This page presents calculations and tables for sample size and poer estimates related to Multiple Regression, using the formula suggested by Cohen (see reference).

The primary algorithm is to estimate the probability of Type II Error (β), based on 3 parameters

• Probability of Type I Error (α, p) used to determine statistical significance
• The number of independent variables in the Multiple Regression equation (u)
• The expected or observed Multiple Correlation Coefficient (R)

Power is estimated from the observations after data is collected. Power = 1 - β

Sample size is estimated by iteratively testing a range of sample size for its β, until the required power (1-β) is approximated.

Sample Size Tables

The tables present sample size requirements for

• Powers of 0.8, 0.9, and 0.95
• Probability of Type I Error (α) of 0.1, 0.05, 0.01 and 0.001
• Number of independent variable (u) of 2 to 15
• Multiple Correlation Coefficient (R) at 0.05 intervals

References

Cohen J. (1988) Statistical Power Analysis for the Behavioural Sciences. Second Edition. Lawrence Erlbaum Associates Publishers. Hillsdale New Jersey USA. ISBN 0-8058-0283-5. p. 407-410; 551.

α = probability of Type I Error
Power = 1 - β, where β is probability of Type II Error
u = the number of independent variables in the Multiple Regression equation
R = the expected Multiple Correlation Coefficient

Input Data

Sample Size :The data is in 4 columns   - Each row a separate calculation   - Col 1 = probability of Type I Error (α)   - Col 2 = Power (1-β)   - Col 3 = number of independent variables u   - Col 4 = expected Multiple Correlation Coefficient R    Power Estimation : The data is in 4 columns.   - Each row a separate calculation   - Col 1 = probability of Type I error α   - Col 2 = sample size in the study   - Col 3 = number of independent variables u   - Col 4 = observed multiple correlation coefficient R

R codes for sample size and power in Multiple Regression.

Ref: Cohen J. (1988) Statistical Power Analysis for the Behavioural Sciences. Second Edition. Lawrence Erlbaum Associates Publishers. Hillsdale New Jersey USA. ISBN 0-8058-0283-5. p. 407-410; 551.

Part 1: subtourine functions to calculate firstly power according to reference, then sample size function by interationg power function with different sample sizes until the correct power is obtained.

# Subroutine functions for power
# alpha = probability of Type I error
# nn = sample size
# u = number of independent variables
# r = Multiple Regression Coefficient
# returns beta, probability of Type II Error
Betamreg <- function(alpha,nn,u,r)
{
   n = round(nn)
   f2 = r^2 /(1 - r^2)         # effect size p. 410 9.2.2.
   lambda = f2 * n             # Lambda p. 415 9.3.3
   v = n - u - 1;              # denominator df p. 550 12.8.5.
   F = qf(1 - alpha, u, v)     # F for alpha
   zb = (sqrt(2.0 * (u + lambda) - (u + 2.0 * lambda) / (u + lambda)) -
              sqrt((2.0 * v - 1) * u * F / v)) /
        sqrt(u * F / v + (u + 2 * lambda) / (u + lambda))
   return(1 - pnorm(zb))       # returns beta, probability of Type II Error
}

# Subroutine function for Sample Size
# alpha = probability of Type I error
# beta =  probability of Type II error
# u = number of independent variables
# r = Multiple Regression Coefficient
Ssmreg <- function(alpha,beta,u,r)
{
   nL = u + 2    # left side lambda
   nR = 1e5;     # right side lambda
   nM = 100;     # starting ssiz value for iteration
   pM = Betamreg(alpha,nM,u,r) - beta;  # diff to beta with current ssiz value
   if(abs(pM)<0.0001)
   {
    return (ceiling(nM)) # correct sample size with first guess (unlikely)
   }
   old_nM = 0   #old sample size
   i = 0        # number of iterations
   while(abs(old_nM-nM)>1.5 & i<100 & abs(pM)>=0.0001)
   {
     old_nM = nM
     if(pM>0)     # adjust range of iteration
     { 
       nL = nM
     }
     else 
     { 
        nR = nM
     }
     nM = (nL + nR) / 2.0   # new sample size value
     pM = Betamreg(alpha,nM,u,r) - beta;
     i = i + 1
   }
   return (ceiling(nM))  # returns sample size
}

Part 2: Main Program

Program 1: Sample Size

# Pgm 1: Sample Size
# data entry
dat = ("
Alpha Power  U    R
0.05    0.8  3	0.2
0.01    0.8  3	0.2
0.05    0.9  3	0.2
0.01    0.9  3	0.2
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    
SSiz <- vector()     # array to sample size
for(i in 1 : nrow(df))
{
   alpha = df$Alpha[i]
   beta = 1 - df$Power[i]
   u = df$U[i]
   r = df$R[i]
   SSiz <- append(SSiz, Ssmreg(alpha, beta, u,r))
}
df$SSiz <- SSiz
df # display data frame of input data and result sample size 

The results are as follows

• Alpha = probability of TypeI Error (p, α)
• Power = 1 - β where β = probability of Type II Error
• U = number of independent variables
• R = expected Multiple Correlation Coefficient

> df # display data frame of input data and result sample size 
  Alpha Power U   R SSiz
1  0.05   0.8 3 0.2  263
2  0.01   0.8 3 0.2  376
3  0.05   0.9 3 0.2  342
4  0.01   0.9 3 0.2  466

Program 2: Power estimate

# Pgm 2: Power
# data entry
dat = ("
Alpha SSiz  U    R
 0.05  263  3	 0.2
 0.01  376  3	 0.2
 0.05  342  3	 0.2
 0.01  466  3	 0.2
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    
Power <- vector()     # array to Power
for(i in 1 : nrow(df))
{
  alpha = df$Alpha[i]
  ssiz = df$SSiz[i]
  u = df$U[i]
  r = df$R[i]
  Power <- append(Power,1 - Betamreg(alpha,ssiz,u,r))
}
df$Power <- Power
df # display data frame of input data and result power 

The results are as follows

• Alpha = probability of TypeI Error (p, α)
• SSiz = sample size of data
• U = number of independent variables
• R = observed Multiple Correlation Coefficient
• Power = 1 - β where β = probability of Type II Error

> df # display data frame of input data and result power
  Alpha SSiz U   R     Power
1  0.05  263 3 0.2 0.8004486
2  0.01  376 3 0.2 0.8027086
3  0.05  342 3 0.2 0.9003614
4  0.01  466 3 0.2 0.9006450