Some results on constructing general minimum lower order confounding <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$2^{n-m}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msup> </math> </EquationSource> </InlineEquation> designs for <InlineEquation ID="IEq2"> ...
Zhang et al. (Stat Sinica 18:1689–1705, <CitationRef CitationID="CR15">2008</CitationRef>) introduced an aliased effect-number pattern for two-level regular designs and proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal designs. All the GMC <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$2^{n-m}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msup> </math> </EquationSource> </InlineEquation> designs with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$N/4+1\le n\le N-1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>N</mi> <mo stretchy="false">/</mo> <mn>4</mn> <mo>+</mo> <mn>1</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation> were constructed by Li et al. (Stat Sinica 21:1571–1589, <CitationRef CitationID="CR8">2011</CitationRef>), Zhang and Cheng (J Stat Plan Inference 140:1719–1730, <CitationRef CitationID="CR14">2010</CitationRef>) and Cheng and Zhang (J Stat Plan Inference 140:2384–2394, <CitationRef CitationID="CR5">2010</CitationRef>), where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$N=2^{n-m}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msup> </mrow> </math> </EquationSource> </InlineEquation> is run number and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>n</mi> </math> </EquationSource> </InlineEquation> is factor number. In this paper, we first study some further properties of GMC design, then we construct all the GMC <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$2^{n-m}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msup> </math> </EquationSource> </InlineEquation> designs respectively with the three parameter cases of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$$n\le N-1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>n</mi> <mo>≤</mo> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation>: (i) <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$$m\le 4$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>m</mi> <mo>≤</mo> <mn>4</mn> </mrow> </math> </EquationSource> </InlineEquation>, (ii) <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$$m\ge 5$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>m</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$$n=(2^m-1)u+r$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>n</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mi>m</mi> </msup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>u</mi> <mo>+</mo> <mi>r</mi> </mrow> </math> </EquationSource> </InlineEquation> for <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$$u>0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>u</mi> <mo>></mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$$r=0,1,2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>r</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math> </EquationSource> </InlineEquation>, and (iii) <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$$m\ge 5$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>m</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$$n=(2^m-1)u+r$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>n</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mi>m</mi> </msup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>u</mi> <mo>+</mo> <mi>r</mi> </mrow> </math> </EquationSource> </InlineEquation> for <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$$u\ge 0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>u</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$$r=2^m-3,2^m-2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>r</mi> <mo>=</mo> <msup> <mn>2</mn> <mi>m</mi> </msup> <mo>-</mo> <mn>3</mn> <mo>,</mo> <msup> <mn>2</mn> <mi>m</mi> </msup> <mo>-</mo> <mn>2</mn> </mrow> </math> </EquationSource> </InlineEquation>. Copyright Springer-Verlag Berlin Heidelberg 2014
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