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Explanations
Javascript Program
R Codes
D
E
F
This page provides 4 programs used in the comparison of two Crnbach Alphas
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 effect size 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 the effect size 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 may have a minimum effect size in mind, and wishes to know if the two Alphas differ enough to stsatisfy this requirement. Under these considerations the one tail model is more appropriate as this requires a smaller sample size. ReferencesBonett D G (2002) Sample Size Requirements for Comparing Two Alpha Coefficients Applied Psychological Measurement, Vol. 27 No. 1, January 2003, 72 - 74Duhachek 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
Sample Size
Power
Confidence Interval
Pilot Studies
E
F
# Program 1: Sample Size (per Crobach Alpha)
# Subroutine
ssAlpha <- function(alpha,beta,k1,k2,delta,tail) # alpha and beta Type I & II error
{ # k1 and k2 number of items delta calculated from a1 and ca2
za = qnorm(alpha / tail)
zb = qnorm(beta)
n = ceiling(2 * (k1 /(k1 - 1) + k2 / (k2 - 1)) * (za + zb)**2 / log(delta)**2 + 2)
return (n)
}
# Main program
# Input Data
# Sig = Probability of Type I Error
# Power = 1 - Probability of Type II Error
# K1 = number of items in Cronbach Alpha 1
# Ca1 = Value of Cronbach Alpha 1
# K2 = number of items in Cronbach Alpha 2
# Ca2 = Value of Cronbach Alpha 2
dat = ("
Sig Power K1 Ca1 K2 Ca2
0.05 0.80 5 0.7 7 0.3
0.01 0.95 5 0.7 7 0.3
0.05 0.80 7 0.8 9 0.2
0.01 0.95 7 0.8 9 0.2 ")
df <- read.table(textConnection(dat),header=TRUE) # conversion to data frame
df # optional display of input data
# vectors for results
Delta <- vector()
SSiz_1 <- vector()
SSiz_2 <- vector()
# Calculations
for(i in 1 : nrow(df))
{
sig = df$Sig[i]
beta = 1 - df$Power[i]
k1 = df$K1[i]
ca1 = df$Ca1[i]
k2 = df$K2[i]
ca2 = df$Ca2[2]
delta = (1 - ca1) / (1 - ca2)
Delta <- append(Delta, delta)
# 1 tail
SSiz_1 <- append(SSiz_1,ssAlpha(sig,beta,k1,k2,delta,1))
# 2 tail
SSiz_2 <- append(SSiz_2,ssAlpha(sig,beta,k1,k2,delta,2))
}
# include into data frame for display
df$Delta <- Delta
df$SSiz_1 <- SSiz_1
df$SSiz_2 <- SSiz_2
df # Input data + results
The results are : Delta = Effect Size, SSiz_1 and SSiz_2 are sample size for each Cronbach Alpha, 1 taii and 2 tail
> df # Input data + results Sig Power K1 Ca1 K2 Ca2 Delta SSiz_1 SSiz_2 1 0.05 0.80 5 0.7 7 0.3 0.4285714 44 55 2 0.01 0.95 5 0.7 7 0.3 0.4285714 109 122 3 0.05 0.80 7 0.8 9 0.2 0.2857143 21 25 4 0.01 0.95 7 0.8 9 0.2 0.2857143 49 55
#Program 2: Power Estimates comparing 2 Cronbach Alphas
# Subroutines
# estimate sample size
ssAlpha <- function(alpha,beta,k1,k2,delta,tail) # alpha and beta Type I & II error
{ # k1 and k2 number of items delta calculated from a1 and ca2
za = qnorm(alpha / tail)
zb = qnorm(beta)
n = ceiling(2 * (k1 /(k1 - 1) + k2 / (k2 - 1)) * (za + zb)**2 / log(delta)**2 + 2)
return (n)
}
# iterate beta until sample size compatible
betaAlpha <- function(sig,n,k1,k2,delta,tail)
{
bL = 0
bR = 1
bM = (bL + bR) / 2
nM = 1
for(i in 1 : 100) # interation should not need more than 100 cycles
{
if(bM<0.00001 | bM>0.99999) return(c(resAr[0],bM))
nM = ssAlpha(sig,bM,k1,k2,delta,tail) - n
if(abs(nM)<0.0001) return(bM)
if(nM>0)
{
bL = bM
}
else
{
bR = bM
}
bM = (bL + bR) / 2;
}
return(bM)
}
# Main Program
# Input Data
# Sig = Probability of Type I Error
# n = sample size each Cronbach Alpha (assuming same sample size for both)
# K1 = number of items in Cronbach Alpha 1
# Ca1 = Value of Cronbach Alpha 1
# K2 = number of items in Cronbach Alpha 2
# Ca2 = Value of Cronbach Alpha 2
dat = ("
Sig n K1 Ca1 K2 Ca2
0.05 55 5 0.7 7 0.3
0.01 55 5 0.7 7 0.3
0.05 10 7 0.8 9 0.5
0.01 10 7 0.8 9 0.5
")
df <- read.table(textConnection(dat),header=TRUE) # conversion to data frame
df # optional display of input data
# vectors for results
Delta <- vector()
Power_1 <- vector()
Power_2 <- vector()
# Calculations
for(i in 1 : nrow(df))
{
sig = df$Sig[i]
n = df$n[i]
k1 = df$K1[i]
ca1 = df$Ca1[i]
k2 = df$K2[i]
ca2 = df$Ca2[i]
delta = (1 - ca1) / (1 - ca2)
Delta <- append(Delta,delta)
Power_1 <- append(Power_1, 1 - betaAlpha(sig,n,k1,k2,delta,1))
Power_2 <- append(Power_2, 1 - betaAlpha(sig,n,k1,k2,delta,2))
}
df$Delta <- Delta
df$Power_1 <- Power_1
df$Power_2 <- Power_2
df # display input data and reaults
The results are: Delta= Effect Size, Power_1 and Power_2 are powers (1 - β) of the model 1 and 2 tail
> df # display input data and reaults Sig n K1 Ca1 K2 Ca2 Delta Power_1 Power_2 1 0.05 55 5 0.7 7 0.3 0.4285714 0.8750000 0.7968750 2 0.01 55 5 0.7 7 0.3 0.4285714 0.6796875 0.5859375 3 0.05 10 7 0.8 9 0.5 0.4000000 0.3125000 0.2187500 4 0.01 10 7 0.8 9 0.5 0.4000000 0.1250000 0.0781250
# Program 3: Confidence Intervals
# Subroutine
CiAlpha <- function(pc,n,k1,k2,delta,tail) # pc = percent ci, n = sample size
{ # k1 k2 = number of items, delta = effect size
logDelta = log(delta)
prob = (1 - pc / 100) / tail
za = qnorm(1 - prob)
logci = abs(logDelta) * za * sqrt(2 * k1 / ((k1 - 1) * (n - 2)) + 2 * k2 / ((k2 - 1) * (n - 2)))
LL = exp(logDelta - logci)
UL = exp(logDelta + logci)
print(c(LL, UL, UL-LL))
return (c(LL, UL, UL-LL))
}
# Main program confidence interval
# Input Data
# Pc = pecent confidence usiually 90 95 or 99
# n = sample size each Cronbach Alpha (assuming same sample size for both)
# K1 = number of items in Cronbach Alpha 1
# Ca1 = Value of Cronbach Alpha 1
# K2 = number of items in Cronbach Alpha 2
# Ca2 = Value of Cronbach Alpha 2
dat = ("
Pc n K1 Ca1 K2 Ca2
95 55 5 0.7 7 0.3
99 55 5 0.7 7 0.3
95 20 7 0.8 8 0.5
99 20 7 0.8 8 0.5
")
df <- read.table(textConnection(dat),header=TRUE) # conversion to data frame
df # optional display of input data
# vectors for results Lower and Upper limit of confidence interval 1 and 2 tail
Delta <- vector()
Lower_1 <- vector()
Upper_1 <- vector()
Lower_2 <- vector()
Upper_2 <- vector()
# Calculations
for(i in 1 : nrow(df))
{
pc = df$Pc[i]
n = df$n[i]
k1 = df$K1[i]
ca1 = df$Ca1[i]
k2 = df$K2[i]
ca2 = df$Ca2[i]
delta = (1 - ca1) / (1 - ca2)
Delta <- append(Delta, delta)
# one tail
resAr = CiAlpha(pc,n,k1,k2,delta,1)
Lower_1 <- append(Lower_1, resAr[1])
Upper_1 <- append(Upper_1, resAr[2])
# two tail
resAr = CiAlpha(pc,n,k1,k2,delta,2)
Lower_2 <- append(Lower_2, resAr[1])
Upper_2 <- append(Upper_2, resAr[2])
}
# Pool results in data frame for disolay
df$Delta <- Delta
df$Lower_1 <- Lower_1
df$Upper_1 <- Upper_1
df$Lower_2 <- Lower_2
df$Upper_2 <- Upper_2
df # show input data and results
The results are as follows
> df # show input data and results. Pc n K1 Ca1 K2 Ca2 Delta Lower_1 Upper_1 Lower_2 Upper_2 1 95 55 5 0.7 7 0.3 0.4285714 0.2813464 0.6528375 0.2595525 0.7076545 2 99 55 5 0.7 7 0.3 0.4285714 0.2363259 0.7772041 0.2217114 0.8284348 3 95 20 7 0.8 8 0.5 0.4000000 0.1864158 0.8582965 0.1610502 0.9934791 4 99 20 7 0.8 8 0.5 0.4000000 0.1358638 1.1776497 0.1210075 1.3222323
# Pilot Studies comparing 2 Cronbach Alphas
# Subroutine function
CiAlpha <- function(pc,n,k1,k2,delta,tail) # pc = percent ci, n = sample size
{ # k1 k2 = number of items, delta = effect size
logDelta = log(delta)
prob = (1 - pc / 100) / tail
za = qnorm(1 - prob)
logci = abs(logDelta) * za * sqrt(2 * k1 / ((k1 - 1) * (n - 2)) + 2 * k2 / ((k2 - 1) * (n - 2)))
LL = exp(logDelta - logci)
UL = exp(logDelta + logci)
print(c(LL, UL, UL-LL))
return (c(LL, UL, UL-LL))
}
# Main Program
# Input Data
pc = 95 # percent confidence
k1 = 5 # number of items in alpha 1
ca1 = 0.7 # number of items in alpha 1
k2 = 7 # number of items in alpha 2
ca2 = 0.3 # number of items in alpha 2
intv = 5 # sample size intervals for testing
maxN = 100 # maximum sample size
delta = (1 - ca1) / (1 - ca2) # delta effect 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,k1,k2,delta,1)
print(resAr)
ci1 = resAr[3] # 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,k1,k2,delta,2)
print(resAr)
ci2 = resAr[3] # 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,k1,k2,delta,1)
ci1 = resAr[3] # 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,k1,k2,delta,2)
ci2 = resAr[3] # 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 table
The results are as follows. Please note:
df # Results SSiz CI1 Diff1 CaseDec1 PcDec1 CI2 Diff2 CaseDec2 PcDec2 1 5 2.4405 3.4754 2 10 1.1211 1.3194 0.2639 10.8 1.4403 2.0351 0.4070 11.7 3 15 0.8193 0.3018 0.0604 5.4 1.0241 0.4162 0.0832 5.8 4 20 0.6742 0.1450 0.0290 3.5 0.8321 0.1920 0.0384 3.7 5 25 0.5856 0.0886 0.0177 2.6 0.7174 0.1147 0.0229 2.8 6 30 0.5245 0.0611 0.0122 2.1 0.6395 0.0780 0.0156 2.2 7 35 0.4792 0.0454 0.0091 1.7 0.5822 0.0573 0.0115 1.8 8 40 0.4438 0.0354 0.0071 1.5 0.5379 0.0443 0.0089 1.5 9 45 0.4152 0.0286 0.0057 1.3 0.5023 0.0356 0.0071 1.3 10 50 0.3915 0.0237 0.0047 1.1 0.4729 0.0294 0.0059 1.2 11 55 0.3715 0.0201 0.0040 1.0 0.4481 0.0248 0.0050 1.0 12 60 0.3542 0.0173 0.0035 0.9 0.4268 0.0213 0.0043 0.9 13 65 0.3392 0.0151 0.0030 0.9 0.4083 0.0185 0.0037 0.9 14 70 0.3259 0.0133 0.0027 0.8 0.3920 0.0163 0.0033 0.8 15 75 0.3140 0.0118 0.0024 0.7 0.3775 0.0145 0.0029 0.7 16 80 0.3034 0.0106 0.0021 0.7 0.3645 0.0130 0.0026 0.7 17 85 0.2937 0.0096 0.0019 0.6 0.3528 0.0117 0.0023 0.6 18 90 0.2850 0.0088 0.0018 0.6 0.3421 0.0107 0.0021 0.6 19 95 0.2769 0.0080 0.0016 0.6 0.3323 0.0098 0.0020 0.6 20 100 0.2695 0.0074 0.0015 0.5 0.3233 0.0090 0.0018 0.5
Contents of E:14
Contents of F:15
Contents of D:3
Contents of E:4
Contents of F:5
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