Economic Trend Resistant2n-(n-k) Designs of Resolutions III and IV Based on Hadamard Matrices

This article utilizes the Normalized Sylvester-Hadamard Matrices of size 2x2and their associated saturated orthogonal arrays OA(2, 2 1, 2, 2) topropose analgorithmbased on factor projection (Backward/Forward) for the construction of three systematic run-after-run2 fractional factorial designs: (i) minimum cost trend free 2designsof resolution III (2≤n≤2– 1 – k)by backward factor deletion (ii) minimum cost trend free 2 designsof resolution III (k+1≤n≤ 2– 2+k ) by forward factor addition (iii) minimum costtrend free 2 designsof resolution IV (2≤n≤2-2) ,where each 2design is economic minimizing the number of factor level changes between the 2successive runs and allows for the estimation of all factor main effects unbiased by the linear time trend,which might be present in the 2sequentially generated responses. The article gives for each 2design: (i) the defining contrast displaying the design’s alias structure(ii) the k independent generators for sequencingthe design’s 2 runs by the Generalized Fold over Scheme and (ii) the minimum total cost of factor level changes between the 2 runs of the design. Proposed designs compete well with existing systematic2 designs (of either resolution) in minimizing the experimental costandin securing factors’ resistance to the nonnegligible time trend.


Introduction
Experiments are carried out in all fields:industrial, educational, agricultural, medical, etc.,where experimentation has led to many innovations and discoveries. Experiments investigate generally the effect of one or several factors on an outcome by manipulating the experimental runs( i.e. humans ,animal, trees, etc.) with these factors, where some multi-factor experiments are called factorial experiments allowing the investigation of the effect of several factors andtheir interactions. Factorial experiments are symmetric or asymmetric, where full factorial 2 k experiments are symmetric and more economical than other full p k factorial experiments ( p>2). Factorial 2 k experiments are mainly used at the start of an experimental investigation in order to identify the most significant factors without the interest ofcharacterizingthese effect precisely ( linear, quadratic, etc). However, factorial 2 k experiments grow in size and complexity as the number of factors get larger, where experimentation becomes costly and unmanageable.Therefore, fractional factorial 2 k-p experiments or orthogonal arrays ( regular or nonregular ) are substitutes in early stages of factorial experimentation , since theyare more economical requiring less experimentation effort and are less costly if high order factor-interactions are negligible.
Full 2 k or fractional 2 k-p factorial experiments are often conducted randomly. However, randomization of allruns of full or fractional factorial experiments may result in large number of factor level changes between runs, rendering experimentation costly and /or impractical, especially if these experiments involve factors with hard-tovary levels, like for instance oven temperature.Therefore, full 2 k or fractional factorial 2 k-p experiments involving difficult-to-vary factors should be carried out sequentially run-after-run or block of runs after block (i.e. not randomly but systematically ) .
The experimental cost when carrying out full 2 k or 2 k-p fractional factorial experiments sequentially run-afterrun involve both the cost of changing factor levels between successive runs as well as the measurement cost of each experimental run. For more on this cost issue, See [25]. We will however concentrate on minimizing the former cost, where we will assume equal cost for changing levels of allthe k the two-level factors (A1 , A2 , A3 , …, Ak) of these 2 k or2 k-p experiments .We will also aim at achieving factor's resistance to the time trend, which might be present in the sequentially generated 2 k or 2 k-p responses,which may bias factor effects. This time trend effect could be smooth of linear /quadratic form or it could be stochastic ofvarying serial correlations. The former time trend form (i.e. the polynomial) will be adopted in this research.
For run-after-run full 2 k factorial experimentation,there are2 k ! run orders (i.e. permutations ) while for runafter-run 2 k-p fractional factorial experimentationthere are 2 k-p ! run orders, where not all these run orders( permutations) of either experimentation scheme are economical with regard to the number of factor level changes nor all are resistant to the time trend effect. One of the 2 k ! run orders of the full 2 k factorial experiment provided for each 2 k-i designbut higher levels of fractionation( i.e. i>3) were not considered and the pattern of factor level changes was not characterized.Reference [10] provided a small catalog of GFS sequenced 2 k-p fractionated experiments [k<16 and p<8], where all factor main effects are robust against the polynomial time trend and where the total number of factor level changes arekept minimum . The (k-p) independent run generators for each 2 k-p design and the total cost of factor level changes were provided but neither the resolution nor the defining contrast were given nor the pattern of factor level changes was characterized. Reference [6] utilized the standard order of the full 2 k experiment in (1.1) laying out all main effects Ai (i =1,2,…,k) and their interaction columns in increasing number of level changes [ from 1 up to (2 k -1) ] then constructed two types of 2 n-(nk) designs:minimum cost 2 n-(n-k) designs of resolution III (2 k-1 ≤n≤2 k -1)and minimum cost 2 n-(n-k) designs of resolution IV (2 k-2 ≤n≤2 k-1 )but regardless of factors' time trend resistance.However, neither the defining relations nor the GFS generator sets were reported nor the minimal total cost of factor level changes wascomputed. [19] elaborated on the work of [6] and constructed minimum cost trend free 2 n-(n-k) designs of resolution IV but without providing the GFS generators. [4]employed an algorithm based on the GFS approach to sequence runs of symmetric orthogonal arrays OA(N,n,q,3) of resolution III in minimum number of factor level but regardless of factors' time trend resistance, where factors have prime number of levels greater than 2 and where the number of factors is constrained to [(N/q-1)/(q-1) +1 ≤n≤(N-1)/(q-1)] to ensure runs non-duplication. Defining relations were not provided, norprovision was made for the total cost of factor level changes. [1] constructed half fractions (i.e. 2 (k+1)-1 ) from the full 2 k factorial experiment having itsk factorsAi (i =1,2,…,k) laid out in minimaltotal number of factor level changes [i.e. ( 2 k -1) ], then incorporated an additional factor Ak+1=A1A2A3…Akrepresented by the interaction of all the k factorsAi (i =1,2,…,k),where the total number of level changes for all (k+1) factors is in increasing pattern totaling =[1+2+2 2 +2 3 +…+2 k-1 +2 k-1 ] =2(2 (k-1) -1),yet not all these (k+1) factorsare time trend free. Higher levels of fractionation ( i.e. 2 (k+i)-i , i>1) were not considered and the GFS approach can not be applied to recover the run order of these 2 (k+1)-1 half fractions.
Extending the scope of the interaction main-effect assignment of [1], reference [3] has provided an algorithm based on the reverse foldover scheme to generate full 2 k factorial experimentin minimal number of factor level changes [i.e. ( 2 k -1) ] then applying the interactions-main effects assignment to create additional two-level factors for the construction of a small catalog of systematic 2 k-p designs ( 4 ≤k≤9 and 1≤p≤5 ), where all factor main effects are linear trend free but regardless of the minimality of the cost of factor level changes. Defining contrasts were given for each systematic2 k-p design but no provision was made for the total cost of factor level changes.These trend free 2 k-p designscan not however be sequenced by the GFS approach. [24] proposed an algorithm based on parity check matrices of binary linear codes to find the GFS independent run generators forsequencing runs of regular orthogonal arrays ( i.e. 2 k-p designs) so that their main effects are time trend freebut regardless of minimality of factor level changes.No catalog is reported and also no provision is made on how to construct these parity check matrices.The algorithmwas however illustrated using some examples from special binary linear codes, namely Reed Muller codes, cyclic codes and BCH codes. Finally , [22] represented experimental runs of regular 2 k-p designs as graph vertices then applied Travelling Salesman Algorithm to locate graph paths ( i.e. run orders) of minimal distance without regard to factors' time trend resistance.These minimallysequenced 2 k-p designs( 4 ≤k≤15 and 1 ≤p≤11) cannot however be sequenced by the GFS approach ,since many of theserun orders do not start with the null treatment (1)=(000…0000.Defining contrasts were provided but neither the factors'pattern of level changes nor the total cost of factor level changes werereported. Having completed this literature review and having seen that it is not yet complete especially forfractional factorial experimentation ( regular or non-regular),where it lacks systematic 2 n-k fractional factorial experiments of resolutions III and IV inminimum cost of factor level changes and resistant to the time trendbut without limiting either the number of factors nor the fractionation level.Therefore, this article addresses this problem utilizing the Normalized Sylvester -Hadamard Matrices of order 2 k and their associated saturated orthogonal arrays OA(2 k , 2 k -1, 2, 2 ) to construct by factor projection threetypes of systematic 2 n-(n-k) fractional factorial designs: (i) minimum cost trend free 2 n-(n-k) designsof resolution III (2 k-1 ≤n≤2 k -1-k) by backward factor deletion (ii) minimum cost trend free 2 n-(n-k) designsof resolution III (k+1≤n≤ 2 k-1 -2+k ) by forward factor addition (iii) minimum costtrend free 2 n-(n-k) designsof resolution IV [2 k-2 ≤n≤2 k-1 -2 ],where each 2 n−(n−k) design ( of either resolution) is economic in minimum number of factor level changes and allows for the estimation of all main effects Ai (i =1,2,…,n) unbiased by the linear time trend.Theoretical reference for thisconstruction will be based on results in [18], [20] and [21].
The rest of this paper proceeds as follows: Section 3 introduces Hadamard matrices and their subclass the Normalized Sylvester-Hadamard matrices of order 2 k then the section examines orthogonality of their columns to the time trend factor. Section 4 discusses (through factor projection )the relationship between the Normalized Sylvester-Hadamard matrices of order 2 k and their associated saturated orthogonal arrays OA(2 k ,2 k -1,2,2) with full 2 k and fractional 2 k-p factorial experiments,where various illustrative factor projections will be given when k=4 . Sylvester -Hadamard matrices of order 2 k and their associated saturated OA(2 k ,2 k -1,2,2) are then utilized in Section 5 for the construction of the three proposed minimum costtrend free 2 n-(n-k) fractional factorial designsbythe Journal of Education and Practice www.iiste.org ISSN 2222-1735 (Paper) ISSN 2222-288X (Online) Vol. 11, No.25, 2020 factor projection process. Section 6 gives a brief discussion and conclusion about run-after-run fractional 2 np factorialexperimentation.

Sylvester-Hadamard matrices of size 2 k x2 k and time trend resistance of their 2 k columns.
This section introduces Hadamard matrices and their subclass the Normalized Sylvester-Hadamard matrices of size 2 k x2 k then examines time trend resistance of their 2 k columns .
ii. TheHadamardmatrixHmwhose all its entries in the first row (column) are+1'sis called Normalized, where any Hadamard matrix Hmis equivalent [by property (i)] to a Normalized Hadamard matrix of the same order.All rows (columns) of the Normalized Hadamard matrixHm(except the first) arepair-wise orthogonal.
(ii) the number of sign changes in these (2 k -1) columnsrange from 1 up to(2 k -1) , where these columns can be rearranged in increasing order,as illustrated in Table (3.2) for k =4. Hence, the (2 k -1) columns of the matrixH2 k (except the first)can be identified in two equivalent ways : either by the number of column sign changes or by their column number,whereeach identification ranges from 1 up to(2 k -1). (iii)the(2 k -1) columns of the matrixH2 k (except the first) have the following properties about linear/quadratic time trend resistance ,whether these columns are arranged in increasing order of sign changesor not: (a)only the n columns having the respective number of level changes{(2 k -1),(2 k--1 1), (2 k-2 -1) , (2 k-3 -1) ,… , (2 k-(k-1) -1) }are not orthogonal to the linear time trend, where their Linear Time Countsare not zeros. (b)all remaining ( 2 k -k-1) columns of thematrixH2 k (except the first)are orthogonal to the linear time trend, where their linear Time Counts are zeros.A subset of these( 2 k -k-1) columns are at least quadratic trend free besides being linear trend free. The exactsize of this subset is [2 k -k-1 -k(k-1)/2] columns. Inductive results in (i) , (ii) and (iii) aboutthe Normalized Sylvester -Hadamardmatrices of size 2 k x2 k and their columns' time trend resistance will be utilized in Sections 4 and 5 for the construction of thethree proposed systematic minimum cost/trend resolution III and IV 2 n-(n-k) designs.

Normalized Sylvester-Hadamard matrices of size 2 k x2 k and their relationship with 2 n-k fractional
designs through factor projections This section discusses the relationship between the Normalized Sylvester-Hadamard matrices H2 k ofsize 2 k x2 k (introduced in Section 3) and the full 2 k and fractional 2 n-k factorial experiments, where it is documented in [20]that deleting the first column of+1's in this matrix results in a saturated orthogonal array OA(2 k , 2 k -1,2,2) in maximum number of two-level factors N=(2 k -1) having level changes from 1 up to N, but not arranged in increasing order. That is, these orthogonal arrays are saturated regular2 N-(N-k) designs of resolution III in N= (2 k -1) factors and in only 2 k =(N+1)experimental runs. However, these saturated 2 N-(N-k) designs are not time trend resistant, where k of their columns (i.e. factors) are not orthogonal to the time effect, as shown in the conclusion at the end of Section 3 and as can be seen from the bottom two rows of Table (3.2), for k=4. Therefore, removing these k non-trend free columns {1,3, 7, 15,31,…,(2 k -1)} from all (2 k -1) columns of theOA(2 k , 2 k -1,2,2)result in a minimum cost trend free resolution III 2 M-(M-k) designof 2 k experimental runsin maximum number of trend free factors,namely M=(2 k -1-k).
Applying factor deletion by deleting columnsof the saturated OA(2 k , 2 k -1,2,2) with large level changes to economize experimentation cost result in a sequence ( or catalog) of unsaturated minimum cost resolution III 2 n-(n-k) fractional factorial designs(2 k-1 ≤n≤2 k -1), where factor bounds ensure that runs are not duplicated. Also reducing the number of factors by deleting columnsfrom the minimum cost trend free resolution III 2 M-(M-k) designin maximum number of trend free factorsM=(2 k -1-k) result in another sequence of unsaturated minimum cost trend free resolution III 2 n-(n-k) fractional factorial experiments (2 k-1 ≤n≤2 k -1-k) .
On the other hand applyingnow factor addition on the minimum cost 2 n-(n-k) design [from the OA(2 k , 2 k -1,2,2) ] with smallest number of factors (n=k+1) and with the smallest number of factor level changesby adding factors sequentially in increasing number of factor level changes produces a sequence of minimum cost resolution III 2 n-(n-k) designs(k+1≤n≤2 k-1 -1+k) without getting into run duplication.Similarly, applying factor addition on the smallest minimum cost trend free 2 m-(m-k) design[from the OA(2 k , 2 k -1,2,2) where (m=k+1) ] by adding trend free factors sequentially in increasing number of factor level changes produces a sequence of minimum cost trend free resolution III 2 m-(m-k) designs(k+1≤m≤2 k-1 -2+k). These two backward and forward factor projectionsof the saturated OA(2 k , 2 k -1,2,2) will be illustrated in the followingsubsections.
In addition, restricting the number of factors to exactly n=k projects the saturated OA(2 k , 2 k -1,2,2) into full 2 k factorial design, where the remaining(2 k -1-k ) columns become factor interactions of all orders (from 2 up to k).When n=k , factor projection of the OA(2 k , 2 k -1,2,2) may however reduce thisOAinto 2 n-(n-k) fractional factorial designsin duplicated runs if the n=k columns chosen are not linearly independent.Fractional factorial 2 n-(n-k) designs in duplicated runs can also be generated if projections involve n<k factors.There are many choices for the (n=k ) generator columns as factor main effects, where some selections (i.e. projections) lead to full 2 k factorial designs in minimum number of factor level changes (i.e. minimum experimentation cost), while other column choices produce full 2 k factorial designs in maximum number of factor level changes (i.e. maximum experimentation cost).Projecting the saturated OA (2 k , 2 k -1, 2, 2) onto its ( n=k) non-trend free columns{(2 k -1),(2 k--1 1), (2 k-2 -1) , (2 k-3 -1) ,… , (2 k-(k-1) -1) } result in the standard order (1.1) of the full 2 k factorial experiment. None of these three projected full 2 k factorial designsare however time trend resistant,since they involve column factors having nonzero Time Counts. Of course, there are other projections of the saturated OA (2 k , 2 k -1, 2, 2) into (n=k) factors producing time trend free full 2 k factorial designs,where this is achieved by avoiding assigning any of the (n=k) non-trend free columns{(2 k -1),(2 k--1 1), (2 k-2 -1) , (2 k-3 -1) ,… , (2 k-(k-1) -1) } as factor main effects and also by avoiding selecting any of the dependent columns of the (2 k -k -1) trend-free columns of this OA.These full 2 k factorial projections will also be illustrated in the following subsections.
It should be noted here that preceding factor projectionsof the saturated OA (2 k , 2 k -1, 2, 2)into unsaturated resolution III 2 n-(n-k) fractional factorial designsor into full 2 k factorial designshave been found through an inductive analysis of the Normalized Sylvester-Hadamard matrices H2 k and their associated OA (2 k , 2 k -1, 2, 2) for k=4, 5,6,7,8,9,10.The following three subsections will illustrate these factor projections( Backward/Forward) utilizing the Normalized Sylvester-Hadamard matrix H16 in Table (3.1) and its associated OA (2 4 , 2 4 -1, 2, 2) in Table (3.2) when (i) time trend is negligible /non-negligibleand when (ii) the projected 2 n-(n-k) design is of resolution III or IV, where Subsection 4.3 will discuss the problem of raising the design's resolution from III into IV while securing minimum factor level and/or factors' time trend resistance.

Discussion and Conclusion
Fractional 2 n-k factorial experiments with factors having levels hard-to-vary should be carried out sequentially ( i.e.not randomly) either run after run or block of runs after block in order to economize the cost of varying factor levels between successive runs. However,systematic fractional 2 n-k factorial experiments suffer from the problem that factor effects may be adversely affected by a time trend which might be present among responses of the successive runs. Therefore,2 n-k fractional factorial experiments should be sequenced but overcome this time trend problem and also economize the experimental cost. There are a total of 2 n-k ! run orders ( i.e. permutations) to carry out fractional 2 n-k factorial experiments run after run but not all these run orders are resistant to the time trend nor economic. Also not all these 2 n-k ! run orders can be sequenced by the GFS technique, yet economic run orders resistant to the time trend can be generated by the GFS approach.
This research has utilized the Normal Sylvester-Hadamard matrices of size 2 k x 2 k and their associated saturated orthogonal arrays OA(2 k ,2 k -1,2,2) to construct (by factor projection) three systematic 2 n-(n-k) fractional factorial designsof resolutions III and IV that are economic regarding the cost of factor level changesand/or resistant to the non-negligible time trend.Proposed 2 n-(n-k) fractional factorial designs have the merit that all their 2 k experimental runs can be sequenced run-after-run by the GFS technique using only k independent run generators,where these k independent generator runs aregiven.The other merit is that all factor effects can be estimated unbiased by the non-negligible time trend. Comparison with existing counterpart systematic 2 n-k designs shows that the proposed2 n-(n-k) designs compete well with and sometimes are better,since they have: (i) smaller cost of factor level changes between successive runs and (ii) securedall factor effects to be orthogonal and unbiased by the non-negligible time trend. All this is done without fixingan upper limit for either the number of factors or the fractionation level, while maintaining resolution III or IV without duplicatingany experimental run. It is however worth to conduct a comparison among existing runs sequencing algorithms for the 2 n-k fractional factorial experiment (run after run or block after block) in terms of thethe following parameters: the total cost of factor level Journal of Education and Practice www.iiste.org ISSN 2222-1735 (Paper) ISSN 2222-288X (Online) Vol. 11, No.25, 2020