invstudrng(P, ngroup, errorDf [,epsilon:eps] [,upper:T or lower:F]), elements of P between 0 and 1, elements of ngroup integers >= 2, elements of errorDf >= 1, eps > 0 small |

invstudrng(P, K, Df) computes the quantiles (probability points, critical values) of the Studentized range based on K normal variates and an independent estimate of variance with Df degrees of freedom. All three arguments must be REAL. The elements of P must be between 0 and 1. K must consist of integers >= 2, and the elements of Df must be >= 1, not necessarily integers. Any of the arguments P, K or Df that are not scalars must be vectors, matrices or arrays all of the same size and shape. invstudrng(P,2,Df) should be the same as sqrt(2)*invstu((1+P)/2,Df) except for computational error. invstudrng(P, K, Df, upper:T) and invstudrng(P, K, Df, lower:F) compute upper tail quantiles. The result is mathematically equivalent to invstudrng(1 - P, K, Df). Many so-called multiple comparison methods are based on these quantiles, among them the Tukey HSD (Honestly Significant Difference) and the SNK (Student-Newman-Keuls) methods. For example, if you have K independent normal samples of size n, all with the same variance, and Ssq is the pooled estimate of the variance, you can compute the 5% HSD as Cmd> q05 <- invstudrng(.05,K,K*(n-1),upper:T);hsd <- q05*sqrt(Ssq/n) In the same situation, you can test the null hypothesis that all means are equal by the studentized range statistic computed as Q <- (max(xbars) - min(xbars))/sqrt(Ssq/n). This is an alternative to the ANOVA F-statistic. You can compute the alpha-level critical value for Q as invstudrng(alpha, K,K*(n-1), upper:T). Here xbars is a vector containing the K sample means and Ssq is the pooled estimate of variance. See cumstudrng() for computing P values for Q. invstudrng(P, K, Df, epsilon:eps [,upper:T]), where eps is a small positive scalar, does the same computation with accuracy influenced by eps. The smaller the value of eps, the more accurate the result should be, but the longer it will take to compute it. The default value of eps is 0.00001. See also cumstudrng(), invstu().

Gary Oehlert 2003-01-15