Multiple regression and standard error of regression MCQs With Answer

Multiple regression and the standard error of regression are essential statistical tools for B. Pharm students analyzing relationships between drug responses and multiple predictors such as dose, age, weight, and biochemical markers. Multiple regression estimates regression coefficients, tests hypotheses, and quantifies model fit with R-squared and adjusted R-squared, while the standard error of regression (standard error of estimate) measures residual variability and precision of predictions. Understanding assumptions—linearity, independence, homoscedasticity, normality—and diagnostics like multicollinearity, Cook’s distance, and variance inflation factor helps ensure valid inference and reliable dosing models. This introduction focuses on interpretation, calculation, hypothesis testing, and practical application in pharmaceutical research. Now let’s test your knowledge with 30 MCQs on this topic.

Q1. What does the standard error of regression (standard error of estimate) primarily measure?

• The average size of residuals around the regression line
• The slope of the regression coefficient
• The correlation between predictors
• The sample mean of the dependent variable

Correct Answer: The average size of residuals around the regression line

Q2. In multiple regression, which formula correctly gives the standard error of the regression (s)?

• s = sqrt(SST / (n – 1))
• s = sqrt(SSE / (n – p)) where p is number of parameters including intercept
• s = sqrt(SSR / p)
• s = SSE / n

Correct Answer: s = sqrt(SSE / (n – p)) where p is number of parameters including intercept

Q3. Which of the following reduces the standard error of a regression coefficient, all else equal?

• Increasing multicollinearity among predictors
• Reducing sample size
• Increasing variation in the predictor variable
• Introducing irrelevant predictors

Correct Answer: Increasing variation in the predictor variable

Q4. Which diagnostic indicates a predictor might be highly collinear with other predictors?

• Low residual standard error
• High variance inflation factor (VIF)
• High adjusted R-squared
• Small p-value for the overall F-test

Correct Answer: High variance inflation factor (VIF)

Q5. What is the primary purpose of the F-test in multiple regression?

• To test whether the residuals are normally distributed
• To test whether all regression coefficients (except intercept) are jointly zero
• To compute confidence intervals for coefficients
• To measure multicollinearity

Correct Answer: To test whether all regression coefficients (except intercept) are jointly zero

Q6. Which assumption, if violated, most directly affects the validity of standard errors and t-tests?

• Linearity between predictors and outcome
• Homoscedasticity (constant variance of residuals)
• Large sample size
• Categorical predictors

Correct Answer: Homoscedasticity (constant variance of residuals)

Q7. How does multicollinearity typically affect standard errors of regression coefficients?

• It decreases standard errors
• It increases standard errors
• It has no effect on standard errors
• It only affects the intercept

Correct Answer: It increases standard errors

Q8. Which metric adjusts R-squared for the number of predictors and sample size?

• Adjusted R-squared
• AIC
• Standard error of regression
• Cook’s distance

Correct Answer: Adjusted R-squared

Q9. When predicting a single new observation, which interval is wider?

• Confidence interval for the mean prediction
• Prediction interval for the individual prediction
• Both have equal width
• Neither—intervals are not used in regression

Correct Answer: Prediction interval for the individual prediction

Q10. Which of these influences the width of a confidence interval for a regression coefficient?

• Standard error of the coefficient
• Sample size
• Chosen significance level (alpha)
• All of the above

Correct Answer: All of the above

Q11. What does a high Cook’s distance for an observation suggest?

• The observation has low leverage and little influence
• The observation is highly influential and may unduly affect coefficients
• The model has perfect fit
• The predictor is categorical

Correct Answer: The observation is highly influential and may unduly affect coefficients

Q12. Which statistic measures the proportion of variance in the dependent variable explained by the model?

• Standard error of regression
• R-squared
• VIF
• Cook’s distance

Correct Answer: R-squared

Q13. Which transformation is commonly used to stabilize variance and improve linearity in pharmacokinetic data?

• Square transformation
• Log transformation
• Reciprocal of dependent variable only
• No transformation is ever useful

Correct Answer: Log transformation

Q14. In regression output, a small p-value for a coefficient indicates what?

• The coefficient is precisely zero
• There is evidence that the predictor has a non-zero association with the outcome
• The model assumptions are violated
• The standard error must be large

Correct Answer: There is evidence that the predictor has a non-zero association with the outcome

Q15. Which change to a model tends to reduce the residual standard error?

• Removing a predictor that explains variance in outcome
• Adding a predictor that explains additional variance in outcome
• Increasing heteroscedasticity
• Reducing sample size

Correct Answer: Adding a predictor that explains additional variance in outcome

Q16. Which degree of freedom is used when computing SSE/(n-p) for the standard error of regression?

• n – 1
• n – p where p is number of estimated parameters
• p – 1
• n + p

Correct Answer: n – p where p is number of estimated parameters

Q17. Why are standardized regression coefficients useful in pharmacology studies?

• They remove units to compare relative effect sizes
• They always reduce multicollinearity
• They increase R-squared artificially
• They eliminate the need for p-values

Correct Answer: They remove units to compare relative effect sizes

Q18. Which method helps detect non-constant variance (heteroscedasticity) in residuals?

• Plotting residuals vs. fitted values
• Computing the intercept only
• Checking R-squared alone
• Calculating standardized coefficients

Correct Answer: Plotting residuals vs. fitted values

Q19. If two predictors are perfectly collinear, what happens to the regression estimation?

• Coefficients are uniquely estimated
• Model estimation fails due to singularity
• Standard error of regression becomes zero
• R-squared becomes negative

Correct Answer: Model estimation fails due to singularity

Q20. Which is true about adjusted R-squared compared to R-squared?

• Adjusted R-squared always increases when adding predictors
• Adjusted R-squared can decrease if predictors do not improve the model
• They are always equal
• Adjusted R-squared ignores sample size

Correct Answer: Adjusted R-squared can decrease if predictors do not improve the model

Q21. What does a 95% confidence interval for a coefficient indicate?

• 95% of the data points lie within this interval
• We are 95% confident the true population coefficient lies within this interval
• The coefficient is significant at any alpha
• The model has 95% predictive accuracy

Correct Answer: We are 95% confident the true population coefficient lies within this interval

Q22. Which effect does increasing sample size have on the standard error of regression coefficients?

• It increases the standard error
• It decreases the standard error
• It has no effect
• It makes coefficients biased

Correct Answer: It decreases the standard error

Q23. When is it appropriate to include interaction terms in a regression for drug response?

• When you suspect the effect of one predictor depends on another predictor
• Always include all possible interactions regardless of theory
• Only when sample size is smaller than number of predictors
• Interactions are never useful in pharmacology

Correct Answer: When you suspect the effect of one predictor depends on another predictor

Q24. Which of the following best distinguishes prediction interval from confidence interval?

• Prediction interval estimates mean response; confidence interval predicts individual response
• Prediction interval includes both model uncertainty and individual variability; confidence interval estimates mean response precision
• They are mathematically identical
• Confidence interval is always wider than prediction interval

Correct Answer: Prediction interval includes both model uncertainty and individual variability; confidence interval estimates mean response precision

Q25. Which practical step can reduce multicollinearity effects?

• Centering predictors (subtracting means)
• Adding more highly correlated predictors
• Reducing sample size
• Removing the dependent variable

Correct Answer: Centering predictors (subtracting means)

Q26. What does a very small residual standard error relative to the outcome range imply?

• Poor model fit
• Good model fit with small average prediction error
• Model is overfitted necessarily
• Heteroscedasticity is present

Correct Answer: Good model fit with small average prediction error

Q27. Which of the following is a correct effect of adding an irrelevant predictor to the model?

• Always increases adjusted R-squared
• May increase R-squared but can decrease adjusted R-squared
• Always decreases R-squared
• Always reduces standard error of regression

Correct Answer: May increase R-squared but can decrease adjusted R-squared

Q28. Which statistic would you examine to assess whether residuals are approximately normally distributed?

• Scatterplot of predictor vs. outcome
• Normal Q-Q plot of residuals
• VIF values
• Adjusted R-squared

Correct Answer: Normal Q-Q plot of residuals

Q29. In a clinical study predicting drug clearance, why is checking leverage important?

• High-leverage points can unduly influence coefficient estimates and predictions
• Leverage only matters for categorical outcomes
• Leverage reduces sample size automatically
• Leverage ensures residuals are zero

Correct Answer: High-leverage points can unduly influence coefficient estimates and predictions

Q30. Which model selection criterion penalizes model complexity and helps choose among competing regression models?

• R-squared
• AIC or BIC
• Standard error of regression alone
• Cook’s distance only

Correct Answer: AIC or BIC

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