---
title: "Overview of Two Co-Primary Endpoints Analysis"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Overview of Two Co-Primary Endpoints Analysis}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

## Introduction

The `twoCoprimary` package provides comprehensive tools for sample size calculation and power analysis in clinical trials with two co-primary endpoints. This package implements methodologies from multiple publications and supports various endpoint types.

```{r setup}
library(twoCoprimary)
```

## What are Co-Primary Endpoints?

In clinical trials, **co-primary endpoints** are multiple primary endpoints that must **all** show statistically significant treatment effects for the trial to be considered successful. This is in contrast to multiple primary endpoints where demonstrating an effect on **any one** endpoint is sufficient.

### Statistical Properties

- **No multiplicity adjustment needed** for Type I error control
- The overall Type I error rate is controlled at level $\alpha$ without Bonferroni correction
- The overall power is the joint probability: Power = $\Pr(\text{Reject } \text{H}_{01} \text{ and Reject } \text{H}_{02})$
- Accounting for **correlation** between endpoints can improve efficiency

### Hypotheses Structure

For two co-primary endpoints, we test:

**Null hypothesis**: $\text{H}_{0} = \text{H}_{01} \cup \text{H}_{02}$ (at least one null hypothesis is true)

**Alternative hypothesis**: $\text{H}_{1} = \text{H}_{11} \cap \text{H}_{12}$ (both alternative hypotheses are true)

We reject $\text{H}_{0}$ only if **both** $\text{H}_{01}$ and $\text{H}_{02}$ are rejected at level $\alpha$.

## Statistical Framework

### Intersection-Union Test (IUT)

The co-primary endpoint framework is based on the intersection-union test principle. Let $Z_{1}$ and $Z_{2}$ be the test statistics for endpoints 1 and 2, respectively.

**Decision rule**: Reject $\text{H}_{0}$ if and only if:
$$Z_{1} > z_{1-\alpha} \text{ and } Z_{2} > z_{1-\alpha}$$

where $z_{1-\alpha}$ is the $(1-\alpha)$-th quantile of the standard normal distribution.

### Type I Error Control

The overall Type I error rate is:

$$\alpha_{\text{overall}} = \Pr(\text{Reject } \text{H}_{0} \mid \text{H}_{0} \text{ true})$$

Under the intersection-union test, this is automatically controlled at level $\alpha$ without adjustment:

$$\alpha_{\text{overall}} \leq \alpha$$

This is because rejecting when **both** statistics exceed the threshold is more conservative than rejecting when **either** statistic exceeds the threshold.

### Overall Power

Under the alternative hypothesis $\text{H}_{1}$, the overall power is:

$$1 - \beta = \Pr(Z_{1} > z_{1-\alpha} \text{ and } Z_{2} > z_{1-\alpha} \mid \text{H}_{1})$$

When $(Z_{1}, Z_{2})$ follow a bivariate normal distribution with correlation $\rho$:

$$1 - \beta = \Phi_{2}(-z_{1-\alpha} + \omega_{1}, -z_{1-\alpha} + \omega_{2} \mid \rho)$$

where:

- $\Phi_{2}(\cdot, \cdot \mid \rho)$ is the bivariate normal cumulative distribution function with correlation $\rho$
- $\omega_{1}$ and $\omega_{2}$ are the non-centrality parameters under $\text{H}_{1}$

### Impact of Correlation

The correlation $\rho$ between test statistics affects the overall power:

- **Positive correlation** ($\rho > 0$): Increases power and reduces required sample size
- **Zero correlation** ($\rho = 0$): Test statistics are independent
- **Negative correlation** ($\rho < 0$): Decreases power and increases required sample size

**Key insight**: Accounting for positive correlation between endpoints can lead to substantial sample size reductions (typically 5-15% for $\rho = 0.5$-0.8) compared to assuming independence.

## Supported Endpoint Types

The package supports five combinations of co-primary endpoints:

### 1. Two Continuous Endpoints

**Use case**: Trials measuring two continuous outcomes (e.g., systolic and diastolic blood pressure)

**Statistical model**: Both endpoints follow normal distributions

**Key functions**:

- `ss2Continuous()`: Sample size calculation
- `power2Continuous()`: Power calculation
- `twoCoprimary2Continuous()`: Unified interface for both

**Reference**: Sozu et al. (2011)

```{r two_continuous_example}
# Example: Two continuous endpoints with correlation rho = 0.5
ss2Continuous(
  delta1 = 0.5,  # Standardized effect size for endpoint 1
  delta2 = 0.5,  # Standardized effect size for endpoint 2
  sd1 = 1,       # Standard deviation for endpoint 1
  sd2 = 1,       # Standard deviation for endpoint 2
  rho = 0.5,     # Correlation between endpoints
  r = 1,         # Balanced allocation
  alpha = 0.025,
  beta = 0.2,
  known_var = TRUE
)
```

### 2. Two Binary Endpoints (Asymptotic Approximation)

**Use case**: Trials with two binary outcomes using normal approximation (large sample)

**Statistical model**: Binary endpoints with asymptotic normal approximation

**Key functions**:

- `ss2BinaryApprox()`: Sample size calculation
- `power2BinaryApprox()`: Power calculation
- `twoCoprimary2BinaryApprox()`: Unified interface for both

**Reference**: Sozu et al. (2010)

**Supported test methods**:

- AN: Asymptotic normal test without continuity correction
- ANc: Asymptotic normal test with continuity correction
- AS: Arcsine transformation without continuity correction
- ASc: Arcsine transformation with continuity correction

**When to use**: Large sample sizes (typically $N > 200$) and probabilities not too extreme ($0.1 < p < 0.9$)

```{r two_binary_approx_example}
# Example: Two binary endpoints
ss2BinaryApprox(
  p11 = 0.7, p12 = 0.6,  # Endpoint 1 and 2 in treatment group
  p21 = 0.4, p22 = 0.3,  # Endpoint 1 and 2 in control group
  rho1 = 0.5,            # Correlation in treatment group
  rho2 = 0.5,            # Correlation in control group
  r = 1,                 # Balanced allocation
  alpha = 0.025,
  beta = 0.2,
  Test = "AN"
)
```

### 3. Two Binary Endpoints (Exact Methods)

**Use case**: Small to medium sample sizes requiring exact inference

**Statistical model**: Binary endpoints with exact tests

**Key functions**:

- `ss2BinaryExact()`: Sample size calculation using exact tests
- `power2BinaryExact()`: Exact power calculation
- `twoCoprimary2BinaryExact()`: Unified interface for both

**Reference**: Homma and Yoshida (2025)

**Supported tests**:

- Chisq: Chi-squared test
- Fisher: Fisher's exact test (conditional test)
- Fisher-midP: Fisher's mid-p test
- Z-pool: Z-pooled exact unconditional test
- Boschloo: Boschloo's exact unconditional test

**When to use**: Small/medium samples ($N < 200$), extreme probabilities ($p < 0.1$ or $p > 0.9$), or when strict Type I error control is required

```{r two_binary_exact_example}
# Example: Exact methods for small samples
ss2BinaryExact(
  p11 = 0.7, p12 = 0.6,
  p21 = 0.4, p22 = 0.3,
  rho1 = 0.5, rho2 = 0.5,
  r = 1,
  alpha = 0.025,
  beta = 0.2,
  Test = "Fisher"  # or "Chisq", "Fisher-midP", "Z-pool", "Boschloo"
)
```

### 4. Mixed Continuous and Binary Endpoints

**Use case**: Trials with one continuous and one binary outcome

**Statistical model**: Normal distribution for continuous endpoint, Bernoulli for binary endpoint, with biserial correlation

**Key functions**:

- `ss2MixedContinuousBinary()`: Sample size calculation
- `power2MixedContinuousBinary()`: Power calculation
- `twoCoprimary2MixedContinuousBinary()`: Unified interface for both

**Reference**: Sozu et al. (2012)

**Supported test methods for binary endpoint**:

- AN: Asymptotic normal test without continuity correction
- ANc: Asymptotic normal test with continuity correction
- AS: Arcsine transformation without continuity correction
- ASc: Arcsine transformation with continuity correction
- Fisher: Fisher's exact test (simulation-based)

**Correlation structure**: Uses biserial correlation between observed continuous variable and latent continuous variable underlying the binary outcome

```{r mixed_cont_binary_example}
# Example: Continuous + Binary endpoints
ss2MixedContinuousBinary(
  delta = 0.5,           # Effect size for continuous endpoint
  sd = 1,                # Standard deviation
  p1 = 0.7,              # Success probability in treatment group
  p2 = 0.4,              # Success probability in control group
  rho = 0.5,             # Biserial correlation
  r = 1,
  alpha = 0.025,
  beta = 0.2,
  Test = "AN"
)
```

### 5. Mixed Count and Continuous Endpoints

**Use case**: Trials with overdispersed count data (e.g., exacerbations) and continuous outcomes (e.g., lung function)

**Statistical model**: Negative binomial distribution for count endpoint, normal distribution for continuous endpoint

**Key functions**:

- `ss2MixedCountContinuous()`: Sample size calculation
- `power2MixedCountContinuous()`: Power calculation
- `twoCoprimary2MixedCountContinuous()`: Unified interface for both
- `corrbound2MixedCountContinuous()`: Calculate valid correlation bounds

**Reference**: Homma and Yoshida (2024)

**Special considerations**:

- The negative binomial distribution accommodates overdispersion (variance $>$ mean) common in count data
- **Treatment effects must be in the negative direction for both endpoints**: Lower event rate (count endpoint) and lower/better continuous values indicate treatment benefit. For example, reduction in exacerbation rate (`r1 < r2`) and improvement in lung function (e.g., `mu1 < mu2` when lower is better, or larger negative change from baseline)

```{r mixed_count_cont_example}
# Example: Count (exacerbations) + Continuous (FEV1)
ss2MixedCountContinuous(
  r1 = 1.0, r2 = 1.25,   # Count rates (events per unit time)
  nu = 0.8,              # Dispersion parameter
  t = 1,                 # Follow-up time
  mu1 = -50, mu2 = 0,    # Continuous means
  sd = 250,              # Standard deviation
  rho1 = 0.5, rho2 = 0.5, # Correlations
  r = 1,
  alpha = 0.025,
  beta = 0.2
)
```

## Impact of Correlation

A key advantage of accounting for correlation between co-primary endpoints is the potential for **sample size reduction**.

The table below illustrates this for two continuous endpoints:

```{r correlation_impact}
# Sample size at different correlation levels
correlations <- c(0, 0.3, 0.5, 0.8)
results <- sapply(correlations, function(rho) {
  ss2Continuous(
    delta1 = 0.5, delta2 = 0.5,
    sd1 = 1, sd2 = 1,
    rho = rho, r = 1,
    alpha = 0.025, beta = 0.2,
    known_var = TRUE
  )$N
})

data.frame(
  Correlation = correlations,
  Total_N = results,
  Reduction = paste0(round((1 - results/results[1]) * 100, 1), "%")
)
```

As correlation increases, the required sample size decreases. At $\rho = 0.8$, approximately 11% reduction in sample size can be achieved compared to $\rho = 0$.

### Why Does Correlation Matter?

The correlation between endpoints affects the joint distribution of test statistics. When endpoints are positively correlated:

1. **Test statistics tend to move together**: If $Z_{1}$ is large, $Z_{2}$ is also likely to be large
2. **Higher probability of rejecting both nulls**: $\Pr(Z_{1} > c, Z_{2} > c)$ increases with $\rho$
3. **Sample size reduction**: Fewer subjects needed to achieve target power

Mathematically, for bivariate normal $(Z_{1}, Z_{2})$ with correlation $\rho$:

$$\Pr(Z_{1} > c, Z_{2} > c \mid \rho) > \Pr(Z_{1} > c, Z_{2} > c \mid \rho = 0)$$

when $\rho > 0$ and both endpoints have positive treatment effects.

## Choosing the Right Method

| Endpoint Types | Sample Size | Method | Key Considerations |
|----------------|-------------|--------|-------------------|
| Both continuous | Any | Asymptotic | Simple, well-established |
| Both binary | Large ($N > 200$) | Asymptotic | Fast computation, probabilities moderate |
| Both binary | Small/Medium | Exact | Better Type I error control, exact inference |
| 1 Continuous + 1 Binary | Any | Asymptotic | Handles mixed types, biserial correlation |
| 1 Count + 1 Continuous | Any | Asymptotic | Accounts for overdispersion |

### Decision Guidelines

**For binary endpoints**:

- Use **asymptotic methods** when: $N > 200$, $0.1 < p < 0.9$, computational efficiency important
- Use **exact methods** when: $N < 200$, extreme probabilities ($p < 0.1$ or $p > 0.9$), regulatory requirements for exact tests

**For mixed endpoint types**:

- Ensure correlation structure is appropriate (e.g., biserial for continuous-binary)
- Consider clinical plausibility of correlation magnitude
- Use conservative estimates if correlation is uncertain

## Sample Size Calculation Approach

All methods in this package follow a similar computational approach:

1. **Specify design parameters**: Effect sizes, probabilities, standard deviations, etc.
2. **Specify correlation**: Between endpoints
3. **Specify error rates**: Type I error $\alpha$ (typically 0.025 for one-sided) and Type II error $\beta$ (typically 0.2 for 80% power)
4. **Calculate sample size**: Using iterative algorithms
5. **Verify power**: Confirm that calculated sample size achieves target power

## Detailed Vignettes

For detailed methodology, examples, and validation against published results, please see:

1. `vignette("two-continuous-endpoints")`: Two continuous endpoints
2. `vignette("two-binary-endpoints-approx")`: Two binary endpoints (asymptotic)
3. `vignette("two-binary-endpoints-exact")`: Two binary endpoints (exact)
4. `vignette("mixed-continuous-binary")`: Mixed continuous and binary
5. `vignette("mixed-count-continuous")`: Mixed count and continuous

## References

Homma, G., & Yoshida, T. (2024). Sample size calculation in clinical trials with two co-primary endpoints including overdispersed count and continuous outcomes. *Pharmaceutical Statistics*, 23(1), 46-59.

Homma, G., & Yoshida, T. (2025). Exact power and sample size in clinical trials with two co-primary binary endpoints. *Statistical Methods in Medical Research*, 34(1), 1-19.

Sozu, T., Sugimoto, T., & Hamasaki, T. (2010). Sample size determination in clinical trials with multiple co-primary binary endpoints. *Statistics in Medicine*, 29(21), 2169-2179.

Sozu, T., Sugimoto, T., & Hamasaki, T. (2011). Sample size determination in superiority clinical trials with multiple co-primary correlated endpoints. *Journal of Biopharmaceutical Statistics*, 21(4), 650-668.

Sozu, T., Sugimoto, T., & Hamasaki, T. (2012). Sample size determination in clinical trials with multiple co-primary endpoints including mixed continuous and binary variables. *Biometrical Journal*, 54(5), 716-729.
