Introduction:
Fitting of regression lines y = a + bx and x = a + by is a key statistical tool for B. Pharm students in analyzing dose-response, pharmacokinetics, and bioequivalence data. This topic covers method of least squares, interpretation of slope and intercept, relationship with correlation, prediction versus estimation, residual analysis, and assumptions underlying linear regression. You will learn how slope reflects change in response per unit change in predictor, how intercept locates the line at mean values, and how inverse regression differs from algebraic inversion. Practical examples include predicting drug concentration from dose and estimating dose from measured response. Now let’s test your knowledge with 30 MCQs on this topic.
Q1. What does the slope b in the regression line y = a + bx represent?
- The change in x per unit change in y
- The change in y per unit change in x
- The average of x values
- The variance of y
Correct Answer: The change in y per unit change in x
Q2. In the line y = a + bx, how is the intercept a calculated from sample means?
- a = x̄ − b ȳ
- a = ȳ + b x̄
- a = ȳ − b x̄
- a = b − x̄ ȳ
Correct Answer: a = ȳ − b x̄
Q3. Which method is used to obtain estimates of a and b in y = a + bx?
- Maximum likelihood with non-linear loss
- Least squares minimization of vertical residuals
- Minimizing horizontal distances from points to line
- Bayesian updating without likelihood
Correct Answer: Least squares minimization of vertical residuals
Q4. The regression line y = a + bx always passes through which point?
- (0,0)
- (x̄, 0)
- (0, ȳ)
- (x̄, ȳ)
Correct Answer: (x̄, ȳ)
Q5. For simple linear regression, the sign of slope b is determined by:
- The difference between x̄ and ȳ
- The correlation r between x and y
- The sample size only
- The units of measurement of x
Correct Answer: The correlation r between x and y
Q6. The slope of regression of x on y (x = a + by) is related to slope of y on x how?
- They are equal always
- Product of the two slopes equals r^2
- Their sum equals 1
- They are negatives of each other
Correct Answer: Product of the two slopes equals r^2
Q7. If correlation r = 0, what is the slope of y = a + bx?
- Undefined
- Zero
- Infinity
- Equal to the intercept
Correct Answer: Zero
Q8. Which statement about x = a + by (inverse regression) is true?
- It is just the algebraic inverse of y = a + bx
- It predicts x from y and uses least squares minimizing horizontal residuals
- It minimizes squared vertical residuals in y
- It always gives identical residuals to y on x regression
Correct Answer: It predicts x from y and uses least squares minimizing horizontal residuals
Q9. In terms of standard deviations, slope b of y on x equals:
- b = r * (s_x / s_y)
- b = r * (s_y / s_x)
- b = s_y + s_x
- b = r^2 * (s_y / s_x)
Correct Answer: b = r * (s_y / s_x)
Q10. Which quantity equals the proportion of variance in y explained by x in simple regression?
- r
- r^2
- Standard error of slope
- Sum of squared residuals
Correct Answer: r^2
Q11. Residuals in regression are defined as:
- Predicted value minus observed value
- Observed value minus predicted value
- Difference between x and y
- Squared difference between observed and mean
Correct Answer: Observed value minus predicted value
Q12. One key assumption of least squares regression is:
- Residuals have constant variance (homoscedasticity)
- Predictor x is random while y is fixed
- No linear relationship exists
- Residuals are always positive
Correct Answer: Residuals have constant variance (homoscedasticity)
Q13. Why is extrapolation beyond observed x-range risky in pharmacokinetics?
- Because slopes become zero outside data
- Model may not hold outside observed range leading to incorrect dose predictions
- Correlation increases outside observed range
- Intercept changes sign automatically
Correct Answer: Model may not hold outside observed range leading to incorrect dose predictions
Q14. Which of the following does NOT change if we rescale x by multiplying by constant c?
- Slope b for y on x
- Intercept a for y on x
- Correlation r between x and y
- Units of slope
Correct Answer: Correlation r between x and y
Q15. In B. Pharm context, using regression to predict plasma concentration from dose is an example of:
- Classification problem
- Descriptive statistics only
- Regression prediction for continuous outcome
- Clustering of patients
Correct Answer: Regression prediction for continuous outcome
Q16. The sum of residuals from least squares regression equals:
- Number of observations n
- Mean of residuals
- Zero
- Sum of predicted values
Correct Answer: Zero
Q17. If r = 0.8, s_y = 10, s_x = 5, what is slope b of y on x?
- 0.8 * (5/10) = 0.4
- 0.8 * (10/5) = 1.6
- 10/5 = 2.0
- 0.8
Correct Answer: 0.8 * (10/5) = 1.6
Q18. Which measure quantifies scatter of points around fitted line?
- Sum of x
- Standard error of estimate (standard deviation of residuals)
- Correlation sign only
- Sample median
Correct Answer: Standard error of estimate (standard deviation of residuals)
Q19. In simple linear regression, R^2 = 0.64. What does this imply?
- 64% of variability in x is explained by y
- 64% of variability in y is explained by x
- Correlation r = 0.64
- Slope b = 0.64
Correct Answer: 64% of variability in y is explained by x
Q20. For prediction of an individual response, compared to prediction of the mean response at a given x, the prediction interval is:
- Narrower
- Wider
- Always identical
- Dependent only on intercept
Correct Answer: Wider
Q21. Which statement about inverse regression x = a + by is correct when used to estimate dose from observed response?
- It uses the same slope as y on x
- It minimizes vertical residuals in y
- It gives unbiased prediction of x given y under model assumptions
- It should never be used in pharmacology
Correct Answer: It gives unbiased prediction of x given y under model assumptions
Q22. The product of slopes b_yx and b_xy equals:
- r
- r^2
- 1
- Zero
Correct Answer: r^2
Q23. If all data points lie exactly on a straight line with positive slope, what is r^2?
- 0
- Between 0 and 1
- Exactly 1
- Negative
Correct Answer: Exactly 1
Q24. Which diagnostic plot is commonly used to assess homoscedasticity?
- Histogram of x
- Residuals versus fitted values plot
- Dot plot of x
- Box plot of y only
Correct Answer: Residuals versus fitted values plot
Q25. In calibration of an assay, which regression is appropriate to estimate unknown concentration from instrument response?
- Regression of response on concentration
- Regression of concentration on response (inverse regression)
- Correlation only
- Cluster analysis
Correct Answer: Regression of concentration on response (inverse regression)
Q26. Changing units of y (e.g., mg to µg) will primarily affect which regression parameter?
- Correlation r
- Slope b and intercept a
- Sample size n
- Positions of x values
Correct Answer: Slope b and intercept a
Q27. The least squares criterion minimizes which of the following?
- Sum of absolute residuals
- Sum of squared residuals
- Maximum residual magnitude
- Product of residuals
Correct Answer: Sum of squared residuals
Q28. When would you prefer x = a + by over y = a + bx in a B. Pharm study?
- When predicting y from x
- When both variables are error-free
- When the goal is to estimate the predictor x from measured response y
- Never; y = a + bx is always preferred
Correct Answer: When the goal is to estimate the predictor x from measured response y
Q29. If residuals show a curved pattern versus x, this suggests:
- Linearity assumption is violated and a non-linear model may be needed
- Homoscedasticity holds perfectly
- No relationship exists between x and y
- Sample size is too large
Correct Answer: Linearity assumption is violated and a non-linear model may be needed
Q30. In the context of drug-response data, which practice improves reliability of regression estimates?
- Using a very narrow range of doses only
- Increasing sample size and covering an appropriate range of predictor values
- Removing the intercept term always
- Ignoring residual diagnostics
Correct Answer: Increasing sample size and covering an appropriate range of predictor values
