Questions & explanations
1. Two judges classify 50 paintings as modern or classical. Their agreement is 80% but kappa is 0.40. Explain why.
The agreement rate of 80% seems high, but kappa is only 0.40 because of chance agreement. If both judges guess randomly, they would agree about 50% of the time (half of all pairs). So the extra agreement beyond chance is only 30 percentage points (80% - 50%), and kappa = 0.30/0.50 = 0.60? Actually, let's recalc: For two categories with equal base rates, chance agreement = 0.5*0.5 + 0.5*0.5 = 0.5? Wait, better: If each judge independently says modern 60% and classical 40%, then expected chance agreement = 0.6*0.6 + 0.4*0.4 = 0.36+0.16=0.52. So observed 0.80, chance 0.52, kappa = (0.80-0.52)/(1-0.52)=0.28/0.48=0.58. Still moderate. The example: if base rates are very skewed, chance agreement can be high. For example, if 90% are classical, both judges will naturally agree often. So kappa corrects for that. The 80% agreement might be partly by chance, leading to lower kappa.
2. You have data on reaction times, which are positively skewed. Why might you consider a reciprocal transformation (1/X), and what does it imply for interpretation?
A reciprocal transformation (1/X) is often used for reaction times or durations because it converts them to rates (e.g., speed = 1/time). Reaction times are typically right-skewed with a lower bound of zero; reciprocal makes the distribution more symmetric and can reduce the influence of extremely slow responses. For example, a transformation of 1/reaction time gives a measure of reaction speed, which may be more normally distributed. Interpretation changes: a unit increase in the predictor is associated with a change in the reciprocal of the outcome. If the reciprocal is used, the relationship becomes nonlinear in the original units. Always interpret coefficients on the transformed scale, and consider back-transforming for practical understanding.
3. What is Cook's distance, and what does a high value tell you?
Cook's distance measures how much the regression coefficients would change if a particular observation were removed from the dataset. A high Cook's distance indicates that the observation is influential, meaning it has a large impact on the fitted model. Typically, a value above 1 or above 4/n (where n is the number of observations) is considered high and worth examining. Influential points can be outliers in the predictors (high leverage), outliers in the outcome (large residual), or both. They may distort the regression results, so you should check if they are data errors or if the model should be changed to accommodate them. A high Cook's distance does not automatically mean the point is bad, but it needs careful investigation.
4. What is a process capability index like Cp or Cpk, and what does a Cpk value of 1.33 indicate?
A process capability index measures how well a process can meet specification limits. Cp compares the spread of the process (usually 6 standard deviations) to the width of the specification range, without considering centering. Cpk accounts for both spread and centering by taking the minimum of the distances from the process mean to the nearest specification limit, divided by 3 sigma. A Cpk of 1.33 indicates that the process mean is at least 4 standard deviations away from the nearest specification limit, so only about 0.0063% of items would be outside that limit (if normally distributed). A Cpk of 1.33 is often considered acceptable, while 1.67 or higher is good. Lower Cpk values suggest a need for process improvement.
5. In a multiple regression, the raw coefficient for education is 2.5 (years of schooling on income in thousands) and the beta is 0.30. The raw coefficient for experience is 0.8 (years on income in thousands) and the beta is 0.25. Which predictor has a larger effect per unit change, and which has a larger relative importance?
Per unit change, education has a larger effect because its raw coefficient (2.5) is bigger than experience's (0.8). That means each additional year of schooling raises income by 2.5 thousand dollars, while each extra year of experience raises it by only 0.8 thousand. However, relative importance (compared to other predictors) is shown by the betas: education's beta is 0.30, experience's is 0.25, so education still has a slightly larger relative importance, but the difference is smaller. The beta sizes depend on the variability of each predictor. If experience has a much larger standard deviation than education, the beta difference might narrow or even reverse. This example shows why both coefficients are informative.
6. What are fitted values in regression, and how are they related to residuals?
Fitted values are the predicted values of the outcome given the predictors, calculated from the estimated regression equation. For each observation, the fitted value is the point on the regression line (or hyperplane) corresponding to its predictor values. The sum of the residuals is always zero (if the model includes an intercept). The relationship is: actual value = fitted value + residual. Fitted values represent the part of the outcome explained by the model, while residuals represent the unexplained part. By plotting fitted values against residuals, you can detect patterns like heteroscedasticity (unequal spread). Good models have randomly scattered residuals with no clear pattern relative to fitted values.
7. How does quantile regression handle heteroscedasticity, and why is it an advantage?
Heteroscedasticity means the variance of the outcome changes across levels of predictors. In OLS, standard errors need correction. Quantile regression naturally handles heteroscedasticity because it models different parts of the distribution separately. For example, if the spread of income increases with education, quantile regression shows that education has a bigger effect on the 90th percentile than on the 10th. This reveals how the entire conditional distribution changes, not just the mean. Quantile regression does not assume constant variance, so it gives a richer picture. It also provides robust standard errors that account for heteroscedasticity. This makes it a powerful tool for data with varying spread.
8. What is a data transformation in regression, and why might you take the logarithm (log) of a variable?
A data transformation applies a mathematical function to a variable to change its scale or distribution. Taking the logarithm (log) is common when the data are right-skewed (many small values and a few large ones). For example, income or house prices often follow a log-normal distribution. Log transformation makes the distribution more symmetric and can help meet the assumption of normality for residuals. It also changes the interpretation: a 1% change in the predictor corresponds to a change of roughly (coefficient/100) units in the outcome when both are log-transformed. Logs can also stabilize variance when the spread increases with the mean. Always check if the transformation improves model diagnostics.
9. Explain the backfitting algorithm for estimating GAMs.
Backfitting is an iterative method that estimates each smooth term one at a time, holding others fixed. Start with initial guesses for all terms. Then, for each predictor, subtract the current sum of other smooth terms from the outcome. Smooth the resulting partial residuals against that predictor. Update the term with this smooth. Repeat this cycle for all predictors until the estimates stop changing. For example, in a model with two smooth terms, you repeatedly refine each. Backfitting is simple and works for additive models. However, modern GAM software often uses a more efficient method called penalized likelihood. But backfitting helps understand the basic idea of iteratively updating each term.
10. What is the difference between quantile regression and robust regression in terms of their goals?
Quantile regression aims to model the entire conditional distribution of the outcome, showing how predictors affect different percentiles. Robust regression aims to get a single estimate of the conditional mean that is not distorted by outliers. For example, quantile regression can tell you that income inequality (the gap between high and low earners) grows with education. Robust regression tells you the typical (average) effect of education on income after removing outlier influence. Quantile regression does not downweight outliers; it just gives separate lines for different quantiles. Robust regression focuses on the mean but makes it resistant. Both handle outliers but answer different questions.
11. What is the Box-Cox transformation, and why might it be better than choosing a transformation by guess?
The Box-Cox transformation is a family of power transformations that includes log, square root, and reciprocal as special cases. It finds the best exponent lambda (λ) to make the data more normally distributed by maximizing the likelihood. For example, if λ=0 it's a log, λ=0.5 is square root, λ=-1 is reciprocal. Using Box-Cox removes guesswork: you let the data choose the optimal transformation. However, it only works for positive data. After transformation, you still need to check model assumptions. Box-Cox can be applied to the outcome variable to fix non-normality and heteroscedasticity. It is a systematic way to find a transformation that makes residuals more normal and constant in variance.
12. A control chart shows a sudden shift in the average to a new level that stays consistent. What does this suggest about the process, and what should you do?
A sudden sustained shift suggests a special cause of variation that changed the process average permanently (or until fixed). For example, a new raw material batch or a machine adjustment could cause this. Even if all points are within control limits after the shift, the shift itself is a signal that the process is not in control because it is not stable over time. You should investigate the time of the shift to find what changed. Look for changes in operators, materials, methods, or environment. Once the cause is found, either correct it or update the control limits to reflect the new baseline if the change is intentional and beneficial. Monitoring with the old limits would give false signals.