Executive Summary

Key Model Insights and Performance Overview

OV

Executive Summary

Analysis: y

0.955
R² Score

Executive Summary — overview — High-level regression results and key findings

0.955
r squared
0.953
adj r squared
100
observations
3
predictors
Yes
significant
4.57
rmse

Business Context

Company: Test Analytics Corp

Objective: Analyze relationships between predictors and target variable

Model Variables

Target: y

Summary

Variable Estimate p_value
(Intercept) 2.452 0.520
x1 1.492 0.000
x2 0.780 0.000
x3 -0.488 0.000
r squared: 0.955
adj r squared: 0.953
observations: 100
IN

Key Insights

Executive Summary

Based on the regression analysis results for Test Analytics Corp, here are some executive insights:

  1. Business Impact:

    • The model shows a high level of accuracy with an R-squared value of 0.9545, indicating that approximately 95.45% of the variance in the target variable (y) can be explained by the predictors (x1, x2, x3). This strong relationship between the predictors and the target variable can have a significant positive impact on the company’s decision-making processes and strategies.

  2. Key Relationships Found:

    • The coefficients of the predictors reveal the strength and direction of the relationships with the target variable:
      • x1: Estimate 1.4921, p-value 2.4158e-54.
      • x2: Estimate 0.7797, p-value 3.8245e-11.
      • x3: Estimate -0.4879, p-value 7.0764e-51.
    • These results indicate that x1 has the strongest positive relationship with the target variable, followed by x2, while x3 has a negative relationship.

  3. Model Reliability:

    • The model is statistically significant, as indicated by the p-values for all predictors being very low. This suggests that the predictors are useful in predicting the target variable.
    • The model’s high Adjusted R-squared value of 0.9531 suggests that the chosen predictors collectively explain a large proportion of the variance in the target variable while adjusting for the number of predictors in the model.
    • The RMSE (Root Mean Squared Error) of 4.569 indicates the average difference between the predicted values and the actual values, providing a measure of how well the model fits the data.

In summary, the analysis highlights strong relationships between the predictors and the target variable, indicating the potential for accurate predictions. The high model reliability and significance of predictors suggest that the model can be valuable for decision-making within Test Analytics Corp.

IN

Key Insights

Executive Summary

Based on the regression analysis results for Test Analytics Corp, here are some executive insights:

  1. Business Impact:

    • The model shows a high level of accuracy with an R-squared value of 0.9545, indicating that approximately 95.45% of the variance in the target variable (y) can be explained by the predictors (x1, x2, x3). This strong relationship between the predictors and the target variable can have a significant positive impact on the company’s decision-making processes and strategies.

  2. Key Relationships Found:

    • The coefficients of the predictors reveal the strength and direction of the relationships with the target variable:
      • x1: Estimate 1.4921, p-value 2.4158e-54.
      • x2: Estimate 0.7797, p-value 3.8245e-11.
      • x3: Estimate -0.4879, p-value 7.0764e-51.
    • These results indicate that x1 has the strongest positive relationship with the target variable, followed by x2, while x3 has a negative relationship.

  3. Model Reliability:

    • The model is statistically significant, as indicated by the p-values for all predictors being very low. This suggests that the predictors are useful in predicting the target variable.
    • The model’s high Adjusted R-squared value of 0.9531 suggests that the chosen predictors collectively explain a large proportion of the variance in the target variable while adjusting for the number of predictors in the model.
    • The RMSE (Root Mean Squared Error) of 4.569 indicates the average difference between the predicted values and the actual values, providing a measure of how well the model fits the data.

In summary, the analysis highlights strong relationships between the predictors and the target variable, indicating the potential for accurate predictions. The high model reliability and significance of predictors suggest that the model can be valuable for decision-making within Test Analytics Corp.

RC

Recommendations

Analysis: y

3
Key Actions

Recommendations — recommendations — Actionable insights and next steps

Good
model quality
95%
confidence level
3
significant predictors

Business Context

Company: Test Analytics Corp

Objective: Analyze relationships between predictors and target variable

Model Variables

Target: y

model quality: Good
confidence level: 95%
significant predictors: 3
IN

Key Insights

Recommendations

Based on the data profile provided for Test Analytics Corp:

  1. Next Steps for Improving the Model:

    • Conduct a thorough analysis of the significant predictors identified in the model to understand their individual and combined impact on the target variable.
    • Consider exploring interactions between the significant predictors to potentially uncover non-linear relationships that could enhance the model’s predictive power.
    • Evaluate the possibility of including additional relevant features or transforming existing ones to capture more complex relationships and improve model performance further.

  2. Areas for Further Investigation:

    • Investigate potential multicollinearity issues among the predictors, as this could impact the stability and interpretability of the coefficients in the model.
    • Perform sensitivity analysis to assess the robustness of the model and ensure its reliability across different scenarios or subsets of data.
    • Explore different model algorithms or techniques to compare and validate the current model’s performance and potentially uncover better fitting approaches.

  3. Business-Relevant Conclusions:

    • Communicate the model’s findings and insights to relevant stakeholders within Test Analytics Corp to guide decision-making processes effectively.
    • Monitor the model’s performance over time and consider implementing regular updates or re-evaluations to account for changing business dynamics and data patterns.
    • Ensure clear documentation of the model development process, assumptions made, and limitations to enhance transparency and facilitate future model enhancements.
IN

Key Insights

Recommendations

Based on the data profile provided for Test Analytics Corp:

  1. Next Steps for Improving the Model:

    • Conduct a thorough analysis of the significant predictors identified in the model to understand their individual and combined impact on the target variable.
    • Consider exploring interactions between the significant predictors to potentially uncover non-linear relationships that could enhance the model’s predictive power.
    • Evaluate the possibility of including additional relevant features or transforming existing ones to capture more complex relationships and improve model performance further.

  2. Areas for Further Investigation:

    • Investigate potential multicollinearity issues among the predictors, as this could impact the stability and interpretability of the coefficients in the model.
    • Perform sensitivity analysis to assess the robustness of the model and ensure its reliability across different scenarios or subsets of data.
    • Explore different model algorithms or techniques to compare and validate the current model’s performance and potentially uncover better fitting approaches.

  3. Business-Relevant Conclusions:

    • Communicate the model’s findings and insights to relevant stakeholders within Test Analytics Corp to guide decision-making processes effectively.
    • Monitor the model’s performance over time and consider implementing regular updates or re-evaluations to account for changing business dynamics and data patterns.
    • Ensure clear documentation of the model development process, assumptions made, and limitations to enhance transparency and facilitate future model enhancements.

Model Performance

Actual vs Predicted Analysis

PF

Model Performance

Actual vs Predicted

0.955
RMSE

Model Performance — performance — Detailed performance metrics and goodness of fit

0.955
r squared
0.953
adj r squared
4.569
rmse
r squared: 0.955
adj r squared: 0.953
rmse: 4.57
IN

Key Insights

Model Performance

The model performance metrics provided are as follows:

  1. R-Squared (Coefficient of Determination): 0.9545

    • Meaning: R-squared represents the proportion of the variance in the dependent variable (y) that is predictable from the independent variables (x1, x2, x3) in the model. In this case, an R-squared of 0.9545 indicates that approximately 95.45% of the variance in the target variable is explained by the model.
    • Interpretation: A higher R-squared indicates better goodness of fit, suggesting that the model captures a significant amount of variability in the data.

  2. Adjusted R-Squared: 0.9531

    • Meaning: Adjusted R-squared takes into account the number of predictors in the model, providing a more accurate assessment of model performance. It penalizes excessive use of variables that do not improve the model’s performance.
    • Interpretation: An adjusted R-squared value of 0.9531 suggests that the independent variables (x1, x2, x3) collectively explain around 95.31% of the variance in the target variable, considering the model’s complexity.

  3. Root Mean Squared Error (RMSE): 4.569

    • Meaning: RMSE is a measure of the differences between predicted values and actual values. It indicates the average magnitude of the residuals in the model.
    • Interpretation: An RMSE of 4.569 signifies that, on average, the model’s predictions deviate from the actual values by approximately 4.569 units. Lower RMSE values indicate better model accuracy.

  4. Mean Absolute Error (MAE): 3.685

    • Meaning: MAE measures the average absolute differences between predicted values and actual values.
    • Interpretation: A MAE of 3.685 implies that, on average, the model’s predictions differ from the actual values by approximately 3.685 units. Like RMSE, lower MAE values indicate better prediction accuracy.

Comparing AIC and BIC:

  • Akaike Information Criterion (AIC): 597.65
  • Bayesian Information Criterion (BIC): 610.68

Lower AIC and BIC values indicate a better model fit with a balance between goodness of

IN

Key Insights

Model Performance

The model performance metrics provided are as follows:

  1. R-Squared (Coefficient of Determination): 0.9545

    • Meaning: R-squared represents the proportion of the variance in the dependent variable (y) that is predictable from the independent variables (x1, x2, x3) in the model. In this case, an R-squared of 0.9545 indicates that approximately 95.45% of the variance in the target variable is explained by the model.
    • Interpretation: A higher R-squared indicates better goodness of fit, suggesting that the model captures a significant amount of variability in the data.

  2. Adjusted R-Squared: 0.9531

    • Meaning: Adjusted R-squared takes into account the number of predictors in the model, providing a more accurate assessment of model performance. It penalizes excessive use of variables that do not improve the model’s performance.
    • Interpretation: An adjusted R-squared value of 0.9531 suggests that the independent variables (x1, x2, x3) collectively explain around 95.31% of the variance in the target variable, considering the model’s complexity.

  3. Root Mean Squared Error (RMSE): 4.569

    • Meaning: RMSE is a measure of the differences between predicted values and actual values. It indicates the average magnitude of the residuals in the model.
    • Interpretation: An RMSE of 4.569 signifies that, on average, the model’s predictions deviate from the actual values by approximately 4.569 units. Lower RMSE values indicate better model accuracy.

  4. Mean Absolute Error (MAE): 3.685

    • Meaning: MAE measures the average absolute differences between predicted values and actual values.
    • Interpretation: A MAE of 3.685 implies that, on average, the model’s predictions differ from the actual values by approximately 3.685 units. Like RMSE, lower MAE values indicate better prediction accuracy.

Comparing AIC and BIC:

  • Akaike Information Criterion (AIC): 597.65
  • Bayesian Information Criterion (BIC): 610.68

Lower AIC and BIC values indicate a better model fit with a balance between goodness of

Coefficient Analysis

Effect Sizes and Statistical Significance

CF

Coefficient Analysis

Effect Sizes & Significance

4
Significant

Regression Coefficients coefficients Coefficient estimates with confidence intervals

Variable Estimate Std_Error t_value p_value CI_Lower CI_Upper
(Intercept) 2.452 3.800 0.645 0.520 -5.090 9.995
x1 1.492 0.045 33.134 0.000 1.403 1.581
x2 0.780 0.104 7.463 0.000 0.572 0.987
x3 -0.488 0.016 -30.251 0.000 -0.520 -0.456
3
significant count
3
total predictors
significant count: 3
total predictors: 3
IN

Key Insights

Coefficient Analysis

Based on the regression coefficients provided, three predictors out of the three total predictors (x1, x2, x3) are statistically significant. Here is the practical interpretation of each significant coefficient:

  1. x1 (Estimate: 1.4921):

    • The coefficient for x1 is 1.4921. This means that for every one unit increase in x1, we expect y to increase by 1.4921 units.
    • The direction of the relationship is positive, indicating that there is a positive linear relationship between x1 and y.
    • Since x1 has the highest coefficient magnitude among the significant predictors, it appears to be the most important predictor in determining y in this regression model.

  2. x2 (Estimate: 0.7797):

    • The coefficient for x2 is 0.7797. This suggests that for every one unit increase in x2, we expect y to increase by 0.7797 units.
    • Similar to x1, the relationship is positive, indicating a positive linear relationship between x2 and y.

  3. x3 (Estimate: -0.4879):

    • The coefficient for x3 is -0.4879. This implies that for every one unit increase in x3, we expect y to decrease by 0.4879 units.
    • The negative sign indicates a negative linear relationship between x3 and y, which means as x3 increases, y is expected to decrease.

In summary, based on the regression coefficients and significance levels, x1 appears to be the most important predictor in predicting y, followed by x2, and then x3.

IN

Key Insights

Coefficient Analysis

Based on the regression coefficients provided, three predictors out of the three total predictors (x1, x2, x3) are statistically significant. Here is the practical interpretation of each significant coefficient:

  1. x1 (Estimate: 1.4921):

    • The coefficient for x1 is 1.4921. This means that for every one unit increase in x1, we expect y to increase by 1.4921 units.
    • The direction of the relationship is positive, indicating that there is a positive linear relationship between x1 and y.
    • Since x1 has the highest coefficient magnitude among the significant predictors, it appears to be the most important predictor in determining y in this regression model.

  2. x2 (Estimate: 0.7797):

    • The coefficient for x2 is 0.7797. This suggests that for every one unit increase in x2, we expect y to increase by 0.7797 units.
    • Similar to x1, the relationship is positive, indicating a positive linear relationship between x2 and y.

  3. x3 (Estimate: -0.4879):

    • The coefficient for x3 is -0.4879. This implies that for every one unit increase in x3, we expect y to decrease by 0.4879 units.
    • The negative sign indicates a negative linear relationship between x3 and y, which means as x3 increases, y is expected to decrease.

In summary, based on the regression coefficients and significance levels, x1 appears to be the most important predictor in predicting y, followed by x2, and then x3.

Residual Diagnostics

Residual Patterns and Homoscedasticity

RA

Residual Analysis

Residuals vs Fitted

0
Pattern

Residual Diagnostics — Analysis of model residuals and assumptions

0
mean residual
4.592
residual std
mean residual: 0
residual std: 4.59
IN

Key Insights

Residual Analysis

Residual Analysis Insights:

  1. Mean Residual and Residual Standard Deviation:

    • The mean residual is close to zero (-1e-06), indicating that, on average, the model is unbiased.
    • The residual standard deviation is 4.592, suggesting that the residuals have a spread around the mean.

  2. Residuals vs Fitted Values:

    • It is essential to check for patterns in the residuals against the fitted values to identify potential issues like heteroscedasticity or non-linearity.
    • If you observe a clear pattern in the residuals (e.g., a funnel shape or a non-random scatter), it could indicate violations of model assumptions.

  3. Heteroscedasticity and Non-Linearity:

    • Heteroscedasticity: Look for a changing spread of residuals as the fitted values change. If the spread systematically increases or decreases, it suggests heteroscedasticity.
    • Non-Linearity: Check for curved or nonlinear patterns in the residuals against the fitted values, which may indicate that the model does not capture the true relationship adequately.

  4. Remedies for Assumption Violations:

    • Heteroscedasticity: Consider transforming the target variable or using weighted least squares to account for varying error variance.
    • Non-Linearity: Explore adding polynomial terms or using more complex modeling techniques to capture the non-linear relationship.

  5. Further Investigation:

    • It would be beneficial to plot the residuals against the fitted values and investigate any visible patterns to determine the extent of heteroscedasticity or non-linearity.
    • Additional diagnostics such as Q-Q plots, residual vs predictor plots, or leverage plots can provide more insights into the model’s performance.

By conducting a thorough analysis of the residuals and addressing any violations of model assumptions, you can enhance the reliability and predictive power of your model.

IN

Key Insights

Residual Analysis

Residual Analysis Insights:

  1. Mean Residual and Residual Standard Deviation:

    • The mean residual is close to zero (-1e-06), indicating that, on average, the model is unbiased.
    • The residual standard deviation is 4.592, suggesting that the residuals have a spread around the mean.

  2. Residuals vs Fitted Values:

    • It is essential to check for patterns in the residuals against the fitted values to identify potential issues like heteroscedasticity or non-linearity.
    • If you observe a clear pattern in the residuals (e.g., a funnel shape or a non-random scatter), it could indicate violations of model assumptions.

  3. Heteroscedasticity and Non-Linearity:

    • Heteroscedasticity: Look for a changing spread of residuals as the fitted values change. If the spread systematically increases or decreases, it suggests heteroscedasticity.
    • Non-Linearity: Check for curved or nonlinear patterns in the residuals against the fitted values, which may indicate that the model does not capture the true relationship adequately.

  4. Remedies for Assumption Violations:

    • Heteroscedasticity: Consider transforming the target variable or using weighted least squares to account for varying error variance.
    • Non-Linearity: Explore adding polynomial terms or using more complex modeling techniques to capture the non-linear relationship.

  5. Further Investigation:

    • It would be beneficial to plot the residuals against the fitted values and investigate any visible patterns to determine the extent of heteroscedasticity or non-linearity.
    • Additional diagnostics such as Q-Q plots, residual vs predictor plots, or leverage plots can provide more insights into the model’s performance.

By conducting a thorough analysis of the residuals and addressing any violations of model assumptions, you can enhance the reliability and predictive power of your model.

Normality Analysis

Q-Q Plot and Distribution Check

NC

Normality Check

Q-Q Plot

0.508
Normality

Normality Assessment — Check normality assumption of residuals

0.508
shapiro p
Passed
normality
shapiro p: 0.508
normality: Passed
IN

Key Insights

Normality Check

The Shapiro-Wilk test resulted in a p-value of 0.5078, indicating that the residuals follow a normal distribution. The Q-Q plot would show the residuals plotted against the theoretical quantiles of a normal distribution. If the points on the plot fall approximately along a straight line, it suggests that the residuals are normally distributed.

If normality assumption is violated, it may indicate issues with the model’s reliability. Non-normal residuals might lead to inaccurate confidence intervals and hypothesis testing results.

If transformation is needed, potential options include:

  1. Log transformation: Useful if the data is right-skewed.
  2. Box-Cox transformation: A generalized power transformation that includes the log transformation as a special case.
  3. Square root transformation: Useful for data with a square root relationship.

It is recommended to re-run the normality assessment after applying transformations to ensure residuals meet the normality assumption.

IN

Key Insights

Normality Check

The Shapiro-Wilk test resulted in a p-value of 0.5078, indicating that the residuals follow a normal distribution. The Q-Q plot would show the residuals plotted against the theoretical quantiles of a normal distribution. If the points on the plot fall approximately along a straight line, it suggests that the residuals are normally distributed.

If normality assumption is violated, it may indicate issues with the model’s reliability. Non-normal residuals might lead to inaccurate confidence intervals and hypothesis testing results.

If transformation is needed, potential options include:

  1. Log transformation: Useful if the data is right-skewed.
  2. Box-Cox transformation: A generalized power transformation that includes the log transformation as a special case.
  3. Square root transformation: Useful for data with a square root relationship.

It is recommended to re-run the normality assessment after applying transformations to ensure residuals meet the normality assumption.

Influence Analysis

Influential Points and Outliers

IP

Influential Points

Cook's Distance

4
Outliers

Influential Observations — Identify observations with high influence on model

4
influential count
0.064
max cooks d
influential count: 4
max cooks d: 0.064
IN

Key Insights

Influential Points

Cook’s distance is a measure used in regression analysis to assess the influence of individual data points on the regression model. It indicates how much the model predictions would change if a particular observation were removed from the dataset.

Influential points are observations that have a significant impact on the regression model due to either their extreme values or their leverage on the model fit. These points can greatly affect the model parameters and predictions.

In your data, there are 4 influential observations identified based on Cook’s distance, with the maximum Cook’s distance being 0.0643. Among the top influential points, indexes 68, 12, 35, and 97 are flagged as having high influence on the model.

Whether to investigate or remove influential points depends on the specific context of the analysis. Investigating these points can help understand why they are influential and whether they are valid data points or potential outliers. Removing influential points can lead to a more stable model, but it is important to assess the trade-off between model accuracy and the risk of removing important information.

Removing influential points can potentially improve model stability by reducing the impact of outliers or data points that excessively influence the model parameters. However, it is essential to carefully evaluate the implications of removing these points on the overall model performance and the validity of the analysis results.

IN

Key Insights

Influential Points

Cook’s distance is a measure used in regression analysis to assess the influence of individual data points on the regression model. It indicates how much the model predictions would change if a particular observation were removed from the dataset.

Influential points are observations that have a significant impact on the regression model due to either their extreme values or their leverage on the model fit. These points can greatly affect the model parameters and predictions.

In your data, there are 4 influential observations identified based on Cook’s distance, with the maximum Cook’s distance being 0.0643. Among the top influential points, indexes 68, 12, 35, and 97 are flagged as having high influence on the model.

Whether to investigate or remove influential points depends on the specific context of the analysis. Investigating these points can help understand why they are influential and whether they are valid data points or potential outliers. Removing influential points can lead to a more stable model, but it is important to assess the trade-off between model accuracy and the risk of removing important information.

Removing influential points can potentially improve model stability by reducing the impact of outliers or data points that excessively influence the model parameters. However, it is essential to carefully evaluate the implications of removing these points on the overall model performance and the validity of the analysis results.

Multicollinearity Check

VIF Analysis and Correlations

MC

Multicollinearity

VIF Analysis

3
Max VIF

Multicollinearity Check multicollinearity Assess multicollinearity among predictors

Variable
x1
x2
x3
NULL
max vif
No
multicollinearity issue
max vif: NULL
multicollinearity issue: No
IN

Key Insights

Multicollinearity

Based on the provided VIF values, there doesn’t seem to be an issue with multicollinearity among the predictors (x1, x2, x3) as the maximum VIF value is within an acceptable range.

Variance Inflation Factor (VIF) measures the correlation and strength of linear association between predictor variables in a regression model. A high VIF value (>10) indicates high multicollinearity, suggesting that the predictors are highly correlated and can lead to inaccurate coefficient estimates and reduced statistical power.

Since the maximum VIF is within an acceptable range, it indicates that the predictors (x1, x2, x3) are not exhibiting severe multicollinearity issues in the model. This implies that the coefficients of the predictors can be interpreted without significant distortion caused by multicollinearity.

In cases where multicollinearity is present, options to mitigate it include:

  1. Removing one or more of the correlated predictors from the model.
  2. Combining the correlated predictors into a single composite variable.
  3. Collecting more data to reduce the impact of multicollinearity.

Given that there is no significant multicollinearity issue based on the VIF values provided, no action is required at this point. It is essential to monitor multicollinearity when conducting regression analysis to ensure the validity and reliability of the model results.

IN

Key Insights

Multicollinearity

Based on the provided VIF values, there doesn’t seem to be an issue with multicollinearity among the predictors (x1, x2, x3) as the maximum VIF value is within an acceptable range.

Variance Inflation Factor (VIF) measures the correlation and strength of linear association between predictor variables in a regression model. A high VIF value (>10) indicates high multicollinearity, suggesting that the predictors are highly correlated and can lead to inaccurate coefficient estimates and reduced statistical power.

Since the maximum VIF is within an acceptable range, it indicates that the predictors (x1, x2, x3) are not exhibiting severe multicollinearity issues in the model. This implies that the coefficients of the predictors can be interpreted without significant distortion caused by multicollinearity.

In cases where multicollinearity is present, options to mitigate it include:

  1. Removing one or more of the correlated predictors from the model.
  2. Combining the correlated predictors into a single composite variable.
  3. Collecting more data to reduce the impact of multicollinearity.

Given that there is no significant multicollinearity issue based on the VIF values provided, no action is required at this point. It is essential to monitor multicollinearity when conducting regression analysis to ensure the validity and reliability of the model results.

AN

ANOVA Results

Analysis of Variance

4
F-Statistic

ANOVA Table anova_results Analysis of variance decomposition

Df Sum Sq Mean Sq F value Pr(>F) Source
1.000 23587.556 23587.556 1084.663 0.000 x1
1.000 322.871 322.871 14.847 0.000 x2
1.000 19900.133 19900.133 915.098 0.000 x3
96.000 2087.659 21.746 NA NA Residuals
672
f statistic
0
f pvalue
f statistic: 672
f pvalue: 0
model significant: Yes
IN

Key Insights

ANOVA Results

The F-statistic in ANOVA tests the overall significance of the model by comparing the variance explained by the model to the variance that is not explained. In this case, the F-statistic is 671.536 with a very low p-value (2.9107e-64), indicating that the model is statistically significant.

The variance decomposition between predictors can be seen in the ANOVA table. Each predictor (x1, x2, and x3) has its own row in the table, showing the sum of squares (Sum Sq), degrees of freedom (Df), mean square (Mean Sq), F-value, and p-value (Pr(>F)).

  • Predictor x1: It explains a significant amount of variance in the target variable y, as indicated by a high F-value of 1084.6626 and a very low p-value (4.1179e-54).
  • Predictor x2: It also has a significant impact on y, with an F-value of 14.8471 and a p-value of 0.0002.
  • Predictor x3: Similar to x1 and x2, x3 significantly contributes to explaining the variance in y, with an F-value of 915.0982 and a very low p-value (7.0764e-51).

The relative importance of predictors can be assessed by looking at the Mean Square values. In this case, x1 has the highest Mean Square (23587.5563), followed by x3 (19900.1332) and then x2 (322.8708). This order suggests that x1 explains the most variance in the target variable, followed by x3 and then x2.

Overall, the ANOVA results indicate that the model is significant, and all three predictors (x1, x2, x3) play a role in explaining the variance in the target variable y, with x1 being the most important predictor followed by x3 and x2.

IN

Key Insights

ANOVA Results

The F-statistic in ANOVA tests the overall significance of the model by comparing the variance explained by the model to the variance that is not explained. In this case, the F-statistic is 671.536 with a very low p-value (2.9107e-64), indicating that the model is statistically significant.

The variance decomposition between predictors can be seen in the ANOVA table. Each predictor (x1, x2, and x3) has its own row in the table, showing the sum of squares (Sum Sq), degrees of freedom (Df), mean square (Mean Sq), F-value, and p-value (Pr(>F)).

  • Predictor x1: It explains a significant amount of variance in the target variable y, as indicated by a high F-value of 1084.6626 and a very low p-value (4.1179e-54).
  • Predictor x2: It also has a significant impact on y, with an F-value of 14.8471 and a p-value of 0.0002.
  • Predictor x3: Similar to x1 and x2, x3 significantly contributes to explaining the variance in y, with an F-value of 915.0982 and a very low p-value (7.0764e-51).

The relative importance of predictors can be assessed by looking at the Mean Square values. In this case, x1 has the highest Mean Square (23587.5563), followed by x3 (19900.1332) and then x2 (322.8708). This order suggests that x1 explains the most variance in the target variable, followed by x3 and then x2.

Overall, the ANOVA results indicate that the model is significant, and all three predictors (x1, x2, x3) play a role in explaining the variance in the target variable y, with x1 being the most important predictor followed by x3 and x2.

Model Validity

Assumptions and Comparison

AS

Assumption Checks

Model Validity

Check residual plots
Linearity

Assumptions Summary assumptions Summary of regression assumptions checks

Check residual plots
linearity
Assumed met
independence
Check residual plots
homoscedasticity
Passed
normality
Failed
no multicollinearity
Linearity
Normality
Homoscedasticity
IN

Key Insights

Assumption Checks

Based on the assumption checks provided in the data profile:

Assumptions Met:

  1. Normality: Passed
  2. Independence: Assumed met

Assumptions Violated:

  1. No Multicollinearity: Failed

Priority of Violations: The violation of the “No Multicollinearity” assumption is the most concerning based on the information provided. Multicollinearity can lead to unreliable regression results, inflated standard errors, and difficulties in interpreting the effects of individual predictors.

Remedial Actions for Multicollinearity:

  1. Check Correlation Matrix: Examine the correlations among the predictor variables. A correlation above a certain threshold (e.g., 0.7) indicates multicollinearity.
  2. VIF (Variance Inflation Factor): Calculate VIF for each predictor variable. VIF values greater than 5 or 10 are often considered problematic.
  3. Address Multicollinearity:
    • Remove one of the highly correlated variables.
    • Use dimensionality reduction techniques like PCA.
    • Combine correlated variables into a single variable.

By addressing the multicollinearity issue, the regression model’s reliability and interpretability can be improved.

IN

Key Insights

Assumption Checks

Based on the assumption checks provided in the data profile:

Assumptions Met:

  1. Normality: Passed
  2. Independence: Assumed met

Assumptions Violated:

  1. No Multicollinearity: Failed

Priority of Violations: The violation of the “No Multicollinearity” assumption is the most concerning based on the information provided. Multicollinearity can lead to unreliable regression results, inflated standard errors, and difficulties in interpreting the effects of individual predictors.

Remedial Actions for Multicollinearity:

  1. Check Correlation Matrix: Examine the correlations among the predictor variables. A correlation above a certain threshold (e.g., 0.7) indicates multicollinearity.
  2. VIF (Variance Inflation Factor): Calculate VIF for each predictor variable. VIF values greater than 5 or 10 are often considered problematic.
  3. Address Multicollinearity:
    • Remove one of the highly correlated variables.
    • Use dimensionality reduction techniques like PCA.
    • Combine correlated variables into a single variable.

By addressing the multicollinearity issue, the regression model’s reliability and interpretability can be improved.

CM

Model Comparison

Performance Metrics

0
Best Model

Model Comparison Metrics model_comparison Metrics for comparing with other models

598
aic
611
bic
aic: 598
bic: 611
r squared: 0.955
IN

Key Insights

Model Comparison

AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are both statistical measures used in model selection. A lower AIC or BIC value indicates a better-fitting model. AIC penalizes model complexity less severely than BIC, meaning it may prefer more complex models compared to BIC.

In this case:

  • AIC: 597.65
  • BIC: 610.68

The AIC is lower than BIC, suggesting that the AIC penalizes model complexity less severely than BIC. Both AIC and BIC are relatively low, indicating a good fit of the model.

Adjusted R-squared adjusts the R-squared value based on the number of predictors in the model, providing a more realistic evaluation of model performance than R-squared alone. It penalizes the inclusion of unnecessary predictors in the model.

In this case:

  • R-squared: 0.9545
  • Adjusted R-squared: 0.9531

The adjusted R-squared is slightly lower than the R-squared, which is expected when there are multiple predictors in the model. The difference is small, indicating that the included predictors are contributing significantly to the model.

Based on the provided metrics, the model seems to be performing well in terms of goodness of fit, with high R-squared and adjusted R-squared values. However, more details on the context of the analysis and the specific data would be needed to determine if the model complexity is appropriate. If the model serves its purpose effectively without unnecessary complexity, then the model complexity can be considered appropriate.

IN

Key Insights

Model Comparison

AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are both statistical measures used in model selection. A lower AIC or BIC value indicates a better-fitting model. AIC penalizes model complexity less severely than BIC, meaning it may prefer more complex models compared to BIC.

In this case:

  • AIC: 597.65
  • BIC: 610.68

The AIC is lower than BIC, suggesting that the AIC penalizes model complexity less severely than BIC. Both AIC and BIC are relatively low, indicating a good fit of the model.

Adjusted R-squared adjusts the R-squared value based on the number of predictors in the model, providing a more realistic evaluation of model performance than R-squared alone. It penalizes the inclusion of unnecessary predictors in the model.

In this case:

  • R-squared: 0.9545
  • Adjusted R-squared: 0.9531

The adjusted R-squared is slightly lower than the R-squared, which is expected when there are multiple predictors in the model. The difference is small, indicating that the included predictors are contributing significantly to the model.

Based on the provided metrics, the model seems to be performing well in terms of goodness of fit, with high R-squared and adjusted R-squared values. However, more details on the context of the analysis and the specific data would be needed to determine if the model complexity is appropriate. If the model serves its purpose effectively without unnecessary complexity, then the model complexity can be considered appropriate.

Model Validation

Comprehensive Performance Metrics

AS

Assumption Checks

Model Validity

Check residual plots
Linearity

Assumptions Summary assumptions Summary of regression assumptions checks

Check residual plots
linearity
Assumed met
independence
Check residual plots
homoscedasticity
Passed
normality
Failed
no multicollinearity
Linearity
Normality
Homoscedasticity
IN

Key Insights

Assumption Checks

Based on the assumption checks provided in the data profile:

Assumptions Met:

  1. Normality: Passed
  2. Independence: Assumed met

Assumptions Violated:

  1. No Multicollinearity: Failed

Priority of Violations: The violation of the “No Multicollinearity” assumption is the most concerning based on the information provided. Multicollinearity can lead to unreliable regression results, inflated standard errors, and difficulties in interpreting the effects of individual predictors.

Remedial Actions for Multicollinearity:

  1. Check Correlation Matrix: Examine the correlations among the predictor variables. A correlation above a certain threshold (e.g., 0.7) indicates multicollinearity.
  2. VIF (Variance Inflation Factor): Calculate VIF for each predictor variable. VIF values greater than 5 or 10 are often considered problematic.
  3. Address Multicollinearity:
    • Remove one of the highly correlated variables.
    • Use dimensionality reduction techniques like PCA.
    • Combine correlated variables into a single variable.

By addressing the multicollinearity issue, the regression model’s reliability and interpretability can be improved.

IN

Key Insights

Assumption Checks

Based on the assumption checks provided in the data profile:

Assumptions Met:

  1. Normality: Passed
  2. Independence: Assumed met

Assumptions Violated:

  1. No Multicollinearity: Failed

Priority of Violations: The violation of the “No Multicollinearity” assumption is the most concerning based on the information provided. Multicollinearity can lead to unreliable regression results, inflated standard errors, and difficulties in interpreting the effects of individual predictors.

Remedial Actions for Multicollinearity:

  1. Check Correlation Matrix: Examine the correlations among the predictor variables. A correlation above a certain threshold (e.g., 0.7) indicates multicollinearity.
  2. VIF (Variance Inflation Factor): Calculate VIF for each predictor variable. VIF values greater than 5 or 10 are often considered problematic.
  3. Address Multicollinearity:
    • Remove one of the highly correlated variables.
    • Use dimensionality reduction techniques like PCA.
    • Combine correlated variables into a single variable.

By addressing the multicollinearity issue, the regression model’s reliability and interpretability can be improved.

CM

Model Comparison

Performance Metrics

0
Best Model

Model Comparison Metrics model_comparison Metrics for comparing with other models

598
aic
611
bic
aic: 598
bic: 611
r squared: 0.955
IN

Key Insights

Model Comparison

AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are both statistical measures used in model selection. A lower AIC or BIC value indicates a better-fitting model. AIC penalizes model complexity less severely than BIC, meaning it may prefer more complex models compared to BIC.

In this case:

  • AIC: 597.65
  • BIC: 610.68

The AIC is lower than BIC, suggesting that the AIC penalizes model complexity less severely than BIC. Both AIC and BIC are relatively low, indicating a good fit of the model.

Adjusted R-squared adjusts the R-squared value based on the number of predictors in the model, providing a more realistic evaluation of model performance than R-squared alone. It penalizes the inclusion of unnecessary predictors in the model.

In this case:

  • R-squared: 0.9545
  • Adjusted R-squared: 0.9531

The adjusted R-squared is slightly lower than the R-squared, which is expected when there are multiple predictors in the model. The difference is small, indicating that the included predictors are contributing significantly to the model.

Based on the provided metrics, the model seems to be performing well in terms of goodness of fit, with high R-squared and adjusted R-squared values. However, more details on the context of the analysis and the specific data would be needed to determine if the model complexity is appropriate. If the model serves its purpose effectively without unnecessary complexity, then the model complexity can be considered appropriate.

IN

Key Insights

Model Comparison

AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are both statistical measures used in model selection. A lower AIC or BIC value indicates a better-fitting model. AIC penalizes model complexity less severely than BIC, meaning it may prefer more complex models compared to BIC.

In this case:

  • AIC: 597.65
  • BIC: 610.68

The AIC is lower than BIC, suggesting that the AIC penalizes model complexity less severely than BIC. Both AIC and BIC are relatively low, indicating a good fit of the model.

Adjusted R-squared adjusts the R-squared value based on the number of predictors in the model, providing a more realistic evaluation of model performance than R-squared alone. It penalizes the inclusion of unnecessary predictors in the model.

In this case:

  • R-squared: 0.9545
  • Adjusted R-squared: 0.9531

The adjusted R-squared is slightly lower than the R-squared, which is expected when there are multiple predictors in the model. The difference is small, indicating that the included predictors are contributing significantly to the model.

Based on the provided metrics, the model seems to be performing well in terms of goodness of fit, with high R-squared and adjusted R-squared values. However, more details on the context of the analysis and the specific data would be needed to determine if the model complexity is appropriate. If the model serves its purpose effectively without unnecessary complexity, then the model complexity can be considered appropriate.

MC

Multicollinearity

VIF Analysis

3
Max VIF

Multicollinearity Check multicollinearity Assess multicollinearity among predictors

Variable
x1
x2
x3
NULL
max vif
No
multicollinearity issue
max vif: NULL
multicollinearity issue: No
IN

Key Insights

Multicollinearity

Based on the provided VIF values, there doesn’t seem to be an issue with multicollinearity among the predictors (x1, x2, x3) as the maximum VIF value is within an acceptable range.

Variance Inflation Factor (VIF) measures the correlation and strength of linear association between predictor variables in a regression model. A high VIF value (>10) indicates high multicollinearity, suggesting that the predictors are highly correlated and can lead to inaccurate coefficient estimates and reduced statistical power.

Since the maximum VIF is within an acceptable range, it indicates that the predictors (x1, x2, x3) are not exhibiting severe multicollinearity issues in the model. This implies that the coefficients of the predictors can be interpreted without significant distortion caused by multicollinearity.

In cases where multicollinearity is present, options to mitigate it include:

  1. Removing one or more of the correlated predictors from the model.
  2. Combining the correlated predictors into a single composite variable.
  3. Collecting more data to reduce the impact of multicollinearity.

Given that there is no significant multicollinearity issue based on the VIF values provided, no action is required at this point. It is essential to monitor multicollinearity when conducting regression analysis to ensure the validity and reliability of the model results.

IN

Key Insights

Multicollinearity

Based on the provided VIF values, there doesn’t seem to be an issue with multicollinearity among the predictors (x1, x2, x3) as the maximum VIF value is within an acceptable range.

Variance Inflation Factor (VIF) measures the correlation and strength of linear association between predictor variables in a regression model. A high VIF value (>10) indicates high multicollinearity, suggesting that the predictors are highly correlated and can lead to inaccurate coefficient estimates and reduced statistical power.

Since the maximum VIF is within an acceptable range, it indicates that the predictors (x1, x2, x3) are not exhibiting severe multicollinearity issues in the model. This implies that the coefficients of the predictors can be interpreted without significant distortion caused by multicollinearity.

In cases where multicollinearity is present, options to mitigate it include:

  1. Removing one or more of the correlated predictors from the model.
  2. Combining the correlated predictors into a single composite variable.
  3. Collecting more data to reduce the impact of multicollinearity.

Given that there is no significant multicollinearity issue based on the VIF values provided, no action is required at this point. It is essential to monitor multicollinearity when conducting regression analysis to ensure the validity and reliability of the model results.

Business Insights

Key Findings and Technical Details

RC

Recommendations

Analysis: y

3
Key Actions

Recommendations — recommendations — Actionable insights and next steps

Good
model quality
95%
confidence level
3
significant predictors

Business Context

Company: Test Analytics Corp

Objective: Analyze relationships between predictors and target variable

Model Variables

Target: y

model quality: Good
confidence level: 95%
significant predictors: 3
IN

Key Insights

Recommendations

Based on the data profile provided for Test Analytics Corp:

  1. Next Steps for Improving the Model:

    • Conduct a thorough analysis of the significant predictors identified in the model to understand their individual and combined impact on the target variable.
    • Consider exploring interactions between the significant predictors to potentially uncover non-linear relationships that could enhance the model’s predictive power.
    • Evaluate the possibility of including additional relevant features or transforming existing ones to capture more complex relationships and improve model performance further.

  2. Areas for Further Investigation:

    • Investigate potential multicollinearity issues among the predictors, as this could impact the stability and interpretability of the coefficients in the model.
    • Perform sensitivity analysis to assess the robustness of the model and ensure its reliability across different scenarios or subsets of data.
    • Explore different model algorithms or techniques to compare and validate the current model’s performance and potentially uncover better fitting approaches.

  3. Business-Relevant Conclusions:

    • Communicate the model’s findings and insights to relevant stakeholders within Test Analytics Corp to guide decision-making processes effectively.
    • Monitor the model’s performance over time and consider implementing regular updates or re-evaluations to account for changing business dynamics and data patterns.
    • Ensure clear documentation of the model development process, assumptions made, and limitations to enhance transparency and facilitate future model enhancements.
IN

Key Insights

Recommendations

Based on the data profile provided for Test Analytics Corp:

  1. Next Steps for Improving the Model:

    • Conduct a thorough analysis of the significant predictors identified in the model to understand their individual and combined impact on the target variable.
    • Consider exploring interactions between the significant predictors to potentially uncover non-linear relationships that could enhance the model’s predictive power.
    • Evaluate the possibility of including additional relevant features or transforming existing ones to capture more complex relationships and improve model performance further.

  2. Areas for Further Investigation:

    • Investigate potential multicollinearity issues among the predictors, as this could impact the stability and interpretability of the coefficients in the model.
    • Perform sensitivity analysis to assess the robustness of the model and ensure its reliability across different scenarios or subsets of data.
    • Explore different model algorithms or techniques to compare and validate the current model’s performance and potentially uncover better fitting approaches.

  3. Business-Relevant Conclusions:

    • Communicate the model’s findings and insights to relevant stakeholders within Test Analytics Corp to guide decision-making processes effectively.
    • Monitor the model’s performance over time and consider implementing regular updates or re-evaluations to account for changing business dynamics and data patterns.
    • Ensure clear documentation of the model development process, assumptions made, and limitations to enhance transparency and facilitate future model enhancements.
OV

Executive Summary

Analysis: y

0.955
R² Score

Executive Summary — overview — High-level regression results and key findings

0.955
r squared
0.953
adj r squared
100
observations
3
predictors
Yes
significant
4.57
rmse

Business Context

Company: Test Analytics Corp

Objective: Analyze relationships between predictors and target variable

Model Variables

Target: y

Summary

Variable Estimate p_value
(Intercept) 2.452 0.520
x1 1.492 0.000
x2 0.780 0.000
x3 -0.488 0.000
r squared: 0.955
adj r squared: 0.953
observations: 100
IN

Key Insights

Executive Summary

Based on the regression analysis results for Test Analytics Corp, here are some executive insights:

  1. Business Impact:

    • The model shows a high level of accuracy with an R-squared value of 0.9545, indicating that approximately 95.45% of the variance in the target variable (y) can be explained by the predictors (x1, x2, x3). This strong relationship between the predictors and the target variable can have a significant positive impact on the company’s decision-making processes and strategies.

  2. Key Relationships Found:

    • The coefficients of the predictors reveal the strength and direction of the relationships with the target variable:
      • x1: Estimate 1.4921, p-value 2.4158e-54.
      • x2: Estimate 0.7797, p-value 3.8245e-11.
      • x3: Estimate -0.4879, p-value 7.0764e-51.
    • These results indicate that x1 has the strongest positive relationship with the target variable, followed by x2, while x3 has a negative relationship.

  3. Model Reliability:

    • The model is statistically significant, as indicated by the p-values for all predictors being very low. This suggests that the predictors are useful in predicting the target variable.
    • The model’s high Adjusted R-squared value of 0.9531 suggests that the chosen predictors collectively explain a large proportion of the variance in the target variable while adjusting for the number of predictors in the model.
    • The RMSE (Root Mean Squared Error) of 4.569 indicates the average difference between the predicted values and the actual values, providing a measure of how well the model fits the data.

In summary, the analysis highlights strong relationships between the predictors and the target variable, indicating the potential for accurate predictions. The high model reliability and significance of predictors suggest that the model can be valuable for decision-making within Test Analytics Corp.

IN

Key Insights

Executive Summary

Based on the regression analysis results for Test Analytics Corp, here are some executive insights:

  1. Business Impact:

    • The model shows a high level of accuracy with an R-squared value of 0.9545, indicating that approximately 95.45% of the variance in the target variable (y) can be explained by the predictors (x1, x2, x3). This strong relationship between the predictors and the target variable can have a significant positive impact on the company’s decision-making processes and strategies.

  2. Key Relationships Found:

    • The coefficients of the predictors reveal the strength and direction of the relationships with the target variable:
      • x1: Estimate 1.4921, p-value 2.4158e-54.
      • x2: Estimate 0.7797, p-value 3.8245e-11.
      • x3: Estimate -0.4879, p-value 7.0764e-51.
    • These results indicate that x1 has the strongest positive relationship with the target variable, followed by x2, while x3 has a negative relationship.

  3. Model Reliability:

    • The model is statistically significant, as indicated by the p-values for all predictors being very low. This suggests that the predictors are useful in predicting the target variable.
    • The model’s high Adjusted R-squared value of 0.9531 suggests that the chosen predictors collectively explain a large proportion of the variance in the target variable while adjusting for the number of predictors in the model.
    • The RMSE (Root Mean Squared Error) of 4.569 indicates the average difference between the predicted values and the actual values, providing a measure of how well the model fits the data.

In summary, the analysis highlights strong relationships between the predictors and the target variable, indicating the potential for accurate predictions. The high model reliability and significance of predictors suggest that the model can be valuable for decision-making within Test Analytics Corp.

TD

Technical Details

Analysis: y

100
Details

Technical Details — Detailed technical information for data scientists

100
observations
3
predictors
96
degrees freedom
y ~ x1+x2+x3
formula

Model Variables

Target: y

Coefficients

Variable Estimate Std_Error t_value p_value CI_Lower CI_Upper
(Intercept) 2.452 3.800 0.645 0.520 -5.090 9.995
x1 1.492 0.045 33.134 0.000 1.403 1.581
x2 0.780 0.104 7.463 0.000 0.572 0.987
x3 -0.488 0.016 -30.251 0.000 -0.520 -0.456
observations: 100
predictors: 3
degrees freedom: 96
IN

Key Insights

Technical Details

Coefficients Interpretation:

  • (Intercept):

    • Estimate: 2.4523
    • This represents the expected value of the target variable when all predictor variables are zero.
    • Since the p-value is high (0.5202), the intercept may not be statistically significant.

  • x1:

    • Estimate: 1.4921
    • For every one-unit increase in x1, there is an estimated increase of 1.4921 units in the target variable, holding other predictors constant.
    • The very low p-value (2.4158e-54) suggests x1 is likely a significant predictor.

  • x2:

    • Estimate: 0.7797
    • For every one-unit increase in x2, there is an estimated increase of 0.7797 units in the target variable, holding other predictors constant.
    • The p-value (3.8245e-11) indicates x2 is likely a significant predictor.

  • x3:

    • Estimate: -0.4879
    • For every one-unit increase in x3, there is an estimated decrease of 0.4879 units in the target variable, holding other predictors constant.
    • The p-value (7.0764e-51) suggests x3 is a significant predictor.

Degrees of Freedom and Sample Size Adequacy:

  • Degrees of Freedom: 96

    • This represents the number of independent values or pieces of information in the estimation of the coefficients.
    • With a sample size of 100 observations and 3 predictors, having 96 degrees of freedom is appropriate for the regression model.

  • Sample Size Adequacy:

    • Given the ratio of observations to predictors (100:3), the sample size is adequate for the number of predictors in the model.
    • However, for more complex models or interactions, increasing the sample size could improve the model’s robustness.

Advanced Modeling Techniques:

  • Regularization Techniques:

    • Consider using techniques like Lasso or Ridge regression to handle multicollinearity or prevent overfitting.

  • Feature Engineering:

    • Explore transformations or interactions among features to capture complex relationships and improve model performance.

  • Model Comparison:

    • Compare the linear model with more advanced models like Random Forest or Gradient Boosting to see if they offer better predictive performance.
IN

Key Insights

Technical Details

Coefficients Interpretation:

  • (Intercept):

    • Estimate: 2.4523
    • This represents the expected value of the target variable when all predictor variables are zero.
    • Since the p-value is high (0.5202), the intercept may not be statistically significant.

  • x1:

    • Estimate: 1.4921
    • For every one-unit increase in x1, there is an estimated increase of 1.4921 units in the target variable, holding other predictors constant.
    • The very low p-value (2.4158e-54) suggests x1 is likely a significant predictor.

  • x2:

    • Estimate: 0.7797
    • For every one-unit increase in x2, there is an estimated increase of 0.7797 units in the target variable, holding other predictors constant.
    • The p-value (3.8245e-11) indicates x2 is likely a significant predictor.

  • x3:

    • Estimate: -0.4879
    • For every one-unit increase in x3, there is an estimated decrease of 0.4879 units in the target variable, holding other predictors constant.
    • The p-value (7.0764e-51) suggests x3 is a significant predictor.

Degrees of Freedom and Sample Size Adequacy:

  • Degrees of Freedom: 96

    • This represents the number of independent values or pieces of information in the estimation of the coefficients.
    • With a sample size of 100 observations and 3 predictors, having 96 degrees of freedom is appropriate for the regression model.

  • Sample Size Adequacy:

    • Given the ratio of observations to predictors (100:3), the sample size is adequate for the number of predictors in the model.
    • However, for more complex models or interactions, increasing the sample size could improve the model’s robustness.

Advanced Modeling Techniques:

  • Regularization Techniques:

    • Consider using techniques like Lasso or Ridge regression to handle multicollinearity or prevent overfitting.

  • Feature Engineering:

    • Explore transformations or interactions among features to capture complex relationships and improve model performance.

  • Model Comparison:

    • Compare the linear model with more advanced models like Random Forest or Gradient Boosting to see if they offer better predictive performance.
Job ID: test_analytics__statistical__regression__linear_20250911_214327
Secret Hash: 36f2d40ab80e