Scatter Plot

Study guide: Exam MAS-I

A Scatter Plot is an Exploratory Data Analysis display of two variables at once: each observation is one point at (x_i, y_i). It is the standard first look at the relationship between a continuous predictor and the response, before any model is fitted.

\{(x_i, y_i)\}_{i=1}^n \quad \text{plotted in the plane}

• What to read off it: direction (up or down), form (straight or curved), strength (tight or diffuse), and outliers that sit away from the cloud
• Strength in the linear direction is summarized by the Correlation coefficient r — but r near 0 only rules out a linear trend, not a U-shape
• Systematic curvature says the predictor needs a transformation (\ln x, x^2, or banding) in the Model Structure; a fan that widens to the right says the variance grows with the mean, which is an argument for a Generalized Linear Model rather than Linear Regression
• A point far out in x has high leverage — on its own it can bend the fitted line, so it deserves a look before it is left in
• With many overlapping points, plot on a log scale, add transparency, or bin the x-axis and plot group means; a solid black blob shows nothing
• A scatter plot matrix (all pairs of predictors) screens for pairwise collinearity, as the correlation matrix does — but it can miss Multicollinearity, collinearity among three or more predictors when no pair is highly correlated; ISLR assesses that with the variance inflation factor, \text{VIF}(\hat\beta_j) = 1/(1 - R^2_{X_j \mid X_{-j}}), from regressing each predictor on all the others

Reading Curvature into the Model A scatter plot of pure premium against driver age shows high values for young drivers, a broad minimum around age 45, and a rise again past 70. The fitted correlation is r = -0.04. Should driver age be dropped?

Answer No. The near-zero Correlation measures only the linear association, and a U-shape has almost no linear component. Age is clearly predictive; it just needs the right structure — band it into a Categorical Predictor, or include a quadratic term \beta_1\,\text{age} + \beta_2\,\text{age}^2. Dropping a variable on the strength of r alone is the classic scatter-plot mistake.

Spotting Non-Constant Variance Claim severity is plotted against insured value. The points form an upward wedge: tight near the origin, spreading to a wide band at high insured values. What does the plot rule out?

Answer It rules out the constant-variance assumption of ordinary Linear Regression. Variance increasing with the mean is the signature of a Gamma response (where the standard deviation is proportional to the mean), so a Gamma Generalized Linear Model with a log Link Function is the natural fit. The same wedge would show up afterwards in a Residual Plot if the ordinary model were used.