Unit 9b. Sample size estimation.ppt

Sample size Calculation
Shakir Rahman
BScN, MScN, MSc Applied Psychology, PhD Nursing (Candidate)
University of Minnesota USA
Principal & Assistant Professor
Ayub International College of Nursing & AHS Peshawar
Visiting Faculty
Swabi College of Nursing & Health Sciences Swabi
Nowshera College of Nursing & Health Sciences Nowshera

Objectives
1. Calculate sample size according to particular
type of research, and purpose.
2. Identify and select various software to calculate
sample size according to particular type of
research, and purpose.

Why to calculate sample size?
 To show that under certain conditions, the
hypothesis test has a good chance of showing a
desired difference (if it exists)
 To show to the IRB committee and funding
agency that the study has a reasonable chance to
obtain a conclusive result
 To show that the necessary resources (human,
monetary, time) will be minimized and well
utilized

Cont.….
 Most Important: sample size calculation is an
educated guess
 It is more appropriate for studies involving
hypothesis testing
 There is no magic involved; only statistical and
mathematical logic and some algebra
 Researchers need to know something about what
they are measuring and how it varies in the
population of interest

 SAMPLE SIZE:
How many subjects are needed to assure a given probability
of detecting a statistically significant effect of a given
magnitude if one truly exists?
 POWER:
If a limited pool of subjects is available, what is the
likelihood of finding a statistically significant effect of a
given magnitude if one truly exists?

Before We Can Determine Sample Size We Need
To Answer The Following:
1. What is the primary objective of the study?
2. What is the main outcome measure?
Is it a continuous or dichotomous outcome?
3. How will the data be analyzed to detect a group difference?
4. How small a difference is clinically important to detect?
5. How much variability is in our target population?
6. What is the desired  and ?
7. What is the anticipated drop out and non-response % ?
4.

Where do we get this knowledge?
 Previous published studies
 Pilot studies
 If information is lacking, there is no good way to
calculate the sample size

 Type I error: Rejecting H0 when H0 is true
 : The type I error rate.
 Type II error: Failing to reject H0 when H0 is false
 : The type II error rate
 Power (1 - ): Probability of detecting group difference
given the size of the effect () and the sample size of the trial
(N)

Estimation of Sample Size by
Three ways:
By using
(1) Formulae (manual calculations)
(2) Sample size tables or Nomogram
(3) Softwares

All studies
Descriptive Hypothesis testing
Sample
surveys
Simple - 2 groups
More than 2
groups &
Complex studies
Scenario 1
Precision
Scenario 2
Power

SAMPLE SIZE FOR ADEQUATE
PRECISION
 In a descriptive study,
 Summary statistics (mean, proportion)
 Reliability (or) precision
 By giving “confidence interval”
 Wider the C.I – sample statistic is not
reliable and it may not give an accurate
estimate of the true value of the
population parameter

 Sample size calculation for cross sectional
studies/surveys
 Cross sectional studies or cross sectional survey are done to
estimate a population parameter like prevalence of some
disease in a community or finding the average value of some
quantitative variable in a population.
 Sample size formula for qualitative variable and quantities
variable are different.
 For qualitative variable Suppose an epidemiologist want to
know proportion of children who are hypertensive in a
population then this formula should be used as proportion is a
qualitative variable.

(Qualitative Variable)
Sample Size = (Za/2)2 p(1-p) / d2
 Z1-a/2 = Is standard normal variate (at 5% type 1
error (P<0.05) it is 1.96 and at 1% type 1 error (P<0.01)
it is 2.58).
 As in majority of studies P values are considered significant
below 0.05 hence 1.96 is used in formula.
 p = Expected proportion in population based on previous
studies or pilot studies.
 d = Absolute error or precision – Has to be decided by
researcher.

Example
let us assume that a researcher wants to
estimate proportion of patients having hypertension in
pediatric age group in a city. According to previously
published studies actual number of hypertensive
may not be more than 15%. The researcher wants to
calculate this sample size with the precision/absolute
error of 5% and at type 1 error of 5%. So if we use the
above formula

196
(0.05)
0.15)
0.15(1
x
1.96
n 2
2



n = (Za/2)2 p(1-p) / d2
p: proportion to be estimated = 15% (0.15)
d: the accuracy of estimate (how close to the true proportion) = 5% (0.05)
Za/2: A Normal deviate reflects the type I error
For 95% the critical value =1.96
Thus, for this cross sectional study researcher has to take
at least 196 subjects. If the researcher want to increase
the error (decrease the precision) then denominator will
increase and hence sample size will decrease.

(Quantitative Variable)
Sample Size = (Z 1-a/2)2 SD2/d2
 Z 1-a/2 = Is standard normal variate as mentioned in previous
section.
 SD = Standard deviation of variable. Value of standard
deviation can be taken from previous studies or pilot studies.
 d = Absolute error or precision – Has to be decided by
researcher.

Example
if the researcher is interested in knowing the average
systolic blood pressure in pediatric age group of that
city at 5% of type I error and precision of 5 mmHg
of either side (more or less than mean systolic BP) and
standard deviation, based on previously done studies, is
25 mmHg

96
(5)
(25)
1.96
n 2
2
2


n = (Z 1-a/2)2 SD2/d2
SD: Standard deviation of variable = 25%
d: the accuracy of estimate (how close to the true proportion) = 5%
Z1-a/2: A Normal deviate reflects the type I error
Thus, the researcher will have to take the blood pressure of at
least 96 children to know average systolic blood pressure
in paediatric age group.

Sample size calculation by Software
• Openepi (www.openepi.com)
• Population Size (rough estimate) leave to one million only
• Anticipated % of frequency (expected proportion = p)
• (if unknown = 0.5 or 50%)
• Precision/ Confidence Interval (assumed as 5% or less)
• Design effect (for SRS = 1 or if n > = 3)

Sample size calculation for Case
Control Studies
• In case control studies cases (the group with disease/
condition under consideration) are compared with controls
(the group without disease/condition under consideration)
regarding exposure to the risk factor under question.
• The formula for sample size calculation for this design
also depends on the type of variable (qualitative or
quantitative).

Sample size calculation for Case
Control Studies (Qualitative Variable)
Suppose a researcher want to see the link between
childhood sexual abuses with psychiatric disorder in
adulthood. He will take a sample of adult persons
with psychiatric disorder and will take another sample
of normal adults having no psychiatric disorders. He
will then go retrospectively to see history of childhood
sexual abuse in both groups. Exposure to both groups
will be compared and odds ratio will be calculated. Here
number of people exposed to childhood sexual abuse
is qualitative variable hence this formula will be used
for such type of design

(Qualitative Variable)
Sample Size = r+1/r X (p*) (1-p*)(Zβ + Z a/2)2 /(p1 - p2)2
r = Ratio of control to cases, 1 for equal number of case andcontrol
p* = Average proportion exposed = proportion of exposed cases +
proportion of control exposed/2
Zβ = Standard normal variate for power = for 80% power it is 0.84
and for 90% value is 1.28. Researcher has to select power for the
study.
Z a/2 = Standard normal variate for level of significance as
mentioned in previous studies.
p1 - p2 = Effect size or different in proportion expected based on
previous studies. p1 is proportion in cases and p2 is proportion in
control.

Example
if the researcher wants to calculate sample size
for the above-mentioned case control study to know
link between childhood sexual abuse with psychiatric
disorder in adulthood and he wants to fix power of
study at 80% and assuming expected proportions in case
group and control group are 0.35 and 0.20 respectively,
and he wants to have equal number cases and control;

9
.
138
)
20
.
0
(0.35
1.96)
(0.84
0.275)
(0.275)(1
1
/
2
n 2
2





n = r+1/r X (p*) (1-p*)(Zβ + Z a/2)2 /(p1 - p2)2
p: average proportion exposed = (0.35 +0.20)/2 = 0.275
Thus, the researcher has to recruit at least 139 subjects in
cases and equal number in control as he wants to have
equal number in both.

Sample size calculation for Case Control
Studies (Quantitative Variable)
Suppose a researcher wants to see the association
between birth weight and diabetes in adulthood. The
birth weight being a quantitative data, the researcher will
select one group i.e. cases that will be diabetic adults and
other group i.e. control will be non-diabetic adults. Both
groups will be traced back for data regarding childhood
weight. The formula for sample size calculation is

(Quantitative Variable)
Sample Size = r+1/r X SD2 (Zβ + Z a/2)2 /d2
 r = Ratio of control to cases, 1 for equal number of case and
control
 Zβ = Standard normal variate for power = for 80% power it
is 0.84 and for 90% value is 1.28. Researcher has to select
power for the study.
 Z a/2 = Standard normal variate for level of significance as
mentioned in previous studies.
 SD = Standard deviation of variable. Value of standard
deviation can be taken from previous studies or pilot studies.
 d = Expected mean difference between case and control
(may be taken from previous studies)

Example
If researcher think that difference in mean weight
between case and control may be around 250 gm and
SD is 1 Kg then considering equal number of cases and
control and 80% power the sample size will be:

8
.
250
0.25
1.96)
(0.84
1
1
/
2
n 2
2
2



n = r+1/r X SD2 (Zβ + Z a/2)2 /d2
SD = 1 kg
d: 0.25 kg
Hence researcher has to take 251 subjects in each group (case and
control).

• Two sided Confidence Interval (assumed as 95%)
• Ratio of control to cases (r) (usually 1 in equal samples)
• % of controls exposed (Previous published studies)
• % of cases exposed (Previous published studies)

Sample size calculation for Cohort Studies
In cohort studies healthy subjects with or without
exposure to some risk factor are observed over a time period to see the
event rate in both groups. If a researcher wants to see the impact of
weight training exercise on cardiovascular mortality then he will select
two groups, one consisting of subjects who do exercise and another
consisting of those who don’t do. These groups will be followed up for a
specific time period to see cardiovascular mortality in both groups. At
the end of the study period both groups will be compared for
cardiovascular mortality.

 Za = Standard normal variate for level of significance
 m = Number of control subject per experimental subject
 Zb = Standard normal variate for power or type 2 error
 p1 = Probability of events in control group
 p2 = Probability of events in experimental group

Example
Suppose the researcher wants to see the impact of
weight training exercise on cardiovascular mortality
and according to previous studies proportion of
cardiovascular death in case may be around 20% and
in control it can be around 40% hence sample size
calculation for 5% of significant level and 80% power
with equal number of case and control will be

     
  41
.
84
2
n 2
20
.
0
40
.
0
20
.
0
1
20
.
0
1
/
40
.
0
1
40
.
0
84
.
0
30
.
0
1
30
.
0
1
1
1
96
.
1

 




















m = Number of control subject per experimental subject
p1 = Probability of events in control group = 0.40
P0 = Probability of events in experimental group = 0.20
p* = 0.60/2 = 0.30
Hence researcher has to take 84 subjects in each group.

• Power = usually 80%
• Effect Size = proportion of event in exposed vs unexposed
(pilot study/ previous studies) if unknown 0.5,50%
• Confidence Interval (assumed as 95%)
• % of unexposed with outcome = previous studies
• % of exposed with outcome = previous studies
• Ratio of sample size = Ratio b/w exposed vs unexposed
• 1:1 means 100 subjects in each group
• 2:1 means 200 in one & 100 in other

Sample size calculation for Interventional
studies
• In this kind of research design researcher wants to see
the effect of an intervention. Suppose a researcher
want to see the effect of an antihypertensive drug
so he will select two groups, one group will be given
antihypertensive drug and another group will be give
placebo.
• After giving these drugs for a fixed time period
blood pressure of both groups will be measured and
mean blood pressure of both groups will be compared
to see if difference is significant or not.
• When the variable is quantitative data like blood pressure,
weight, height, etc., then the following formula can be used for
calculation of sample size for comparison between two groups.

(2 Independent Samples)
• Test H0: 1 = 2 vs. HA: 1  2
• Two-sided alternative
• Assume outcome normally distributed with:
S= standard deviation; d=difference between two means;
Zα= 1.96 for 95% confidence level; Zβ= 1.28 for 90% power
or 0.84 for 80% power
  
2
2
2
/
2
d
z
z
S
n group
per

 


Example
Suppose a researcher wants to see the
effect of a potential antihypertensive drug and He wants
to compare the new drug with placebo. Researcher
thinks that if this new drug reduces this blood pressure
by 10 mmHg as compared to placebo then it should
be considered as clinically significant. Let us assume
standard deviation found in previously done studies
was 25 mmHg. Suppose the researcher selects the level
of significance at 5% and the power of study at 80%.,
and he thinks suitable statistical test in this condition will be two
tailed unpaired t test. The effect size in this condition is 10 mmHg.

Hence, the researcher needs 98 subjects per group.
    98
10
84
.
0
96
.
1
25
2
2
2
2
/ 


group
per
n

Comparison between two groups when
endpoint is qualitative
When the endpoint of a clinical intervention study is
qualitative like alive/dead, diseased/non diseased, male/
female etc.
Suppose the researcher is interested in knowing
protective effect of a drug on mortality in patients
of myocardial infarction. He selected two groups of
patients of myocardial infarction one group was given
that drug and another group was given placebo. The
both groups were kept under observation and at the end of study
death in both groups were compared.

Example
Za/2 = 0.05/2 = 0.025 = 1.96 (From Z table) at type 1 error of 5%
Zb = 0.20 = 0.842 (From Z table) at 80% power
p1−p2 = Difference in proportion of events in two groups
P = Pooled prevalence = [prevalence in case group (p1)
+ prevalence in control group (p2)]/2
In above example, let us assume that previous study says that 20% of
patient of myocardial infarction die within a specified time. The
researcher feels that if the drug being tested increases survival to 30%
then the finding can be considered as clinically significant. Effect size
will be difference between proportions. 0.2 – 0.3= –0.1. At 5% of
significance level and 80% power sample size will be Pooled prevalence
= (0.20 + 0.30)/2 = 0.25

Hence, the researcher needs 298 subjects per group.
    298
)
1
.
0
(
25
.
0
1
25
.
0
84
.
0
96
.
1
2
2
2
2
/ 



group
per
n

• Power = usually 80%
• Confidence Interval (assumed as 95%)
• % of unexposed with outcome = previous studies
• % of exposed with outcome = previous studies
• Ratio of sample size = Ratio b/w Interventional vs Control
• (Usually = 1)

Unit 9b. Sample size estimation.ppt

Unit 9b. Sample size estimation.ppt

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Unit 9b. Sample size estimation.ppt