Determination of Sample

Size

Dr.Deepak Langade

Objectives

List the factors influencing the sample size

Appreciate importance of incorrect sample

size in research

Calculate the sample size using

appropriate formulae

Factors affecting sample size

Size of population

Resources – subjects, financial, manpower

Method of Sampling- random, stratified

Degree of difference to be detected

Variability (S.D.) – pilot study, historical

Degree of Accuracy (or errors)

- Type I error (alpha) p<0.05

- Type II error (beta) less than 0.2 (20%)

- Power of the test : more than 0.8 (80%)

Statistical Formulae

Correct () decisions and Types

of Errors (X) in hypothesis testing

X

X

Difference exists (H1) No Difference (H0)

Difference exists

(H1)

No Difference

Do not reject (H0)

TRUE Situation

CONCLUSION hypothesis test

(Power or 1-beta)

Type II error or

Beta error

Type I error or

Alpha error

Approach to sample sizeDetermine the expected difference

Find out the Standard deviations of both groups

Set alpha error to be tolerated viz. P = 0.05

Decide the power of the study desired viz. 80%, beta error 0.2

Select the appropriate formula

Calculate the sample size using the formula

Give allowance for drop-out rate

Give allowance for non-compliance of treatment if possible

Incorrect sample size

Wrong conclusions

Poor quality research (Errors)

Type II error can be minimized by increasing the sample

size

Waste of resources

Loss of money

Ethical problems

Delay in completion

Formulae for Sample Size

Comparison of means

(two groups)

Alpha=0.05,

Beta=0.2, (power 80%)

Between group comparison (Unpaired)

n = 16 X (S.D./M1-M2)2

Within group comparison (Paired)

n = 8 X (S.D. of differences/M1-M2)2

2

2

/2

2

difference

)Z(2

Zn

Formula for difference in

means

Sample size in each

group (assumes equal

sized groups)

Represents the desired

power (typically .84

for 80% power).

Represents the desired

level of statistical

significance (typically

1.96).

Standard deviation of

the outcome variable Effect Size (the

difference in

means)

Comparison of percentages

(two groups)

Alpha=0.05,

Beta=0.2, (power 80%)

n = 8 X p1q1 + p2q2

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

(p1-p2)2

2

21

2

/2

)(p

)Z)(1)((2

p

Zppn

Formula for difference in

proportions

Sample size in each

group (assumes equal

sized groups)

Represents the desired

power (typically .84

for 80% power).

Represents the

desired level of

statistical

significance

(typically 1.96).

A measure of

variability (similar to

standard deviation)

Effect Size (the

difference in

proportions)

Comparison of one mean only

Alpha=0.05,

Beta=0.2, (power 80%)

n = 8 X (S.D./M1-M0)2

Sample Size Example

Effect on sleep

Sleep Aid Example : 1 Sample

Study the effect of new sleep aid

1 sample test

Parameter – Sleep time after taking

the medication for one week

Two-sided test, α = 0.05, power =

90%

Difference = 1 (4 hours of sleep to 5)

Standard deviation = 2 hr

Sleep Aid Example

1 sample test

2-sided test, α = 0.05, 1-β = 90%

σ = 2hr (standard deviation)

δ = 1 hr (difference of interest)

2 2 2 21 / 2 1

2 2

( ) (1.960 1.282) 242.04 43

1

Z Zn

Effect of Difference

Change difference of interest from 1hr to 2 hr

n goes from 43 to 11

2 2

2

(1.960 1.282) 210.51 11

2n

Effect of Power

Change power from 90% to 80%

n goes from 11 to 8

(Small sample: start thinking about using the t distribution)

2 2

2

(1.960 0.841) 27.85 8

2n

Effect of S.D.

Change the standard deviation from 2

to 3

n goes from 8 to 18

2 2

2

(1.960 0.841) 317.65 18

2n

Sleep Aid Example: 2 Sample

2 2 2 21 / 2 1

2 2

2( ) 2(1.960 1.282) 284.1 85 170 total!

1

Z Zn

Original design (2-sided test, α = 0.05, 1-β = 90%, σ = 2hr, δ = 1 hr)

Two sample randomized parallel design

Needed 43 in the one-sample design

In 2-sample need twice that, in each group!

4 times as many people are needed in this design

Conclusion

2 2

1 / 2 1

2

4( )2

Z ZN

Changes in the detectable difference have HUGE impacts on sample size

20 point difference → 25 patients/group

10 point difference → 100 patients/group

5 point difference → 400 patients/group

Changes in α, β, σ, number of samples, if it is a 1- or 2-sided test can all have a large impact on your sample size calculation

Group Activity

Group Task 1 - Question

The cure rate of disease is 20% with a known drug

treatment. It is claimed that yoga is better than the

drug and a trial is to be conducted find out the

truth. It is decided that a even 10% increase in

cure rate would be clinically important.

The alpha and beta were set at 0.05 and 0.2.

The results will be analysed using Chi Square test.

How many patients would be required for the trial?

Task 1 - Answer

Aim – To see whether yoga is better than standard drug Rx in curing the pt.

Analysis type- comparison of proportion

Parameters- cure rate 20% vs 30%

No. of groups – 2

p1=20 q1=80, p2=30 q2=70

Set alpha=0.05, beta=0.2, Power=0.8

Statistical formula to be used

n = p1q1 + p2q2 X 8

(p1-p2)2 Ans. 296

Group Task No. 2 - Question

The mean(+SD) hospital stay of patients after a

conventional surgical procedure (CP) is 12.3

(4.8) days. A modified procedure (MP) is to be

tried to reduce the hospital stay.

Their hospital stay will be compared using

unpaired t test at p<0.05 with power of 80%.

The minimum clinically important difference in the

duration of hospital stay is expected to be 3.

Calculate the sample size for each group ?

Task 2 - Answer

Aim – To see whether modified procedure reduces the hospital stay as compared to conventional proc

Analysis type- comparison of mean, unpaired data

Parameters- duration of hospital stay 12.3 vs 9.3

No. of groups-2

Given M1=12.3, M2=9.3, SD= 4.8

Set alpha=0.05, beta=0.2, Power=0.8

Statistical formula to be used

n = 16 X (S.D./M1-M2)2

Ans 40.96

Group Task 3 - Question

The mean fruit juice consumption in the population

is 5 oz./day.

Dennison and colleagues wanted to know whether

mean juice consumption in 2 year old children is

different from 5 oz./day – either more or less by1

oz/day.

SD is 3 oz/day.

Calculate the sample size required ?

Task 3 - Answer

Aim – To see whether fruit juice consumption differs by 1 from the population (Normal standard) mean of 5oz./day

Analysis type- comparison of mean, paired data

Parameters- fruit juice/day 5 vs 6 or 4

No. of groups-1

Given M1= 4 or 6, M0=5, SD= 3

Set alpha=0.05, beta=0.2, Power=0.8

Statistical formula to be used

n = 8 X (S.D./M1-M0)2

Ans 72

Demonstration

Thank You !