write notes on "hypothesis and underlying principles to estimating behind sample size and power". structured exam oriented notes for rguhs mpt paper. simple indian englsih self explanatory with headings, subheadings, bullet points, key words. flowchart and one diagram.

Reading File

Finding Sources

Searching PubMed

"sample size" AND "statistical power"

Searching the Web

sample size power hypothesis testing type I type II error NIH

Reading File

Reading Web Page

https://pmc.ncbi.nlm.nih.gov/articles/PMC7745163

Hypothesis and Underlying Principles Behind Estimating Sample Size and Power

1. Introduction

Sample size estimation means deciding the minimum number of participants needed before starting a research study.

Power means the ability of a study to detect a real difference or real effect, if it actually exists.

In MPT research, proper sample size is needed to avoid:

Very small study with unreliable results
Missing a useful physiotherapy intervention effect
Wasting time, money and participants by recruiting unnecessarily large numbers
Ethical problems due to exposing subjects to research without adequate chance of getting meaningful results

Key idea: Sample size is calculated from the study hypothesis, expected effect, variability, chosen significance level and desired power.

2. Hypothesis

Definition

A hypothesis is a clear, testable statement or prediction about the relationship or difference between variables.

Example

“Task-specific balance training improves Berg Balance Scale score more than conventional exercise in stroke patients.”

Types of Hypothesis

A. Null Hypothesis (H₀)

The null hypothesis states that there is no real difference, no association or no treatment effect.

Example:

H₀: There is no difference in Berg Balance Scale score between task-specific balance training and conventional exercise.

B. Alternative Hypothesis (H₁ or Ha)

The alternative hypothesis states that a real difference, association or effect exists.

Example:

H₁: Task-specific balance training produces greater improvement in Berg Balance Scale score than conventional exercise.

Types of Alternative Hypothesis

1. Non-directional hypothesis

States that a difference exists, but does not say which group will be better.

H₁: There is a difference in pain reduction between ultrasound therapy and TENS.

This usually needs a two-tailed test.

2. Directional hypothesis

States the expected direction of difference.

H₁: Ultrasound therapy reduces pain more than TENS.

This may use a one-tailed test, if justified before data collection.

Exam point: In most clinical studies, a two-tailed test is preferred because the intervention may produce benefit or harm in either direction.

3. Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether the observed study result is likely due to a real effect or only due to chance.

Basic Steps

State H₀ and H₁
Select the significance level, usually α = 0.05
Collect data from the calculated sample
Apply the appropriate statistical test
Obtain the p value
Compare p value with α
Reject or fail to reject H₀

Interpretation

If p < 0.05, result is statistically significant. Reject H₀.
If p ≥ 0.05, result is not statistically significant. Fail to reject H₀.

Important: Correct wording is “fail to reject H₀”, not “prove H₀ is true.”

A null hypothesis usually represents the default condition of no difference, while H₁ represents a difference or effect (Schwartz's Principles of Surgery, p. 1710).

4. Errors in Hypothesis Testing

When a sample is used to make conclusions about a population, errors can occur.

Actual situation	Research decision	Result
H₀ is true	Fail to reject H₀	Correct decision
H₀ is true	Reject H₀	Type I error
H₀ is false	Fail to reject H₀	Type II error
H₀ is false	Reject H₀	Correct decision

A. Type I Error (α error)

Definition

Type I error occurs when the researcher rejects H₀ even though H₀ is actually true.

It is a false-positive result.

Example

A researcher concludes that a new physiotherapy protocol improves knee pain, but in reality it has no extra benefit.

Alpha level

Represented by α
Usually set at 0.05
α = 0.05 means accepting a 5% risk of false-positive conclusion

B. Type II Error (β error)

Definition

Type II error occurs when the researcher fails to reject H₀ even though H₀ is false.

It is a false-negative result.

Example

A new gait-training programme is truly effective, but the study concludes that it is not effective because the sample was too small.

Beta level

Represented by β
Commonly set at 0.20
β = 0.20 means 20% risk of missing a true effect

Type I error is rejection of a true H₀, whereas Type II error is failure to reject a false H₀ (Schwartz's Principles of Surgery, p. 1711-1713).

5. Statistical Power

Definition

Statistical power is the probability that a study will correctly detect a true effect.

[ \textbf{Power = 1 - β} ]

Common Values

Beta (β)	Power (1 - β)	Meaning
0.20	80%	Common minimum power
0.10	90%	Better power, needs larger sample
0.05	95%	Very high power, needs still larger sample

Example

If a study has:

[ β = 0.20 ]

Then:

[ Power = 1 - 0.20 = 0.80 = 80% ]

This means there is an 80% chance that the study will detect the treatment effect if that effect truly exists.

Power is defined as 1 minus β and is affected by sample size, effect size and variance (The Harriet Lane Handbook, p. 958).

6. Diagram: Relationship Between H₀, Errors and Power

                      TRUE SITUATION IN POPULATION
                   H₀ is true              H₀ is false
              ------------------------------------------------
Decision:       | Correct decision       | Type II error
Fail to reject  | True negative          | False negative
H₀              |                       | β
              ------------------------------------------------
Decision:       | Type I error           | Correct decision
Reject H₀       | False positive         | True positive
                | α                      | Power = 1 - β
              ------------------------------------------------

Memory point:
α = False positive
β = False negative
Power = Ability to find a true positive result

7. Sample Size

Definition

Sample size is the number of participants required to answer the research question with acceptable precision, significance level and power.

It should be calculated before data collection.

Why Sample Size Estimation is Important

Ensures adequate power
Reduces Type II error
Gives reliable and interpretable findings
Prevents underpowered studies
Avoids unnecessary recruitment
Protects participant time and safety
Helps in ethical approval and research proposal writing
Supports proper budgeting and planning

8. Underlying Principles of Sample Size Estimation

Sample size is not selected randomly. It depends on the following main factors.

1. Research Question and Primary Outcome

The first step is to clearly define:

Population
Intervention
Comparison group
Outcome measure
Study design
Main or primary outcome

Example

Component	Example
Population	Patients with chronic low back pain
Intervention	Core stabilization exercise
Comparison	Conventional exercise
Outcome	Oswestry Disability Index score
Design	Randomized controlled trial

Sample size should be calculated mainly for the primary outcome, not for every outcome measure.

2. Type of Outcome Variable

The formula changes according to the type of data.

Type of outcome	Example	Common statistical approach
Continuous	VAS pain score, ROM, 6-minute walk distance	t test, ANOVA
Categorical	Improved/not improved, fall/no fall	Chi-square test, proportion test
Correlation	Relationship between pain and disability	Correlation analysis
Time-to-event	Time to return to work	Survival analysis

3. Level of Significance (α)

Alpha is the probability of Type I error.

Usually α = 0.05
In highly sensitive studies, α may be 0.01
Lower alpha means stricter evidence is required
Lower alpha generally increases required sample size

Relationship

Lower α → stricter significance level → larger sample required

4. Power (1 - β)

Power indicates the probability of detecting a true effect.

Commonly selected power: 80%
Higher power: 90%
Higher desired power requires a larger sample

Relationship

Higher power → lower β → larger sample required

5. Effect Size

Definition

Effect size is the magnitude of difference expected between groups or before-after measurements.

Example

If expected mean reduction in VAS pain is:

Experimental group: 4 cm
Control group: 2 cm

Then expected difference:

[ \Delta = 4 - 2 = 2 \text{ cm} ]

This difference is the expected effect.

Important Rule

Small effect size → large sample needed
Large effect size → small sample may be enough

Effect size should be clinically meaningful and preferably based on:

Previous studies
Pilot study
Published literature
Expert opinion
Minimum clinically important difference, or MCID

A sample-size calculation should use a realistic effect size and should state the primary outcome, variability, beta and expected difference. Arbitrary assumptions can make the calculation misleading (Rheumatology, p. 252).

6. Variability or Standard Deviation (SD)

Variability means how widely individual values differ from the mean.

High variability means participants are very different from each other.
Low variability means participants are more similar.

Relationship

High SD / high variability → larger sample required
Low SD / low variability → smaller sample required

Example

Pain score may vary widely due to:

Severity of condition
Age
Duration of symptoms
Psychological factors
Compliance with exercise
Different medication use

Therefore, studies with highly variable pain scores need more participants.

7. One-tailed or Two-tailed Test

One-tailed test

Tests effect in only one direction
Requires a comparatively smaller sample
Used only when opposite direction is not considered relevant

Two-tailed test

Tests effect in both directions
More commonly used in health research
Usually requires a larger sample than one-tailed testing

MPT exam point: A two-tailed test is generally safer and more acceptable in clinical research.

8. Study Design

Required sample size differs according to design.

Study design	Sample size consideration
Cross-sectional study	Prevalence, allowable error
Case-control study	Exposure difference, odds ratio
Cohort study	Incidence and expected risk
RCT	Difference between intervention and control groups
Pre-post study	Expected mean change and SD of change
Correlation study	Expected correlation coefficient
Diagnostic study	Expected sensitivity, specificity and prevalence

9. Number of Groups

More groups usually need more participants.

Example

Two-group RCT: exercise versus control
Three-group RCT: exercise A versus exercise B versus control

If there are more groups, enough subjects should be present in each group for meaningful comparison.

10. Allocation Ratio

Allocation ratio means the distribution of participants between groups.

Equal allocation

[ 1:1 ]

Example: 30 participants in experimental group and 30 in control group.

This is usually statistically efficient.

Unequal allocation

[ 2:1 ]

Example: 40 participants in experimental group and 20 in control group.

Unequal allocation may be used when:

Treatment is expensive
Control data are easily available
Ethical reasons require more participants to receive the better intervention

However, unequal allocation often increases the total sample required.

11. Dropout or Attrition

Some participants may leave the study due to:

Loss to follow-up
Non-compliance
Travel difficulty
Adverse event
Withdrawal of consent
Change in health status

Therefore, sample size should be increased for expected dropout.

Formula for dropout adjustment

[ \text{Final sample size} = \frac{\text{Calculated sample size}} {1 - \text{Expected dropout proportion}} ]

Example

Calculated sample size = 50 participants
Expected dropout = 20% = 0.20

[ \text{Final sample size} = \frac{50}{1-0.20} ]

[ = \frac{50}{0.80} = 62.5 ]

Therefore, recruit approximately 63 participants.

12. Cluster Sampling and Design Effect

In cluster studies, participants are recruited in groups such as:

Hospitals
Villages
Schools
Physiotherapy centres
Wards

Participants within one cluster may be similar. Therefore, the usual sample size is multiplied by the design effect.

[ \text{Adjusted sample} = \text{Usual sample size} \times \text{Design effect} ]

9. Important Relationships for Examination

Sample size ↑  →  Power ↑  →  β ↓

Effect size ↓  →  Sample size ↑

Variability / SD ↑  →  Sample size ↑

Desired power ↑  →  Sample size ↑

Alpha ↓  →  Sample size ↑

Dropout expected ↑  →  Recruitment target ↑

10. Flowchart: Steps in Sample Size Estimation

Start
  ↓
Define research problem and primary outcome
  ↓
Frame H₀ and H₁
  ↓
Choose study design and statistical test
  ↓
Decide α level, usually 0.05
  ↓
Decide desired power, usually 80% or 90%
  ↓
Estimate clinically meaningful effect size
  ↓
Estimate SD / variability or expected proportion
  ↓
Calculate sample size using formula or software
  ↓
Adjust for number of groups, design effect and dropout
  ↓
Round up to nearest whole number
  ↓
Final recruitment target

11. Common Formulae

A. Sample Size for Estimating a Single Proportion

Used in prevalence studies.

[ n = \frac{Z^2pq}{d^2} ]

Where:

n = required sample size
Z = standard normal value, 1.96 for 95% confidence level
p = expected prevalence
q = 1 - p
d = allowable error or precision

B. Sample Size for Comparing Two Means

Used when comparing continuous outcomes, such as VAS, ROM, walking distance or disability score.

[ n = \frac{2\sigma^2 (Z_{\alpha/2} + Z_{\beta})^2}{\Delta^2} ]

Where:

n = sample required per group
σ = standard deviation
Δ = minimum meaningful difference between groups
Zα/2 = Z value for selected alpha level
Zβ = Z value for selected beta level

For α = 0.05 and power = 80%:

[ Z_{\alpha/2} = 1.96 ]

[ Z_{\beta} = 0.84 ]

Note: The formula should be selected according to study design and outcome variable. In practical research, G*Power, OpenEpi, nMaster or consultation with a statistician may be used.

12. Worked Example

Research Question

Does a 6-week core stabilization programme reduce pain more than conventional physiotherapy in chronic low back pain?

Assumptions

Primary outcome: VAS pain score
Expected difference between groups: 2 cm
Standard deviation: 2.5 cm
Alpha: 0.05
Power: 80%
Two groups
Expected dropout: 15%

Interpretation

A difference of 2 cm is the effect size of clinical interest
SD represents expected variability in pain score
Alpha protects against false-positive conclusion
Power protects against missing a true treatment benefit
Final sample should be increased to compensate for dropout

13. Underpowered and Overpowered Studies

Underpowered Study

An underpowered study has insufficient sample size.

Problems

High chance of Type II error
Real treatment benefit may be missed
Non-significant result may be wrongly interpreted as “no effect”
Results may be inconclusive

Overpowered Study

An overpowered study has unnecessarily very large sample size.

Problems

Wastes resources and time
May detect a statistically significant but clinically unimportant difference
May expose more subjects than needed

Key point: Statistical significance does not always mean clinical significance.

For example, a very large study may show a statistically significant 0.2 cm reduction in VAS pain. This difference may not be important for the patient.

14. Difference Between Statistical Significance and Clinical Significance

Statistical significance	Clinical significance
Based mainly on p value	Based on benefit to patient
Indicates result is unlikely due to chance	Indicates change is meaningful in practice
Influenced by sample size	Influenced by MCID and patient outcome
Large sample can make very small effects significant	A small effect may still be unimportant clinically

Example

An intervention improves walking distance by 3 metres with p < 0.05.

It may be statistically significant
But may not be clinically meaningful for a stroke patient

15. Short Note: A Priori and Post Hoc Power

A Priori Power Analysis

Done before the study begins.

Purpose:

To calculate required sample size
To plan adequate recruitment
To avoid underpowered research

This is the preferred method.

Post Hoc Power Analysis

Done after completion of the study.

Purpose:

To estimate observed power based on obtained results

Limitation

Post hoc power has limited value when interpreting a non-significant study result. A confidence interval and effect estimate are usually more useful.

16. Common Mistakes in Sample Size Estimation

Selecting sample size by convenience only
Not identifying a primary outcome
Using an unrealistic effect size
Ignoring variability or SD
Not accounting for dropout
Using a one-tailed test without justification
Not mentioning alpha and power
Using a formula not suitable for study design
Confusing statistical significance with clinical significance
Concluding “no effect” only because p > 0.05 in a small study

17. One-Minute Revision Box

H₀ = No difference / no association

H₁ = Difference / association exists

Type I error = False positive = α

Type II error = False negative = β

Power = 1 - β

Common alpha = 0.05

Common power = 80% or 90%

Small effect size = larger sample needed

High variability = larger sample needed

Expected dropout = increase final recruitment target

Sample size must be calculated before data collection.

18. Conclusion

Sample-size estimation is based on the study hypothesis and aims to obtain a study with adequate statistical power. The investigator must decide the primary outcome, effect size, variability, alpha, beta, power, type of test, study design and expected dropout. A properly calculated sample size improves the scientific, ethical and clinical value of physiotherapy research.

For further reading, the NIH review on sample size and power explains the relation among alpha, beta, effect size, power and required participant number.

This is a shared conversation. Sign in to Orris to start your own chat.