library(statlingo)
# Fit a simple linear model
fm <- lm(dist ~ speed, data = cars)
context <- "
The data give the speed of cars (mph) and the distances taken to stop
(ft). Note that the data were recorded in the 1920s!
"Get started (R)
Installation
Install from r-universe (once the build finishes syncing after registration):
install.packages("statlingo", repos = c("https://bgreenwell.r-universe.dev", "https://cloud.r-project.org"))Or install the development version directly from GitHub:
# install.packages("pak")
pak::pak("bgreenwell/statlingo/r")You’ll also need the ellmer package to create an LLM client:
install.packages("ellmer")Quick example
First, load the package, fit a linear model, and define the context of your data:
This fits a basic linear regression of stopping distance on speed using the built-in cars dataset and captures the research context.
Next, initialize the ellmer chat client using Google Gemini:
# Create an ellmer chat client using Google Gemini
client <- ellmer::chat_google_gemini()Using model = "gemini-2.5-flash".This creates a chat client instance configured to use Gemini models. The client is passed to explain() without mutating the original client state.
Generate the natural language explanation using explain(), targeting a student audience:
# Get an explanation tailored for a student audience
ex <- explain(
fm,
client = client,
context = context,
audience = "student",
verbosity = "detailed"
)
print(ex)Here is a detailed explanation of your linear regression model output.
## Linear Regression Model Interpretation
You have fitted a **Linear Regression Model** to examine the relationship between a car's `dist` (stopping distance in feet) and its `speed` (speed in miles per hour). The goal of this model is to understand how changes in speed are linearly associated with changes in the mean stopping distance.
### Model Appropriateness and Context
Given that `dist` (stopping distance) is a continuous variable measured in feet, a linear regression model is generally an appropriate choice for exploring its relationship with a continuous predictor like `speed` (in mph). The research question implicitly assumes a linear relationship, which the model attempts to capture. The context that the data is from the 1920s is important for understanding the *real-world implications* of the model's coefficients (e.g., stopping distances were likely different then due to car technology and road conditions), but it does not invalidate the use of a linear model itself for *this specific dataset*. However, when interpreting, we should be mindful that these findings are specific to the conditions of the 1920s.
### Model Specification (Call)
The `Call` section shows the command used to fit the model:
`lm(formula = dist ~ speed, data = cars)`
* `lm()`: This is the function in R used to fit linear models.
* `formula = dist ~ speed`: This specifies the model. `dist` is the **response variable** (or dependent variable), which we are trying to predict or explain. `speed` is the **predictor variable** (or independent variable). The `~` symbol means "is modeled by" or "is predicted by."
* `data = cars`: This indicates that the variables `dist` and `speed` come from a dataset named `cars`.
This model estimates the stopping distance as a linear function of speed:
$\text{Stopping Distance (ft)} = \beta_0 + \beta_1 \times \text{Speed (mph)} + \epsilon$
where $\beta_0$ is the intercept, $\beta_1$ is the coefficient for speed, and $\epsilon$ represents the error term.
### Residuals Summary
`Residuals:`
`Min 1Q Median 3Q Max`
`-29.069 -9.525 -2.272 9.215 43.201`
**Residuals** are the differences between the observed stopping distances and the stopping distances predicted by your model. They represent the errors or unexplained variation in the model. This summary provides a five-number summary of these errors:
* **Min (-29.069 ft):** The smallest (most negative) residual. This means that for one observation, the model *over-predicted* the stopping distance by approximately 29.07 feet.
* **1Q (-9.525 ft):** The first quartile. 25% of the residuals are less than or equal to -9.525 feet.
* **Median (-2.272 ft):** The middle value of the residuals. A median close to zero (which -2.272 ft is, relative to the range of stopping distances) suggests that the model's predictions are, on average, not systematically too high or too low.
* **3Q (9.215 ft):** The third quartile. 75% of the residuals are less than or equal to 9.215 feet.
* **Max (43.201 ft):** The largest (most positive) residual. For one observation, the model *under-predicted* the stopping distance by approximately 43.20 feet.
The spread of these residuals (from -29.069 to 43.201 ft) gives you an idea of the typical magnitude of the model's prediction errors. The fact that the median is close to zero is a good sign that the model isn't systematically biased in its predictions. However, the range between the min and max residuals (and the somewhat skewed distribution, where the max is further from the median than the min) indicates there are still some substantial errors.
### Coefficients Table
The `Coefficients` table is the core of the linear regression output, providing the estimated values for the intercept and the predictor, along with statistics to assess their significance.
| Term | Estimate | Std. Error | t value | Pr(>|t|) | Signif. |
| :------------ | :------- | :--------- | :------ | :------- | :------ |
| (Intercept) | -17.5791 | 6.7584 | -2.601 | 0.0123 | `*` |
| speed (mph) | 3.9324 | 0.4155 | 9.464 | 1.49e-12 | `***` |
**Signif. codes:** `0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1` These codes provide a quick visual indicator of the p-value's magnitude. More stars mean stronger evidence against the null hypothesis.
#### Interpretation of (Intercept)
* **Estimate (-17.5791 ft):** This is the estimated expected stopping distance when `speed` is 0 mph.
* **In context:** An estimated stopping distance of -17.5791 feet when speed is 0 mph doesn't make physical sense (you can't have a negative stopping distance). This is often the case when a zero value for the predictor is outside the range of observed data or not physically meaningful in the context of the relationship. In your dataset, the lowest speed is 4 mph. Therefore, interpreting the intercept directly might be an **extrapolation** and not practically useful. It primarily serves to adjust the regression line to best fit the observed data.
* **Std. Error (6.7584):** This is the standard deviation of the intercept's estimate, representing the uncertainty in our estimate of the true intercept. A smaller standard error indicates a more precise estimate.
* **t value (-2.601):** This is the test statistic for the null hypothesis that the true intercept is zero. It's calculated as `Estimate / Std. Error`.
* **Pr(>|t|) (0.0123):** This is the p-value associated with the intercept.
* **Interpretation:** Assuming the true intercept is zero (the null hypothesis), there is a 1.23% probability of observing an intercept estimate as extreme as -17.5791 or more extreme, purely by random chance. Since this p-value (0.0123) is less than common significance levels (e.g., 0.05), we would conclude there is statistically significant evidence that the true intercept is not zero.
#### Interpretation of speed (mph)
* **Estimate (3.9324 ft/mph):** This is the most crucial coefficient. It tells us the estimated change in the mean stopping distance for every one-unit increase in speed, holding all other predictors constant (though in this model, there are no other predictors).
* **In context:** For every 1 mph increase in speed, the estimated stopping distance increases by 3.9324 feet. This is a very direct and interpretable finding, showing a positive linear relationship.
* **Std. Error (0.4155):** This is the standard deviation of the speed coefficient's estimate, indicating the precision of our estimate of the true effect of speed. This value is relatively small compared to the estimate, suggesting good precision.
* **t value (9.464):** This is the test statistic for the null hypothesis that the true coefficient for `speed` is zero (i.e., that `speed` has no linear relationship with `dist`). It's calculated as `Estimate / Std. Error`.
* **Pr(>|t|) (1.49e-12):** This is the p-value associated with the `speed` coefficient. The notation `1.49e-12` means $1.49 \times 10^{-12}$, which is an extremely small number (essentially 0).
* **Interpretation:** Assuming the true coefficient for `speed` is zero (the null hypothesis), there is an extremely small probability (close to 0%) of observing a coefficient estimate as extreme as 3.9324 or more extreme, purely by random chance. Since this p-value is much smaller than any common significance level (e.g., 0.05, 0.01, 0.001), we have very strong statistically significant evidence that `speed` is a significant linear predictor of `dist`. The faster a car goes, the longer it takes to stop.
### Residual Standard Error
`Residual standard error: 15.38 on 48 degrees of freedom`
* **Residual standard error (15.38 ft):** This value is an estimate of the standard deviation of the residuals. It represents the typical distance that the observed stopping distances fall from the regression line (the predicted values), in the units of the response variable.
* **In context:** On average, the model's predictions for stopping distance are off by about 15.38 feet. This gives you a practical sense of the model's accuracy.
* **Degrees of freedom (48):** This is calculated as `n - p - 1`, where `n` is the number of observations (50 in your `cars` dataset) and `p` is the number of predictor variables (1 for `speed`). So, $50 - 1 - 1 = 48$. It represents the number of independent pieces of information available to estimate the error variance.
### R-squared Values
`Multiple R-squared: 0.6511, Adjusted R-squared: 0.6438`
* **Multiple R-squared (0.6511):** This statistic, often simply called R-squared ($R^2$), represents the proportion of the variance in the response variable (`dist`) that can be explained by the predictor variable (`speed`) in the model.
* **In context:** Approximately 65.11% of the variability in stopping distances can be explained by the car's speed. This indicates that speed is a moderately strong predictor of stopping distance. The remaining 34.89% of the variability is unexplained by this model and is attributed to other factors or random error.
* **Adjusted R-squared (0.6438):** The Adjusted R-squared is a modified version of R-squared that accounts for the number of predictors in the model and the number of observations. It is generally preferred when comparing models with different numbers of predictors because it penalizes models for including unnecessary predictors.
* In this case, with only one predictor, the adjusted R-squared is very close to the multiple R-squared, as expected. It tells a similar story: about 64.38% of the variance in stopping distance is explained by speed, accounting for model complexity.
### F-statistic
`F-statistic: 89.57 on 1 and 48 DF, p-value: 1.49e-12`
* **F-statistic (89.57):** This is a test statistic for the overall significance of the linear regression model. It tests the null hypothesis that *all* of the regression coefficients (excluding the intercept) are equal to zero.
* **Null Hypothesis ($H_0$):** All regression coefficients are zero (i.e., `speed` has no linear relationship with `dist`).
* **Alternative Hypothesis ($H_A$):** At least one regression coefficient is not zero (i.e., `speed` *does* have a linear relationship with `dist`).
* **Degrees of freedom (1 and 48 DF):** The first number (1) is the number of predictor variables in the model. The second number (48) is the residual degrees of freedom.
* **p-value (1.49e-12):** This is the p-value associated with the F-statistic. Again, `1.49e-12` is an extremely small number, essentially 0.
* **Interpretation:** Assuming the null hypothesis is true (that speed has no effect on stopping distance), there is an extremely small probability (close to 0%) of observing an F-statistic as large as 89.57 or larger. Since this p-value is far below common significance levels (e.g., 0.05), we strongly reject the null hypothesis. This provides very strong evidence that the overall model is statistically significant, meaning that `speed` *is* indeed a significant predictor of `dist`.
## Suggestions for Checking Model Assumptions
While the model output provides estimates and tests of significance, it's crucial to check the underlying assumptions of linear regression to ensure the validity and reliability of these results. Here are the key assumptions and ways to check them, primarily using graphical methods:
1. **Linearity:** The relationship between the predictor (`speed`) and the mean of the response (`dist`) should be linear.
* **Check:** Create a scatter plot of `dist` versus `speed`. Look for a straight-line pattern.
* **Also check:** Plot **residuals versus fitted values**. If the relationship is linear, the residuals should be randomly scattered around zero with no discernible pattern (no curves, no fanning out/in).
* **What to look for:** A clear pattern (e.g., a curve) in the residuals vs. fitted values plot suggests that the linear model might not be appropriate, and a non-linear relationship might exist.
2. **Independence of Errors:** The errors (residuals) should be independent of each other. This is often violated in time-series data or clustered data.
* **Check:** Since this is cross-sectional data (not time-series), independence is often assumed if data collection was randomized. If there's a specific order to the `cars` data, you could plot residuals against that order.
* **What to look for:** No patterns in the residuals over time or order of data collection. A Durbin-Watson test can formally check for autocorrelation, but visual inspection is a good first step.
3. **Homoscedasticity (Constant Variance of Errors):** The variance of the residuals should be constant across all levels of the predictor variable.
* **Check:** Plot **residuals versus fitted values**. Look for a random scatter of points.
* **Also check:** A **Scale-Location plot** (square root of standardized residuals vs. fitted values).
* **What to look for:** No "fanning out" (heteroscedasticity) or "fanning in" pattern in these plots. If the spread of residuals changes as fitted values increase, this assumption is violated. Formal tests like the Breusch-Pagan test or White test can be used, but graphical checks are often more informative.
4. **Normality of Errors:** The residuals should be approximately normally distributed.
* **Check:** Create a **Normal Q-Q plot of the residuals**.
* **Also check:** A histogram or density plot of the residuals.
* **What to look for:** In the Q-Q plot, the points should fall approximately along the diagonal line. Deviations from the line, especially at the tails, indicate non-normality. The histogram should resemble a bell-shaped curve. A Shapiro-Wilk test can formally test for normality.
5. **No Severe Multicollinearity:** This refers to high correlation among predictor variables.
* **Check:** Since you only have one predictor (`speed`), multicollinearity is **not an issue** in this specific model. If you had multiple predictors, you would calculate Variance Inflation Factors (VIFs).
6. **Outliers and Influential Observations:** Data points that are far from the general trend or that have a disproportionate impact on the regression line.
* **Check:** Plot **Residuals versus Leverage**.
* **Also check:** The residuals vs. fitted values plot can also help identify outliers.
* **What to look for:** Points that stand out significantly from the rest, especially those with high leverage (far from the mean of the predictors) and large residuals.
By performing these diagnostic checks, you can gain confidence in your model's reliability or identify areas where the model might need adjustment (e.g., transforming variables, including additional predictors, or using a different model type).
***
*Caution: This explanation was generated by a Large Language Model. Users should critically review the output and consult additional statistical resources or experts to ensure correctness and a full understanding.* The returned explanation object prints out a structured, student-oriented summary of the regression model, detailing coefficients, overall fit, and checkable assumptions.
Finally, request diagnostic code recommendations based on the generated explanation using suggest_code():
# Suggest next diagnostic code steps based on the explanation
suggest_code(ex)## Next Steps: Suggested Coding Diagnostics
# 1. Plot Residuals vs Fitted (Linearity & Homoscedasticity)
plot(model, which = 1)
# 2. Plot Normal Q-Q (Normality of residuals)
plot(model, which = 2)
# 3. Test for Multicollinearity (requires package 'car')
if (requireNamespace("car", quietly = TRUE)) {
car::vif(model)
} else {
message("Install package 'car' to run: car::vif(model)")
}
# 4. Test for Autocorrelation (requires package 'car')
if (requireNamespace("car", quietly = TRUE)) {
car::durbinWatsonTest(model)
}
# 5. Test for Heteroscedasticity (requires package 'lmtest')
if (requireNamespace("lmtest", quietly = TRUE)) {
lmtest::bptest(model)
}
***
*Caution: These code suggestions were generated by a Large Language Model. Users should critically review the code to ensure correctness and applicability before executing.*This prints recommended R code snippets to check residual diagnostics, multicollinearity, autocorrelation, and constant variance.