Checking Regression Assumptions - Biostatistics

1. Examine the normal Q-Q plot of residuals shown. What does this pattern suggest about the normality assumption?
   Answer: Normality is violated due to lighter tails than expected in a normal distribution The Q-Q plot shows points that curve inward at both extremes, staying closer to the reference line than expected, indicating lighter tails than a normal distribution. Choice A is wrong because the systematic curvature indicates departure from normality. Choice C is wrong because heavy tails would curve away from the line. Choice D is wrong because this pattern doesn't indicate skewness but tail behavior. Choice E is wrong because this shows a systematic pattern, not isolated outliers.
2. In a regression analysis, the residuals versus predictor plot shows a distinct megaphone shape, spreading out as the predictor variable increases. However, after applying a square root transformation to the response variable, the residuals versus fitted values plot shows random scatter with constant spread. What conclusion is most appropriate?
   Answer: The original model violated equal variance, which was stabilized by the transformation When you encounter residual plots in regression analysis, you're diagnosing which fundamental assumptions of linear regression might be violated. The key is interpreting the pattern you observe and understanding what different transformations can fix. The "megaphone shape" described here is a classic indicator of heteroscedasticity—where the variance of residuals increases as the predictor increases. This violates the equal variance (homoscedasticity) assumption of regression. When you apply a square root transformation to the response variable and see the residuals become randomly scattered with constant spread, you've successfully stabilized the variance. This is exactly what variance-stabilizing transformations like square root, logarithmic, or reciprocal transformations are designed to do. Looking at the wrong answers: A) is incorrect because linearity violations would show up as curved patterns in residual plots, not megaphone shapes. If linearity were the issue, the transformation might not have produced random scatter. C) is wrong because independence violations relate to the correlation structure between observations (often addressed through time series methods or mixed models), not variance patterns that transformations can fix. D) is incorrect because megaphone patterns are definitely not acceptable—they indicate heteroscedasticity, which violates regression assumptions and can lead to inefficient estimates and invalid inference. Study tip: Remember the "shape-to-violation" pattern: curved residuals suggest non-linearity, megaphone/funnel shapes indicate unequal variance, and systematic patterns over time suggest dependence issues. Variance-stabilizing transformations specifically address the megaphone problem.
3. Based on the residual plots shown, which regression assumption appears to be most clearly satisfied?
   Answer: Linearity assumption, because residuals show random scatter around zero in the fitted values plot Random scatter of residuals around zero in the residuals vs. fitted values plot indicates the linearity assumption is satisfied. Choice B is wrong because Q-Q plots assess normality, not independence. Choice C is wrong because residuals vs. fitted plots assess linearity and equal variance, not normality. Choice D is wrong because Q-Q plots don't assess variance equality. Choice E requires examining all diagnostic evidence, which the question asks for the most clearly satisfied assumption.
4. A researcher fits a linear regression model predicting systolic blood pressure from age. The residual plot shows a curved pattern with negative residuals for young and old subjects, and positive residuals for middle-aged subjects. Additionally, the Shapiro-Wilk test for normality of residuals yields p = 0.03. Which regression assumption violation should be addressed FIRST?
   Answer: Non-linearity, since curvature in residuals typically indicates model misspecification that affects other assumptions Non-linearity should be addressed first because when the fundamental model form is incorrect (linear when the true relationship is curved), it creates systematic patterns in residuals that can masquerade as violations of other assumptions. The curved residual pattern described (negative-positive-negative across age range) is classic evidence of non-linearity. Once the correct functional form is specified, apparent violations of normality and homoscedasticity often resolve. The Shapiro-Wilk result may be driven by the systematic bias from model misspecification rather than true non-normality.
5. A regression model includes data collected from multiple hospitals over several years. When examining residuals, the researcher notices that observations from the same hospital tend to have similar residual values, while the overall residual pattern appears random. This suggests a violation of which assumption?
   Answer: Independence, because observations within hospitals are more similar than expected by chance When you encounter regression problems involving clustered data (like patients within hospitals, students within schools, or repeated measurements on the same subjects), always consider whether the independence assumption might be violated. The independence assumption requires that each observation's residual should be unrelated to all other residuals. However, this researcher observes that residuals from the same hospital cluster together—they're more similar to each other than to residuals from other hospitals. This indicates that observations within hospitals share some unmeasured characteristics, creating correlation between what should be independent error terms. This pattern directly violates the independence assumption because observations from the same hospital are clearly not behaving independently—they're systematically more similar than random chance would predict. Looking at the incorrect options: Choice A misinterprets the clustering as a linearity issue, but linearity violations would show curved patterns in residuals versus fitted values, not grouping by data source. Choice B suggests unequal variance, but the question describes similar residual values within hospitals, not different variance levels—unequal variance would mean some hospitals show much wider residual spreads than others. Choice C incorrectly links clustering to normality violations, but normality concerns the distribution shape of residuals overall, not their correlation structure. Remember this pattern: whenever you see data collected from natural groups (hospitals, schools, families) and residuals cluster by group membership, think independence violation first. This is why methods like mixed-effects models or cluster-robust standard errors exist—to handle correlated observations within groups.
6. In a multiple regression with predictors X1 and X2, residual analysis reveals that while residuals versus fitted values shows random scatter, the partial residual plot for X1 shows a curved relationship. The partial residual plot for X2 shows a linear relationship. What is the most appropriate interpretation?
   Answer: The relationship between Y and X1 may be non-linear, requiring model modification When evaluating multiple regression assumptions, you need to distinguish between different types of residual plots and what each reveals about model adequacy. The key insight here is that different plots diagnose different aspects of your model. The residuals versus fitted values plot provides an overall assessment of model adequacy, checking for linearity, homoscedasticity, and outliers across the entire model. However, partial residual plots (also called component-plus-residual plots) are specifically designed to detect non-linear relationships between individual predictors and the response variable. These plots show $e_i + \beta_j x_{ij}$ versus $x_{ij}$ for each predictor, allowing you to see the relationship between each predictor and the response after accounting for other variables. Answer B correctly identifies that curvature in the partial residual plot for X1 suggests a non-linear relationship between Y and X1, even though the overall model diagnostics appear acceptable. This requires model modification, perhaps by adding polynomial terms or transforming X1. Answer A misses the point entirely – while overall linearity might seem satisfied, partial residual plots reveal predictor-specific issues that the global residual plot can mask. Answer C confuses the diagnostic purpose; partial residual plot curvature indicates non-linearity, not heteroscedasticity (unequal variance). Answer D incorrectly interprets the curvature as an independence violation, when independence issues typically appear as temporal or spatial patterns in residuals. Study tip: Remember that partial residual plots are your best tool for detecting non-linear relationships with individual predictors in multiple regression – they can reveal problems that overall residual plots miss.
7. A researcher examines residual plots from a linear regression analysis. The plot of residuals versus fitted values shows a clear funnel shape, with residuals spreading out as fitted values increase. Additionally, a normal Q-Q plot of residuals shows points that closely follow a straight line. Which regression assumption is most clearly violated?
   Answer: Equal variance assumption, because residual spread increases with fitted values When analyzing regression assumptions through residual plots, you're looking for specific patterns that reveal which assumptions might be violated. The key is understanding what each plot tells you about different assumptions. The funnel-shaped pattern in the residuals versus fitted values plot is the classic signature of heteroscedasticity (unequal variances). When residual spread increases systematically with fitted values, it means the variance of errors isn't constant across all levels of the predictor variable. This directly violates the equal variance (homoscedasticity) assumption. The Q-Q plot showing points closely following a straight line actually confirms that normality is satisfied, not violated. Looking at the wrong answers: Choice A misinterprets the funnel pattern. Non-linearity would show up as a curved pattern in the residuals versus fitted values plot, not just changing spread. The funnel shape indicates changing variance, not a non-linear relationship. Choice B confuses systematic variance patterns with independence violations. Independence issues typically arise from correlated observations (like time series data) and wouldn't necessarily create this funnel pattern. Choice C directly contradicts the given information—the Q-Q plot shows points following a straight line, which indicates the normality assumption is met, not violated. Remember this pattern: funnel shapes in residual plots almost always point to variance problems. A helpful memory aid is "funnel = unequal variance." Also, always read Q-Q plot descriptions carefully—points following a straight line means normality is satisfied, while points deviating from the line suggest normality violations.
8. After fitting a linear regression model, a researcher creates a scatterplot of residuals versus the predictor variable X, which shows a curved pattern resembling a parabola. The residuals are negative for small and large values of X, but positive for middle values of X. Which assumption is violated and what is the most appropriate next step?
   Answer: Linearity is violated; consider adding a quadratic term to the model When analyzing residual plots in linear regression, you're checking whether the model's assumptions hold. The pattern you observe in residuals versus predictors reveals which assumptions might be violated. The curved, parabolic pattern described—where residuals are negative at both extremes of X but positive in the middle—is a classic signature of non-linearity. This U-shaped or inverted U-shaped pattern indicates that the true relationship between X and Y contains curvature that your straight-line model cannot capture. The systematic pattern shows that your linear model consistently under-predicts (negative residuals) in certain regions and over-predicts (positive residuals) in others. Adding a quadratic term ($X^2$) to create the model $Y = \beta_0 + \beta_1X + \beta_2X^2 + \varepsilon$ would likely eliminate this systematic pattern, making answer C correct. Answer A is wrong because equal variance violations typically show residuals that spread out more (or less) as X increases—a fan or funnel shape, not a curved pattern. Answer B misidentifies the issue; independence violations usually require knowledge about data collection methods and wouldn't create this specific residual pattern. Answer D incorrectly focuses on normality, but the curved pattern in residuals versus predictors indicates a model specification problem, not a distributional issue with the error terms. Study tip: Remember that residual plots against predictors primarily check linearity assumptions. If you see systematic curves or patterns (not just random scatter), think about missing terms in your model before considering other assumption violations or alternative methods.
9. A researcher fits a linear regression model and finds that 95% of standardized residuals fall within ±1.96, and the residuals versus fitted values plot shows random scatter with constant spread. However, the Durbin-Watson statistic is 1.2, and critical values are $d_L = 1.08$ and $d_U = 1.36$. What conclusion is most appropriate?
   Answer: The Durbin-Watson test is inconclusive, requiring additional assessment of independence When evaluating regression assumptions, you need to systematically check normality, equal variance (homoscedasticity), and independence of residuals. The Durbin-Watson test specifically assesses independence by detecting autocorrelation in residuals. The Durbin-Watson statistic of 1.2 falls between the critical values $d_L = 1.08$ and $d_U = 1.36$. This creates an inconclusive zone where you cannot definitively determine whether positive autocorrelation exists. Values below $d_L$ indicate positive autocorrelation, values above $d_U$ suggest no autocorrelation, but values between these bounds require additional testing or alternative methods to assess independence. Option A is incorrect because it ignores the Durbin-Watson results entirely. You cannot conclude all assumptions are satisfied when the independence test is inconclusive. Option B is wrong because the test statistic doesn't fall below $d_L$, so you cannot definitively conclude positive autocorrelation exists. Option D is incorrect because it states independence "cannot be determined conclusively" as a final conclusion, but the Durbin-Watson test does provide information—it's just inconclusive and requires further investigation. Option C correctly recognizes that the Durbin-Watson test falls in the inconclusive zone, acknowledging that additional assessment methods (like examining residual plots over time, runs tests, or Breusch-Godfrey tests) are needed to properly evaluate independence. Study tip: Remember the three-zone interpretation of Durbin-Watson: below $d_L$ (positive autocorrelation), above $d_U$ (no autocorrelation), and between $d_L$ and $d_U$ (inconclusive—need more testing).
10. In a longitudinal study, patients are measured at baseline, 6 months, and 12 months. The regression model predicts cholesterol level from treatment group and time. When examining residuals ordered by measurement sequence, alternating positive and negative residuals are observed within each patient. Which assumption is most likely violated?
    Answer: Independence, since the systematic pattern suggests autocorrelation in repeated measures within patients When analyzing residual patterns in longitudinal studies, you're looking for violations of regression assumptions that are particularly problematic with repeated measures data. The key insight here is recognizing what an alternating pattern of residuals within patients actually indicates. The systematic alternating positive-negative pattern within each patient strongly suggests autocorrelation - a violation of the independence assumption (D). In repeated measures, observations from the same patient are naturally correlated because they share unmeasured patient-specific factors. When residuals alternate systematically, it indicates the model isn't accounting for this temporal correlation structure, and consecutive measurements are influencing each other in predictable ways. Let's examine why the other options miss the mark. Option A incorrectly assumes alternating residuals indicate a cyclical relationship requiring non-linear terms, but this pattern is about correlation structure, not functional form. Option B misinterprets the issue - while residuals might not be normal, the alternating pattern specifically points to dependence, not distributional shape. Option C mentions a real concern in longitudinal studies (heteroscedasticity over time), but increasing variance would show up as expanding residual spread, not systematic alternating patterns. The alternating pattern is the telltale sign: if residuals were truly independent, you wouldn't see this systematic sequencing within patients. Instead, you'd see random scatter. Study tip: In longitudinal data questions, always consider correlation between repeated measures first. Look for temporal patterns in residuals as evidence of violated independence - it's one of the most common issues in repeated measures analysis.
11. In a study examining the relationship between study hours and exam scores, the Durbin-Watson test statistic equals 0.8. The sample size is 30, and there is one predictor variable. Given that the critical values are $d_L = 1.35$ and $d_U = 1.49$, what conclusion can be drawn about the regression assumptions?
    Answer: Independence assumption is violated due to positive autocorrelation in residuals The Durbin-Watson test evaluates whether regression residuals are independent by detecting autocorrelation - when residuals are correlated with previous residuals in the sequence. This is crucial because regression assumes independent errors. The test statistic ranges from 0 to 4, where values near 2 indicate no autocorrelation, values near 0 suggest positive autocorrelation (residuals tend to have the same sign as preceding ones), and values near 4 indicate negative autocorrelation. With a test statistic of 0.8, which is well below the lower critical value of $d_L = 1.35$, you can conclude there is significant positive autocorrelation in the residuals. This violates the independence assumption of regression analysis, making answer B correct. Looking at the wrong answers: A incorrectly states the independence assumption is satisfied - when the statistic falls below $d_L$, independence is actually violated, not confirmed. C claims the test is inconclusive, but this would only be true if the statistic fell between $d_L$ and $d_U$ (the indeterminate region of 1.35 to 1.49). Since 0.8 is below this range, the test gives a definitive conclusion. D confuses the purpose of the Durbin-Watson test - it specifically tests for independence, not equal variance (homoscedasticity). Remember this decision rule for Durbin-Watson: below $d_L$ means positive autocorrelation exists, between $d_L$ and $d_U$ is inconclusive, and above $d_U$ suggests no positive autocorrelation. The test specifically targets the independence assumption, not other regression assumptions.
12. In examining regression diagnostics, a researcher finds that the standardized residuals range from -2.8 to +3.1, with 5% of residuals falling outside the ±2 range. The residuals versus fitted values plot shows random scatter, and the Q-Q plot shows slight deviation from linearity at the extremes. What is the most appropriate interpretation?
    Answer: Outliers are present and should be investigated, but normality assumption appears reasonably satisfied When analyzing regression diagnostics, you're evaluating whether your model meets the key assumptions: linearity, independence, normality, and equal variance. Each diagnostic tool provides specific information about these assumptions. The evidence here points to a reasonably well-behaved model with some outliers worth investigating. Standardized residuals typically follow a standard normal distribution, so about 5% should fall outside ±2 - this matches exactly what you're seeing. The random scatter in the residuals vs. fitted plot indicates equal variance (homoscedasticity) and linearity assumptions are met. The Q-Q plot showing only slight deviation at extremes suggests the normality assumption is reasonably satisfied, with those extreme points likely representing outliers rather than systematic non-normality. Answer A incorrectly assumes that any residuals exceeding ±2 or Q-Q deviations indicate violated normality. Actually, 5% outside ±2 is expected, and slight Q-Q deviations at extremes are common and acceptable. Answer C misinterprets what violates equal variance - that would show up as funnel shapes or patterns in residuals vs. fitted plots, not in the magnitude of residuals. Answer D incorrectly links extreme residuals to independence violations, which would appear as systematic patterns (like trends or cycles) in residuals vs. fitted plots, not random scatter. Answer B correctly recognizes that while outliers (residuals around ±3) merit investigation, the overall pattern suggests model assumptions are reasonably met. Study tip: Remember the 5% rule for standardized residuals - having about 5% outside ±2 is normal, not problematic. Focus on patterns, not just magnitude.
13. A researcher applies the Breusch-Pagan test for heteroscedasticity and obtains a test statistic of 8.2 with 1 degree of freedom. The critical value at α = 0.05 is 3.84. Simultaneously, the residuals versus fitted values plot shows what appears to be random scatter. How should these results be interpreted?
    Answer: Equal variance assumption is violated because the test statistic exceeds the critical value When you encounter questions about heteroscedasticity testing, you're dealing with one of the key assumptions of linear regression: constant variance of residuals (homoscedasticity). The Breusch-Pagan test provides a formal statistical assessment of this assumption. The Breusch-Pagan test statistic of 8.2 with 1 degree of freedom exceeds the critical value of 3.84 at α = 0.05, indicating statistical significance. This means you reject the null hypothesis of homoscedasticity and conclude that heteroscedasticity is present. Formal statistical tests take precedence over visual inspection because they provide objective, quantitative evidence. Therefore, B is correct—the equal variance assumption is violated. A is incorrect because visual interpretation of residual plots is subjective and can miss subtle patterns that statistical tests detect. What appears as "random scatter" to one observer might still contain systematic variance patterns. C misinterprets the situation. While the visual and statistical results seem contradictory, this isn't uncommon—statistical tests are more sensitive and reliable than visual assessment. The test isn't inappropriate; it's revealing what the eye cannot easily detect. D confuses heteroscedasticity with independence. The Breusch-Pagan test specifically examines variance assumptions, not independence of observations. Independence violations would be detected through different diagnostic tools like the Durbin-Watson test. Study tip: In biostatistics, always trust formal statistical tests over visual inspection when they conflict. Statistical tests are designed to detect patterns that aren't visually obvious, especially in complex biological datasets where subtle heteroscedasticity can significantly impact your conclusions.
14. A researcher applies both the Kolmogorov-Smirnov test and Shapiro-Wilk test for normality to regression residuals from a sample of n = 40. The Kolmogorov-Smirnov test yields p = 0.12, while the Shapiro-Wilk test yields p = 0.03. How should these conflicting results be interpreted?
    Answer: The Shapiro-Wilk result should be trusted more due to higher power for this sample size When you encounter conflicting results from different normality tests, you need to understand that these tests have different statistical properties, particularly regarding their power to detect departures from normality. The Shapiro-Wilk test is generally considered the gold standard for testing normality, especially for smaller to moderate sample sizes (n < 50). It has superior power compared to the Kolmogorov-Smirnov test, meaning it's better at detecting non-normality when it actually exists. With n = 40, the Shapiro-Wilk test's higher sensitivity makes its significant result (p = 0.03) more trustworthy than the non-significant Kolmogorov-Smirnov result (p = 0.12). This makes answer C correct. Answer A is wrong because you shouldn't simply accept normality just because one test is non-significant – this ignores the crucial difference in test power. Answer B takes the opposite extreme, assuming any significant test automatically means violation, without considering which test is more reliable. Answer D suggests the tests measure different aspects of normality, but both are comprehensive normality tests; the issue here is statistical power, not different measurement focuses. The Kolmogorov-Smirnov test is less powerful, particularly for detecting deviations in the tails of distributions, which makes it more likely to miss departures from normality that the Shapiro-Wilk test catches. Study tip: For normality testing with sample sizes under 50, prioritize Shapiro-Wilk results over Kolmogorov-Smirnov when they conflict. Remember that higher statistical power means better ability to detect true departures from normality.
15. A normal probability plot (Q-Q plot) of residuals from a linear regression shows points that follow the reference line closely in the middle but curve away from the line at both extremes, creating an S-shaped pattern. The residuals versus fitted values plot shows random scatter. What assumption violation does this indicate?
    Answer: Normality violation due to heavier tails than expected in normal distribution When analyzing regression diagnostics, you need to interpret different plots to identify specific assumption violations. The key is understanding what each plot reveals about your model's performance. The S-shaped pattern in a Q-Q plot is a classic indicator of non-normality in residuals. When residuals follow a normal distribution, points should align closely with the reference line throughout. The described pattern—where points curve away at both extremes—specifically indicates that your residuals have heavier tails than a normal distribution would predict. This means you're observing more extreme values (both high and low) than normal theory expects, creating the characteristic S-curve as the actual quantiles deviate from theoretical normal quantiles at the extremes. Option A is incorrect because independence violations typically show up as patterns in residuals versus fitted values or residuals versus order plots, not in Q-Q plots. Option B misidentifies the issue—linearity problems would appear as systematic patterns (curves, trends) in the residuals versus fitted values plot, but you're told that plot shows random scatter. Option C confuses the diagnostic tool; equal variance (homoscedasticity) violations are detected through residuals versus fitted values plots showing funnel shapes or other variance patterns, not through Q-Q plot curvature. Study tip: Remember the "one plot, one assumption" rule for regression diagnostics. Q-Q plots specifically test normality of residuals—any systematic deviation from the reference line indicates non-normality. S-shaped curves always mean heavy tails, while other patterns (like gentle curves) suggest skewness.
16. A time series regression analysis yields residuals that show no pattern when plotted against fitted values, but when plotted sequentially over time, residuals alternate between positive and negative values in a regular pattern. The Durbin-Watson statistic is 2.8. What does this indicate about regression assumptions?
    Answer: Negative autocorrelation is present, violating the independence assumption When analyzing time series data, you must check whether regression assumptions hold, particularly the independence of residuals. The key diagnostic tools here are residual plots and the Durbin-Watson statistic. The Durbin-Watson statistic tests for autocorrelation in residuals, ranging from 0 to 4. Values near 2 indicate no autocorrelation, values near 0 suggest positive autocorrelation (consecutive residuals are similar), and values near 4 indicate negative autocorrelation (consecutive residuals alternate between positive and negative). The given value of 2.8 is significantly above 2, pointing toward negative autocorrelation. This interpretation aligns with the sequential plot showing residuals alternating between positive and negative values in a regular pattern – a classic sign of negative autocorrelation where each residual tends to have the opposite sign of the previous one. Answer C correctly identifies negative autocorrelation violating the independence assumption. Answer A is wrong because while residuals vs. fitted values may look fine, the sequential time plot reveals the violation – you need both plots for time series analysis. Answer B incorrectly identifies the type of autocorrelation; a Durbin-Watson of 2.8 indicates negative, not positive autocorrelation. Answer D mischaracterizes the problem as heteroscedasticity (unequal variance) when it's actually an autocorrelation issue affecting independence. Remember: For time series regression, always examine residuals both against fitted values AND sequentially over time. The Durbin-Watson statistic quantifies what you see in the sequential plot – values much different from 2.0 signal autocorrelation problems.
17. A researcher conducts the Shapiro-Wilk test for normality on regression residuals and obtains a p-value of 0.03. The sample size is 50. Separately, a histogram of the residuals appears roughly bell-shaped with slight right skewness. How should the researcher interpret these results regarding the normality assumption?
    Answer: Normality assumption is likely violated, but regression is robust to moderate departures from normality When evaluating normality assumptions in regression, you need to balance statistical test results with practical considerations and the robustness of your analytical method. The Shapiro-Wilk test with p = 0.03 suggests departure from normality, but this must be interpreted alongside the histogram showing only slight skewness and the sample size of 50. Answer D correctly recognizes that while normality may be violated statistically, regression analysis is remarkably robust to moderate departures from normality, especially with reasonable sample sizes. Answer A is wrong because it ignores the statistical test results entirely. Visual inspection alone isn't sufficient for assumption checking—you need both graphical and statistical evidence. Answer B takes the opposite extreme, treating the p-value as definitive regardless of context. This rigid interpretation ignores that statistical significance doesn't always equal practical significance, especially with normality tests that can be overly sensitive. Answer C incorrectly suggests the results are contradictory when they're actually complementary—a significant test with only slight visual skewness often indicates minor departures that don't severely impact regression validity. The key insight is that normality tests become increasingly sensitive with larger sample sizes, sometimes detecting trivial departures that don't meaningfully affect regression results. With n = 50 and only slight skewness, the violation is likely minor enough that standard regression procedures remain valid. Study tip: Remember that assumption violations exist on a spectrum. Always consider the degree of violation, sample size, and method robustness together—don't rely solely on p-value thresholds for assumption checking.
18. A researcher fits a model predicting log(hospital costs) from patient age, comorbidity count, and length of stay. The residuals show no clear pattern when plotted against fitted values, but when plotted against length of stay, a clear fan-shaped pattern emerges. What should be concluded about the regression assumptions?
    Answer: Heteroscedasticity is present and may require a different model specification or variance structure When evaluating regression assumptions, you need to examine residual plots carefully to detect violations of homoscedasticity (constant variance). The key insight here is that different residual plots can reveal different assumption violations. The fan-shaped pattern in the residuals versus length of stay plot is a classic indicator of heteroscedasticity—where the variance of residuals changes systematically with the predictor variable. Even though the residuals versus fitted values plot looks fine, the individual predictor plot reveals that variance increases (or decreases) as length of stay changes. This violates the constant variance assumption and suggests you need either a different variance structure (like weighted least squares) or a model transformation. Option A is wrong because you can't conclude all assumptions are satisfied based on just one residual plot. The length of stay plot clearly shows a problem that the fitted values plot missed. Option B incorrectly identifies this as an independence violation—independence refers to observations being unrelated to each other, not systematic patterns in variance. Option C misinterprets the fan pattern as indicating non-linearity, but fan shapes specifically signal changing variance, not curved relationships (which would appear as curved patterns, not fans). The correct answer is D because heteroscedasticity is exactly what fan-shaped residual patterns indicate, and the solution involves either modeling the variance structure explicitly or using robust standard errors. Study tip: Always examine residuals against both fitted values AND individual predictors. Different plots can reveal different assumption violations that might be masked in other views.
19. In checking assumptions for a multiple regression model, a researcher finds that the Breusch-Pagan test yields p = 0.03, while visual inspection of residual plots shows no clear pattern. Additionally, the sample size is n = 450. How should these results be interpreted?
    Answer: Mild heteroscedasticity may be present but could be due to large sample size; consider practical significance With large sample sizes, statistical tests for assumption violations can detect very small departures that may not be practically meaningful. A significant Breusch-Pagan test with no visible pattern in residual plots suggests mild heteroscedasticity that may not substantially affect inference. The researcher should consider the practical impact, potentially use robust standard errors as a precaution, but not necessarily abandon the analysis. Both the test and visual inspection provide valuable but different information.
20. Based on the diagnostic plot shown, what can be concluded about the regression assumptions?
    Answer: Linearity assumption is violated due to the curved pattern in residuals across fitted values The curved pattern in residuals vs. fitted values indicates non-linearity, where the relationship is not adequately captured by a straight line. Choice B is wrong because the spread appears relatively constant, not increasing. Choice C is wrong because this pattern indicates model misspecification, not independence issues. Choice D is wrong because this plot doesn't assess normality. Choice E is wrong because only linearity appears clearly violated in this plot.

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