The main Cronbach Alpha program is presented in CronbachAlpha.php. This page provides 4 programs related to sample size for Cronbach's Alpha

• Sample Size: calculates sample size required from 5 parameters
  • Probability of Type I Error (α, p) to determine statistical significance. In most cases this would be p=0.05
  • Power (1 - β). In most cases this would be 0.8, corresponding to Probability of Type II Error (β) of 0.2
  • The number of items of measurements in the correlation matrix
  • The Cronbach Alpha the model is set up to detect
  • The reference Cronbacj Alpha. The algorithm is to compare two alpha, but in most cases the reference alpha = 0
• Power ( 1 - β): Calculates power of the data already collected from 5 parameters
  • Probability of Type I Error (α, p) to determine statistical significance. In most cases this would be p=0.05
  • The sample size in the data
  • The number of items of measurements in the correlation matrix
  • The Cronbach Alpha fround from the data
  • The reference Cronbacj Alpha. In most cases the reference alpha = 0
• Confidence Interval: Calculate the confidence interval of Cronbach Alpha from data already collected using 4 parameters
  • Percent of confidence. In most cases the 95% condfidence interval is used
  • The sample size in the data
  • The number of items of measurements in the correlation matrix
  • The Cronbach Alpha fround from the data
• Pilot Study: produces a table of confidence intervals from varying sample sizes. This requires 5 parameters
  • Percent of confidence. In most cases the 95% condfidence interval is used
  • The number of items of measurements in the correlation matrix
  • The Cronbach Alpha fround from the data
  • The interval of sample size to be tested. In most cases 2 - 5 is used
  • The maximum sample size to be tested. In most cases it is 100, as the precision required seldom need more than 30-70 cases in a pilot study

Use of the 4 calculations

The pilot study and sample size estimation are used in the planning stage of a research projecr. The sample size estimate is used when the researcher has some backgorund information for the estimate.

In a completely new project, where the environment of the project is unknown, the pilot study is required to compare the outcomes of different sample size, and determine when further increase in sample size no longer further improves the confidnce interval in a meaningful way

The power estimate and confidence interval are carried out at the end of a project, using the data already collected. The power estimate provides an index whether the sample size was adequate for the requirements of the model. The confidence intrval provides an estimate of the precision of the Alpha thus obtained.

One and two tail models

The programs provides results in both the one and two tail model. The two tail model is commonly presented, and this describes the range of Alpha for precision estimates and for comparison with other data.

In many cases, when Cronbach Alpha is used as a tool during the development of a multivariate measurement (such as when Factor Analysis or Multiple Regression is used), the researcher merely wish to know if a minimum value of Alpha can be achieved by the model. In these circumstances, the one tail model is more appropriate as this requires a smaller sample size.

References

Bonett D G (2002) Sample Size Requirements for Testing and Estimating Coefficient Alpha Journal of Educational and Behavioral Statisitcs 27: 335-340

Duhachek A and Iacobucci D (2004) Alpha's Standard Error (ASE): An Accurate and Precise Confidence Interval Estimate. Journal of Applied Psychology Vol. 89, No. 5, 792-808

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

Data

Sample Size : Data input a table of 5 columns   - Each row contains data from a separate study   - Col 1 = Probability of Type I Error α   - Col 2 = Power (1 - β)   - Col 3 = number of items in the group (questions in a factor)   - Col 4 = Cronbach Alpha required or anticipated   - Col 5 = Reference or hypothetical Cronbach Alpha, usually 0 for null hypothesis    Power : Data input a table of 5 columns   - Each row contains data from a separate study   - Col 1 = Probability of Type I Error α   - Col 2 = Sample size used   - Col 3 = number of items in the group (questions in a factor)   - Col 4 = Cronbach Alpha found   - Col 5 = Reference or hypothetical Cronbach Alpha, usually 0 for null hypothesis    Confidence Interval : Data input a table of 4 columns   - Each row contains data from a separate study   - Col 1 = Percent confidence (usually 95 or 99)   - Col 2 = Sample size used   - Col 3 = number of items in the group (questions in a factor)   - Col 4 = Cronbach Alpha found    Pilot Study : Data is for a single plan, a single column with 5 rows   - Row 1 : Percent confidence required, usually 95 or 99   - Row 2 : Number of items in the data   - Row 3 : Cronbach's Alpha to be detected   - Row 4 : Sample size interval for tabulation of changing width of the confidence interval.       This is usually established by trial and error, starting at about 5   - Row 5 : Maximum sample size or tabulation of changing width of the confidence interval.       This is usually established by trial and error. For most clinical studies however,       the sample size is between 10 and 200, and this range can be used as starting levels

Sample Size

# Program 1: Sample size
# subroutine functions to calculate delta, and ssiz (1 and 2 tail)
ssAlpha <- function(alpha, beta, k, calpha, ralpha, tail) # alpha and beta Type I & II error
{                          # k= number of items r alpha = proposed alpha calpha= hypothetical alpha
  delta = (1 - ralpha) / (1 - calpha)
  za = qnorm(alpha/tail)
  zb = qnorm(beta)
  n = ceiling((2 * k/(k - 1)) * (za + zb)^2 / log(delta)**2 + 2)
  ar <- c(delta,n)
  return (ar)
}
# Main program
# Sig = Probability of Type I Error
# Power = 1 - Probability of Type II Error
# k = number of items
# Alpha = Cronbach Alpha to detect
# Refalpha = reference alpha for comparison (0 unless specified otherwise)
dat = ("
Sig   Power k  Alpha Refalpha 
0.05  0.80  5  0.7   0.0
0.01  0.95  5  0.7   0.0
0.05  0.80  7  0.8   0.0
0.01  0.95  7  0.8   0.0
       ") 
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    
df   # optional display of input data 
delta <- vector()
ssiz1tail <- vector()
ssiz2tail <- vector()
for(i in 1 : nrow(df))
{
  sig = df$Sig[i]
  beta = 1 - df$Power[i]
  k = df$k[i]
  calpha = df$Alpha[i]
  ralpha = df$Refalpha[i]
  ssAr <- ssAlpha(sig,beta,k,calpha,ralpha,1)
  delta <- append(delta, ssAr[1])
  ssiz1tail <- append(ssiz1tail,ssAr[2])
  ssAr <- ssAlpha(sig,beta,k,calpha,ralpha,2)
  ssiz2tail <- append(ssiz2tail,ssAr[2])
}
delta
ssiz1tail
ssiz2tail
df$Delta <- delta
df$ssiz1Tail <- ssiz1tail
df$ssiz2Tail <- ssiz2tail
df  # show input data and ssiz results

Results are as follows

> df  # show input data and ssiz results
   Sig Power k Alpha Refalpha    Delta ssiz1Tail ssiz2Tail
1 0.05  0.80 5   0.7        0 3.333333        13        16
2 0.01  0.95 5   0.7        0 3.333333        30        33
3 0.05  0.80 7   0.8        0 5.000000         8        10
4 0.01  0.95 7   0.8        0 5.000000        17        19

Power

# Program 2 Power
# subroutine
betaAlpha <- function (alpha,n,k,calpha,ralpha,tail)  # alpha Type I error
{        # k= number of items r alpha = proposed alpha calpha= hypothetical alpha
  delta = (1 - ralpha) / (1 - calpha)
  za = abs(qnorm(alpha/tail))
  zb = log(delta) / sqrt(2 * k/((k - 1) * (n - 2))) - za
  pw = pnorm(abs(zb))
  ar <- c(delta,pw)
  return (ar)
}
# Main Power Program
# Sig = Probability of Type I Error
# n = sample size
# k = number of items
# Alpha = Cronbach Alpha to detect
# Refalpha = reference alpha for comparison (0 unless specified otherwise)
dat = ("
Sig   n   k  Alpha  Refalpha  
0.05  16  5  0.7    0.0
0.01  33  5  0.7    0.0
0.05  10  7  0.8    0.0
0.01  19  7  0.8    0.0
       ") 
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    
df   # optional display of input data
delta <- vector()
pw1tail <- vector()
pw2tail <- vector()
for(i in 1 : nrow(df))
{
  alpha = df$Sig[i]
  n = df$n[i]
  k = df$k[i]
  calpha = df$Alpha[i]
  rAlpha = df$Rfalpha[i]
  resAr <- betaAlpha(alpha,n,k,calpha,ralpha,1)
  delta <- append(delta, resAr[1])
  pw1tail <- append(pw1tail, resAr[2])
  resAr <- betaAlpha(alpha,n,k,calpha,ralpha,2)
  pw2tail <- append(pw2tail, resAr[2])
}
df$Delta <- delta
df$Pw1tail <- pw1tail
df$Pw2tail <- pw2tail
df # display input data and power results

The results of power analysis are

> df # display input data and power results
   Sig  n k Alpha Refalpha    Delta   Pw1tail   Pw2tail
1 0.05 16 5   0.7        0 3.333333 0.8857566 0.8130403
2 0.01 33 5   0.7        0 3.333333 0.9721438 0.9519234
3 0.05 10 7   0.8        0 5.000000 0.9091021 0.8461680
4 0.01 19 7   0.8        0 5.000000 0.9781969 0.9615009

Confidence Intervals

# Program 3 Confidence Interval
# subroutine function
ciAlpha <- function(pc,n,k,calpha,tail)  # pc = percent ci
{      # k= number of items cralpha = proposed alpha 
   za = abs(qnorm((1 - pc / 100) / tail))
   ll = 1 - exp(log(1 - calpha) + za * sqrt(2 * k / ((k - 1) * (n - 2))))
   ul = 1 - exp(log(1 - calpha) - za * sqrt(2 * k / ((k - 1) * (n - 2))))
   ar <- c(ll,ul)
   return (ar)
}
# Main Program
# Pc = Percent confidence (usually 95 or 99)
# n = Sample size used

# k = number of items in the group (questions in a factor)
# Alpha = Cronbach Alpha found
dat = ("
Pc  n   k  Alpha
95  16  5  0.7 
99  33  5  0.7 
95  10  7  0.8 
99  19  7  0.8
       ") 
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    
df   # optional display of input data                                                   # check input
low1tail <- vector()
high1tail <- vector()
low2tail <- vector()
high2tail <- vector()
for(i in 1 : nrow(df))
{
  pc = df$Pc[i]
  n = df$n[i]
  k = df$k[i]
  alpha = df$Alpha[i]
  resAr = ciAlpha(pc,n,k,alpha,1)
  low1tail <- append(low1tail,resAr[1])
  high1tail <- append(high1tail,resAr[2])
  resAr = ciAlpha(pc,n,k,alpha,2)
  low2tail <- append(low2tail,resAr[1])
  high2tail <- append(high2tail,resAr[2])
}
df$Low1Tail <- low1tail
df$High1Tail <- high1tail
df$Low2Tail <- low2tail
df$High2Tail <- high2tail
df # display input data and confidence interval results

The results are

> df # display input data and confidence interval results
  Pc  n k Alpha  Low1Tail High1Tail  Low2Tail High2Tail
1 95 16 5   0.7 0.3988407 0.8502893 0.3132170 0.8689542
2 99 33 5   0.7 0.4191916 0.8450436 0.3765498 0.8556420
3 95 10 7   0.8 0.5137904 0.9177310 0.4235902 0.9306049
4 99 19 7   0.8 0.5264862 0.9155252 0.4806340 0.9229830

Pilot Studies

# Pilot studies
# subroutine function
ciAlpha <- function(pc,n,k,calpha,tail)  # pc = percent ci
{      # k= number of items cralpha = proposed alpha 
  za = abs(qnorm((1 - pc / 100) / tail))
  ll = 1 - exp(log(1 - calpha) + za * sqrt(2 * k / ((k - 1) * (n - 2))))
  ul = 1 - exp(log(1 - calpha) - za * sqrt(2 * k / ((k - 1) * (n - 2))))
  ar <- c(ll,ul)
  return (ar)
}

# Main Program
#                      Input Data
pc = 95       # percent confidence
k = 5         # number of items
alpha = 0.7   # Cronbach Alpha to detect
intv = 5      # sample size intervals for testing
maxN = 100    # maximum sample size
#                 Vectors of results
SSiz <- vector()         # sample size
# 1 tail
CI1 <- vector()          # confidence interval 1 tail
Diff1 <- vector()        # difference from previous row 1 tail
CaseDec1 <- vector()  # decrease per case 1 tail
PcDec1 <- vector()       # % decrease per case 1 tail
# 2 tail
CI2 <- vector()          # confidence interval 2 tail
Diff2 <- vector()        # difference from previous row 2 tail
CaseDec2 <- vector()  # decrease per case 2 tail
PcDec2 <- vector()       # % decrease per case 2 tail
#                    First row 
n = intv
SSiz <- append(SSiz,n)
# 1 tail
resAr <- ciAlpha(pc,n,k,alpha,1)  # ll and ul of ci 1 tail
ci1 = resAr[2] - resAr[1]          # ci 1 tail
CI1 <- append(CI1,sprintf(ci1, fmt="%#.4f"))         # confidence interval 1 tail
Diff1 <- append(Diff1,"")       # difference from previous row 1 tail
CaseDec1 <- append(CaseDec1,"")  # decrease per case 1 tail
PcDec1 <- append(PcDec1,"")       # % decrease per case 1 tail

# 2 tail
resAr <- ciAlpha(pc,n,k,alpha,2)  # ll and ul of ci 2 tail
ci2 = resAr[2] - resAr[1]          # ci 2 tail
CI2 <- append(CI2,sprintf(ci2, fmt="%#.4f"))         # confidence interval 2 tail
Diff2 <- append(Diff2,"")       # difference from previous row 2 tail
CaseDec2 <- append(CaseDec2,"")  # decrease per case 2 tail
PcDec2 <- append(PcDec2,"")       # % decrease per case 2 tail
#                   Subsequent rowa
while(n < maxN)
{
  n = n + intv
  SSiz <- append(SSiz,n)
  # 1 tail
  oldci1 = ci1
  resAr <- ciAlpha(pc,n,k,alpha,1)  # ll and ul of ci 1 tail
  ci1 = resAr[2] - resAr[1]          # ci 1 tail
  CI1 <- append(CI1,sprintf(ci1, fmt="%#.4f"))         # confidence interval 1 tail
  diff = oldci1 - ci1
  Diff1 <- append(Diff1,sprintf(diff, fmt="%#.4f")) # difference from previous row 1 tail
  decpercase = diff / intv
  CaseDec1 <- append(CaseDec1,sprintf(decpercase, fmt="%#.4f"))  # decrease per case 1 tail
  pcdec = decpercase / oldci1 * 100
  PcDec1 <- append(PcDec1,sprintf(pcdec, fmt="%#.1f"))       # % decrease per case 1 tail
  # 2 tail
  oldci2 = ci2
  resAr <- ciAlpha(pc,n,k,alpha,2)  # ll and ul of ci 2 tail
  ci2 = resAr[2] - resAr[1]          # ci 2 tail
  CI2 <- append(CI2,sprintf(ci2, fmt="%#.4f"))         # confidence interval 2 tail
  diff = oldci2 - ci2
  Diff2 <- append(Diff2,sprintf(diff, fmt="%#.4f"))       # difference from previous row 2 tail
  decpercase = diff / intv
  CaseDec2 <- append(CaseDec2,sprintf(decpercase, fmt="%#.4f"))  # decrease per case 2 tail
  pcdec = decpercase / oldci2 * 100
  PcDec2 <- append(PcDec2,sprintf(pcdec, fmt="%#.1f"))       # % decrease per case 2 tail
}
#        Combine all result vectors into single data frame for display
df <- data.frame(SSiz,CI1,Diff1,CaseDec1,PcDec1,CI2,Diff2,CaseDec2,PcDec2)
df # Results

The results are as follows. Please note:

• SSiz = sample size
• CI1 and CI2 = confidence intervals (1 and 2 tail)
• Diff1 and Diff 2 = difference in confidence interval from the previous row (1 and 2 tail)
• CaseDec1 and CaseDec2 = change in confidence interval per case increase (1 and 2 tail)
• PcDec1 and PcDec2 = % change in confidence interval per case increase (1 and 2 tail)

> df # Results
   SSiz    CI1  Diff1 CaseDec1 PcDec1    CI2  Diff2 CaseDec2 PcDec2
1     5 1.2797                        1.7453                       
2    10 0.6328 0.6469   0.1294   10.1 0.7970 0.9482   0.1896   10.9
3    15 0.4713 0.1615   0.0323    5.1 0.5816 0.2155   0.0431    5.4
4    20 0.3913 0.0800   0.0160    3.4 0.4783 0.1033   0.0207    3.6
5    25 0.3416 0.0497   0.0099    2.5 0.4153 0.0630   0.0126    2.6
6    30 0.3069 0.0346   0.0069    2.0 0.3718 0.0434   0.0087    2.1
7    35 0.2810 0.0259   0.0052    1.7 0.3396 0.0322   0.0064    1.7
8    40 0.2607 0.0203   0.0041    1.4 0.3145 0.0251   0.0050    1.5
9    45 0.2443 0.0165   0.0033    1.3 0.2942 0.0203   0.0041    1.3
10   50 0.2306 0.0137   0.0027    1.1 0.2774 0.0168   0.0034    1.1
11   55 0.2189 0.0116   0.0023    1.0 0.2632 0.0142   0.0028    1.0
12   60 0.2089 0.0100   0.0020    0.9 0.2509 0.0122   0.0024    0.9
13   65 0.2001 0.0088   0.0018    0.8 0.2403 0.0107   0.0021    0.9
14   70 0.1924 0.0077   0.0015    0.8 0.2308 0.0094   0.0019    0.8
15   75 0.1855 0.0069   0.0014    0.7 0.2224 0.0084   0.0017    0.7
16   80 0.1793 0.0062   0.0012    0.7 0.2149 0.0075   0.0015    0.7
17   85 0.1736 0.0056   0.0011    0.6 0.2081 0.0068   0.0014    0.6
18   90 0.1685 0.0051   0.0010    0.6 0.2018 0.0062   0.0012    0.6
19   95 0.1638 0.0047   0.0009    0.6 0.1961 0.0057   0.0011    0.6
20  100 0.1594 0.0043   0.0009    0.5 0.1909 0.0052   0.0010    0.5