Scaled Minimal Detectable Slope for a t-Test for Linear Trend

Compute the scaled minimal detectable slope associated with a t-test for liner trend, given the sample size or predictor variable values, power, and significance level.

linearTrendTestScaledMds(n, x = lapply(n, seq), alpha = 0.05, power = 0.95, 
    alternative = "two.sided", two.sided.direction = "greater", approx = FALSE, 
    tol = 1e-07, maxiter = 1000)

Arguments

n

numeric vector of sample sizes. All values of n must be positive integers larger than 2. This argument is ignored when x is supplied. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.

x

numeric vector of predictor variable values, or a list in which each component is a numeric vector of predictor variable values. Usually, the predictor variable is time (e.g., days, months, quarters, etc.). The default value is x=lapply(n,seq), which yields a list in which the i'th component is the seqence of integers from 1 to the i'th value of the vector n. If x is a numeric vector, it must contain at least three elements, two of which must be unique. If x is a list of numeric vectors, each component of x must contain at least three elements, two of which must be unique. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.

alpha

numeric vector of numbers between 0 and 1 indicating the Type I error level associated with the hypothesis test. The default value is alpha=0.05.

power

numeric vector of numbers between 0 and 1 indicating the power associated with the hypothesis test. The default value is power=0.95.

alternative

character string indicating the kind of alternative hypothesis. The possible values are "two.sided" (the default), "greater", and "less".

two.sided.direction

character string indicating the direction (positive or negative) for the scaled minimal detectable slope when alternative="two.sided". When
two.sided.direction="greater" (the default), the scaled minimal detectable slope is positive. When two.sided.direction="less", the scaled minimal detectable slope is negative. This argument is ignored if alternative="less" or alternative="greater".

approx

logical scalar indicating whether to compute the power based on an approximation to the non-central t-distribution. The default value is approx=FALSE.

tol

numeric scalar indicating the toloerance to use in the uniroot search algorithm. The default value is tol=1e-7.

maxiter

positive integer indicating the maximum number of iterations argument to pass to the uniroot function. The default value is maxiter=1000.

Details

If the argument x is a vector, it is converted into a list with one component. If the arguments n, x, alpha, and power are not all the same length, they are replicated to be the same length as the length of the longest argument.

Formulas for the power of the t-test of linear trend for specified values of the sample size, scaled slope, and Type I error level are given in the help file for linearTrendTestPower. The function linearTrendTestScaledMds uses the uniroot search algorithm to determine the minimal detectable scaled slope for specified values of the power, sample size, and Type I error level.

Value

numeric vector of computed scaled minimal detectable slopes. When alternative="less", or alternative="two.sided" and two.sided.direction="less", the computed slopes are negative. Otherwise, the slopes are positive.

References

See the help file for linearTrendTestPower.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

See the help file for linearTrendTestPower.

Examples

  # Look at how the scaled minimal detectable slope for the t-test for linear 
  # trend increases with increasing required power:

  seq(0.5, 0.9, by = 0.1) 
#> [1] 0.5 0.6 0.7 0.8 0.9
  #[1] 0.5 0.6 0.7 0.8 0.9 

  scaled.mds <- linearTrendTestScaledMds(n = 10, power = seq(0.5, 0.9, by = 0.1)) 

  round(scaled.mds, 2) 
#> [1] 0.25 0.28 0.31 0.35 0.41
  #[1] 0.25 0.28 0.31 0.35 0.41

  #----------

  # Repeat the last example, but compute the scaled minimal detectable slopes 
  # based on the approximate power instead of the exact:

  scaled.mds <- linearTrendTestScaledMds(n = 10, power = seq(0.5, 0.9, by = 0.1), 
    approx = TRUE) 

  round(scaled.mds, 2) 
#> [1] 0.25 0.28 0.31 0.35 0.41
  #[1] 0.25 0.28 0.31 0.35 0.41

  #==========

  # Look at how the scaled minimal detectable slope for the t-test for linear trend 
  # decreases with increasing sample size:

  seq(10, 50, by = 10) 
#> [1] 10 20 30 40 50
  #[1] 10 20 30 40 50 

  scaled.mds <- linearTrendTestScaledMds(seq(10, 50, by = 10), alternative = "greater") 

  round(scaled.mds, 2) 
#> [1] 0.40 0.13 0.07 0.05 0.03
  #[1] 0.40 0.13 0.07 0.05 0.03

  #==========

  # Look at how the scaled minimal detectable slope for the t-test for linear trend 
  # decreases with increasing values of Type I error:

  scaled.mds <- linearTrendTestScaledMds(10, alpha = c(0.001, 0.01, 0.05, 0.1), 
    alternative="greater") 

  round(scaled.mds, 2) 
#> [1] 0.76 0.53 0.40 0.34
  #[1] 0.76 0.53 0.40 0.34

  #----------

  # Repeat the last example, but compute the scaled minimal detectable slopes 
  # based on the approximate power instead of the exact:
 
  scaled.mds <- linearTrendTestScaledMds(10, alpha = c(0.001, 0.01, 0.05, 0.1), 
    alternative="greater", approx = TRUE) 

  round(scaled.mds, 2) 
#> [1] 0.70 0.52 0.41 0.36
  #[1] 0.70 0.52 0.41 0.36

  #==========

  # Clean up
  #---------
  rm(scaled.mds)