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Noncentral t Noncentrality (delta) Solver

Back out the noncentrality delta of a noncentral t distribution from a t value, lower probability and degrees of freedom for power analysis.

Input

Enter a t value, the target lower probability and the degrees of freedom nu to solve the noncentrality delta of the noncentral t distribution by bisection.

Observed or critical t statistic

Target probability for P(T ≤ t), between 0 and 1

Enter a positive value

Result

Noncentrality delta for t=2.086, nu=20, p=0.18

3.02322241

Degrees of freedom nu

20

Target lower probability p

0.18

Power (1 minus p)

0.82

Upper probability P(T ≥ t)

0.82

Distribution mean

3.14282404

Check: achieved lower probability

0.18

Noncentrality delta versus lower probability P(T ≤ t)

Approximate density shape at the solved noncentrality delta

In power analysis the upper probability, one minus the lower probability, corresponds to the statistical power for that effect.

How it works

  • The noncentral t cumulative distribution function P(T ≤ t ; nu, delta) is evaluated with Lenth series using the regularized incomplete beta function and Poisson type weights.
  • Because this function is monotonically decreasing in the noncentrality delta, the delta matching a target lower probability p is found uniquely by bisection.
  • In power analysis you take the critical t value and significance level, find the corresponding noncentrality, and read statistical power as one minus the lower probability.
  • The mean is computed as delta times the square root of nu over 2, times Gamma of (nu minus 1) over 2 divided by Gamma of nu over 2, when nu is greater than 1.
  • The lower probability p must be between 0 and 1. The calculator cannot return a value when p is out of range or when no delta satisfies the given t and nu.

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