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Sample size estimation: Part I
Sasivimol Rattanasiri, Ph.D
Section for Clinical Epidemiology and Biostatistics
Ramathibodi Hospital, Mahidol University
E-mail: [email protected]
Outline
Estimation
•Proportion
•Diagnostic study
•Mean
Hypothesis testing
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Outline
Hypothesis testing
Test for difference of proportions • One proportions
• Two independent proportions
• Two dependent proportions
• More than two groups of proportions
Test for difference of means • Two independent means
• Two dependent means
• More than two groups of means
Outline
Hypothesis testing
Test for equivalence• Two independent means
• Two dependent means
• Cross-over trial
Test for non-inferiority • Two independent means
• Cross-over design
• Two independent proportion
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Can not study in the whole population
•Time, financial, resources
Study plan
•Manpower, budget, time
Ethical consideration
•Studies that are too small or too large may be judged as unethical studies
Why do we need to estimate sample size?
Too large sample size
Study could have met the objective before
trial is ended, thus some participants are un-neccessarily enroll to the study.
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Too small sample size
Study will have a little chance to meet
the study’s objective, participants may be
put through the potential risk of the trial rather benefit.
Required information
The primary research objective: PICO
Types of statistic inferences • Estimation
• Hypothesis test
Study design
Type of outcome/variable
Basic information e.g. • Prevalence or incidence
• Mean and SD
• Size of difference
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Estimation proportion
Example 1
Aim to estimate prevalence of hepatitis B positive in blood donors Previous studies reported 5% to 15%Set
• Alpha=0.05 • Confidence interval width=0.03 • Assign P=.15
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Example 2
Aim to estimate prevalence of steroid abuse in Thailand.
The stratified-cluster random sampling were applied, with strata refers to 5 regions and clusters were those provincial hospitals within each strata.
Previous study reported the proportion of using steroid abuse was 575/100,000 patients/year.
Investigator had set confidence interval width of 300/100,000 patients/year.
Example 2
Since the design was a multi-centre-hospital base study, sample size estimation should be take into account the design effect.
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Example 2
The design effect is equal to 1/intra-class correlation.
Intra-class correlation is usually set at 0.2-0.5.
Sample size for multi-stage sampling is therefore equal to sample size from simple random sampling multiplied by the design effect.
For this study, if intra-class correlation was 0.5, the design effect = 1/0.5 = 2.
Sample size in this study was 2441x2=4882 subjects.
N for diagnostic study
Cross-sectional design
Parameters• Sensitivity• Specificity• PV+• PV-• LR+• LR-• Area under ROC
New test Standard test
Total
+ -
+ a b R1
- c d R2
Total C1 C2 n
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Example 3
• Aim to estimate sensitivity of visual inspection acetic acid (VIA) vs Pap smear
• Previous study reported sensitivity = 70% and ∆ = 5%
• Prevalence of cervical cancer =14%
Estimation mean
•As for a range estimation
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Example 4
• Aim to estimate mean systolic blood pressure in Thai adults.
• Study design was cross-sectional study
• Previous study reported variance of SBP = 100 mmHg.
• ∆ = 2 mmHg
Hypothesis testing
Required components α (FP, regulator’s error) β (FN, investigator’s error) Ratio Information in control (reference) group
•Event rate (proportion) •Mean (SD)
Delta •Size of detectable •Clinically important difference •Effect size
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Test for difference of one proportion with reference population
Prevalence of CKD in Thailand versus Asian countries
Prevalence of breast cancer in Thailand versus European countries
Prevalence of cardiovascular disease in Asian versus Western countries
Test for difference of one proportion with reference population
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As for the CKD prevalence study (Ingsathit et al.), investigator would like to compare the CKD prevalence in Thailand with the prevalence in Asian population.
The pooled prevalence of CKD stage III or higher in Asian population was 8.3% (95% CI: 4.3%, 12.4%) (Anothaisintawee et al., 2009).
How many subjects are required to detect the difference?
Example 5
•Required information
•Type I and II errors are set at 5% and 20%,
•Size of difference that they want to detect is ±5%.
•Size of detectable = 5%
•Prevalence for the reference population = 8.3%
Example 5
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Example 5
Two independent proportions
•Compare proportions of CKD between patients whose HbA1C < 6.5 and ≥ 6.5
•Compare incidence of CV events between patients do/do not receive Rosigitazone
•Compare incidence of micro- or macro-albuminuriabetween patients who received ACEI and other hypertensive drugs
•Compare rates of breast cancer between HRT vs Non-HRT
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Two independent proportions
Investigators would like to assess whether receiving calcium supplement 500 mg/day would be able to reduce osteoporotic fracture comparing with receiving placebo.
The rate of fracture in general population was 0.1 and reducing the rate to be 0.05 would be clinical significant.
• Set• Type I & II errors were set at 5% and 20%• Detectable size = 5%
Example 6
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Example 6
Take into account 20% loss follow up, total sample size should be 434+(434x0.20)=521 subjects/group
Sample size estimation based on testing a magnitude of effect size
For case-control or cross-sectional study:
For cohort or randomized control trial:
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• In the case-control study of risk factors of steroid abuse, investigators want to assess whether using traditional medicine is associated with adrenal insufficiency or adrenal crisis.
•Previous studies reported prevalence of using traditional medicine in general population was about 15%.
•The odds ratio that they want to detect is set at 1.5.
Example 7
•Since the case(i.e., adrenal insufficiency or adrenal crisis) is quite rare, the ratio of case versus controls is set of 1:4.
•False positive & false negative rates are set at 5% and 20%, respectively. The estimated P2 can be estimated as
Example 7
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Two dependent proportions
Matched case-control study
Matched cohort study
Matched few characteristics between case and control, or expose vs non-expose
• The equation for sample size calculation for m discordant pair is
• Ψ = Odds ratio, R = risk ratio,
• P = Proportion of expose-discordant pair
= (n10+n01)/m
Two dependent proportions
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Total M pairs is needed to adjust with discordant proportion as
Two dependent proportions
•As for case-control study of factors associated with hip fracture, authors wanted to perform a pair-match by matching case and control with age.
•One interested factor was history of taking traditional medicine in which previous literature reported about 30% in general population.
•Ratio of case per control, type I, type II errors, and OR were set at 1:1, 0.05, 0.2, and 2.5 respectively.
Example 8
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Correlation of exposure between case and control should be take into account in estimating sample size.
Correlation depends on proportion of expose in controls and OR that wants to detect.
Two dependent proportions
Equation for estimate correlation:
where,
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Example 8
More than two proportions
Compare incidence of gastro-intestinal ulcer between Celecoxib, Valdecoxib, and Naproxen in Arthritis patients;
Compare recovery rates between Acyclovir plus Prednisolone, Acyclovir alone, and Prednisolone alone in Bell’s palsy;
Comapre remission rates between MMF plus Prednisolone, Cyclophosphamide plus Prdenisolone, and MMF alone in LN
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• Investigators wish to estimate sample size for a randomized control trial of Treatments in Bell’s Palsy.
• Three treatments of interest,
• Acyclovir plus Pednisolone,
• Acyclovir, and
• Prednisolone alone.
• Review previous studies showed that recovery rate within 3 months after receiving Prednisolone(standard treatments) was about 40%-60%.
Example 9
• If Acyclovir works well, it should at least 10% higher recovery rates compare with Prednisolone.
•Set •a ratio between treatment groups is 1:1:1, • alpha and beta = 0.05 and 0.20
Example 9
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•Since Acyclovir was much more expensive than Prednisolone, investigator may not be able to support and thus
•Re-set
•a ratio between treatment groups is 1:1:2,
• Increasing recovery rate as 15%.
•How much sample size that they need?
Example 9
•Suppose that investigators could later enroll subjects only 500 because time and budget were running out,
•How was the power of the test based on sample size 500?
Example 9
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Two independent means
Bone mineral density (BMD) between calcium supplement versus placebo
Systolic/diastolic blood pressure between angiotensin-receptor blocker (ARB) and angiotensin-converting enzyme inhibitor (ACEI) in diabetic patients
Pain scores of Arthritis patients who receive Celecoxib and Ibuprofen
•Ho: 1 -2= 0•Ha: 1-2 0
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Investigators would like to conduct a randomized control trial to compare efficacy of Mycophenolate Mofetil (MMF) and Cyclophosphamide (CYC) in induction therapy of Lupus Nephritis.
One of their interested outcome is serum creatinine within 6 months.
Example 10
As for previous systematic review(Kamanamoolet al., 2010), serum creatinine after receiving CYC was 0.902 (SD=0.179)
If the MMF is efficacious, it should be able to decrease serum creatinine at least 0.1 comparing with CYC
•Set
•MMF: CYC= 1:3
•Type I and II errors are set at 5% and 20%
Example 10
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Two dependent means
Before-after study
Matched case-control
Mean systolic blood pressure before and after receiving treatment
Mean pain scores before/after receiving treatments
• Ho: before = after
• Ha: beforeafter
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From the previous example, suppose that investigators are interested in comparing change of serum creatinine within groups, i.e., before and after receiving treatment for 6 months.
Example 11
More than two means
Comparison of BMI between AA, TA, and TT genotype of fat mass and obesity-associated (FTO) polymorphism.
Comparison of pain, function, and quality of life scores between glucosamine sulphate, diacerein, glucosamine plus diacerein in knee osteoarthritis patients.
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Compare mean VAS score (ranges 0 -100) between •Celecoxib, •Valdecoxib, and •Naproxen after receiving treatments for 7 days.
Exploring previous studies showed that mean and SD of VAS score in Naproxen group were 85 and 25, respectively.
Example 12
Investigators thought that effects of Celecoxib and Valdecoxib should be similar and they should be able to reduce pain scores at least 15 units compare with pain score in Naproxen group.
•Assume further that SDs for the three groups are common (i.e., 25), i.e., variance = 625
•A ratio of Celecoxib: Valdecoxib: Naproxen is assigned as 1:1:1.
Example 12
