Download Genetic Algorithm Using SAS/IML

GENETIC ALGORITHM USING SAS IML®
David Steenhard, LexisNexis, Louisville, KY
Nan-ting Chou, University ofLouisville, Louisville, KY
ABSTRACT
The genetic algorithm is a search technique which borrows ideas
from natural evolution to effectively find good solutions for
optimization problems. What is particularly appealing about the
technique is that it is robust at finding good solutions for a large
variety of problems. In this paper we provide an introduction to
genetic algorithms and use a genetic algorithm written in SAS IML®
to solve a combinatorial optimization problem - the traveling
salesman problem. Sufficient details are provided to enable the
readers to easily use genetic algorithm written in SAS IML® as a tool
for solving additional optimization problems.
INTRODUCTION
A genetic algorithm (GA) is an adaptive search technique based on
the principles and mechanisms of natural selection and 'survival of
the fittest' from natural evolutions. GA grew out of Holland's 1967
study of adaptation in artificial and natural systems. By simulating
natural evolution in this way, a GA can effectively search the
problem domain and easily solve complex problems. However,
Researchers were uncomfortable with genetic algorithms because of
their dependence on random choices; these choices seemed arbitrary
and unpredictable. However, in the 1970s, Holland developed a solid
theoretical foundation for the technique. His theocy of schemata gives
insight into why genetic algorithm work [Goldberg 1989].
The genetic algorithm operates as an iterative procedure on a fixed
size population or a pool of candidate solutions. The candidate
solutions represent an encoding of the problem in a form that is
analogous to the chromosomes of biological systems. Each
chromosome represents a possible solution for a given objective
function. Associated with each chromosome is a fitness value, which
is found by evaluating the chromosome with the objective function. It
is the fitness of a chromosome that determines its ability to survive
and produce offspring.
other traditional techniques. The formulation of the
problem can be more flexible: The constraints and the
objective function can be non-linear or discontinuous; GAs
doesn't find these problems any more difficult than the
linear or continuous problems. This is what gives GAs their
advantages over traditional methods of optimization. But
by using a heuristic technique like a genetic algorithm you
can't guarantee an optimal solution. Users must often settle
for "near optimal" solutions. These solutions, while usually
not perfect, can suffice for a broad range of problems.
This paper illustrates how a genetic algorithm written in
SAS IML® can be used to solve a combinatorial
optimization problem (traveling salesman) and outlines the
mechanics of a genetic algorithm in detail.
TRAVELING SALESMAN PROBLEM
The Traveling Salesman Problem (TSP) is a well-known
combinatorial optimization problem. The TSP has been
proven to be one large set of problems termed ''NP
complete" (nondeterministic polynomial). NP complete
problems have no known method of solution better than
trying all possibilities. Because such an exhaustive search
is generally impractical for more than a few cities, heuristic
methods such as genetic algorithms have been applied to
find solutions that are acceptable, if not optimal.
In this application we used a IS-city tour where the
distance measure is the flying distance between the cities
(see Appendix B). The following are the cities we used in
this application: Atlanta, Anchorage, Chicago, Cleveland,
Dallas, Denver, Houston, Kansas City, Los Angeles,
Miami, New York City, Phoenix, San Diego, Seattle, and
Washington DC. We assigned each city a gene value with a
number l through 15 (e.g. Atlanta=l, Anchorage=2, ... ,
Washington DC= IS).
The basic genetic algorithm relies on four genetic operators:
selection, crossover, mutation, and replacement. The selection
operators use the fitness values to select a portion of the population
to be parents for the next generation. Parents are combined using the
crossover and mutation operators to produce offspring. This process
combines the fittest chromosomes and passes superior genes to the
next generation, thus providing new points in the solution space. The
replacement operators ensure that the 'least fit' or weakest
chromosomes of the population are displaced by more fit
chromosomes.
The smallest unit of a GA is called a gene. A gene
represents a unit of information in your problem domain.
For example, in the traveling salesman problem a gene
would be the name or a number representation of a city
that you are traveling to. A series of genes, or a
chromosome, represent one possible complete solution to
the problem.
Genetic algorithms are different from other normal optimization and
search procedures in three ways [Goldberg 1989].
Chromosome
I.
GAs search from a population of points, not a single point
2.
GAs use payoff(objective function) information, not
derivatives or other auxiliacy knowledge.
3.
GAs use probabilistic transition rules, not deterministic
rules.
THE BASICS OF A GENETIC
ALGORITHM
1~3J10J2J8l1~5l
t
Gene
Unlike many mathematical techniques, solution times with GAs are
usually highly predictable. Also, solution time is usually not radically
affected as the problem gets larger, which is not always the case with
196
In order to make use of a chromosome the GA needs to
decode it and determine how good a chromosome's
solution is for a particular problem. This operation is
carried out by the objective function. For the TSP the
objective function is used to find the minimum distance closed path
that visits each city exactly once. The objective function is simply the
total distance of the trip.
A genetic algorithm first creates a random population of
chromosomes where each chromosome is unique. By creating a
diverse population instead of a single solution, the GA is tiying many
solutions at once. After an initial random population is set up, the GA
begins an iterative process of refining the initial solutions so that the
better ones are more likely to become the top solution.
The OA experiments with new solutions by combining and refining
the information in the chromosomes using four basic operations:
sekction, crossover, mutation, and replacement. This combining and
refining process continues until the genetic algorithm converges to an
optimal value. Appendix A illustrates the complete process of a basic
genetic algorithm .
SELECTION METHODS
Selection is the process of choosing two parents from the population
for crossing. Determining which individuals should be allowed to
interchange their genetic information bas great bearing on the rate of
convergence specifically and the success of the search in general.
There are numerous different selection techniques used in genetic
algorithm software, the ones we used in the SAS IML® genetic
algorithm are: roulette, tournament, and remainder stochastic
sampling without replacement.
Roulette Seleetiou
Roulette selection is one of the traditional GA selection techniques.
The principle of roulette selection is a linear search through a roulette
wheel with the slots in the wheel weighted in proportion to the
individual's chromosome fitness value. A target value is generated,
which is a random proportion of the sum of the fitness scores in the
population. The selection method then iterates through the population
until the target value is reached. A fit individual (high fitness score)
will have a higher probability of contributing one or more offspring
in the next generation. This process is carried out until the desired
population size is met. The following SAS IML® code performs
roulette selection.
start roulette(mate,popsize,fitness,nchrome,
iparent);
/* Find the cumulative sum of the fitness
scores and select a member of the
population by using a weighted (by fitness
score) random number. */
Remainder Stocbastie Sampling Without Replacement
Selection
For this method the probabilities of selection are calculated
by preselect; =jj
jj where jj is the fitness score
/L
of the ith chromosome. Then the expected number of each
chromosome c 1 is calculated c1 = preselect1 • n where
n is the total number of chromosomes in the population.
Each chromosome is allocated to the population according
to the integer part of the c1 values. The fractional parts of
the expected number values are treated as probabilities.
One by one, weighted coin tosses (Bernoulli trials) are
performed using the fractional parts as success
probabilities. This procedure continues until the desired
population size is reached.
CROSSOVER OPERATORS
After selection, crossover is then perform on the selected
parents (chromosomes). Crossover which occurs in nature,
takes two chromosomes and basically swap some of their
information gene for gene. The resulting chromosomes,
called children, have a piece inherited from each of their
parents. Applying crossover to the pairs of chromosomes
proceeds by choosing a random number between 0 and 1 to
determine whether they cross over. If this random number
is greater than a specified crossover probability then
crossover is performed. The crossover probability is usually
chosen between 0.6 and 0.8 [Mitchell1996].
There are several crossover methods found in the GA
literature [Mitchell 1996, Goldberg 1989], but there is no
clear method that works the best in all situations. The
success or failure of a crossover method depends on
complexity of the fitness or objective function, the
encoding of the chromosomes, and other details of the GA.
The two crossover methods that we used in the SAS IML®
genetic algorithm are: fu:ed-position sequential cross(IVer
andjixed-position cross(IVer. Traditional crossover
methods will not work for order-based problems, since
order is not preserved which is needed for the traveling
salesman problem. The fixed-position sequential and
fixed-position crossover operators are similar to the
partially matched crossover (PMX) operator. The following
diagrams outline the fixed-point sequential and fixed-point
crossover operators.
Fixed-Point Sequential Crossover
partsum~cusum(fitnessJ;
rand=ranuni(O)*(sum(fitness));
k~O;
do until((partsum(k,] >=rand) I (k=popsize));
k=k+l;
mate==k;
end;
finish;
Tournament Selection
Tournament selection is a method where successive pairs of
individuals are randomly drawn from the population. After drawing a
pair, the chromosome with the higher fitness score is declared the
winner, inserted into the new population, and another pair is drawn.
This process continues until the desired population size is reached.
197
One at random selected part (consisting of consecutive
genes) of parent I chromosome is copied into offspring 2 at
exactly the same location. The order and location is
preserved. The genes that are not selected from parent I
are copied from parent 2 to fill out the empty spaces in
offspring 2. The order of the genes in parent 2 is preserved
but the exact locations are not. Perform similar steps to find
the chromosome for offspring I.
ichild[,j)-pmxl;
if nchild=O then ichild[,j+l)=pmx2 ;
end;
nextl:
free pmxl pmx2;
finish;
Fixed-Point Crossover
::::
MUTATION OPERATOR
Selection according to fitness scores combined with
crossover operators gives genetic algorithms the majority of
their processing power and the mutation operator plays a
secondary role. Mutation rarely occurs in nature and is the
result of a miscoded genetic material being passed from the
parent to the offspring. This mutation of the chromosome
may represent an improvement in the fitness scores, or
may cause harmful results. The mutation rate is very low in
nature and is usually kept quite low in genetic algorithms.
Usually the mutation probability is Pm = I/ population.
The following diagram outlines the mutation operator.
The genes of parent 1 are selected at random and copied into
offspring 2 at exactly the same location as they appear in parent 1.
Thus, both order and position of the genes in parent l are preserved
in offspring 2. The genes that are not selected from parent l are
copied from parent2 to fill out the empty spaces in offspring 2. The
orders of the genes in parent 2 are preserved but not the exact
locations. Perform similar steps to find the chromosome for
offspring l. The following SAS IML® code performs fixed-point
crossover.
start xoverfp(pcross,n chrome,ichild,ip arent,
nchild,j,matel,m ate2);
*
Mutation
Mallen
Perfonn fixed position crossover between
the randomly selected parents;
M1171ii2M 9l4l3l&t-il117l5l211 9!413!&1
B!iaaMlalicn
pmxl=repeat(O,nc hrome,l);
pmx2=repeat(O,nc hrome,l);
ncross=O;
if ranuni(O) > pcross then goto nextl;
-
Mlalicn
Select at random one gene from an offspring and swap it with
another randomly selected gene in the same offspring.
do n=l to nchrome;
if ranuni(O) <= 0.5 then do;
ncross=ncross+l;
pmxl[n,)=iparent [n,matel];
pmx2[n,)=iparent [n,mate2];
end;
end;
REPLACEMENT MEmODS
Replacement is the last stage of the cycle. Two parents are
selected from a fixed size population, they crossover
fonning one or two children, and there is mutation,
however not all children and parents can return to the
population, so some of them must be replaced. The
technique used to decide which individuals stay in the
population and which ones are replaced are very important
in the convergence of the genetic algorithm.
There are numerous different replacement techniques used
in genetic algorithm software, but the ones we use in the
SAS IML® genetic algorithm are: both parents, and
weakest individual replacement
* If there is only one child produced;
if ncross > 0 then do;
do n-1 to nchrome;
check=iparent[n,m ate2);
do m=l to nchrome;
test=any(pmxl= check);
if pmxl[m,)=O & test=O then do;
pmxl [m, J =check;
goto found;
end;
end;
found:
end;
Both Parents
Both parents replacement is simple, the children replace the
parents regardless of their fitness scores. Under this method
each individual only gets to breed once. This keeps the
population and genetic material moving around
* If there is two children produced;
Weakest Individual
This replacement method replaces the weakest individuals
in the population with the children, given the children are
fitter. The parents are included in the search for the weakest
individuals. This technique rapidly illlproves the overall
frtness of the population, and works very w,ell with large
populations.
if nchild-2 then do;
do n=l to nchrome;
check=iparent[n, matel];
do m=l to nchrome;
test=any(pmx2= check);
if pmx2[m,]=0 & test=O then do;
pmx2 [m, ) =check;
goto foundl;
end;
end;
foundl:
end;
end;
SEARCH TERMINATION
The termination, or convergence criteria is what brings the
genetic algorithm to a halL The tennination techniques in
the SAS IML® genetic algorithm are quite basic, but they
work adequately. The following are brief descriptions of
198
the convergence criteria used in the SAS IML® genetic algorithm.
interc =0;
end;
Max Iterations
This convergence criterion compares the current iteration to a
specified number of iterations. lfthe current iteration is less than the
requested number of iterations it continues. Otherwise, it ends
execution.
I* Find the slope and the intercept that
will be used to re-scale the
objective scores.*/
else do;
if fmin > (fmult*avgsc r- fmax)/(fmult 1)
then do;
delta-fmax - avqscr;
slope-(fmu1t -1)*avgscr/d elta;
interc•avqscr *(fmaxfmu1t*avgsc r)/delta;
end;
else do;
delta=avgscr - fmin;
slope-avgscr /delta;
interc- -fmin*avgsc r/delta;
end;
end;
Terminate Upon Convergence
This convergence criteria checks to see if the current best-ofgeneration (best fitness score) is equal to the n previous best-ofgeneration. When using the terminate upon convergence as a
stopping criterion you must specify the number of previous
generations against which to compare.
Terminate Upon Convergence of Average
This convergence criteria compares the average fitness scores of the
current population with that of the best individual in the population.
lfthe population average is within a specified percentage of the best
individual's score the algorithm terminates.
* Rescale the objective function;
fitness= interc + slope*object ;
do i=l to popsize;
if fitness[i,] < 0 then
fitness[i,]-0 ;
end;
finish;
ADVANCED OPERATORS
This section discusses a few advanced operators that have been
implemented to the basic genetic algorithm over time. The advanced
operators discussed here is by no means a exhaustive list of the
possible improvements that have been made through the years. The
following are the advanced operators we implemented in the SAS
IML® genetic algorithm.
Sigma Truncation- Linear scaling works well except when
negative fitness scores prevent its use. To prevent this
sealing problem, Forrest [Forrest 1985] suggested using
population variance information to scale the raw fitness
scores by the following:
Fitness Sealing
Fitness scaling has become a widely accepted practice. Scaling is
done to keep appropriate levels of competition throughout the
optimization process. Without scaling. early on there is a tendency
for a few "outlier'' individuals to dominate the selection process. In
this case the objective function values or fitness scores must be
scaled back to prevent takeover of the population by these
extraordinary individuals. Later on when the population has more or
less converged, competition among population members is less
strong and the experiment tends to wander.In this case the fitness
scores must be sealed up to continue to reward the best performers.
The most common used fitness sealing methods include:
l.
2.
!'= f -(j -co-)
The constant c is chosen (user specified) as a reasonable
multiple of the standard deviation and negative sealed
fitness scores (f' < 0) are set to 0.
Elitism
Elitism is a method that is used to retain the best overall
individual at each generation. This individual could be lost
if it was not selected for reproduction or destroyed by
crossover or mutation. Elitism has been shown to improve
performance of genetic algorithms in many different
situations.
Linear Scaling
Sigma {a) truncation
Linear sealing- Calculates a scaled fitness score f' from the raw
fitness score using a linear equation of the form:
!' =af +b
to enforce equality of the raw and
chosen
are
b
and
a
The coefficients
sealed average fitness scores and cause maximum sealed fitness
scores to be a specified multiple of the average fitness score. The
fOllowing SAS IML® code performs linear sealing.
start lscale(objec t,fitness,pop size,fmult);
if fmult <= 1.2 then fmult =1.2;
avqscr = sum(object)/ popsize;
fmin = min(object);
fmax = max(object);
if fmax = avqscr then do;
slope =1;
199
Overlapping Populations (Steady-State GA)
The steady-state genetic algorithm uses overlapping
populations with a user specified amount of overlap. For
each generation, the algorithm creates new offspring that
are added to the population, and then the worst individuals
(based on fitness scores) are removed in order to return the
population to its original size.
Micro-Genetic Algorithm Operation
The micro-GA is a technique used on a smaller population
size (micro population). It first checks for convergence of
this micro population. If converged, it then starts a new
micro population with the best individual and fills in the
remaining micro population with new randomly selected
parents (chromosomes).
To determine if the micro population has converged, for
two replacement methods is probably due to the fact that
we used a constant population size (n=2SO). Worst
individual replacement works better with a larger
population.
each individual in the micro population (excluding the best
individual) count the number of genes that are not the same as the
best individual. If this total is less than S% of the total number of
genes, then consider that the micro population bas converged. The
following SAS IML® code performs the micro-GA operation.
CONCLUSIONS
start gamicro(popsize,nchrome ,iparent,
ibest,icount,tsp);
rand= repeat(O,nchrome,l);
icount =0;
do j=l to popsize;
do n=l to nchrame;
if iparent[n,j) A= ibest[n,) then
icount=icount =l;
end;
end;
Genetic algorithms are ideally suited for solving
combinatorial optimization problems like the traveling
salesman problem. We obtained good results for a 15-city
TSP using a genetic algorithm written in SAS IML®. By
developing an efficient genetic algorithm one can find good
solutions for a large variety of problems and are very
helpful when other mathematical techniques break down.
REFERENCES
/* If icount less than 5% of number of genes,
then consider the population to be
converged. */
Bagley, J.D. (1967). The behavior of adaptive systems
which employ genetic and correlation algorithms.
Dissertation Ab.ltracts Inrernational, 28(12),51068.
Berry, Michael J. A. and Linoff, Gordon (1997). Data
Mining Techniques for Marketing, Sales, and Customer
Support. John Wiley & Sons, Inc.
Chambers, Lance (editor) (1995). Practical Handbook of
Genetic Algorithms. Applications Volume I, CRC Press.
diffrac = icount/((popsize-l)*nchro me);
if diffrac < 0.05 then do;
/* Add the best individual to the new
population and randomly select new parents
to fill out the new population. */
do n-1 to nchrome;
iparent[n,l)=ibest[n,];
end;
do j=2 to popsize;
do n=l to nchrome;
rand[n,)=ranuni(O);
end;
rcity=randl ltsp[,l];
tt=rcity;
rcity[rank(rcity[,l)),)= tt;
iparent[,j)=rcity[,l];
end;
end;
free tt rcity;
finish;
RESULTS
To test the effectiveness of the SAS IML® genetic algorithm and
compare the TSP results at different parameter settings we ran 50
simulations for 12 different parameter settings. The twelve different
parameter settings consisted of changing the selection methods
(roulette, tournament, and remainder stochastic sampling w/o
replacement), crossover operators (fixed-point and fiXed-point
sequential), and replacement method (both parent and worst
individual). The population size (n=250), and fitness scaling (sigma
truncation) were held constant for each run. Table I shows the
summary statistics of the 600 simulation runs. The optimal solution
for this problem is 10,173 miles. In this case, the simulations with
tournament selection, fixed position crossover, and both parent
replacement performed the best among the 12 combinations. To
compare the statistical significance of the average fitness scores
between the combinations we conducted a 3x2x2 fuctorial design.
Table 2 shows the results of this factorial design. From this table we
conclude that all the main effects (selection method, crossover,
operator, and replacement method) are statistically different from one
another. Table 3 displays all the pairwise comparisons using Fishers
LSD.
One important note here is that the statistical difference between the
200
Dhar, Vasant and Stein, Roger Stein (1997). Seven
Methods for Transforming Corporate Data Into Business
Intelligence. Prentice Hall Inc.
DeJong, K. A. (1975). An analysis of the behavior of a
class of genetic adaptive systems. Dissertation Ab.ltracts
International 36( 10) 51408.
Forrest, S. (1985). Scaling Fitnesses in the genetic
algorithm. In Documentation for PRISONERS DILEMMA
and NORMS Programs That Use the Genetic Algorithm.
Unpublished manuscript.
Gillies, A. M ( 1985). Machine learning procedures for
generating image domain feature detectors. Unpublished
doctoral dissertation, University of Michigan, Aon Arbor.
Goldberg, D. E. (1989). Genetic Algorithms in Search,
Optimization, and Machine Learning. Addison-Wesley.
Mitchell, Melanie (1996). An Introduction to Genetic
Algorithms. MIT Press.
SAS Institute Inc. (1999). SASIIML User's Guide, Version
8, Cary NC: SAS Institute Inc.
CONTACT INFORMATION
Your comments and questions are valued and encouraged.
Contact the authors at:
David Steenbard
LexisNexis
7107 Mallgate Place
Louisville, KY 40207
Work Phone: (502) 721-8502
E-mail: david.steenhard@!exis-nexis com
SAS is a registered trademark of SAS Institute Inc. in the
USA and other countries. ® indicates USA registration.
SAS IML® GENETIC ALGORITHM PROCESS
(APPENDIX A)
Step 1
Randomly initialize a population
of parent chromosomes
(of size n)
...
1
Stepl
Evaluate each chromosome in
the population by calculating
their fitness score (perfonn
fitness scaling if desired)
Step3
Select parent chromosomes
for reproduction based on
their fitness score.
•
_t_
Step4
Stop and return the
best chromosome
(solution set)
StepS
Yes
Step6
Apply crossover and mutation as
the parent chromosomes
reproduce
•
Replace selected members of the
population (based on replacement
method) with the new members.
~
No
or maximum generation
reached
FLYING DISTANCE BETWEEN 15 U.S. CITIES
(APPENDIX B)
Adam Anfuag::
Anfuag::
599
<le\elmxi
548
2,1111
3,030
llllla;
72!J
1,190
2,998
2;?Rl
680
3,218
1,7Jl
1,311
3,945
Mllli
NYC
Phlenix
Smlleg:l
Seattle
oc
~ llllla;
Il:nver H:liSal
KC
lA
Mllli
NYC PIJJeoix Sat lleg:l Seattle
3,36S
OJica&o
Il:nver
H:liSal
KC
lA
ClJil:¥
670
1,917
S89
751
1,562
1,862
1,149
540
3,322
2,515
2,417
1,416
3,322
310
i92
886
915
398
1,718
1,183
1ll
1,418
1,(98
1,@3
ro2
um
1,194
l,f117
681
2,022
1,00!
412
1,711
1,995
1,989
284
636
222
44S
1,216
1,106
1,368
8SS
1,152
1,636
1,174
853
S43
837
1,8)2
1,606
582
828
1,003
1,464
62S
1,358
950
1,348
t;m
1,~
994
1,283
l,&l8
1,191
1,031
1,317
1,478
930
For a 15-city tour there are 151 (1,307,674,368,000), possible tours
and 15!130 (43,589,145,600) of the tours are unique.
201
1,223
1,317
2,432
36S
108
942
1,275
l,Cl!4
1,942
2,233
1,685
910
2,116
1,403
2,375
211
300
1,002
1,948
1,037
2,2111
2,292
SAS IML® GENETIC ALGORITHM RESULTS
(APPENDIX C)
Table 1: Genetic Algorithm 15-City Tour Overall Performanee
10,408
10,505
10,415
10,655
10,635
10,769
10,445
10,616
10,412
Stochastic w/o Replacement
Stochastic w/o Replacement
Stochastic w/o Replacement
10,381
10,381
10,358
10,655
10,531
10,681
10,412
10,445
10,358
11,033
11,224
11,116
11,636
11,505
12,671
11,631
12,036
11,454
38%
26%
36%
18%
18%
14%
32%
28%
40%
16%
Table 2: Full Three-Factor Factorial Design Model
Source
Selection Method
Crossover Operator
Selection*Crossover
Replacement Method
Selection*Replaeement
Crossover*Replaeement
Selection*Crossover*Replaeement
Error
corrected Total
Table 3: Pair wise Comparisons
GA Parameter Settings Comparisons
Selection Method Comparison
Tournament - Roulette
Tournament - Remainder Stochastic w/o Replacement
Remainder Stochastic w/o Replacement - Roulette
Crossover Ooerator Comparision
Fixed Point Crossover - Fixed Point Sequential
Renlaeement Method Comparision
Both Parents - Worst lndividual
DF
AnovaSS
4,018,841
1,420,872
367,215
5,310,628
545,910
6,977
569,883
76,258,973
88,499,299
2
1
2
1
2
1
2
588
599
Ddterenee Between
Means
Mean Sguare
2,009,420
1,420,872
183,608
5,310,628
272,955
6,977
284,942
129,692
95% Confidence Limits
-195.49
-136.22
-59.27
-266.48 to -124.49 •••
-207.21 to -65.23 •••
-130.26 to 11.73
-97.33
-155.29 to -39.36 •••
-188.16
-246.13 to -130.19 •••
*** Significant at 0.05 level.
202
FValue
15.49
10.96
1.42
40.95
2.10
0.05
2.20
Pr>F
0.0000
0.0010
0.2436
0.0000
0.1228
0.8167
0.1120