• Confidence intervals show the interval in which true mean of a population most likely lies.
• Most common confidence intervals are 90%, 95%, and 99%
• Let’s say we calculate a 95% two-sided confidence interval and get the result (3, 6). This means that:
  • Based on the sample data, the probability that the interval (3, 6) contains the true mean of the population, \(\mu\), is 0.95.

n: sample size (number of observations)

\(x_i\): each individual observation - (ex. if the observation values are 1, 5, 7, 3, 2, then \(x_2 = 5\))

sample mean: \(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\)

sample standard deviation: \(\sigma = \sqrt {\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2} {n}}\)

critical value: \(1 - \alpha = \text{Confidence Level}\)

We will be focusing on two-sided confidence intervals. Here are the necessary equations:

Lower bound: \(\hat{L} = \bar{x} - z_\frac{\alpha}{2}\frac{\sigma}{\sqrt{n}}\)

Upper bound: \(\hat{U} = \bar{x} + z_\frac{\alpha}{2}\frac{\sigma}{\sqrt{n}}\)

We will be using data regarding survival in patients with advanced lung cancer from the North Central Cancer Treatment Group.

First, let’s load the dataset we’ll be using:

library(survival)
data(cancer)

Find a 95% two-sided confidence interval for the mean survival time of cancer patients in this data set.

\(\bar{x}\) = 305.2324561

\(\sigma\) = 210.6455431

n = 228

\(1 - \alpha = .95\), so \(\alpha = .05\)

\(\hat{L}\) = 305.2324561 - \(z_\frac{\\.05}{2}\frac{210.6455431}{\sqrt{228}} = 277.7437328\)

\(\hat{U}\) = 305.2324561 + \(z_\frac{\\.05}{2}\frac{210.6455431}{\sqrt{228}} = 332.7211795\)

Conclusion: Based on the sample data, the probability that the interval (277.7437, 332.7212) contains the true mean, \(\mu\), of survival time of cancer patients, is 0.95.

mean = mean(cancer$time)
std = sd(cancer$time)
n = nrow(cancer)
result = t.test(cancer$time)
L = result$conf.int[1]
U = result$conf.int[2]

The Karnofsky performance score is a scale of 0 to 100, where bad=0, and good=100. In this boxplot, you can see the physicians’ scores of patients.

ph.ecog is the ECOG performance score as rated by the physician

• 0 = asymptomatic
• 1 = symptomatic but completely ambulatory
• 2 = in bed <50% of the day
• 3 = in bed > 50% of the day but not bedbound
• 4 = bedbound

It is important to note that there are no patients who scored a 4 in this dataset.

ggplot(cancer, aes(x=ph.ecog)) + 
  geom_bar(stat="count", width=0.7, fill="pink") +
  theme_minimal() + 
  labs(title = "Number of Patients per ECOG Performance Score",
       x = "ECOG Performance Score",
       y = "Number of Patients")