1. CALCULATING SAMPLE SIZE
DR ISMARULYUSDA ISHAK

2. OBJECTIVE Calculate sample size for UNKNOWN population
Calculate sample size for KNOWN population

3. Determining Sample Size
An important step in planning a statistical study.
Part of designing a high-quality study.
A study must be of adequate size, relatives to the goal of the study.
An under-sized study is a waste of resources for not having the capability to produce useful result.
An over-sized study uses more resources than necessary.

4. THE APPROACH
To answer the question:
“How many samples would I need to produce a result with a certain level of confidence?”

5. Strategies
Using a census for a small population. A census eliminates sampling error and provides data on all individuals in the population.
Using a sample size of a similar study. Without reviewing procedures employed in the study, you may run the risk of repeating errors that were made in determining the sample size for another study.
Using formula to calculate a sample size.

6. Pitfalls for a study having smaller sample size than it should:
Inadequate power to detect the hypothesized effect.
Small sample size leads to P value being higher & Confidence interval wider.
Thereby concluding no difference between groups.
An under-sized study can be a waste to resources because it is incapable of producing useful study.
“ Absence of evidence is not evidence of absence”

7. Larger than necessary sample size
The larger the sample the more accurate is the study but there must be a balance between sample size and the cost.
Adequate sample size protected some subjects from unnecessarily having to endure treatments, especially those that might have unwanted or grievous outcomes. Larger than necessary sample size put people in jeopardy unnecessarily.

8. Many top international journals recommends publications to report on DSS!
Sample size determination
Directionality. Usually two-tailed.
Alpha level. Usually 5%.
Statistical power used. Usually 80%.
DELTA and reason for this delta. Usually 5-15%.
Statistical tests
Confidence intervals - define the upper and lower range which provide an estimate of precision and can give some indication of clinical significance of the results (especially when the results are not statistically significant).

9. Basic Formula for DSS
n1 = n2 = 2 [zα/2 + z (1 –β ) ]2 ________________ 2

10. Four Factors must be known or estimated to calculate sample size using numerical scale data
The “effect size”: the minimum difference between the two groups to detect significance. “Precision”. - delta
Population standard deviation
Power of experiment to detect postulated effect.
Significance level or Confidence Interval

11. Values of Zα and Z 1-
α Zα Z 1- one-sided two sided 1-β Z 1-

12. Delta Symbol =  Minimum worthwhile difference.
Based on clinical importance, and not on statistical ground.
It should not be confused with statistically significant treatment effect.
The value is DECIDED BY THE RESEARCHER. The smallest difference that is assumed as the most significant clinically. Usually between 5-15% difference. FOR OUR FACULTY IT IS ADVISABLE TO USE 10%.
The researcher must give the justification for choosing the delta value.
Error can also be interpreted as “margin of error between sample mean and population mean”.

13.  Margin of error /effect size 3%, 5% , 10%
The smaller the difference specified, the more the number of subjects needed.
Sample size is inversely proportional to delta square.
Sample size could be reduced by increasing delta, but would run into the risk of having the result of your statistics to be not significant or “negative-result”.
Margin of error /effect size
3%, 5% , 10%

14. Cohen’s d = effect size
Comparing two independent group d= M1-M2 Sp Sp= (n1-1) s1 2 + (n2-1)s22 n1+n2-2

15. Cohen’s d = effect size
Comparing two dependent group d= M1-M2 Sp Sp= s1 + s2 2

16. CALCULATING SAMPLE SIZE FOR SURVEY (LARGE POPULATION)
SINGLE MEAN/ CONTINUOUS

17. CONTINUOUS DATA & SINGLE MEAN
n = [Z /2 x ]2 ______________ 2 Note: You have to know the  value from literature or simulation, and  from your own need.
Cochran, W. G. (1977). Sampling techniques (3rd ed.).
New York: John Wiley & Sons.

18.  DELTA Margin of error /effect size Example: SBP 120 +10 mmHg
3%, 5% , 10%
Example: SBP mmHg
10% x 120 = 12 mmHg : mmHg, n=3
5% x120 = 6 mmHg: mmHg, n=11
3% x 120 = 3.6 mmHg : mmHg, n=178

19. Calculate
A researcher would like to know the average diastolic blood pressure for graduate students. It is estimated that the mean diastolic blood pressure for Malaysian youth is 80mm Hg with s.d. of 10mm Hg. Level of confidence wanted is 95% and the estimate for accuracy is within 2mmHg from the mean blood pressure. Determine the sample size needed for this study.
Effect size/margin of error = 2/80=0.025= 2.5%

20. Answer
x + 2 (95% confidence interval) mmHg n = [Z /2 x ]2 ______________ 2 = (1.96 x 10)2 = 96 22

21. Calculate
A sample of 10 age-3 fish has the following lengths (inches): 2.3, 4.1, 3.9, 3.7, 3.0, 2.5, 3.0, 2.7, 2.9, 3.0. The mean length from the sample is 3.1 and the standard deviation is 2.4. We wish to sample enough fish to achieve precision ± 25% of the mean with 95% certainty.

22. Answer
x + 25% pop mean (95% confidence interval) n = [Z /2 x ]2 ______________ 2 = (1.96 x 2.4 )2 (3.1 x 0.25)2 =36.8, Thus a minimum of 37 age – fish should be measured

23. CALCULATING SAMPLE SIZE
FOR
SINGLE PROPORTION

24. Sample size for nominal data estimating for single proportion for large population
no = (Z /2 )2 p (1 – p) _____________ 2
Cochran, W. G  Sampling Techniques, 2nd Ed., New York: John Wiley and Sons, Inc.

25. How about if we don’t know the proportion?
Assume p = 0.5 and q = 0.5
P = 0.5 is the maximum variability (proportion) for the population.

26. Calculate
A researcher would like to study on Malaysian male population who smoke, especially those pertaining to their smoking habits. It is known that 30% of Malaysian male smoke. Confidence level needed is 95% and the researcher wants the estimation to be accurate within 10%. Determine the sample size needed for this study.

27. Answer
p+ 0.1 ( within 10% and 95% confidence interval) no = (Z /2 )2 p (1 – p) = (1.96) 2 0.3(1-0.3)  = = 81

28. Calculate
Let us assume this time we don’t know the proportion that smoke. What would be your sample size?

29. Comparing Two Group Means (independent groups)
n1 = n2 = 2 C  2
1 + ________________ 2
(Snedecor and Cochran 1989)

30. Comparing two group means (dependent groups)
n1 = n2 = ____ 2
(Snedecor and Cochran 1989)

31. Comparing two proportions. Eg. Between treatment and control.
n1 = n2 = C [p1 (1 –p1) + p2 (1 – p2)] _____________________ 2  P1 treatment p2 control
Fleiss JL 1981 Statistical Methods for rates and proportions. 2nd Ed. New York: Wiley

32. =83.57 = 84 for each group. So total= 170
If the observed p1=0.5 and investigator wishes to detect a rate of 0.25 (p2=0.25), then = =0.25, with alpha =0.05 and power 0.9, C=10.5. So sample size=..
n1 = n2 = C [p1 (1 –p1) + p2 (1 – p2)]
_____________________
 
n1 = n2 = 10.5 [0.5 (1 –0.5) (1 – 0.25)]
_____________________
= = 84 for each group. So total= 170

33. C values for 2 tails Power  5% 1% 95% 13.0 17.8 90% 10.5 14.9 80% 7.8
11.7

34. CALCULATE
A dietary consultant want to test the efficacy of one weight-reduction program to certain age adult. Previous study showed that SD of body weight for that age was 4kg. Assume that the dietician would like to be able to detect 2 kg reduction in weight between control and treated subject with power of 80% and significance level of 5%. Calculate the sample size subjects needed for this experiment.

35. n1 = n2 = 1 + 2 C  2 ________ 2 =1+ 2 (7. 8) (4)2 2 2 = 63
n1 = n2 = C  2 ________ 2 =1+ 2 (7.8) (4)2 2 2 = 63.4 =64 Total = 64 x 2=128 subjects

36. Calculating sample size for SURVEY ONLY!! -KNOWN population

37. according to
Krejcie R V & Morgan D W Determining sample size for research activities. Educational And Psychological Measurement. 30:

38. Value of Chi Square at df=1α
0.1
0.05
0.01
0.001 χ2
2.71
3.84
6.64
10.83

39. calculate
You need to do a research about health awareness among first year UKM medic’s students. The population for all first year UKM FSK’s student are 200. Calculate the sample size that needed to do this survey with significance level of 5% and degree of accuracy is 5%.

40. n = χ2 NP (1 –P) 2 (N-1) + χ2P (1-P) = 3. 84 (200) 0. 5 (1-0. 5) 0

41. BUT USING THIS TABLE- P=0
BUT USING THIS TABLE- P=0.5 – IT’S ALREADY OVERESTIMATE, SO NO NEED TO ADD FOR DROP OUT.

42. The relationship between sample size and total population is illustrated in figure 1.
As the population increase, the sample size increases at a diminishing rate and remains relatively constant at slightly more than 380 cases.

43. Figure 1

44. In determining sample size it would be wise to have extra samples
(in case of occurrence of missing data, outliers and attrition)

45. Withdrawals, missing data and losses to follow up…..
It may be necessary to calculate the number of subjects that need to be approached in order to achieve the final desired sample size. A proportion would drop out and this percentage can be estimated from pilot study or literature. Suppose it is expected that 10% would drop out, and we have previously determine the sample size needed is 52.

46. The formula to make up for the losses of samples:
n* = =  58
( )

47. What to do if you have no choice about sample size?
Often a study has a limited budget.
It is hard to argue with budgets and supervisors.
Note: Sample size is one of several quality characteristics of a statistical study.
We need to focus also on other aspects of research quality.
Asking bigger budget? Better instrument?, Using stratification? Narrow the scope of the study (so that budget can be used for samples), making sure the research has external and internal validity etc.etc.

48. Sample size does matter!
Bartlett J.E., Kotrlik J.W., Higgins C.C Organizational Research: Determining
Appropriate Sample Size in Survey Research
Information Technology, Learning, and Performance Journal, Vol. 19, No. 1, Spring
THANK YOU
Sample size does matter!

49. Exercise Choose any value for mean and standard deviation
Eg- BMI= kg/m2
MDA= nmol/l protein
What is your sigma value and calculate delta value for these effect size
(3%, 5% and 10%)
Calculate sample size using formula for single mean