Why Use a Scale-Location Plot?
A residuals-versus-fitted plot is useful for checking whether a regression model captures the mean pattern. A different diagnostic question is whether the spread of the residuals remains roughly the same across the range of fitted responses. If the spread grows or shrinks, the constant-variance assumption may be questionable, even when the residuals remain centered near zero.
A scale-location plot, also called a spread-location plot, makes changes in residual magnitude easier to see. It places fitted values on the horizontal axis and a transformed residual magnitude on the vertical axis. The transformation is designed to display spread rather than the sign of the residual. This plot is a visual diagnostic, not a formal hypothesis test.
The ideal reference pattern is a roughly horizontal band: the vertical spread of points and any smooth trend should remain broadly similar as fitted values increase. A smooth curve is often added to help reveal the overall pattern, but it does not replace inspection of the points. The ordinate is nonnegative, so the plot deliberately does not show whether an observation is above or below its fitted value.
Constructing and Reading the Plot
The model context matters. Under the classical linear regression assumptions, the response has a linear mean given the predictors, errors have mean zero, errors are uncorrelated, and their conditional variance is a common value \(\sigma^2\). The design matrix is assumed to have full column rank so the coefficients can be estimated. Under these assumptions, residual magnitudes should not show a systematic change with fitted response, aside from sampling variation and the residual adjustment for leverage.
For each observation, calculate or obtain its fitted value and standardized residual, take the absolute value of the standardized residual, and then take its square root. Plot the resulting pair \((\hat{y}_i,q_i)\). In particular, do not take the square root of the signed residual: negative residuals would not have real square roots.
For each observation, identify \(\hat{y}_i\), \(e_i\), and, if calculating standardization yourself, \(h_{ii}\) and \(s\).
Use \(r_i=e_i/[s\sqrt{1-h_{ii}}]\) under the stated internal-standardization convention.
Calculate \(q_i=\sqrt{|r_i|}\). Retain this nonnegative value even when \(e_i\) is negative.
Plot \(q_i\) against \(\hat{y}_i\). Look for roughly stable spread, a widening or narrowing band, or another systematic change.
Worked Example: Transforming Standardized Residuals
A hypothetical regression model produces the fitted values and internally standardized residuals below for six observations. Construct the vertical coordinates for the scale-location plot and describe the pattern.
| Observation | Fitted value \(\hat{y}_i\) | Standardized residual \(r_i\) | \(q_i=\sqrt{|r_i|}\) |
|---|---|---|---|
| 1 | 12 | -0.36 | 0.60 |
| 2 | 18 | 0.49 | 0.70 |
| 3 | 24 | -0.64 | 0.80 |
| 4 | 30 | 0.81 | 0.90 |
| 5 | 36 | -1.00 | 1.00 |
| 6 | 42 | 1.21 | 1.10 |
Method. Apply \(q_i=\sqrt{|r_i|}\) to each standardized residual. For observation 1, \(q_1=\sqrt{|-0.36|}=\sqrt{0.36}=0.60\). For observation 2, \(q_2=\sqrt{|0.49|}=\sqrt{0.49}=0.70\). In the same way, the remaining ordinates are \(\sqrt{0.64}=0.80\), \(\sqrt{0.81}=0.90\), \(\sqrt{1.00}=1.00\), and \(\sqrt{1.21}=1.10\).
Interpretation. The plotted ordinates increase steadily from 0.60 to 1.10 as fitted values rise from 12 to 42. That sequence suggests a widening residual spread at higher fitted values, rather than a horizontal band. Six observations are not enough to establish a population pattern, so this is a signal to investigate the variance assumption, not proof that it fails.
Common mistake avoided. The negative signs do not make the first and third points invalid. The absolute value is taken before the square root, so the plot displays magnitude, not residual direction.
Why the Transformation Helps
The scale-location ordinate is not itself an estimate of the error standard deviation. Its purpose is to reveal changes in magnitude on a less compressed scale than the squared residuals. The following approximation explains why an increasing pattern is informative.
Under constant variance, the conditional variance of an ordinary least-squares residual is \(\operatorname{Var}(e_i\mid X)=\sigma^2(1-h_{ii})\). The factor \(1-h_{ii}\) accounts for the smaller residual variance at higher-leverage observations under the model. Standardizing by the corresponding residual standard deviation therefore puts residuals on a more comparable scale. If the errors instead have standard deviations \(\sigma_i\) that vary across observations, a useful approximation is \(r_i\approx(\sigma_i/s)Z_i\), where \(Z_i\) has a common, roughly stable standardized distribution. Taking absolute values and square roots gives the justification below.
If the distribution of \(Z_i\) is broadly similar across fitted values, the factor \(E[\sqrt{|Z_i|}]\) does not systematically change with \(\hat{y}_i\). The expected plotted height then changes with the square root of the local residual scale. A change in spread can consequently appear as a change in the height or dispersion of points. This is a diagnostic approximation: estimated coefficients, finite samples, and the dependence among regression residuals mean the plotted points are not independent observations from that simple expression.
Worked Example: Calculating the Plot Coordinates from Residuals
A hypothetical model has \(n=20\) observations and \(p=3\) fitted coefficients. Its sum of squared errors is \(680\). For four selected observations, the fitted values, residuals, and leverages are shown below. Calculate \(s\), then each standardized residual and scale-location ordinate.
| Observation | \(\hat{y}_i\) | \(e_i\) | \(h_{ii}\) |
|---|---|---|---|
| A | 40 | 4.5 | 0.19 |
| B | 55 | -6.4 | 0.36 |
| C | 70 | 8.4 | 0.51 |
| D | 85 | -9.6 | 0.64 |
Step 1: Find the residual standard error. The residual degrees of freedom are \(n-p=20-3=17\). Thus \(s=\sqrt{680/17}=\sqrt{40}\approx6.325\). (The unrounded value is used in the calculations below.)
Step 2: Standardize each residual. For A, \(\sqrt{1-h_{ii}}=\sqrt{0.81}=0.9\), so \(r_A=4.5/(\sqrt{40}\times0.9)\approx0.791\). For B, the denominator is \(\sqrt{40}\times0.8\), giving \(r_B=-6.4/(\sqrt{40}\times0.8)\approx-1.265\). For C, \(\sqrt{1-0.51}=0.7\), giving \(r_C=8.4/(\sqrt{40}\times0.7)\approx1.897\). For D, \(\sqrt{1-0.64}=0.6\), giving \(r_D=-9.6/(\sqrt{40}\times0.6)\approx-2.530\).
Step 3: Transform the magnitudes. The ordinates are \(q_A=\sqrt{0.791}\approx0.889\), \(q_B=\sqrt{1.265}\approx1.125\), \(q_C=\sqrt{1.897}\approx1.377\), and \(q_D=\sqrt{2.530}\approx1.591\). The plotted points are therefore approximately \((40,0.889)\), \((55,1.125)\), \((70,1.377)\), and \((85,1.591)\). Values are rounded to three decimals only at the final step.
Check. Each ordinate is nonnegative, including those from negative residuals, and each increases with the fitted value. The selected observations suggest increasing spread. The table contains only four selected cases, however, so interpretation should use the complete plot rather than treating this subset as decisive.
Common mistake avoided. Dividing each residual by \(s\) alone would omit the leverage adjustment \(\sqrt{1-h_{ii}}\). The standardized-residual definition used here includes that factor.
Interpreting More Than a Simple Funnel
A funnel-shaped band is a familiar warning pattern: points become more vertically dispersed as fitted values increase, or less dispersed as they increase. But changing variance need not be monotone. A U-shaped or inverted-U-shaped smooth can indicate that residual spread is larger in one part of the fitted range than in another. The important question is whether the vertical spread changes systematically, not whether the pattern matches one particular geometric shape.
A single high point is different from a broad pattern. It may reflect one unusually large residual, but by itself it does not show that residual variance changes throughout a range of fitted values. Conversely, a cloud of points can have no dramatic outlier and still show a gradual funnel. Look at the overall distribution of heights and the smooth trend together.
| Pattern in the scale-location plot | What it suggests | What not to conclude automatically |
|---|---|---|
| Roughly even vertical spread with a near-horizontal smooth | No clear visual evidence of changing spread | That constant variance has been proven |
| Increasing or decreasing spread across fitted values | A possible funnel-shaped variance pattern | That every observation follows a perfectly monotone trend |
| Spread larger in the middle or at both ends | A possible nonmonotone change in residual spread | That a straight-line trend is the only pattern worth checking |
| One isolated high ordinate without a broader trend | A potentially unusual residual magnitude | That the entire model has changing variance |
Worked Example: Recognizing a Nonmonotone Spread Pattern
A hypothetical scale-location plot has the following standardized residuals at five increasing fitted values. Calculate the ordinates and decide whether the pattern is consistent with a simple horizontal band.
| \(\hat{y}_i\) | \(r_i\) | \(q_i=\sqrt{|r_i|}\) |
|---|---|---|
| 20 | 1.44 | 1.20 |
| 30 | -0.64 | 0.80 |
| 40 | 0.25 | 0.50 |
| 50 | -0.64 | 0.80 |
| 60 | 1.44 | 1.20 |
Calculation. At fitted value 20, \(q=\sqrt{|1.44|}=1.20\). At 30, \(q=\sqrt{0.64}=0.80\); at 40, \(q=\sqrt{0.25}=0.50\); at 50, \(q=\sqrt{0.64}=0.80\); and at 60, \(q=\sqrt{1.44}=1.20\).
Interpretation. The ordinates fall from 1.20 to 0.50 and then rise to 1.20. This is a U-shaped pattern, not a simple increasing funnel and not a roughly horizontal band. If the full plot shows this pattern consistently, residual spread may be smaller in the middle of the fitted range than at either end.
Common mistake avoided. Looking only for a steadily widening funnel would miss this pattern. The plot should be inspected for any systematic change in spread, including a change that reverses direction.
Common Traps and Practical Limits
The plot is useful only when its axes and purpose are read correctly. The horizontal coordinate is the fitted response, not automatically one chosen predictor. A changing pattern against fitted values indicates that spread varies across the model’s fitted range; it does not by itself identify which predictor or mechanism caused the change. The earlier material on residuals versus predictors addresses patterns associated with particular predictors and misspecified mean relationships.
- Taking the square root before the absolute value. For \(r_i=-0.64\), \(\sqrt{r_i}\) is not real. The correct ordinate is \(\sqrt{|-0.64|}=0.80\).
- Keeping the sign as the focus. The plot intentionally removes the residual sign. Use a residuals-versus-fitted plot when the direction of residuals is central to the question.
- Calling a flat-looking plot proof of constant variance. A small sample or a narrow fitted-value range may conceal a pattern. A visual diagnostic can fail to reveal a real departure.
- Treating one high point as a funnel. An isolated ordinate may be worth investigating, but a variance trend requires a broader pattern across fitted values.
- Confusing variance with the mean pattern. The scale-location plot focuses on residual magnitude. It is not, on its own, a complete check that the regression mean is correctly specified.
A scale-location plot is a useful first check, but it cannot establish the cause of a pattern or prove that the model assumptions hold. It is aligned with the SOA Exam SRM (Statistics for Risk Modeling) syllabus because candidates should be able to construct and interpret regression diagnostic plots, including distinguishing a roughly stable spread from changing residual spread.
Check Your Understanding
For each problem, calculate or interpret the scale-location information as requested.
- A fitted model reports a standardized residual of \(-1.21\) for an observation with fitted value 25. What is its scale-location ordinate, and what does the sign contribute to that ordinate? (Answer: \(q=1.10\); the negative sign is removed by the absolute value.)
- For two observations, \((\hat{y}_i,r_i)=(30,0.81)\) and \((60,-1.44)\). Find both plotted points and state whether these two points alone establish a funnel. (Answer: \((30,0.90)\) and \((60,1.20)\); no, two points alone do not establish a systematic pattern.)
- A regression has \(n=14\), \(p=2\), and \(\mathrm{SSE}=432\). For one case, \(e_i=6\) and \(h_{ii}=0.36\). Find \(s\), \(r_i\), and \(q_i\), rounding the final two values to three decimals. (Answer: \(s=6\), \(r_i=1.250\), and \(q_i=1.118\).)
- Across increasing fitted values, a plot’s smooth declines, is lowest near the middle, and then rises; points are also more vertically spread at both ends. What pattern is suggested, and why might a funnel-only inspection miss it? (Answer: A nonmonotone, roughly U-shaped change in spread is suggested; it does not widen in just one direction.)
- A scale-location plot is roughly horizontal except for one very high ordinate, while the remaining points have similar spread throughout. What is the more careful interpretation? (Answer: The isolated point has a large residual magnitude and may merit investigation, but it alone does not establish changing variance across fitted values.)