NIST/SEMATECH e-Handbook of Statistical Methods · 2020

NIST/SEMATECH e-Handbook of Statistical Methods (NIST Handbook 151), Chapter 5: Process Improvement

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Factorial gives the source's answers, with a few differences. Where they differ, this page says why: some are misprints in the source, and for some the reason is not known yet.

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Examples

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Misprints

Misprints found in the source, by page. Each shows what the source prints and what its own data give.

Coefficient: X1·X2p. 5.4.7.1
Book -14.467Data 14.467
The fit gives +14.467, as the page's R code prints (sr 14.467): the page reverses the sign of five two-factor interactions, X1·X2, X1·X3, X1·X4, X2·X4 and X3·X4, while X1·X2·X4 and the main effects agree. The page's own optimum (speed and feed rate high, grit and direction low) is the 11-effect model's maximum over the 16 corners only with the fitted signs; with the printed ones it would put table speed low.
Coefficient: X1·X3p. 5.4.7.1
Book 7.339Data -7.339
The fit gives -7.339, as the page's R code prints (sg -7.339): the page reverses the sign of five two-factor interactions, X1·X2, X1·X3, X1·X4, X2·X4 and X3·X4, while X1·X2·X4 and the main effects agree. The page's own optimum (speed and feed rate high, grit and direction low) is the 11-effect model's maximum over the 16 corners only with the fitted signs; with the printed ones it would put table speed low.
Coefficient: X1·X4p. 5.4.7.1
Book -7.188Data 7.189
The fit gives +7.18854, as the page's R code prints (ds 7.189): the page reverses the sign of five two-factor interactions, X1·X2, X1·X3, X1·X4, X2·X4 and X3·X4, while X1·X2·X4 and the main effects agree. The page's own optimum (speed and feed rate high, grit and direction low) is the 11-effect model's maximum over the 16 corners only with the fitted signs; with the printed ones it would put table speed low. Its 7.188 also drops the rounding: 7.18854 is 7.189.
Coefficient: X2·X4p. 5.4.7.1
Book -9.160Data 9.160
The fit gives +9.160, as the page's R code prints (rd 9.160): the page reverses the sign of five two-factor interactions, X1·X2, X1·X3, X1·X4, X2·X4 and X3·X4, while X1·X2·X4 and the main effects agree. The page's own optimum (speed and feed rate high, grit and direction low) is the 11-effect model's maximum over the 16 corners only with the fitted signs; with the printed ones it would put table speed low.
Coefficient: X3·X4p. 5.4.7.1
Book 12.965Data -12.966
The fit gives -12.965, as the page's R code prints (gd -12.965): the page reverses the sign of five two-factor interactions, X1·X2, X1·X3, X1·X4, X2·X4 and X3·X4, while X1·X2·X4 and the main effects agree. The page's own optimum (speed and feed rate high, grit and direction low) is the 11-effect model's maximum over the 16 corners only with the fitted signs; with the printed ones it would put table speed low.

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Regression mean squarep. 5.4.7.2
Book 3691.6Data 3691.4
22148.55 / 6 = 3691.4, the library's 3691.42; the page's 3691.6 is not its own sum of squares over its degrees of freedom, and its F, 3691.6 / 162.1 = 22.77, follows from it (R's summary prints 22.78).
Number of runsp. 5.4.7.3
Book 13Data 11
The printed design has 11 runs: 4 corners, 4 axial runs and 3 centre runs. 13 runs is the size of the CCI with 5 centre runs that the page says is typically recommended (checked beside), not of this one.
Terms kept: statistical significancep. 5.6.1.8
Book 0.14875Data 0.06375
0.14875 is the X2*X3 coefficient, as the page's summary 'X1 X2 X2*X3' says; X1*X2's is 0.06375
Average at -1: X1·X2·X3p. 5.6.1.1
Book 2.5100Data 2.5875
Row 7 should be the X1*X2*X3 averages (the Youden plot labels it x1:x2:x3), 2.5875 at -1; the script computes q7 from fx23 instead of fx123, so it repeats row 6, the X2*X3 averages
Average at +1: X1·X2·X3p. 5.6.1.1
Book 2.8075Data 2.7300
Row 7 should be the X1*X2*X3 averages (the Youden plot labels it x1:x2:x3), 2.73 at +1; the script computes q7 from fx23 instead of fx123, so it repeats row 6, the X2*X3 averages
Effect: X2·X3p. 5.6.2.3
Book 63.5Data -63.5
The effect is -63.4625: the page drops the minus sign that the legends (23 : -63.5) print
Effect: X1·X5p. 5.6.2.6
Book -63.4626Data -0.5625
The table's confounding tag is wrong: this is the 23 = 17 = 45 effect (-63.4625; the axis below its spike reads 23), while the 15 group's effect is -0.5625; Dataplot's confounding labels for the 23, 34 and 24 columns are shifted
Effect: X1·X7p. 5.6.2.6
Book -16.3375Data -63.4625
The table's confounding tag is wrong: this is the 34 = 16 = 25 effect (-16.3375; the axis below its spike reads 34), while the 17 group's effect is -63.4625; Dataplot's confounding labels for the 23, 34 and 24 columns are shifted
Effect: X1·X6p. 5.6.2.6
Book 1.6875Data -16.3375
The table's confounding tag is wrong: this is the 24 = 35 = 67 effect (1.6875; the axis below its spike reads 24), while the 16 group's effect is -16.3375; Dataplot's confounding labels for the 23, 34 and 24 columns are shifted
Coefficient: X1p. 5.6.2.8
Book 33.106245Data 33.106300
A typo: the X1 coefficient is 33.10625 (half of 66.2125, printed 66.21249 in the |effects| plot)
Coefficient: X1·X5p. 5.6.2.8
Book -31.7313Data -0.2812
The term is mislabelled: -31.7313 is the coefficient of the X2*X3 = X4*X5 = X1*X7 group (-31.73125; the R script calls it x17), the sixth term of the cumulative model; X1*X5's coefficient is -0.28125. The |effects| plot's table tags the X2*X3 row 15+26+47, which the page copied
Effect: X3p. 5.6.2.10
Book 63.5Data 63.8
The X3 effect is 63.8125 (63.8 in the legends and the |effects| table); 63.5 is the size of the next group, X2*X3
R-squaredp. 5.5.3.1.3
Book 0.6580Data 0.6504
5.5.3.1.1 gives R2 = 0.6504 for the same first-order fit (503.3035/773.8660); 0.6580 matches no fit of these runs (x1 + x2 + I(x1**2) gives 0.6611), so it looks like a typo
Optimal timep. 5.5.3.1.4
Book 367.32Data 367.34
The page prints 367.32 here and 367.36 below; 350 + 50(0.3472) = 367.36, and the library's x*2 = 0.3468 gives 367.34
Canonical axisp. 5.5.3.1.4
Book 0.8830Data 0.3830
With v2 = (0.6851, 0.7285), w2 = 0.6851(x1 + 0.9278) + 0.7285(x2 - 0.3468) = 0.3830 + 0.6851x1 + 0.7285x2, as the library finds (0.3830); the page writes -0.7285(x2 - 0.3468) in the middle, and its 0.8830 follows from neither sign (the other gives 0.8883)
Coefficient: Y2: x2²p. 5.5.3.2.2
Book -124.92Data -124.82
The fit gives -124.82 (and -125.59 with the axial runs at the printed 1.63); the page's -124.92 differs in one digit, likely a transcription slip
Number of levels: X1p. 5.3.3.2.2
Book 3Data 4
The printed 4-level design has 4 levels of X1: the page repeats the line of the 3-level design above it
Number of levels: X2p. 5.3.3.2.2
Book 3Data 4
The printed 4-level design has 4 levels of X2: the page repeats the line of the 3-level design above it
Number of levels: X3p. 5.3.3.2.2
Book 3Data 4
The printed 4-level design has 4 levels of X3: the page repeats the line of the 3-level design above it
Number of levels: X4p. 5.3.3.2.2
Book 3Data 4
The printed 4-level design has 4 levels of X4: the page repeats the line of the 3-level design above it
Number of levels: X1p. 5.3.3.2.2
Book 3Data 5
The printed 5-level design has 5 levels of X1: the page repeats the line of the 3-level design above it
Number of levels: X2p. 5.3.3.2.2
Book 3Data 5
The printed 5-level design has 5 levels of X2: the page repeats the line of the 3-level design above it
Number of levels: X3p. 5.3.3.2.2
Book 3Data 5
The printed 5-level design has 5 levels of X3: the page repeats the line of the 3-level design above it
Number of levels: X4p. 5.3.3.2.2
Book 3Data 5
The printed 5-level design has 5 levels of X4: the page repeats the line of the 3-level design above it
Standard orderp. 5.3.3.6.3
Book 1, 2, 3, 4, 5, 6, 7, 8Data 1, 2, 3, 4, 5, 6, 5, 8
Run 7 is printed 12, 12, 18 (X2 low), the settings of run 5, so it codes to standard order 5; standard order 7 is 12, 18, 18, as the CCC beside it (10, 20, 20): a typo in Table 3.25
Aliases: X2·X3p. 5.6.2.6
Book 15, 26, 47Data 123457, 1246, 1356, 17, 2567, 3467, 45
Dataplot's confounding label for column 23 is wrong: 23's aliases are 17, 45; the labels of the 23, 34 and 24 columns are shifted by one group, here and in the Youden, half-normal and cumulative residual SD plots
Aliases: X3·X4p. 5.6.2.6
Book 17, 23, 45Data 123456, 1247, 1357, 16, 2367, 25, 4567
Dataplot's confounding label for column 34 is wrong: 34's aliases are 16, 25; the labels of the 23, 34 and 24 columns are shifted by one group, here and in the Youden, half-normal and cumulative residual SD plots
Aliases: X2·X4p. 5.6.2.6
Book 16, 25, 34Data 1236, 1257, 1347, 1456, 234567, 35, 67
Dataplot's confounding label for column 24 is wrong: 24's aliases are 35, 67; the labels of the 23, 34 and 24 columns are shifted by one group, here and in the Youden, half-normal and cumulative residual SD plots
Number of words in the defining relationp. 5.3.3.2.4
Book 3Data 7
5.3.3.2.4 calls the three generating relations the defining relation and counts their words; the defining relation they generate has 2^3 - 1 = 7 words, as 5.3.3.4.4 says
Resolutionp. 5.3.3.4.4
Book VData 4
No 2^(8-3) fraction has resolution V: the best, of minimum aberration, is this resolution IV fraction (5.3.3.4.7's Table 3.17 lists 2_IV^(8-3) and 2_V^(8-2), 64 runs). The page's \( 2_{V}^{8-3} \) comes from the HTML comment it replaced, an image res5.gif with alt text '2(V)**(8-3)'
Resolutionp. 5.3.3.4.4
Book VData 4
No 2^(8-3) fraction has resolution V: the best, of minimum aberration, is this resolution IV fraction (5.3.3.4.7's Table 3.17 lists 2_IV^(8-3) and 2_V^(8-2), 64 runs). The page's \( 2_{V}^{8-3} \) comes from the HTML comment it replaced, an image res5.gif with alt text '2(V)**(8-3)'
Resolutionp. 5.3.3.4.4
Book VData 4
Its defining relation, as the page itself prints it, has five words of length four, so its resolution is IV; no 2^(8-3) fraction has resolution V
Number of clear two-factor interactionsp. 5.3.3.4.4
Book 28 - 23Data 4
27 is not estimable: the word 2467, which the page lists in the defining relation, aliases 27 with 46. 24 of the 28 two-factor interactions are confounded, and four (12, 14, 23 and 34) are clear
Aliases: 1·5p. 5.3.3.4.7
Book 25Data 1234, 12356, 146, 245, 26, 3, 3456
15 is aliased with 3 (word 135) and 26 (word 1256), not 25: the file's own chains print 3 + 15 + 26 and 26 + 15, so 25 is a typo for 26
Aliases: 8p. 5.3.3.4.7
Book 1234568Data 1234567
8 times the word 12345678 is 1234567, the DEFINITION the file gives for 8; 1234568 is a typo
Aliases: 3·4p. 5.3.3.4.7
Book 57Data 1237, 1246, 13568, 14578, 2345678, 258, 67
34 is aliased with 67 through the word 3467, as the file's own chain 67 + 34 says, and 57 is clear (its chain prints 57 + Higher); 57 is a typo for 67
Resolutionp. 5.3.3.4.4
Book VData 4
Figure 3.8 itself labels this fraction 2_IV^(8-3); no 2^(8-3) fraction has resolution V: the best, of minimum aberration, is this resolution IV fraction (5.3.3.4.7's Table 3.17 lists 2_IV^(8-3) and 2_V^(8-2), 64 runs). The page's \( 2_{V}^{8-3} \) comes from the HTML comment it replaced, an image res5.gif with alt text '2(V)**(8-3)'
Defining relationp. 5.3.3.4.7
Book I +Data 12346789, 12359, 12458, 1267, 13457, 1368, 1469, 15789, 23456, 2378, 2479, 25689, 3489, 35679, 45678
The file prints 'I +' and no word: the defining relation of its generators (6 = 2345, 7 = 1345, 8 = 1245, 9 = 1235) has 15 words, which the library lists
Defining relationp. 5.3.3.4.7
Book 169, 279, 389, 459, 1456, 2457, 3458, 1267, 1368, 2379, 3562, 2568, 1578, 4678, 1235, 2346, 1347, 1248, 12349, 23569, 13579, 12589, 34678, 24689, 14789, 56789, 12345678, 1245679, 1345689, 2345789, 1236789Data 12345678, 12349, 1235, 1236789, 1245679, 1248, 12589, 1267, 1345689, 1347, 13579, 1368, 1456, 14789, 1578, 169, 2345789, 2346, 23569, 2378, 2457, 24689, 2568, 279, 3458, 34679, 3567, 389, 459, 4678, 56789
Three of the 31 words are misprinted: 2379 for 2378, 3562 for 3567 and 34678 for 34679; the other 28 are words of the relation of 5 = 123, 6 = 234, 7 = 134, 8 = 124, 9 = 1234
Generatorp. 5.3.3.2.7
Book C=ABCDData ABCDG, ABEFH
The image prints C = ±ABCD, which defines the base factor C by itself; the 2^(8-2) of Box, Hunter and Hunter and of 2to8m2.txt (5.3.3.4.7) has G = ABCD (7 = 1234), so C is a typo for G
Resolutionp. 5.3.3.2.7
Book VData 4
With J = ACDE and K = ACDF the product JK = EFJK is a word of length four, so the fraction is of resolution IV; 2to10m3.txt (5.3.3.4.7) has 9 = 2345, J = BCDE, which gives resolution V, so ACDE is a typo for BCDE
Runs of the foldover: 5p. 5.3.3.8.2
Book 1, 2, 3, 4, 5, 6, 7, 8Data
No run of TABLE 3.36 has X5 = -X1X3: every run has X5 = X1X3, so the column heading is a typo

Not yet

Answers the source prints that Factorial cannot compute yet, by what it lacks.

Box and Draper's fraction of directions excluded by the confidence cone; no check computes it (0.71 verified while cataloguing: 1 - T_1(0.7753) = 0.7099)p. 5
Box and Draper's fraction of directions excluded by the confidence cone; no check computes itp. 5
The cone's angle, 0.29 x 360 degrees; no check computes itp. 5
Half the cone's angle; no check computes itp. 5
The library cannot build a 3^k for it (3^k = 9)p. 5

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The library cannot build a 3^k for it (3^k = 27)p. 5
The library cannot build a 3^k for it (3^k = 81)p. 5
The library cannot build a 3^k for it (3^k = 243)p. 5
The library cannot build a 3^k for it (3^k = 729)p. 5
The size of the 3^4 full factorial, which design_property cannot build (two-level factors only)p. 5
N times the prediction variance d(x) averaged over the 20 candidate points of Table 5.1 (12 x 0.388542 = 4.6625, verified while cataloguing); the library averages d(x) over the continuous cube (average_prediction_variance 0.2555, 3.07 times N), and no option averages over a candidate setp. 5
A resolution claimed for a non-regular 12-run design, from its two-factor interaction columns being all different; the library's resolution metric gives None for runs without a defining relationp. 5