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Regression-Based Machine Learning Models for Estimating Follow-Up Renal Biomarkers in Lupus Nephritis Patients with Lymphopenia: A/C Ratio, 24h Protein, and Albumin


**Introduction**

Lupus nephritis (LN), a severe renal manifestation of systemic lupus erythematosus (SLE), remains a formidable challenge in clinical practice due to its intricate pathophysiology, characterized by immune dysregulation, renal injury, and systemic inflammation. As one of the most common and debilitating complications of SLE, LN affects approximately 40-50% of patients, often leading to significant morbidity and mortality if not managed effectively. The clinical management of LN hinges on the accurate monitoring of disease activity and treatment response, for which renal biomarkers play a pivotal role. Among these, the albumin-to-creatinine (A/C) ratio, 24-hour urinary protein excretion (24h protein), and serum albumin levels stand out as critical indicators of glomerular function, tubular integrity, and overall renal health. These biomarkers not only facilitate the diagnosis of active nephritis but also guide therapeutic decisions and predict long-term renal outcomes, making them indispensable tools in the clinician's arsenal.

Despite their clinical utility, the interpretation of these biomarkers is often complicated by the dynamic and heterogeneous nature of LN. The disease's progression is influenced by a multitude of factors, including genetic predispositions, environmental triggers, and the patient's response to immunosuppressive therapies. This variability underscores the limitations of traditional monitoring approaches, which often rely on static assessments and fail to capture the nuanced interplay between clinical variables and disease trajectories. Moreover, the presence of lymphopenia—a hallmark hematological abnormality in LN—further complicates the clinical picture. Lymphopenia, characterized by reduced lymphocyte counts, is associated with heightened disease activity, immune dysregulation, and an increased risk of renal flares, making it a critical factor to consider in the management of LN patients.

In this context, the integration of advanced predictive models into clinical practice offers a promising avenue to enhance the precision and efficacy of LN management. Machine learning, with its capacity to handle complex, high-dimensional datasets and uncover non-linear relationships, provides a powerful framework for predicting renal biomarker trajectories and informing personalized treatment strategies. By leveraging a comprehensive dataset from a clinical cohort of lupus nephritis patients with lymphopenia, this study aims to develop regression-based machine learning models for estimating follow-up values of key renal biomarkers: the A/C ratio, 24h protein, and serum albumin. These models will incorporate a wide array of clinical, laboratory, and treatment-related variables, including demographic characteristics, baseline and follow-up measurements of hematological and renal function markers, and details of immunosuppressive regimens.

The dataset utilized in this study is derived from a longitudinal clinical study of patients with chronic autoimmune diseases, specifically focusing on those with lupus nephritis and lymphopenia. It encompasses baseline and follow-up measurements of essential clinical parameters, such as hematological values (e.g., hemoglobin, white blood cell count, lymphocytes), renal function indicators (e.g., A/C ratio, 24h protein, serum creatinine), and markers of systemic inflammation (e.g., erythrocyte sedimentation rate, complement levels). Additionally, the dataset includes detailed information on the use of common immunosuppressive agents, such as prednisone, Plaquenil, azathioprine, and mycophenolate mofetil, enabling a thorough evaluation of their impact on renal biomarkers and disease progression.

The primary objective of this research is to construct robust, interpretable regression models capable of predicting the future values of the A/C ratio, 24h protein, and serum albumin, thereby providing clinicians with actionable insights into the trajectory of renal function and proteinuria in LN patients with lymphopenia. To achieve this, the study employs a range of advanced machine learning techniques, including feature selection methods (e.g., F-value, mutual information, recursive feature elimination), regression algorithms (e.g., Poisson regression, LassoLars, LightGBM), and hyperparameter optimization tools (e.g., Optuna, HyperOpt). These methodologies are designed to identify the most predictive features, mitigate overfitting, and maximize model performance, ensuring the reliability and clinical relevance of the predictions.

The significance of this work extends beyond the development of predictive models; it represents a step toward bridging the gap between clinical data and data-driven decision-making in the management of lupus nephritis. Accurate prediction of renal biomarker changes could enable early identification of impending renal flares, allowing for timely therapeutic adjustments and potentially preventing irreversible kidney damage. For instance, an elevated A/C ratio or 24h protein level may signal the need for intensified immunosuppression, while a decline in serum albumin could indicate the onset of nephrotic syndrome, necessitating targeted interventions. By focusing on LN patients with lymphopenia, this study also offers a deeper understanding of the interplay between immune dysfunction and renal outcomes, shedding light on how lymphopenia influences disease progression and treatment response.

The integration of machine learning into the clinical management of lupus nephritis aligns with the broader trend of precision medicine in autoimmune diseases. While traditional statistical approaches have provided foundational insights into LN, machine learning offers the advantage of handling complex, non-linear relationships and uncovering subtle patterns in clinical data that may otherwise go unnoticed. However, the application of these models must be approached with caution, recognizing the need for large, representative datasets, the risk of overfitting, and the importance of validating predictions in real-world clinical settings. By addressing these challenges, this study aims to contribute to the growing body of evidence supporting the use of machine learning in autoimmune disease management, ultimately improving patient outcomes and advancing the field of precision nephrology.
This dataset is derived from a clinical study focused on patients with chronic autoimmune diseases, potentially lupus or related conditions. It provides comprehensive information about patient demographics, disease duration, and laboratory test results collected at baseline and follow-up visits. The data also includes details of prescribed treatments, such as Prednisone, Plaquenil, Azathioprine, and Mycophenolate mofetil, which are commonly used in managing autoimmune disorders.
The dataset encompasses a range of clinical parameters, including hematological values (e.g., hemoglobin, white blood cell count, platelet count), kidney function markers (e.g., albumin-to-creatinine ratio, 24-hour protein in urine, creatinine levels), and indicators of systemic inflammation (e.g., erythrocyte sedimentation rate). It also tracks specific diagnostic tests, such as antinuclear antibody (ANA) and anti-dsDNA tests, along with complement levels, which are crucial for diagnosing and monitoring autoimmune activity.
By offering both baseline and follow-up measurements, the dataset provides a unique opportunity to analyze disease progression, evaluate the effectiveness of treatments, and investigate relationships between clinical markers and patient outcomes. It is well-suited for predictive modeling, statistical analysis, and exploratory research aimed at improving the understanding and management of autoimmune diseases.
1.1	Challenges
The primary challenge in managing Lupus Nephritis (LN) lies in its complex and variable nature.  LN, a severe kidney complication of Systemic Lupus Erythematosus (SLE), presents differently in each patient due to a combination of genetic factors, environmental influences, and individual responses to treatment. This variability makes it difficult to predict disease progression and tailor treatment effectively using traditional, static assessments of renal biomarkers.  Furthermore, the presence of lymphopenia (low lymphocyte count), a common feature in LN, adds another layer of complexity. Lymphopenia is linked to increased disease activity, immune dysregulation, and a higher risk of renal flares, making it crucial to understand its impact on kidney function.  The traditional clinical challenge is made even more substantial due to a dataset full of numerous and highly correllated variables. Feature engineering, reduction, and selection are necessary to ensure useful and actionable output from the models.

1.2	Problem Statement and Research Gap

The problem is that current methods for monitoring LN and predicting its course, particularly in patients with lymphopenia, are insufficient.  Clinicians rely heavily on renal biomarkers like the A/C ratio, 24-hour protein, and serum albumin to assess kidney function and guide treatment. However, these biomarkers are often interpreted in isolation and don't fully capture the dynamic interplay of factors influencing disease progression.  There's a critical need for more sophisticated tools that can integrate diverse clinical data, including the presence of lymphopenia, to accurately predict changes in renal function and identify patients at risk of flares.  The research gap is the lack of robust, validated predictive models that can leverage machine learning to forecast the trajectory of key renal biomarkers in LN patients with lymphopenia, thus enabling more personalized and proactive management. Furthermore, there are numerous possible combinations of regression methods and feature selection that may provide optimal results, and narrowing the field to a preferred method is essential.

1.3	Research Question
The central research question is: Can regression-based machine learning models, trained on a comprehensive clinical dataset, accurately predict follow-up values of A/C ratio, 24-hour protein, and serum albumin in lupus nephritis patients with lymphopenia, and how do these predictions correlate with clinical outcomes and treatment responses?

1.4	Contributions 
This research aims to make several key contributions:
•	Development of Novel Predictive Models:  The study will create and validate regression-based machine learning models to predict future values of A/C ratio, 24-hour protein, and serum albumin in LN patients with lymphopenia.
•	Enhanced Understanding of Lymphopenia's Role: By focusing on patients with lymphopenia, the research will provide deeper insights into how this specific immune dysfunction influences renal outcomes and interacts with other clinical variables.
•	Personalized Medicine Approach: The predictive models will pave the way for more personalized treatment strategies, allowing clinicians to anticipate changes in renal function and adjust therapy proactively.
•	Improved Clinical Decision-Making: The study will equip clinicians with a data-driven tool to identify patients at high risk of renal flares, enabling timely interventions and potentially preventing irreversible kidney damage.
•	Advancement of Machine Learning in Nephrology: The research will contribute to the growing body of evidence supporting the use of machine learning in autoimmune disease management, specifically in the field of precision nephrology.
•	Guidance on Optimal Feature Selection and Regression Methods: The best methods among the options tested will be selected, with justification provided as to why they were chosen.

1.5	Research Outline:
The research will be conducted in the following phases:


1.6	Summary of Findings from Provided Data
The provided data analysis, including correlation tables and regression results, offers valuable preliminary insights:
•	Strong Correlations in Renal Markers:  The high positive correlations between A/C ratio, 24-hour protein, albumin, and markers of kidney damage (casts, RBCs) confirm the importance of these biomarkers in assessing renal function and pathology in LN.
•	Age and Weight Influence:  The weak but notable correlations between age/weight and several renal parameters suggest that these demographic factors should be considered in clinical assessments and model development.
•	Lymphopenia's Potential Impact:  The negative correlation between A/C ratio and lymphocytes warrants further investigation, as it might reflect a link between immune dysregulation and renal impairment.
•	Treatment Effects:  The correlations between immunosuppressive medications and inflammatory markers (e.g., prednisone and ESR) suggest treatment efficacy but also highlight the need for ongoing monitoring.
•	Regression Model Performance:  The regression results indicate that optimization methods like Optuna and HyperOpt, when combined with appropriate regression algorithms, can achieve high R-squared values and low error rates, demonstrating the potential for accurate prediction of renal biomarkers. ExtraTreesRegressor and TweedieRegressor showed particular promise for Albumin 2 prediction.  LassoLarsCV did well for predicting Albumin 2 and would be considered as well. For the A/C Ratio 2 and 24h Protein 2, Optuna and HyperOpt performed significantly better than other methods.
•	Feature Selection Efficacy: The top features selected varied between the various methods used. Variance Thresholding provided the largest set of potential features, but did not significantly narrow the field, as 27 of the 39 columns were identified. The F-value, Mutual Information, and RFE with Random Forest methods did prove helpful at narrow selection.

This research has the potential to significantly impact the clinical management of lupus nephritis, particularly in patients with lymphopenia. By developing and validating robust predictive models, the study aims to bridge the gap between clinical data and data-driven decision-making, ultimately improving patient outcomes and advancing the field of precision nephrology.

I. Data Preparation and Initial Steps
1.	Gather the Complete Dataset: Obtain all correlation data, including coefficients and p-values, as these are essential for assessing statistical significance.
2.	Data Cleaning and Formatting: Ensure consistency in the data to prevent interpretation errors.
3.	Clarify "2" Suffix: Determine the meaning of the "2" suffix (e.g., second measurement, alternative method) for accurate interpretation.
4.	Correlation Matrix: Construct a complete correlation matrix to visualize all pairwise correlations, helping to identify patterns and clusters of related markers.

The breakdown of its structure can be summarized as follows:
Columns Description:
1.	Demographics and Baseline Characteristics:
o	Age: Patient's age in years.
o	Sex: Patient's sex (m for male, f for female).
o	Wt: Weight in kilograms.
o	Duration: Duration of illness in years.
2.	Treatments:
o	Prednison: Dosage of prednisone (mg/day).
o	Plaqunil: Dosage of Plaquenil (mg/day).
o	Azathioprine: Dosage of azathioprine (mg/day).
o	Micofenolate mofetile: Dosage of mycophenolate mofetil (mg/day).
3.	Diagnostics and Autoantibodies:
o	ANA: Anti-nuclear antibody test result (positive or negative).
o	Anti-DNA: Anti-double-stranded DNA antibody test result (positive or negative).
4.	Laboratory Parameters (Initial and Follow-up Measurements):
o	Hb, Hb 2: Hemoglobin level (g/dL) at baseline and follow-up.
o	WBCs, WBCs 2: White blood cell count (cells/µL) at baseline and follow-up.
o	PLT, PLT 2: Platelet count (x10³/µL) at baseline and follow-up.
o	Lymphocytes, Lymphocytes 2: Lymphocyte count (cells/µL) at baseline and follow-up.
o	A/C ratio, A/C ratio 2: Albumin/creatinine ratio (mg/g) at baseline and follow-up.
o	24h protein in urine, 24h protein 2: 24-hour urine protein excretion (g/day) at baseline and follow-up.
o	Pus cells, Pus cells 2: Presence of pus cells in urine at baseline and follow-up.
o	RBCs, RBCs 2: Presence of red blood cells in urine at baseline and follow-up.
o	Casts, Casts 2: Presence of casts in urine at baseline and follow-up.
o	Albumin, Albumin 2: Serum albumin level (g/dL) at baseline and follow-up.
o	Urea, Urea 2: Blood urea level (mg/dL) at baseline and follow-up.
o	Creatinine, Creatinine 2: Serum creatinine level (mg/dL) at baseline and follow-up.
5.	Inflammation Markers:
o	ESR 1, ESR 2: Erythrocyte sedimentation rate (mm/hr) at baseline and follow-up.
o	Low complement: Status of complement levels (positive or negative).
Observations:
•	Heterogeneous Population: The dataset includes patients of varying ages and weights.
•	Longitudinal Data: Measurements are recorded at two time points, reflecting baseline and follow-up.
•	Clinical Focus: Emphasis on autoimmune markers (e.g., ANA, Anti-DNA), kidney function (e.g., A/C ratio, 24h protein, pus cells), and inflammation markers (e.g., ESR, low complement).
•	Treatment Diversity: Different treatment regimens and doses are included.
•	Outcomes: Laboratory markers, urine analysis, and inflammatory parameters track disease progression or treatment efficacy.
Potential Applications:
•	Predictive Modeling: Predicting treatment outcomes or disease progression.
•	Comparative Analysis: Assessing the impact of various treatments.
•	Cluster Analysis: Identifying patient subgroups based on clinical and laboratory features.
•	Longitudinal Studies: Evaluating changes over time in response to treatment.

Potential Use Cases:
1.	Disease Progression Analysis: Study how clinical markers change over time.
2.	Treatment Effectiveness: Evaluate how different medications affect health outcomes.
3.	Risk Prediction Models: Develop machine learning models to predict complications or treatment responses.
4.	Exploratory Analysis: Analyze correlations between markers like proteinuria and kidney function deterioration.


Table 1: Strong Positive Correlations in Renal Parameters
Feature 1	Feature 2	Pearson Correlation Coefficient	Clinical Significance
A/C Ratio	A/C Ratio 2	+0.830	Indicates high consistency in measurements, reinforcing reliability in diagnosis.
A/C Ratio 2	Pus Cells 2	+0.828	Suggests significant association between renal inflammation/infection and A/C ratios.
24h Protein 2	Albumin 2	+0.905	Strongly correlates, indicating high proteinuria relates to serum albumin; critical for nephrology diagnosis.
24h Protein 2	RBCs 2	+0.851	Elevation in proteinuria parallels RBCs, suggesting underlying kidney damage/inflammation.
Casts	Casts 2	+0.889	Strong correspondence in cast types reinforces diagnostic accuracy for kidney pathology.
Table 2: Weak Positive Correlations in Renal Parameters
Feature 1	Feature 2	Pearson Correlation Coefficient	Clinical Significance
A/C Ratio 2	Age	+0.230	Mild association suggests age might influence renal function.
RBCs 2	Age	+0.234	Weak correlation that may necessitate age-related RBC level evaluations.
Albumin 2	Weight	+0.182	Suggests a weak connection that could indicate metabolic status related to body composition.
PLT 2	Age	+0.308	Moderate correlation indicating awareness of age-related platelet changes.
WBCs 2	Duration	+0.110	Minimal association hints at ongoing infections or inflammatory processes over time.
Table 3: Negative Correlations in Renal and Hematological Parameters
Feature 1	Feature 2	Pearson Correlation Coefficient	Clinical Significance
A/C Ratio 2	Lymphocytes 2	-0.315	Inverse relationship suggesting immune response variability correlated with urine A/C ratios.
Duration	Micofenolate Mofetile	-0.069	Suggests a weak decline in treatment effectiveness over time.
Hb 2	RBCs	-0.042	Minor significance indicating potential issues with anemia.
PLT 2	WBCS 2	-0.232	Indicates compensatory mechanisms in hematological profiles during disease processes.
Lymphocytes 2	WBCS 2	-0.293	Points to potential immune system dysregulation needing clinical evaluation.
Table 4: Key Correlations for Clinical Monitoring
Parameter	Key Positive Correlations	Clinical Implications
A/C Ratio	Pus Cells, Protein Levels	Essential for diagnosing renal disease and infection.
Albumin	Protein Levels, RBC Counts	Useful in assessing nephrotic syndrome and overall kidney function.
24h Protein	Albumin, RBCs, Casts	Critical for evaluating kidney function and damage.
Age	Multiple Measures	Suggests age is a variable needing individualized assessment.
Lymphocyte Count	A/C Ratios, WBC Counts	Indicates possible immune dysfunction warranting attention.
Table 5: Recommendations for Diagnostic Use
Insight	Recommendation
Strong correlations underscore key biomarkers for diagnosis.	Focus on A/C ratios, protein levels, and inflammatory markers to guide clinical decisions.
Weak correlations hint at multifactorial disease influences.	Employ a comprehensive approach considering demographic factors like age and weight.
Negative correlations may indicate concurrent health issues.	Investigate underlying causes for negative trends in lymphocyte and RBC counts to inform treatment strategies.
Regular assessments of proteinuria are crucial.	Routine evaluation in at-risk patients enhances the early detection of renal issues.
Integrated analysis of findings into clinical practice can optimize patient outcomes.	Regular updates to diagnostic criteria aligning with emerging research will strengthen clinical efficacy.
Table 6: Age-Related Correlation Insights
Feature 1	Feature 2	Pearson Correlation Coefficient	Clinical Significance
Age	24h Protein 2	+0.267	Indicates potential age-related protein excretion changes.
Age	Urea 2	+0.258	Suggests a link between renal function and aging, impacting diagnosis.
Age	Albumin	+0.237	Relationship suggests that aging may affect serum albumin levels.
Age	ESR 2	+0.211	Weak correlation could indicate age-related alterations in inflammation.
Age	Creatinine	+0.339	A stronger link suggesting that creatinine should be interpreted considering the patient's age.
Table 7: Weight-Associated Correlations
Feature 1	Feature 2	Pearson Correlation Coefficient	Clinical Significance
Weight	24h Protein 2	+0.157	Suggests body mass may influence protein levels in the urine.
Weight	Albumin 2	+0.196	A mild correlation hinting at weight-related issues such as obesity impacting protein status.
Weight	Urea	+0.196	Suggests potential metabolic alterations connected to weight.
Weight	ESR 2	+0.182	Points to the impact of body weight on inflammatory markers.
Weight	WBCs	+0.135	Indicates potential associations between body mass and immune response.
Table 8: Treatment Correlations
Feature 1	Feature 2	Key Correlations	Clinical Implications
Prednison	ESR	Strong correlation	Indicates effectiveness against inflammatory conditions.
Azathioprine	Creatinine	Related to protein levels	Shows therapeutic impact on kidney functions over time.
Micofenolate Mofetile	Lymphocytes	Weak correlations	Suggests the need for further investigation into patient adherence.
Anti-DNA	Albumin	Strongly correlates	Valuable for diagnosing systemic lupus erythematosus and other autoimmune disorders.
Casts	24h Protein	Highly correlated	Indicates the presence of renal damage that requires monitoring.
Table 9: Hematological Correlations
Feature 1	Feature 2	Key Correlations	Clinical Implications
Hemoglobin (Hb)	WBCs	Correlates positively	Highlights the importance of balanced evaluation in anemia cases.
Platelet Count (PLT)	RBCs	Mild negative correlation	Indicates hematologic interdependencies that may suggest malignancy or inflammation.
Lymphocytes	WBCs	Strong link	Vital in monitoring immune response, especially in systemic diseases.
RBCs	Albumin	Positively correlated	Indicates broader implications in hemolytic disorders and renal dysfunction.
WBCs	Inflammatory Markers	Positively correlates	Key indicators in assessing patient’s immune response and infection status.
Table 10: Summary and Future Directions
Summary Insight	Future Directions
Strong correlations among key renal and hematologic parameters have diagnostic implications.	Further research is needed to explore the causative factors in these correlations.
Weak correlations suggest complexity in disease presentations.	Clinical trials should test interventions considering patient demographics and comorbidities.
Negative correlations highlight potential underlying pathologies.	Enhanced clinical protocols focusing on comprehensive assessments of lymphocyte activity and renal function.
Continued integration of findings into clinical practice can optimize patient outcomes.	Regular updates to diagnostic criteria aligning with emerging research will strengthen clinical efficacy.
Holistic understanding of patient health through correlation analysis can guide better decision-making.	Multidisciplinary teams should collaborate for optimal patient management strategies, focusing on individual patient profiles.
These tables provide structured insights into the correlation analysis focused on clinical diagnosis and treatment, effectively splitting correlation pairs into specific features for better readability and application.

II. In-Depth Analysis by Category
1. Renal Function
A. Glomerular Filtration Rate (GFR) Markers: Interrelationships and Clinical Implications
Correlation	Feature 1	Feature 2	Clinical Significance
+0.926	Creatinine 2	Urea 2	Extremely high correlation suggests robust consistency in GFR assessment across measurements.
+0.839	Creatinine 2	Creatinine	High correlation indicates excellent stability in creatinine measurement over time.
+0.835	Urea	Urea 2	High correlation indicates excellent stability in urea measurement over time.
+0.788	Creatinine 2	Urea	Strong relationship between creatinine and urea reflects accurate kidney function assessment.
+0.775	Creatinine	Urea	Reinforces strong correlation between creatinine and urea in assessing kidney filtration.
+0.773	Creatinine	Urea 2	Further strengthens the relationship across different time points or methods.
B. Tubular Integrity Markers: Assessing Damage and Dysfunction
Correlation	Feature 1	Feature 2	Clinical Significance
+0.889	RBCs	Albumin	Strong positive correlation suggests glomerular damage or inflammation impacting both hematuria and proteinuria.
+0.889	Casts	Casts 2	Strong correlation indicates reliable assessment of tubular damage/dysfunction.
+0.843	24h Protein 2	Albumin	Reinforces the relationship between total protein and albumin measurements, supporting their utility in evaluating proteinuria.
+0.851	24h Protein 2	RBCs 2	Indicates possible glomerular damage as a source of both proteinuria and hematuria.
C. Comprehensive Proteinuria Assessment: Evaluating Different Measurement Methods
Correlation	Feature 1	Feature 2	Clinical Significance
+0.905	24h Protein 2	Albumin 2	Very high correlation suggests strong agreement between these two methods for assessing proteinuria.
+0.843	24h Protein 2	Albumin	High correlation supports the use of albumin as a surrogate marker for total protein excretion.
2. Inflammation and Immune Response
A. Inflammatory Markers: Assessing Acute and Chronic Inflammation in Kidney Disease
Correlation	Feature 1	Feature 2	Clinical Significance
+0.863	A/C Ratio	Pus Cells	Strong correlation indicates inflammation (pyuria) driving increased A/C ratios.
+0.650	ESR 2	Low Complement	Moderate correlation suggests simultaneous activation of multiple inflammatory pathways.
B. Immunosuppression Efficacy and Disease Activity: Monitoring Treatment Response
Correlation	Feature 1	Feature 2	Clinical Significance
+0.791	ESR 2	Prednison	Strong positive correlation could suggest incomplete suppression of inflammation despite treatment.
-0.262	Lymphocytes 2	Prednison	Negative correlation might indicate prednisone's effectiveness in reducing lymphocyte activation.
3. Hematological Markers
A. Anemia Assessment: Hemoglobin and Related Measures
Correlation	Feature 1	Feature 2	Clinical Significance
+0.769	Hb	Hb 2	Excellent agreement between hemoglobin measurements.
+0.592	Hb	WBCs	Suggests anemia may be related to infection/inflammatory processes, requiring more data for comprehensive analysis.
B. White Blood Cell Dynamics: Response to Infection and Inflammation
Correlation	Feature 1	Feature 2	Clinical Significance
+0.814	WBCs	Lymphocytes	Indicates immune response dynamics; essential for understanding infectious vs. autoimmune etiology.
C. Platelet Involvement: Clinical Significance in Renal Health
Correlation	Feature 1	Feature 2	Clinical Significance
-0.262	Lymphocytes 2	Azathioprine	Suggests potential interaction between immune suppression and platelet counts.
4. Patient Characteristics and Disease Progression
A. Influence of Patient Characteristics and Disease Course on Clinical Markers
Correlation	Feature 1	Feature 2	Clinical Significance
+0.776	Age	Duration	Suggests longer disease duration is associated with older age, indicating a need for age-specific management strategies.
5. Negative Correlations: Identifying Counter-Regulatory Mechanisms and Diagnostic Patterns
Correlation	Feature 1	Feature 2	Clinical Significance
-0.315	A/C Ratio 2	Lymphocytes 2	Negative correlation may indicate that higher A/C ratios correlate with lower lymphocytes, suggesting different immune responses.
-0.232	PLT 2	WBCS 2	Indicates a potential counter-regulatory mechanism in hematological responses.
-0.185	A/C Ratio 2	Hb 2	Suggests that as A/C ratio increases, hemoglobin levels may decrease, indicating potential underlying anemia.
III. Advanced Analytical Techniques
(No tables presented; describe the chosen methods and their results/interpretations.)
IV. Interpretation and Reporting
o	Focus on Clinical Relevance: Prioritize correlations that have clear clinical interpretations and potential diagnostic value.
o	Consider P-values: Only report statistically significant correlations and correct for multiple comparisons if necessary.
o	Visualizations: Use tables, heatmaps, and other visualizations to effectively communicate the key findings.
o	Limitations: Acknowledge the limitations of the analysis, such as sample size, missing data, and the correlational nature of the study (correlation does not imply causation).

Regression Results Summary
________________________________________
1. Regression for A/C Ratio 2
Optimization Method	Mean Absolute Error	R² Score	Mean Squared Error	Execution Time (s)
Optuna	9.0241	0.8830	20.8022	529.76
HyperOpt	9.0241	0.8830	202.8022	362.12
PoissonRegressor	1.0440	0.8680	12.0904	0.1224
LassoLarsIC	1.0738	0.7786	15.6604	0.0913
LGBMRegressor	1.0744	0.7767	15.7264	0.0899
________________________________________
2. Regression for 24h Protein 2
Optimization Method	Mean Absolute Error	R² Score	Mean Squared Error	Execution Time (s)
Optuna	0.1181	0.9633	0.0342	779.83
HyperOpt	0.1181	0.9632	0.0343	735.84
GammaRegressor	1.0642	0.8074	0.2563	0.1296
ExtraTreesRegressor	1.0781	0.7657	0.2827	0.5408
PoissonRegressor	1.0796	0.7611	0.2855	0.0624
________________________________________
3. Regression for Albumin 2
Optimization Method	Mean Absolute Error	R² Score	Mean Squared Error	Execution Time (s)
Optuna	0.1885	0.9698	0.0630	400.07
HyperOpt	0.1779	0.9731	0.0561	638.29
LassoLarsCV	1.0213	0.9362	0.2720	0.3678
ExtraTreesRegressor	1.0238	0.9285	0.2880	0.5078
TweedieRegressor	1.0284	0.9147	0.3146	0.0676
________________________________________
Let me know if you'd like adjustments!


Column	Minimum	Maximum	Mean	Median	Standard Deviation	Variance	Skewness
age	20	52	34.373	34	8.471	71.758	0.344
sex	0	1	0.039	0	0.196	0.038	4.893
Wt	64	90	79.941	80	5.515	30.416	-0.584
duration	3	12	6.667	6	2.260	5.107	0.563
prednison	5	60	26.863	20	16.522	272.961	0.679
azathioprine	0	500	73.529	100	87.951	7735.294	2.265
micofenolate mofetile	0	3000	470.588	0	896.726	804117.647	1.780
ANA	1	2	1.020	1	0.140	0.020	7.141
Anti-DNA	0	3	1.314	2	0.990	0.980	-0.551
Hb	7.200	15.200	10.702	10.400	1.715	2.941	0.168
WBCs	1430	9400	4572.098	4200	1991.657	3966696.490	0.454
PLT	126	451	255.176	247	81.194	6592.388	0.210
lymphocytes	567	3500	1630.373	1430	730.576	533741.918	0.854
A/C ratio	11.200	140.300	51.018	30.700	37.694	1420.859	0.860
pus cells	0	60	13.294	10	15.658	245.172	1.535
RBCs	0	35	7.647	4	9.226	85.113	1.309
casts	0	3	0.686	0	0.990	0.980	0.809
albumin	0	4	1.078	1	1.278	1.634	0.925
urea	12	42	22.980	21	8.339	69.540	0.844
creatinine	0.500	1.800	0.792	0.600	0.376	0.142	1.445
Hb 2	7.500	16.100	10.763	10.600	1.525	2.325	0.614
WBCS 2	2100	8900	4272.549	3780	1620.417	2625751.373	1.128
PLT 2	140	393	234.157	231	64.054	4102.895	0.597
Lymphocytes 2	642	3200	1521.353	1360	535.524	286785.913	1.211
A/C ratio 2	10.400	146.200	57.627	45.200	42.053	1768.479	0.655
24h protien 2	0.024	3.641	0.994	0.632	0.974	0.949	0.888
pus cells 2	0	55	16.196	10	16.675	278.041	0.749
RBCs 2	0	50	10.490	5	12.449	154.975	1.380
casts 2	0	3	0.804	0	1.020	1.041	0.529
albumin 2	0	4	1.412	1	1.458	2.127	0.484
urea 2	12	45	23.431	20	9.741	94.890	1.172
creatinin 2	0.500	2.300	0.857	0.600	0.510	0.260	1.667
ESR 1	10	129	45.118	45	24.658	608.026	0.953
ESR 2	10	135	47.961	35	33.027	1090.758	0.864
low complement	0	3	1.196	2	1.020	1.041	-0.294

Here’s a table summarizing the top features selected by each method in your code:
Feature Selection Method	Top Features
F-value Selector	age, Wt, micofenolate mofetile, Hb, Hb 2
Mutual Information Selector	age, Wt, plaqunil, ANA, Lymphocytes 2
RFE with Logistic Regression	age, azathioprine, pus cells 2, RBCs 2, ESR 2
Select from Model with Random Forests	Wt, Hb, WBCs, A_C ratio, Hb 2
Variance Thresholding	age, Wt, duration, prednison, azathioprine, micofenolate mofetile, Anti-DNA, Hb, WBCs, PLT, lymphocytes, A_C ratio, 24h protein in urine, pus cells, RBCs, albumin, urea, creatinine, Hb 2, WBCS 2, PLT 2, Lymphocytes 2, A_C ratio 2, 24h protein 2, pus cells 2, RBCs 2, casts 2, albumin 2, urea 2, creatinin 2, ESR 1, ESR 2, low complement
RFE with Random Forests	Wt, Hb, 24h protein in urine, creatinine, Hb 2
Feature Importance with Random Forests	Hb 2, Wt, Hb, pus cells, 24h protein in urine