x̄ - > Confounding factors example in R

 In clinical trials, **confounding factors** may affect both the treatment and outcome, making it challenging to attribute the effects directly to the treatment. In an example trial where we are testing the effectiveness of penicillin, confounders might include **age, baseline health status, or prior antibiotic use**. Below is an example of R of controlling for confounding factors in analyzing the effectiveness of the treatment.

For this example, we will simulate a dataset where a confounder (e.g., age) influences both the likelihood of receiving penicillin treatment and the recovery outcome.

### Step 1: Simulate Data

We’ll simulate a dataset where:

- Patients receive either penicillin treatment or no treatment.

- The outcome variable indicates recovery.

- Age acts as a confounding factor affecting both treatment and recovery likelihood.

```r

# Load necessary library

library(dplyr)

# Set seed for reproducibility

set.seed(42)

# Simulate data

n <- 200  # Number of patients

age <- rnorm(n, mean = 50, sd = 12)  # Age as a confounding variable

treatment <- rbinom(n, 1, prob = 0.5 + 0.01 * (age - mean(age)))  # Treatment influenced by age

recovery <- rbinom(n, 1, prob = 0.4 + 0.1 * treatment + 0.01 * (age - mean(age)))  # Outcome influenced by treatment and age

# Create data frame

trial_data <- data.frame(

  age = age,

  treatment = factor(treatment, labels = c("No", "Yes")),

  recovery = factor(recovery, labels = c("No", "Yes"))

)

# Inspect the data

head(trial_data)

```

### Step 2: Examine Confounding Using Descriptive Statistics

To check if age is associated with both the treatment assignment and recovery, we can examine the average age by treatment and recovery groups.

```r

# Mean age by treatment

trial_data %>%

  group_by(treatment) %>%

  summarize(mean_age = mean(age), .groups = 'drop')

# Mean age by recovery

trial_data %>%

  group_by(recovery) %>%

  summarize(mean_age = mean(age), .groups = 'drop')

```

### Step 3: Assess Treatment Effect without Controlling for Confounding

First, we estimate the effect of treatment on recovery without adjusting for age. This approach could give biased results.

```r

# Fit a logistic regression model without confounders

model_unadjusted <- glm(recovery ~ treatment, data = trial_data, family = binomial)

# Display results

summary(model_unadjusted)

```

### Step 4: Control for Confounding Using Multivariable Logistic Regression

Now, we’ll include age as a covariate in the logistic regression model to control for its confounding effect.

```r

# Fit a logistic regression model with age as a confounder

model_adjusted <- glm(recovery ~ treatment + age, data = trial_data, family = binomial)

# Display results

summary(model_adjusted)

```

The coefficient for `treatment` in the adjusted model now represents the effect of penicillin on recovery, holding age constant.

### Step 5: Interpret the Results

Compare the coefficients for `treatment` in the unadjusted vs. adjusted models. If age is a true confounder, we may see a significant difference between the two models' `treatment` coefficients.

### Step 6: Visualize the Effects

To better understand the adjusted and unadjusted treatment effects, you can visualize them.

```r

# Load library for visualization

library(ggplot2)

# Predictions for visualization

trial_data$pred_unadjusted <- predict(model_unadjusted, type = "response")

trial_data$pred_adjusted <- predict(model_adjusted, type = "response")

# Plot

ggplot(trial_data, aes(x = age, color = treatment)) +

  geom_point(aes(y = as.numeric(recovery) - 1), alpha = 0.5) +

  geom_line(aes(y = pred_unadjusted, linetype = "Unadjusted")) +

  geom_line(aes(y = pred_adjusted, linetype = "Adjusted")) +

  labs(

    y = "Probability of Recovery",

    title = "Effect of Penicillin on Recovery with and without Adjustment for Age"

  ) +

  theme_minimal()

```

### Summary

1. **Simulate and explore** the relationship between treatment, recovery, and age.

2. **Fit unadjusted and adjusted models** to assess the confounding effect of age.

3. **Interpret results and visualize** the adjusted vs. unadjusted effects.

This approach helps clarify the true impact of penicillin on recovery by accounting for age as a confounding factor.