Power Calculations 

in the Predictor 

Sort Computer 

Program 

Steve P. Verrill 

David W. Green 

Victoria L. Herian 

United States

Department of 

Agriculture 

Forest Service 

Forest 

Products 

Laboratory 

Research 

Note 

FPL−RN−0277 
Abstract 

In a series of papers Verrill, Green, and Herian have developed theory and a computer program to aid in the design and 

analysis of predictor sort experiments. In this paper we provide the mathematical justification for the power calculations 

that the predictor sort program implements. 

Keywords: power calculations, predictor sort sampling, 

noncentral F

November 2000 

Verrill, Steve P.; Green, David W.; Herian, Victoria L. Power calculations 

in the predictor sort computer program. Res. Note FPL–RN–0277. 

Madison, WI: U.S. Department of Agriculture, Forest Service, Forest 

Products Laboratory. 4 p. 

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Power Calculations in the

Predictor Sort Computer Program

Steve P. Verrill, Mathematical Statistician

David W. Green, Supervisory Research General Engineer

Victoria L. Herian, Statistician

Forest Products Laboratory, Madison, Wisconsin

1 Introduction

In recent years wood strength researchers have replaced experimental unit allocation via random

sampling with allocation via sorts based on non-destructive measurements of strength predictors

such as modulus of elasticity and specic gravity. Although this procedure has the potential

of greatly increasing experimental sensitivity, improperly implemented it can reduce sensitivity. Verrill (1993), Verrill and Green (1996), and Verrill (1999) address this problem. Verrill and others

(1997) provide annotated input to and output from a computer program 1 | the \predictor sort"

program | that implements some of the methods discussed in the 1996 paper. In this note we

discuss the power calculations that are implemented in the predictor sort computer program.

2 Case 1 | Two Levels for One Factor

In the case of two levels for one factor, we rst determine whether we can interpolate in the

power tables obtained via simulation (tables 19{54, Verrill and Green, 1996). For interpolation

in the tables to be possible, the sample size, the treatment dierence divided by the coecient

of variation ratio, and the correlation between the predictor and the response must all lie within

certain bounds. If they do not lie within the appropriate bounds, an approximate approach is

taken. The reported power value is then

Power = Prob F1;2k2;
2 > F1;2k2(1  )

(the probability that a noncentral F distribution with 1 numerator degree of freedom, 2k  2 denominator degrees of freedom, and noncentrality parameter 
2

lies above the 100(1)th percentile

of a central F distribution with 1 numerator degree of freedom and 2k  2 denominator degrees

of freedom). Here k is the number of replicates for each of the two levels of the factor. The

noncentrality parameter is




2 =



(=y ) .q

2(1  2)=k

2

where  is the dierence in the mean response for the two levels, y is the variance of the response

(for a xed level), and  is the correlation between the predictor and the response. The CDFT

and CDFFNC functions of DCDFLIB2 were used to calculate the central F critical value and the

noncentral F probability. 1This program can be run over the World Wide Web at http://www1.fpl.fs.fed.us/ttweb.html. This web page

also provides links to LATEX and postscript versions of this note's references. 2DCDFLIB is a public domain library of \routines for cumulative distribution functions, their inverses, and their

parameters." It was produced by Barry Brown (bwb@odin.mda.uth.tmc.edu), James Lovato, and Kathy Russell of

the Department of Biomathematics, M.D. Anderson Cancer Center, The University of Texas. The DCDFLIB source

code can be obtained at http://odin.mdacc.tmc.edu/anonftp/source.html.

1
3 Case 2 | Multiple Factors/Multiple Levels

To explain our approach in the case of multiple factors and/or multiple levels, we use a combination

of the notation in the appendix to Verrill (1993) and in section 4 of Verrill (1999).

Assume that the predictor variable and the variable of interest, Y , have a joint bivariate normal

distribution with correlation . Denote the variance of Y by 2

Y . Suppose that there are I blocks

and F factors with K1;:::;KF levels. Let the allocation of samples be as described in Section 1 of

Verrill (1999). (For a multiple factor case, enough adjacent experimental units would be chosen at

a time to provide one additional observation for each cell.) Let Y

j1::: be the standard estimator

of mean response for the j1th level of factor 1.

Approximate distribution of the test statistic numerator

sum of squares when the null hypothesis does not hold

Suppose that we are interested in a test of the hypothesis that there is no dierence in the eects

of the K1 levels of factor 1. (The argument is the same for the other factors.) The numerator sum

of squares of the relevant F statistic is then

NumSS = X

K1

j1=1

(I K2 ::: KF )(Y

j1:::  Y:::)

2

where

Yij1:::jF = E(Yij1:::jF ) + Y Xij1:::jF + q

1  2Zij1:::jF ;

the Xij1:::jF 's, j1 2 f1;:::;K1g, ..., jF 2 f1;:::;KF g, are a randomization of the ith group of

order statistics from I K1 ::: KF iid N(0,1)'s, the Zij1:::jF 's are iid N(0,1), and the X's and

Z's are independent. Thus

NumSS = X

K1

j1=1

(I K2 ::: KF )





j1::: + Y X

j1::: + q

1  2Z

j1:::  :::  Y 

X

::: + q

1  2Z

:::2

= X

K1

j1=1

(I K2 ::: KF )2

Y 

2

(X

j1:::  X:::)

2

(1)

+2 X

K1

j1=1

(I K2 ::: KF )Y (X

j1:::  X:::)





Y q

1  2(Z

j1:::  Z:::) + j1:::  :::

+ X

K1

j1=1

(I K2 ::: KF )2

Y (1  

2

)



Z

j1:::  Z::: + (j1:::  :::)=Y q

1  2 2

where j1::: = E(Y

j1:::) and ::: = E(Y:::). Now by the same reasoning as that used in section

A.1.1 of Verrill (1993), the rst term in (1) converges in probability to zero.

2
The last term in (1) is of the form

(I K2 ::: KF )2

Y (1  

2

) X

K1

j1=1

(Bj1  B)

2

where

Bj1  N

j1:::.Y q

1  2 ; 1=(I K2 ::: KF ): Thus

NumSS  2

Y (1  

2

) X

K1

j1=1

(Cj1  C)

2

where

Cj1  N(pI K2 ::: KF j1:::.Y q1  2 ; 1)

and the Cj1's are independent. It is well known that in this case PK1

j1=1(Cj1  C

)

2

is distributed as

a noncentral 2 with K1  1 degrees of freedom and noncentrality parameter




2 = X

K1

j1=1

(j1:::  :::)

2 (I K2 ::: KF )=(2

Y (1  

2

)) (2)

Finally, by the Cauchy{Schwarz inequality and the work above, the second term in (1) is

asymptotically negligible with respect to the third term in (1). Thus for larger I we have

NumSS  2

Y (1  

2

)2

K11;
2 where 
2

is given by (2).

Approximate distribution of the denominator

sum of squares in the blocked case

The distributions of the blocked and unblocked F denominators discussed in the appendix to

Verrill (1993) do not change under a non-null hypothesis. By following the argument made in

appendix A.1.3 of Verrill (1993) one can show that in the blocked case, for larger sample sizes, the

distribution of the denominator sum of squares is approximately 2

Y (1  2

) times a central 2 with

I K1 :::KF (I +K1 +:::+KF F ) degrees of freedom, and it is approximately independent

of the numerator sum of squares.

Approximate power 3

We conjecture from the asymptotic results that small sample powers can be approximated in the

blocked case by treating the blocked statistic as a noncentral F statistic with K1  1 numerator

degrees of freedom, I K1 ::: KF  (I + K1 + ::: + KF  F ) denominator degrees of freedom,

and noncentrality parameter given by (2). 3The entry point to the power programs on the web is

http://www1.fpl.fs.fed.us/ttweb.html

Sample HTML, Perl, and FORTRAN code to perform these power calculations can be found at

http://www1.fpl.fs.fed.us/power.code.html

3
The power value reported by the predictor sort program is thus

Power = Prob FDOFN;DOFD;
2 > FDOFN;DOFD(1  )

(the probability that a noncentral F distribution with DOFN numerator degree of freedom, DOFD

denominator degrees of freedom, and noncentrality parameter 
2lies above the 100(1  )th percentile of a central F distribution with DOFN numerator degree of freedom and DOFD denominator

degrees of freedom). Here

DOFN = K1  1

DOFD = I K1 ::: KF  (I + K1 + ::: + KF  F )

and 
2is given by (2). The CDFF and CDFFNC functions of DCDFLIB were used to calculate

the central F critical value and the noncentral F probability. Power simulations

Because the \asymptotic" power results are only approximate, the predictor sort computer program

permits a user to check the asymptotic results with small sample simulations. The program generates bivariate normal random variables (predictor/response pairs) with a user-specied correlation.

It then allocates the responses to the treatments via a predictor sort. It produces design matrices

that are appropriate for \pooled," \paired," and analysis of covariance analyses (see Verrill and

Green (1996)). It makes use of both null hypothesis response vectors and non-null response vectors

(to obtain simulation estimates of both size and power). It then performs the pooled, paired, and

ANOCOV analyses on the generated data to detect statistically signicant results. This process is

repeated for the number of trials specied by the user.

The program performs one further theoretical check on the simulations by calculating an analytical estimate of the power in the ANOCOV case. This estimate is the average of the theoretical

power values obtained for all the trials. (The power of an ANOCOV will depend on the column

composed of predictor values, and these change from trial to trial.)

The numerical techniques used to perform the simulations will be described in a later publication4. References

Verrill, S.P. 1993. Predictor Sort Sampling, Tight T's, and the Analysis of Covariance. Journal

of the American Statistical Association. 88: 119{124.

Verrill, S. 1999. When Good Condence Intervals Go Bad: Predictor Sort Experiments and

ANOVA. The American Statistician. 53: 38{42.

Verrill, S.; Green, D. 1996. Predictor Sort Sampling, Tight t's, and the Analysis of Covariance:

Theory, Tables, and Examples. Res. Pap. FPL-RP-558. Madison, WI: U.S. Department of

Agriculture, Forest Service, Forest Products Laboratory. Verrill, S.; Green, D.; Herian, V. 1997. TT: A Program That Implements Predictor Sort Design

and Analysis. Gen. Tech. Rep. FPL-GTR-101. Madison, WI: U.S. Department of Agriculture,

Forest Service, Forest Products Laboratory.

4\Rolling Your Own: Linear Model Hypothesis Testing and Power Calculations via the Singular Value

Decomposition"

4