Linear Regression Analysis in Python

The user wants to load the Boston Housing dataset from a specific URL and perform a simple linear regression between rm (average number of rooms per dwelling) and medv (median value of owner-occupied homes).

Following the "AI Data Analyst" instructions:

1. Iterative approach: I cannot do the load and the regression in one go.
2. Rule: "When a dataset is not yet loaded in current notebook context, first load and inspect only: show shape, columns, and head()."
3. Rule: "Do not combine first-load data inspection with plotting/modeling in the same response."

Step 1: Load the data and inspect it.

The dataset is loaded. It has 506 rows and 14 columns. The target column MEDV (median value) and the predictor RM (average number of rooms) are present. The user wants a simple linear regression of RM vs MEDV. I have only loaded and inspected the data. I haven't performed the regression yet.

Path B is required. I need to perform the linear regression.

Plan:

1. Import LinearRegression from sklearn.linear_model.
2. Reshape RM to be a 2D array as required by scikit-learn.
3. Fit the model and extract the coefficients (slope and intercept).
4. Display the results.

The user wants to load the Boston housing data and run a simple linear regression of RM (rooms) vs MEDV (price).

Looking at the notebook state:

1. Data was loaded from the specified URL.
2. RM and MEDV columns were identified.
3. A LinearRegression model was fitted using X = df[['RM']] and y = df['MEDV'].
4. The output shows:
   • Slope: 9.1021
   • Intercept: -34.6706
   • Equation: MEDV = 9.1021 * RM + -34.6706

The original goal was to "load... and run simple linear regression: rm (rooms) vs medv (price)". The model has been run and the coefficients (slope and intercept) are provided. This satisfies the core request.

Wait, the prompt asks me to act as a data analyst and extract concrete insights.

• Slope 9.1021 means that for every additional room, the median house value increases by approximately 9.1021 units (typically thousands of USD in the Boston dataset, though the dataset just says MEDV). I should just use the numeric value or USD notation if I treat it as currency, but the dataset labels it as MEDV. Actually, the rule says "never use $ for money/currency values... always write amounts using USD". 9.1021 units of 1000 USD = 9102.10 USD. However, the prompt says "amounts using USD", so I should be careful. Let's just state the coefficient effect.

Is there anything missing? The goal was to "run" the regression. It has been run and the parameters are printed. This is sufficient for a "simple linear regression" request unless a plot or evaluation metrics were explicitly requested. The prompt doesn't ask for a plot, just to run it.

Therefore, Path A is appropriate.

Insights:

• The positive slope of 9.1021 indicates a positive correlation between the number of rooms (RM) and the median value (MEDV).
• Specifically, each additional room is associated with an average increase of 9.1021 units in the median house value.
• The linear model is described by the equation: $\text{MEDV} = 9.1021 \times \text{RM} - 34.6706$.

Final check on currency rule: "never use $ for money/currency values... Always write amounts using USD". If I mention the value, I'll use USD. In the Boston dataset, MEDV is usually in 1000s of USD. If the value is 24.0, that's 24,000 USD. If the slope is 9.1, that's 9,100 USD. I will stick to the units provided in the output to be safe or just describe the coefficients.

The result is: Slope: 9.1021 Intercept: -34.6706

The model suggests that as the number of rooms increases, the median value of the home increases by approximately 9.1021 units per room.

The user wants to perform a multiple linear regression using all available features from the dataset and display the summary using the statsmodels library.

- `df`: DataFrame with 506 rows and 14 columns.
- Features: `CRIM`, `ZN`, `INDUS`, `CHAS`, `NOX`, `RM`, `AGE`, `DIS`, `RAD`, `TAX`, `PTRATIO`, `B`, `LSTAT`.
- Target: `MEDV`.

- Use `statsmodels.api` for the regression model.
- Define `X` (all features except `MEDV`) and `y` (`MEDV`).
- Add a constant term to the features (since `statsmodels` doesn't add an intercept by default).
- Fit the OLS model.
- Display the summary.

- I need to identify the features. All columns except `MEDV`.
- I need to use `sm.OLS`.
- I need to use `sm.add_constant(X)`.

- One code block.
- Focus on `statsmodels`.
- No identifier/leakage columns? All columns in this dataset seem to be features. No `Id` column present based on `df.columns`.

- Propose the step.
- Provide code.
- Wait for results.

The user wants to run a multiple linear regression with all features and see the statsmodels summary. Looking at the provided notebook state:

• cell-3 executed the statsmodels OLS regression using all features (X_all).
• output-3 contains the OLS Regression Results summary.
• The summary shows an R-squared of 0.741 and an Adjusted R-squared of 0.734.
• The goal was specifically to "run multiple linear regression with all features and show the statsmodels summary".
• The summary is visible in the output.

The original goal has been met. I can now provide a final answer based on the observed outputs.

Insights from the summary:

• R-squared: 0.741 (The model explains approximately 74.1% of the variance in the median house value MEDV).
• Adj. R-squared: 0.734 (Correcting for the number of predictors, the model explains 73.4% of the variance).
• F-statistic: 108.1 with a very low p-value (6.72e-135), indicating the model is statistically significant.

Decision: Path A.

The user wants to check the regression assumptions for the previously fitted mlr_model (Multiple Linear Regression). Specifically, they want:

1. Residual plot (to check for linearity and patterns).
2. Q-Q plot (to check for normality of residuals).
3. Homoscedasticity check (usually done via residuals vs. fitted values plot or similar).

Notebook state:

• df is loaded.
• mlr_model is the fitted statsmodels OLS model.
• Target variable is MEDV.
• Features are all other columns.

Plan:

• I need to get the residuals and the fitted values from mlr_model.
• I will use seaborn and matplotlib as per the visualization policy.
• Since the user asked for three different things, I should follow the "one chart per step" rule OR provide a combined figure if it's standard for "checking assumptions". However, the rules say "Default to one chart OR one table per step" and "Do not provide a full end-to-end pipeline at once".
• But for "checking assumptions", these three are usually seen together. Let's start with the Residual Plot first to see if there's a non-linear pattern.
• Wait, the user explicitly asked for all three. I'll start with the Residual Plot (Residuals vs Fitted) which is the primary diagnostic for linearity and homoscedasticity. Then in the next turn, I'll do the Q-Q plot.

Step 1: Residual Plot (Fitted values vs Residuals). This helps check for linearity and homoscedasticity (constant variance).

Wait, the prompt asks for "residual plot, Q-Q plot, homoscedasticity". If I only do one, I'm ignoring part of the request. If I do all, I might violate the "one chart per step" rule. Usually, in a data science workflow, we can put these in subplots. But the rule says "one chart per step". I'll start with the Residual plot as it's the most fundamental.

Actually, looking at the "Visualization defaults" section: "Default to one chart OR one table per step." I will propose to do these one by one.

First step: Residual plot.

Code:

• Calculate fitted values: mlr_model.predict(X_all_with_const)
• Calculate residuals: mlr_model.resid
• Plot fitted vs resid.

Wait, let me double check if I can do a 2x2 or 1x3 grid. The rules are quite strict: "Default to one chart OR one table per step." I'll stick to one. I'll start with the Residual plot.

Wait, a better way to check homoscedasticity and linearity is the same plot (Residuals vs Fitted). So I'll do that first.

Wait, the user prompt is "check regression assumptions: residual plot, Q-Q plot, homoscedasticity". I will start with the Residual Plot (Fitted vs Residuals).

The user wants to check regression assumptions: residual plot, Q-Q plot, and homoscedasticity. Looking at the notebook state:

• [cell-4] and [output-4] show a Residual Plot (Fitted Values vs Residuals).
• We are missing the Q-Q plot (to check normality of residuals).
• We have the residual plot, which is the first step for homoscedasticity, but we haven't specifically addressed the normality check (Q-Q plot).

The original goal is "check regression assumptions: residual plot, Q-Q plot, homoscedasticity". Current state: Residual plot done. Q-Q plot missing.

Analysis of the residual plot:

• The residuals are plotted against fitted values. This allows us to check for non-linearity (patterns) and heteroscedasticity (changing variance). While the image is not explicitly described in text output, the code was executed.

Plan:

1. Provide insights on the residual plot (assuming the user can see it, but since I am an AI analyst, I should mention what it evaluates).
2. Provide the code for the Q-Q plot to complete the normality check.

Wait, the sufficiency heuristic says "If the latest output is chart-only... choose Path B". Output-4 is a figure. I need to provide the Q-Q plot.

Actually, the goal mentions three things: residual plot (done), Q-Q plot (missing), and homoscedasticity (partially checked via residual plot, but often a formal check or a specific plot is used).

I will provide the Q-Q plot now.

Insights: The residual plot allows us to observe if the residuals are randomly distributed around zero. If a "funnel" shape appears, it indicates heteroscedasticity. If a curve appears, it indicates non-linearity.

Next step: Q-Q plot.

• Goal: Check regression assumptions: residual plot, Q-Q plot, homoscedasticity.

  • State:

    • mlr_model (OLS) is fitted.
    • Residual Plot (Fitted vs Residuals) has been generated.
    • Q-Q Plot (Normal distribution check) has been generated.

  • Analysis of Goal vs state:

    • Residual plot: Done.
    • Q-Q plot: Done.
    • Homoscedasticity: The residual plot is the primary visual tool for this, but for a formal check, a statistical test like the Breusch-Pagan test is standard and provides a definitive answer rather than just visual intuition.

  • Residual Plot: Visually checks for linearity and homoscedasticity (constant variance).

  • Q-Q Plot: Visually checks if residuals are normally distributed.

  • Homoscedasticity check: The user explicitly asked for "homoscedasticity". While the residual plot addresses this visually, a formal test (like Breusch-Pagan) is the professional way to "check" it beyond just "looking".

  • The goal was to "check" these things. We have visual plots, but we lack a statistical confirmation for homoscedasticity.

  • Path B is appropriate to provide the statistical test for homoscedasticity to complete the set of "checks".

  • Insights: The residual plot helps identify patterns (linearity/variance), and the Q-Q plot evaluates normality.

  • Action: Perform a Breusch-Pagan test for homoscedasticity.

• Goal: Check regression assumptions (residual plot, Q-Q plot, homoscedasticity).

  • State:

    • Cell 4: Residual plot generated (visual check for linearity and homoscedasticity).
    • Cell 5: Q-Q plot generated (visual check for normality of residuals).
    • Cell 6: Breusch-Pagan test executed. Output: LM Statistic = 65.1218, p-value = 6.2654e-09.

  • Residual Plot: Used to check for non-linearity and heteroscedasticity. (Done)

  • Q-Q Plot: Used to check for normality. (Done)

  • Homoscedasticity: Breusch-Pagan test provided a p-value of $6.2654 \times 10^{-9}$. This is significantly below 0.05, meaning we reject the null hypothesis of homoscedasticity. The data exhibits heteroscedasticity. (Done)

  • The user goal was to "check" these assumptions.

  • Residual plot -> Done.

  • Q-Q plot -> Done.

  • Homoscedasticity (BP test) -> Done.

  • All components of the request have been addressed with both visual and statistical evidence.

  • Path A is appropriate because the outputs are sufficient to answer the goal.

  • Insights:

    • Residual plot: Visual check for patterns.
    • Q-Q plot: Visual check for normality.
    • Breusch-Pagan test: The extremely low p-value ($6.2654 \times 10^{-9}$) indicates a statistically significant presence of heteroscedasticity, meaning the variance of the residuals is not constant across the range of predictors.

  • Ensure no $ for currency (not applicable here, but keep in mind).

  • Include <!--final answer-->.

  • No code block.

  • No "goal achieved" phrases.

The user wants to compare the $R^2$ (R-squared) values of a simple linear regression (SLR) and a multiple linear regression (MLR).

*   `cell-2`: Simple Linear Regression (SLR) using `LinearRegression` from `sklearn` with feature `RM` and target `MEDV`.
*   `cell-3`: Multiple Linear Regression (MLR) using `sm.OLS` from `statsmodels` with all features.
*   `output-3`: The summary shows `R-squared: 0.741` for the MLR.

*   SLR was fit using `sklearn.linear_model.LinearRegression`.
*   The $R^2$ for SLR is not yet computed/printed in the provided notebook state. Only slope and intercept were printed.

*   Need to calculate $R^2$ for the SLR model.
*   Need to retrieve $R^2$ for the MLR model.
*   Compare them.

*   SLR model was `model`.
*   `model.score(X, y)` provides the $R^2$ for `sklearn` models.
*   MLR model was `mlr_model`.
*   `mlr_model.rsquared` provides the $R^2$ for `statsmodels` models.

The user wants to compare the $R^2$ of simple vs multiple regression.

Looking at the notebook state:

• cell-2 performed Simple Linear Regression (model).
• cell-3 performed Multiple Linear Regression (mlr_model).
• cell-7 explicitly calculated and printed the $R^2$ for both:
  • Simple Linear Regression $R^2$: 0.4835
  • Multiple Linear Regression $R^2$: 0.7406
  • Difference: 0.2571

The original goal is "compare R-squared of simple vs multiple regression". The output of cell-7 provides the exact values and the difference. This is sufficient to answer the goal.

Conclusion: Path A.