REGRESSION - Complete 10-Mark Exam Notes

(Biostatistics | Based on Mahajan's Methods in Biostatistics & Supplementary Sources)

MARK DISTRIBUTION STRATEGY (10 Marks)

SECTION 1 - DEFINITION AND KEY CONCEPTS (1.5 marks)

• The variable being estimated is called the Dependent variable (Y)
• The variable that is known is called the Independent variable (X)

Memory trick: Think of regression as a "prediction machine" - you feed it X, it gives you Y.

Example: If height (X) increases by 1 cm, weight (Y) increases by b kg. This b value is the regression coefficient.

SECTION 2 - TYPES OF REGRESSION (1.5 marks)

A. Simple Linear Regression

• One independent variable (X) predicts one dependent variable (Y)
• Relationship is a straight line
• Example: Predicting blood pressure (Y) from age (X)

B. Multiple Linear Regression

• Two or more independent variables predict one dependent variable
• Example: Predicting weight gain (Y) from caloric intake + physical activity + age (X1, X2, X3)
• Examiner tip: Always mention that multiple regression controls for confounding variables - this shows higher-order understanding²

C. Logistic Regression

• Used when the dependent variable is binary (yes/no, 0/1)
• Example: Predicting whether a patient has a disease (yes/no) based on risk factors
• Gives output as probability or odds ratio (OR)
• Commonly used in case-control studies and epidemiology²

D. Polynomial Regression

• Used when the relationship between X and Y is curved (non-linear)
• Less common in medical statistics

Key examiner point: Know when to use each type - the choice depends on the nature of the dependent variable

SECTION 3 - REGRESSION EQUATION AND COEFFICIENT (2 marks)

The Regression Equation (Line of Best Fit):

• Y_c = calculated (predicted) value of Y
• a = intercept (value of Y when X = 0)
• b = regression coefficient (slope of the line)
• X = known independent variable

Calculating the Regression Coefficient (b):

There are TWO regression coefficients:

**Critical insight: The two regression coefficients are NOT reciprocals of each other. This is a commonly tested point!**¹

Constant 'a' (Intercept):

Worked Example (from textbook):

• a = 5.5 - 0.75 × 7 = 0.25
• Regression equation: Y_c = 0.25 + 0.75X
• For X = 8: Y_c = 0.25 + 0.75(8) = 6.25¹

SECTION 4 - REGRESSION LINE (1.5 marks)

Properties of the Regression Line:

1. Always passes through the point (X̄, Ȳ) - the intersection of the two means
2. When r = +1 or -1 (perfect correlation): both regression lines coincide into one line
3. When r = 0 (no correlation): the two regression lines are perpendicular to each other
4. When correlation is partial: two separate lines diverge forming an acute angle
5. The steeper the lines, the stronger the correlation

How to Draw the Regression Line:

1. Calculate Y_c for multiple X values using Y_c = a + bX
2. Plot the (X, Y_c) points on the scatter diagram
3. Join them - this is the regression line
4. About half the scatter points fall on each side of this line (if correlation is significant)

Examiner tip: The regression line acts as a graphic test of significance - scatter points clustering closely around the line = strong correlation¹

SECTION 5 - STANDARD ERROR OF ESTIMATE (1 mark)

Formula:

Interpretation of Se:

• Within ±1 Se of the regression line: 68% of scatter points will lie
• Within ±2 Se of the regression line: 95% of scatter points will lie

Critical clinical example: If X = height and Y = weight, any person whose weight falls outside ±2 Se lines is either overweight or underweight at 95% confidence. This is how regression is used to assess proportionality in clinical practice.¹

Examiner-valued note: Se is analogous to standard deviation - but instead of deviation from the mean, it measures deviation from the regression line.

SECTION 6 - MULTIPLE AND LOGISTIC REGRESSION (1.5 marks)

Multiple Linear Regression:

• Each b is a partial regression coefficient - it shows the effect of one predictor holding others constant
• R-squared (R²): measures what proportion of variation in Y is explained by all X variables together
  • R² = 0.928 means 92.8% of variation in Y is explained by X
• Adjusted R²: corrects R² for the number of predictors (more conservative)
• F-test (ANOVA): tests if the overall regression model is significant
• VIF (Variance Inflation Factor): detects multicollinearity (when predictors are too correlated with each other)¹

Logistic Regression:

• Output is expressed as Odds Ratio (OR)
• OR > 1: exposure increases the odds of the outcome
• OR < 1: exposure decreases the odds of the outcome
• Widely used in epidemiology to find risk factors for disease²

High-value examiner point: Logistic regression is the go-to method in case-control studies; multiple linear regression is used when the outcome is continuous

SECTION 7 - CORRELATION vs REGRESSION (Comparison Table) (1 mark)

Critical Insight: Correlation tells you IF and HOW MUCH two variables are related. Regression tells you WHAT VALUE to expect. Both are needed for a complete analysis.¹

Common examiner trap: "Can regression exist without correlation?" - NO. Regression is only meaningful when the two variables are significantly correlated.

SECTION 8 - QUICK REVISION BULLETS (Memory Anchors)

• b = slope of regression line
• a = intercept (Y when X = 0)
• Two regression lines exist: Y on X, and X on Y
• The two lines meet at (X̄, Ȳ)
• When r = 0, lines are perpendicular
• When r = ±1, lines overlap
• Within ±2 Se: 95% of points lie
• R² = proportion of variance explained
• Logistic regression for binary outcomes → gives Odds Ratio
• Multiple regression for multiple predictors → gives partial coefficients

SECTION 9 - IMPORTANT FORMULAS AT A GLANCE

REFERENCES (Vancouver Format)

1. Mahajan BK. Methods in Biostatistics for Medical Students and Research Workers. 7th ed. New Delhi: Jaypee Brothers Medical Publishers; 2010. Chapter 12: Correlation and Regression. pp. 222-238.
2. Schober P, Boer C, Schwarte LA. Correlation coefficients: appropriate use and interpretation. Anesth Analg. 2018 May;126(5):1763-1768. Available from: https://pubmed.ncbi.nlm.nih.gov/29481436/
3. Midi H, Sarkar SK, Rana S. Collinearity diagnostics of binary logistic regression model. J Interdiscip Math. 2010;13(3):253-267.
4. Vittinghoff E, Glidden DV, Shiboski SC, McCulloch CE. Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models. 2nd ed. New York: Springer; 2012.

Final Examiner Tip: A complete 10-mark answer must include: definition, at least 3 types with examples, the regression equation with a worked calculation, the regression line and its properties, standard error of estimate, and a comparison with correlation. Diagrams (scatter plot with regression line) will earn bonus credit.

REGRESSION

Complete Notes for 10-Mark University Examination

HOW THESE NOTES COVER 10 MARKS

SECTION 1 - DEFINITION AND KEY TERMS (1.5 marks)

What is Regression?

Simple memory trick: Regression = "Prediction machine." You PUT IN what you know (X) → You GET OUT what you want to know (Y).

Physiotherapy Example (use this in your answer):

A physiotherapist wants to predict quadriceps muscle strength (Y) from a patient's knee flexion range of motion (X). She has measured both in 50 patients and found they are correlated. Regression helps her build a formula: "For every 10° increase in knee flexion range, muscle strength increases by 2 kg." She can now predict strength just from knowing the range of motion.

Key Terms:

Historical Note (Examiner bonus point):

SECTION 2 - TYPES OF REGRESSION (2 marks)

A. Simple Linear Regression

• One independent variable (X) predicts one dependent variable (Y)
• Relationship is a straight line
• Physiotherapy example: Predict walking speed (Y) from leg length (X) in stroke rehabilitation patients

B. Multiple Linear Regression

• Two or more independent variables predict one dependent variable
• Formula: Y = a + b₁X₁ + b₂X₂ + b₃X₃ ...
• Physiotherapy example: Predict functional recovery score (Y) after total knee replacement from age (X₁) + pre-surgery muscle strength (X₂) + number of physiotherapy sessions (X₃)
• Each b-value is a "partial regression coefficient" - it shows the effect of one variable while keeping others constant¹

Critical Examiner Insight: Multiple regression is the most used regression in physiotherapy research because recovery always depends on MORE than one factor. Always mention this!²

C. Logistic Regression

• Used when the dependent variable is binary (yes/no, 0/1)
• Does NOT give a predicted number - gives a PROBABILITY or ODDS RATIO
• Physiotherapy example: Predict whether a patient will be able to return to work after back injury rehabilitation (Yes=1 / No=0) based on pain levels, functional score and age
• Output = Odds Ratio (OR). If OR = 2.5, the patient is 2.5 times more likely to return to work for each unit improvement in functional score³

Examiner Tip: Logistic regression is the method of choice when the outcome is categorical (yes/no, pass/fail, discharged/not discharged). Linear regression cannot be used for such outcomes.

D. Polynomial Regression (brief mention)

• Used when the relationship is curved, not a straight line
• Less common in clinical physiotherapy settings

When to Use Which? (Must-remember rule)

SECTION 3 - REGRESSION EQUATION AND COEFFICIENT (2 marks)

The Regression Equation (Line of Best Fit):

• Y_c = Calculated (predicted) value of Y
• a = Intercept (value of Y when X = 0)
• b = Regression coefficient (slope - how steep the line is)
• X = Known independent variable

Physiotherapy Example of the Equation:

A study predicts VAS pain score (Y) after physiotherapy from number of treatment sessions (X): Y_c = 8.0 - 0.5X

• When X = 0 sessions: predicted pain = 8.0 (high, before treatment starts)
• When X = 10 sessions: predicted pain = 8.0 - 0.5(10) = 3.0 (significant reduction)
• The regression coefficient b = -0.5 means every extra session reduces pain by 0.5 points on the VAS

How to Calculate Regression Coefficient b:

How to Calculate the Intercept a:

Two Regression Coefficients Exist:

Critical Examiner Point: b_yx and b_xy are NOT equal to each other and NOT reciprocals of each other. This is one of the most commonly tested points.¹

Worked Example (adapted for physiotherapy):

• a = 5.5 - 0.75 × 7 = 0.25
• Regression equation: Y_c = 0.25 + 0.75X
• For a patient who has attended 8 sessions (X = 8): Y_c = 0.25 + 0.75(8) = 6.25 (predicted pain score)¹

SECTION 4 - THE REGRESSION LINE (1.5 marks)

What is a Regression Line?

5 Key Properties of the Regression Line (all commonly tested):

1. Always passes through (X̄, Ȳ) - the point where both means meet
2. When r = +1 or -1 (perfect correlation): both regression lines coincide into one line
3. When r = 0 (no correlation): the two regression lines are perpendicular (90°) to each other
4. When correlation is partial: two lines diverge forming an acute angle
5. Steeper the lines = stronger the correlation

How to Draw the Regression Line:

Physiotherapy Example of Regression Line Use:

Plot shoulder abduction angle (X-axis) against muscle strength in Newtons (Y-axis) for 30 post-surgical patients. Draw the regression line. Now, for any new patient whose angle you measure, you can read off the predicted strength directly from the graph - no calculation needed.

Examiner-valued note: The regression line is also called a "graphic test of significance." If scatter points cluster tightly around the line, it means strong correlation and reliable prediction.¹

SECTION 5 - STANDARD ERROR OF ESTIMATE (Se) (1 mark)

What is It?

Interpretation - The ±1 Se and ±2 Se Rules:

Physiotherapy Clinical Example:

In a study predicting gait speed (Y_c) from hip extension range (X), Se = 0.3 m/s.

• A patient's actual gait speed falls within ±2 Se of the predicted value → their speed is proportionate to their hip extension → normal finding
• A patient's gait speed falls OUTSIDE ±2 Se → something abnormal is present (weakness, pain, neurological deficit) even though their ROM seems adequate
• This helps the physiotherapist say: "This patient's walking speed is NOT proportionate to their hip mobility at the 95% confidence level."

Critical clinical insight: Se is used not just to check model accuracy, but to flag patients who are outliers - those who need extra investigation or a different treatment approach.¹

SECTION 6 - CORRELATION vs REGRESSION (1 mark)

Most important distinction to write in an exam: Correlation says "how related." Regression says "what value." You need correlation first to justify doing regression.¹

Common examiner trap: "Can you do regression without significant correlation?" Answer: No. Regression is only meaningful when the variables are significantly correlated.

SECTION 7 - CLINICAL APPLICATIONS IN PHYSIOTHERAPY (1 mark)

1. Predicting Functional Recovery

• Multiple regression is used to predict FIM (Functional Independence Measure) scores at discharge from age, diagnosis severity, and number of PT sessions received.⁴
• Helps plan discharge timing and intensity of rehabilitation.

2. Predicting Return to Sport / Work

• Logistic regression predicts whether an athlete will successfully return to sport after ACL reconstruction (Yes/No) based on quadriceps strength ratio, psychological readiness score, and weeks of rehabilitation.³

3. Outcome Prediction in Stroke Rehabilitation

• Simple linear regression predicts walking speed at 6 months from lower limb muscle strength at 2 weeks post-stroke.
• Helps set realistic goals and justify intensive physiotherapy early.

4. Assessing Proportionality

• A patient's grip strength is measured along with their forearm circumference (muscle bulk).
• If the grip strength falls outside ±2 Se of the predicted value from the regression line → the patient has disproportionate weakness → warrants further neurological testing.¹

5. Physiotherapy Research

• Regression is used to identify which factors (exercise dose, session frequency, patient age) are the strongest predictors of pain reduction in chronic low back pain studies.²

6. National Exam Prediction (real study)

• Logistic regression correctly classified 90% of cases predicting whether physiotherapy assistant students would pass their national board examination based on practice exam scores.⁵

Examiner-valued statement to include: "Regression analysis allows physiotherapists to move from observation to prediction - from describing what they see in a group of patients to making evidence-based predictions for individual patients."

QUICK REVISION - MEMORY ANCHORS

• A - a = intercept (Y when X = 0)
• B - b = slope = regression coefficient
• C - Y_c = Calculated/predicted value of Y
• D - Dependent variable = Y (what you want to predict)
• E - Equation: Y_c = a + bX

ALL FORMULAS AT A GLANCE

MODEL 10-MARK ANSWER STRUCTURE

REFERENCES (Vancouver Format)

1. Mahajan BK. Methods in Biostatistics for Medical Students and Research Workers. 7th ed. New Delhi: Jaypee Brothers Medical Publishers; 2010. Chapter 12: Correlation and Regression. pp. 231-238.
2. Jones LD, Golan D, Hanna S, Ramachandran M. Artificial intelligence, machine learning and the evolution of healthcare: A bright future or cause for concern? Bone Joint Res. 2018;7(3):223-225.
3. McNall WRP. Predicting success on the National Physical Therapy Examination for graduates of a Physical Therapist Assistant Program: A correlational, logistic regression study [Doctoral dissertation]. Liberty University; 2025. Available from: https://digitalcommons.liberty.edu/doctoral/7227
4. Frontera WR, Bean JF, Damiano D, Ehrlich-Jones L, Fried-Oken M, Jette A, et al. Rehabilitation research at the National Institutes of Health. Neurorehabil Neural Repair. 2017;31(4):304-309.
5. Schober P, Boer C, Schwarte LA. Correlation coefficients: appropriate use and interpretation. Anesth Analg. 2018;126(5):1763-1768.
6. Wikipedia contributors. Regression analysis. Wikipedia, The Free Encyclopedia [Internet]. 2024 [cited 2026 Jul 31]. Available from: https://en.wikipedia.org/wiki/Regression_analysis

Final tip for the exam: Always start with a definition, include at least one physiotherapy-specific worked example or diagram, and end with clinical applications. Examiners reward students who show they understand the CLINICAL PURPOSE of the statistical method - not just the formulas.

REGRESSION

Complete Notes for 10-Mark University Examination

MARK DISTRIBUTION GUIDE

SECTION 1 - DEFINITION AND KEY TERMINOLOGY (1.5 marks)

Formal Definition (from textbook - write this in exam)

*"In experimental sciences, after having understood the correlation between two variables, there are situations when it is necessary to estimate or predict the value of one character (variable, say Y) from the knowledge of another character (variable, say X). This is possible when the two are linearly correlated. The variable to be estimated (Y) is called the dependent variable and the variable which is known (X) is called the independent variable. This is done by finding a constant called the regression coefficient (b)."*¹

Origin of the term: The word "regression" was coined by Sir Francis Galton (1886). He observed that very tall parents tend to have children slightly shorter than themselves - the children's heights "regressed" back toward the population mean. This phenomenon is called regression toward the mean.

Key Terms - Know All of These

SECTION 2 - TYPES OF REGRESSION (2 marks)

A. Simple Linear Regression

• Uses one independent variable (X) to predict one dependent variable (Y)
• The relationship is represented as a straight line
• Physiotherapy example: Predict a patient's grip strength (Y) from their forearm circumference (X) in hand rehabilitation patients
• Also used to predict VO₂ max from a 6-minute walk test distance in cardiac rehabilitation⁴

B. Multiple Linear Regression

• Uses two or more independent variables to predict one dependent variable
• Formula: Y = a + b₁X₁ + b₂X₂ + b₃X₃ ...
• Each b-value is called a partial regression coefficient - it tells the effect of one predictor while keeping all other predictors constant
• R² (R-squared): Shows what percentage of variation in Y is explained by all the X variables together. For example, R² = 0.928 means 92.8% of variation in Y is explained by X¹
• Adjusted R²: A corrected version of R² that accounts for the number of predictors used (more conservative and reliable)

Physiotherapy example: A study used multiple linear regression to predict Functional Movement Screen (FMS) total score (Y) from shoulder ROM (X₁), hip ROM (X₂), knee ROM (X₃), and isometric strength (X₄). ROM and strength together were found to significantly predict functional movement capacity.²

Critical Examiner Insight: Multiple regression is the most commonly used regression in physiotherapy research because patient outcomes depend on MORE than one variable. Mentioning this shows higher-order understanding.

C. Logistic Regression

• Used when the dependent variable is binary (yes/no / 0/1 / success/failure)
• Does NOT give a predicted number - it gives a probability or an Odds Ratio (OR)
• Formula (logit): log(p / 1-p) = a + bX
• If OR > 1: the predictor increases the chance of the outcome
• If OR < 1: the predictor decreases the chance of the outcome

Physiotherapy examples:

• Predicting whether a patient will successfully return to sport after ACL reconstruction (Yes = 1 / No = 0) based on quadriceps strength ratio and psychological readiness score
• A logistic regression study correctly classified 90% of cases in predicting whether physiotherapy assistant students would pass their national board examination³

Examiner Tip: Logistic regression is used when the outcome is a category, not a number. Linear regression cannot be used for yes/no outcomes.

D. Polynomial Regression (brief mention)

• Used when the relationship between X and Y follows a curve, not a straight line
• Example: Regression-based models used to predict changes in trunk, hip, and knee ROM over a 12-week physiotherapy treatment programme for low back pain, where the change is non-linear over time⁵

When to Use Which Type - Summary Table

SECTION 3 - REGRESSION COEFFICIENT AND EQUATION (2 marks)

The Regression Equation

• Y_c = Predicted (calculated) value of Y
• a = Intercept (value of Y when X = 0; constant part of the equation)
• b = Regression coefficient (the slope of the line)
• X = Known value of the independent variable

"Y - bX being constant may be denoted by a, then Y_c = a + bX; 'a' is an intercept of the line, i.e. value of Y when X is 0, and b indicates the slope of the line."¹

Three Formulae to Calculate Regression Coefficient (b_yx)

*"NB: X and Y are the original measurements or deviations from any assumed mean."*¹

Two Regression Coefficients Exist

**Critical Examiner Point: b_yx and b_xy are NOT equal to each other and NOT reciprocals of each other. This is a frequently tested distinction.**¹

Calculating the Intercept (a)

Worked Example (Physiotherapy context)

A physiotherapist studies 8 patients and records number of physiotherapy sessions (X) and pain score on VAS (Y).

Given: Mean Y (Ȳ) = 5.5, Mean X (X̄) = 7, Regression coefficient b = 0.75

Step 1 - Find intercept: a = 5.5 - 0.75 × 7 = 5.50 - 5.25 = 0.25

Step 2 - Write regression equation: Y_c = 0.25 + 0.75X

Step 3 - Predict pain scores for different session numbers:

Sessions (X)	Predicted Pain Score (Y_c)
4	0.25 + 0.75(4) = 3.25
6	0.25 + 0.75(6) = 4.75
8	0.25 + 0.75(8) = 6.25
10	0.25 + 0.75(10) = 7.75

(Adapted from textbook calculation, p. 234)¹

SECTION 4 - THE REGRESSION LINE (1.5 marks)

What is a Regression Line?

*"When corresponding values Y_c1, Y_c2 ... Y_cn are plotted on a graph, a straight line called the regression line or the mean correlation line (Y on X) is obtained."*¹

How to Draw the Regression Line - Step by Step

5 Key Properties of the Regression Line (all examinable)

1. Always passes through the intersection of means (X̄, Ȳ) - the two regression lines intersect at the point where perpendiculars drawn from the means of X and Y meet¹
2. Perfect correlation (r = +1 or r = -1): The two regression lines coincide and become one single line¹
3. No correlation (r = 0): The two regression lines intersect at right angles (90°) to each other¹
4. Partial correlation: The two lines diverge and form an acute angle at their meeting point - the lesser the correlation, the greater the divergence¹
5. Steeper the regression lines = stronger the correlation - closeness of correlation is indicated by the steepness of the lines¹

Diagram - Regression Line with Scatter Points

Y (Strength, kg)
|                            * (actual point)
|                      *  /  
|                   * / *   ← Regression line (Y_c = a + bX)
|              *  /
|           * /
|        * /
|______/_________________________ X (ROM, degrees)
       ↑
  Passes through (X̄, Ȳ)

Physiotherapy Application of the Regression Line

A physiotherapist plots shoulder abduction angle (X-axis) against shoulder abductor strength in Newtons (Y-axis) for 30 post-surgical patients. After drawing the regression line:

• For any new patient whose angle is 90°, they can read the predicted strength directly off the graph without recalculating
• If the patient's actual strength falls close to the line = their strength is proportionate to their ROM (normal)
• If the actual point falls far from the line = disproportionate weakness requiring further investigation

SECTION 5 - STANDARD ERROR OF ESTIMATE (Se) (1 mark)

What is It?

*"The regression line is a graphic test of significance for correlation and can be made use of in finding the extent of correlation between any two observed values from the normal distribution of scatter points around the regression line."*¹

Formula for Se

The ±1 Se and ±2 Se Rule (Must know for exams)

*"Now draw on a graph paper two parallel lines at a distance of 1 SD regression from the regression line on either side. Within these lines, 68% of the scatter points will lie. Draw two more parallel lines at a distance of 2 SD regression. Within these lines, 95% of scatter points will lie."*¹

Physiotherapy Clinical Example of Se

In a study predicting gait speed (m/s) from hip extension range (°) in post-hip replacement patients, Se = 0.3 m/s

• Patient A: Predicted gait speed = 1.2 m/s. Actual speed = 1.1 m/s. This falls within ±1 Se → Gait speed is proportionate to hip extension → Expected finding
• Patient B: Predicted gait speed = 1.2 m/s. Actual speed = 0.5 m/s. This falls beyond ±2 Se → Gait speed is NOT proportionate to hip ROM at 95% confidence level → Investigate for weakness, pain, neurological deficit, or fear of movement

Critical clinical insight: Se is not just a statistical concept - it is a clinical decision tool. It tells the physiotherapist when a patient's actual performance deviates significantly from what their ROM or strength predicts.¹

SECTION 6 - CORRELATION vs REGRESSION (1 mark)

Most important distinction to write: "Correlation gives the degree and direction of relationship between the two variables, whereas regression analysis enables us to predict the values of one variable on the basis of the other variable."¹

Critical exam trap: Can you perform regression without first establishing correlation? No. Regression is only meaningful when a statistically significant correlation exists between the two variables.

SECTION 7 - APPLICATIONS OF REGRESSION IN PHYSIOTHERAPY (1 mark)

1. Predicting Cardiorespiratory Fitness (VO₂ max)

• Simple linear regression is used to predict VO₂ max from sub-maximal exercise test data (e.g., 6-minute walk distance, step test heart rate) in cardiac and pulmonary rehabilitation
• Direct measurement of VO₂ max requires expensive equipment; regression-based prediction makes it clinically accessible⁴

2. Predicting Functional Movement Capacity

• Multiple linear regression was used to predict FMS (Functional Movement Screen) total score from ROM and isometric strength measurements across multiple joints
• ROM and strength together significantly predicted functional movement capacity - helping physiotherapists identify high-risk patients before injury occurs²

3. Predicting 1-Repetition Maximum (1RM) Strength

• In musculoskeletal physiotherapy, regression equations predict maximum strength (1RM) from sub-maximal effort tests (e.g., 7-10RM test)
• Equation: 1RM = 8.841 + (1.183 × 10RM weight), r = 0.969 - a highly accurate prediction⁶
• This avoids the risk of injury from direct maximum-effort testing in rehabilitation patients

4. Predicting ROM Change During Treatment

• Regression models have been used to predict changes in trunk, hip, and knee ROM at 6 and 12 weeks from initial ROM measurements in low back pain patients undergoing physiotherapy
• This allows therapists to set realistic, data-driven targets for patients at the start of treatment⁵

5. Assessing Proportionality in Clinical Practice

• If X = height and Y = weight (or X = ROM and Y = muscle strength), regression lines are drawn. **"If a point falls beyond 2 Se lines, the body is either bulky or thin and the weight in 95 cases out of 100 is not proportional to height."**¹
• Physiotherapy equivalent: A patient's strength is NOT proportionate to their ROM → flag for further assessment

Key clinical statement to include in your answer: Regression analysis allows physiotherapists to move from group-level observations to individual-level predictions - enabling evidence-based, personalized treatment planning.

QUICK REVISION - COMPLETE MEMORY ANCHORS

"Predicted outcome = Starting point + (How much it changes per unit of input × the input)"

• A = 'a' is the intercept (Y when X is zero)
• B = 'b' is the slope (how steep the line is)
• C = Y_c is the Calculated/predicted value
• D = Dependent variable = Y (what you are predicting)

• When r = ±1: lines merge (one line)
• When r = 0: lines are perpendicular (90°)
• Partial r: lines diverge (angle depends on correlation)

ALL FORMULAE IN ONE PLACE

MODEL ANSWER STRUCTURE FOR 10 MARKS

1. Definition (formal) from textbook + dependent/independent variable explanation + PT example (1.5 marks)
2. Types - Simple, Multiple, Logistic with one PT example each + "when to use which" rule (2 marks)
3. Formula - Write Y_c = a + bX, define all terms, write one formula for b, calculate a worked PT example (2 marks)
4. Regression line - Short definition, 4-5 properties as numbered list, brief diagram (1.5 marks)
5. Se - Define, one formula, ±1Se/±2Se rule, one PT clinical example (1 mark)
6. Comparison table - Correlation vs Regression, minimum 5 rows (1 mark)
7. Applications - List 3-4 PT applications in 3-4 lines (1 mark)

REFERENCES (Vancouver Format)

1. Mahajan BK. Methods in Biostatistics for Medical Students and Research Workers. 7th ed. New Delhi: Jaypee Brothers Medical Publishers; 2010. Chapter 12: Correlation and Regression. p. 231-238, 390-395.
2. Ergin E, Karabicak GO, Akdeniz Kudubes A, Selekler HM. Predicting functional movement capacity in adults: the effect of range of motion and isometric strength. BMC Sports Sci Med Rehabil. 2024;16:200.
3. McNall WRP. Predicting success on the National Physical Therapy Examination for graduates of a Physical Therapist Assistant Program: A correlational, logistic regression study [Doctoral dissertation]. Liberty University; 2025. Available from: https://digitalcommons.liberty.edu/doctoral/7227
4. Bradshaw DI, George JD, Hyde A, LaMonte MJ, Vehrs PR, Hager RL, et al. An accurate VO₂max nonexercise regression model for 18-65-year-old adults. Res Q Exerc Sport. 2005;76(4):426-432.
5. Stetter BJ, Ringhof S, Krafft FC, Sell S, Stein T. Regression-based machine learning for predicting lifting movement pattern change in people with low back pain. Sensors (Basel). 2024;24(4):1337.
6. Mayhew JL, Johnson BD, LaMonte MJ, Lauber D, Kemmler W. Does a regression equation to predict maximal strength in untrained lifters remain valid when the subjects are technique trained? J Strength Cond Res. 2004;18(3):469-473.

Final exam tip: The examiner expects the formal definition word-for-word or very close to it. Use the textbook wording for the definition, then explain everything else simply. Always end with a clear physiotherapy application - this is what separates a 7/10 answer from a 10/10 answer.

REGRESSION

Complete 10-Mark University Examination Notes

HOW THESE NOTES EARN FULL MARKS

SECTION 1 - DEFINITION, ORIGIN AND KEY TERMS (2 marks)

1.1 Formal Definition (from authoritative sources - write exactly in exams)

"Regression analysis is a statistical method for estimating the relationship between a dependent variable (the outcome or response variable) and one or more independent variables (predictors, covariates, or explanatory variables). Its core purpose is twofold: to understand and quantify how changes in predictor values are associated with changes in the expected value of the outcome, and to generate predictions for new observations based on those learned relationships." ²

"Regression analysis is a type of statistical evaluation that enables: (i) Description - relationships among dependent and independent variables can be statistically described; (ii) Estimation - values and their confidence intervals can be estimated; and (iii) Prediction - the value of a dependent variable can be predicted from the value of one or more independent variables." ⁵

"In experimental sciences, after having understood the correlation between two variables, there are situations when it is necessary to estimate or predict the value of one character (variable, say Y) from the knowledge of another character (variable, say X). This is possible when the two are linearly correlated. The variable to be estimated (Y) is called the dependent variable and the variable which is known (X) is called the independent variable. This is done by finding a constant called the regression coefficient (b)." ¹

1.2 Origin of the Term

Physiotherapy parallel: A group of patients with extremely poor balance scores will, on re-testing, show slightly better scores even without intervention - partly due to regression toward the mean. Physiotherapy researchers must account for this when interpreting treatment results.

1.3 Key Terms - Build Your Foundation First

Memory trick - DIPR: Dependent variable is what you predict. Independent variable is what you know. Put them in the equation Y_c = a + bX. Read the answer off the regression line.

SECTION 2 - TYPES OF REGRESSION (2 marks)

2.1 Simple Linear Regression

• One independent variable (X) predicts one dependent variable (Y)
• The relationship is represented as a straight line
• Assumes the relationship between X and Y is linear (both increase or decrease together in a straight-line pattern)

Predict quadriceps muscle strength (Y) from knee range of motion (X) in patients recovering from total knee arthroplasty. For every 10° increase in active knee flexion, strength increases by a measurable amount that the regression equation can quantify and predict.

In cardiac and pulmonary rehabilitation, simple linear regression is used to predict VO₂max from sub-maximal step test heart rate - avoiding the need for expensive and risky maximal exercise testing.⁶

2.2 Multiple Linear Regression

• Uses two or more independent variables (X₁, X₂, X₃...) to predict one dependent variable (Y)
• Formula: Y = a + b₁X₁ + b₂X₂ + b₃X₃ ...
• Each b-value is a partial regression coefficient - it shows the unique effect of one predictor while holding all others constant
• R² (R-squared / Coefficient of Determination): Tells what percentage of the variation in Y is explained by all the X variables together
  • Example: R² = 0.928 means 92.8% of variation in Y is accounted for by the predictors ¹
• Adjusted R²: A more conservative version of R² - adjusts for the number of predictors used; preferred in research reporting ¹
• F-test (ANOVA table): Tests whether the overall regression model is statistically significant. A high F-value with low p-value (p < 0.05) confirms the model is valid ¹

A study used multiple linear regression to predict Functional Movement Screen (FMS) total score (Y) from shoulder ROM (X₁), hip ROM (X₂), knee ROM (X₃), and isometric strength (X₄). Both ROM and strength were significant independent predictors of functional movement capacity. ⁴

Multiple regression models predicting VO₂max from age, resting heart rate, exercise heart rate, and body fat percentage achieved R² = 0.689, meaning these four predictors together explained 69% of the variance in aerobic capacity.⁷

Critical Examiner Insight: Multiple regression is the most frequently used regression in physiotherapy research because clinical outcomes always depend on MORE than one variable. Always mention this. The ability to control for confounding variables is its greatest strength.

2.3 Logistic Regression

• Used when the dependent variable is binary (yes/no, 0/1, success/failure)
• Does NOT give a continuous predicted value - gives a probability or Odds Ratio (OR)
• Formula (logit transformation): log(p / 1-p) = a + bX
  • Where p = probability of the event occurring
• Odds Ratio (OR) interpretation:
  • OR > 1 → the predictor increases the likelihood of the outcome
  • OR < 1 → the predictor decreases the likelihood of the outcome

• Predicting whether a patient will return to sport after ACL reconstruction (Yes = 1 / No = 0) based on quadriceps strength symmetry, hop test results, and psychological readiness score. Logistic regression is used in 64% of return-to-sport predictive studies in sports physiotherapy.³
• Predicting whether a post-stroke patient will achieve independent ambulation at discharge (Yes/No) based on initial functional score, age, and lesion side.

Examiner Tip: Logistic regression is selected when the outcome is a category (yes/no, discharged/not, walks/doesn't walk). Using linear regression for a binary outcome is a statistical error. Logistic regression converts the outcome into log-odds to allow linear modeling.

2.4 Polynomial / Non-linear Regression (brief mention)

• Used when the relationship between X and Y forms a curve rather than a straight line
• Physiotherapy example: Predicting changes in trunk, hip, and knee ROM over a 12-week course of physiotherapy for low back pain - the rate of improvement is faster in early weeks and slows toward the end, forming a curve rather than a straight line.⁸

2.5 Decision Guide - When to Use Which Type

SECTION 3 - REGRESSION COEFFICIENT AND EQUATION (2 marks)

3.1 The Regression Equation (Line of Best Fit)

From the textbook: "Y - bX being constant may be denoted by 'a', then Y_c = a + bX; 'a' is an intercept of the line, i.e. the value of Y when X is 0, and b indicates the slope of the line." ¹

3.2 Three Formulae to Calculate Regression Coefficient (b_yx)

From the textbook: "NB: X and Y are the original measurements or deviations from any assumed mean." ¹

3.3 Two Regression Coefficients - A Critical Point

Critical Examiner Point: b_yx and b_xy are NOT equal. They are NOT reciprocals of each other. This is among the most commonly tested distinctions in regression. The relationship between them is: b_yx × b_xy = r² ¹

3.4 How to Find the Intercept (a)

3.5 Worked Example (Physiotherapy context)

A physiotherapy researcher studies 8 patients and records the number of treatment sessions (X) and their functional outcome score (Y, maximum = 10).

SECTION 4 - THE REGRESSION LINE (1.5 marks)

4.1 What is the Regression Line?

From the textbook: "When corresponding values Y_c1, Y_c2 ... Y_cn are plotted on a graph, a straight line called the regression line or the mean correlation line (Y on X) is obtained." ¹

4.2 How to Construct the Regression Line - Step by Step

4.3 Five Key Properties of the Regression Line (All examinable)

4.4 Diagram - Regression Line with Physiotherapy Data

Y-axis: Quadriceps Strength (kg)
│                                    * actual point
│                             *   ╱
│                       * ╱  * ← Regression line (Y_c = a + bX)
│                  * ╱
│             *  ╱
│        *  ╱  ← Line passes through (X̄, Ȳ)
│      ╱
└────────────────────────────────── X-axis: Knee Flexion ROM (°)
      60°  70°  80°  90° 100° 110°

   ........ ±2 Se line (95% of points inside)
   - - - -  ±1 Se line (68% of points inside)
   ________ Regression line

4.5 Physiotherapy Interpretation of the Regression Line

A physiotherapist plots hip abductor strength (Y) against hip abduction ROM (X) for 40 patients with hip osteoarthritis. After drawing the regression line:

• The line passes through the mean values of both variables
• Any new patient's predicted strength can be read directly off the line for their measured ROM - no recalculation needed
• If a patient's actual strength point plots close to the line: their strength is proportionate to their ROM (expected)
• If the actual point plots far above or below the line: the physiotherapist knows something else is limiting this patient (pain, neurological issue, disuse atrophy)

SECTION 5 - STANDARD ERROR OF ESTIMATE (Se) (1 mark)

5.1 Definition

From the textbook: "The regression line is a graphic test of significance for correlation and can be made use of in finding the extent of correlation between any two observed values from the normal distribution of scatter points around the regression line." ¹

5.2 Formula for Se

5.3 The ±1 Se and ±2 Se Rule (Must know - always asked)

From the textbook: "Within ±1 SD regression lines, 68% of scatter points will lie. Within ±2 SD regression lines, 95% of scatter points will lie." ¹

Why this works: The actual Y values are assumed to follow a normal distribution around each predicted Y_c. This is the same logic as the 68-95-99.7 rule for normal distributions - but applied around the regression line instead of the mean.

5.4 Clinical Physiotherapy Application of Se

In a study predicting gait speed (m/s) from lower limb muscle strength (N) in stroke rehabilitation patients, Se = 0.25 m/s.

• The regression equation predicts Patient A's gait speed as 1.0 m/s based on their muscle strength.
• Patient A's actual gait speed = 1.1 m/s → This falls within ±1 Se → Speed is proportionate to strength → expected result → No additional investigation needed
• Patient B's actual gait speed = 0.3 m/s → This falls outside ±2 Se → Speed is NOT proportionate to strength at 95% confidence → Unexplained impairment exists → Investigate for spasticity, pain, fear of falling, or perceptual deficits

From the textbook: "If the point falls beyond 2 SD regression lines, the body is either bulky or thin and the weight in 95 cases out of 100 is not proportional to height." ¹ (Applied to PT: "If the patient's functional score falls beyond ±2 Se, their function is not proportionate to their measured impairment - investigate further.")

Critical clinical insight: Se transforms regression from a research tool into a clinical decision support tool. It helps physiotherapists identify the patient who is performing below expectation and needs deeper assessment.

SECTION 6 - CORRELATION vs REGRESSION (1 mark)

Most important distinction (write this sentence in your exam): "Correlation gives the degree and direction of relationship between two variables, whereas regression analysis enables us to predict the values of one variable on the basis of the other variable. Thereby, the cause-and-effect relationship between two variables is understood very precisely." ¹

Critical exam trap: Can regression be performed without first demonstrating significant correlation? No. Regression is meaningless if the two variables are not significantly correlated. The correlation coefficient (r) and its significance must be established before the regression equation is used for prediction.

SECTION 7 - APPLICATIONS OF REGRESSION IN PHYSIOTHERAPY (0.5 mark)

7.1 Predicting Cardiorespiratory Fitness (VO₂max)

7.2 Predicting Functional Movement Capacity

7.3 Return to Sport Prediction

7.4 Predicting Return to Work / Discharge Outcomes

7.5 Assessing Proportionality (Outlier Detection)

QUICK REVISION - ALL KEY POINTS IN ONE PLACE

• A = 'a' is the starting point (intercept - Y when X = 0)
• B = 'b' is the slope (how much Y changes per 1 unit of X)
• C = Y_c is the Calculated / Predicted Y value
• D = Dependent variable (Y) is what you Predict
• R = Regression line always passes through (X̄, Ȳ)

• r = ±1 → two lines merge into one
• r = 0 → two lines are perpendicular (90°)
• Partial r → two lines form an acute angle; smaller r = larger angle

• Continuous Y, 1 predictor → Simple linear
• Continuous Y, 2+ predictors → Multiple linear
• Binary Y (yes/no) → Logistic

ALL FORMULAE AT A GLANCE

MODEL 10-MARK ANSWER BLUEPRINT

REFERENCES (Vancouver Format)

1. Mahajan BK. Methods in Biostatistics for Medical Students and Research Workers. 7th ed. New Delhi: Jaypee Brothers Medical Publishers; 2010. Chapter 12: Correlation and Regression. p. 222-238, 390-395.
2. Editage Insights. What is regression analysis? Types, assumptions, modeling, steps [Internet]. 2024 [cited 2026 Jul 31]. Available from: https://www.editage.com/blog/what-is-regression-and-types-of-regression-for-biomedical-researchers
3. Rambaud AJM, Semay B, Samozino P, Morin JB, Testa R, Philippot R, et al. Machine learning models predicting return to sport in athletes: a systematic review. BMJ Open Sport Exerc Med. 2025 [cited 2026 Jul 31]. Available from: https://pmc.ncbi.nlm.nih.gov/articles/PMC12775306
4. Ergin E, Karabicak GO, Akdeniz Kudubes A, Selekler HM. Predicting functional movement capacity in adults: the effect of range of motion and isometric strength. BMC Sports Sci Med Rehabil. 2024;16:200.
5. Schneider A, Hommel G, Blettner M. Linear regression analysis - part 14 of a series on evaluation of scientific publications. Dtsch Arztebl Int. 2010;107(44):776-782. Available from: https://pmc.ncbi.nlm.nih.gov/articles/PMC2992018
6. Bradshaw DI, George JD, Hyde A, LaMonte MJ, Vehrs PR, Hager RL, et al. An accurate VO₂max nonexercise regression model for 18-65-year-old adults. Res Q Exerc Sport. 2005;76(4):426-432.
7. Dong Z, Liu Y, Zheng Y, Chen H, Zhang H. Establishing VO₂max prediction models based on exercise and body parameters from the step test. Med Sci Monit. 2025;22:2676. Available from: https://www.medsci.org/v22p2676.htm
8. Stetter BJ, Ringhof S, Krafft FC, Sell S, Stein T. Regression-based machine learning for predicting lifting movement pattern change in people with low back pain. Sensors (Basel). 2024;24(4):1337.

Final Examiner Note: Examiners award highest marks to students who: (1) use formal definitions from authoritative sources, (2) correctly write the regression equation and solve a numerical example, (3) draw the regression line diagram with Se boundaries labeled, and (4) demonstrate clinical application through physiotherapy-specific examples. These notes cover all four.