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**Sample size calculation**

Dr Sa’ed H. Zyoud

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**Testing hypothesis (Outcomes)**

Decision Errors Two types of errors can result from a hypothesis test. Type I error. occurs when the researcher rejects a null hypothesis when it is true. The probability of committing a Type I error is called the significance level. This probability is also called alpha (α). Type II error. A Type II error occurs when the researcher fails to reject a null hypothesis that is false. The probability of not committing a Type II error is called the Power of the test. The probability of committing a Type II error is called Beta (β). .

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**Factors that affect the sample size**

1. Power of the study The power of a study is the probability of detecting a real difference. In other words, it is same as the probability of making a positive diagnosis when the disease is present. For a good study one should have to take at least 80% power. An increase in the sample size or a greater difference between the control and the test groups leads to an increase in the power of the test.

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2. Level of significance Level of significance is the probability of rejecting null hypothesis when it is true. This is the most important factor in determination of sample size. Therefore, the level of significance must be fixed before the testing of hypothesis, estimation and sample size calculation. In a standard situation, the probability can be taken as 0.05 or 0.01.

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**Estimating a proportion**

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**1. Raosoft sample size calculator**

An automated software program, (Raosoft sample size calculator: ( is used to calculate the required sample size for this study.

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**Raosoft sample size calculator**

The estimated sample size is …….. patients out of the ………. eligible patients registered in the public health care clinics.

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**Raosoft sample size calculator**

In order to minimize erroneous results and increase the study reliability, the target sample size will be increased 5% to 10%.

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**2. Danial sample size formula**

The sample size formula (Daniel, 1999) is used, which is If your population more than Where Z: statistic for a level of confidence. (For the level of confidence of 95%, which is conventional, Z value is 1.96). P: expected prevalence or proportion. (P is considered 0.5) d: precision. (d is considered 0.05 to produce good precision and smaller error of estimate)

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A hospital administrator wishes to know what proportion of discharged patients is unhappy with the care received during hospitalization. If 95% Confidence interval is desired to estimate the proportion within 5%, how large a sample should be drawn? n = Z2 p(1-p)/d2 =(1.96)2(.5×.5)/(.05)2 =384.2 ≈385 patients

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**Adjusted sample for population less than 10.000**

n / (1+ (n/population)) = adjusted sample Where n= calculated sample size from Danial formula

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**3. Pocock’s sample size formula**

Pocock’s sample size formula which is a common formula for comparing two proportions can be used. This equation provides a continuity corrected value intended to be used with Chi-square or Fischer’s exact tests (Pocock, 1983) .

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**Pocock’s sample size formula**

n = [P1 (1-P1) + P2 (1-P2)] ( Zα/2 + Z β) 2 (P1-P2)2 Where: n: required sample size P1: estimated proportion of study outcome in the exposed group P2: estimated proportion of study outcome in the unexposed group α: level of statistical significance Zα/2: Represents the desired level of statistical significance (typically 1.96 for α = 0.05) Z β: Represents the desired power (typically 0.84 for 80% power) n for each group *2= total sample (i.e. for the 2 groups)

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**Typical values for significance level and power**

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**4. Altman’s nomogram (Altman 1980) n = total sample (i. e**

4. Altman’s nomogram (Altman 1980) n = total sample (i.e. for the 2 groups)

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P = (P1 +P2) 2 Where: P: mean of the two proportions P1: estimated proportion of study outcome in the exposed group P2: estimated proportion of study outcome in the unexposed group Standardised difference = (P1 – P2) √ [P (1- P)]

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**Pocock’s sample size formula**

This equation assumes that the comparison is to be made across two equally sized groups. However, comparisons in observational studies are mainly made across two unequally sized groups. In this case, the sample size should be adjusted according to the actual ratio of the two groups in order to reflect the inequality (Whitley and Ball, 2002).

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**The adjusted sample size (nadjusted) depends on the following formula (Whitley and Ball 2002):**

nadjusted = n × (k +1)2 4 k where: n: sample size = total sample (i.e. for the 2 groups) k: actual ratio of the two groups

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**Estimating a mean n = Z2 σ2 / d2 Where “**

Z: statistic for a level of confidence. (For the level of confidence of 95%, which is conventional, Z value is 1.96). σ: estimated standard deviation from previous studies d: precision. (d is considered 0.05 to produce good precision and smaller error of estimate)

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