Prediction intervals

Introduction

• A prediction interval is a lower and an upper limit within which a future measurement of a sample taken from a population will fall (given a certain probability)
• It is based on what has been observed in a sample taken from the given population
• Prediction intervals therefor predict the distribution of individual future measurements
• Confidence intervals predict the distribution of estimates of parameters in the population (i.e. the mean), which can not inherently be observed (the whole population cannot be included in a sample so as to calculate the true parameter)
• Therefor, under the assumption of a normal distribution of a variable in a population, a confidence interval is used to estimate a true mean or standard deviation, or at least, the limits in between which it may be located
  • For a given confidence level, c %, if an experiment is repeated an infinite number of times, the true population parameter will be found within the intervals c% of the time
• A prediction intervals is simply concenrned with limits within which a future measurement for that variable will fall and expresses this as a percentage probability

Example explanation

• A serum total cholesterol value is taken from \(500\) patients after treatment with a new cholesterol-lowering drug
• The sample mean is \(180.7\) mg/dL, with a standard deviation of \(19.4\)
• The \(95\)%confidence interval for the mean is \(179\) to \(182.4\)
  • This means that repeating this experiments an infinite many times, with a random sample taken from the population each time, that in 95% of the cases, the true population parameter would correctly fall within the given limits
• The prediction interval for the mean is \(142.5\) to \(218.9\)
  • This means that with a confidence of \(95\)% a new measurement taken from a random sample wouyld be within these limits

The code

• Set pseudo-random
• Take \(500\) samples from a normal distribution with \(\mu = 180\) and \(\sigma^2 = 20\)
• Save the dataset as a data.frame

set.seed(123)
df <- data.frame(Cholesterol = round(rnorm(500,mean = 180,sd = 20),digits = 0))

• Mean and standard deviation

mean(df$Cholesterol)

## [1] 180.694

sd(df$Cholesterol)

## [1] 19.44285

• Confidence intervals

ci <- predict(lm(df$Cholesterol ~ 1),
              interval = "confidence")
ci[1,]

##      fit      lwr      upr 
## 180.6940 178.9856 182.4024

• Prediction interval (a warning will be stated)

pi <- predict(lm(df$Cholesterol ~ 1),
              interval = "predict")

## Warning in predict.lm(lm(df$Cholesterol ~ 1), interval = "predict"): predictions on current data refer to _future_ responses

pi[1,]

##      fit      lwr      upr 
## 180.6940 142.4559 218.9321