SAMPLE SIZE ESTIMATION

DR. SHRIRAM V. GOSAVIMODERATED BY: BHARAMBE SIR

FRAMEWORK

• What is sample size? & Why it required?

• Practical issues in determining sample sizes• Determining sample size• Sample size calculation by different ways• Sample size estimation for descriptive studies• Sample size estimation for hypothesis testing• Summary • References

WHAT IS SAMPLE SIZE? & WHY IT REQUIRED?

• Sample size means “n”• After planning for any research it is important to know

that how many subjects should be included in their study i.e. sample size & how these subjects should be selected (sampling methods).

• If a study does not have an optimum sample size, the significance of the results in reality (true differences) may not be detected.

• This implies that the study would lack power to detect the significance of differences because of inadequate sample size.

How Big a SampleDo You Need?

• Small sample size (less than the optimum sample size) – May fail to detect a clinically important difference, – or may estimate those effects or associations too imprecisely,

– Even the most rigorously executed study may fail to answer its research question

• Very large sample size (more than the optimum size):– Involve extra patients– Costs more – Difficult to maintain high data quality

NOT VERY SMALL AND NOT VERY LARGE

Practical issues in Determining Sample Sizes

• Importance of the Research Issue: If the results of the survey research are very critical, then the sample size should be increased. As sample size increases, the width of the confidence interval decreases.

• Heterogeneity of the population: If there is likely to be wide variations in the results obtained from various respondents, the sample size should be increased

Practical issues in Determining Sample Sizes

• Funding: quite often, budgetary constraints limit the sample size for the study

• Number of sub-groups to analyze: If multiple sub-groups in a population are going to be analyzed, the sample size should be increased to ensure that adequate numbers are obtained for each sub-group

Determining sample sizeThe things you need to know:• Random Error:• Systematic Error:• Validity & Precision:• Null Hypothesis:• Alternate Hypothesis:• Hypothesis Testing:• Type I & II Error:• Power:• Effect Size:• Design Effect:

Random error

• It describes the role of chance, • Sources of random error include:

- sampling variability, - subject to subject differences & - measurement errors.

• It can be controlled and reduced to acceptably low levels by:

- Averaging, - Increasing the sample size &- Repeating the experiment

Systematic error (Bias)

• It describes deviations that are not a consequence of chance alone.

• Several factors, including:- Patient selection criteria, might contribute to

it. • These factors may not be amenable to measurement, • Removed or reduced by good design and conduct of

the experiment. • A strong bias can yield an estimate very far from the

true value.

Validity and Precision (1)

Fundamental concern: avoidance and/or control of error

Error = difference between true values and study results

Accuracy= lack of error

Precision= lack of random

error

Validity= lack or

control of systematic

error

Validity and Precision (2)

validity

target estimator

actual estimator

results

Precision Precision

Any possibility of errors?• Since our decision is based on the sample we

chose from the population, there is a possibility that we make a wrong decision

• A type I error occurs when Null hypothesis is rejected when it is in fact true

• A type II error occurs when Null hypothesis is not rejected when it is false

Summary of possible results of any hypothesis test

Researcher’s Decision

Reality of hypothesis

Accepted Rejected

True CorrectPower = (1-Beta)

Type I error (or) alpha error

False Type II error (or) beta error

Correct

Type I error / α error• The probability of making a error is called as

level of significance i.e. consider as 0.05 (5%). • For computing the sample size its specification

in terms of Zα is required. • The quantity Zα is a value from the standard

normal distribution corresponding to α• Sample size is inversely proportional to type

I error.

Type II error / β error• For computing the sample size its specification

in terms of Zb is required. • The quantity Zb is a value from the standard

normal distribution corresponding to β• A type II error is frequently due to small

sample sizes• The exact probability of a type II error is

generally unknown

Power of the study:

• Probability that the test will correctly identify a significant difference or effect or association in sample should one exist in the population

• 1- β corresponds to sensitivity of a diagnostic test, i.e. probability of making a positive diagnosis when disease is present

• Thus, sample size directly related to the power of study.

• A well designed trial should have a power of at least 0.8

Effect size

– It should represent smallest difference that would be of clinical or biological significance.

– If the effect size is increased, the type II error decreases

– A large sample size is needed for detection of a minute difference.

– Thus, the sample size is inversely related to the effect size.

Variability of the measurement:

– The variability of measurements is reflected by the standard deviation or the variance.

– The higher the standard deviation, larger sample size is required.

– Thus, sample size is directly related to the SD

Types of Problems in Medical Research

Estimation: (Prevalence/Descriptive Study)- Given proportion of prevalence- Given mean & standard deviation

Testing hypothesis: (Cohort/Case Control/Clinical Trial)

- Given two proportion or incidence rates- Given two group means and standard deviations

SAMPLE SIZE CALCULATION BY DIFFERENT WAYS

• By use of Formulae• Computer Soft wares• Readymade tables,• Nomograms

Formulae &

Problems

Sample size

Quantitative Qualitative

2d

2σ2Zn

2

2

d

p)p(1Zn

Descriptive studyWhen proportion is the parameter of our study

n = Z2α

* p * q/d2

where z = standardized normal deviate (Z value)p = Proportion or prevalence of interest (from pilot study or literature survey) expressed in percentage formq = 100-pd = clinically expected variation (precision)

Example From a pilot study it was reported that among

headache patients 28% had vascular headache. It was decided to have 95% CI and 10% variability in the estimated 28%. How many patients are necessary to conduct the study.

ANSWER

p = 28%, q = 72%Z α = 1.96 for α at 0.05d = 10% of 28% = 2.8

n = (1.96)2 * 28* 72 /(2.8)2 = 987.8

B. When mean is the parameter of our study

n = Z2α* S2/d2

Where Z = Standardized Normal Deviate (Z value)S = Sample standard deviationd = Clinically expected variation

Example In a Health survey of school children it is found

that the mean hemoglobin level of 55 boys is 10.2/100 ml with a standard deviation of 2.1 & Clinically meaningful difference is 0.8Mean = 10.2Standard Deviation = 2.1Z α = 1.96 for α at 0.05d = 0.8

n = (1.96)2 * 2.12/(0.8)2 = 26

Testing Hypothesis

Formulae&

Problems

When mean is the parameter of our study

n = (Zα + Zβ)2 *S 2 * 2/d2

WhereZα = Z value for α error

Zβ = Z value for β error

S = Common standard deviation between two groups

d = Clinically meaningful difference

Example: Quantitative

• An investigator compares the change in blood pressure due to placebo with that due to a drug. If the investigator is looking for a difference between groups of 5 mmHg, then with a between – subject, SD as 10 mmHg, how many patients should he recruit?

ANSWER

n = (Zα + Zβ)2 *S 2 * 2/d2

Zα = 1.96 at α = 5%

Zβ = 1.28 at β = 10%

S = 10

d = 5

Hence, n = 85

When Proportion is the parameter of our study

• Formula:n = Z2α[P1(1-P1) + P2(1-P2)]/d2,

where,n = sample sizeZ2α = confidence interval

P1 = estimated proportion (larger)

P2 = estimated proportion (smaller)

d = Clinically meaningful difference

EXAMPLE

• What sample size to be selected from each of two groups of people to estimate a risk difference to be within 3 percentage points of true difference at 95% confidence when anticipated P1 & P2 are 40% & 32% respectively.

ANSWER

Available information:zα = 1.96P1 = .40P2 = .32d = 0.03

n = (1.96)2[ .40(1-.40) + .32(1-.32)] / (.03)2

n = 1953

SUMMARY: Steps in Estimating Sample Size

• 1. Identify the major study variables.• 2. Determine the types of estimates of study

variables, such as means or proportions.• 3. Select the population or subgroups of interest

(based on study objectives and design).• 4a. Indicate what you expect the population

value to be.• 4b. Estimate the standard deviation of the

estimate.

SUMMARY: Steps in Estimating Sample Size

• 5. Decide on a desired level of confidence in the estimate (confidence interval).

• 6. Decide on a tolerable range of error in the estimate (desired precision).

• 7. Compute sample size, based on study assumptions.

COMPUTER SOFTWARE USED IN ESTIMATION OF SAMPLE SIZE

REFERNCES• Lwanga SK, Lemeshow S. Sample size determination in health

studies - A practical manual. 1st ed. Geneva: World Health Organization; 1991.

• Zodpey SP, Ughade SN. Workshop manual: Workshop on Sample Size Considerations in Medical Research. Nagpur: MCIAPSM; 1999

• Rao Vishweswara K. Biostatistics A manual of statistical methods for use in health , nutrition and anthropology. 2nd edition. New Delhi: Jaypee brothers;2007

• VK Chadha . Sample size determination in health studies. NTI Bulletin 2006,42/3&4, 55 - 62