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SAMPLE SIZE ESTIMATION DR. SHRIRAM V. GOSAVI MODERATED 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 Sample Do 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 size The 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 Validity Precision = lack or control of systematic error = lack of random error Validity and Precision (2) results validity actual estimator target estimator 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 Accepted Rejected True Correct Power = (1-Beta) Type I error (or) alpha error False Type II error (or) beta error Correct Reality of hypothesis 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 Z 2σ 2 n d2 Qualitative Z p(1 p) n d2 2 Descriptive study When 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 form q = 100-p d = 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 0.05 d = 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 deviation d = 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 0.8 Mean = 10.2 Standard Deviation = 2.1 Z α = 1.96 for α at 0.05 d = 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 Where Zα = 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 size Z2α = 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 = 1.96 P1 = .40 P2 = .32 d = 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