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Power & Sample Size Calculations

Presented by: Ms. Kamla Kumari D Maharaj MSc Nursing CON JPMC KarachiCourse facilitator: Ms. Rabia Riaz Dated: 28th Sep, 2010

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Sample size

• Sample size refer to the number of subjects or participants studied in a trial, including the treatment and control group, where applicable.

• Sample size refers to the specific size of the group or groups being studied in research.

• Four criteria are used to estimate the appropriate sample size for a study. Sometime called power analysis.

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Criteria for estimating sample size

Level of Significance (alpha)

Statistical power (beta-1)

Expected difference (effect size).

Standard deviation

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Level of significance (alpha)

This is the threshold for finding statistical significance. Normally this is set out at .05, a 5% chance of rejecting null hypothesis when there is no fact no significance difference or relationship between underling population. As alpha get smaller, sample size requirement increase.

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Power (beta-1)

Power is 1-Beta and is defined as the probability of correctly finding statistical significance. A common value for power is .80.

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Expected difference (effect size)

This is the expected difference or relationship between two independent samples. Also known as the effect size. the effect size is determined by literature review, logical assertion, and conjecture.

Formula:

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Basic Terms and Concepts involved inSample Size and Power Estimation

Null Hypothesis:

is the supposition that the effect which we are checking for does not exist.

comparing treatment to no-treatment:

H0 = treatment has no effect

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Basic Terms and Concepts involved inSample Size and Power Estimation

Alternative Hypothesis:

is the supposition that the effect which we are checking

for does exist.

comparing treatment to no-treatment:

H1 = treatment has effect

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Hypothesis Testing in Sample Size or PowerEstimation

Hypothesis tests are inferential procedures, they

concern the inferences from sample to population.

It involves the calculation of some test statistics.

The most commonly used test statistics are

Z (normal), χ2 (Chi-square), and t.

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Two Possible Errors of Hypothesis Testingin Sample Size or Power Estimation

The Type I Erroroccurs when we conclude from an experiment that adifference between groups exists when in truth it does

not.Rejecting H0 when H0 is in Fact TrueProbability of making type I error is denoted by “α” ,Investigators reject H0 and declare that a real effect

exists. when the chance of this decision being wrong is less than 5%. This is what is meant when it is claimed that the result is statistically significant at p<.05

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Two Possible Errors of Hypothesis Testingin Sample Size or Power Estimation cont…

The Type II Error

occurs when we conclude that there is no difference

between treatments when in truth there is a difference

fail to reject H0 when H0 is in Fact False

probability of making type II error is denoted by β

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Errors and Probabilities in Hypothesis Testing

Type-I Error Power

Power Type-II Error

H0 True H0 False

Rej

ect

H0

Not

Rej

ect

H0

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Basic Terms and Concepts involved inSample Size and Power Estimation cont…

Two-tailed testWhen the investigator is interested in determiningwhether treatment A is different from treatment B(either better or worse) a 2 tailed test is indicated.Usually a 2 tailed test is performed with the risk ofmaking aType-1 error set at α / 2 in each tail.For a 2 tailed test at α = .05 and equal allocation oftype-1 error to each tail Zα = 1.96

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Basic Terms and Concepts involved inSample Size and Power Estimation cont…

One-tailed testSometimes an investigator is only interested in adifference between treatments in one direction.This is appropriate when either1. the scientific reasoning behind the experiment leadsto a prediction in one direction or2. a new treatment will be used if it is better than thestandard but abandoned if it is worse or the sameFor a 1 tailed test at α = .05 Zα = 1.65

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SAMPLE SIZE CALCULATIONS :SYMBOLS

Z α is the “Standard Normal Deviate” corresponding to

the probability α

Z β is the “Standard Normal Deviate” corresponding to

the probability β

Common α, β= .2 .1 .05 .025 .01 .005

value Zα β= .84 1.28 1.65 1.96 2.33 2.58

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Mathematical Symbols used to denotesome common summary Statistics

Population

Parameters

Mean Variance Standard

deviation

Proportion Correlation

Greek

lettersμ σ2 σ π ϒ

Roman

letters_

×

s2 s p r

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Sample Size and PowerCalculations

The calculation of sample size depends onthe summary statistics chosen. The most commonchoices areTreatment meane.g. average blood pressure, average cholesterol,average days in hospitalTreatment proportione.g. % of patients who die, recover, achieve sometherapeutic goal or any defined state

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Most Common Sample Size Calculations

Comparing 2 independent groups- means

Comparing 2 related groups- means

Comparing 2 independent groups- proportions

Comparing 2 related groups- proportions

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Sample size estimation for tests between twoindependent sample proportions

Formula:

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Sample size estimation for tests between twoindependent sample proportions cont…

Where asN= the sample size estimateZcv=Z critical value for alpha (.05 alpha has a Zcv of1.96)Z power=Z value for 1-beta (.80 power has a Z of0.842)P1=expected proportion for sample 1P2=expected proportion for sample 2

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Sample size estimation for tests between twoindependent sample proportions cont…

Proportion Example

Alpha=.05

Power=.80

P1=.70

P2=.80

p= .75

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Sample size estimation for tests between twoindependent sample means

whereN= the sample size estimateZcv=Z critical value for alpha (.05 alpha has aZcv of 1.96)Zpower=Z value for 1-beta (.80 power has a Zof 0.842)s=standard deviationD=the expected difference between the twomeans.

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Sample size estimation for tests between twoindependent sample means cont…

Mean Example

Alpha=.05

Power=.80

D=10

S=20

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Reference

• Germann, E. (2003). Sample Size and Statistical Power in the Planning of Experiments retrieved from www.nihtraining.com/cc/ippcr/current/.../Johnson111 505bw.ppt on dated 20/08/2010.

• Johnson,L,L.(2005). Sample Size and Power retrieved from http://www.nihtraining.com/cc/ippcr/current/downloads/Johnson111505bw.ppt on dated 22/08/2010.

• Thalheimer, W. Cook, S.(2002). How to calculate effect sizes from published research articles: A simplified methodology retrieved from www.work-learning.com/effect_sizes.htm. 0n dated 21/08/2010