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cumstudrng(x, ngroup, errorDf [,epsilon:eps] [,upper:T or lower:F])
  where x is REAL, elements of ngroup integers >= 2, elements of errorDf
  >= 1, eps > 0 small

Keywords: probabilities, comparisons
cumstudrng(x, K, Df) computes the probability that Q <= x, where Q is a
Studentized range based on K normal variates and an independent estimate
of variance with Df degrees of freedom.  All three arguments must be
REAL.  K must consist of integers >= 2, and the elements of Df must be
>= 1, not necessarily integers.  The value is 0 for any x <= 0.  For any
element of Df > 1000, the asymptotic value (Df = infinity) is used.

Any of the arguments x, K or Df that are not scalars must be vectors,
matrices or arrays all of the same size and shape.

cumstudrng(x,2,Df) should be the same as 2*cumstu(x/sqrt(2),Df) - 1
except for computational error.

cumstudrng(x, K, Df, upper:T) and cumstudrng(x, K, Df, lower:F) compute
the upper tail probability P(Q >= x).  The result is equivalent to 1 -

When you have K independent normal samples of size n, all with the same
variance, 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

You can compute the P-value based on Q as cumstudrng(Q,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 invstudrng() for computing
critical values for Q.

cumstudrng(x, K, Df, epsilon:eps [,upper:T or lower:F]), 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 0.0000001.

See also invstudrng(), cumstu().

Gary Oehlert 2003-01-15