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Introduction to two-levelexperimentation
1. Two-factor two-level experiments
Let us study the impact of two factors on a single variable.Example: (direct mail offering study)
Variable: response rateTwo factors: envelope size, postageSimple situation: each factor at two fixed levels
Low level High level
Factor A: envelope #10 9x12Factor B: postage 3 rd class 1 st class
It does not matter which level of a factor is labeled low andwhich labeled high; low and high are arbitrary labels. But once
this decision is made, the labels must be retained.
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There are four possible experimentalcombinations. These are called treatments,or treatment combinations:
a 0b 0 A at low level, B at low levela 0b 1 A at low level, B at high levela 1b o A at high level, B at low levela 1b 1 A at high level, B at high level
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For now we will imagine that each of the fourtreatments is run once; the experiment will becalled a two-factor two-level factorial design(without replication).
For each treatment the response rate is computed.
The four observed response rates are called yields or responses . The symbols used for treatments are
also used for yields (or yield m eans fo r wi threpl icat ions case ). Thus, a 0b 0 also represents theyield when the treatment a 0b 0 is run.
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2. Estimating effects in two-factor
two-level experimentsEstimate of the effect of Aa 1b 1 - a 0b 1 estimate of effect of A at high Ba 1b 0 - a 0b 0 estimate of effect of A at low B
sum/2 estimate of effect of A over all B
Estimate of the effect of Ba 1b 1 - a 1b 0 estimate of effect of B at high Aa 0b 1 - a 0b 0 estimate of effect of B at low A
sum/2 estimate of effect of B over all A
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Estimate the interaction of A and B
a 1b 1 - a 0b 1 estimate of effect of A at high Ba 1b 0 - a 0b 0 estimate of effect of A at low Bdifference/2 estimate of the effect of B on the effect of A
Called th e in teract ion of A and B
a 1b 1 - a 1b 0 estimate of effect of B at high Aa 0b 1 - a 0b 0 estimate of effect of B at low A
difference/2 estimate of the effect of A on the effect of BCalled th e in teract ion of B and A
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Note that the two differences in theinteraction estimate are identical; bydefinition, the interaction of A and Bis the same as the interaction of B
and A. In a given experiment one ofthe two literary statements ofinteraction may be preferred by the
experimenter to the other; but bothhave the same numerical value.
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Remarks on interactionMany people feel the need for experiments which
will reveal the effect, on the variable under study, of factorsacting jointly. This is what we have called in teract ion . Thesimple experimental design discussed here provides a wayof estimating such interaction, with the latter defined in away which corresponds to what many scientists and
managers have in mind when they think of interaction.
It is us eful to note that in teract ion w as not inv entedby s ta t is t ic ians . It is a jo int effect exis t ing, of tenpro m inent ly, in the real wo rld .
Statisticians have (wonderfully enough!!) providedways and means to measure it.
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4. Symbolism and languageA is called a main effect. Our estimate of A is
often simply written A.B is called a main effect. Our estimate of B is
often simply written B.AB is called an interaction effect. Our estimate of
AB is often simply written AB.So the same letter is used, generally without
confusion, to describe the factor, to describe itseffect, and to describe our estimate of its effect.
Keep in mind that it is only for economy in speakingand writing that we sometimes speak/write about aneffect rather than an estimate of the effect." Weshould always remember that all quantities formedfrom the yields are, OF COURSE, estimates.
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5. Table of signsThe following table is useful:
A B ABa 0b 0 - - +a 0b 1 - + -a 1b 0 + - -a 1b 1 + + +
Notice than in estimating A, the two treatments with A athigh level are compared to the two treatments with A at low level.Similarly B. This is, of course, logical.
Note also that the signs of treatments in the estimate ofAB are the products of the signs of the corresponding treatmentsof A and B.
Note, finally, that in each estimate, plus and minus signsare equal in number.
Effect = Ave of + Ave of - .
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6B
low high
10 1213 15
lowhigh
A
Example 1 Blow high
10 1515 15
lowhigh
A
Example 2
Blow high
10 1313 10
lowhigh
A
Example 3 B
low high
12 1212 12
lowhigh
A
Example 4
A B AB1 3 2 02 2.5 2.5 -2.53 0 0 -3
4 0 0 0
Discussion of examples:Notice that in Examples 2 & 3interaction is as large as orlarger than main effects.
?
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Change of scale, by multiplying each yield
by a constant (3 inches 3x2.54 cm),
multiplies each estimate by the constant
but does not affect the relationship of
estimates to each other. Addition of a
constant to each yield does not affect the
estimates. The numerical magnitude of
estimates is not important here; it is their
relationship to each other.
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Earlier we formed:
estimate of 2A
estimate of 2B
estimate of 2AB
1 a b ab
-1
-1
1
1
-1
-1
-1
1
-1
1
1
1
A = (-1+a-b+ab)/2
B = (-1-a+b+ab)/2
AB = (1-a-b+ab)/2
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Which for present purposes we replace by:
1 a b ab Z12
1
2
12
.
.
.
.
.
.
.
.
12
1
2
12
12
1
2
12
12
1
2
12
-
-
-
-
--
Now we can see that these coefficients of the threecontrasts are orthogonal and thus A, B and AB constituteorthogonal estimates and their SSs can be foundaccordingly. (For example, SSA = RxZ A^2.)
A
B
AB
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8. Three factors each at two levelsThe dependent variable is response rate of a direct mailoffering.
low highA postage 3rd class 1st classB price $9.95 $12.95
C envelope size #10 9 x 12
Treatments (also yields) (a) old notation (b) new notation.(a) a 0b 0c 0 a 0b 0c 1 a 0b 1c 0 a 0b 1c 1 a 1b 0c 0 a 1b 0c 1 a 1b 1c 0 a 1b 1c 1(b) 1 c b bc a ac ab abc
Yates (standard) order : (add factors one after one)1 a b ab c ac bc abc
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9. Estimating effects in three-factor
two-level designsEstimate of A
(1) a - 1 estimate of A, with B low and C low(2) ab - b estimate of A, with B high and C low(3) ac - c estimate of A, with B low and C high(4) abc - bc estimate of A, with B high and C high
= (a+ab+ac+abc-1-b-c-bc)/4,
= (-1+a-b+ab-c+ac-bc+abc)/4,(in Yates order)
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Estimate of AB (the effect of B on the effect of A)
effect of A with B high - effect of A with B low, all at C high
plus
effect of A with B high - effect of A with B low, all at C low
Note that interaction are averages. Just as ourestimate of A is an average of response to A over allB and all C, so our estimate of AB is an average
response to AB over all C.AB = {[(4)-(3)] + [(2) - (1)]}/4
= {1-a-b+ab+c-ac- bc+abc)/4, in Yates order.
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Estimate of ABC (the effect of C on AB)
interaction of A and B, at C high
minus
interaction of A and B at C low
ABC = {[(4) - (3)] - [(2) - (1)]}/4
= (-1+a+b-ab+c-ac- bc+abc)/4, in Yates order.
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This is our first encounter with a three-factorinteraction. It measures the impact on the responserate of interaction AB as C (envelope size) goes from#10 to 9 x 12. Or, it measures the impact on responserate of interaction AC as B (price) goes from $9.95 to
$12.95. Or, finally, it measures the impact on theresponse rate of interaction BC as A (postage) goesfrom 3rd class to 1st class.
As with two-factor two-level factorial designs,the formation of estimates in three-factor two-levelfactorial designs can be summarized in a table:
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A B AB C AC BC ABC
1 - - + - + + -a + - - - - + +
b - + - - + - +
ab + + + - - - -
c - - + + - - +
ac + - - + + - -
bc - + - + - + -
abc + + + + + + +
Plus-Minus Table
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10. DATA ANALYSIS
1 a b ab c ac bc abc.062 .074 .010 .020 .057 .082 .024 .027
A = main effect of postage = .0125B = main effect of price = -.0485AB = interaction of A and B = -.0060C = main effect of envelope size = .0060AC = interaction of A and C = .0015BC = interaction of B and C = .0045ABC = interaction of A, B, and C = -.0050
NOTE: ac = largest yield; AC = smallest effect.
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We describe several of these estimates, though onlater analysis of this example, taking into accountthe unreliability of estimates based on a smallnumber (eight) of data values, some estimates mayturn out to be so small in magnitude as not to
reject the null hypothesis that the correspondingtrue effect is zero. The largest estimate is -.0485,the estimate of B; an increase in price, from $9.95to $12.95, is associated with a decline in response
rate. The interaction AB = -.0060; an increase inprice from $9.95 to $12.95 reduces the effect of A,whatever it is (A = .0125), on response rate. Orequivalently,
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an increase in postage from 3rd class to 1st classreduces (makes more negative) the alreadynegative effect (B = -.0485) of price. Finally, ABC =-.0050. Going from #10 to 9 x 12 envelope, thenegative interaction effect AB on response rate
becomes even more negative. Or, going from lowto high price, the positive interaction effect AC isreduced. Or, going from low to high postage, thepositive interaction effect (BC) is reduced. All three
descriptions of ABC have the same numericalvalue, but the direct marketer would select one ofthem, and then say it better!
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11. Number and kinds of effects
We introduce the notation 2 k. This means a k-factor design with each factor at two levels. The
number of treatments in an unreplicated 2k
design is 2 k.
The following table shows the number of eachkind of effect for each of the six two-leveldesigns shown across the top.
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22 2 3 2 4 2 5 26 27
2 3 4 5 6 71 3 6 10 15 21
1 4 10 20 351 5 15 35
1 6 211 7
1
main effect2 factor interaction3 factor interaction4 factor interaction5 factor interaction6 factor interaction7 factor interaction
In a 2 k design the number of r-factor effects is C k = k!/[r!(k - r)!]r
3 7 15 31 63 127
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Notice that the total number of effects estimated in anydesign is always one fewer than the number oftreatments:
One need not repeat the earlier logic to determine theforms of estimates in 2 k designs for higher values of k.
A table going up to 2 5 is on P.265, Table 9.4.
in a 2 2 design there are 2 2=4 treatments; we estimate 2 2-1=3 effects,
in a 2 3 design there are 2 3=8 treatments; we estimate 2 3-1=7 effects.
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Exercise:
Write down the plus-minus table for the24 design.
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Note: for 2k designs with replications
All terms (1, a, b, ) are treatment combinations or
yield means of the treatment combinations.
In the previous example, yields are response rates andthus (1, a, b, ) are average response rates for the
corresponding treatment combinations.
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12. Yates forward algorithm 1. Applied to Complete Factorials (Yates, 1937)
A systematic method of calculating estimates
of effects. For complete factorials firstarrange the yields in Yates (standard) order.Addition, then subtraction of adjacent yields.The addition and subtraction operations arerepeated until 2 k terms appear in each line:for a 2 k there will be k columns ofcalculations.
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Example: 2 3 Yield 1st. Column 2nd. Column 3rd. Column
1
a
b
ab
c
acbc
abc
a+1
ab+b
ac+c
abc+bc
a-1
ab - bac - c
abc -bc
ab+b +a+1
abc+bc+ac+ c
ab-b+a-1
abc-bc+ ac- c
ab+b-a-1
abc+bc- ac-cab-b-a + 1
abc-bc-ac+ c
abc+ bc+ac+ c+ab +b+a+1
abc - bc+ac - c+ab - b+a -1
abc+ bc- ac - c+ab+ b -a -1
abc - bc- ac+ c+ab - b -a+1
abc+ bc+ac+ c -ab - b -a-1
abc - bc+ac - c-ab+ b -a+1abc+ bc- ac - c-ab- b+a+1
abc - bc- ac+ c-ab+ b+a-1
Checking in our 2 3 table of signs, entries in the third columnestimate, respectively,
=1) A B AB C AC BC ABCNote the line-by-line correspondence between yields (lower case lettersin the left column of the table) and factors estimated (upper case lettersdirectly above). Treatments and estimates of effects are in Yates order.
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Yates Forward AlgorithmEXAMPLE:
23 already used.
1 .062 .136 .166 .356 -
a .074 .030 .190 .050 estimate of 4Ab .010 .139 .022 -.194 estimate of 4Bab .020 .051 .028 -.024 estimate of 4ABc .057 .012 -.106 .024 estimate of 4Cac .082 .010 -.088 .006 estimate of 4AC
bc .024 .025 -.002 .018 estimate of 4BCabc .027 .003 -.022 -.020 estimate of 4ABC
Again, note the line-by-line correspondence between treatmentsand estimates; both are in Yates order.
Yield 1 st Col 2 nd Col 3 rd ColTr.
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2k
1 a b ab c ac bc abc
.062 .074 .010 .020 .057 .082 .024 .027
.062 1.00 1.00 1.00
.074 2.00 1.00 1.00
.010 1.00 2.00 1.00
.020 2.00 2.00 1.00
.057 1.00 1.00 2.00
.082 2.00 1.00 2.00
.024 1.00 2.00 2.00
.027 2.00 2.00 2.00
SPSS is not oriented toward providing output in a form traditionally associated
with tw o-level exper imentation . In fact, the ou tpu t does not, literally, provid e the
effects. For examp le, for factor A (imm ediately below , VAR0002), the ou tpu t tells
us th at th e mean is .0507 for high A , .0383 for low A. The d ifference between th e
two values, .0124, is the effect of A. The value resulting from Yates algorithm in
the p rev ious section was .0125 (i.e., 4 A = .05, A = .0125); the d ifferen ce is
rounding error, as the .0383 below is actually .03825, while the .0507 is below
actually .05075.
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- - Description of Subpop ulations - -
Summa ries of VAR00001
By levels of A
Variable Value Label Mean Std Dev Cases
For Entire Pop u lation .0445 .0274 8
A 1.00 .0383 .0253 4A 2.00 .0507 .0318 4 Total Cases = 8
- - Description of Subp opu lations - -
Summa ries of VAR00001By levels of B
Variable Value Label Mean Std Dev Cases
For Entire Pop u lation .0445 .0274 8
B 1.00 .0688 .0114 4 B 2.00 .0203 .0074 4
Total Cases = 8
D i ti f S b l ti
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- - Description of Subpop ulations - -
Summa ries of VAR00001By levels of C
Variable Value Label Mean Std Dev Cases
For Entire Pop u lation .0445 .0274 8
C 1.00 .0415 .0313 4 C 2.00 .0475 .0274 4 Total Cases = 8
DESIGN EASE A B C1 1 -1.000000 -1.000000 -1.000000 0.0620002 1 -1.000000 -1.000000 1.000000 0.057000
3 1 1.000000 -1.000000 -1.000000 0.0740004 1 -1.000000 1.000000 -1.000000 0.0100005 1 -1.000000 1.000000 1.000000 0.0240006 1 1.000000 1.000000 1.000000 0.0270007 1 1.000000 -1.000000 1.000000 0.0820008 1 1.000000 1.000000 -1.000000 0.020000
(first column is coun ter, second colum n = # rep licates)
INTERCEPT 0.0445000A 0.0062500B -0.0242500C 0.0030000AB -0.0030000AC 0.0007500BC 0.0022500ABC -0.0025000
NOTE: the va lues a re half of what we call th e effects
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13. Main effects in the face
of large interactionsSeveral writers have cautioned againstmaking statements about main effectswhen the corresponding interactionsare large; interactions describe thedependence of the impact of one factoron the level of another; in the presenceof large interaction, main effects maynot be meaningful.
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EXAMPLE Yields are purchase intent for cigarettes.
low level high levelSex Male FemaleBrand Frontiersman April
The yields are1 = 4.44 s = 2.04 b = 3.50 sb = 4.52The estimates areS = -.69 B = +.77 NP = +1.71.
In the face of such high interaction we now specialize the main
effect of each factor to particular levels of the other factor.Effect of B at high level S = sb - s = 4.52 - 2.04 = 2.48Effect of B at low level S = b - 1 = 3.50 - 4.44 = -.94,which appear to be more valuable for branding strategy thanthe mean (.77) of such disparate numbers.
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Note that answers to these specializedquestions are based on fewer than 2 k yields. Inour numerical example, with interaction SBprominent, we have only two of the four yields inour estimate of B at each level of S.
In general we accept high interactionswherever found and seek to explain them; in theprocess of explanation, main effects (and lower-order interactions) may have to be replaced in ourinterest by more meaningful specialized effects.
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14. Levels of factorsThe responses or yields are conjec tured to follow the curves
1815
20222930
47
58
t1 t2 t3 t4
Yield
Temperature
p 1
p 2
Compare P effect at (t 1, t 2) vs (t 3, t 4) or others.
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At P effect (P 2-P 1)t1 22 - 30 = -8t2 29 - 58 = -29t3 47 - 20 = +27t4 18 - 15 = +3
at (t 1, t 3) ?
P= 27+3 = 15
T= (18-47)+(15-20) = -17PT= 3-27 = -29-(-5) = -12
2
2
2 2
at (t 3, t 4):
P= -8-29 = -18.5
T=(29-22)+(58-30)
= +17.5PT= -29-(-8) = 7-28 = -10.5
2
2
2 2
at (t 1, t 2):
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It is only when the conjecturedresponses in the diagram are in fact linear
and parallel that choice of levels isunimportant.
One must acknowledge the essentiallycircular nature of the discussion. Oneneeds to have a good idea of the responsecurves in order to fix the levels of anexperiment which seeks essentially todiscover the response curves. But this kindof circularity characterizes all experimentalscience.
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15. Factorial designs vs. designs
varying one factor at a timeExample: Variable: ProfitabilityTwo factors each at two levels:
Time Frame: Past Year FutureMode: Numerical Non-numerical
Vary one factor at a time. Hold Time Frame at
past year and take two observations onprofitability at each mode; we take twoobservations to facilitate comparison with afactorial design. Then we take two more
observations at (Numerical, Future):
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Mode
Time Frame
Num.
Non-N.
Past Future
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Now consider an unreplicated 2 2 factorial design.
Mode
Time Frame
Num.
Non-N.
Past Future
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Comparison of the 2 2 factorial design and the one-factor-at-a-time design:
In the factorial design each estimate of a main effectis based on all four yields. Each estimate has asmuch supporting data (is as reliable) as the
corresponding estimate from the more costly six-yield one-factor-at-a-time design; the latter was ableto use only four of its six yields in each estimate.
In the factorial design, interaction = ( w hatever i t i s ),an effect not estimable from the one-factor-at-a-time design.
a.
b.
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In the factorial design, each main effect is estimatedover both levels of the other factor, not at one levelas in the case of the one-factor-at-a-time design; thisincreased generality is usually, though, not always,attractive. If interaction is high, we may, as we have
seen, want the effect of each factor at each level ofthe other factor; this the one-factor-at-a-time designcan provide at two points (the Time Frame effect atNum. and the Mode effect at Past ) better than thefactorial design. But the one-factor-at-a-time designwill not reveal the magnitude of interaction in thefirst place!!
c.
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An estimate of the effect of factors other than thetwo factors studied is possible in the 6-yieldexperiment. Thus, the differences in yields at agiven treatment combination cannot be due to TimeFrame, Mode, or their interaction since Time Frame
and Mode were fixed throughout each difference.These differences must be due to other factors.However, a replicated factorial experiment
can, of course, provide such an estimate.
d.
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One-factor-at-a-time designs are less vulnerableto missing yields.
The general judgment, particularly in recentyears, is that factorial designs are definitelysuperior to one-factor-at-a-time experimentation.
f.
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In a complete 2 5 design, we have 32 treatmentcombinations and, without replication, 32data values. Each data value contributes to
the estimate of each Effect. Thus, each Effecthas the reliability of 32 data values.
To achieve the same reliability doing one -at-a- time experimentation, we would need 96(NINETY-SIX!) data values:
Another Example
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AL, B L, C L, DL, E L AH, B L, C L, DL, E L AL, B H, C L, DL, E L AL, B L, C H, DL, E L
AL, B L, C L, DH, E L AL, B L, C L, DL, E H
AB
C
D
E
Having 16 of each of these 6 treatment combinations
( = 96 data values in total) would give us estimates ofeach main Effect with the same reliability of 32 datavalues.
BUT, WHAT ABOUT INFORMING US ABOUT THEPRESENCE OF INTERACTIONS??
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16. Factors not studied
In any experiment factors other than thosestudied may be influential. Their presence issometimes acknowledged under the title error.
They may be neglected, but the cost of neglectcould be high.It is important to deal explicitly with them;
even more, it is important to measure their impact.How?
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1. Hold them constant.2. Randomize their effects.3. Estimate their magnitude by replicating
the experiment.4. Estimate their magnitude via side or
earlier experiments.5. Confound certain non-studied factors.
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2. Effect of the number of factors on
the error of an estimateWhat is the variance of an estimate of aneffect? In a 2 k design, 2 k treatments go into eachestimate; the signs of the treatments are + or -,
depending on the effectbeing estimated. So any estimate
= 1 [generalized (+ or -) sum of 2 k treatments]
(any estimate) = 1 [2k 2] = 2 /2 k-2 .22k-2
NOTE: 2(kx)=k2 2(x)
2k-1
2
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3. Effect of replication on the error ofan estimate
What is the effect of replication onthe error of an estimate? Consider a 2 k design with each treatment replicated rtimes.
1 a b abc d-
---
-
---
-
---
-
---
-
---
... ...
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any estimate = 1 [sums of 2k
terms, all of themmeans ]
2
(any estimate) =1
[2k
2
= 2
/(r x 2k-2
) ;the larger the replication per treatment,the smaller the error of each estimate.
2k-1 based on samples of size r
22k-2 r
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So, the error of an estimate depends onk (the number of factors studied) and r(the replication per treatmentcombination). It also (obviously)depends on 2.
The variance 2 can be reduced byholding some of the non-studied factors
constant. But, as has been noted, thisgain is offset by reduced generality ofany conclusions.