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Gary E. Erickson 

Langley Research Center, Hampton, Virginia 

Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802

 

Acknowledgments 

The author would like to thank Mr. Richard Deloach of the Aeronautics Systems Engineering Branch, Systems 

Engineering Directorate, NASA Langley Research Center for his pioneering efforts in advocating, developing, and 

validating Modern Design of Experiments techniques in aerospace testing. Special thanks are extended to Dr. 

James R. Simpson, Professor, Industrial Engineering Department, Florida A&M University and Florida State 

University, and Dr. Drew Landman, Professor, Aerospace Engineering Department, Old Dominion University, for 

their significant contributions to the graduate-level Design of Experiments courses at NASA Langley Research 

Center and for their efforts in developing, managing, and operating the Old Dominion University low-speed wind 

tunnel as a tool for the advancement of statistically-designed experiments. 
v

Table of Contents 

Table of Contents .......................................................................................................................................... v

Abstract ......................................................................................................................................................... 1

Introduction ................................................................................................................................................... 1

Nomenclature ................................................................................................................................................ 2

Experimental Design Method and Procedures .............................................................................................. 3

Test Information ............................................................................................................................................ 5

Discussion of Results .................................................................................................................................... 6

First-Stage Analysis .................................................................................................................................. 6

Second-Stage Analysis .............................................................................................................................. 9

First Step ............................................................................................................................................... 9

Second Step ......................................................................................................................................... 11

Third-Stage Analysis .............................................................................................................................. 11

Concluding Remarks ................................................................................................................................... 13

References ................................................................................................................................................... 14

Tables.................................................................................................................................16

Figures.............................................................................................................................................18 
1

Abstract

A procedure to analyze a split-plot wind tunnel experimental design 

featuring two input factors, two levels of randomization, and two error 

structures is described in this report. Standard commercially-available 

statistical software was used to analyze the test results obtained in a 

randomization-restricted environment often encountered in wind tunnel 

testing. Experimental data were obtained in a statistically-designed, 

low-speed wind tunnel test using a small-scale model of a fighter 

airplane configuration. The input factors were differential horizontal 

stabilizer incidence and the angle of attack. The response variables were 

the aerodynamic coefficients of lift, drag, and pitching moment. Using 

split-plot terminology, the whole plot, or difficult-to-change, factor was 

the differential horizontal stabilizer incidence, and the subplot, or easyto-change, factor was the angle of attack. The whole plot factor and 

subplot factor were each tested at three levels. Degrees of freedom for 

whole plot error were provided by replication in the form of three blocks, 

or replicates, which were intended to simulate three consecutive days of 

wind tunnel facility operation. The analysis was conducted in three 

stages. The first stage provided an assessment of statistical significance 

of a combined whole plot term representing the aggregate of linear and 

quadratic terms at the whole plot level, a combined subplot term 

representing the aggregate of linear and quadratic terms at the subplot 

level, and the interaction of the combined whole plot and combined 

subplot terms. The results of this preliminary assessment justified a 

second-stage analysis, which was performed in two steps to 

systematically separate out the individual terms at the whole plot and 

subplot levels. This analysis yielded the correct error terms at the whole 

plot and subplot levels, correct mean squares and multiple regression 

coefficient estimates of all terms at the whole plot and subplot levels, and 

valid tests of statistical significance at the subplot level. Statistical tests 

of significance at the whole plot level using the correct whole plot error 

terms were performed using manual input in a third stage. The 

combined analyses from the second and third stages yielded the 

estimated mean squares, multiple regression function coefficients, and 

corresponding tests of significance for all individual terms at the whole 

plot and subplot levels for the three aerodynamic response variables. 

Statistically significant terms included main effects and two-factor 

interaction terms for the lift coefficient response; main effects, two-factor 

interaction, and pure quadratic terms for the drag coefficient response; 

and only main effects for the pitching moment coefficient response. 

Introduction

The application of modern design of 

experiments (MDOE) techniques to wind tunnel 

testing of aerospace vehicles (references 1-3) 

improves the data quality and data uncertainty 

quantification and provides mathematical 

models describing the complex aerodynamic 

responses to the selected input factors. Key 

concepts that are used in designed experiments 

are randomization, replication, and blocking, 

which are described in detail in references 4-9. 

Typical responses in wind tunnel tests are the 

six-component longitudinal and lateral-
2

directional aerodynamic coefficients. Common 

input factors include the model incidence 

relative to the free stream (for example, angles 

of attack and sideslip), the model configuration, 

and the test conditions (for example, Mach 

number and Reynolds number). Support 

systems providing movement of the model in the 

wind tunnel test section are frequently 

automated, which facilitates the control of the 

angles of attack and sideslip. However, input 

factors that are difficult or time-consuming to 

change are often encountered. For example, 

scale models of military and civilian aircraft, 

missiles, manned and unmanned launch 

vehicles, and atmospheric re-entry vehicles 

typically require lengthy configuration changes 

and attendant access to the model in the test 

section. In addition, changes to the Mach 

number and Reynolds number might only be 

achieved through complex tunnel configuration 

modifications to obtain the desired expansion 

and compression ratios. Although it is well 

understood that complete randomization of the 

experimental run order is desired to average out 

the effects of all uncontrolled and unknown 

sources of variability, sometimes it is 

unreasonable, impractical, or extremely 

inefficient to conduct experiments in such a 

fashion. These factors represent a restriction on 

randomization in statistically-designed 

experiments as well as a constraint on the 

operational productivity. The use of a split-plot 

experimental design (references 10 and 11) is an 

effective means of experimenting with hard-tochange factors in a randomization-restricted 

environment. 

The objective of this report is to describe a 

split-plot analysis of a statistically-designed lowspeed wind tunnel experiment on a small-scale 

model of a fighter airplane configuration. The 

three response variables were the aerodynamic 

lift, drag, and pitching moment coefficients, and 

the two input factors were differential horizontal 

stabilizer incidence and angle of attack. 

Variation of the horizontal stabilizer incidence 

was the hard-to-change factor, since the wind 

tunnel was brought off-line to allow access to 

the test section for the manual configuration 

modifications. In contrast, the model angle of 

attack could be easily and rapidly varied during 

wind-on operation. Consequently, a simple 

split-plot design featuring two input factors, two 

levels of randomization, and two different error 

structures (references 10 and 11) was used. A 

three-stage analysis and statistical modelbuilding method derived from reference 12 was 

used, which was augmented by commerciallyavailable statistical software (reference 13) and 

spreadsheet software with statistical tools 

(reference 14). 

Nomenclature

a number of levels of the whole 

plot factor 

ANOVA analysis of variance 

AoA subplot factor, angle of attack, 

degrees

b number of levels of the subplot 

factor

cref reference wing chord, 

 3.46 inches 

Coef estimated regression coefficient 

CD drag force coefficient,

   ref

Drag Force

q S

CL lift force coefficient,

   ref

Lift Force

q S

Cm pitching moment coefficient, 

   ref ref

PitchingMoment

qS c

C. Total calculated total 

DF degrees of freedom 

DF Num degrees of freedom used in the 

numerator of the F-test

DOE design of experiments 

EMS expected mean squares 

F ratio ratio of the mean squares (MS)

for the whole model or factor of 

interest to the error mean 

square; also F
3

F-test statistical test of significance in 

ANOVA to test if the variances 

of two populations are equal 

LaRC Langley Research Center

MDOE modern design of experiments 

MS mean squares, sum of squares 

(SS) divided by its associated 

degrees of freedom (DF)

MS Num mean squares used in the 

numerator of the F-test

MSE mean square error 

NASA National Aeronautics and Space 

Administration 

ODU Old Dominion University 

Prob > F probability the test statistic 

(F ratio) will take on a value at 

least as extreme as the observed 

value of the statistic, assuming 

the null hypothesis is true (e.g. 

no effect due to a factor of 

interest); also p-value or P

Prob >|t| two-tailed probability that the 

test statistic, t, will take on a 

value at least as extreme as the 

observed value of the statistic, 

assuming the null hypothesis is 

true (e.g. no effect due to a 

factor of interest); also twotailed p-value

psf pounds per square foot 

q free stream dynamic pressure, 

6.4 pounds per square foot (psf)

r number of replicates 

Re unit Reynolds number, 0.48 

million per foot 

Rec Reynolds number based on 

reference wing chord, 0.14 

million

Rep replicate

Run experimental run in wind tunnel 

 test 

SE Coef standard error of regression 

coefficient 

ref S reference wing area, 

 36.08 square inches 

sptreat combined subplot factor 

containing linear and quadratic 

terms in AoA

Source source of variation in linear 

statistical model 

SP subplot 

SS sum of squares 

t t-ratio or test statistic from 

Student’s t distribution, 

obtained by dividing the 

regression coefficient Coef by 

the standard error of the 

coefficient, SE Coef 

Term statistical model term at the 

 whole plot or subplot level 

Tail_Angle whole plot factor, horizontal 

stabilizer incidence, degrees 

wptreat combined whole plot factor 

containing linear and quadratic 

terms in Tail_Angle 

WP whole plot 

Y response variable 

s level of statistical significance, 

acceptable probability of error 

(assumed level of significance is 

0.05)

Experimental Design Method and 

Procedures

A split-plot experimental design strategy was 

used because of the randomization restriction 

imposed by changes to the differential horizontal 

stabilizer incidence. Figure 1 illustrates the 

split-plot experimental design strategy used in 

the current investigation. Figure 2 presents 

photographs of the small-scale fighter airplane 

model used in the low-speed wind tunnel test, 

including a close-up view of the right-hand 

horizontal stabilizer. Using standard design of 

experiments (DOE) terminology (references 10 

and 11), the hard-to-change factor, differential 

horizontal stabilizer incidence (Tail_Angle), was 

designated the whole plot (WP) factor, and it 

was tested at three levels (0, +10, and +20 

degrees). In the present application, differential 

stabilizer incidence was obtained by changing 

the right-hand stabilizer incidence while 

maintaining the left-hand stabilizer at 0-degree 

incidence. An incidence angle with the 

stabilizer leading edge up is defined as positive. 
4

The easy-to-change factor, angle of attack 

(AoA), was designated the subplot (SP) factor, 

which was run at three levels (0, 4, and 8 

degrees). Positive angle of attack is “nose up” 

relative to the free stream. The simple split-plot 

design consisting of a single hard-to-change 

input factor and a single easy-to-change input 

factor can be viewed as two experiments 

superimposed on each other with two different 

randomization schemes and error structures 

(reference 12). In a split-plot experiment, the 

subplot error is often less than the whole plot 

error because the subplots are generally more 

homogeneous than the whole plots 

(reference 12). In addition, more degrees of 

freedom are usually available for the subplot 

error, which allows the subplot factors to be 

analyzed with greater precision. A challenge in 

a split-plot design is to have sufficient degrees 

of freedom for the whole plot error to perform 

credible statistical tests of significance on the 

whole plot terms. In the current experimental 

design, this was provided by replication in the 

form of three blocks or replicates (Rep)

(references 4-9). All three levels of the whole 

plot factor were tested in random order within 

each complete block or replicate. The three 

levels of the subplot factor were tested in 

random order at each level of the whole plot 

factor, which comprised an incomplete block. 

The whole plot and subplot factors and their 

corresponding levels in run order are shown in 

the split-plot experimental design strategy 

previously illustrated in figure 1. The whole 

plot and subplot factors and their respective 

levels in actual factor settings and in coded units 

(where the low, mid, and high levels of each 

factor are coded -1, 0, and +1, respectively) are 

also presented in Table I. The use of coded units 

in the analysis of a statistically-designed 

experiment can avoid spurious statistical results 

due to different measurement scales for the input 

factors, collinearity among the terms in the 

model, inflated variability in the regression 

coefficient estimates, and reduced precision in 

the tests of statistical significance (references 4 

and 5). 

 A listing of the experimental observations 

in run order is provided in Table II, which was 

also the data table used as input to the statistical 

software package in reference 13. A 30-minute 

time period was allocated between complete 

blocks or replicates, during which the wind 

tunnel facility was shut down and facility 

personnel vacated the area. Consequently, the 

blocks (replicates) represented an approximation 

to three consecutive days of facility operation. 

Blocking defends against unwanted “day-today” variability, but also represents a restriction 

on randomization (references 4-9). 

Numerous commercially-available software 

packages are available for the design and 

analysis of experimental data. The software 

cited in reference 13 was the primary analysis 

tool in the current investigation. The analysis 

was augmented by spreadsheet software with 

statistical tools in reference 14. 

The lift, drag, and pitching moment 

coefficient responses to the selected input 

factors were analyzed in three stages using a 

split-plot analysis procedure similar to that 

described in reference 12. The analysis 

procedure is illustrated in figure 3. An initial, 

somewhat qualitative, analysis in the first stage 

assessed the statistical significance of a 

combined, or aggregate, whole plot factor 

containing linear and quadratic terms in the 

differential horizontal stabilizer incidence 

(Tail_Angle and Tail_Angle*Tail_Angle), an 

aggregate subplot factor containing linear and 

quadratic terms in the angle of attack (AoA and 

AoA*AoA), and a two-factor interaction of the 

aggregate whole plot and subplot factors 

(Tail_Angle*AoA and mixed higher-order terms) 

for each of the three aerodynamic response 

variables. The initial analysis was a screening 

procedure to verify the statistical significance of 

the aggregate input factors before committing to 

a more detailed quantitative analysis of 

individual whole plot and subplot terms that 

were supported by the assumed linear statistical 

models. The decomposition of the individual 

terms at the whole plot and subplot levels was 
5

conducted in a two-step, second-stage analysis 

in which the individual whole plot and subplot 

terms featuring main effects, two-factor 

interactions, and quadratic terms were 

systematically separated out to estimate their 

respective mean squares and regression function 

coefficients and the error terms at the whole plot 

and subplot levels. Valid tests of statistical 

significance at the subplot level were also 

performed at this stage of the analysis. Tests of 

significance of the estimated mean squares and 

regression function coefficients using the correct 

error term at the whole plot level were 

conducted in a third analysis stage. These 

results were combined with the output from the 

two-step, second-stage analyses to assemble 

final listings of the estimated mean squares, 

multiple regression coefficients, and tests of 

statistical significance at the whole plot and 

subplot levels for the three response variables 

CL, CD, and Cm.

Test Information 

A diagnostic wind tunnel test was performed 

as part of a split-plot experimental design 

investigation of the effects of differential 

horizontal stabilizer incidence and the angle of 

attack on the lift, drag, and pitching moment 

responses of a small-scale fighter airplane model 

previously shown in figure 2. The rolling 

moment, yawing moment, and side force 

responses were also measured, since differential 

stabilizer incidence is a means of augmenting 

the roll, yaw, and side force control of air 

combat vehicles. However, the current report 

focuses on the lift, drag, and pitching moment to 

demonstrate an analysis procedure of a simple 

split-plot experimental design implemented in a 

wind tunnel environment with restrictions on 

randomization. 

A 0.025-scale precision metal model of the 

selected fighter airplane configuration was tested 

in an Old Dominion University (ODU) 

low-speed diagnostic wind tunnel facility 

(figure 2). The model overall length and span 

were 16.80 inches and 11.23 inches, 

respectively. The model incorporated horizontal 

stabilizers that were manually adjusted to a 

selected incidence. Scribe lines along the model 

fuselage provided a visual reference for setting 

the incidence angle. Differential incidence 

angles were obtained by changing only the right 

stabilizer incidence and retaining the incidence 

of the left stabilizer at 0 degrees. A close-up 

view of the right stabilizer at +20-degrees 

incidence (leading-edge up) was previously 

shown in figure 2.

The ODU facility was an open-circuit, 

atmospheric wind tunnel and a 1/15th-scale

version of the NASA Langley Research Center 

(NASA LaRC) 30- by 60-Foot Full-Scale Wind 

Tunnel (references 15 and 16). The wind tunnel 

was used primarily for diagnostic experiments 

and as a research laboratory to support joint 

NASA- and ODU-sponsored graduate-level 

courses in experimental design for the local 

technical and academic communities 

(reference 12). The model was attached to the 

wind tunnel main support system, which 

provided remote control of the angle of attack 

relative to the incident air flow. 

The six aerodynamic forces and moments 

arising from the air flow about the model were 

measured using a NASA LaRC strain-gage 

balance designated IR-15 mounted inside the 

model. The balance was attached to a 

cylindrical “sting” mounted to the main support 

system. 

The free stream dynamic pressure, q, in the 

test section was approximately 6.4 pounds per 

square foot (approximately 50 miles per hour) 

for all measurements. The corresponding unit 

Reynolds number, Re, was 0.48(106

). The 

Reynolds number based on a reference wing 

chord, cref, of 3.46 inches was Rec = 0.14(106

).

 The model attitude was corrected for 

deflections of the balance, sting, and support 

system due to aerodynamic loads. Since the 

testing was exploratory in nature and focused on 

aerodynamic trends, however, there were no 

corrections applied to the data to account for 
6

tunnel flow angularity, model base pressures, 

blockage, buoyancy, or wall interference. 

Each balance measurement represented the 

average of 12 five-second samples of data, that 

is, 60-second data acquisition time for each 

point. The balance measurements were reduced 

to coefficient form using an on-line computer, 

and the data were stored to disk for off-line 

analysis. 

The two quantitative input variables were 

differential deflection of the horizontal 

stabilizers (deflection of right tail only), 

Tail_Angle, and the model angle of attack, AoA.

The three aerodynamic responses, CL, CD, and 

Cm, were nondimensional (coefficients) and 

corresponded to the measured lift force and drag 

force divided by the product of the reference 

wing area, Sref, and the free-stream dynamic 

pressure, q, and the measured pitching moment 

divided by the product of Sref, cref, and q.

No limitations or problems were encountered 

during the wind tunnel experiment. The entire 

experiment was completed in one morning over 

a 4-hour period of time. 

Discussion of Results 

First-Stage Analysis 

Table II is a listing of the experimental data 

which shows the input factor settings in coded 

units and the three aerodynamic coefficient 

responses in run order. The listing is divided 

into the three blocks, or “work days.” The run 

order was randomized using the statistical 

software in reference 13. The whole plot and 

subplot factors are designated Tail_Angle and 

AoA, which correspond to the differential 

horizontal stabilizer incidence in degrees and the 

model angle of attack in degrees, respectively. 

The column labeled Rep is identical to Block and 

is assigned values of 1, 2, and 3 corresponding 

to the three blocks or replicates in the current 

split-plot experimental design. Rep is a term 

included in the statistical models assumed in this 

analysis. The three responses are CL, CD, and

Cm which correspond, respectively, to the 

aerodynamic coefficients of lift, drag, and 

pitching moment. Two columns labeled wptreat

and sptreat are also included in the data listing 

in Table II. Since the whole plot factor was 

tested at three levels, wptreat is assigned values 

of 1, 2, and 3 and represents a combination, or 

aggregate, of linear and quadratic terms in 

Tail_Angle as previously described in the 

section “Experimental Design Method and 

Procedures.” The subplot factor was also tested 

at three levels. Consequently, sptreat is 

assigned values of 1, 2, and 3 and represents the 

aggregate of linear and quadratic terms in AoA.

Initial statistical models were constructed using 

the aggregate terms wptreat, sptreat, and the 

two-factor interaction wptreat*sptreat declared 

as nominal factors in order to identify 

statistically significant effects on the 

aerodynamic responses of the combined whole 

plot factor, combined subplot factor, and 

combined two-factor interaction. 

The assumed linear statistical model in the 

first stage of the analysis is 

  ijk i j ij k ijk jk

Y           	 	 


where each term in the model is defined as 

Yijk response in i

th block (replicate), jth level 

of the whole plot factor, k

th level of the 

subplot factor 


 mean response 

i effect of the i

th block (replicate) on the 

response (a random effect) 

j effect of the j

th level of the combined 

whole plot factor (wptreat)

ij whole plot error, calculated from the 

interaction ()ij

	k effect of the k

th level of the combined 

subplot factor (sptreat)

	jk interaction between the combined 

whole plot and subplot factors


ijk subplot error or mean square error 

(MSE)
7

Figure 4 is a screen capture from the “Fit

Model” platform in the statistical analysis 

software package in reference 13, which shows 

the model specification dialog. The “Select

Columns” box includes the column names in the 

split-plot experimental design listing in Table II. 

The columns labeled CL, CD, and Cm (specified 

as continuous variables) are input to the “Pick

Role Variables” box as the “Y” response 

variables. The linear model is specified in the 

“Construct Model Effects” box by adding the 

appropriate terms from the “Select Columns”

box. For example, the first term in the model is 

Rep, which is declared as a random effect, 

Rep&Random, using the “Attributes” pop-up 

menu. The software appends &Random to any 

term that is declared a random effect. The 

inclusion of the term Rep in the statistical model 

acknowledges the blocking scheme used in the 

current experimental design. This will assign 

degrees of freedom to Rep, which accounts for 

the restriction on randomization due to the 

blocking. The second term is the nominal 

aggregate whole plot factor wptreat. The third 

term corresponds to Rep crossed with wptreat,

which is declared as a random effect, 

Rep*wptreat&Random. The latter term 

represents the error at the whole plot level, and 

its inclusion in the model ensures the aggregate 

whole plot factor, wptreat, is tested against the 

correct whole plot error. The fourth term that is 

added to the model is the nominal aggregate 

subplot factor, sptreat. The fifth term is wptreat

crossed with sptreat, wptreat*sptreat, which 

represents the two-factor interaction between the 

aggregate whole plot and subplot factors. The 

latter two terms are tested against the subplot 

error, which is the mean square error or MSE.

The “Personality” button indicates that the 

linear model will be fit using standard least 

squares. The “Emphasis” button specifies a 

minimal report, which suppresses all plots and 

arranges the whole model and effect details 

tables in a vertical format. The “Method” button 

indicates that the expected mean squares (EMS)

method is used to fit the linear model with 

random effects. 

The first stage of the analysis was intended to 

test for overall significance of the whole plot 

and subplot factors before initiating a more 

detailed quantitative assessment of the 

individual linear and higher-order terms at both 

levels. A portion of the “Fit Model” platform 

output from the first-stage analysis is shown in 

figure 5. The analysis of variance (ANOVA) 

listing for each aerodynamic response 

corresponds to the whole model with two 

sources labeled “Model” and “Error” under 

“Source”. The source “Error” corresponds to 

the residual error, or mean square error, MSE.

There are 3 levels of wptreat (a = 3), 3 levels of 

sptreat (b = 3), and 3 replicates, Rep (r = 3). 

Consequently, the calculated total (C. Total)

degrees of freedom (DF) is r*a*b – 1 = 26. The 

MSE DF is a*(r-1)*(b-1) = 12. Consequently, 

there are 14 DF for the overall model. The 

estimated mean squares (MS) for a given term is 

obtained by dividing the sum of squares (SS) by 

the corresponding DF. The results from the 

ANOVA indicate that the overall models for the 

three aerodynamic responses are statistically 

significant, since the corresponding probability 

levels or p-values (labeled Prob > F) of the test 

statistic, F ratio, are less than the prescribed 

level of significance of 0.05 (s = 0.05). The 

F ratio is obtained by dividing the estimated 

mean squares of the model term of interest by 

the appropriate error mean squares. A 

probability value of the F ratio that is less than 

0.05 is considered unlikely in the presence of 

random error but, instead, consistent with 

systematic variation in the data caused by 

changes in the levels of the input factor. For 

example, the mean squares for the whole model 

of the response CL (using higher-precision 

values from the statistical software output) is: 

MS(model) = SS(Model)/DF(model) = 

1.16554482/14 = 0.0832532014 

The mean square error is: 

MSE = SS(Error)/DF(error) =

0.0007395953/12 = 0.0000616329 

Consequently, the F Ratio is: 

MS(model)/MSE =

0.0832532014/.0000616329 = 1350.791 
8

Given an F distribution with 14 DF in the 

numerator and 12 DF in the denominator, the 

probability value of this F statistic is 

approximately 0 (references 4-9). Since this 

value is less than the cutoff value of 0.05, the 

whole model is declared statistically significant. 

A more detailed analysis at the whole plot 

and subplot levels is provided in the output 

labeled “Tests wrt Random Effects” in figure 5. 

Table III shows the corresponding breakdown of 

the degrees of freedom at both levels. Note that 

the combined whole plot factor, wptreat, is 

tested against the correct whole plot error term, 

Rep*wptreat&Random, and the combined 

subplot factor, sptreat, and the two-factor 

interaction wptreat*sptreat are correctly tested 

against the subplot error term or mean square 

error, MSE. The combined whole plot factor 

and the combined subplot factor, wptreat and 

sptreat, respectively, have statistically 

significant effects on the lift and pitching 

moment coefficients, since their corresponding 

p-values (Prob > F) are less than the cutoff 

value of 0.05. In the analysis of the CL

response, for example, the mean squares 

estimate (listed in the column labeled MS Num)

for wptreat is: 

MS(wptreat ) = SS(wptreat)/DF(wptreat) = 

0.0680479905/2 = 0.0340239953 

The whole plot error mean squares estimate 

corresponding to Rep*wptreat&Random is: 

MS(WP error)= SS(WP error)/DF(WP error) = 

0.000421438/4 = 0.0001053595

The F statistic for wptreat is: 

MS(wptreat)/MS(WP error) = 

0.0340239953/0.0001053595 = 322.9324 

The corresponding p-value from an 

F-distribution with 2 DF in the numerator and 

4 DF in the denominator is approximately 

0.00003789. 

The mean squares estimate in the CL response 

analysis for sptreat is: 

MS(sptreat) = SS(sptreat)/DF(sptreat) = 

1.0966651483/2 = 0.5483325742 

The subplot error mean squares estimate is the 

MSE from the ANOVA table and is: 

MSE = SS(Error)/DF(Error) =

0.0007395953/12 = 0.0000616329 

Note that the error estimate at the whole plot 

level is higher than the corresponding error 

estimate at the subplot level. This is common in 

split-plot experimental designs, since there are 

typically more degrees of freedom to estimate 

the subplot error. Consequently, the statistical 

tests of significance are conducted with more 

precision at the subplot level. 

The F statistic for sptreat is: 

MS(sptreat)/MSE = 

0.5483325742/0.0000616329 = 8896.7457

The corresponding p-value from an 

F- distribution with 2 DF in the numerator and 

12 DF in the denominator is essentially zero. It 

is noted that the mean squares estimate for the 

whole plot error term Rep*wptreat&Random is 

listed in the column labeled MS Num, since the 

statistical software also tests this estimate 

against the MSE. However, this test is not 

relevant to the current analysis. In contrast to 

the statistical significance of the aggregate 

whole plot and subplot terms, the interaction 

term wptreat*sptreat is statistically insignificant 

in the lift and pitching moment coefficient 

responses, since the p-values exceed the cutoff 

value of 0.05. The analysis of the drag 

coefficient response, CD, shows that wptreat,

sptreat, and wptreat*sptreat are all statistically 

significant. It is noted, however, that the firststage analysis does not isolate potentially 

significant two-factor interactions and quadratic 

terms, for example, since they are embedded in 

the terms wptreat, sptreat, and wptreat*sptreat.

The decomposition of individual terms at the 

whole plot and subplot levels is conducted in the 

second-stage analysis. 
9

Second-Stage Analysis 

A second analysis derived from reference 12 

featuring a two-step approach was performed to 

decompose the whole plot and subplot terms into 

quantitative effects by calculating the individual 

mean squares associated with linear, two-factor 

interaction, and quadratic terms supported by the 

model. For example, Table III previously 

indicated 2 DF for the aggregate whole plot 

term, wptreat; 2 DF for the aggregate subplot 

term, sptreat; and 4 DF for the aggregate whole 

plot term crossed with the aggregate subplot 

term. As a result, statistical models for the CL,

CD, and Cm responses will support linear and 

pure quadratic terms in Tail_Angle, linear and 

pure quadratic terms in AoA, and two-factor 

interactions involving Tail_Angle and AoA (for 

example, Tail_Angle*AoA, Tail_Angle*AoA^2,

Tail_Angle^2*AoA, and Tail_Angle^2*AoA^2).

Inclusion of two-factor interaction terms of 

higher-order than Tail_Angle*AoA is 

problematic, however, since this promotes 

collinearity of the estimated terms in the 

statistical models (references 4-9). As a result, 

only Tail_Angle*AoA will be included in the 

models in the current analyses. It is expected 

based on previous testing experience 

(reference 17) that the linear-by-linear term is 

the “heavy hitter” among the two-factor 

interaction terms. The estimated sums of 

squares of the three higher-order terms and their 

associated degrees of freedom will be pooled 

into the estimate of the subplot error, or mean 

square error (MSE).

First Step 

In the initial step of the second-stage 

analysis, the whole plot main effect and 

quadratic terms corresponding to Tail_Angle and 

Tail_Angle*Tail_Angle were again combined in 

the term wptreat and specified as a nominal 

factor as in the first-stage analysis. However, 

the subplot terms were separated out and 

declared continuous. Here, sptreat from the 

preceding analysis was broken out into the 

individual terms AoA and AoA*AoA 

corresponding to the linear and quadratic terms 

in the subplot factor AoA. In order to avoid a 

programmatic error message for a 

nonhierarchical model (references 4-9), a 

column was included in the experimental design 

listing shown previously in Table II to represent 

the interaction between the whole plot and 

subplot factors, Tail_Angle*AoA. Terms 

corresponding to the subplot factor main effect, 

two-factor interaction between the whole plot 

and subplot factors, and subplot factor pure 

quadratic effect were supported by the assumed 

statistical model, since there were sufficient 

degrees of freedom (as described at the outset of 

this section) by testing the subplot factor at three 

levels within each level of the whole plot factor 

and in three replicates or blocks. The added 

column for Tail_Angle*AoA in Table II uses 1 

of the 4 available DF from the wptreat*sptreat

term shown previously in the DF breakdown in 

Table III. The model specification dialog in the 

“Fit Model” platform is shown in figure 6. 

Similar to the first-stage analysis in the previous 

section, the first step in the current analysis stage 

also estimates the correct whole plot error term, 

Rep*wptreat&Random, without performing 

statistical tests of the individual terms at the 

whole plot level. The aggregate term wptreat is 

tested against the correct whole plot error. The 

key element in the first step of the second 

analysis is the decomposition of terms at the 

subplot level, which allows the estimation of the 

mean squares and multiple regression 

coefficients of all individual subplot terms, 

which were tested for statistical significance 

against the correct subplot error term (MSE).

The whole model ANOVA and the tests with 

respect to random effects from the first-step 

analyses for each of the three aerodynamic 

responses are shown in figure 7. The ANOVA 

for the whole model contains the mean square 

error (MSE) that is used to test all terms at the 

subplot level. There are 15 DF for the subplot 

error in figure 7 compared to the 12 DF shown 

previously in figure 5 from the first-stage 

analysis. This reflects the 3 DF for the whole 

plot and subplot interaction terms that were 

excluded from the statistical models for the three 

response variables and, instead, pooled with the 

MSE.
10 

Comparison of the “Tests wrt Random 

Effects” output in figure 7 with the 

corresponding output in figure 5 from the firststage analysis indicates the sums of squares, 

mean squares, error estimates, and statistical 

tests for Rep&Random and wptreat at the whole 

plot level are unchanged. In addition, the sums 

of squares for AoA and AoA*AoA in figure 7 add 

to the corresponding values of the aggregate 

subplot factor sptreat in figure 5. The inclusion 

of only the Tail_Angle*AoA interaction in the 

statistical models in figure 7 accounts for 

approximately 75%, 100%, and 50% of the sums 

of squares attributed to the aggregate 

wptreat*sptreat interaction term in the CL, CD,

and Cm responses, respectively, in figure 5. The 

smaller percentage of the overall interaction SS

accounted for by Tail_Angle*AoA in the Cm

response is not an issue, however, since the firststage analysis in figure 5 indicated the aggregate 

wptreat*sptreat term is highly insignificant. 

These results support the pooling of the sums of 

squares of the three higher-order interaction 

terms and their corresponding DF with the 

estimate of the subplot mean square error in the 

second-stage analysis. 

The “Tests wrt Random Effects” output 

shown in figure 7 indicates the main effect of 

AoA has a highly significant effect on the CL

response, since its p-value is much less than the 

cutoff value of 0.05. The two-factor interaction 

Tail_Angle*AoA is marginally significant with a 

p-value slightly less than the cutoff value. In 

contrast, the quadratic effect of AoA is highly 

insignificant with a p-value nearly an order of 

magnitude greater than the cutoff value. It is 

noted that the estimates of the subplot terms are 

uncorrelated, such that the removal of any term 

from the statistical model does not affect the 

estimates of terms that are retained in the model. 

In addition, the use of coded units allows an 

assessment of the relative contribution of each 

model term to the total systematic variation of 

the aerodynamic response. For example, the 

sum of squares for AoA represents nearly 100% 

of the variability in the CL response attributed to 

the three subplot terms. This suggests that the 

exclusion of Tail_Angle*AoA and AoA*AoA

would not have a practical effect on the 

predictive capability of the assumed statistical 

model for the CL response. 

The subplot terms AoA, Tail_Angle*AoA, and 

AoA*AoA all have statistically significant effects 

on the CD response as shown in the “Tests wrt 

Random Effects” output shown in figure 7. The 

term AoA is again the “heavy hitter” in the 

statistical model at the subplot level, since it 

accounts for over 90% of the variability 

accounted for by the three subplot terms. An 

interpretation of the statistical significance of the 

two-factor interaction and quadratic terms is that 

the effect of AoA on the CD response depends 

on the levels of Tail_Angle and AoA. Drag 

coefficient is typically the most sensitive 

component in aerodynamic force and moment 

wind tunnel testing (reference 18). 

Only the main effect of AoA has a 

statistically significant effect on the pitching 

moment coefficient response in figure 7. The 

term AoA represents approximately 94% of the 

variability in the Cm response accounted for by 

the three subplot terms. The term 

Tail_Angle*AoA is highly insignificant, since its 

p-value is nearly an order of magnitude greater 

than the cutoff value. This indicates that the 

effect of AoA on the Cm response does not 

depend on the level of Tail_Angle. The 

quadratic term AoA*AoA is marginally 

insignificant, since its p-value is slightly greater 

than 0.05. 

The dominance of the main effect of AoA at 

the subplot level on CL, CD, and Cm is 

consistent with the current experimental design 

space featuring angles of attack from 0 degrees 

to 8 degrees where linear aerodynamics are 

expected to prevail (reference 19). Significant 

two-factor interaction and quadratic effects on 

all three aerodynamic responses would be 

expected at high angles of attack where the flow 

field about the current fighter airplane model 

would be dominated by flow separation, vortex 

flows, and flow-field interactions between the 

various airframe components (reference 20). 

The output from the analyses of the linear 

statistical models for the CL, CD, and Cm
11 

responses also included valid estimates of the 

multiple regression model coefficients computed 

by least squares method (references 4-9) and the 

corresponding statistical tests of significance for 

all terms at the subplot level. However, 

presentation of these results is deferred to the 

discussion of the third-stage analysis. 

Second Step 

The next step in the second-stage analysis 

was to build statistical models for the CL, CD,

and Cm responses having only the whole plot 

main effect and quadratic terms separated out 

from wptreat. This part of the analysis provided 

the correct estimates of the mean squares and 

regression coefficients for the linear and 

quadratic effects of the whole plot factor, which 

are designated Tail_Angle and 

Tail_Angle*Tail_Angle, respectively. The 

model specification dialog in the “Fit Model”

platform is shown in figure 8, and partial output 

from the analyses of the three response variables 

is presented in figure 9. The valid mean squares 

estimates at the whole plot level are shown in 

“Tests wrt Random Effects” in figure 9. The 

p-values for all terms at the whole plot level 

exceed the cutoff value of 0.05 for the CL

response. Only the main effect of Tail_Angle is 

statistically significant in the CD and Cm

responses. However, the statistical tests are not

correct since the terms Tail_Angle and 

Tail_Angle*Tail_Angle (and Rep) are tested 

against the overall model error shown in the 

corresponding ANOVA output instead of the 

whole plot error mean square, 

Rep*wptreat&Random, which was obtained 

from the first step. Similarly, the estimates of 

the regression coefficients for the terms at the 

whole plot level, as presented in the next 

section, are valid at this stage of the analysis, but 

the statistical tests of significance are incorrect. 

The correct tests of statistical significance for 

the individual terms at the whole plot level are 

performed in the third and final stage of the 

analysis. 

Third-Stage Analysis 

The only missing element in the analysis of 

the current split-plot experimental design is the 

application of the correct error terms in the 

statistical tests of significance of the mean 

squares estimates of the whole plot terms and 

the corresponding regression coefficients for the 

three response variables. The results obtained in 

the two-step second analysis were combined in a 

third stage featuring manual input to the 

spreadsheet software with statistical tools in 

reference 14 to ensure valid tests of significance 

at the whole plot level. Results from the third 

stage of the analysis featuring a modified 

ANOVA for the lift, drag, and pitching moment 

coefficient responses are shown in figure 10, 

figure 11, and figure 12, respectively. Each 

figure contains three separate listings: (1) results 

from the first step of the second-stage analysis 

containing the correct error terms at the whole 

plot and subplot levels and correct mean squares 

estimates and statistical tests of all model terms 

at the subplot level, (2) results from the second 

step of the second-stage analysis containing 

valid mean squares estimates of the model terms 

at the whole plot level but incorrect statistical 

tests of significance, and (3) modified ANOVA 

combining the correct mean squares estimates of 

all terms at the whole plot level (including Rep)

and at the subplot level and the correct error 

terms for statistical tests at each level. The 

decomposition of the aggregate whole plot term 

wptreat is also included in the listings from the 

third-stage analysis. In the modified third-stage 

analyses, the subplot error with 15 DF is smaller 

than the whole plot error with 4 DF for all three 

responses. Consequently, the precision of the 

statistical tests is greater at the subplot level, 

which is typical of split-plot experimental 

designs (references 4-9). 

To illustrate the modified calculations in the 

third-stage analysis, consider the CL response in 

figure 10. The correct F statistic for the whole 

plot term Tail_Angle is obtained by dividing the 

mean squares estimate for Tail_Angle (with 

1 DF) obtained in step 2 of the second-stage 

analysis (MS(Tail_Angle) = 0.06804609) by the 

whole plot error (with 4 DF) obtained in 

step 1 of the second-stage analysis 

(Rep*wptreat&Random = 0.00010536). The 

corresponding F statistic is 645.8437 as shown 

in the modified analysis in figure 10. The 
12 

p-value for this F statistic from an F distribution 

with 1 DF in the numerator and 4 DF in the 

denominator is approximately 0.00001424. 

Similar calculations for the whole plot term 

Tail_Angle*Tail_Angle yield an F statistic of 

0.01803341 and a corresponding p-value of 

0.89966021. Consequently, at the whole plot 

level, only the main effect of Tail_Angle has a 

statistically significant effect on the lift 

coefficient response. Modified tests of 

significance of the pitching moment coefficient 

response at the whole plot level in figure 12 also 

reveal the main effect of Tail_Angle as the only 

statistically significant term at this level. Both 

the main effect and quadratic effect of 

Tail_Angle are statistically significant in the 

drag coefficient response in figure 11, although 

the term Tail_Angle represents nearly 97% of 

the variability in the CD response accounted for 

by the two whole plot terms. The valid tests of 

statistical significance of all terms at the subplot 

level from the step 1 second-stage analyses of 

CL, CD, and Cm are retained in the modified 

ANOVA listings in figure 10, figure 11, and 

figure 12, respectively. Statistically significant 

terms at the whole plot and subplot levels are 

highlighted in figures 10-12. The main effects 

of Tail_Angle and AoA and their two-factor 

interaction Tail_Angle*AoA are significant terms 

in the model for the CL response. Whole plot 

and subplot main effects, their two-factor 

interaction, and both quadratic effects 

(Tail_Angle*Tail_Angle and AoA*AoA) have 

statistically significant effects on the drag 

coefficient response. Only the whole plot and 

subplot main effects are significant in the model 

for the Cm response. For all three responses, the 

main effects are the dominant terms. The sum 

of squares due to the subplot main effect, AoA,

represents approximately 94.1% of the total 

variability in the CL response accounted for by 

the statistical model. By comparison, the sum of 

squares due to the whole plot main effect, 

Tail_Angle, represents approximately 5.8% of 

the total systematic variation accounted for by 

the model. Although statistically significant, the 

sum of squares due to the two-factor interaction, 

Tail_Angle*AoA, represents only about 0.02% of 

the estimated variability of the CL response. 

The subplot main effect AoA and the whole plot 

main effect Tail_Angle contribute approximately 

60.5% and 32.1%, respectively, to the total 

variability in the CD response accounted for by 

the model. The remaining terms, AoA*AoA,

Tail_Angle*AoA, and Tail_Angle*Tail_Angle 

contribute approximately 4.7%, 1.6%, and 1.0%, 

respectively. Of the two significant terms in the 

model for the Cm response, the whole plot main 

effect Tail_Angle contributes about 98.6% of the 

total variability accounted for by the statistical 

model, whereas the subplot main effect AoA

contributes 1.2%. 

The valid estimates of the regression 

coefficients in coded units for the CL, CD, and 

Cm responses that were obtained by least 

squares method in the second-stage analysis are 

listed in figure 13. Modified calculations for the 

significance tests of the regression coefficients 

at the whole plot level were conducted in the 

third-stage analysis, and the results are reflected 

in figure 13. The column labeled “Coef” lists 

the estimated regression coefficient for the linear 

model found by least squares. “SE Coef” is the 

standard error of the regression coefficient, or 

estimate of the standard deviation of the 

distribution of the coefficient estimate. The 

column labeled “t” is the t-ratio obtained by 

dividing Coef by SE Coef, and this statistic is 

assumed to have a Student’s t-distribution

(references 4-9). The column labeled 

“Prob > |t|” is the two-tailed p-value or the 

probability the absolute value of the test statistic, 

t, will take on a value at least as extreme as the 

observed value of the statistic, assuming the null 

hypothesis is true (e.g. the regression coefficient 

does not contribute to the prediction capability 

of the statistical model). In the step 2, secondstage analysis of the CL response, for example, 

the regression coefficient estimate (previously 

not shown) for Tail_Angle was 0.06148446 with 

a SE Coef of 0.05266038. The t-ratio was 

0.06148466/0.05266038 = 1.16756962 with a 

corresponding p-value of 0. 0.25548085 from a 

Student’s t distribution with 22 DF. However, 

this test incorrectly used the overall model error 

(MS(Overall Model) = 0.04991609) with 22 DF 

instead of the correct whole plot error term 

Rep*wptreat&Random = 0.00010536 with 4 DF

(see figure 10 for the corresponding error 
13 

estimates and DF). The correct value for 

SE Coef is obtained by using the whole plot 

error mean squares estimate. For example, 

multiplying the original value of SE Coef by 

(MS(WP Error)/MS(Overall Model))^0.5 yields: 

0.05266038*(0.00010536/0.04991609)^0.5 = 

0.00241935 

The t-ratio is then 0.06148466/0.00241935 = 

25.41364233. The corresponding two-tailed 

p-value is 0.00001424 based on a Student’s 

t-distribution with 4 DF (references 4-9). 

Similar calculations were performed for the 

significance tests of the terms at the whole plot 

level for the CD and Cm responses. Statistically 

significant estimated regression coefficients are 

highlighted in figure 13. As discussed in the 

section on the second-stage analysis, the 

removal of insignificant terms does not affect 

the estimates of the regression coefficients that 

are retained in the statistical model. 

 Statistically significant coefficients in the 

estimated regression function for the mean lift 

coefficient response correspond to the main 

effects of the whole plot factor (Tail_Angle) and 

the subplot factor (AoA) and their two-factor 

interaction (Tail_Angle*AoA). The estimated 

regression function for the mean drag coefficient 

response includes statistically significant terms 

corresponding to the whole plot and subplot 

main effects (Tail_Angle, AoA), whole 

plot and subplot quadratic effects 

(Tail_Angle*Tail_Angle, AoA*AoA), and whole 

plot and subplot interaction (Tail_Angle*AoA).

The least complex regression function 

corresponds to the pitching moment coefficient 

response, which features only the main effects of 

Tail_Angle and AoA. The terms that are 

highlighted in the estimated regression functions 

in figure 13 are statistically significant based on 

the assumed level of significance of 0.05. 

However, the practical significance of the 

individual terms would require engineering 

judgment of the relative magnitudes of the 

regression coefficients and the corresponding 

test statistics. For example, the largest 

coefficient in the regression function for the lift 

coefficient response corresponds to the main 

effect of the subplot factor AoA. The magnitude 

of the next most significant coefficient 

corresponding to the main effect of the whole 

plot factor Tail_Angle is smaller by a factor of 

four. The interaction of the whole plot and 

subplot factors Tail_Angle*AoA marginally 

satisfies the criteria of statistical significance. 

However, the magnitude of the regression 

coefficient is smaller than the coefficient for the 

main effect of the subplot factor AoA by over a 

factor of 50. 

Concluding Remarks 

A procedure to analyze a split-plot 

experimental design implemented in a low-speed 

diagnostic wind tunnel was described in this 

report. A statistically-designed experiment was 

conducted in an ODU facility to obtain the lift, 

drag, and pitching moment coefficient responses 

of a small-scale precision metal model of a 

fighter airplane configuration. The two input 

factors were differential horizontal stabilizer 

incidence and the angle of attack relative to the 

incident air flow. The differential horizontal 

stabilizer incidence was a difficult-to-change 

factor and, therefore, a restriction on 

randomization, since each level of this input 

factor required physical access to the test section 

to perform the required model change. The 

angle of attack was an easy-to-change input 

factor, since its levels were remotely controlled 

during wind-on operation. This investigation 

featured a simple split-plot experimental design 

consisting of two input factors, where the hardto-change factor was tested in incomplete blocks 

called whole plots, and all levels of the easy-tochange factor were run in subplots within each 

incomplete block. The experimental design and 

analysis methods accounted for testing in a 

randomization-restricted environment with two 

different error structures. The key design-ofexperiment concepts of replication, 

randomization, and blocking were used to 

defend against known and unknown sources of 

variation and to provide sufficient degrees of 

freedom to perform valid tests of statistical 

significance at both the whole plot and subplot 

levels. A three-stage analysis procedure was 
14 

used to estimate the mean squares of all terms 

supported by the statistical models, to estimate 

the error terms at the whole plot and subplot 

levels, to perform valid tests of statistical 

significance at both levels, and to construct 

multiple regression models of the three response 

variables. Statistically significant terms in the 

estimated regression function for the lift 

coefficient response included the main effects of 

the whole plot factor (differential horizontal 

stabilizer incidence) and the subplot factor 

(angle of attack) and their two-factor interaction. 

The estimated regression function for the mean 

drag coefficient response included statistically 

significant terms corresponding to the whole 

plot and subplot main effects, whole plot and 

subplot quadratic effects, and whole plot and 

subplot two-factor interaction. The only 

significant terms in the estimated regression 

function for the mean pitching moment 

coefficient response corresponded to the main 

effects of the differential horizontal stabilizer 

incidence and the angle of attack. The more 

complex regression function for the drag 

coefficient response was consistent with 

aerodynamic testing experience, where the drag 

component typically exhibits a higher-order 

dependency on input factors related to the 

vehicle configuration and the flight conditions. 

More complex regression functions would be 

expected if the experimental design space was 

expanded to include high angles of attack where 

the vehicle aerodynamics are dominated by 

separated flows and flow-field interactions. 

References 

1. DeLoach, R., “Applications of Modern 

Experiment Design to Wind Tunnel Testing 

at NASA Langley Research Center,” 

AIAA-1998-0713, January 1998. 

2. Luner, J. J. and Healey, M. D., “Modern 

Design of Experiments Techniques to 

Optimize a Leading Edge Extension,” 

AIAA-2003-0655, January 2003. 

3. Deloach, R., “Improved Quality in 

Aerospace Testing Through the Modern 

Design of Experiments (Invited),” 

AIAA-2000-0825, January 2000. 

4. Montgomery, D. C., Design and Analysis of 

Experiments, 7th edition, John Wiley & 

Sons, Inc., New York, 2009. 

5. Myers, R. H. and Montgomery, D. C., 

Response Surface Methodology: Process 

and Product Optimization Using Designed 

Experiments, 2nd edition, John Wiley & 

Sons, Inc., New York, 2002. 

6. Kuehl, R. O., Design of Experiments: 

Statistical Principles of Research Design 

and Analysis, 2nd edition, Duxbury Press, 

1999. 

7. Wu, C. F. J. and Hamada, M., Experiments:

Planning, Analysis, and Parameter Design 

Optimization, John Wiley & Sons, Inc., 

2000. 

8. Box, G. E. P., Hunter, W. G., and 

Hunter, J. S., Statistics for Experimenters,

John Wiley & Sons, Inc., 1978. 

9. Lentner, M. and Bishop. T., Experimental 

Design and Analysis, 2nd edition, Valley 

Book Company, 1993. 

10. Kowalski, S. M. and Potcner, K. J., “How to 

Recognize a Split-Plot Experiment,” 

Quality Progress, Vol. 36, Issue: 11, 

pp. 60-66, November 2003. 

11. Potcner, K. J. and Kowalski, S. M., “How to 

Analyze a Split-Splot Experiment,” Quality

Progress, Vol. 37, Issue: 12, pp. 67-74, 

December 2004. 

12. Simpson, J. R., Kowalski, S. M., and 

Landman, D., “Experimentation with 

Randomization Restrictions: Targeting 

Practical Implementation,” Quality and 

Reliability Engineering International 2004; 

20: 481-495. 

13. Sall, J., Creighton, L., and Lehman, A., 

JMP® Start Statistics: A Guide to Statistics 

and Data Analysis Using JMP®, 4th edition, 

SAS Institute Inc., 2007. 

14. Microsoft, Microsoft Excel, Redmond, 

Washington, Microsoft, 2010. 

15. Landman, D., Simpson, J. R., Vicroy, D., 

and Parker, P. A., “Efficient Methods for 

Complex Aircraft Configuration 

Aerodynamic Characterization using 

Response Surface Methodologies,” 

AIAA-2006-0922, January 2006. 

16. Landman, D., Simpson, J. R., Mariani, R., 

Ortiz, F., and Britcher, C., “Hybrid Design 
15 

for Aircraft Wind-Tunnel Testing Using 

Response Surface Methodologies,” 

Journal of Aircraft, Vol. 44, No. 4, 

pp. 1214-1221, 2007. 

17. Moore, W. A., Erickson, G. E., Lorincz, D. 

L., and Skow, A. M., “Effects of Forebody, 

Wing, and Wing-Body-LEX Flowfields on 

High Angle of Attack Aerodynamics,” SAE 

Technical Paper 791082, February 1979. 

18. Küchemann, D., The Aerodynamic Design of 

Aircraft, Pergamon Press, June 1978. 

19. Hoerner, S. F. and Borst, H. V., FluidDynamic Lift, Hoerner Fluid Dynamics, 

1975. 

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of Vortex Flows on F/A-18 Configuration at 

Subsonic through Transonic Speeds,” 

NASA Technical Paper 3111, December 

1991. 
16 

Table I. Whole plot and subplot factors and levels in actual factor settings and coded units. 

Table II. Data table showing the experimental observations in run order and the input factors and 

 factor levels in coded units for the split-plot design. 

Factor Levels

 (Actual Factor Settings) 

Levels 

(Coded Units) 

Whole plot 

factor

Differential 

horizontal stabilizer 

incidence 

0, 10, 20 degrees -1, 0 1 

Subplot

factor Angle of attack 0, 4, 8 degrees -1, 0, 1 
17 

Table III. Degrees of freedom breakdown for the first-stage analysis. 

Source Degrees of Freedom (DF) 

Replicate

(Rep&Random) r-1 = 2 

Aggregate whole plot factor 

(wptreat) a-1 = 2 

Whole plot error 

(Rep*wptreat&Random) (r-1)*(a-1) = 4 

Aggregate subplot factor 

(sptreat) b-1 = 2 

Aggregate whole plot factor 

crossed with aggregate 

subplot factor 

(wptreat*sptreat)

(a-1)*(b-1) = 4 

Subplot error 

(MSE) a*(r-1)*(b-1) = 12 

Total r*a*b – 1 = 26 

Number of replicates (Rep), r = 3

Levels of aggregate whole plot factor (wptreat), a = 3 

Levels of aggregate subplot factor (sptreat), b = 3 
18 

Figure 1. Split-plot design in run order for the low-speed wind tunnel experiment. The design contains 

 nine whole plots (three whole plots within each block or replicate (Rep)), three subplot runs 

 within each whole plot, and a total of 27 runs. 

 Tail_Angle = 0 degrees Tail_Angle = 20 degrees Tail_Angle = 10 degrees 

Rep 1

AoA = 0 degrees 

AoA = 4 degrees 

AoA = 8 degrees 

AoA = 4 degrees 

AoA = 8 degrees 

AoA = 0 degrees 

AoA = 0 degrees 

AoA = 8 degrees 

AoA = 4 degrees 

 Tail_Angle = 20 degrees Tail_Angle = 10 degrees Tail_Angle = 0 degrees 

Rep 2

AoA = 0 degrees 

AoA = 4 degrees 

AoA = 8 degrees 

AoA = 4 degrees 

AoA = 8 degrees 

AoA = 0 degrees 

AoA = 4 degrees 

AoA = 0 degrees 

AoA = 8 degrees 

Rep 3

AoA = 4 degrees 

AoA = 8 degrees 

AoA = 0 degrees 

AoA = 4 degrees 

AoA = 0 degrees 

AoA = 8 degrees 

AoA = 4 degrees 

AoA = 0 degrees 

AoA = 8 degrees 

 Tail_Angle = 0 degrees Tail_Angle = 10 degrees Tail_Angle = 20 degrees 

WP Level 

WP Level 

WP Level 

SP Level 

SP Level 

SP Level 

Whole plot; incomplete block 
19 

(a) overall view 

(b) close-up view of the right horizontal stabilizer at +20-degrees incidence 

Figure 2. Photographs of the 0.025-scale fighter airplane model installed in the ODU low-speed 

 diagnostic wind tunnel facility. 
20 

(a) stage 1 analysis and stage 2, step 1 analysis 

Figure 3. Three-stage analysis approach for the split-plot experimental design. 

Stage 1 Analysis

Objective: Verify the overall statistical 

significance of linear models of CL, CD, and 

Cm responses at the whole plot and subplot 

levels

Approach: Construct and analyze linear 

statistical models containing aggregate 

whole plot and subplot terms and their twofactor interaction (all terms declared 

nominal) with correct error terms and 

statistical tests at the whole plot and subplot 

levels. Verification of statistical 

significance warrants follow-on analyses. 

Stage 2 Analysis – Step 1

Objective: Decompose all terms, estimate the 

mean squares and multiple regression 

coefficients, and perform correct statistical 

tests at the subplot level.

Approach: Construct and perform 

correct analysis of linear statistical 

models and estimated multiple 

regression functions with continuous 

individual terms at the subplot level; 

retain aggregate, nominal whole plot 

factor wptreat in the analysis. 

WP Level wptreat (nominal) 

WP error 

sptreat (nominal) 

wptreat*sptreat (nominal) 

SP error 

AoA

Tail_Angle*AoA

AoA*AoA

SP error 

(all terms continuous) 

Model Specification 

SP Level 

WP Level wptreat (nominal) 

WP error 

Model Specification 

SP Level 
21 

 (b) stage 2, step 2 analysis and stage 3 final analysis 

Figure 3. Concluded. 

Stage 2 Analysis – Step 2

Objective: Decompose all terms and estimate the 

mean squares and multiple regression 

coefficients at the whole plot level

Approach: Construct linear statistical models 

and estimated multiple regression functions 

with continuous individual terms at the whole 

plot level; tests of significance are not valid at 

this level because of incorrect error term 

Tail_Angle

Tail_Angle*Tail_Angle

Overall model error 

(all terms continuous) 

Objective: Obtain correct tests of significance of the individual model 

terms at the whole plot level and assemble final analyses with estimated 

regression coefficients for the three response variables. 

Approach: Use the correct whole plot error term from 

stage 2, step 1 to perform correct tests of significance of 

the estimated mean squares and regression coefficients 

at the whole plot level obtained in stage 2, step 2. 

Assemble final listings of estimated mean squares and 

regression coefficients with corresponding tests of 

significance at the whole plot and subplot levels. 

Tail_Angle

Tail_Angle*Tail_Angle

WP error 

(all terms continuous) 

WP Level 

Model Specification 

AoA

Tail_Angle*AoA

AoA*AoA

SP error 

(all terms continuous) 

WP Level 

Model Specification 

SP Level 

Stage 3 Final Analysis
22 

Figure 4. Model specification with the combined whole plot term (wptreat) and the

 combined subplot term (sptreat) for the first-stage analysis. 

(a) analysis of variance 

Figure 5. Partial output from the first-stage analysis of the split-plot design with the combined whole 

 plot and combined subplot terms. 

Source DF Sum of Squares Mean Square F Ratio Prob > F

Model 14 1.16554482 0.08325320 1350.79074000 5.9895E-17

Error 12 0.00073960 0.00006163

C. Total 26 1.16628442

Source DF Sum of Squares Mean Square F Ratio Prob > F

Model 14 0.02498387 0.00178456 458.83201900 3.8463E-14

Error 12 0.00004667 0.00000389

C. Total 26 0.02503055

Source DF Sum of Squares Mean Square F Ratio Prob > F

Model 14 0.10519991 0.00751428 334.97972900 2.5207E-13

Error 12 0.00026918 0.00002243

C. Total 26 0.10546910

 Cm Response

 Analysis of Variance

 CL Response

 Analysis of Variance

 Analysis of Variance

 CD Response
23 

(b) tests with respect to random effects 

Figure 5. Concluded. 

Source SS MS Num DF Num F Ratio Prob > F

Rep&Random 0.00008245 0.00004123 2 0.39128300 0.69951662

wptreat 0.06804799 0.03402400 2 322.93238300 0.00003789

Rep*wptreat&Random 0.00042144 0.00010536 4 1.70946748 0.21234330

sptreat 1.09666515 0.54833257 2 8896.74571000 9.3705E-20

wptreat*sptreat 0.00032779 0.00008195 4 1.32961523 0.31459038

Source SS MS Num DF Num F Ratio Prob > F

Rep&Random 0.00002260 0.00001130 2 2.06360525 0.24223504

wptreat 0.00827416 0.00413708 2 755.40947900 0.00000697

Rep*wptreat&Random 0.00002191 0.00000548 4 1.40810040 0.28979894

sptreat 0.01626442 0.00813221 2 2090.88669000 5.4886E-16

wptreat*sptreat 0.00040078 0.00010020 4 25.76154780 0.00000821

Source SS MS Num DF Num F Ratio Prob > F

Rep&Random 0.00081175 0.00040587 2 2.26741461 0.21964955

wptreat 0.10236051 0.05118026 2 285.91833000 0.00004825

Rep*wptreat&Random 0.00071601 0.00017900 4 7.97979326 0.00223038

sptreat 0.00129345 0.00064672 2 28.83035090 0.00002613

wptreat*sptreat 0.00001819 0.00000455 4 0.20275193 0.93199657

 CD Response

 Tests wrt Random Effects

 Cm Response

 Tests wrt Random Effects

 Tests wrt Random Effects

 CL Response
24 

Figure 6. Model specification for the first step of the second-stage analysis of the split-plot design. 

(a) analysis of variance 

Figure 7. Partial output from the first step of the second-stage analysis of the split-plot design. 

Source DF Sum of Squares Mean Square F Ratio Prob > F

Model 11 1.16546906 0.10595173 1949.17699000 1.4033E-21

Error 15 0.00081536 0.00005436

C. Total 26 1.16628442

Source DF Sum of Squares Mean Square F Ratio Prob > F

Model 11 0.02498310 0.00227119 718.00054400 2.4782E-18

Error 15 0.00004745 0.00000316

C. Total 26 0.02503055

Source DF Sum of Squares Mean Square F Ratio Prob > F

Model 11 0.10519399 0.00956309 521.42181500 2.7077E-17

Error 15 0.00027511 0.00001834

C. Total 26 0.10546910

 CD Response

 Cm Response

 Analysis of Variance

 CL Response

 Analysis of Variance

 Analysis of Variance
25 

(b) tests with respect to random effects 

Figure 7. Concluded. 

 

Source SS MS Num DF Num F Ratio Prob > F

Rep&Random 0.00008245 0.00004123 2 0.39128300 0.69951662

wptreat 0.06804799 0.03402400 2 322.93238300 0.00003789

Rep*wptreat&Random 0.00042144 0.00010536 4 1.93828183 0.15623996

AoA 1.09663735 1.09663735 1 20174.66110000 6.9027E-25

Tail_Angle*AoA 0.00025203 0.00025203 1 4.63655851 0.04797679

AoA*AoA 0.00002779 0.00002779 1 0.51132859 0.48554603

Source SS MS Num DF Num F Ratio Prob > F

Rep&Random 0.00002260 0.00001130 2 2.06360525 0.24223504

wptreat 0.00827416 0.00413708 2 755.40947900 0.00000697

Rep*wptreat&Random 0.00002191 0.00000548 4 1.73134181 0.19546921

AoA 0.01509471 0.01509471 1 4771.95070000 3.3690E-20

Tail_Angle*AoA 0.00040001 0.00040001 1 126.45602100 0.00000001

AoA*AoA 0.00116971 0.00116971 1 369.784266 5.5695E-12

Source SS MS Num DF Num F Ratio Prob > F

Rep&Random 0.00081175 0.00040587 2 2.26741461 0.21964955

wptreat 0.10236051 0.05118026 2 285.91833000 0.00004825

Rep*wptreat&Random 0.00071601 0.00017900 4 9.76003571 0.00042926

AoA 0.00122056 0.00122056 1 66.55024270 0.00000068

Tail_Angle*AoA 0.00001227 0.00001227 1 0.66906420 0.42618129

AoA*AoA 0.00007289 0.00007289 1 3.97420462 0.06471406

 CD Response

 Cm Response

 Tests wrt Random Effects

 Tests wrt Random Effects

 Tests wrt Random Effects

 CL Response
26 

Figure 8. Model specification for the second step of the second-stage analysis of the split-plot design. 

 

(a) analysis of variance 

Figure 9. Partial output from the second step of the second-stage analysis of the split-plot design. 

Source DF Sum of Squares Mean Square F Ratio Prob > F

Model 4 0.06813044 0.01703261 0.34122485 0.84717633

Error 22 1.09815397 0.04991609

C. Total 26 1.16628442

Source DF Sum of Squares Mean Square F Ratio Prob > F

Model 4 0.00829677 0.00207419 2.72695193 0.05539259

Error 22 0.01673378 0.00076063

C. Total 26 0.02503055

Source DF Sum of Squares Mean Square F Ratio Prob > F

Model 4 0.10317226 0.02579307 247.05607200 6.14E-18

Error 22 0.00229684 0.00010440

C. Total 26 0.10546910

 CD Response

 Cm Response

 Analysis of Variance

 CL Response

 Analysis of Variance

 Analysis of Variance
27 

(b) tests with respect to random effects 

Figure 9. Concluded. 

 

Source SS MS Num DF Num F Ratio Prob > F

Rep&Random 0.00008245 0.00004123 2 0.00082589 0.99917448

Tail_Angle 0.06804609 0.06804609 1 1.36320952 0.25548085

Tail_Angle*Tail_Angle 0.00000190 0.00000190 1 0.00003810 0.99513098

Source SS MS Num DF Num F Ratio Prob > F

Rep&Random 0.00002260 0.00001130 2 0.01485822 0.98526149

Tail_Angle 0.00801514 0.00801514 1 10.53755030 0.00370563

Tail_Angle*Tail_Angle 0.00025902 0.00025902 1 0.34054096 0.56545251

Source SS MS Num DF Num F Ratio Prob > F

Rep&Random 0.00081175 0.00040587 2 3.88762149 0.03582952

Tail_Angle 0.10221366 0.10221366 1 979.04245800 0.00000000

Tail_Angle*Tail_Angle 0.00014685 0.00014685 1 1.40658721 0.24827524

 CD Response

 Cm Response

 Tests wrt Random Effects

 Tests wrt Random Effects

 Tests wrt Random Effects

 CL Response
28 

Figure 10. Combined second- and third-stage analyses for the lift coefficient response. 

Source SS DF MS F Ratio Prob > F

Rep&Random 0.00008245 2 0.00004123 0.39128300 0.69951662

wptreat 0.06804799 2 0.03402400 322.93238300 0.00003789

Rep*wptreat&Random 0.00042144 4 0.00010536 1.93828183 0.15623996

AoA 1.09663735 1 1.09663735 20174.66110000 6.9027E-25

Tail_Angle*AoA 0.00025203 1 0.00025203 4.63655851 0.04797679

AoA*AoA 0.00002779 1 0.00002779 0.51132859 0.48554603

Error 0.00081536 15 0.00005436

Total 1.16628442 26

Source SS DF MS F Ratio Prob > F

Rep&Random 0.00008245 2 0.00004123 0.00082589 0.99917448

Tail_Angle 0.06804609 1 0.06804609 1.36320952 0.25548085

Tail_Angle*Tail_Angle 0.00000190 1 0.00000190 0.00003810 0.99513098

Error 1.09815397 22 0.04991609

Total 1.16628442 26

Source SS DF MS F Ratio Prob > F

Whole Plot Level

Rep&Random 0.00008245 2 0.00004123 0.39128300 0.69951662

wptreat 0.06804799 2 0.03402400 322.93238300 0.00003789

 Decomposition of wptreat

 Tail_Angle 0.06804609 1 0.06804609 645.84367882 0.00001424

 Tail_Angle*Tail_Angle 0.00000190 1 0.00000190 0.01803341 0.89966021

Rep*wptreat&Random 0.00042144 4 0.00010536 1.93828183 0.15623996

Subplot Level

AoA 1.09663735 1 1.09663735 20174.66110000 6.9027E-25

Tail_Angle*AoA 0.00025203 1 0.00025203 4.63655851 0.04797679

AoA*AoA 0.00002779 1 0.00002779 0.51132859 0.48554603

Error 0.00081536 15 0.00005436

Total 1.16628442 26

Whole Plot and Subplot Levels

Analysis of Lift Coefficient

Note: Statistically significant terms are highlighted

First Step, Second-Stage Analysis

Second Step, Second-Stage Analysis

Modified ANOVA - Third-Stage Analysis

Decomposition of Subplot Terms

Decomposition of Whole Plot Terms

Combined Analyses from Second-Stage, Steps 1 and 2 with Correct Tests of Signficance at
29 

Figure 11. Combined second- and third-stage analyses for the drag coefficient response. 

Source SS DF MS F Ratio Prob > F

Rep&Random 0.00002260 2 0.00001130 2.06360525 0.24223504

wptreat 0.00827416 2 0.00413708 755.40947900 0.00000697

Rep*wptreat&Random 0.00002191 4 0.00000548 1.73134181 0.19546921

AoA 0.01509471 1 0.01509471 4771.95070000 3.3690E-20

Tail_Angle*AoA 0.00040001 1 0.00040001 126.45602100 0.00000001

AoA*AoA 0.00116971 1 0.00116971 369.78426600 5.5695E-12

Error 0.00004745 15 0.00000316

Total 0.02503055 26

Source SS DF MS F Ratio Prob > F

Rep&Random 0.00002260 2 0.00001130 0.01485822 0.98526149

Tail_Angle 0.00801514 1 0.00801514 10.53755030 0.00370563

Tail_Angle*Tail_Angle 0.00025902 1 0.00025902 0.34054096 0.56545251

Error 0.01673378 22 0.00076063

Total 0.02503055 26

Source SS DF MS F Ratio Prob > F

Whole Plot Level

Rep&Random 0.00002260 2 0.00001130 2.06360525 0.24223504

wptreat 0.00827416 2 0.00413708 755.40947900 0.00000697

Decomposition of wptreat

 Tail_Angle 0.00801514 1 0.00801514 1462.61678832 0.00000279

 Tail_Angle*Tail_Angle 0.00025902 1 0.00025902 47.26642336 0.00234501

Rep*wptreat&Random 0.00002191 4 0.00000548 1.73134181 0.19546921

Subplot Level

AoA 0.01509471 1 0.01509471 4771.95070000 3.37E-20

Tail_Angle*AoA 0.00040001 1 0.00040001 126.45602100 0.00000001

AoA*AoA 0.00116971 1 0.00116971 369.78426600 5.5695E-12

Error 0.00004745 15 0.00000316

Total 0.02503055 26

Analysis of Drag Coefficient

Decomposition of Subplot Terms

Decomposition of Whole Plot Terms

Combined Analyses from Second-Stage, Steps 1 and 2 with Correct Tests of Significance at

Whole Plot and Subplot Levels

Note: Statistically significant terms are highlighted

First Step, Second-Stage Analysis

Second Step, Second-Stage Analysis

Modified ANOVA - Third-Stage Analysis
30 

Figure 12. Combined second and third-stage analyses for the pitching moment coefficient response. 

Source SS DF MS F Ratio Prob > F

Rep&Random 0.00081175 2 0.00040587 2.26741461 0.21964955

wptreat 0.10236051 2 0.05118026 285.91833000 0.00004825

Rep*wptreat&Random 0.00071601 4 0.00017900 9.76003571 0.00042926

AoA 0.00122056 1 0.00122056 66.55024270 0.00000068

Tail_Angle*AoA 0.00001227 1 0.00001227 0.66906420 0.42618129

AoA*AoA 0.00007289 1 0.00007289 3.97420462 0.06471406

Error 0.00027511 15 0.00001834

Total 0.10546910 26

Source SS DF MS F Ratio Prob > F

Rep&Random 0.00081175 2 0.00040587 3.88762149 0.03582952

Tail_Angle 0.10221366 1 0.10221366 979.04245800 9.8149E-20

Tail_Angle*Tail_Angle 0.00014685 1 0.00014685 1.40658721 0.24827524

Error 0.00229684 22 0.00010440

Total 0.10546910 26

Source SS DF MS F Ratio Prob > F

Whole Plot Level

Rep&Random 0.00081175 2 0.00040587 2.26741461 0.21964955

wptreat 0.10236051 2 0.05118026 285.91833000 0.00004825

Decomposition of wptreat

 Tail_Angle 0.10221366 1 0.10221366 571.02603352 0.00001819

 Tail_Angle*Tail_Angle 0.00014685 1 0.00014685 0.82039106 0.41629108

Rep*wptreat&Random 0.00071601 4 0.00017900 9.76003571 0.00042926

Subplot Level

AoA 0.00122056 1 0.00122056 66.55024270 0.00000068

Tail_Angle*AoA 0.00001227 1 0.00001227 0.66906420 0.42618129

AoA*AoA 0.00007289 1 0.00007289 3.97420462 0.06471406

Error 0.00027511 15 0.00001834

Total 0.10546910 26

Analysis of Pitching Moment Coefficient

Decomposition of Whole Plot Terms

Decomposition of Subplot Terms

Combined Analyses from Second-Stage, Steps 1 and 2 with Correct Tests of Significance at

Whole Plot and Subplot Levels

Note: Statistically significant terms are highlighted

First Step, Second-Stage Analysis

Second Step, Second-Stage Analysis

Modified ANOVA - Third-Stage Analysis
31 

Figure 13. Final regression model coefficients (in coded units) and statistical tests of significance 

 for the lift, drag, and pitching moment coefficient responses. 

Term Coef SE Coef t Prob > |t|

Intercept 0.31597778 0.00245758 128.57288900 3.0713E-24

Tail_Angle 0.06148446 0.00241935 25.41364233 0.00001424

Tail_Angle*Tail_Angle -0.000563 0.00419044 -0.13435337 0.89961212

AoA 0.24682848 0.00173777 142.03753400 6.9027E-25

Tail_Angle*AoA -0.0045828 0.00212832 -2.15326690 0.04797679

AoA*AoA 0.0021523 0.00300990 0.71507244 0.48554603

Term Coef SE Coef t Prob > |t| 

Intercept 0.05444769 0.00059285 91.84093160 4.7463E-22

Tail_Angle 0.02110179 0.00055180 38.24155595 0.00000279

Tail_Angle*Tail_Angle 0.00657044 0.00095566 6.87529109 0.00234472

AoA 0.02895851 0.00041921 69.07930730 3.3690E-20

Tail_Angle*AoA 0.00577356 0.00051342 11.24526660 0.00000001

AoA*AoA 0.01396249 0.00072609 19.22977550 5.5695E-12

Term Coef SE Coef t Prob > |t|

Intercept -0.08041760 0.00142752 -56.33362700 7.1019E-19

Tail_Angle -0.07535610 0.00315306 -23.89932687 0.00001818

Tail_Angle*Tail_Angle -0.00494720 0.00546156 -0.90582231 0.41625914

AoA -0.00823460 0.00100941 -8.15783320 0.00000068

Tail_Angle*AoA -0.00101120 0.00123627 -0.81796340 0.42618129

AoA*AoA 0.00348541 0.00174835 1.99354072 0.06471406

CL Regression Coefficients

CD Regression Coefficients

Cm Regression Coefficients

Note: Statistically significant terms are highlighted
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