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1. Type I and Type II Error

1.1 Hypothesis test review

  • \(H_0\): \(\theta\)=\(\theta_0\)
  • \(H_a\): \(\theta \ne \theta_0\).
  • Given a test statistic, the null distribution of the statistic, and a significant level, we are able to determine if we have enough evidence in the data to reject the null hypothesis.


1.2 Errors

  • When testing hypothesis, there are two possible errors that we can make:
    • Type I error: Rejecting the null when the null hypothesis is true.
    • Type II error: Not rejecting the null when the null hypothesis is false.


Test vs. truth \(H_0\) is true \(H_0\) is false
Reject \(H_0\) Type I error Correct
Do not reject \(H_0\) Correct Type II error


  • In classical statistics, we want to minimize the type I error, by setting a threshold to limit it to a pre-specified acceptable value, say 5%.
  • The significance level of a test is the \(\alpha\) that we find acceptable for the probability of a type I error.
  • Good scientific and statistical practice is to set this value at the time you are designing the study, well before you look at any study data.
  • The definition of type I error (or the level of significance) is

\[ \alpha=\operatorname{Pr}\left(\text { Reject } H_{0} \mid H_{0} \text { is true }\right) \]

  • We can control the probability of a type I error by setting the significance level, \(\alpha\).

  • The definition of type II error is:

\[ \beta=\operatorname{Pr}\left(\text { Do not reject } H_{0} \mid H_{0} \text { is false }\right) \]

  • If we are just given a set of data, we can control the type I errors by setting the \(\alpha\) as we are planning the analysis, before we look at the data.
  • However in this case, we do not have control over the probability of a type II error and we just accept its value.
    • However, if we are designing a study from scratch, we can set both the type I error and type II error and determine the sample size needed to achieve these values.

Study design 단계에서부터 필요한 sample size를 구하고 들어오면, type I and type II error 모두 control 가능함


1.3 Power

  • Usually, we are not interested directly in the type II error, but in its complement,

\[ 1-\beta=\operatorname{Pr}\left(\text { Reject } H_{0} \mid H_{0} \text { is false }\right)=\text { power } \]

  • This is called the power of the test: the probability of correctly rejecting the null hypothesis when it is false.
Test vs. Truth \(H_0\) is true \(H_0\) is false
Reject \(H_0\) \(\alpha\) (Type I error) \(1-\beta\) (Power)
Do not reject \(H_0\) \(1-\alpha\) \(\beta\) (Type II error)


1.4 Illustration of type I and type II error

  • Can use density plots to illustrate the relationship of \(\alpha\), power \((1-\beta)\) and the null and alternative hypotheses.
curve(dnorm, from = -3, to = 3,
      xlim = c(-3, 6),
      lwd = 2, col = 4,
      axes = F, xlab = NA, ylab = NA)
curve(dnorm(x, mean = 3.2), 
      add = TRUE, col = 2, lwd = 2, from = 0, to = 6)
coord.x <- c(qnorm(0.95), seq(qnorm(0.95), 6, by = 0.01), 6)
coord.y <- c(0, dnorm(coord.x[-c(1, length(coord.x))], mean = 3.2), 0)
polygon(coord.x, coord.y, col = rgb(1, 0, 0, 0.5))
coord.x <- c(qnorm(0.95), seq(qnorm(0.95), 3, by = 0.01), 3)
coord.y <- c(0, dnorm(coord.x[-c(1, length(coord.x))]), 0)
polygon(coord.x, coord.y, col = rgb(0, 0, 1, 0.5))
abline(v = qnorm(0.95), lty = 2, lwd = 2)
text(3.2, 0.2, labels = expression(1 - beta))
text(2, 0.02, labels = expression(alpha), col = "white")
axis(1, at = c(0, 3.2), labels = c(expression(theta[0]), expression(theta[1])))
axis(2)
box()
legend("topleft",
       c("Distribution under H0",
         "Distribution under H1",
         "Critical value"), 
       bg = "transparent", 
       lty = c(1, 1, 2),
       col = c(4, 2, 1), lwd = 2)

curve(dnorm, from = -3, to = 3, 
      xlim = c(-3, 6), 
      lwd = 2, col = 4, axes = FALSE, 
      xlab = NA, ylab = NA)
curve(dnorm(x, mean = 3.2), 
      add = TRUE, col = 2, lwd = 2,
      from = 0, to = 6)
coord.x <- c(qnorm(0.95), seq(qnorm(0.95), 0, by = -0.01), 0)
coord.y <- c(0, dnorm(coord.x[-c(1, length(coord.x))], mean = 3.2), 0)
polygon(coord.x, coord.y, col = rgb(1, 0, 0, 0.5))
coord.x <- c(qnorm(0.95), seq(qnorm(0.95), 3, by = 0.01), 3)
coord.y <- c(0, dnorm(coord.x[-c(1, length(coord.x))]), 0)
polygon(coord.x, coord.y, col = rgb(0, 0, 1, 0.5))
abline(v = qnorm(0.95), lty = 2, lwd = 2)
text(1, 0.02, labels = expression(beta))
text(2, 0.02, labels = expression(alpha), col = "white")
axis(1, at = c(0, 3.2),
     labels = c(expression(theta_0), expression(theta_a)))
axis(2)
box()
legend("topleft",
       c("Type I error", "Type II error"),
       fill = c(rgb(0, 0, 1, 0.5),
                rgb(1, 0, 0, 0.5)))

curve(dnorm, from=-3, to=3, 
      xlim=c(-3, 6), lwd=2, col=4,
      axes=FALSE, xlab=NA, ylab=NA) 
curve(dnorm(x, mean=3.2), 
      add=TRUE, col=2, lwd=2,
      from=0, to=6) 
coord.x <- c(qnorm(0.9), seq(qnorm(0.9), 0, by=-0.01), 0) 
coord.y <- c(0, dnorm(coord.x[-c(1, length(coord.x))], mean=3.2), 0) 
polygon(coord.x, coord.y, col=rgb(1, 0, 0, 0.5)) 
coord.x <- c(qnorm(0.9), seq(qnorm(0.9), 3, by=0.01), 3) 
coord.y <- c(0, dnorm(coord.x[-c(1, length(coord.x))]), 0) 
polygon(coord.x, coord.y, col=rgb(0, 0, 1, 0.5)) 
abline(v = qnorm(0.9), lty=2, lwd=2) 
text(1, 0.02, labels = expression(beta)) 
text(2, 0.02, labels = expression(alpha), col = "white") 
axis(1, at = c(0, 3.2), labels = c(expression(theta_0), expression(theta_a))) 
axis(2) 
box() 
legend("topleft", 
       c("Type I error", "Type II error"), 
       fill = c(rgb(0, 0, 1, 0.5), 
                rgb(1, 0, 0, 0.5)))

curve(dnorm, 
      from=-3, to=3, 
      xlim=c(-3, 6), lwd=2, col=4, 
      axes=FALSE, xlab=NA, ylab=NA) 
curve(dnorm(x, mean=3.2), 
      add=TRUE, col=2, lwd=2, 
      from=0, to=6) 
coord.x <- c(qnorm(0.99), seq(qnorm(0.99), 0, by=-0.01), 0) 
coord.y <- c(0, dnorm(coord.x[-c(1, length(coord.x))], mean=3.2), 0) 
polygon(coord.x, coord.y, col=rgb(1, 0, 0, 0.5)) 
coord.x <- c(qnorm(0.99), seq(qnorm(0.99), 3, by=0.01), 3) 
coord.y <- c(0, dnorm(coord.x[-c(1, length(coord.x))]), 0) 
polygon(coord.x, coord.y, col=rgb(0, 0, 1, 0.5)) 
abline(v = qnorm(0.99), lty=2, lwd=2) 
text(1.6, 0.02, labels = expression(beta)) 
text(2.5, 0.015, labels = expression(alpha), col = "white") 
axis(1, at = c(0, 3.2), labels = c(expression(theta_0), expression(theta_a))) 
axis(2) 
box() 
legend("topleft", 
       c("Type I error", "Type II error"), 
       fill = c(rgb(0, 0, 1, 0.5), 
                rgb(1, 0, 0, 0.5)))



2. Example: Binomial distribution

  • simple random sample from a population with two outcomes (success and failure), with unknown probability of success \(\pi\)
y <- rbinom(100, 1, 0.5)
  • to test
    • \(H_0: \pi= \pi_0=0.5\)
    • \(H_a: \pi= \pi_a>0.5\) (one-sided)
  • Under \(H_0\), \(P\) has a rescaled \(B(100, \pi_0)\) distribution.
  • Under \(H_a\), \(P\) has a rescaled \(B(100, \pi_a)\) distribution.
  • (One-sided test) Rejection region: \((c, +\infty)\)
    • where \(c\) is called the critical value
  • Since the sampling distribution of \(P\) is a rescaled binomial, we can express \(c\) either on the proportion scale or on the number of successes scale.
  • Suppose we fix \(\alpha\), then the critical value can be computed from the binomial distribution:

\[ \operatorname{Pr}\left(Y>y_{1-\alpha} \mid H_{0}\right)=\alpha \]

  • When \(Y~B(n, \pi_0)\), then \(P=Y/n\)
  • The rejection region is 95th percentile of the binomial distribution.
(q <- qbinom(0.95, 100, 0.5))
## [1] 58
  • Therefore, the **rejection region on the number of successes scale is \(Y>58\) or on the sample proportion scale (\(P\) scale) is:

\[ \begin{array}{c}Y>58 \\ Y / 100>58 / 100 \\ P>0.58\end{array} \]

  • The plot of the null distribution and the critical value on the sample proportion scale
pi_0 <- 0.5 
n <- 100 
plot((20:80)/100,dbinom(20:80, n, pi_0),
     type = "h",lwd=2,
     main="Rejection region",
     xlab = "sample proportion",
     ylab="PMF") 
abline(v = 0.58, lty=2, lwd=2, col=3) 
lines((58:80)/100, dbinom(58:80,n, pi_0), 
      type = "h", lwd=4, col=4)
legend("topright", 
       c("Null distribution", "Critical value"), 
       lwd=2, lty=c(1, 2), col=c(1, 3))

  • The probability of the blue region is approximately 0.05.


2.1 Power

  • Power = \(1-\beta\)
  • In order to compute the power, we have to assume a specific value for the alternative distribution.
    • In this case, presume that the researchers believe that \(\pi_a=0.6\).
    • This corresponds to a sample proportion of 0.60 or a \(Y\) value of \(Y=100 \times \pi_a=60\).

\[ 1-\beta=\operatorname{Pr}\left(\text { Reject } H_{0}|H_{a})=\operatorname{Pr}\left(Y>58 \mid \pi_{a}=0.6\right)\right. \]

  • R: pbinom()
(1 - pbinom(58, 100, 0.6)) %>% round(3)
## [1] 0.623

  • Therefore, the chance we will get a statistically significant result (meaning a \(p\)-value < \(\alpha=0.05\)) **if the alternative hypothesis is true (i.e. under the alternative hypothesis) (e.g. \(\pi_a=0.60\)) is 0.623.

  • The plot of power, the sum of the red bars for the \(H_a\) PMF is the power

pi.0 <- 0.5
pi.a <- 0.6
n <- 100
plot((20:80)/100,dbinom(20:80, n, pi.0),
     type = "h",lwd=5, main="Power", 
     xlab = "sample proportion", ylab="PMF")
lines((20:80)/100,
      dbinom(20:80,n, pi.a),
      type = "h", lwd=2, col="orange")
lines((59:80)/100,
      dbinom(59:80,n, pi.a),
      type = "h", lwd=2, col="orangered3")
abline(v = 0.58, lty=2, lwd=4, col=3)
legend("topleft",
       c("Null distribution",
         "Alternative distribution",
         "Critical value"),
       lwd=2, lty=c(1, 1, 2),
       col=c(1, "orange", 3))

  • Conclusion: We have 62.3% power to reject \(H_0: \pi=0.5\) when the true population proportion is \(\pi=0.60\) with a sample size of \(n\) and a 5% (one-sided) level of significance.

  • R function to calculate the power: propTestPower() in the EnvStats package.

library(EnvStats)
## 
## Attaching package: 'EnvStats'
## The following objects are masked from 'package:stats':
## 
##     predict, predict.lm
## The following object is masked from 'package:base':
## 
##     print.default
  • To get the power using an exact binomial distribution: set the argument to approx = F in propTestPower().
    • approx = T will make the function use a normal approximation to the binomial to determine the power.
samplePower <- propTestPower(100,0.6, 0.5,
                             alpha=0.05,
                             sample.type = "one.sample",
                             alternative = "greater",
                             approx=FALSE)
samplePower
## $power
## [1] 0.6225327
## 
## $alpha
## [1] 0.04431304
## 
## $q.critical.upper
## [1] 58
  • Due to the discreteness of the binomial distribution, the \(\alpha\) is not exactly equal to the specified value.
  • Conclusion: Using one-sided level of significance of 0.05 with a sample size of \(n\), there is 62.3% probability to detect a \(\pi_a=0.60\). The critical value is 58, meaning that if we observe more than this many successes, 58, we would reject the null hypothesis, \(H_0\).
  • We DO NOT use the values of the sample in any of our calculations.
  • This means we can compute the power of a test BEFORE performing the experiment.
  • The power is actually determined during study design.
  • It is not particularly useful to determine power after you have your data.
  • The power associated with a sample can be better conveyed by presenting the confidence interval for your estimate.
  • If the confidence interval is wide, this means that the study did not have very high power.



3. Power

3.1 Power curve

  • So far, we have been interested in testing hypotheses, \(H_a: \pi>\pi_0\), where there are many values of \(\pi\) under \(H_a\).
    • Also, \(H_a:\pi<\pi_0\).
  • R: the power of a range of values of \(\pi_a\)
pi.a <- seq(0, 1, by = 0.01) 
power <- 1 - pbinom(qbinom(0.95,100,0.5), 100, pi.a)

# Power curve with the sample size of n=100 for a population proportion
plot(pi.a, power, 
     type='s', lwd=2,
     ylab=expression(1-beta),
     xlab=expression(pi))

  • Features of the power curve:
    • <1> the curve converges to \(\alpha=0.05\) when \(\pi\) gets closer to \(\pi_0=0.5\)
    • <2> the power increases with the distance between \(\pi\) and \(\pi_0\)
    • <3> the curve converges to 1 as \(\pi\) gets further from \(\pi_0\)


3.2 Power and sample size

  • Power depends on the sample size.
  • By increasing the number of observations in the sample, we can increase the power of the test (i.e., \(1-\beta\) or decreasing the Type II error).
  • The power of the test is important in order to understand what magnitude of effects we will be able to test.
  • In particular, this is important for studies that do not have a statistically significant result.
    • Because the studies that have a statistically significant reuslt already have enough power.
  • (\(p\)-value > \(\alpha\)) the result is said not to be statistically significant.
  • The question is
    • whether this is because \(H_0\) is true, or
    • because we did not have enough power to detect a difference of interest.
  • Often, researchers decide what power they need to test a given effect size and plan to collect a sample of a size \(n\) that will give them enough power.
  • R: the plot of the power of a test as a function of sample size
pi.a50 <- seq(0, 1, by = 0.02)
pi.a100 <- seq(0, 1, by = 0.01) 
pi.a200 <- seq(0, 1, by = 1/200) 

power50 <- 1 - pbinom(qbinom(0.95,50,0.5), 50, pi.a50)
plot(pi.a50, power50, type='s', ylab=expression(1-beta), xlab=expression(pi), lwd=2) 

power100 <- 1 - pbinom(qbinom(0.95,100,0.5), 100, pi.a100)
lines(pi.a100, power100, type='s', col=3, lwd=2) 

power200 <- 1 - pbinom(qbinom(0.95,200,0.5), 200, pi.a200) 
lines(pi.a200, power200, type='s', col=6, lwd=2) 

legend("bottomright", paste0("n = ", c(50,100,200)), lwd=2, col=c(1,3,6))


3.3 What do we know?

  • In general we know \(\pi_0\) and \(\alpha\).
  • \(\pi_0\) is often based on historical data and the research team determines the \(\alpha\) they want to use.
  • In order to compute this power, we need to specify two of the following three:
    • \(n\), \(\beta\), \(\pi_a\)
    • Usually, we want to know what is the power for a given \(\pi_a\) and \(n\) or
    • we want to know what sample size \(n\) we need to obtain a certain power, \(1-\beta\), and \(\pi_a\).


3.4 Effect size

  • Power depends only on:
    • \(n\), the sample size
    • the difference between the null and alternative proportions also known as effect size


3.5 Power and sample size (given \(\pi_a\) and \(n\))

  • What is the samllest sample size \(n\) that we need to achieve a given power?
    • Very common question in biostatistics
    • Because when we are testing experimental treatments on patients, we do not want to harm patients unnecessarily.


It is a moral obligation to:

  • <1> collect a sample size large enough to be able to prove an effect if there is one.
  • <2> collect a sample size small enough to avoid potentially harming many people.
  • This process is generally refeered to as power calculation
  • It involves making decisions on:
    • \(\alpha\): a significance level
    • \(\pi_a\): we want to be able to detect
    • \(1-\beta\): the power of the test


  • Example: suppose that we want a 80% chance to reject \(H_0: 0.5\) at \(\alpha=0.01\) if the true proportion is \(\pi_a=0.65\) with one-sided test. What is the smallest sample size \(n\) that we need?
  • The given information is:
    • \(1-\beta=0.80\)
    • \(\pi_a=0.5\)
    • \(\alpha=0.01\)
    • \(\pi_a=0.65\)
  • R: propTestN
sampleSize <- propTestN(0.65, 0.5, alpha = 0.01, power = 0.80,
                        sample.type = "one.sample",
                        alternative = "greater",
                        approx=FALSE) 
sampleSize
## $n
## [1] 114
## 
## $power
## [1] 0.817283
## 
## $alpha
## [1] 0.009405374
## 
## $q.critical.upper
## [1] 69
  • Set approx=F: the function compute the sample size using the exact binomial.
    • approx=T: the function uses a normal approximation to the binomial.
  • Since the function uses an exact binomial, it is not possible sometimes to get the exact power and \(\alpha\) that are stated.
    • This is because of the discreteness of binomial distribution.
  • The program determines the value as close as possible doing a search.
  • Conclusion: the required sample size would be 114 with one-sided \(alpha=0.01\) and power of 81.7%.


3.6 What is the minimum detectable difference? (given \(n\), power)

  • propTestMdd in the EnvStats package is the function to determine the minimum detectable difference if the sample size \(n\) is known and for a desired level of power.
  • e.g. Suppose we have a sample size 100 and want 90% power with \(alpha=0.05\), one-sided significance.
propDetectDiff <- propTestMdd(100, 0.50, alpha = 0.05, power = .9,
                              sample.type = "one.sample",
                              alternative = "greater",
                              approx=FALSE)
propDetectDiff
## $delta
## [1] 0.1465346
## 
## $power
## [1] 0.9
## 
## $alpha
## [1] 0.04431304
## 
## $q.critical.upper
## [1] 58
  • Since the function is using an exact binomial, it is not possible sometimes to get the exact \(alpha\) that was stated.
    • This is because of the discreteness of a binomial distribution.
  • The program determines the value as close as possible doing a search.
  • Conclusion: for a power of 90% and a sample size of 100, the minimum detectable difference is 0.15 with one-sided \(\alpha=0.01\).


3.7 Summing up

  • As the difference between means increases, power increases.
  • As the sample size increases, power increases.
  • As the significance level increases, power increases.


  • To compute the sample size, for a single population proportion using an exact binomial, use propTestN().
    • You need power, significance level \(\alpha\), and the difference to detect.
  • To compute the power for a sample of a single population proportion using an exact binomial, use propTestPower().
    • You need the sample size \(n\), significance level \(\alpha\), and the difference to detect.
  • To compute the minimal detectable difference for a single population proportion using an exact binomal, use propTestMdd().
    • You need the sample size \(n\), significance level \(\alpha\), and the power.




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