Introduction

On this page however, the sample size estimations for these proportions are modified to suit the needs of the prediction environment, and they are as follows

Sample size for a Single Group

• Type I Error (α), commonly 0.05 is used
• Power, 1 - Type II Error (1 - β), commonly 0.8 is used
• The expected or nominated rate, and this should be > 0.5

Please Note: If required, the sample size for False Positev Rate (FPR = 1 - TNR) is the same as that for True Negative Rate (TNR), and for False Negative Rate (FNR = 1 - TPR) the same as that for True Positve Rate (TPR)

Sample Size for comparing rates from two groups

More recently Casagrande et.al. suggested an improved sample size calculation that provides greater precision, which allows both paired and unpaired comparisons. This algorithm is presented on this page.

Unpaired comparisons is used to compare two groups of unrelated individuals. An example may be to compare the True Positive Rate (TPR) of the mother feeling decreased fetal movement as a predictor of impending stillbirth between one group with first pregnancies and another group who had a baby before. The sample size calculated is the number of subjects that are outcome positive (OPos, stillbirths in this case) needed in each of the groups.

Paired comparisons is used to compare two tests or predictors, when both are administered to the same individual to predict the same outcome. An example is to compare the True Positive Rates (TPR) of the mother feeling decreased fetal movement, and that of an ultrasound detection of abnormal blood flow pattern, as predictors of impending stillbirth. The pair of tests can be administered to the same pregnant woman, and the qualities of the tests compared against the outcome.

Paired comparison is very much more powerful, as it reduces or eliminates variations between individuals. The sample size required pertains to the number of subjects that received both tests and are outcome positive, the number of matched pairs.

Two sample sizes are calculated for paired comparisons, the minimum and the maximum. In theory, the correct sample size is somewhere between the minimum and the maximum, depending on the correlation (agreeing with each other) between the tests. In practice, a conclusion that a statistically significant difference exists can be drawn if this is demonstrated when the sample size reaches or exceeds the minimum, but a conclusion that there is no significant difference can only be drawn after the maximum sample size has been reached.

The sample size using Casagrande's algorithm is based on the two tail mode

References

Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 18-20, p. 254-255

Beam, C. A. (1992), "Strategies for Improving Power in Diagnostic Radiology Research," American Journal of Radiology, 159, 631-637.

Casagrande, J. T., Pike, M. C., and Smith, P. G. (1978), "An Improved Approximate Formula for Calculating Sample Sizes for Comparing Two Binomial Distributions," Biometrics, 34, 483-486.

# Pgm1: Single geoup TPR or TNR > 0.5
#Note: Rate must be >0.5
myDat = ("
Alpha Power Rate
0.05  0.8   0.55
0.01  0.8   0.60
0.01  0.95  0.85
         ") 
myDat
df <- read.table(textConnection(myDat),header=TRUE)  # conversion to data frame    
df      # display input data (true positive, false positive, false negative, and true negative)

SSiz <- vector()     # sample size
for(i in 1:nrow(df))
{
  za = qnorm(df$Alpha[i])
  zb = qnorm(1 - df$Power[i])
  pn = df$Rate[i]
  p0 = 0.5;
  top = za * sqrt(p0*(1-p0)) + zb * sqrt(pn*(1-pn));
  bot = pn - p0;
  SSiz <- append(SSiz, ceiling((top*top) / (bot*bot)))
}
SSiz
df$SSiz <- SSiz
df    # display data frame including calculated sample size

> df    # display data frame including calculated sample size
  Alpha Power Rate SSiz
1  0.05  0.80 0.55  617
2  0.01  0.80 0.60  249
3  0.01  0.95 0.85   26

# Pgm2: SSiz comparing two rates
myDat = ("
Alpha  Power Rate1  Rate2
0.05   0.8   0.8    0.7
0.01   0.8   0.8    0.7
0.05   0.9   0.8    0.7
0.01   0.9   0.8    0.7        ") 
myDat
df <- read.table(textConnection(myDat),header=TRUE)  # conversion to data frame    
df      # display True Positive Rateate and False Positive Rate
SSizU <- vector()   # Sample Size per group for unpaired comparison
SSizPMin <- vector()   # Minimum Sample Size (pairs) for paired comparison
SSizPMax <- vector()   # Maximum Sample Size (pairs) for paired comparison
for(i in 1:nrow(df))
{
  za = qnorm(df$Alpha[i])
  zb = qnorm(1 - df$Power[i])
  r1 = df$Rate1[i]
  r2 = df$Rate2[i]
  Phat = (r1 + r2) / 2.0
  q1 = 1.0 - r1
  q2 = 1.0 - r2
  Qhat = (q1 + q2) / 2.0
  a = za * sqrt(2.0 * Phat * Qhat) + zb * sqrt(r1 * q1 + r2 * q2)
  a = a * a
  dif = abs(r1-r2);
  SSizU <- append(SSizU, ceiling(a * (1 + sqrt(1 + 4 * dif / a))^2 / (4 * dif * dif)))
  phimax = r1 * (1 - r2) + r2 * (1 - r1);
  SSizPMin <- append(SSizPMin, ceiling((za * sqrt(dif) + zb * sqrt(dif - dif * dif))^2 / (dif * dif)))
  SSizPMax <- append(SSizPMax, ceiling((za * sqrt(phimax) + zb * sqrt(phimax - dif * dif))^2 / (dif * dif)))
}
df$SSizU <- SSizU
df$SSizPMin <- SSizPMin
df$SSizPMax <- SSizPMax
df      # display data frame plus sample sizes for unpairec xomparison, plus min and max of paired comparison

> df      # display data frame plus sample sizes for unpairec xomparison, plus min and max of paired comparison
  Alpha Power Rate1 Rate2 SSizU SSizPMin SSizPMax
1  0.05   0.8   0.8   0.7   251       60      233
2  0.01   0.8   0.8   0.7   395       98      379
3  0.05   0.9   0.8   0.7   339       82      322
4  0.01   0.9   0.8   0.7   506      126      491