This page presents 4 programs related to sample size requirements when comparing 2 proportions, and tables of these sample size.

The programs and tables on this page assumes that the difference between 2 proportions has a binomial distribution which can be converted into an approximate normal distribution, so is mainly for comparing risk difference using the Standard model.

As the risk difference (Exact model), risk ratio, and odds ratio have similar powers, the results from this page are also applicable to these models.

However, the use of odds ratio in the retrospective matched pair controlled studies is different, and sample size for these are separately calculated in sample size for matched paired control studies page

if a researcher wishes to use a smaller sample size than those produced by the programs on this page or its tables, he/she should use either the Chi Squares Test for the 2x2 contingency table or the Fisher's Exact Probability in the Compare two proportions page, as these do not assume a normal transformation of the difference between proportions. The requirements of these program are discussed in their own page

The following 4 programs are available on this page

• Sample Size requires the following input
  • The probability of Type I Error (p, α) that will be used to determine significance. This is usually 0.05, but 0.1, 0.01, 1nd 0.001 are also used
  • The power (1 - β) required. Usually this is 0.8, and sometimes 0.9
  • The expected proportions in the two groups (P1 and P2)
  The results is the sample size in each of the 2 groups, assuming that they are equal. Sample size for the 1 and 2 tail models are produced.

  For example, if α of 0.05 and power of 0.8 are required, and the two proportions are expected to be 5% (0.05) and 10% (0.1), then the sample size of 343 cases per group is required for the 1 tail model, and 435 for the 2 tail model

• Power requires the following imput
  • The probability of Type I Error (p, α) that will be used to determine significance. This is usually 0.05, but 0.1, 0.01, 1nd 0.001 are also used
  • The sample size and proportion found in group 1 (N1, P1) and group 2 (N2, P2)
  The results are the powers of the comparison, 1 and 2 tail models.

  For example, if α of 0.05 is to be used to determine statistical significance, the sample size in the two groups are (N1=76 and N2=78), and the two proportions found are 5% (P1=0.05) ans 17% (P2=0.17), then the powers of the comparison are 0.93 for the 1 tail model and 0.89 for the 2 tail model

• Confidence Interval (CI) reuires the following input
  • The percentage representing the level of confidence reqwuired. This is usually 95%, but sometimes 99% is used
  • The sample size and proportion found in group 1 (N1, P1) and group 2 (N2, P2)
  The results are the confidence interval of the difference, 1 and 2 tail

  For example, if the sample size in the two groups are (N1=76 and N2=78), and the two proportions found are 5% (P1=0.05) ans 17% (P2=0.17), then the 95% confidence intervals are

  • For the 1 tail model, > -0.20 (-20%) or <-0.04 (-4%)
  • For the two tail model, -0.22 (-22%) to -02 (-2%)
• Pilot Studies requires the following parameters "

  	1 Tail				2 Tail
  Ssiz	CI1	Diff	Dec/case	%Dec/cas	CI	Diff	Dec/case	%Dec/cas
  5	0.999				1.1904
  10	0.7064	0.2926	0.0585	6.0	0.8417	0.3487	0.0697	6.0
  15	0.5768	0.1296	0.0259	4.0	0.6873	0.1545	0.0309	4.0
  20	0.4995	0.0773	0.0155	3.0	0.5952	0.0921	0.0184	3.0
  25	0.4468	0.0527	0.0105	2.0	0.5324	0.0628	0.0126	2.0
  30	0.4078	0.0389	0.0078	2.0	0.486	0.0464	0.0093	2.0
  35	0.3776	0.0303	0.0061	1.0	0.4499	0.036	0.0072	1.0
  40	0.3532	0.0244	0.0049	1.0	0.4209	0.0291	0.0058	1.0
  45	0.333	0.0202	0.004	1.0	0.3968	0.0241	0.0048	1.0
  50	0.3159	0.0171	0.0034	1.0	0.3764	0.0204	0.0041	1.0

  • The % of confidence, which is usually 95 or 99%. This is converted to probability of Type I error (α)
  • The expected proportions in the two groups (P1 and P2)
  • The interval of sample size (intv) to exaamine changes in confidence interval as sample sizw=es increase. Usually this is between 3 and 10 cases
  • The maximum sample size per group for the estimates. In most cases, pilot studies end in 30 to 40 cases, and there is no point having a pilot study with more than 100 cases per group. A common value used is 50
  The program produces a table as shown to the right, listing the confidence intervals (1 and 2 tails). With increasing sample size, the reduction in confidence interval decreases, and it can be seen that beyond 30 cases per group, the further decrease in confidence interval become trivial. A conclusion can therefore be made that a sample size of 30 cases per group would be suitable for a pilot study, to define the expected proportions, and provide information on the research environment, so that a formal comparison can be planned.

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

Johanson GA and Brooks GP (2010) Initial Scale Development: Sample Size for Pilot Studies. Educational and Psychological Measurement Vol.70,Iss.3;p.394-400

This subpanel presents tables of sample size (per group) comparing two unpaired proportions, for the following combinations.

• Powers ( 1 - β) of 0.8, 0.9, and 0.99
• Probability of Type I Error (α) of 0.1, 0.05, 0.01, and 0.001
• One and two tail models
• p1 and p2 are the proportions in the two groups to be detected

Sample size calculated to be less than 1/proportion (ssiz<1/p1 or ssiz<1/p2) are not included in the table, as a single positive case in the group will then exceed the proportion being compared.

Sample size for two proportions are used mostly in the Risk Difference model, although it is also used for Risk Ratio comparisons.
Power=0.8

Power=0.9 Power=0.95

Input Data

Data Input for Sample Size Estimation: Table in 4 columns   - Each row contains data from a separate study   - Col 1 = probability of Type I error (alpha)   - Col 2 = power (1-beta)   - Col 3, 4 = proportion expected in grp 1 & 2    Data Input for Power Estimation Table in 5 columns   - Each row contains data from a separate study   - Col 1 = probability of Type I error (alpha)   - Col 2, 4 = sample size in group 1 & 2   - Col 3, 5 = proportion found in group 1 & 2    Data Input for Confidence Interval Study Table in 5 columns   - Each row contains data from a separate study   - Col 1 = percent confidence (usually 95)   - Col 2, 4 = sample size in group 1 & 2   - Col 3, 5 = proportion found in group 1 & 2    Data Input for Pilot Study is for a single plan. Single column, 5 rows   - Row 1 : Percent Confidence required, usually 95 or 99   - Row 2, 3 : Planned Proportion in Group 1 & 2   - Row 4 : Sample Size (per group) Interval, usually 5   - Row 5 : Maximum Sample size (per group), usually 50

Sample Size

#  Sample Size
# data entry
dat = ("
Alpha Power    P1    P2
 0.05   0.8  0.05   0.1
 0.01   0.8  0.05   0.1
 0.05   0.9  0.05   0.1
 0.01   0.9  0.05   0.1
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame 
# vectors for results   
SSiz1Tail <- vector()
SSiz2Tail <- vector()
# Calculate sample size
for(i in 1 : nrow(df))
{
  alpha = df$Alpha[i]
  beta = 1 - df$Power[i]
  p1 = df$P1[i]
  p2 = df$P2[i]
  ra = 1
  pm = (p1 + ra * p2) / (1 + ra) # mean prop
  delta = abs(p1 - p2)
  zb = qnorm(beta)
  za = qnorm(alpha / 1)  # 1 tail za
  top = za * sqrt((1 + ra) * pm * (1 - pm)) + zb * sqrt(ra * p1 * (1 - p1) + p2 * (1 - p2))
  SSiz1Tail <- append(SSiz1Tail, ceiling(top^2 / (ra * delta^2)))
  za = qnorm(alpha / 2)  # 2 tail za
  top = za * sqrt((1 + ra) * pm * (1 - pm)) + zb * sqrt(ra * p1 * (1 - p1) + p2 * (1 - p2))
  SSiz2Tail <- append(SSiz2Tail, ceiling(top^2 / (ra * delta^2)))
}
df$SSiz1Tail <- SSiz1Tail
df$SSiz2Tail <- SSiz2Tail
df # input data and sample size per group

The results are sample size per group for 1 and 2 tail models

> df # input data and sample size per group
  Alpha Power   P1  P2 SSiz1Tail SSiz2Tail
1  0.05   0.8 0.05 0.1       343       435
2  0.01   0.8 0.05 0.1       556       647
3  0.05   0.9 0.05 0.1       474       582
4  0.01   0.9 0.05 0.1       721       824

Power

#           Power
# data entry
dat = ("
Alpha N1  P1    N2  P2
0.05	76	0.05	78	0.17
0.01	113	0.05	115	0.22
0.05	101	0.05	100	0.21
0.01	143	0.05	140	0.19
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame  
df
# Vectors for results
Power1Tail <- vector()
Power2Tail <- vector()
# Calculations
for(i in 1 : nrow(df))
{
  alpha = df$Alpha[i]
  n1 = df$N1[i]
  p1 = df$P1[i]
  n2 = df$N2[i]
  p2 = df$P2[i]
  ra = n2 / n1
  pm = (p1 + ra * p2) / (1 + ra)   # mean prop
  delta = abs(p1 - p2)
  za = abs(qnorm(alpha / 1))  # 1 tail
  zb = (delta * sqrt(ra * n1) - za * sqrt((1 + ra) * pm * (1 - pm))) / sqrt(ra * p1 * (1 - p1) + p2 * (1 - p2))
  Power1Tail <- append(Power1Tail,pnorm(zb))
  za = abs(qnorm(alpha / 2))  # 2 tail
  zb = (delta * sqrt(ra * n1) - za * sqrt((1 + ra) * pm * (1 - pm))) / sqrt(ra * p1 * (1 - p1) + p2 * (1 - p2))
  Power2Tail <- append(Power2Tail,pnorm(zb))
}
# include results to data frame for display
df$Power1Tail <- Power1Tail
df$Power2Tail <- Power2Tail
df # dataframw with data entry and results (power 1 and 2 tail)

The results are as follows

• Alpha = probability of Type I Error (α, p)
• N1, P1 = sample size and probability of group 1
• N2, P2 = sample size and probability of group 2
• Power = power, 1 and 2 tail

> df # dataframw with data entry and results (power 1 and 2 tail)
  Alpha  N1   P1  N2   P2 Power1Tail Power2Tail
1  0.05  76 0.05  78 0.17  0.7720891  0.6637205
2  0.01 113 0.05 115 0.22  0.9297388  0.8878224
3  0.05 101 0.05 100 0.21  0.9623354  0.9271517
4  0.01 143 0.05 140 0.19  0.9084133  0.8592515

Confidence Intervals

#         Confidence Interval
# data entry
dat = ("
Pc    N1    P1   N2       P2
95    76  0.05	 78	0.17
99   113  0.05	115	0.22
95   101  0.05	100	0.21
99   143  0.05	140	0.19
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame  
# df  # optional display of entry data
# Vectors for results
LL1 <- vector()  # lower limit 1 tail
UL1 <- vector()  # upper limit 1 tail
LL2 <- vector()  # lower limit 2 tail
UL2 <- vector()  # upper limit 2 tail
# Calculations
for(i in 1 : nrow(df))
{
  pc = df$Pc[i]        # % confidence
  prob = 1 - pc / 100  # alpha
  n1 = df$N1[i]
  p1 = df$P1[i]
  n2 = df$N2[i]
  p2 = df$P2[i]
  diff = p1 - p2
  se = sqrt(p1 * (1 - p1) / n1 + p2 * (1 - p2)/ n2)
  za = abs(qnorm(prob / 1))  # 1 tail
  LL1 <- append(LL1, diff - za * se)
  UL1 <- append(UL1, diff + za * se)
  za = abs(qnorm(prob / 2))  # 2 tail
  LL2 <- append(LL2, diff - za * se)
  UL2 <- append(UL2, diff + za * se)
}
df$LL1 <- LL1
df$UL1 <- UL1
df$LL2 <- LL2
df$UL2 <- UL2
df # Display entry data and result confidence intervals

The results are as follows

• Pc = % confidence
• N1, P1 = sample size and proportion of group 1
• N2, P2 = sample size and proportion of group 2
• LL1 and UL1 = lower and upper limits of confidence interval, 1 tail
• LL2 and UL2 = lower and upper limits of confidence interval, 2 tail

> df # Display entry data and result confidence intervals
  Pc  N1   P1  N2   P2        LL1         UL1        LL2         UL2
1 95  76 0.05  78 0.17 -0.2011494 -0.03885061 -0.2166955 -0.02330454
2 99 113 0.05 115 0.22 -0.2717369 -0.06826309 -0.2826473 -0.05735265
3 95 101 0.05 100 0.21 -0.2359006 -0.08409937 -0.2504412 -0.06955882
4 99 143 0.05 140 0.19 -0.2280163 -0.05198366 -0.2374554 -0.04254464

Pilot Studies

#              Pilot study
# Parameters
pc = 95         # % confidence
p1 = 0.677      # proportion grp 1
p2 = 0.413      # proportion grp 2
intv = 5        # interval
maxN = 100      # maximum sample size
# vectors for results
SSiz <- vector()     # sample size
CI1 <- vector()      # confidence interval 1 tail
Diff1 <- vector()    # difference in CI from previous row 1 tail
DecCase1 <- vector() # decrease in CI per case increase 1 tail
PDCase1 <- vector()  # % decrease in CI per case increase 1 tailCI1 <- vector()      # confidence interval 1 tail
CI2 <- vector()      # confidence interval 2 tail
Diff2 <- vector()    # difference in CI from previous row 2 tail
DecCase2 <- vector() # decrease in CI per case increase 2 tail
PDCase2 <- vector()  # % decrease in CI per case increase 2 tail
# Calculations
prob = 1 - pc / 100
z1 = abs(qnorm(prob / 1))  # 1 tail
z2 = abs(qnorm(prob / 2))  # 2 tail
# first row
n = intv
SSiz <- append(SSiz,n)
se = sqrt(p1 * (1 - p1) / n + p2 * (1 - p2)/ n)
ci1 = z1 * se * 2
CI1 <- append(CI1,sprintf(ci1, fmt="%#.4f"))      # confidence interval 1 tail
Diff1 <- append(Diff1,0)    # difference in CI from previous row 1 tail
DecCase1 <- append(DecCase1,0) # decrease in CI per case increase 1 tail
PDCase1 <- append(PDCase1,0)  # % decrease in CI per case increase 1 tailCI1 <- vector()      # confidence interval 1 tail
ci2 = z2 * se * 2
CI2 <- append(CI2,sprintf(ci2, fmt="%#.4f"))      # confidence interval 1 tail
Diff2 <- append(Diff2,0)    # difference in CI from previous row 1 tail
DecCase2 <- append(DecCase2,0) # decrease in CI per case increase 1 tail
PDCase2 <- append(PDCase2,0)  # % decrease in CI per case increase 1 tailCI1 <- vector()      # confidence interval 1 tail
# subsequent rows
while(n < maxN)
{
  n = n + intv
  SSiz <- append(SSiz,n)
  se = sqrt(p1 * (1 - p1) / n + p2 * (1 - p2)/ n)
  oldci1 = ci1
  ci1 = z1 * se * 2
  CI1 <- append(CI1,sprintf(ci1, fmt="%#.4f"))      # confidence interval 1 tail
  diff1 = oldci1 - ci1
  Diff1 <- append(Diff1,sprintf(diff1, fmt="%#.4f"))    # difference in CI from previous row 1 tail
  decCase1 = diff1 / intv
  DecCase1 <- append(DecCase1,sprintf(decCase1, fmt="%#.4f")) # decrease in CI per case increase 1 tail
  pDCase1 = sprintf(decCase1 / oldci1 * 100, fmt="%#.1f")
  PDCase1 <- append(PDCase1,pDCase1)  # % decrease in CI per case increase 1 tail

  oldci2 = ci2
  ci2 = z2 * se * 2
  CI2 <- append(CI2,sprintf(ci2, fmt="%#.4f"))      # confidence interval 2 tail
  diff2 = oldci2 - ci2
  Diff2 <- append(Diff2,sprintf(diff2, fmt="%#.4f"))    # difference in CI from previous row 2 tail
  decCase2 = diff2 / intv
  DecCase2 <- append(DecCase2,sprintf(decCase2, fmt="%#.4f")) # decrease in CI per case increase 2 tail
  pDCase2 = sprintf(decCase2 / oldci2 * 100, fmt="%#.1f")
  PDCase2 <- append(PDCase2,pDCase2)  # % decrease in CI per case increase 2 tail
}
df <- data.frame(SSiz,CI1,Diff1,DecCase1,PDCase1,CI2,Diff2,DecCase2,PDCase2)
df # display results in data frame

The results are as follows

• SSiz = sample size used (per group)
• CI1 and CI2 = confidence interval for that sample size (1 and 2 tail)
• Diff1 and Diff2 = difference in CI from previous row (1 and 2 tail)
• DecCase1 and DecCase2 = decrease in CI per case increase (1 and 2 tail)
• PDCase1 and PDCase2 = % decrease in CI per case increase, based on the previous row (1 and 2 tail)

> df # display results in data frame
   SSiz    CI1  Diff1 DecCase1 PDCase1    CI2  Diff2 DecCase2 PDCase2
1     5 0.9990      0        0       0 1.1904      0        0       0
2    10 0.7064 0.2926   0.0585     5.9 0.8417 0.3487   0.0697     5.9
3    15 0.5768 0.1296   0.0259     3.7 0.6873 0.1545   0.0309     3.7
4    20 0.4995 0.0773   0.0155     2.7 0.5952 0.0921   0.0184     2.7
5    25 0.4468 0.0527   0.0105     2.1 0.5324 0.0628   0.0126     2.1
6    30 0.4078 0.0389   0.0078     1.7 0.4860 0.0464   0.0093     1.7
7    35 0.3776 0.0303   0.0061     1.5 0.4499 0.0360   0.0072     1.5
8    40 0.3532 0.0244   0.0049     1.3 0.4209 0.0291   0.0058     1.3
9    45 0.3330 0.0202   0.0040     1.1 0.3968 0.0241   0.0048     1.1
10   50 0.3159 0.0171   0.0034     1.0 0.3764 0.0204   0.0041     1.0
11   55 0.3012 0.0147   0.0029     0.9 0.3589 0.0175   0.0035     0.9
12   60 0.2884 0.0128   0.0026     0.9 0.3436 0.0153   0.0031     0.9
13   65 0.2771 0.0113   0.0023     0.8 0.3302 0.0135   0.0027     0.8
14   70 0.2670 0.0101   0.0020     0.7 0.3181 0.0120   0.0024     0.7
15   75 0.2579 0.0091   0.0018     0.7 0.3074 0.0108   0.0022     0.7
16   80 0.2498 0.0082   0.0016     0.6 0.2976 0.0098   0.0020     0.6
17   85 0.2423 0.0075   0.0015     0.6 0.2887 0.0089   0.0018     0.6
18   90 0.2355 0.0068   0.0014     0.6 0.2806 0.0081   0.0016     0.6
19   95 0.2292 0.0063   0.0013     0.5 0.2731 0.0075   0.0015     0.5
20  100 0.2234 0.0058   0.0012     0.5 0.2662 0.0069   0.0014     0.5