Statistical experimental designs are presented for various numbers of organisms and agar solutions pertinent to the experiment, ""colony growth in zero gravity''. Missions lasting 7 and 30 days are considered. For the designs listed, the statistical analysis of the observations obtained on the space shuttle are outlined

Hinkelmann, K.

Myers, R. H.

English

NASA Technical Reports Server

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https://ntrs.nasa.gov/search.jsp?R=19760020726 2020-03-22T14:59:41+00:00Z
(NASA-CF.- 148311)
 SMISTICAL SUPPORT FCR
	 576-27814
THE ATL PRCGFAI Final Report, 1 Jul. 1974 -
15 Jun. 1976 (virgicia Polytechnic lost, and
State Univ.) 21 p VC $3.50
	 CSCL 061	 Unclas
G3/51 44609
FINAL REPORT
GRANT NO. NSG 1071
"STATISTICAL SUPPORT
FOR THE ATL PROGRAM"
July 1, 1974 June 15, 1976
Principal Investigators,;
K. Hinkelmann, Professor
R. H. Myers, Professor
Department of Statistics
Virginia Polytechnic Institute
and State University
Blacksburg, Virginia
INTRODUCTION AND SUMMARY
The investigations "Statistical Support for the ATL" were done in essentially
three phases:
Phase I: Preliminary survey of all experiments listed in Technical Report
NASA TMX - 2813;
Phase II: Interviews with Principal Investigators of some experiments;
Phase III: Development of statistical designs and analysis for the
experiment "Colony Growth in Zero Gravity".
This report gives detailed statistical experimental designs for various
numbers of organisms and agar solutions pertinent to the experiment. 7-day and
30-day missions have been considered. For the designs listed, the statistical
analysis of the observations obtained on the space shuttle has been outlined.
-2 -
PHASE I:
In order to determine which experiments from the ATL program would lend
themselves to statistical applications, we considered and studied in as much
detail as possible all the experiments as described in the technical report
NASA :MX - 2813. We tried to determine from the descriptions given there,
whether an experiment could possibly benefit from a statistical design as to
how the experiment should be performed on the space shuttle and/or to what
extent more detailed statistical analysis might help in the interpretation
of the data once they are collected.
PHASE II
After completion of Phase I, it was decided, in cooperation with Dr. Janet
W. Campbell, Technical Monitor of the grant, to consider and discuss in more
detail the following four experiments:
1. "Lidar Measurements of Cirrus Clouds and Lower Straetospheric
Aerosols", Dr. Ellis Remsburg, Principal Investigator.
2. "Colony Growth in Zero Gravity", Dr. Judd Wilkins, Principal
Investigator.
3. "Microwave Radiometer Measurements", Dr. Calvin Swift,
Principal Investigator.
4. "Environmental Effects on Nonmetallic Materials", Mr.
Wayne S. Slemp, Principal Investigator.
The purpose of these discussions was to get a clearer understanding of the
experiments and thereby determine to what extent these experiments or parts
of them were statistical in nature. We looked for applications of statistical
experimental design techniques for the data gathering phase and statistical
analysis techniques for the evaluation phase of each experiz:ent.
Based upon the information obtained we found that, at the present time,
only the experiment "Colony Growth in Zero Gravity" would call for statistical
input. Our findings and contributions concerning this experiment are reported
in detail under Phase III.
-3-
PHASE III.
The objectives and description of the experiment,"Colony growth
in zero gravity" are given in Technical Report NASA TM X-2813, p. 109-
111. The only difference between the equipment described there and
the one actually used is that there will be 10 culture tubes instead
of 8.
The following basic situations were considered:
1. A 7-day mission with a 5-day observation period;
2. a30 -day mission with a 25-day observation period.
Since the principal investigator wants to study the growth patterns
of different organisms in different agar solutions we considered
accordingly the following possibilities: For the 7-day mission:
1.1 2 organisms, 5 agar solutions
1.2 3 organisms, 5 agar solutions
1.3 4 organisms, 5 agar solutions
1.4 5 organisms, 5 agar solutions;
and for the 30 -day mission:
2.1 6 organisms, 5 agar solutions
2.2 7 organisms, 5 agar solutions
2.3 8 organisms, 5 agar solutions
2.4 9 organisms, 5 agar solutions
2.5 10 organisms, 5 agar solutions.
A. THE STATISTICAL DESIGNS:
For each combination 1.1 - 1.4 and 2.1 - 2.5 we have constructed
appropriate statistical experimental designs. These are listed in
Tables 1.1 - 1.4 and 2.1 - 2.5, respectively.
In Tables 1.1 - 1.4 we have listed the actual (randomized) assign-
ments of all organisms -agar solution (0, A) combinations to the positions
on the culture tube rack for each of the five days of experimentation.
The incomplete block designs were constructed in such a way that they
have the following properties: (i) they are nearly balanced in the
sense that each (0,A)-combination appears with almost equal frequency,
(ii) they are pseudo-globally connected, i.e. give as much information
as possible with respect to comparisons of different (0,A)-combinations,
and (iii) they are nearly resistant in the sense that the effect of
the loss of the data for any one day is minimized.
In Tables 2.1 - 2.5 we give the arrangement of (0,A) -combinations
for each day of experimentation in a slightly different way. Because
of the larger number of (0,A)-combinations and hence the increase in
"complexity" of the experiment, it seems preferable to use any given
organism together with the five agar solutions on the same day. As
a result, only two organisms are used each day, indicated by "X" in
the appropriate table. For the actual experiment the ten designated
(0,A)-combinations have to be randomly assigned to the ten positions
on the culture tube rack. These incomplete block designs are (i) nearly
balanced over the whole duration of the experiment, (ii) nearly balanced
over each five-day period, (iii) pseudo-globally connected, and (iv)
nearly resistant.
B. THE ANALYSIS
Each of the designs given under A is an incomplete block design
for which an appropriate model is of the form
yijk ^ u+Tij +dk +eijk .	 (1)
(i	 1,	 2,....
	 P;	 j	 ' 1 9	2,..., 5; k - 1,	 2,...,	 d)
-5-
where yijk = response of the i-th organism in the j-th agar solution
obtained on the k-th day of experimentation (we suggest
that some form of index be developed to characterize the
growth patterns numerically, denoted by y),
U = overall mean
Tij = effect of the i-th organism and j-th agar solution
combination,
d  = effect of the k-th day
E
ijk = random error.
Let rij denote the number of times that the combination of the i-th
organism and the j-th rgar solution occur in the experment and let
rll	 0
r12
R =
r15
0 rp5
where p denotes the number of organisms used in a particular experiment.
Furthermore, let N denote the incidence matrix. N is a 5pxd matrix,
where d denotes the number of days of experimentation, consisting of
1's and 0's, a 1 in the [(i,j),k]th position indicating that the
combination if the i-th organisms and the j-th agar solution is being
observed on the k-th day of experimentation, a 0 indicating that the
particular (0,A)-combination is not being observed on that day.
RITRODUCMTI I' `,' a T' , r
ORIGiNAI j = ; ifs I't _^^
way:
-6-
The reduced normal equations for estimating the "treatment" effects
Tij are then given as
(R
-1 NN') T 4	 (2)
where
T' • (T11 , T12 ,..., T15 ,..., Tp5)
and
S' a (Q11 , Q12 ,..., Q15 ,..., Qp5)
with
Qij M (Sum of all observations for (i,j)-th treatment combination)
_ 1Yo-
 
(Sum of all day totals on which the (i,j)-th
combination is being observed).
A solution to the equations (2) is given by
T 
(R - 1 NN , + J)-1 4	 (3)
where J is a 5px5p matrix of unity elements. Let
(R - 10 NN , + J)
-1 - V
	 (4)
and denote a typical element of V by v, j „ where 1,1' - 1, 2,...,
p : j , j '	 1, 2, 3, 4,•5.
Since in the present study the "treatments" represent combinations
of organisms and agar-solutions, it is appropriate to take this
factorial structure into account by partitioning Tij in the following
-7-
where Oi= effect of i-th organism
a  = effect of j-th agar solution,
(0a) ij = interaction between i-th organism aad j -th agar solution.
In terms of the solutions (3) these effects can be estimated as
1 5
	
Oi 5 E Tij	 (5)j-1
	
aj - 
P 
E Tij	 (6)
i=1
(0a) iiTij - 0  - a 	 (7)
The following comparisons are of interest;
W Growth pattern of i-th organism versus growth pattern of
i-th organism;
(ii) growth pattern of i-th organism in j-th agar solution
versus growth pattern of i-th organism in j' -th agar solution;
(iii) growth pattern of i-th organism in j -th agar solution versus
growth pattern of i'-th organism in j -th agar solution.
Using (5), (6), (7) these comparisons are estimated as
(i) 0  - Oi l	 (8)
(ii) a  + (09) ij - aj , - (0a) ij r	 Tij - T ij r 	 (9)
(iii) 0  + (0a) ij - Oi l - (0a) i , j = T ij - T i ,j 	 (10)
The variances of these comparisons are given by
(i) var(Oi
 - Oil) - 25
	 E , vij,ij' + E	 virj,irj,
	
jr j	 j.j
	
- 2 E	 vij^irjrl62 (11)j l
i_
-8-
(ii)	 2var (T i j - Tip) - IV ii- ii + vij , , i j , - 2vij , ij'   Q (12)
2var (T i j	 Ti, j ) - IV ii- ii + vij , i' j - 1vi j ' i' j c 	(13)
In (11), (12), (13) a2 has to be estimated from the usual intra-block
analysis of variance table for incomplete block designs. For the
designs given in Tables 1.2 - 1.4, the elements vij.i , j , are listed
in Tables 3.2 - 3.4, respectively, where the rows and columns are
labeled in the order (1,1), (1,2),..., (1,5), (2,1), (2,2),..., (2,5),
etc. The design of Table 1.1 is, of course, a randomized complete
block design, and hence all diagonal elements vij,ij - 1/5 and all
off-diagonal elements vij,i , j , - 0. For the designs given in Tables
2.1 - 2.5 the elements vij,i , j , will have to be computed, using the
def inition of V as given in (4).
1. WERIMENTAL DESIGNS FOR SEVEN-DAY MISSION.
1.1 2 ORGANISMS. 5 AGAR SOLUTIONS
DAY 1: (2 , 1 ) (2 ,3) (1,3) (2 . 2 ) (2,5)
(1,1) (1.4) (2.4) (1.2) (1.5)
DAY 2: (1.3) (1.1) (1.2) (2.1) (2,2)
(2 , 4 ) (2.3) (2 .5) (1,5) (1,4)
DAY 3: (1.2) (2,3) (2,4) (2,1) (2,5)
(1 , 1 ) (1 .3) (2 . 2 ) (1 . 5) (1.4)
DAY 4: (2,5) (2,3) (1.3) (1,1) (2,2)
(1.5) (2.4) (1.2) (1,4) (2.1)
DAY 5: (1 , 1) (1.4) (1. 2) (2 , 2 ) (1.5)
(2,4) (2.1) (2,3) (1.3) (2,5)
1.2 3 ORGANISMS, S AGAR SOLUTIONS
DAY 1: (1,3) (2,5) (1,S) (2,1) (1,4)
(2 . 4 ) (1 . 1 ) (1.2) (2 . 2 ) (2.3)
DAY 2: (3,1) (1,3) (3,2) (1,4) (3,5)
(1 . 1 ) (1 . 2 ) (1 ► 5 ) (3 ► 3) (3.4)
DAY 3: (2,1) (3 . 4 ; (3 . 2 ) (2.2) (3,5)
(3.3) (2 . 3) (2 ► 5) (3.1) (2.4)
DAY 4: (2,1) (3,4) (2,3) (3,2) (3,1)
(1 . 1 ) (3.3) (1 .3) (1. 2) (2 ► 2)
DAY 5: (1,3) (3,4) (2,5) (2,3) (2,4)
(3.3) (3,5) (3,2) (1,4) (1.5)
1.3 4 ORGANISMS, 5 AGAR SOLUTIONS
DAY 1: (3,1) (2,1) (4,3) (4,2) (2,2}
(3.3) (1 ► 2) (4 ► 1) (3 ► 2) (1.1)
DAY 2: (1,5) (1,4) (1,3) (4,4) (2,5)
(3.4) (2 . 3 ) (2.4) (3.5) (4.5)
DAY 3: (1,1) (1,3) (2,1) (3,1) (3,3)
(1.2) (4.1) (4.2) (3,2) (2,2)
DAY 4: (4,4) (2,3) (4,3) (1,5) (3,4)
(3 .5) (4.5) (2.5) (1.4) (2.4)
DAY 5: (2,2) (1,3) (3,3) (4,4) (4.3)
(1 . 4 ) (3. 2 ) (3 ► 4) (2 .3) (1.2)
1.4
	 5 ORGANISMS, 5 AGAR SOLUTIONS
DAY 1: (1,2) (4,1) (1,1) (5.1) (3.1)
(4, 2) (2. 2 ) (3. 2 ) (5, 2 ) (2.1)
DAY 2: (3,3) (4,2) (1,3) (4,3) (5,3)
(5. 2) (2 , 2 ) (1. 2 ) (2 .3) (3.2)
DAY 3: (5,4) (2,3) (3,4) (4,3) (1,4)
(5.3) (3.3) (2, 4 ) (4. 4 ) (1.3)
DAY 4: (3,4) (2.5) (5. 4 ) (1, 4) (2.4)
(4.5) (4,4) (3.5) (5,5) (1.5)
DAY 5: (5.1) (1,1) (4,1) (2,5) (3,5)
(3.1) (4.5) (2.1) (1.5) (5.5)
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Statistical support for the ATL program

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