Minimum Risk Point Estimation of the Population Standardized Mean Difference

A function for the sequential estimation of the standardized mean difference with minimum risk. The function implements the ideas of Chattopadhyay and Kelley (2017), which considers study cost and accuracy of the estimated standardized mean difference simultaneously. This is important to specify that mr_smd was developed under the assumption of normally distributed data with equal sample size and equal cost of sampling per observation for each group.

Usage

mr_smd(
  A,
  structural_cost,
  epsilon,
  d,
  n,
  sampling_cost,
  pilot = FALSE,
  m0 = 4,
  gamma = 0.49
)

Arguments

A: The price one is willing to pay in order to have a maximum allowable difference of \(epsilon^2\) between the estimate of the standardized mean difference and its corresponding parameter
structural_cost: The structural cost of what one is willing to pay in a study
epsilon: The maximum desired difference between the estimated standardized mean difference and the population value
d: The current estimate of the standardized mean difference
n: Current sample size per group (thus total sample size is \(2n\)); requires equal sample size per group
sampling_cost: The sampling cost to collect an additional observation. For example, if each survey costs 10 dollars to distribute and score, sampling_cost would be 10 dollars per additional observation
pilot: TRUE or FALSE based on whether the users is using the function to plan a pilot sample size (TRUE) or if it is being used to assess if the optimization criterion has been satisfied (FALSE)
m0: The minimum bound on the initial pilot sample size
gamma: A correction factor in which we suggest .49; see the two Chattopadhyay & Kelley articles for more details (ignorable for most users)

Value

risk: The value of the risk function.
n1: Sample size for group 1 (echos the input value)
n2: Sample size for group 2 (echos the input value)
d: Observed value of the standardized mean difference (i.e., d; echos the input value)
is_satisfied: A TRUE or FALSE statement of that evaluates a stopping rule using the risk function to determine if the optimization criterion has been satisfied (based on the goals of the researcher and current information available)

Details

The standardized mean difference is a widely used measure effect size. In this article, we developed a general theory for estimating the population standardized mean difference by minimizing both the mean square error of the estimator and the total sampling cost. This function implements our ideas discussed in Chattopadhyay and Kelley (2017). See also Kelley and Rausch (2006) for additional information on the standardized mean difference.

Note

When pilot=TRUE the function returns the size of the pilot sample size, per group, that should be used (thus, the total sample size is twice the pilot sample size).

References

Chattopadhyay, B., & Kelley, K. (2016). Estimation of the coefficient of variation with minimum risk: A sequential method for minimizing sampling error and study cost. Multivariate Behavioral Research, 51(5), 627–648. doi:10.1080/00273171.2016.1203279

Chattopadhyay, B., & Kelley, K. (2017). Estimating the standardized mean difference with minimum risk: Maximizing accuracy and minimizing cost with sequential estimation. Psychological Methods, 22(1), 94–113. doi:10.1037/met0000089

Kelley, K., Darku, F. B., & Chattopadhyay, B. (2018). Accuracy in parameter estimation for a general class of effect sizes: A sequential approach. Psychological Methods, 23, 226–243. doi:10.1037/met0000127

Kelley, K., & Rausch, J. R. (2006). Sample size planning for the standardized mean difference: Accuracy in parameter estimation via narrow confidence intervals. Psychological Methods, 11(4), 363–385. doi:10.1037/1082-989X.11.4.363

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge.

Author

Ken Kelley kkelley@nd.edu

Examples

# To obtain pilot sample size in a situation in which A=10000. Note that 'A' is
# 'structural_cost' divided by the square of 'epsilon'.

# From Chattopadhyay and Kelley (2017)
mr_smd(pilot = TRUE, A = 10000, sampling_cost = 2.4, gamma = .49)
#>  term     value
#>  pilot_ss 13   

High.SLS <- c(11, 7, 22, 13, 6, 9, 11, 16, 12, 17, 14, 8, 16)
Low.SLS <- c(3, 6, 10, 8, 14, 5, 12, 10, 6, 8, 13, 5, 9)

mr_smd(d = 1.021484, n = 13, A = 10000, sampling_cost = 2.40, gamma = .49)
#>  term value
#>  risk 1800 
#>  n1   13   
#>  n2   13   
#>  d    1.02 

# Or, using the smd() function:
mr_smd(d = smd(group_1 = High.SLS, group_2 = Low.SLS)[1,2], n = 13, A = 10000,
       sampling_cost = 2.40, gamma = .49)
#>  term value
#>  risk 1800 
#>  n1   13   
#>  n2   13   
#>  d    1.02 

# Here, for this situation, the stopping rule is satisfied:
mr_smd(d = 1.00, n = 75, A = 10000, sampling_cost = 2.40, gamma = .49)
#>  term value
#>  risk 660  
#>  n1   75   
#>  n2   75   
#>  d    1