drug-discovery-prompts/upstream/mims-harvard-ToolUniverse/skills/tooluniverse-statistical-modeling/references/common_patterns.md

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title	task	lineage_type	upstream_source	upstream_sha	imported_at	prompt_class	upstream_changes	author	validated
Statistical Question Patterns		import	https://github.com/mims-harvard/ToolUniverse/blob/e2520a96/skills/tooluniverse-statistical-modeling/references/common_patterns.md	e2520a96	2026-06-26	prompt	accepted	upstream	false

Statistical Question Patterns

Common statistical question patterns with solutions.

Pattern 1: Odds Ratio from Binary Logistic Regression

Question format: "What is the odds ratio of [outcome] associated with [exposure]?"

Example: "What is the odds ratio of disease associated with exposure to chemical X?"

Solution:

import statsmodels.formula.api as smf
import numpy as np

# Fit logistic regression
model = smf.logit('disease ~ exposure + age + sex', data=df).fit(disp=0)

# Extract odds ratio
or_exposure = np.exp(model.params['exposure'])
ci = np.exp(model.conf_int())
ci_lower = ci.loc['exposure', 0]
ci_upper = ci.loc['exposure', 1]
p_val = model.pvalues['exposure']

# Answer
print(f"Odds Ratio: {or_exposure:.4f}")
print(f"95% CI: ({ci_lower:.4f}, {ci_upper:.4f})")
print(f"p-value: {p_val:.6f}")

---

Pattern 2: Odds Ratio from Ordinal Logistic Regression

Question format: "What is the odds ratio of [ordinal outcome] associated with [exposure] in ordinal logistic regression?"

Example: "What is the odds ratio of COVID-19 severity associated with BCG vaccination?"

Solution:

from statsmodels.miscmodels.ordinal_model import OrderedModel
import pandas as pd
import numpy as np

# Set up ordered outcome
severity_order = ['Mild', 'Moderate', 'Severe']
df['severity'] = pd.Categorical(df['severity'], categories=severity_order, ordered=True)
y = df['severity'].cat.codes

# Prepare predictors
X = pd.get_dummies(df[['bcg_vaccination', 'age', 'sex']], drop_first=True, dtype=float)

# Fit ordinal logit
model = OrderedModel(y, X, distr='logit').fit(method='bfgs', disp=0)

# Extract OR for BCG vaccination (first predictor)
or_bcg = np.exp(model.params[0])
ci = np.exp(model.conf_int())
ci_lower = ci.iloc[0, 0]
ci_upper = ci.iloc[0, 1]
p_val = model.pvalues[0]

# Answer
print(f"Odds Ratio (BCG): {or_bcg:.4f}")
print(f"95% CI: ({ci_lower:.4f}, {ci_upper:.4f})")
print(f"p-value: {p_val:.6f}")

---

Pattern 3: Percentage Reduction in Odds Ratio

Question format: "What is the percentage reduction in odds ratio for [outcome] after adjusting for [confounders]?"

Example: "What is the percentage reduction in OR for disease after adjusting for age and sex?"

Solution:

# Unadjusted model
model_crude = smf.logit('disease ~ exposure', data=df).fit(disp=0)
or_crude = np.exp(model_crude.params['exposure'])

# Adjusted model
model_adj = smf.logit('disease ~ exposure + age + sex', data=df).fit(disp=0)
or_adj = np.exp(model_adj.params['exposure'])

# Percentage reduction
pct_reduction = (or_crude - or_adj) / or_crude * 100

# Answer
print(f"Crude OR: {or_crude:.4f}")
print(f"Adjusted OR: {or_adj:.4f}")
print(f"Percentage reduction: {pct_reduction:.1f}%")

---

Pattern 4: Hazard Ratio from Cox Regression

Question format: "What is the hazard ratio for [exposure] in a Cox proportional hazards model?"

Example: "What is the hazard ratio for treatment in a Cox model adjusting for age and stage?"

Solution:

from lifelines import CoxPHFitter

# Fit Cox model
cph = CoxPHFitter()
cph.fit(df[['time', 'event', 'treatment', 'age', 'stage']],
        duration_col='time', event_col='event')

# Extract HR
hr_treatment = cph.hazard_ratios_['treatment']
summary = cph.summary
ci_lower = summary.loc['treatment', 'exp(coef) lower 95%']
ci_upper = summary.loc['treatment', 'exp(coef) upper 95%']
p_val = summary.loc['treatment', 'p']

# Answer
print(f"Hazard Ratio: {hr_treatment:.4f}")
print(f"95% CI: ({ci_lower:.4f}, {ci_upper:.4f})")
print(f"p-value: {p_val:.6f}")
print(f"Concordance: {cph.concordance_index_:.4f}")

---

Pattern 5: Kaplan-Meier Survival Estimate

Question format: "What is the Kaplan-Meier survival estimate at time T?"

Example: "What is the 5-year survival probability in the treatment group?"

Solution:

from lifelines import KaplanMeierFitter

# Subset treatment group
treatment_df = df[df['treatment'] == 1]

# Fit KM
kmf = KaplanMeierFitter()
kmf.fit(treatment_df['time'], treatment_df['event'])

# Survival at 5 years (60 months)
survival_5yr = kmf.predict(60)

# Answer
print(f"5-year survival probability: {survival_5yr:.4f}")
print(f"Median survival time: {kmf.median_survival_time_:.1f} months")

---

Pattern 6: Interaction Effect

Question format: "What is the odds ratio associated with the interaction between [A] and [B]?"

Example: "What is the OR for the interaction between treatment and biomarker status?"

Solution:

# Fit model with interaction
model = smf.logit('outcome ~ treatment * biomarker + age', data=df).fit(disp=0)

# Interaction term
interaction_coef = model.params['treatment:biomarker']
interaction_or = np.exp(interaction_coef)
interaction_p = model.pvalues['treatment:biomarker']

# Answer
print(f"Interaction OR: {interaction_or:.4f}")
print(f"p-value: {interaction_p:.6f}")

# Interpretation
if interaction_p < 0.05:
    print("Significant interaction: effect of treatment varies by biomarker status")

---

Pattern 7: Linear Regression Coefficient

Question format: "What is the coefficient for [predictor] in a linear regression model?"

Example: "What is the coefficient for BMI in a linear regression of blood pressure?"

Solution:

import statsmodels.formula.api as smf

# Fit OLS
model = smf.ols('blood_pressure ~ bmi + age + sex', data=df).fit()

# Extract coefficient
coef_bmi = model.params['bmi']
ci = model.conf_int()
ci_lower = ci.loc['bmi', 0]
ci_upper = ci.loc['bmi', 1]
p_val = model.pvalues['bmi']

# Answer
print(f"Coefficient (BMI): {coef_bmi:.4f}")
print(f"95% CI: ({ci_lower:.4f}, {ci_upper:.4f})")
print(f"p-value: {p_val:.6f}")
print(f"R-squared: {model.rsquared:.4f}")

# Interpretation
print(f"Interpretation: Each 1-unit increase in BMI is associated with {coef_bmi:.2f} mmHg change in blood pressure")

---

Pattern 8: Mixed-Effects Model Coefficient

Question format: "What is the coefficient for [predictor] in a mixed-effects model with random intercepts for [grouping]?"

Example: "What is the treatment effect in a mixed model with random intercepts for patient?"

Solution:

import statsmodels.formula.api as smf

# Fit LMM
model = smf.mixedlm('outcome ~ treatment + time', data=df, groups=df['patient_id']).fit(reml=True)

# Extract fixed effect
coef_treatment = model.fe_params['treatment']
se = model.bse_fe['treatment']
p_val = model.pvalues['treatment']
ci = model.conf_int()
ci_lower = ci.loc['treatment', 0]
ci_upper = ci.loc['treatment', 1]

# Answer
print(f"Coefficient (treatment): {coef_treatment:.4f}")
print(f"95% CI: ({ci_lower:.4f}, {ci_upper:.4f})")
print(f"p-value: {p_val:.6f}")

# ICC
group_var = float(model.cov_re.iloc[0, 0])
resid_var = float(model.scale)
icc = group_var / (group_var + resid_var)
print(f"ICC: {icc:.4f}")

---

Pattern 9: Model Comparison with AIC/BIC

Question format: "Which model has better fit according to AIC?"

Example: "Compare models with and without interaction term using AIC"

Solution:

# Fit models
model1 = smf.logit('outcome ~ A + B', data=df).fit(disp=0)
model2 = smf.logit('outcome ~ A * B', data=df).fit(disp=0)

# Compare
print("Model 1 (no interaction):")
print(f"  AIC: {model1.aic:.2f}")
print(f"  BIC: {model1.bic:.2f}")

print("\nModel 2 (with interaction):")
print(f"  AIC: {model2.aic:.2f}")
print(f"  BIC: {model2.bic:.2f}")

# Answer
if model2.aic < model1.aic:
    print("\nModel 2 preferred (lower AIC)")
else:
    print("\nModel 1 preferred (lower AIC)")

---

Pattern 10: Proportional Odds Assumption Test

Question format: "Is the proportional odds assumption met?"

Example: "Test if proportional odds assumption holds for ordinal severity model"

Solution:

import statsmodels.api as sm

# Fit binary logits at each cutpoint
severity_order = ['Mild', 'Moderate', 'Severe']
df['severity'] = pd.Categorical(df['severity'], categories=severity_order, ordered=True)
y_codes = df['severity'].cat.codes

X = pd.get_dummies(df[['exposure', 'age']], drop_first=True, dtype=float)
X_const = sm.add_constant(X)

# Fit at each cutpoint
coef_by_cutpoint = {}
for k in range(len(severity_order) - 1):
    y_binary = (y_codes > k).astype(int)
    model = sm.Logit(y_binary, X_const).fit(disp=0)
    coef_by_cutpoint[k] = model.params['exposure']

# Check if coefficients similar
coefs = list(coef_by_cutpoint.values())
coef_range = max(coefs) - min(coefs)

print(f"Coefficients by cutpoint: {coefs}")
print(f"Range: {coef_range:.4f}")

# Answer
if coef_range < 0.5:
    print("Proportional odds assumption likely satisfied")
else:
    print("Proportional odds assumption may be violated")

---

Pattern 11: Log-Rank Test

Question format: "Is there a significant difference in survival between groups?"

Example: "Test if survival differs between treatment and control groups"

Solution:

from lifelines.statistics import logrank_test

# Split by group
treatment_group = df['treatment'] == 1
control_group = df['treatment'] == 0

# Log-rank test
result = logrank_test(
    df.loc[treatment_group, 'time'],
    df.loc[control_group, 'time'],
    df.loc[treatment_group, 'event'],
    df.loc[control_group, 'event']
)

# Answer
print(f"Log-rank test statistic: {result.test_statistic:.4f}")
print(f"p-value: {result.p_value:.6f}")

if result.p_value < 0.05:
    print("Survival curves are significantly different")
else:
    print("No significant difference in survival")

---

Pattern 12: R-squared Interpretation

Question format: "What is the R-squared of the model?"

Example: "What proportion of variance is explained by the linear model?"

Solution:

model = smf.ols('outcome ~ predictor1 + predictor2 + age', data=df).fit()

# R-squared
r2 = model.rsquared
adj_r2 = model.rsquared_adj

# Answer
print(f"R-squared: {r2:.4f}")
print(f"Adjusted R-squared: {adj_r2:.4f}")
print(f"Interpretation: {r2*100:.1f}% of variance in outcome is explained by predictors")

---

Pattern 13: Concordance Index Interpretation

Question format: "What is the concordance index of the Cox model?"

Example: "How well does the Cox model discriminate between patients?"

Solution:

# Already fitted cph model
c_index = cph.concordance_index_

# Answer
print(f"Concordance index: {c_index:.4f}")

# Interpretation
if c_index > 0.7:
    print("Good discrimination (C > 0.7)")
elif c_index > 0.6:
    print("Acceptable discrimination (0.6 < C < 0.7)")
else:
    print("Poor discrimination (C < 0.6)")

---

Pattern 14: Coefficient Change After Adjustment

Question format: "How does the coefficient change after adjusting for confounders?"

Example: "Compare exposure coefficient before and after adjusting for age"

Solution:

# Unadjusted
model_crude = smf.ols('outcome ~ exposure', data=df).fit()
coef_crude = model_crude.params['exposure']

# Adjusted
model_adj = smf.ols('outcome ~ exposure + age + sex', data=df).fit()
coef_adj = model_adj.params['exposure']

# Change
absolute_change = coef_adj - coef_crude
pct_change = (coef_adj - coef_crude) / coef_crude * 100

# Answer
print(f"Crude coefficient: {coef_crude:.4f}")
print(f"Adjusted coefficient: {coef_adj:.4f}")
print(f"Absolute change: {absolute_change:.4f}")
print(f"Percentage change: {pct_change:.1f}%")

---

Pattern 15: Stratified Analysis

Question format: "What is the odds ratio stratified by [variable]?"

Example: "What is the OR for exposure separately in men and women?"

Solution:

# Stratify by sex
results_by_sex = {}

for sex in ['M', 'F']:
    df_subset = df[df['sex'] == sex]
    model = smf.logit('outcome ~ exposure + age', data=df_subset).fit(disp=0)

    or_exposure = np.exp(model.params['exposure'])
    ci = np.exp(model.conf_int())

    results_by_sex[sex] = {
        'OR': or_exposure,
        'CI_lower': ci.loc['exposure', 0],
        'CI_upper': ci.loc['exposure', 1],
        'p_value': model.pvalues['exposure']
    }

# Answer
for sex, result in results_by_sex.items():
    print(f"\n{sex}:")
    print(f"  OR: {result['OR']:.4f}")
    print(f"  95% CI: ({result['CI_lower']:.4f}, {result['CI_upper']:.4f})")
    print(f"  p-value: {result['p_value']:.6f}")

---

Quick Reference Table

Pattern	Model Type	Key Output	Formula Example
1	Binary Logistic	Odds Ratio	logit('y ~ x + z')
2	Ordinal Logistic	Odds Ratio	OrderedModel(y, X)
3	Logistic (2 models)	% Reduction	Compare crude vs adjusted
4	Cox PH	Hazard Ratio	cph.fit(..., duration_col, event_col)
5	Kaplan-Meier	Survival Prob	kmf.predict(time)
6	Logistic + Interaction	Interaction OR	'y ~ A * B'
7	Linear Regression	Coefficient	ols('y ~ x + z')
8	Mixed-Effects	Fixed Effect	mixedlm(..., groups=...)
9	Model Comparison	AIC/BIC	Compare .aic values
10	Ordinal Assumption	PO Test	Binary logits at cutpoints
11	Survival Comparison	Log-rank	logrank_test(...)
12	Linear Regression	R²	.rsquared
13	Cox PH	C-index	.concordance_index_
14	Any Regression	Coef Change	Compare models
15	Stratified Analysis	Stratum-specific OR	Subset + fit

---

Common Mistakes to Avoid

1. Forgetting to exponentiate: Logistic/Cox coefficients are log-odds/log-hazards. Must use np.exp() for ORs/HRs.

2. Wrong variable type: Ordinal outcomes need OrderedModel, not Logit.

3. Missing confounders: Always check if question specifies "adjusting for" variables.

4. Interpretation direction: HR > 1 = worse outcome (higher hazard), OR > 1 = higher odds.

5. Precision: Round to requested decimal places (typically 4 for ORs/HRs, 6 for p-values).

6. CI extraction: Use conf_int() method, not conf_int attribute.

7. Formula syntax: Interactions use *, not +. E.g., 'y ~ A * B' includes A, B, and A:B.

8. Duration/event cols: Cox models require explicit duration_col and event_col parameters.