truly different.

➢ And also at the end of the study, you get the result as "two

group means are significantly different" (one is more than

the other).

➢ It means that "you detect the difference".

➢ Let's say, you get the result "the difference is not

significant" ... meaning that "you fail to detect it".

Important Concepts

How to decide Detectable Difference?

It should reflect the “Clinically Significant

Difference” (CSD).

We should be able to detect the “CSD”.

In other words, we should design a study to

detect CSD.

Comparing means of two (2) POPULATIONS

• Comparing 2 means

• Comparing 2 proportions

Using PS software ….

http://biostat.mc.vanderbilt.edu/twiki/bin/view/Main/PowerSampleSize

• Comparing 2 means


Researchers want to compare duration of exercise

between male and female college students.

Objective: To compare mean duration of exercise

between male and female students

( )2

2

βα/2

2 ZZ2σn

Δ

+=

Alpha (0.05)

Power (start with 80%)σ = SD (within group SD of duration of

exercise)

∆ = Detectable Difference (Clinically important

difference)

1

2

3


SD from other study or pilot study

Ratio between 2 groups (m=1 means 1:1)

4. Fill all 5 inputs5


SD from other study or pilot study


4. Fill all 5 inputs

With the

sample size

33 in each

group, we

will achieve

80% power

to detect the

difference of

3 mins/day

(DEx.) with

the Alpha at

0.05.

With the sample size 33 in each group,

we will achieve 80% power

to detect the difference of 3 mins/day (DEx.)

with the Alpha at 0.05.

Example:

Say, in really, the difference is 5 mins/day between male

and female.

With this sample size, you have at least 80% chance to

get the ‘significant’ or ‘positive’ result. (You have at least

80% power to reject the Null).

Say, if the difference is 1 mins/day. So, this sample size

will fail to detect this difference. But it’s OK, we don’t

want to detect this small size. It is not clinically/

practically important.

IN SUMMARY, for comparing 2 means

We need to decide ….

Alpha (0.05; consensus – 0.05)

Power (80%=0.8)

SD (variable of interest – from previous study or pilot study)

Detectable Difference (should reflect clinical/practical importance)

Ratio of sample size between 2 groups (m = 1 “1:1”; m=2 “2:1”)

--------------------------------------------------------------------------

How to report?We use PS software (Dupont & Plummer, 1997) to calculate the

sample size based on comparing 2 means.

To detect the difference of 3 mins/day (Duration of Exercise) with

80% power and alpha 0.05, we need 33 students in each study

group (SD was estimated as 4.3, reference?).

We have decided to take 40 male and 40 female students

(additional 20%) with the anticipation of some non-responses.

Exercise

• Researchers want to compare the SBP between treated and

untreated hypertension patients.

• A recent study revealed that the SD of SBP among hypertension

patients was 10 mmHg (state ref.??).

• The researchers feel that it is important to detect the difference

of 5 mmHg between 2 study groups.

• They plan to take equal sample size (1:1) for 2 study groups

(m=1).

• Set alpha at 0.05 as usual.

• Calculate the sample size to achieve the power of 80%.

64 in

each

group

• Researchers want to compare the SBP between treated and

untreated hypertension patients.

• SD was 10 mmHg (state reference??).

• DD sets at 5 mmHg.

• They plan to take 1:2 ratio for untreated: treated (m=2) (because

difficult to find ‘untreated’).

• Set alpha at 0.05 as usual.

• Calculate the sample size to achieve the power of 80%.

48 untreated

and 96 treated

hypertensive

patients



Researchers want to compare prevalence of obesity

between male and female students.

Objective: To compare the prevalence of obesity


Remember … We have to set Alpha, Power and

Detectable Difference.

Alpha = 0.05

Power = 80% (0.8)

Detectable Difference = ???





Alpha (0.05)

Power (80% = 0.8)

∆ = Detectable Difference (Clinically important difference) (P1-P0)

P0 = Prevalence of obesity among male (Get from literature)

P1 = Prevalence of obesity among female (Set based on

desired DD)

m = 1 (equal ratio between male and female)







Alpha (0.05)

Power (80% = 0.8)

∆ = Detectable Difference (Clinically important difference) (P1-P0)

P0 = 0.27 (Say, we get from literature)

P1 = 0.37 (This is our decision based of DD. Here, we put

0.37. It means that we are setting the DD in this study as 0.10

or 10%)

m = 1 (equal ratio between male and female)



1

2

(P1-P0) is Detectable Difference.


4. Fill all 5 inputs

5

P0 – from previous or pilot study

3

IN SUMMARY, for comparing 2 proportions

We need to decide ….

Alpha (0.05; consensus – 0.05)

Power (80% = 0.8)

Po (Prevalence of obesity among male, 27% (reference?)

Detectable difference (should reflect clinical/practical importance – in this

example, 10% difference is decided. Therefore, P1 = 37%)

Ratio between 2 groups (m=1 “1:1”; m=2 “2:1”)

--------------------------------------------------------------------------

How to report?

We use PS software (Dupont & Plummer, 1997) to calculate the sample size

based on comparing 2 proportions.

To detect the difference of 10% in prevalence of obesity (P0 27% versus P1

37%) between the 2 study groups with 80% power and alpha 0.05, we need

340 male and 340 female students. (Po, the prevalence of obesity among

male was estimated as 27%, ref?) (You may add some e.g. 10%)

Exercise!• Calculate the sample size for the detectable

difference of prevalence 20%. It means P0 = 27% and

P1 = 47%.90 students in

each group.

• Calculate the sample size for the detectable

difference of prevalence 20% (as above).

• And male: female ratio as 2:1 (m=2).

66 female & 132

male students.

Comparing 3 means

Using G power….

Correlation…using online http://www.medic.usm.my/biostat/

C) DIAGNOSTIC TESTS –eg :sensitivity and specificity

http://www.medic.usm.my/biostat/C) DIAGNOSTIC TESTS –eg :kappa aggrement

http://www.medic.usm.my/biostat/D) Validation

–eg :Cronbach alpha

http://www.medic.usm.my/biostat/D) Validation

–eg :intraclass correlation

• For each specific objective, should calculate the

sample size.

• Sometimes, in one objective, more than one variables

of interest (multiple linear regression).

• In this case, we need to calculate for each

variable of interest.

• Then, the biggest sample size will be “the sample size

of the study”.

• We need to add-up 10-20% because we may get non-

response, loss of follow up, or any other loss.

Final COMMENTS

Acknowledgement

Special thank to :

Assoc Prof Dr Mohd Ayub Saddiq

Thank You.

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