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StatsToDo: Sample Size for Estimating Pearson's Correlation Coefficient ρ
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Explanations and References
This page provides programs and sample size table for estimating Pearson's Correlation Coefficient rho (ρ). The calculations are based on the Fisher's Z transformation of ρ so it becomes normally distributed. The 4 programs are
Sample Size Table
Fisher's Z TransformationCorrelation coefficients (ρ) are constrained in values between -1 and 1, so its distribution is only symmetrical when ρ=0. As values betmes closer to -1 or 1, the distribution becomes increasingly assymetrical, with a longer tail towards the value 0 and shorter tail towards the extremes of -1 or 1In order to estimate sample size, power, and confidence intervals, ρ is transformed into a normally distributed value (Fisher's Z). At the end of the calculation Z is reverse transformed back to ρ values.
ρ=exp(2Z-1) / exp(2Z+1) 1 or 2 tail ModelsIn most cases, the researcher merely wish to know if a statistically significant ρ exists. In this situation, the 1 tail model suffices, as a significant ρ is one where either the lower limit of the confidence interval is >0 (for a positive ρ) or the upper limit <0 (for a negative ρ). The advantage of using the 1 tail model is that a smaller sample size is required.If a precise ρ and its full range of confidence is requird, then the 2 tail model should be used. Sample Size for Correlation and RegressionAlthough correlation and regression analysis are oftern carried out on the same set of data, the sample size required for the two are similar but different.In correlation (ρ), both x and y variables are assumed to be normally distributed. The distribution of correlation coefficient (ρ) is assymmetric, and its calculation requires a prior transformation to Fishers's Z. In Regression analysis, only the dependent variable (y) needs to be normally distributed, while the independent variable (x) need only be at least ordinal. The regression coefficient (b) is assumed to be normally distributed. The estimation for sample size in regression analysis are presented in the program for sample size for multiple regression (using the number of independent variable = 1). Sample size for non-parametric Correlation Coefficients
ReferencesMachin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 168-172Altman DG, Machin D, Bryant TN and Gardner MJ. (2000) Statistics with Confidence Second Edition. BMJ Books ISBN 0 7279 1375 1. p. 89-92 Siegel S and Castellan Jr. NJ (2000) Nonparametric Statistics for the Behavioral Sciences. Second Edition. McGraw Hill, Sydney. ISBN0-07-100326-6 p. 244 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
# subroutine
SSizRho <- function(alpha,beta,r,tail)
{
r = abs(r)
if(r<0.00001 | r>0.99999)
{
return (0)
}
za = qnorm(alpha / tail);
zb = qnorm(beta);
gamma = (za + zb)^2
n = 0.5 * log((1.0 + r) / (1.0 - r))
oldn = 0
iterate = 0
while((iterate<1000) & (abs(oldn - n)>0.00001))
{
oldn = n
mu = 0.5 * log((1.0 + r) / (1.0 - r)) + (r /(2.0 * (n-1)))
n = gamma / mu^2 + 3.0
iterate = iterate + 1
}
if(iterate>=1000)
{
return (0)
}
return (ceiling(n))
}
# main ssiz program
# data entry
dat = ("
Alpha Power Rho
0.05 0.8 0.6
0.01 0.8 0.6
0.05 0.9 0.6
0.01 0.9 0.6
")
df <- read.table(textConnection(dat),header=TRUE) # conversion to data frame
# vectors for sample size results
SSiz1Tail <- vector()
SSiz2Tail <- vector()
# calculations
for(i in 1 : nrow(df))
{
alpha = df$Alpha[i]
beta = 1 - df$Power[i]
rho = df$Rho[i]
SSiz1Tail <- append(SSiz1Tail, SSizRho(alpha,beta,rho,1)) # 1 tail
SSiz2Tail <- append(SSiz2Tail, SSizRho(alpha,beta,rho,2)) # 2 tail
}
# results to data frame for display
df$SSiz1Tail <- SSiz1Tail
df$SSiz2Tail <- SSiz2Tail
df # data frame with input data and resukts
The results are as follows
> df # data frame with input data and resukts Alpha Power Rho SSiz1Tail SSiz2Tail 1 0.05 0.8 0.6 16 19 2 0.01 0.8 0.6 24 27 3 0.05 0.9 0.6 21 25 4 0.01 0.9 0.6 30 34
# Program 2: Power
# data entry
dat = ("
Alpha SSiz Rho
0.05 16 0.6
0.01 24 0.6
0.05 21 0.6
0.01 30 0.6
")
df <- read.table(textConnection(dat),header=TRUE) # conversion to data frame
# vectors for results
Power1Tail <- vector()
Power2Tail <- vector()
# Calculations
for(i in 1 : nrow(df))
{
alpha = df$Alpha[i]
ssiz = df$SSiz[i]
rho = df$Rho[i]
if(ssiz<4 | rho<0.00001 | rho>0.99999) # not calculable
{
Power1Rail <- append(Power1Tail,0)
Power2Rail <- append(Power2Tail,0)
}
else
{
mu = 0.5 * log((1.0 + rho) / (1.0 - rho)) + (rho / (2.0 * (ssiz - 1.0)))
za = abs(qnorm(alpha)) # 1 tail
Power1Tail <- append(Power1Tail,pnorm(mu * sqrt(ssiz - 3) - za))
za = abs(qnorm(alpha / 2)) # 2 tail
Power2Tail <- append(Power2Tail,pnorm(mu * sqrt(ssiz - 3) - za))
}
}
# combine results into data frame for display
df$Power1Tail <- Power1Tail
df$Power2Tail <- Power2Tail
df # show data input and power results
The results are as follows
> df # show data input and power results Alpha SSiz Rho Power1Tail Power2Tail 1 0.05 16 0.6 0.8228900 0.7295077 2 0.01 24 0.6 0.8185424 0.7454831 3 0.05 21 0.6 0.9130155 0.8518614 4 0.01 30 0.6 0.9080931 0.8598448
program 3 : Confidence Interval
Firstly the subroutine used by both 1 and 2 tail estimates
ConfIntv <- function(pc,n,r,tail) # %conf, ssiz, rho, tail
{
alpha = (1 - pc/100)
za = abs(qnorm(alpha / tail))
z = log((1 + r) / (1 - r)) / 2
se = sqrt(n - 3)
f = z - za / se # lower limit
ll = (exp(2 * f) - 1) / (exp(2 * f) + 1)
g = z + za / se # upper limit
ul = (exp(2 * g) - 1) / (exp(2 * g) + 1)
return (c(ll, ul))
}
Now the main program
#Main Program 3: Confidence Interval
# data entry
dat = ("
Pc SSiz Rho
95 16 0.6
99 24 0.6
95 21 0.6
99 30 0.6
")
df <- read.table(textConnection(dat),header=TRUE) # conversion to data frame
# vectors for results
LL1Tail <- vector() # lower limit 1 tail
UL1Tail <- vector() # upper limit 1 tail
LL2Tail <- vector() # lower limit 2 tail
UL2Tail <- vector() # upper limit 2 tail
# Calculations
for(i in 1 : nrow(df))
{
pc = df$Pc[i]
ssiz = df$SSiz[i]
rho = df$Rho[i]
resAr = ConfIntv(pc,ssiz,rho,1) # 1 tail
LL1Tail <- append(LL1Tail, resAr[1]) # lower limit 1 tail
UL1Tail <- append(UL1Tail, resAr[2]) # upper limit 1 tail
resAr = ConfIntv(pc,ssiz,rho,2) # 2 tail
LL2Tail <- append(LL2Tail, resAr[1]) # lower limit 2 tail
UL2Tail <- append(UL2Tail, resAr[2]) # upper limit 2 tail
}
# combine input data and results for display
df$LL1Tail <- LL1Tail
df$UL1Tail <- UL1Tail
df$LL2Tail <- LL2Tail
df$UL2Tail <- UL2Tail
df # Input data and confidence interval results
The results are as follows
> df # Input data and confidence interval results Pc SSiz Rho LL1Tail UL1Tail LL2Tail UL2Tail 1 95 16 0.6 0.2326099 0.8175378 0.1484459 0.8445242 2 99 24 0.6 0.1833978 0.8338978 0.1303100 0.8497463 3 95 21 0.6 0.2962935 0.7935115 0.2271470 0.8194415 4 99 30 0.6 0.2406287 0.8147011 0.1949028 0.8302268
Program 4: Pilot study
Firstly the subroutine for confidence interval, which is the same as that for Program 3
ConfIntv <- function(pc,n,r,tail) # %conf, ssiz, rho, tail
{
alpha = (1 - pc/100)
za = abs(qnorm(alpha / tail))
z = log((1 + r) / (1 - r)) / 2
se = sqrt(n - 3)
f = z - za / se # lower limit
ll = (exp(2 * f) - 1) / (exp(2 * f) + 1)
g = z + za / se # upper limit
ul = (exp(2 * g) - 1) / (exp(2 * g) + 1)
return (c(ll, ul))
}
Now the main program
# Pgm 4 : Pilot studies
# Parameters
pc = 95 # % confidence
rho = 0.6 # correlation coefficient rho
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 tail
CI1 <- 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
n = intv
SSiz <- append(SSiz,n)
resAr = ConfIntv(pc,n,rho,1) # 1 tail
ci1 = resAr[2] - resAr[1]
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 tail
resAr <- ConfIntv(pc,n,rho,2) # 2 tail
ci2 = resAr[2] - resAr[1]
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 tail
# subsequent rows
while(n < maxN)
{
n = n + intv
SSiz <- append(SSiz,n)
oldci1 = ci1
resAr = ConfIntv(pc,n,rho,1) # 1 tail
ci1 = resAr[2] - resAr[1]
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
resAr = ConfIntv(pc,n,rho,2) # 2 tail
ci2 = resAr[2] - resAr[1]
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
The results are as follows
> df # display results in data frame SSiz CI1 Diff1 DecCase1 PDCase1 CI2 Diff2 DecCase2 PDCase2 1 5 1.3905 0 0 0 1.5690 0 0 0 2 10 0.7942 0.5963 0.1193 8.6 0.9401 0.6289 0.1258 8.0 3 15 0.6087 0.1855 0.0371 4.7 0.7241 0.2160 0.0432 4.6 4 20 0.5116 0.0971 0.0194 3.2 0.6094 0.1147 0.0229 3.2 5 25 0.4497 0.0619 0.0124 2.4 0.5359 0.0735 0.0147 2.4 6 30 0.4059 0.0438 0.0088 1.9 0.4837 0.0521 0.0104 1.9 7 35 0.3728 0.0331 0.0066 1.6 0.4443 0.0394 0.0079 1.6 8 40 0.3466 0.0261 0.0052 1.4 0.4132 0.0312 0.0062 1.4 9 45 0.3253 0.0213 0.0043 1.2 0.3878 0.0254 0.0051 1.2 10 50 0.3075 0.0178 0.0036 1.1 0.3665 0.0212 0.0042 1.1 11 55 0.2923 0.0152 0.0030 1.0 0.3484 0.0181 0.0036 1.0 12 60 0.2792 0.0131 0.0026 0.9 0.3327 0.0157 0.0031 0.9 13 65 0.2677 0.0115 0.0023 0.8 0.3190 0.0137 0.0027 0.8 14 70 0.2575 0.0102 0.0020 0.8 0.3069 0.0122 0.0024 0.8 15 75 0.2483 0.0091 0.0018 0.7 0.2960 0.0109 0.0022 0.7 16 80 0.2401 0.0082 0.0016 0.7 0.2862 0.0098 0.0020 0.7 17 85 0.2327 0.0074 0.0015 0.6 0.2773 0.0089 0.0018 0.6 18 90 0.2259 0.0068 0.0014 0.6 0.2692 0.0081 0.0016 0.6 19 95 0.2197 0.0062 0.0012 0.6 0.2618 0.0074 0.0015 0.6 20 100 0.2139 0.0057 0.0011 0.5 0.2550 0.0068 0.0014 0.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||