Confidence Intervals for Mean Values in Clinical Research

Introduction

The ciSingle function in ClinicoPath provides comprehensive confidence interval calculations for mean values of continuous variables. This is essential in clinical and pathological research where researchers need to quantify the uncertainty around sample means and make statistical inferences about population parameters.

Clinical Motivation

In clinical research, confidence intervals are crucial for:

Biomarker levels: Establishing normal ranges and treatment thresholds
Patient characteristics: Age distributions, measurement precision
Treatment effects: Quantifying uncertainty in outcome measures
Laboratory values: Quality control and reference ranges
Survival times: Time-to-event analysis and prognosis
Diagnostic accuracy: Performance metrics with uncertainty bounds

Confidence intervals provide more information than point estimates alone, helping clinicians understand the precision of their measurements and the reliability of their conclusions.

When to Use Confidence Intervals

Use confidence intervals when:

Quantifying uncertainty around sample means
Making population inferences from sample data
Comparing groups (though formal hypothesis testing may be preferred)
Establishing reference ranges for clinical parameters
Planning sample sizes for future studies
Reporting clinical trial results with appropriate uncertainty bounds

Package Setup

library(ClinicoPath)

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##   fortify.kmeans ggfortify
##   fortify.ts     ggfortify

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library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(knitr)

# Load the histopathology dataset
data("histopathology")

# Display dataset structure for continuous variables
numeric_vars <- sapply(histopathology, is.numeric)
cat("Available continuous variables:\n")

## Available continuous variables:

print(names(histopathology)[numeric_vars])

##  [1] "ID"               "Age"              "Grade"            "TStage"          
##  [5] "Anti-X-intensity" "Anti-Y-intensity" "OverallTime"      "Outcome"         
##  [9] "Rater 1"          "Rater 2"          "Rater 3"          "Rater A"         
## [13] "Rater B"          "New Test"         "Golden Standart"  "MeasurementA"    
## [17] "MeasurementB"     "Measurement1"     "Measurement2"

Basic Confidence Interval Analysis

Example 1: Single Variable Analysis

Let’s start with calculating confidence intervals for patient age:

# Basic confidence interval for age
# Note: Function available in jamovi interface
ciSingle(
  data = histopathology,
  deps = "Age",
  ciWidth = 95,
  method = "t",
  showPlot = FALSE,
  showDiagnostics = FALSE
)

The results would show: - Mean: Average age in the sample
- Std. Dev: Standard deviation measuring variability - Std. Error: Standard error of the mean - Lower/Upper Bound: 95% confidence interval limits - CI Width: Total width of the confidence interval

Interpretation

The results show: - Mean: Average age in the sample - Std. Dev: Standard deviation measuring variability - Std. Error: Standard error of the mean - Lower/Upper Bound: 95% confidence interval limits - CI Width: Total width of the confidence interval

Example 2: Multiple Variables

Analyzing multiple continuous variables simultaneously:

# Multiple variables analysis
ciSingle(
  data = histopathology,
  deps = c("Age", "OverallTime", "MeasurementA", "MeasurementB"),
  ciWidth = 95,
  method = "t",
  showPlot = FALSE,
  showDiagnostics = FALSE
)

Clinical Significance

This analysis helps: - Compare variability across different measurements - Identify which variables have more precise estimates - Plan future studies based on observed variance

Confidence Interval Methods

Understanding Different Methods

t-Distribution Method (Default)

Best for: Normal or approximately normal data
Assumptions: Data follows normal distribution
Use when: Sample size is moderate, normality is reasonable

ciSingle(
  data = histopathology,
  deps = "Age",
  ciWidth = 95,
  method = "t",
  showDiagnostics = TRUE
)

Bootstrap Method

Best for: Non-normal data or small samples
Assumptions: Minimal distributional assumptions
Use when: Normality is questionable or robust estimates needed

ciSingle(
  data = histopathology,
  deps = "MeasurementA",
  ciWidth = 95,
  method = "bootstrap",
  bootstrapSamples = 1000,
  showDiagnostics = TRUE
)

Normal Approximation

Best for: Large samples (n > 30)
Assumptions: Central limit theorem applies
Use when: Sample size is large, faster computation needed

ciSingle(
  data = histopathology,
  deps = "OverallTime",
  ciWidth = 95,
  method = "normal",
  showDiagnostics = TRUE
)

Method Comparison

# Compare all three methods for the same variable
cat("Method Comparison for Age Variable:\n\n")

cat("1. t-Distribution Method:\n")
ciSingle(
  data = histopathology,
  deps = "Age",
  method = "t",
  showPlot = FALSE
)

cat("\n2. Bootstrap Method:\n")
ciSingle(
  data = histopathology,
  deps = "Age",
  method = "bootstrap",
  bootstrapSamples = 1000,
  showPlot = FALSE
)

cat("\n3. Normal Approximation:\n")
ciSingle(
  data = histopathology,
  deps = "Age",
  method = "normal",
  showPlot = FALSE
)

Group Analysis with Split-By

Example 1: Gender Differences

Comparing confidence intervals between groups:

# Analyze by gender
ciSingle(
  data = histopathology,
  deps = c("Age", "OverallTime"),
  splitBy = "Sex",
  ciWidth = 95,
  method = "t",
  showDiagnostics = TRUE
)

Clinical Interpretation

This analysis reveals: - Gender-specific mean values and confidence intervals - Whether confidence intervals overlap (suggesting similar means) - Differences in precision between groups - Sample size effects on interval width

Example 2: Treatment Group Analysis

Comparing treatment and control groups:

# Analyze by treatment group
ciSingle(
  data = histopathology,
  deps = c("Age", "MeasurementA", "MeasurementB"),
  splitBy = "Group",
  ciWidth = 95,
  method = "t",
  showDiagnostics = TRUE
)

Research Implications

Results help determine: - Baseline characteristics between treatment groups - Biomarker differences between groups - Whether randomization was successful - Need for covariate adjustment in analysis

Different Confidence Levels

Choosing Appropriate Confidence Levels

# Compare different confidence levels
confidence_levels <- c(80, 90, 95, 99)

for (level in confidence_levels) {
  cat(paste0("\n", level, "% Confidence Interval:\n"))
  result <- ciSingle(
    data = histopathology,
    deps = "Age",
    ciWidth = level,
    method = "t",
    showPlot = FALSE
  )
}

Level Selection Guidelines

80%: Exploratory analysis, screening studies
90%: Clinical decision-making, quality control
95%: Standard research reporting, most publications
99%: High-stakes decisions, regulatory submissions

Trade-offs in Confidence Level

# Demonstrate width changes with confidence level
cat("Effect of Confidence Level on Interval Width:\n")

## Effect of Confidence Level on Interval Width:

cat("Variable: Age\n\n")

## Variable: Age

levels_demo <- data.frame(
  Level = c("80%", "90%", "95%", "99%"),
  Width_Factor = c("Narrowest", "Narrow", "Standard", "Widest"),
  Use_Case = c("Screening", "Clinical", "Research", "Regulatory")
)

kable(levels_demo, caption = "Confidence Level Selection Guide")

Confidence Level Selection Guide
Level	Width_Factor	Use_Case
80%	Narrowest	Screening
90%	Narrow	Clinical
95%	Standard	Research
99%	Widest	Regulatory

Diagnostic Features

Normality Assessment

Understanding when different methods are appropriate:

# Comprehensive diagnostics for multiple variables
ciSingle(
  data = histopathology,
  deps = c("Age", "OverallTime", "MeasurementA", "Anti-X-intensity"),
  ciWidth = 95,
  method = "t",
  showDiagnostics = TRUE
)

Interpreting Diagnostics

Shapiro-Wilk Test: - p > 0.05: Normal distribution (t-method appropriate) - p ≤ 0.05: Non-normal distribution (consider bootstrap)

Assumptions Met: - “Met”: Normal distribution, t-method recommended - “Violated”: Non-normal distribution, bootstrap recommended - “Unknown”: Cannot determine (very small samples)

Method Selection Based on Data

# Example of choosing methods based on data characteristics
cat("Adaptive Method Selection Example:\n\n")

# For normal data - use t-distribution
cat("1. Normal Data (Age) - t-distribution method:\n")
ciSingle(
  data = histopathology,
  deps = "Age",
  method = "t",
  showDiagnostics = TRUE
)

# For potentially non-normal data - use bootstrap
cat("\n2. Potentially Non-normal Data - bootstrap method:\n")
ciSingle(
  data = histopathology,
  deps = "MeasurementA",
  method = "bootstrap",
  bootstrapSamples = 1000,
  showDiagnostics = TRUE
)

Visualization Features

Basic Confidence Interval Plots

# Note: Plotting functionality available in jamovi interface
ciSingle(
  data = histopathology,
  deps = c("Age", "OverallTime", "MeasurementA"),
  ciWidth = 95,
  method = "t",
  showPlot = TRUE
)

Group Comparison Plots

# Visualize group differences
ciSingle(
  data = histopathology,
  deps = c("Age", "OverallTime"),
  splitBy = "Sex",
  ciWidth = 95,
  method = "t",
  showPlot = TRUE
)

Plot Interpretation

Confidence interval plots help visualize: - Point estimates: Central values for each variable/group - Uncertainty: Width of intervals indicating precision - Group differences: Overlapping vs. non-overlapping intervals - Sample size effects: Larger samples have narrower intervals

Clinical Applications and Case Studies

Case Study 1: Biomarker Reference Ranges

Establishing reference ranges for a new biomarker:

# Establish reference ranges for biomarkers
ciSingle(
  data = histopathology,
  deps = c("MeasurementA", "MeasurementB"),
  ciWidth = 95,
  method = "t",
  showDiagnostics = TRUE
)

Clinical Utility

This analysis provides: - Mean values for healthy/disease populations - Confidence intervals for population means - Assessment of measurement variability - Basis for clinical decision thresholds

Case Study 2: Treatment Effect Precision

Quantifying precision of treatment effects:

# Analyze treatment effect precision
ciSingle(
  data = histopathology,
  deps = "OverallTime",
  splitBy = "Group",
  ciWidth = 95,
  method = "t",
  showDiagnostics = TRUE
)

Research Implications

Results inform: - Precision of treatment effect estimates - Sample size adequacy for detecting differences - Power calculations for future studies - Clinical significance of observed differences

Case Study 3: Quality Control Analysis

Laboratory quality control and method validation:

# Quality control for measurement precision
ciSingle(
  data = histopathology,
  deps = c("Anti-X-intensity", "Anti-Y-intensity"),
  ciWidth = 99,  # Higher confidence for QC
  method = "t",
  showDiagnostics = TRUE
)

Quality Metrics

Analysis provides: - Measurement precision assessment - Control limits for quality control - Method comparison data - Validation of analytical procedures

Advanced Applications

Bootstrap Method Applications

When to use bootstrap confidence intervals:

# Bootstrap for potentially skewed data
ciSingle(
  data = histopathology,
  deps = "OverallTime",  # Often skewed in survival data
  ciWidth = 95,
  method = "bootstrap",
  bootstrapSamples = 2000,
  showDiagnostics = TRUE
)

Bootstrap Advantages

Robust: Works with any distribution
Flexible: No normality assumptions
Accurate: Especially for skewed data
Modern: Computationally intensive but reliable

Sample Size Considerations

Understanding how sample size affects confidence intervals:

# Demonstrate sample size effects
# Using subsets of different sizes
sample_sizes <- c(25, 50, 100, 250)

cat("Effect of Sample Size on Confidence Interval Width:\n\n")

for (n in sample_sizes) {
  subset_data <- histopathology[1:n, ]
  cat(paste0("Sample size n = ", n, ":\n"))
  
  result <- ciSingle(
    data = subset_data,
    deps = "Age",
    ciWidth = 95,
    method = "t",
    showPlot = FALSE
  )
  cat("\n")
}

Sample Size Guidelines

n < 30: Use t-distribution or bootstrap
30 ≤ n < 100: t-distribution usually appropriate
n ≥ 100: Normal approximation acceptable
Large n: Narrower intervals, more precise estimates

Best Practices and Guidelines

Method Selection Flowchart

cat("Confidence Interval Method Selection Guide:\n\n")

## Confidence Interval Method Selection Guide:

cat("1. Check sample size:\n")

## 1. Check sample size:

cat("   - n < 30: Consider bootstrap or t-distribution\n")

##    - n < 30: Consider bootstrap or t-distribution

cat("   - n ≥ 30: Any method acceptable\n\n")

##    - n ≥ 30: Any method acceptable

cat("2. Test normality (if using t-distribution):\n")

## 2. Test normality (if using t-distribution):

cat("   - Normal: Use t-distribution\n")

##    - Normal: Use t-distribution

cat("   - Non-normal: Use bootstrap\n\n")

##    - Non-normal: Use bootstrap

cat("3. Consider computational constraints:\n")

## 3. Consider computational constraints:

cat("   - Fast computation: Normal approximation\n")

##    - Fast computation: Normal approximation

cat("   - Robust results: Bootstrap\n")

##    - Robust results: Bootstrap

cat("   - Standard approach: t-distribution\n\n")

##    - Standard approach: t-distribution

cat("4. Match method to research context:\n")

## 4. Match method to research context:

cat("   - Exploratory: Bootstrap\n")

##    - Exploratory: Bootstrap

cat("   - Confirmatory: t-distribution\n")

##    - Confirmatory: t-distribution

cat("   - Large studies: Normal approximation\n")

##    - Large studies: Normal approximation

Reporting Guidelines

Essential Elements to Report

cat("Confidence Interval Reporting Checklist:\n\n")

## Confidence Interval Reporting Checklist:

cat("✓ Sample size for each group\n")

## ✓ Sample size for each group

cat("✓ Mean and standard deviation\n")

## ✓ Mean and standard deviation

cat("✓ Confidence level (e.g., 95%)\n")

## ✓ Confidence level (e.g., 95%)

cat("✓ Method used (t-distribution, bootstrap, normal)\n")

## ✓ Method used (t-distribution, bootstrap, normal)

cat("✓ Confidence interval bounds\n")

## ✓ Confidence interval bounds

cat("✓ Missing data handling\n")

## ✓ Missing data handling

cat("✓ Assumption checking results\n")

## ✓ Assumption checking results

cat("✓ Clinical interpretation\n")

## ✓ Clinical interpretation

Example Results Section

cat("Example Results Reporting:\n\n")

## Example Results Reporting:

cat("Patient age showed a mean of 52.3 years (SD = 14.7, n = 250).\n")

## Patient age showed a mean of 52.3 years (SD = 14.7, n = 250).

cat("The 95% confidence interval for the population mean age was\n")

## The 95% confidence interval for the population mean age was

cat("50.5 to 54.1 years (t-distribution method). Normality\n")

## 50.5 to 54.1 years (t-distribution method). Normality

cat("assumption was met (Shapiro-Wilk p = 0.12). The confidence\n")

## assumption was met (Shapiro-Wilk p = 0.12). The confidence

cat("interval width of 3.6 years indicates adequate precision\n")

## interval width of 3.6 years indicates adequate precision

cat("for clinical decision-making.\n")

## for clinical decision-making.

Common Pitfalls and Solutions

Avoiding Common Mistakes

cat("Common Pitfalls in Confidence Interval Analysis:\n\n")

## Common Pitfalls in Confidence Interval Analysis:

cat("1. WRONG: 'There's a 95% chance the true mean is in this interval'\n")

## 1. WRONG: 'There's a 95% chance the true mean is in this interval'

cat("   RIGHT: 'We're 95% confident the true mean is in this interval'\n\n")

##    RIGHT: 'We're 95% confident the true mean is in this interval'

cat("2. WRONG: Using normal approximation for small samples\n")

## 2. WRONG: Using normal approximation for small samples

cat("   RIGHT: Use t-distribution for n < 30\n\n")

##    RIGHT: Use t-distribution for n < 30

cat("3. WRONG: Ignoring normality assumptions\n")

## 3. WRONG: Ignoring normality assumptions

cat("   RIGHT: Test assumptions, use bootstrap if violated\n\n")

##    RIGHT: Test assumptions, use bootstrap if violated

cat("4. WRONG: Confusing confidence intervals with prediction intervals\n")

## 4. WRONG: Confusing confidence intervals with prediction intervals

cat("   RIGHT: CI estimates population mean, PI predicts individual values\n\n")

##    RIGHT: CI estimates population mean, PI predicts individual values

cat("5. WRONG: Using overlapping CIs to test for differences\n")

## 5. WRONG: Using overlapping CIs to test for differences

cat("   RIGHT: Use appropriate statistical tests for comparisons\n")

##    RIGHT: Use appropriate statistical tests for comparisons

Integration with Other Analyses

Combining with Hypothesis Testing

cat("Confidence Intervals and Hypothesis Testing:\n\n")

## Confidence Intervals and Hypothesis Testing:

cat("• CIs provide effect size information\n")

## • CIs provide effect size information

cat("• Hypothesis tests provide p-values\n")

## • Hypothesis tests provide p-values

cat("• Together: Complete statistical picture\n")

## • Together: Complete statistical picture

cat("• Non-overlapping 95% CIs ≈ p < 0.05 for group differences\n")

## • Non-overlapping 95% CIs ≈ p < 0.05 for group differences

cat("• CI width indicates practical significance\n")

## • CI width indicates practical significance

Power Analysis Applications

cat("Using CIs for Power Analysis:\n\n")

## Using CIs for Power Analysis:

cat("• CI width indicates current precision\n")

## • CI width indicates current precision

cat("• Narrow CIs suggest adequate sample size\n")

## • Narrow CIs suggest adequate sample size

cat("• Wide CIs may indicate need for larger samples\n")

## • Wide CIs may indicate need for larger samples

cat("• Bootstrap CIs help with non-normal power calculations\n")

## • Bootstrap CIs help with non-normal power calculations

Conclusion

The ciSingle function provides comprehensive confidence interval analysis essential for clinical and pathological research. Key takeaways:

Choose appropriate methods based on data characteristics and sample size
Use diagnostics to validate assumptions and guide method selection
Report comprehensively including methods, assumptions, and interpretations
Consider clinical context when interpreting interval width and precision
Integrate with other analyses for complete statistical understanding

Confidence intervals are fundamental to statistical inference, providing not just point estimates but crucial information about uncertainty and precision that guides clinical decision-making and research planning.

References

Altman, D. G. (2000). Statistics in medical journals: Developments in the 1990s. Statistics in Medicine, 19(23), 3275-3289.
Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman and Hall.
Gardner, M. J., & Altman, D. G. (1986). Confidence intervals rather than P values. British Medical Journal, 292(6522), 746-750.
Morey, R. D., et al. (2016). The fallacy of placing confidence in confidence intervals. Psychonomic Bulletin & Review, 23(1), 103-123.

This vignette was created for the ClinicoPath jamovi module. For more information and updates, visit https://github.com/sbalci/ClinicoPathJamoviModule.

Comprehensive Guide to ciSingle Function for Statistical Inference

ClinicoPath Module

2025-07-13

Introduction

Clinical Motivation

When to Use Confidence Intervals

Package Setup

Basic Confidence Interval Analysis

Example 1: Single Variable Analysis

Interpretation

Example 2: Multiple Variables

Clinical Significance

Confidence Interval Methods

Understanding Different Methods

t-Distribution Method (Default)

Bootstrap Method

Normal Approximation

Method Comparison

Group Analysis with Split-By

Example 1: Gender Differences

Clinical Interpretation

Example 2: Treatment Group Analysis

Research Implications

Different Confidence Levels

Choosing Appropriate Confidence Levels

Level Selection Guidelines

Trade-offs in Confidence Level

Diagnostic Features

Normality Assessment

Interpreting Diagnostics

Method Selection Based on Data

Visualization Features

Basic Confidence Interval Plots

Group Comparison Plots

Plot Interpretation

Clinical Applications and Case Studies

Case Study 1: Biomarker Reference Ranges

Clinical Utility

Case Study 2: Treatment Effect Precision

Research Implications

Case Study 3: Quality Control Analysis

Quality Metrics

Advanced Applications

Bootstrap Method Applications

Bootstrap Advantages

Sample Size Considerations

Sample Size Guidelines

Best Practices and Guidelines

Method Selection Flowchart

Reporting Guidelines

Essential Elements to Report

Example Results Section

Common Pitfalls and Solutions

Avoiding Common Mistakes

Integration with Other Analyses

Combining with Hypothesis Testing

Power Analysis Applications

Conclusion

References