From Regression Output to a Model Evaluation
In “Writing a Complete r-squared Interpretation,” you practiced turning an \(r^2\) value into a sentence that names the response, explanatory variable, and observed cases. An AP-style free-response question may ask you to do that and then evaluate the model. The second part requires more than repeating the \(r^2\) interpretation: consider whether a linear model is suitable and whether its prediction errors seem useful for the task.
Computer output might label the coefficient of determination \(R\)-sq or \(R\)-squared. In simple linear regression, read that value as \(r^2\). To evaluate a model, consider the evidence in combination: what fraction of response variation is accounted for, whether the residual plot has a systematic pattern, and how large the residuals typically are in response units. Each piece addresses a different question.
The residual plot matters because an \(r^2\) value summarizes the amount of variation accounted for by the fitted line, but it does not show whether the line misses in an organized way. As covered in “Reading a Residual Plot for Random Scatter” and “Curved Patterns in a Residual Plot,” random scatter around zero supports a linear form, while a systematic curve is evidence that a line is missing structure. A high \(r^2\) cannot make a curved residual pattern disappear.
The residual standard deviation \(s\), discussed in “Using s to Describe Prediction Accuracy,” gives a typical scale for prediction errors in the response variable’s units. It is not a percentage, and its practical importance depends on the context. An error of a few minutes might be acceptable for one use but too large for another. Likewise, a low \(r^2\) does not automatically make a model useless: the purpose and accuracy requirements matter.
A Free-Response Plan
A clear answer separates the requested interpretation from the evaluation. First state what the \(r^2\) percentage describes. Then assess the evidence about the linear form and typical prediction error. Finish with a limited conclusion about how useful the model appears for the stated setting.
Determine which variable is the response, which is explanatory, and what observations the data represent.
Convert the displayed value to a percentage and describe variation in the response accounted for by its linear relationship with the explanatory variable.
State whether the residuals show random scatter or a systematic pattern, and explain what that suggests about using a line.
Interpret \(s\) in response units when it is provided. Say whether the evidence supports using the model for the intended purpose, without claiming more than the data show.
A model evaluation is not a search for a universal “good \(r^2\)” cutoff. Instead, explain what the reported value says, what the residual plot adds, and whether the prediction-error scale is acceptable for the particular use. Keep any conclusion limited to the observed cases and predictor range unless the question gives a basis for going beyond them.
Worked Examples: Interpret and Evaluate
Worked Example: A Linear Model With Random Residual Scatter
A fictional project records weekly volunteer hours and the kilograms of produce harvested from each of 48 community garden plots during one growing season. Harvest weight is the response, and volunteer hours are the explanatory variable. A regression output reports \(R\)-sq \(=0.684\) and \(S=4.8\) kilograms. The residual plot shows residuals scattered above and below zero without a clear pattern. Write a complete interpretation and evaluate the model for describing the relationship in these plots.
State. The requested interpretation is about variation in harvest weight among the 48 plots. The output’s \(R\)-sq is the coefficient of determination for the fitted linear model.
Plan. Convert \(0.684\) to a percentage and name both variables and the cases. For the evaluation, use the residual plot to comment on the linear form and interpret \(S\) in kilograms. Do not treat \(r^2\) as a percentage of accurate predictions or as evidence of causation.
Do. The conversion is
The residuals are scattered without a clear systematic pattern, which supports using a linear model to describe the relationship in these data. The output’s \(S=4.8\) kilograms means that the typical size of a residual around the fitted line is about 4.8 kilograms. Whether that error size is acceptable for making decisions would depend on the purpose of those decisions.
Conclude. About 68.4% of the variation in harvest weight among the 48 community garden plots is accounted for by the linear relationship between harvest weight and weekly volunteer hours. The residual plot supports a linear model for describing this relationship in the observed plots, and the typical prediction error is about 4.8 kilograms. This evidence does not show that volunteer hours caused the harvest weights or guarantee accurate predictions for every plot.
Worked Example: A High r-squared With a Curved Residual Pattern
A fictional greenhouse project records daily hours under a grow light and the heights, in centimeters, of 36 seedlings after three weeks. Seedling height is the response and light hours are the explanatory variable. The output reports \(R\)-sq \(=0.940\) and \(S=2.1\) centimeters. The residual plot has a curved pattern: residuals tend to be positive at low and high light durations and negative in the middle. Should the linear model be considered adequate? Explain.
State. The \(r^2\) interpretation concerns variation in seedling height among the 36 seedlings. The question also asks whether a straight-line model is an adequate description of the relationship.
Plan. Convert the reported \(R\)-sq to a percentage, then evaluate it alongside the residual pattern. A high percentage indicates that the line accounts for much of the response variation, but a curved residual pattern signals systematic misses by the line.
Do. The percentage conversion is
About 94.0% of the variation in seedling height is accounted for by its linear relationship with light hours in these observations. However, the residuals do not show random scatter around zero. Their curved pattern means that the line tends to overpredict in the middle and underpredict at low and high light durations: negative residuals mean observed height is below predicted height, while positive residuals mean observed height is above predicted height. The \(S\) value describes a typical residual size, but it does not remove this systematic pattern.
Conclude. The linear model accounts for a large percentage of the variation in seedling height, but it does not appear to be an adequate description of the pattern across the observed light durations because its residuals follow a curve. The high \(r^2\) alone is not enough to justify using a straight line for this relationship. The conclusion is about the observed seedlings and range of light durations, not all possible growing conditions.
Worked Example: A Modest r-squared and a Purpose-Dependent Judgment
A fictional delivery service records route distance, in kilometers, and delivery time, in minutes, for 65 orders. Delivery time is the response, and route distance is the explanatory variable. The output gives \(R\)-sq \(=0.290\) and \(S=6.2\) minutes. The residual plot shows random scatter around zero, with no obvious curve or fan shape. A dispatcher says the model may be useful for rough scheduling but not for promising a customer an exact arrival time. Assess that claim using the output.
State. The \(r^2\) interpretation concerns variation in delivery times among the 65 orders. The model’s proposed use is rough scheduling, not precise individual arrival-time promises.
Plan. Convert \(R\)-sq to a percentage. Then consider the residual plot’s support for a linear form and interpret \(S\) in minutes. Evaluate whether those features fit the dispatcher’s limited claim; do not infer that a modest \(r^2\) automatically makes the model useless.
Do. The conversion is
About 29.0% of the variation in delivery time among these orders is accounted for by its linear relationship with route distance. The random residual scatter supports using a linear model to describe the relationship in the observed orders. However, the typical residual size is about 6.2 minutes, so an individual prediction may differ from the actual delivery time by a meaningful amount. Whether that is acceptable depends on the scheduling purpose.
Conclude. The output is consistent with using the model as a rough description for scheduling: the residual plot shows no obvious systematic pattern, while the modest \(r^2\) and typical residual size of about 6.2 minutes caution against treating individual predictions as exact. About 29.0% of the variation in delivery times among the 65 orders is accounted for by the linear relationship between delivery time and route distance. This does not establish that distance is the only factor affecting delivery time or that the model will work equally well outside the observed data.
Common Mistakes and Full-Credit Communication
A strong free-response answer makes separate claims for separate evidence. The \(r^2\) interpretation describes the fraction of response variation accounted for by the linear relationship. The residual plot helps assess whether a linear form is reasonable. The residual standard deviation describes typical error size in response units. Blending these ideas can turn a correct output reading into an unsupported conclusion.
- “The model is good because \(r^2\) is high.” A high \(r^2\) does not reveal whether the residuals follow a curve or another systematic pattern. Refer to the residual plot before judging the linear form.
- “The model predicts 68% of observations correctly.” This misreads \(r^2\) as a success rate. Say instead that the stated percentage of variation in the named response is accounted for by its linear relationship with the explanatory variable.
- Ignoring a visible residual pattern. A curve or changing spread is relevant even when \(r^2\) is large. Describe the pattern and explain that it weakens the case for the fitted line as an adequate model.
- Calling a model useless just because \(r^2\) is modest. Explain what the value says and consider the task. A model used for rough planning may have a different standard from one used for precise individual predictions.
- Using \(s\) as if it were a percentage. \(s\) is measured in the response variable’s units. For delivery time, for example, report a typical residual size in minutes, not as a percent of orders.
- Making a causal or universal claim. Regression output alone does not show that the explanatory variable caused changes in the response. Keep the evaluation tied to the observed cases and the predictor range represented by the data.
For full credit, do not merely say “the model fits well” or “the model is poor.” State the evidence and its meaning: for example, “The residuals show random scatter around zero, supporting a linear form for these observations,” or “The curved residual pattern suggests that a straight line misses the relationship systematically.” Then connect the \(r^2\) and, when supplied, \(s\) to the context. These statements let the reader see how the judgment follows from the output rather than from an unexplained label.
Check Your Understanding
For each item, distinguish what \(r^2\) says from what the additional evidence says about the linear model.
- A project relates weekly screen time to sleep duration for 42 students. Sleep duration is the response, and \(R\)-sq is 0.52. Write a complete interpretation of \(r^2\), including the observed cases.
- A regression of water depth on distance downstream has \(R\)-sq \(=0.88\), but the residual plot shows a clear curve. What does the output say about variation, and what does the residual pattern suggest about using a line?
- A model of commute distance and commute time has random residual scatter and \(S=5.5\) minutes. Interpret \(S\) in context and state why the model’s usefulness depends on its intended use.
- Explain why a high \(r^2\) does not by itself establish that a linear model is appropriate.
- A student writes, “The model explains 35% of the delivery orders.” Identify the interpretation error and describe the kind of variation the \(r^2\) percentage should refer to.