Recently, I was asked to clarify the difference between relative precision and absolute precision in sample size calculation. This post aims to shed some light on the two concepts.

**Review of Terms**

**True Population value**: The actual value of a population parameter (e.g. prevalence). This is what investigators wish to capture by conducting studies.

**Confidence Interval**: A range of values likely to contain the (true but unknown) value of the population parameter of interest.

Now let us consider the two terms in turn using two examples:

**Example 1 (Absolute Precision)**

A local health department wishes to estimate the prevalence of tonsillitis

among children under five years of age in its locality. It is known that the true rate is unlikely to exceed 20%.

The department wants to estimate the prevalence to * within 5 percentage points of the true value*, with 95% confidence.

How many children should be included in the sample?

**Solution**:

(a) **Anticipated population proportion 20%**

(b) Confidence level 95%

(c)** Absolute precision (15%-25%)** **5 percentage points**

The sample size formula for estimating a population proportion with a given absolute precision is:

Here, the first element (just after the ‘=’ sign, and before ‘P’) refers to the value of Z corresponding to a 95% confidence interval, and is 1.96 according to the Z table. P is 20% or 0.20 in decimal terms; (1-P) is therefore 80% or 0.80; d is 5% or 0.05 in decimal terms. Substituting the decimal values for each component in the formula, we get:

n = 1.96*1.96*0.20*0.80/(0.05*0.05)

= 245.86 ~ 246

Therefore, for P = 0.20 and d = 0.05 a sample size of 246 would be needed.

**Example 2 (Relative Precision)**

An investigator working for a national programme of immunization seeks

to estimate the proportion of children in the country who are receiving Measles vaccinations.

The vaccination coverage is not expected to be below 50%.

How many children must be studied if the resulting estimate is to fall * within 10%* (not 10 percentage points)

*with 95% confidence?*

**of the true proportion****Solution**:

(a) **Anticipated population proportion 50%**

(conservative choice)

(b) Confidence level 95%

(c) **Relative precision (E) (from 45% to 55%) 10% **

The formula for estimating population proportion with a specified relative precision is:

Here, the first element (just after the ‘=’ sign, and before ‘P’) refers to the value of Z corresponding to a 95% confidence interval, and is 1.96 according to the Z table. P is 50% or 0.50 in decimal terms; (1-P) is therefore 50% or 0.50; E is 10% or 0.10 in decimal terms. Substituting the decimal values for each component in the formula, we get:

n = 1.96*1.96*0.50/(0.10*0.10*0.5)

= 384.16 ~ 384

Therefore, for P = 0.50 and E = 0.10 a sample size of 384 would be needed.

As can be seen from the examples above, the difference is subtle, but discernible:

The term **Absolute precision** is used when one wishes to estimate the population parameter to within defined percentage points of the true value. This is described as **‘Estimating P to within “d” percentage points’ **

On the other hand, the term **Relative precision** is used when one wishes to estimate the population parameter to within a defined percentage of the population parameter itself. This is described as * ‘Estimating P to within “E” of P’*.

Although using either approach will likely yield similar looking output, the difference lies in how the output was generated.

Useful links:

Click to access lemeshow_samplesize.pdf

http://apps.who.int/iris/bitstream/10665/41607/1/0471925179_eng.pdf?ua=1

The examples described in this post have been adapted from ‘Sample Size Determination in Health Studies- A Practical Manual’, by Lwanga and Lemeshow (first link).

JoelThanks for the info!

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Hassen IbrahimHi I am asked to calculate a sample size for my 2nd objective of finding risk factors for irrational drugs use study by using double population proportion in cross sectional study. So how can I calculate it.

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drroopeshPost authorDear Hassen,

I am not sure I fully understand your question. Could you please rephrase it for clarity? Thanks!

Regards,

Dr. Roopesh

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Vishal MandhanHi I have an assignment to measure the Basal Metabolic Rate of the students of a specific Medical college currently there are five batches studying, each batch comprises of almost 300-350 students so that will make some 1500 students for the population size from which I need to calculate the sample size. The problem is that I am not sure which technique should be used to calculate it. Although supervisors have guided me to use the convenient purposive technique but can’t use that either, then I asked to some higher authorities they said to use single group mean( On the W.H.O calculator)

Now the main problem I am facing is to estimate the:

1) Population mean

2) Standard Deviation of it

3) Population Variance

I’ll be waiting for your positive feedback. Thanks

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drroopeshPost authorDear Vishal,

The determination of sample size will be influenced by the following:

The outcome measure(s) of your objective(s).

In turn, this is influenced by your research question in the PICO format.

You mention purposive technique, which is a non-probabilistic sampling method, and comes into consideration only after the sample size has been calculated.

Please state your research question in PICO format, and the objective(s) of your study.

With each objective, you must mention the outcome measure as well.

Typically, one estimates sample size for each objective, then chooses the largest sample size from them. This ensures adequate power for each objective.

I hope this helps.

Regards,

Dr. Roopesh

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sathyaseelanwhat is the maximum level of absolute precision for cluster sampling for cross sectional study

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drroopeshPost authorDear Sathyaseelan,

There is no definite answer to your question. Several factors would influence this decision.

Please read the following:

Click to access eb69_bp.pdf

Click to access WaltherAndMoore2005.pdf

https://ump.pnnl.gov/showthread.php/5106-2.3-Confidence-and-Precision

Click to access KS-RA-13-029-EN.PDF

Please read the first 30 odd pages of the last document.

I hope this helps.

Regards,

Dr. Roopesh

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MarvinHow did you arrived at the 246 and 384 number in your Example 1 and Example 2 above?

Not everyone here is a genius! Please show how you computed those numbers.

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drroopeshPost authorDear Marvin,

Thank you for pointing out that the calculation was missing!

I have revised the article to include the calculations.

Hopefully this will help clarify your doubt(s).

Regards,

Dr. Roopesh

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Halimah binti JalilDear Dr. Roopesh

How to calculate the sample size which needed 250 in control and 250 in intervention group using Epi info TM7. I have 7 clinics (clusters). Another thing how to write-up the formula in my research study?

Thank you

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drroopeshPost authorDear Halimah,

I suspect you want a tutorial on sample size calculation for case control studies using Epi Info 7.

Please indicate if this is the case.

Regards,

Dr. Roopesh

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Jen GDear Dr Roopesh,

Thank you for the very clear explanation of relative versus absolute precision. However, one point about the outcome seems counter-intuitive to me, for the same confidence interval and level of precision why does the sample size decrease with decreasing anticipation pop proportion? I would have expected the sample size to increase when something is less prevalent (and thus harder to detect) in the population. Thanks.

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