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Functions and CALL Routines

PROBMC Function



Returns a probability or a quantile from various distributions for multiple comparisons of means.
Category: Probability

Syntax
Arguments
Details
Formulas and Parameters
Computing the Analysis of Means
Many-One t-Statistics: Dunnett's One-Sided Test
Many-One t-Statistics: Dunnett's Two-sided Test
Computing the Partitioned Range
The Studentized Range
The Studentized Maximum Modulus
Williams' Test
Comparisons
Examples
Example 1: Computing Probabilities by Using PROBMC
Example 2: Computing the Analysis of Means
Example 3: Comparing Means
Example 4: Computing the Partitioned Range
Example 5: Computing Confidence Intervals
Example 6: Computing Williams' Test
See Also
References

Syntax

PROBMC(distribution, q, prob, df, nparms<, parameters>)


Arguments

distribution

is a character constant, variable, or expression that identifies the distribution. The following are valid distributions:

Distribution Argument
Analysis of Means
ANOM

One-sided Dunnett
DUNNETT1
Two-sided Dunnett
DUNNETT2
Maximum Modulus
MAXMOD
Partitioned Range
PARTRANGE

Studentized Range
RANGE
Williams
WILLIAMS

q

is the quantile from the distribution.

Restriction: Either q or prob can be specified, but not both.
prob

is the left probability from the distribution.

Restriction: Either prob or q can be specified, but not both.
df

is the degrees of freedom.

Note:   A missing value is interpreted as an infinite value.  [cautionend]

nparms

is the number of treatments.

Note:   For DUNNETT1 and DUNNETT2, the control group is not counted.  [cautionend]

parameters

is an optional set of nparms parameters that must be specified to handle the case of unequal sample sizes. The meaning of parameters depends on the value of distribution. If parameters is not specified, equal sample sizes are assumed, which is usually the case for a null hypothesis.


Details

The PROBMC function returns the probability or the quantile from various distributions with finite and infinite degrees of freedom for the variance estimate.

The prob argument is the probability that the random variable is less than q. Therefore, p-values can be computed as 1- prob. For example, to compute the critical value for a 5% significance level, set prob= 0.95. The precision of the computed probability is O(10--8) (absolute error); the precision of computed quantile is O(10--5).

Note:   The studentized range is not computed for finite degrees of freedom and unequal sample sizes.  [cautionend]

Note:   Williams' test is computed only for equal sample sizes.  [cautionend]


Formulas and Parameters

The equations listed here define expressions used in equations that relate the probability, prob, and the quantile, q, for different distributions and different situations within each distribution. For these equations, let [nu] be the degrees of freedom, df.

[equation]

[equation]

[equation]


Computing the Analysis of Means

Analysis of Means (ANOM) applies to data that is organized as k (Gaussian) samples, the ith sample being of size ni. Let [equation]. The distribution function [1, 2, 3, 4, 5] is the CDF for the maximum absolute of a k-dimensional multivariate [Computing the Analysis of Means] vector, with [equation] degrees of freedom, and an associated correlation matrix [equation]. This equation can be written as

[equation]

where

[equation]

where [equation], [equation], and [equation], are the gamma function, the density, and the CDF from the standard normal distribution, respectively.

For [equation], the distribution reduces to:

[equation]

where

[equation]

For the balanced case, the distribution reduces to:

[equation]

where

[equation]

and [equation]

The syntax for this distribution is:

x=probmc('anom', q,p,nu,n,<alpha1 ,..., alphan>);

where

x

is a numeric value with the returned result.

q

is a numeric value denoting the quantile.

p

is a numeric value denoting the probability. One of p and q must be missing.

nu

is a numeric value denoting the degrees of freedom.

n

is a numeric value denoting the number of samples.

alphai, i=1,...,k

are optional numeric values denoting the alpha values from the first equation of this distribution (see Computing the Analysis of Means.


Many-One t-Statistics: Dunnett's One-Sided Test


Many-One t-Statistics: Dunnett's Two-sided Test


Computing the Partitioned Range

RANGE applies to the distribution of the studentized range for n group means. PARTRANGE applies to the distribution of the partitioned studentized range. Let the n groups be partitioned into k subsets of size n1+ ...+ nk= n. Then the partitioned range is the maximum of the studentized ranges in the respective subsets, with the studentization factor being the same in all cases.

[equation]

The syntax for this distribution is:

x=probmc('partrange', q,p,nu,k,n1,...,nk);

where

x

is a numeric value with the returned result (either the probability or the quantile).

q

is a numeric value denoting the quantile.

p

is a numeric value denoting the probability. One of p and q must be missing.

nu

is a numeric value denoting the degrees of freedom.

k

is a numeric value denoting the number of groups.

ni i=1,...,k

are optional numeric values denoting the n values from the equation in this distribution (see Computing the Partitioned Range.


The Studentized Range

Note:    The studentized range is not computed for finite degrees of freedom and unequal sample sizes.  [cautionend]


The Studentized Maximum Modulus


Williams' Test

PROBMC computes the probabilities or quantiles from the distribution defined in Williams (1971, 1972) (See References). It arises when you compare the dose treatment means with a control mean to determine the lowest effective dose of treatment.

Note:   Williams' Test is computed only for equal sample sizes.  [cautionend]

Let X1, X2, ..., Xk be identical independent N(0,1) random variables. Let Yk denote their average given by

[equation]

It is required to compute the distribution of

[equation]

where

Yk

is as defined previously

Z

is an N(0,1) independent random variable

S

is such that ½[nu]S2 is a [chi]2 variable with [nu] degrees of freedom.

As described in Williams (1971) (See References), the full computation is extremely lengthy and is carried out in three stages.

  1. Compute the distribution of Yk. It is the fundamental (expensive) part of this operation and it can be used to find both the density and the probability of Yk. Let Ui be defined as

    [equation]

    You can write a recursive expression for the probability of Yk > d, with d being any real number.

    [equation]

    To compute this probability, start from an N(0,1) density function

    [equation]

    and recursively compute the convolution

    [equation]

    From this sequential convolution, it is possible to compute all the elements of the recursive equation for [equation], shown previously.

  2. Compute the distribution of Yk - Z. This computation involves another convolution to compute the probability

    [equation]

  3. Compute the distribution of (Yk - Z)/S. This computation involves another convolution to compute the probability

    [equation]

The third stage is not needed when [nu] = [infinity]. Due to the complexity of the operations, this lengthy algorithm is replaced by a much faster one when k [le] 15 for both finite and infinite degrees of freedom [nu]. For k [ge] 16, the lengthy computation is carried out. It is extremely expensive and very slow due to the complexity of the algorithm.


Comparisons

The MEANS statement in the GLM Procedure of SAS/STAT Software computes the following tests:


Examples


Example 1: Computing Probabilities by Using PROBMC

This example shows how to compute probabilities.

data probs;
   array par{5};
      par{1}=.5;
      par{2}=.51;
      par{3}=.55;
      par{4}=.45;
      par{5}=.2;
   df=40;
   q=1;
   do test="dunnett1","dunnett2", "maxmod";
      prob=probmc(test, q, ., df, 5, of par1-par5);
      put test $10. df q e18.13 prob e18.13;
   end;
run; 

SAS writes the following results to the log:

Probabilities from PROBMC

DUNNETT1  40  1.00000000000E+00 4.82992196083E-01
DUNNETT2  40  1.00000000000E+00 1.64023105316E-01
MAXMOD    40  1.00000000000E+00 8.02784203408E-01

Example 2: Computing the Analysis of Means

data _null_;
   q1=probmc('anom',.,0.9,.,20);                      put q1=;
   q2=probmc('anom',.,0.9,20,5,0.1,0.1,0.1,0.1,0.1);  put q2=;
   q3=probmc('anom',.,0.9,20,5,0.5,0.5,0.5,0.5,0.5);  put q3=;
   q4=probmc('anom',.,0.9,20,5,0.1,0.2,0.3,0.4,0.5);  put q4=;
run;

SAS writes the following output to the log:

q1=2.7892895753
q2=2.4549773558
q3=2.4549773558
q4=2.4532130238


Example 3: Comparing Means

This example shows how to compare group means to find where the significant differences lie. The data for this example is taken from a paper by Duncan (1955) (See References) and can also be found in Hochberg and Tamhane (1987) (See References).

The following values are the group means:

49.6

71.2

67.6

61.5

71.3

58.1

61.0

For this data, the mean square error is s2 = 79.64 (s = 8.924) with [nu] = 30.
data duncan;
   array tr{7}$;
   array mu{7};
   n=7;
   do i=1 to n;
      input tr{i} $1. mu{i};
   end;
   input df s alpha;
   prob= 1-alpha;
      /* compute the interval */
   x = probmc("RANGE", ., prob, df, 7);
   w = x * s / sqrt(6);
      /* compare the means */
   do i = 1 to n;
      do j = i + 1 to n;
         dmean = abs(mu{i} - mu{j});
         if dmean >= w then do;
            put tr{i} tr{j} dmean;
         end;
      end;
   end;
   datalines;
A 49.6
B 71.2
C 67.6
D 61.5
E 71.3
F 58.1
G 61.0
 30 8.924 .05
;
run;

SAS writes the following output to the log:

Group Differences

A B 21.6
A C 18
A E 21.7

Example 4: Computing the Partitioned Range

data _null_;
   q1=probmc('partrange',.,0.9,.,4,3,4,5,6);  put q1=;
   q2=probmc('partrange',.,0.9,12,4,3,4,5,6); put q2=;
run;

SAS writes the following output to the log:

q1=4.1022395729
q2=4.788862411


Example 5: Computing Confidence Intervals

This example shows how to compute 95% one-sided and two-sided confidence intervals of Dunnett's test. This example and the data come from Dunnett (1955) (See References) and can also be found in Hochberg and Tamhane (1987) (See References). The data are blood count measurements on three groups of animals. As shown in the following table, the third group serves as the control, while the first two groups were treated with different drugs. The numbers of animals in these three groups are unequal.

Treatment Group: Drug A Drug B Control

9.76 12.80 7.40

8.80 9.68 8.50

7.68 12.16 7.20

9.36 9.20 8.24


10.55 9.84



8.32
Group Mean 8.90 10.88 8.25
n 4 5 6

The mean square error s2 = 1.3805 (s = 1.175) with [nu] = 12.

data a;
   array drug{3}$;
   array count{3};
   array mu{3};
   array lambda{2};
   array delta{2};
   array left{2};
   array right{2};

      /* input the table */
   do i = 1 to 3;
      input drug{i} count{i} mu{i};
   end;

      /* input the alpha level,    */
      /* the degrees of freedom,   */
      /* and the mean square error */
   input alpha df s;
   
      /* from the sample size, */
      /* compute the lambdas   */
   do i = 1 to 2;
      lambda{i} = sqrt(count{i}/
        (count{i} + count{3}));
   end;

      /* run the one-sided Dunnett's test */
   test="dunnett1";
      x = probmc(test, ., 1 - alpha, df, 
                 2, of lambda1-lambda2);
      do i = 1 to 2;
         delta{i} = x * s * 
            sqrt(1/count{i} + 1/count{3});
         left{i} = mu{i} - mu{3} - delta{i};
      end;
   put test $10. x left{1} left{2};

      /* run the two-sided Dunnett's test */
   test="dunnett2";
      x = probmc(test, ., 1 - alpha, df, 
                 2, of lambda1-lambda2);
      do i=1 to 2;
         delta{i} = x * s * 
            sqrt(1/count{i} + 1/count{3});
         left{i} = mu{i} - mu{3} - delta{i};
         right{i} = mu{i} - mu{3} + delta{i};
      end;
   put test $10. left{1} right{1};
   put test $10. left{2} right{2};
   datalines;
A 4 8.90
B 5 10.88
C 6 8.25
0.05 12 1.175
;
run;

SAS writes the following output to the log:

Confidence Intervals

DUNNETT1  2.1210786586 -0.958751705 1.1208571303
DUNNETT2  -1.256411895 2.5564118953
DUNNETT2  0.8416271203 4.4183728797

Example 6: Computing Williams' Test

In the following example, a substance has been tested at seven levels in a randomized block design of eight blocks. The observed treatment means are as follows:

Treatment Mean
X0 10.4
X1 9.9
X2 10.0
X3 10.6
X4 11.4
X5 11.9
X6 11.7

The mean square, with (7 - 1)(8 - 1) = 42 degrees of freedom, is s2 = 1.16.

Determine the maximum likelihood estimates Mi through the averaging process.

Now the order restriction is satisfied.

The maximum likelihood estimates under the alternative hypothesis are:

M0 = M1 = M2 = X0,1,2 = 10.1

M3 = X3 = 10.6

M4 = X4 = 11.4

M5 = M6 = X5,6 = 11.8

Now compute [equation], and the probability that corresponds to k = 6, [nu] = 42, and t = 2.60 is .9924467341, which shows strong evidence that there is a response to the substance. You can also compute the quantiles for the upper 5% and 1% tails, as shown in the following table.

SAS Statements Results
prob=probmc("williams",2.6,.,42,6);
0.99244673
quant5=probmc("williams",.,.95,42,6);
1.80654052
quant1=probmc("williams",.,.99,42,6);
2.49087829


See Also

Functions:

CDF Function

LOGCDF Function

LOGPDF Function

LOGSDF Function

PDF Function

SDF Function


References

Guirguis, G. H. and R. D. Tobias. 2004. "On the computation of the distribution for the analysis of means." Communications in Statistics: Simulation and Computation 33: 861-887.

Nelson, P. R. 1981. "Numerical evaluation of an equicorrelated multivariate non-central t distribution." Communications in Statistics: Part B - Simulation and Computation 10: 41-50.

Nelson, P. R. 1982. "Exact critical points for the analysis of means." Communications in Statistics: Part A - Theory and Methods 11: 699-709.

Nelson, P. R. 1982a. "An Approximation for the Complex Normal Probability Integral." BIT 22(1): 94-100.

Nelson, P. R. 1988. "Application of the analysis of means." Proceedings of the SAS Users Group International Conference 13: 225-230.

Nelson, P. R. 1991. "Numerical evaluation of multivariate normal integrals with correlations [equation]." The Frontiers of Statistical Scientific Theory and Industrial Applications 2: 97-114.

Nelson, P. R. 1993. "Additional Uses for the Analysis of Means and Extended Tables of Critical Values." Technometrics 35: 61-71.

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