QUESTION_1

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University of Toronto *

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3000F

Subject

Industrial Engineering

Date

Dec 6, 2023

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pdf

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2

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Question 1 #a pow.z.test <- function (r,n,alpha,theta,sigma){ n2 <- n/(r+ 1 ) x <- qnorm( 1 -alpha/ 2 )-abs(theta)/(sigma*sqrt( 1 /(r*n2)+ 1 /n2)) pow <- 1 -pnorm(x) return(pow)} #b plot( x = seq(. 1 , 5 , by= 0.1 ), y = pow.z.test(seq(. 1 , 5 , by= 0.1 ), 168 ,. 05 , 1 , 2 ), type= "l" , xlab= "Allocation ratio" , ylab= "Power" , main= "Power vs. allocation ratio with total sample size fixed at \n n=168 alpha=0.05" ) #c total_sample_size <- 168 power <- 0.80 alpha <- 0.05 difference_to_detect <- 1 standard_deviation <- 2
# Define the function pow.z.test <- function (r, n, alpha, theta, sigma){ n2 <- n / (r + 1 ) x <- qnorm( 1 - alpha / 2 ) - abs(theta) / (sigma * sqrt( 1 / (r * n2) + 1 / n2)) pow <- 1 - pnorm(x) return(pow) } # Find the value of 'r' using the optimize function r <- optimize( function (r) abs(pow.z.test(r, total_sample_size, alpha, difference_to_detect, standard_deviation) - power), interval = c( 0 , 100 ))$minimum # Print the value of 'r' cat( "The ratio of patients in the experimental group to the control group (r) is:" , r, " \n " ) ## The ratio of patients in the experimental group to the control group (r) is: 3.019929 #d To increase power, the statistician should recommend modifying the allocation ratio. The current allocation ratio of 4:1 between the experimental and control groups may not be the most efficient distribution of subjects. The power can be increased by expanding the number of patients in the experimental group. The statistician can perform simulations with various allocation ratios to determine the optimal one that meets or exceeds the desired 80% power while keeping the total sample size constant at 168.
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