Two-Factor StudiesDefining (Fitted) Factorial EffectsMain EffectsInteractionsConfidence Intervals for Two-Way Factorial EffectsPerspectiveIE 361 Module 21Design and Analysis of Experiments: Part 2(Two-Way Studies and Analyses)Reading: Section 6.2, Statistical Quality Assurance Methods forEngineersProf. Steve Vardeman and Prof. Max MorrisIow a State UniversityVardeman and Morris (I owa State University) IE 3 61 Module 21 1 / 22Two-Way Factorial StudiesIn this module we start to consider what can be learned about the actionof several factors on a response variable, based on r samples collec tedunder di¤erent sets of process conditions, i.e. under di¤erent combinationsof levels of those factors. We begin with the simplest such situation,where there are two factors of interest, some Factor A has I levels, anotherFactor B has J levels, and samples of a response y are obtained undereach combination of a level i of Factor A and a level j of Factor B. Thisresults in what can be thought of as a two-way table of datayijk= the kth response at level i of Factor Aand level j of Factor Billustrated in the table on panel 3. There, the sample size in the ith rowand jth column is denoted as nij.Vardeman and Morris (I owa State University) IE 3 61 Module 21 2 / 22Two-Way Factorial DataFactor B1 2 J1y111, y112,. . . , y11n11y121, y122,. . . , y12n12  y1J 1, y1J 2,. . . , y1Jn1J2y211, y212,. . . , y21n21y221, y222,. . . , y22n22...Factor A......IyI 11, yI 12,. . . , yI 1nI 1  yIJ 1, yIJ 2,. . . , yIJnIJThis layout is complete in the sense that there are data in every cell.The terminology factorial used in the name of this section means that thecombinations of levels of the two factors are considered, and the jargon"I  J factorial" (naming the number of levels of each factor) is common.Vardeman and Morris (I owa State Un iversity) IE 361 Module 21 3 / 22Two-Way Factorial DataExample 21-1 (Example 20-1 Revisited)The glass-phosphor study of Module 20 has 2  3 factorial structure. Werepeat the summary statistics, this time emphasizing the the naturaltwo-way structure through the use of double subscripts indicating glass(row) and phosphor (column) in the table. The table adds row andcolumn averages of the cell means ¯yij(the ¯yi.’s and ¯y.j’s respectively).These prove useful for de…ning important summaries of two-way factorialsin the balance of this section.Phosphor1 2 31¯y11= 285s211= 25¯y12= 301.67s212= 58.33¯y13= 281.67s213= 108.33¯y1.= 289.44Glass2¯y21= 235s221= 25¯y22= 245s222= 175¯y23= 225s223= 25¯y2.= 235¯y.1= 260 ¯y.2= 273.33 ¯y.3= 253.33 ¯y..= 262.22Vardeman and Morris (I owa State Un iversity) IE 361 Module 21 4 / 22Two-Way Factorial DataExample 21-1 continuedA useful plot in two-way studies is that of sample means versus level ofone factor, connecting plotted points for a given level of the second factorwith line segments. Such a plot is commonly called an interaction plot.The …gure below is such a plot (with the raw data values also shown).Figure: Interaction Plot for the Glass-Phosphor Study (With Raw Data Plotted)Vardeman and Morris (I owa State Un iversity) IE 361 Module 21 5 / 22Two-Way Factorial DataExample 21-1 continuedA better interaction plot adds "error bars." In Module 20 we saw that95% con…dence limits for the tube type means are (using the doublesubscript notation) ¯yij 10.44. Below is the interaction plot for theglass-phosphor study enhanced with 10.44 error bars indicating theprecision with which the means are known.Figure: Interaction Plot for the Glass-Phosphor Study Enhanced With Error BarsFrom 95% Con…dence LimitsVardeman and Morris (I owa State Un iversity) IE 361 Module 21 6 / 22Two-Way Factorial DataExample 21-1 continuedThe qualitative story told by the interaction plots is that:Glass 1 current requirements are clearly higher than those for Glass 2,Phosphor 2 current requirements seem to be somewhat higher thanthose for Phosphors 1 and 3,the "Glass" di¤erences in current requirements seem to be larger than"Phosphor" di¤erences, andthe pattern in current requirement means for Glass 1 is similar to thepattern in current requirement means for Glass 2Vardeman and Morris (I owa State Un iversity) IE 361 Module 21 7 / 22Two-Way Factorial E¤ectsMain E¤ectsA way to quantify insights like those made in the Glass-Phosphor study isto de…ne so-called (…tted) factorial e¤ects. To begin, so-called maine¤ects of the factors are de…ned as appropriate row or column average ¯y ’sminus the grand average ¯y. That isai= ¯yi. ¯y..= ( the row i average ¯y )  (the grand average ¯y)= the (…tted) main e¤ect of the i th level of Factor Aandbj= ¯y.j ¯y..= ( the column j average ¯y)  (the grand average ¯y )= the (…tted) main e¤ect of the jth level of Factor BVardeman and Morris (I owa State Un iversity) IE 361 Module 21 8 / 22Fitted Main E¤ectsExample 21-1 continuedUsing the row and column averages of cell sample means, we havea1= 289.44  262.22 = 27.22 and a2= 235  262.22 = 27.22andb1= 260  262.22  2.22 and b2= 273.33  262.22 = 11.11and b3= 253.33  262.22 = 8.88The fact that A main e¤ects are larger in absolute value than B maine¤ects is consistent with the fact that the gap between the top and bottompro…les on panel 6 is more pronounced than the up-then-down patternseen in them. The fact that a1> 0 indicates that current requirementsfor Glass 1 are larger than for Glass 2 (that has a2< 0). (Similarly, thefact that b2> 0 indicates that current requirements for Ph osphor 2 arelarger than for Phosphors 1 and 3 that have b1< 0 and b3< 0.)Vardeman and Morris (I owa State Un iversity) IE 361 Module 21 9 / 22Fitted Main E¤ectsIt is no accident that in the glass-phosphor example the (2) Factor A maine¤ects add to 0 and the (3) Factor B main e¤ects also add to 0. This isan algebraic consequence of the de…nitions of these quantities and can beused as a check on one’s calculations.In some cases the …tted main e¤ects in a two-way factorial essentiallycapture the entire story told in the data set, in the sense that for eachcombination of a level i of Factor A and a level j of Factor B¯yij ¯y..+ ai+ bj(the sample means can essentially be reconstructed from an overall meanand Factor A and Factor B main e¤ects). The glass-phosphor example issuch a case.Vardeman and Morris (I owa State Un iversity) IE 361 Module 21 10 / 22Fitted Main E¤ectsExample 21-1