United States 

Department of 

Agriculture 

forest Service 

Southern 

Research Station 

Research Paper 

SRS-5 

Choosing a Transformation 

in Ana yses of Insect Counts 

from Contagious Distributions 

with Low BIIeans 
Choosing a Transformation in Analyses 

of Insect Counts from Contagious 

Distributions with Low Means 

W.D. Pepper, S.J. Zarnoch, G.L. DeBarr, 

P. de Groot, and C.D. Tangren 

Abstract guidelines are available on how to choose a transfomtion in 

parametric analyses of data where insect counts are low or 

Guidelines based on computer simulation are suggested for choosing a 

ttansformation of insect counts from negative binomial distributions with how to determine the influence of various controllable 

low mean counts and high levels of contagion. Typical values and ranges factors on the method of analyses. Thus, our objective was 

of negative binomial model parameters were determined by fitting the to develop guidelines to help researchers choose a 

model to data from 19 entomological field studies. Random sampling of transformation and to develop a simple program1 that negative binomial distributions was simulated and ANOVA's were 

performed on simulated data for randomized complete block designs with evaluates various tradeoffs and identifies an affordable 

treatment means corresponding to means of negative binomial experimental design with acceptable statistical test attributes. 

distributions. The influence of analysis variable, treatment-mean 

configuration, range in treatment means, significance level of statistical 

tests, level of contagion, number of blocks, and number of replications in 

time on observed power and Type I error of F-tests was studied. A Methods 

computer program was developed to recompute observed power of F-tests 

for any combination of these factors. The program facilitates choosing a Field Data 

transformation and may also be used to evaluate tradeoffs and determine 

affordable experimental factors for a future design with a given set of 

statistical attributes. Preliminary observations on frequency distributions of insect 

counts were obtained from data sets collected in 19 field 

Keywords: Contagious distribution, entomology, transformation, studies conducted on various pheromone "treatments" for the 

variance-mean relationship, variance stabilization. white pine cone beetle, Conophthorus coniperda (Schwarz), 

Introduction 

Entomological studies involving the field testing of different 

treatments require consideration of a number of controllable 

factors in the planning and analysis phases of the work. In 

the planning stage, factors common to many studies, such as 

the number of replicates in space or time and the 

significance level for tests, must be considered. In some 

studies, the effect of low insect counts and the skewness of 

their dis~btltion must also be considered. A common 

problem with insect counts is that the frequency distribution 

is frequently skewed to the right, resembling a nonnoml 

discrete distribution where the variance is related to the 

mean. Thus, when treatmen& differ, heterogeneous 

in the United States and Canada (Birgersson and others 

1995, DeBarr and others 1995). These pheromone 

treatments were field tested by installing a trap on a selected 

tree and baiting it with a preparation (usually chemical 

formulations); the procedure is replicated in additional trees 

for each treatment and blocking may or may not be used. 

Thus, the trap on an individual tree is the experimental or 

observational unit. The traps are revisited after some period 

of time and the insects are removed and counted. If the 

procedure is repeated (usually), the traps are rebaited with 

the same pheromone treatment and reinstalled on the same 

tree, or more commonly, new random assignments of 

treatments are used to reduce tree position bias. The 

observational response for a treatment is the insect count at 

the end of the designated time period or the mean count per 

variance is guaranteed, Parmetric analyses of such data 

tend to produce too many significant results in F-tests and ttests (Snedecor and Cochran 1967). Therefore, in the 

analysis phase of a study, two questions must be answered: 

(1) does the data require transformation, and (2) what type 

of transformation is required to produce a variate with a 

more stable variance and a frequency distribution that is 

approximately normal? The data analyst also decides 

whether to rank the response from experimental units by 

treatment and perform nonparametric analyses or perform 

parmetric analyses on the ranks (Conover 1980). Few 

'The computer program described in this paper is available on request with 

the understanding that the U.S. Department of Agriculture cannot assure its 

accuracy, completeness, =liability, or suitability for any other purpose than 

that reported. The recipient may not assert any propriety rights thereto nor 

represent it to anyone as other than a Government-produced computer 

program. The program is written in SAS code but is not a packaged SAS 

procedure and is not available in other common statistical software packages 

such as BMDP and Systat. The program (SIMPWR) is available from the 

senior author upon request: W.D. Pepper, U.S. Depattment of Agriculture, 

Forest Service, Southern Research Station, Forestry Sciences Laboratory, 

320 Green Street, Athens, GA 30602. 
trap where the average is calculated over several time 

periods. In this paper, the observational response is the 

mean count per trap per week. The range, average value of 

insect counts and degree of contagion with respect to counts 

were observed. LRvel of contagion is the degree to which 

the frequencies of high counts and counts near zero exceed 

those expected in a random process such as the Poisson. 

ate number of pheromone treatments and the number of 

replications in space and time were observed to determine 

typical values and ranges (table 1). These field observations 

were a prerequisite to pedoMng computer simulations 

comparing alternative methods of analysis on computergenerated data sets with attributes similar to those observed 

in the 19 field studies. 

Statistical Distributions and Transformations 

To transform data, its true distribution must be determined. 

First, a basic statistical model that fits data from a wide 

range of field conditions should be identified. atis model is 

then used to derive a transformation for analysis. Because 

ecologists almost exclusively use the negative binomial 

(NB) model to describe contagious distributions (Kuno 

1991), we chose to fit this model to insect counts obtained 

from the United States-Canadian field studies. 

With the NB model, the probability, P,, that an 

observational unit will contain x = 0, 1, 2, ... insects is 

where 

P = pk, 

Q = l+P, 

p = the population mean, and 

k -. the contagion parameter. 

The I\ITB distribution converges to other distributions as k 

varies over its range (Anscombe 1949). When k is 

considerably large, the counts begin to approach 

randomess and a Poisson model may fit the data. Indeed, 

as k becomes infinitely large, the NB distribution converges 

to the Poisson distribution. When k = 1, the NB distribution 

reduces to the discrete geometric distribution. Values of k 

in the interval 0 < k < 1 are associated with high levels of 

contagion which is characteristic of the NB dis~bution. As 

k approaches zero, the count distribution approaches the 

logarithmic series. 

The population mean p is estimated as the ariwetic average 

per trap per experimntal unit (m) where the average is 

calculated over replications in time. The variance of 

individual counts is 

Var (x) = p (1 + p1k) . (2) 

Thus, variances associated with different treamnts with a 

comon value of k are heterogeneous by definition. The 

moment estimator for Var(x) is the sample variance s2. The 

parameter k is estimated with Anscombe's (1949) method 2 

which uses successive approximtion to choose a value of E 

that makes the following expression approximately equal: 

In this expression, f, is the proportion of the total number of 

traps with a zero count and (1 + m/ li)-' is the estimated 

probability of a zero count computed with the 1VlB model, 

This is the most efficient estimation method proposed by 

Anscombe (1949) for estimting a single population value of 

kwhenk<landl<p<lO. 

Transformations of counts were discussed in reports 

published over 60 years ago. Among the first were the 

square root transformation (Bartlett 1936) and the 

logarithmic transformation (Williams 1937). These early 

derivations were based on an assumed linear relationship 

between the mean (p) and variance (a ') of x, a = ap, where 

a is constant. From the linear relationship, a new variate, y = 

f(x), was derived where the variance of y is constant. Beall 

(1942) extended this relationship to the nonlinear case, 

which is required to describe some Pgrpes of field data. This 

relationship is characteristic of the NB distribution where llb 

= k, the contagion parameter in the NB model. From this 

nonlinear relationship, Beall (1942) derived the variate, 

where the vasiance of y is consmt. 

Thus, by estimting b in terms of the sample man and 

frequency of zero counts and applying the transformation, 

one obtains the variate y that is theoretically appropriate for 

use in parmtric statistical analyses. It can be shown that 

the square root and logarithmic trans 

cases of Beall's (1942) transformation. 
Table l4tatistics from Weld experiment in the United States and Canada 

Treat- Replicaeon Replication 

Test ments in space in time ma s2/m gb Pr >X2 

a m is sample mean. 

k" is estimate of contagion parameter. 

Every trap had 0 or 1. 

Located in Ontario, Canada. 

Kuno (1991) cited several comprehensive textbooks and where 

review articles in a recent review of methodology for 

sampling and analyzing insect populations. Work has a and b = constants. 

continued on the use of variance-mean relationships to 

describe insect populations. Two types of relationships, From this, Bliss (1941) derives the variance stabilizing 

empirical and deductive, have been used to describe the transformation, 

variance-mean relationship. One is exemplified by the 

empirical power equation, 

where 

(6) b = estimate from data fitted to the variance-mean model. 
A second variance-mean relationship is Iwao and Kuno's 

(1968) deductive quadratic equation, 

where 

A and B = parameters. 

This equation includes Beall's (1942) formula as a special 

case. The variance stabilizing transfomtion is also quite 

similar to Beallfs (1942) transformation. Because we were 

confident that the NB model would fit our data, Beall's 

(1942) transformation seemed the most appropriate. We 

saw no need for the generality of the deductive equation of 

Iwao and Kuno (1968). For comparison, we chose logtransformed counts, the power bransfomtion, ranked 

values, and untransforrned counts as additional analysis 

variables. Alternative approaches for optimizing power 

functions are described elsewhere (Bliss 1941, Box and Cox 

1964, Perry 1987). 

Computer Simulation 

A desirable design and analysis controls Type I error and 

has an acceptable level of power for F-tests of treatments. 

Our intent was to evaluate alternative designs and 

transformations with respect to these criteria. However, an 

analytical solution would be very complex because it would 

require deriving the distribution of a pseudo F-statistic when 

the underlying transformation does not have a normal 

distribution. Thus, we elected to simulate a range of 

experimental and data-related conditions and compute 

observed power and Type I error rates for F-tests of 

treatments. 

Fixed and Variable Factors 

We used a randomized complete block (RCB) design to 

compare alternative methods of analyses because this design 

was used most often in the 19 United States-Canadian 

studies . The number of treatments was constant at five 

because this was a typical value. Number of blocks was 

two, four, or six. It was assumed that insects were collected 

and empty traps reinstalled on three separate occasions 

1 week apart in each simulated experiment and on six 

occasions 1 week apart in a second replication of these 

experiments. 

Differential treatment effects were produced in simulated 

studies by generating data for a treament or a group of 

treatments from a specified NB distribution. 

Random NB counts were generated by first defining a 

gamma variable (Boswell and Patil 1970) with shape 

parameter 

(the NB contagion parameter) and scale parameter 

p=pn. 

where 

p = the mean of the NB distribution. 

Using SAS function RANGAM (SAS Institute Inc. 1988a), 

we generated values of the gamma variable, 

Subsequently, the value of x was used as the mean of a 

Poisson distribution (Johnson and Kotz, 1969) and SAS 

function RANPOI was used to generate a random value of 

Boswell and Patil(1970) show that y has a NB distribution 

with mean p and contagion parameter k . Thus, when 

selected treatments had different means, so did the NB 

distribution treatments. The parameter k was varied to 

include the range of estimates obtained from analyses of the 

United States-Canadian studies, 

k = 0.1, 0.5, and 0.75. (1 3) 

The likelihood of a significant F-test depends on the range 

of treatment means, 6 = maximum mean-minimum mean, 

and the configuration (spacing) of means within the range. 

In this simulation study, we used treatment-man 

configurations described by Young and Young (1991) plus 

one configuration where all means were equal (fig. 1). 
Figure 1-Treatment-mean configurations used in computer simulations. 

(A) Two subsets of two or more treatments-i.e., treatment means are 

equal within subsets but different between subsets; number of NB 

populations = 2. This configuration maximizes the power of the F-test of 

treatments. (B) All treatment means equal except the smallest and largest 

which differ from one another; number of NB populations = 3. This 

configuration minimizes the power of the F-test of treatments. (C) All 

treatment means equal, except one; number of NB populations = 2. This 

configuration results in intermediate power. 

The smallest mean in each configuration has a value of 0.5. 

This mean is intended to simulate the value of a control 

treatment. For a given value of k, the control-treatment 

mean will have a much smaller variance than a treatment 

with a mean count of six because the variance of a mean 

count is 

where 

n = number of replications. 

When no contrasts involving the control treatment are 

planned, some data analysts exclude the control treatment 

from analysis and reduce overall heterogeneity of variance. 

However, we elected to judge the perfomance of selected 

transfomations in the presence of maximum heterogeneity. 

In the simulations, each configuration in figure 1 was 

associated with each of the three values of k to define 

hypothetical NI3 populations. Additional populations were 

defined by using ranges of 2,4, and 6 and sampled 1,000 

times for each RCB design with both 3 and 6 replications in 

time. 

Comparing Transformations in Analyses 

Each simulated study data set was subjected to: (1) A 

parametric ANOVA with SAS procedure GLM (SAS 

Institute Inc. 1988b) on mean count per trap by treatment 

and block with means based on untransfomed data 

averaged over both three and six collection dates; (2) 

parametric ANOVA's on mean count per trap with means 

based on the transfomations, 

y = log (x t- 1) 

where 

2 = the mean count for a given trap, 

k^ = the estimated NB contagion parameter, and 

b = the estimated parameter for the power function. 

and (3) an ANOVA on ranks by block of the mean counts 

described in (1). We elected to forego classic nonparametric analyses. Friedman's test would be the 

nonparametric method of choice, but Conover (1980) states 

that the parametric F-test on ranks performs as well or better 

than Friedman's test. Our comparisons of analysis variables 

were based on the ANOVA F-tests. We conducted F-tests 

at the 1 percent, 5 percent, and 10 percent levels and tallied 

conclusions as correct or incorrect. When treatment sets 

were completely homogenous in our simulated studies, we 

computed an observed Type I error rate by calculating the 

proportion of F-tests where significance was declared at the 

nominal level. These observed proportions derived from the 

five methods of transfodng counts were compared. The 

best analysis variable will yield a proportion that is the 

smallest and does not exceed the nominal rate. 

In simulated studies with differential treatment effects, we 

computed observed power of Ptests. This is the proportion 

of F-tests at 1 percent, 5 percent, and 10 percent that 

correctly reject the null hypothesis and detect treatment 

differences. Observed power is used as a second criterion 

for comparing performance of the five analysis variables. 
Table 2-Factors studied in simulation expe~ment 

Factors used Levels of factors 

Analysis variables Y" RY~ we BY^ PY~ 

Significance level for tests (a) 0.01,0.05,0.10 

Contagion parameter (k) 0.10,0.50,0.75 

Replication in space (B) 2,4,6 

Replication in time (R) 3,6 

Range, maximum man-minimum mean, (6) 2,4,6 

Treatment-mean configuration A, B, Cf 

a Untransfomed counts, 

Ranked values of counts. 

Logarithmic transformation, log (x+l). 

Beall's (1942) transformation, sinh -' (C /x)Os' (8)O.'. 

Power transformation, x l-bn. 

Configurations A, B, and C are described in figure 1. 

We judged the relative performance of each analysis 

variable and checked for consistency among designs of 

different size, different levels of contagion in NB 

populations, different treatment-mean configurations, and 

different ranges of treatment-mean values and between 

different numbers of replication in time (table 2). 

Results 

Modeling Field Data 

Table 1 shows estimates of NB population parameters p and 

k for the field data. The estimated mean, (m), is defined as 

the mean insect count per trap averaged over collection 

dates, block, and treatments. Experimental estimates range 

from slightly more than seven to less than one. Treatment 

means (not shown) within experimnts range from 14 to less 

than 1. The low experimental estimates of k, C s 0.67, 

indicate a high degree of contagion. The high variancemean ratios are characteristic of the NB distribution. 

Goodness-of-fit tests based on Pearson's (1900) chi-square 

statistic were performed with data from each field test. This 

test cowares observed frequencies in count categories 

0,1,2, ... with expected frequencies detedned by the 

proposed model. Categories with expected frequencies < 1 

are combined (Snedecor and Cochran 1967). The NB 

model fit well, 0.10 I, P I, 0.87, in 15 of the 19 field studies. 

Of the remaining four tests, three had P 5 0.02 and the 

fourth only had two count categories, 0 and 1, prohibiting 

the goodness-of-fit test. 

Computer Simulation Results 

Type, f Error-Type I errxs occuxed when &eatmemts in 

the homogeneous group were declared significantly 

different. Type I error rates were observed in simulated 

analyses with each transfomtion for all combinations of 

k-values, replication in time, number of blocks, and a-levels 

of 0.01,0.05, and 0.10. Results were good for all analysis 

variables. For all transformations except ranks, the 

observed Type I error rate never exceeded the nominal value 

by more than 1 percent. Occasionally, observed rates in 

analyses of ranked values exceeded the nominal value by 

2 percent- still not a serious discrepancy. 

Treatment Mean Configuration A-Results for 

treatment-mean configuration A are shown in figures 2 to 4 

for various combinations of k, replications (R) in time and 

range (6) in treatment means. Figure 2 (A-F) shows 

observed power versus number of blocks when testing at the 

0.10 level for the highest level of contagion in the data (k = 

0.1). Men testing at the 0.05 level (k = 0. I), observed 

power was generally well below 0.80 regardless of required 

test precision (size of 6) and amount of replication (not 

shown in figure 2). In these figures, graphs for ranked 

values, log-transfomed values, and Beall's (1942) 

transformation generally dominate. 

Men k = 0.1, testing at the 0.10 level still does not result in 

a satisfactory analysis in most cases. The number of blocks 

needed to detect a treatmnt-man difference of 6 = 2 

appears to be at least 10. 
(A) R=3, b=2, k=0.1, a=O. 1 0 (B) R=6,Zi=2,k=0.iya=.0.i0 

0.3 r 0.6 r 

(C) R=3, b=4, k=0.1, a-0.1 0 

OS5 1 

(E) R=3, b=6, k=0.1, a=0.10 

0a6 i 

(D) R=6, b=4, k=0.1, a=0.10 

0.8 r 

/ 

(F) R=6, 6=6, k=O. 1, a=0.10 

Number of Blocks Number of Blocks 

Figure 2 (A-F)--Relationships between observed power of F-tests and number of blocks when treatment-mean configuration = A, k = 0.1, and a = 0.10 for 

five functions of insect counts: ranks (. . . . .), logarithmic transfornation ( - - ), power transfornation ( - - - ), Beall's (1942) transfornation 

(-), and the untransformed count (- . . - *). 
(A) R=3, 6=2, k=0.5, az.05 

0.8 

(C) R=3, 6=4, k=0.5, a=.05 

2 4 6 

(D) R=6, 6=4, k=0.5, a=.05 

(E) R=3,6=6,k=0.5,a=.05 (F) R=6, 6=6, k=0.5, a=.05 

1.1 r 

Number of Blocks Number of Blocks 

Figure 3 (A-WRelationship between observed power of F-tests and number of blocks when treatment-means configuration = A, k = 0.5, and a = 0.05 for 

five functions of insect counts: ranks (. . . . .), logarithmic transformation ( - - ), power transformation ( - - - ), Beall's (1942) transformation 

(-1, and the untransfonned count (- . - . .). 
(A) R=3, b=2, kz0.75, a=.05 

1.0 r (B) R=6,b=2,k=0.75,a=.05 

(C) R=3, b=4, k=0.75, a=.05 (D) R=6, b=4, k=0.75, a=.05 

(E) R=3,6=6,k=0.75,a=.05 

2 4 6 

(F) R=6,b=6,k=0.75,a=.05 

I-' i 

Number of Blocks Number of Blocks 

Figure 4 (A-%Relationships between observed power of F-tests and number of blocks when treatment-man configuration = A, k = 0.75, and a = 0.05 for 

five functions of insect counts: ranks (. . . . .), logarithmic transformation ( - - ), power transformation ( - - - ), Beall's (1942) transfornation 

(-1, and the untransfod count (- . . - -1. 
When 0.5 s k I 0.75 and a = 0.05 (figures 3 to 41, four or 

five blocks are generally needed to keep observed power at 

or above 0.80 regardless of mount of replication in time and 

value of 6. Occasionally, three blocks suffice if 6 2 4 with 

six replications in time. When testing at the 0.01 level, the 

same trends are observed, but observed power shifts 

downward too much. 

In most comparisons, analysis variables of choice are ranks 

and the log transfomation. In many cases, the choice 

between these two variables in of power is not clear, 

and other factors such as relative performance with respect to 

Type I error and ease of analysis must be considered. 

Treatment-Mean Configuration B-Of the three 

configurations tested, configuration B analyses result in the 

lowest values of observed power. In figure 1, this is 

intuitively clear because the range involves only two means 

resulting in minimal replication. The low power poses a 

serious problem to the researcher with limited resources. 

When k = 0.1 and a = 0.10, the experiment with the most 

replication will not provide sufficient power to detect the 

largest value of 6 considered in this study. 

When k = 0.50 and a = 0.05, the largest experiments do not 

provide sufficient power unless R > 3,6 > 4, or both. When 

R = 3, 6 = 6, and B = 6, observed power is near 0.8 for tests 

of all analysis variables except the untransfonned count. If 

we increase a to 0.10, observed power can be kept at or 

above 0.8 with four to six blocks depending on values of R 

and 6. 

When k = 0.75 and a = 0.05, observed power can be kept 

above 0.8 with four, five, or six blocks depending on values 

of R and 6. If R = 6 and 6 = 6, three blocks will be 

sufficient. 

Treatment-Mean Configuration C-When means are in a 

C configuration, observed power in analyses of treatments is 

between maimum-power-A-configuration and minimumpower-B-configuration values. However, when k = 0.1 and 

a = 0.10, the experiments with the most replication provided 

values of observed power below 0.8. When 0.5 5 k 5 0.75 

and a = 0.05, observed power can be maintained at 0.8 or 

better with four to six blocks depending on the value of R 

and 6. 

A Numerical Example with the Computer Program 

We assume that users will want to know the value of 

observed power for various combinations of controllable 

factors. Because our tables and graphs are cumbersome, we 

developed a partidfly interactive SAS computer program 

(SIMPm) that computes observed power for any 

combination of experimental factors used in our study. 

However, this is not a prediction procedure and the user 

must choose input values only from those used in our 

simulation studies. The program simply allows the user to 

determine a tabulated entry of power without handling 

volumes of computer simulation output. 

The following example illustrates the use of SIMPWR with 

some basic input from Test 7b data, table 1. From table 1, 

we obtain and input a value of 4.5 for m, the experimental 

mean. We also input the proportion of traps with zero 

counts, f, = 0.35 (not shown). Finally, we must input the 

values shown in figure 5. In this example, we choose to test 

at the 5 percent level, a = 0.05. The values of m and f, are 

used by the program to estimate k = 0.43. The program 

determines that this estimate is nearer the value of k = 0.5 

than 0.1 or 0.75. Thus k = 0.5 is used in the program. We 

choose B = 4, the number of proposed blocks. Three 

replications in time are proposed for each treatment, and we 

wish to detect differences between means as close ;bs two 

units apart, 6 = 2. Furthermore, we assume that true 

treatment means have the A configuration. Information 

required by the window shown in figure 5 is entered, and 

used to calculate the observed power. Both input values 

and observed power are printed as output. In this case 

POWER = 0.49 and 0.56 for the logarithmic and rank 

transformations, respectively. These values are 

unacceptably low and we must change experimental or test 

factors to achieve POWER 2 0.8. 

If the design is installed, we are restricted to changes in 

statistical test attributes. In this second run, we set a = 

0.10 with additional factors unchanged. This results in 

POWER = 0.66 and 0.69 for the logarithmic and rank 

transformations, respectively-still unacceptably low. A 

third possibility is to set 6 = 4 and set other factors at the 

original values. This results in POWER = 0.78 and 0.81, 

respectively, for the logarithmic and rank transformations. 

These values may be regarded as acceptable, but the 

analysis will be less sensitive in detecting treatment-mean 

differences. 

Suppose we have estimates f , and m from a preliminary 

sample and are at liberty to choose quantity of replication in 

space and time to control power. We run SIMPWR again 

with all factors at their originally proposed values except 

R = 6. This results in POWER = 0.82 and 0.88 for the 

logarithmic and rank transformations, respectively. Linear 
:I~RO 

Comand ===> 

WLCOPJZE: TO THE SIMULATED POWER PROG 

ENTER VALUES FOR THE VARIRBLES ALPHA, FO, &I, B, R, DELTA 

RMD COWIGURATION. AFTER DATA IS ENTERED FOR THE LAST 

VARImLE THE SIMUUTED POWR OF THE F-TEST WILL BE COMPUTED. 

PRESS ENTER TO START DATA ENTRY. 

:1m 

Comand ===> 

ALPHA .05 FO .35 M 4.5 B 4 I 

I 

I 

R 3 DELTA 2 CONFIGURATION A 

SMULAED POWER PROGRAM FOR F-TESTS 

THE INPUT DATA AND FOUR SETS OF OUTPUT 

CONFIG TRANSFRM ALPHA FO M KHAT K B R DELTA POWR 

A RANK 0.05 0.35 4.5 0.43 0.5 4 3 2 0.56237 

A LOG 0.05 0.35 4.5 0.43 0.5 4 3 2 0.48549 

A RANK 0.1 0.35 4.5 0.43 0.5 4 3 2 0.68612 

A LOG 0.1 0.35 4.5 0.43 0.5 4 3 2 0.66366 

A RANK 0.05 0.35 4.5 0.43 0.5 4 3 4 0.80943 

A LOG 0.05 0.35 4.5 0.43 0.5 4 3 4 0.77834 

A K 0.05 0.35 4.5 0.43 0.5 4 6 2 0.88025 

A LOG 0.05 0.35 4.5 0.43 0.5 4 6 2 0.82 186 

Figure 5-Input window and four sets of output from SAS program SIMPWR. 
interpolation is possible, but must be used with caution 

because relationships are not linear. In this example, we 

may average the values of POmR corresponding to the 

rank transfomtion at R =3 and R = 6, (0.81 + 0.88)/2 = 

0.845, which corresponds to a value of R between 3 and 6. 

If POVVlER = 0.845 is acceptable, R = 5 would be a prudent 

choice. 

Discussion and Conclusions 

For a given treatment-mean configuration, observed power 

of the F-test for a given analysis variable is an increasing 

function of a, k, number of blocks, number of replications 

in time, and range in treatment means to be detected. The 

volumes of graphs and tables produced with computer 

simulations present a complex picture. This occurs in part 

because most independent factors interact with at least one 

other independent factor to exert an influence on observed 

power. For instance, the effect of increasing number of 

blocks depends on the level of k (figs. 2 to 4). However, the 

following generalizations can be made: 

The level of power relationships for all analysis variables 

depends on treatment-mean configuration. For a given 

relationship, configuration A produces maximum power. 

For the same relationship, configuration B produces 

minimum power and configuration C intermediate power. 

All three configurations occur in pheromone field tests. 

However, if a researcher cannot predict which 

configuration will occur, we recommend using 

configuration B. This is a conservative approach and 

should help the researcher avert an overestimation of 

power for a proposed experiment. 

Regardless of other factors, the analysis variables 

cosfesponding to rank, logarithmic, and Beall's (1942) 

transformations generally dominate the other two with 

respect to observed power relationships. The 

transfomtion of choice is generally between ranked 

values and log-transform& values and in many cases, 

choosing will depend on how one views ease of analysis. 

It is not surprising that the logarithmic transfomtion has 

a stabilizing effect on x. The discrete variable x has a 

frequency distribution which is skewed to the right and 

resembles the lognormal distribution of the continuous 

variable z where log (z) has a normal distribution and the 

estimator for the mean of z is, 

where 

2 = the sample mean, and 

2 = the sample variance (Johnson and Kotz 1970). 

This estimator should be used for the discrete variable x, 

where 

m = a treatment mean, and 

2 = the ANOVA error mean square on the log scale. 

Beall's (1942) transformation was not competitive as a 

function of k. This transformation will perform best when 

the true value of k is known, which never occurs in practice. 

So, even though we knew the value of k for our 

distributions, we estimated the parameter with our data. We 

believe this estimation error in k causes Beall's (1942) 

transfornation to be less competitive. In our initial 

simulations, we inadvertently used the true value of k and 

Beall's (1942) transformation performed well. 

Ranked values competed well for several reasons. These 

NB distributions have a lot of positive skewness which 

results in low power for the parametric F-test on raw counts. 

Skewness is not as much an issue with ranked values 

because the test statistic is not a function of the raw data, 

which reflect the magnitude of the skewness directly. Test 

statistics may be identical with normally distributed data or 

with highly skewed data as long as the ranks are identical, 

so power is increased by using the rank transformation, 

which apparently alleviates the problem of skewness as well 

as log transfornation. 

* The level of contagion in the data has a dramatic effect on 

observed power. This is explained by the influence of k 

on the variance of individual NB counts for a given mean, 

and the inverse relationship between power and the 

variance. For k = 0.1, 0.5, and 0.75, the variance of x is 

respectively p + 10 p2, p + 2p2, and p + 4p2/3. When k r, 

0.1, mntaining a satisfactory level of power is difficult 

without making compromises such as increasing the level of 

significance for testing, increasing 6 = maximum mean 

difference to be detected, or increasing replication. 
As shown in the numerical example, SMPM can be used 

to facilitate the choice of analysis for existing data. 

Furthemre, if preli~nary estimtes f, and m are 

available, the program can be used to evaluate tradeoffs and 

detedne affordable experimntal factors for a future 

design with a given set of statistical test attributes. 

Acknowledgments 

We thank John C. Schneider, Department of Entomology 

and Plant Pathology, Mississippi State University and Floyd 

Bridgwater, USDA Forest Service, Department of Genetics, 

North Carolina State University for helpful reviews of an 

early draft of the manuscript. 

Literature Citations 

Anscornbe, FJ. 1949. The statistical analysis of insect counts based on the 

negative binomial distribution. Biornetrics. 5: 165-173. 

Bartlett, M.S. 1936. Square root transformation in an analysis of variance. 

Journal of the Royal Statistical Society. 3: 68-78. 

Beall, G. 1942. The transfornation of data from entomological field 

experiments so that the analysis of variance becomes applicable. 

Biometrika. 32: 243-262. 

Birgersson, G.; DeBarr, G.L.; de Groot, P. [and others]. 1995. 

Pheromones in white pine cone beetle, Conophthorus coniperda 

(Schwarz) (Co1eoptera:Scolytidae). Journal of Chemical Ecology. 21 : 

143-167. 

Bliss, C.I. 1941. Statistical problems in estimating populations of Japanese 

beetle larvae. Journal of Economic Entomology. 34: 221-232. 

Boswell, M.T.; Patil, G.P. 1970. Chance mechanisms generating the 

negative binomial distributions. In: Patil, G.P., ed. Random counts in 

scientific work, volume 1. University Park, PA: The Pennsylvania State 

University Press. 

Conover, WJ. 1980. Practical nonparmetric statistics. 2d ed. New Yo&: 

John Wiley. 493 p. 

DeBarr, G.L.; de Groot, P.; Birgemon, G. [and others]. [In press]. 

Field bioassays of spthetic pheromones and host monoterpena for 

Conophthorus coniperda (Co1eoptera:Scolytidae). Environmental 

Entomology. 

Iwao, S.; Kuno, F. 1968. Use of the regression of mean crowding on mean 

density for estimting sample size and the transfomation data for the 

analysis of variance. Researches on Population Ecology. 10: 2 10-214. 

Johnson, N.L.; Kotz, S. 1969. Discrete distributions. Distributions in 

statistics. New York: John Wiley. 328 p. 

Johnson, N.L.; Kotz, S. 1970. Continuous univariate distributions-1. 

Distributions in statistics. New York: John Wiley. 300 p. 

Kuno, Eizi. 1991. Sampling and analysis of insect populations. Annual 

Review of Entomology. 36: 285-304. 

Pearson, K. 1900. On a criterion that a given system of deviations from 

the probable in the case of a correlated system of variables is such that it 

can be reasonably supposed to have arisen in random sampling. 

Philosophical Magazine. 5(50): 157. 

Perry, J.N. 1987. Iterative improvement of a power transfornation to 

stabilize variance. Applied Statistics. 36: 15-21. 

SAS Institute Inc. 1988a. SAS language guide for personal computers. 

Release 6.03 ed. Cary, NC: SAS Institute Inc. 558 p. 

SAS Institute Inc. 1988b. SASfSTAT user's guide. Release 6.03 ed. Cary, 

NC: SAS Institute Inc. 1,028 p. 

Snedecor, G.W.; Cochran, W.G. 1967. Statistical methods, 6th ed. Ames, 

LA: The Iowa State University Press. 593 p. 

Williams, C.B. 1937. The use of the logarithms in the interpretation of 

certain entomological problems. Annals of Applied Biology. 24: 404- 

414. 

Young, LJ.; Young, J.H. 1991. Alternative view of statistical hypothesis 

testing. Environmental Entomology. 20: 1241 - 1245. 

Box, G.E.P.; Cox, D.R 1964. An analysis of transfomations. Journal of 

the Royal Statistical Society B. 26: 21 1-252. 
The use of trade or firm names in this publication is for reader infomation and does not imply 

endorsement by the U.S. Department of Agriculturt: of any product or service. 

This publication reports research involving pesticides. It does not contain recommendations for 

their use, nor does it imply that the uses discussed here have been registered, All uses of 

pesticides must be registered by appropriate State and/or Federal agencies before they can be 

CAUTION: Pesticides can be injurious to humans, domestic animals, desirable plants, and fish 

or other wildlife-if they are not handled or applied properly. Use all hepbicides selectively and 

carefully. Follow recommended practices for the disposal of surplus pesticides and their 

containers. 
Pepper, W.D.; Zarnoch, S.J.; DeBarr, G.L.; de Groot, P.; and Tangren, C.D. 1997. Choosing a 

Transformation in Analyses of Insect Counts from Contagious Distributions with Low Means. Res. 

Pap. SRS-5. Asheville, NC: U.S. Department of Agriculture, Forest Service, Southern Research 

Station. 13 p. 

Guidelines based on computer simulation are suggested for choosing a transformation of insect counts 

from negative binomial distributions with low mean counts and high levels of contagion. Typical values 

and ranges of negative binomial model parameters were determined by fitting the model to data from 

19 entomologicai field studies. Random sampling of negative binomial distributions was simulated and 

ANOVA's were performed on simulated data for randomized complete blocks designs with treatment 

means corresponding to means of negative binomial distributions. The influence of analysis variable, 

treatment-mean configuration, range in treatment means, significance level of statistical tests, level of 

contagion, number of blocks, and number of replication in time on observed power and Type I error of 

F-tests was studied. A computer program was developed to recompute observed power of F-tests for any 

combination of these factors. The program facilitates choosing a transformation and may also be used to 

evaluate tradeoffs and determine affordable experimental factors for a future design with a given set of 

statistical attributes. 

Keywords: Contagious distribution, entomology, transformation, variance-mean relationship, variance 

stabilization.