ANOVA TABLEFactorial ExperimentCompletely Randomized Design

Anova table for the 3 factor Experiment

SourceSSdfMSFp -valueASSAa - 1MSAMSA/MSError

BSSBb - 1MSBMSB/MSError

CSSCc - 1MSCMSC/MSError

ABSSAB(a - 1)(b - 1)MSABMSAB/MSError

ACSSAC(a - 1)(c - 1)MSACMSAC/MSError

BCSSBC(b - 1)(c - 1)MSBCMSBC/MSError

ABCSSABC(a - 1)(b - 1)(c - 1)MSABCMSABC/MSError

ErrorSSErrorabc(n - 1)MSError

Sum of squares entriesSimilar expressions for SSB , and SSC.Similar expressions for SSBC , and SSAC.

Sum of squares entriesFinally

The statistical model for the 3 factor Experiment

Anova table for the 3 factor Experiment

SourceSSdfMSFp -valueASSAa - 1MSAMSA/MSError

BSSBb - 1MSBMSB/MSError

CSSCc - 1MSCMSC/MSError

ABSSAB(a - 1)(b - 1)MSABMSAB/MSError

ACSSAC(a - 1)(c - 1)MSACMSAC/MSError

BCSSBC(b - 1)(c - 1)MSBCMSBC/MSError

ABCSSABC(a - 1)(b - 1)(c - 1)MSABCMSABC/MSError

ErrorSSErrorabc(n - 1)MSError

The testing in factorial experiments Test first the higher order interactions.If an interaction is present there is no need to test lower order interactions or main effects involving those factors. All factors in the interaction affect the response and they interactThe testing continues with lower order interactions and main effects for factors which have not yet been determined to affect the response.

Random Effects and Fixed Effects Factors

So far the factors that we have considered are fixed effects factorsThis is the case if the levels of the factor are a fixed set of levels and the conclusions of any analysis is in relationship to these levels.If the levels have been selected at random from a population of levels the factor is called a random effects factorThe conclusions of the analysis will be directed at the population of levels and not only the levels selected for the experiment

Example - Fixed EffectsSource of Protein, Level of Protein, Weight GainDependentWeight Gain IndependentSource of Protein,BeefCerealPork Level of Protein,HighLow

Example - Random EffectsIn this Example a Taxi company is interested in comparing the effects of three brands of tires (A, B and C) on mileage (mpg). Mileage will also be effected by driver. The company selects b = 4 drivers at random from its collection of drivers. Each driver has n = 3 opportunities to use each brand of tire in which mileage is measured.DependentMileage IndependentTire brand (A, B, C),Fixed Effect FactorDriver (1, 2, 3, 4),Random Effects factor

The Model for the fixed effects experimentwhere m, a1, a2, a3, b1, b2, (ab)11 , (ab)21 , (ab)31 , (ab)12 , (ab)22 , (ab)32 , are fixed unknown constants And eijk is random, normally distributed with mean 0 and variance s2.Note:

The Model for the case when factor B is a random effects factorwhere m, a1, a2, a3, are fixed unknown constants And eijk is random, normally distributed with mean 0 and variance s2. bj is normal with mean 0 and varianceand (ab)ij is normal with mean 0 and varianceNote:This model is called a variance components model

The Anova table for the two factor model

SourceSSdfMSASSAa -1SSA/(a 1)BSSAb - 1SSB/(a 1)ABSSAB(a -1)(b -1)SSAB/(a 1) (a 1)ErrorSSErrorab(n 1)SSError/ab(n 1)

The Anova table for the two factor model (A, B fixed) EMS = Expected Mean Square

SourceSSdfMSEMSFASSAa -1MSAMSA/MSErrorBSSAb - 1MSBMSB/MSErrorABSSAB(a -1)(b -1)MSABMSAB/MSErrorErrorSSErrorab(n 1)MSError

The Anova table for the two factor model (A fixed, B - random) Note: The divisor for testing the main effects of A is no longer MSError but MSAB.

SourceSSdfMSEMSFASSAa -1MSAMSA/MSABBSSAb - 1MSBMSB/MSErrorABSSAB(a -1)(b -1)MSABMSAB/MSErrorErrorSSErrorab(n 1)MSError

Rules for determining Expected Mean Squares (EMS) in an Anova TableSchultz E. F., Jr. Rules of Thumb for Determining Expectations of Mean Squares in Analysis of Variance,Biometrics, Vol 11, 1955, 123-48. Both fixed and random effectsFormulated by Schultz[1]

The EMS for Error is s2.The EMS for each ANOVA term contains two or more terms the first of which is s2.All other terms in each EMS contain both coefficients and subscripts (the total number of letters being one more than the number of factors) (if number of factors is k = 3, then the number of letters is 4)The subscript of s2 in the last term of each EMS is the same as the treatment designation.

The subscripts of all s2 other than the first contain the treatment designation. These are written with the combination involving the most letters written first and ending with the treatment designation. When a capital letter is omitted from a subscript , the corresponding small letter appears in the coefficient.For each EMS in the table ignore the letter or letters that designate the effect. If any of the remaining letters designate a fixed effect, delete that term from the EMS.

Replace s2 whose subscripts are composed entirely of fixed effects by the appropriate sum.

Example: 3 factors A, B, C all are random effects

Example: 3 factors A fixed, B, C random

Example: 3 factors A , B fixed, C random

Example: 3 factors A , B and C fixed

Example - Random EffectsIn this Example a Taxi company is interested in comparing the effects of three brands of tires (A, B and C) on mileage (mpg). Mileage will also be effected by driver. The company selects at random b = 4 drivers at random from its collection of drivers. Each driver has n = 3 opportunities to use each brand of tire in which mileage is measured.DependentMileage IndependentTire brand (A, B, C),Fixed Effect FactorDriver (1, 2, 3, 4),Random Effects factor

The Data

Generator

s30

453

55-10

-0.4314131274-2.9177726901-5.8357909438-6.402915460131.650814182428.732795928713.165671412529.5685868726

2.082276751-0.748086677-3.39830876331.717298800936.33419007433.683967987723.79957555232.082276751

2.71862745650.6079380912.72083525494.659077603738.326565547540.439462711427.377705060232.7186274565

3.6222218114-3.27846464641.79550625030.960671968635.34375716540.417728061724.5828937833.6222218114

3528.925.91429.1

33.934.524.130.7

42.242.727.430.7

35.136.626.432.4

35.128.319.931.3

35.538.528.337.9

35.840.626.533.7

40.135.114.625.8

Data

DriverTireMileage

1A39.6

1A38.6

1A41.9

1B18.1

1B20.4

1B19

1C31.1

1C29.8

1C26.6

2A38.1

2A35.4

2A38.8

2B18.2

2B14

2B15.6

2C30.2

2C27.9

2C27.2

3A33.9

3A43.2

3A41.3

3B17.8

3B21.3

3B22.3

3C31.3

3C28.7

3C29.7

4A36.9

4A30.3

4A35

4B17.8

4B21.2

4B24.3

4C27.4

4C26.6

4C21

Data (2)

DriverTireMileageDriverTireMileage

1A39.63A33.9

1A38.63A43.2

1A41.93A41.3

1B18.13B17.8

1B20.43B21.3

1B193B22.3

1C31.13C31.3

1C29.83C28.7

1C26.63C29.7

2A38.14A36.9

2A35.44A30.3

2A38.84A35

2B18.24B17.8

2B144B21.2

2B15.64B24.3

2C30.24C27.4

2C27.94C26.6

2C27.24C21

Sheet2

Sheet3

Asking SPSS to perform Univariate ANOVA

Select the dependent variable, fixed factors, random factors

The OutputThe divisor for both the fixed and the random main effect is MSABThis is contrary to the advice of some texts

The Anova table for the two factor model (A fixed, B - random) Note: The divisor for testing the main effects of A is no longer MSError but MSAB.References Guenther, W. C. Analysis of Variance Prentice Hall, 1964

SourceSSdfMSEMSFASSAa -1MSAMSA/MSABBSSAb - 1MSBMSB/MSErrorABSSAB(a -1)(b -1)MSABMSAB/MSErrorErrorSSErrorab(n 1)MSError

The Anova table for the two factor model (A fixed, B - random) Note: In this case the divisor for testing the main effects of A is MSAB . This is the approach used by SPSS.References Searle Linear Models John Wiley, 1964

SourceSSdfMSEMSFASSAa -1MSAMSA/MSABBSSAb - 1MSBMSB/MSABABSSAB(a -1)(b -1)MSABMSAB/MSErrorErrorSSErrorab(n 1)MSError

Crossed and Nested Factors

The factors A, B are called crossed if every level of A appears with every level of B in the treatment combinations.Levels of BLevels of A

Factor B is said to be nested within factor A if the levels of B differ for each level of A. Levels of BLevels of A

Example: A company has a = 4 plants for producing paper. Each plant has 6 machines for producing the paper. The company is interested in how paper strength (Y) differs from plant to plant and from machine to machine within plant PlantsMachines

Machines (B) are nested within plants (A)The model for a two factor experiment with B nested within A.

The ANOVA tableNote: SSB(A ) = SSB + SSAB and a(b 1) = (b 1) + (a - 1)(b 1)

SourceSSdfMSFp - valueASSAa - 1MSAMSA/MSErrorB(A)SSB(A)a(b 1)MSB(A)MSB(A) /MSError

ErrorSSErrorab(n 1)MSError

Example: A company has a = 4 plants for producing paper. Each plant has 6 machines for producing the paper. The company is interested in how paper strength (Y) differs from plant to plant and from machine to machine within plant. Also we have n = 5 measurements of paper strength for each of the 24 machines

The Data

Plant

1

2

machine

1

2

3

4

5

6

7

8

9

10

11

12

98.7

59.2

84.1

72.3

83.5

60.6

33.6

44.8

58.9

63.9

63.7

48.1

93.1

87.8

86.3

110.3

89.3

84.8

48.2

57.3

51.6

62.3

54.6

50.6

100.0

84.1

83.4

81.6

86.1

83.6

68.9

66.5

45.2

61.1

55.3

39.9

Plant

3

4

machine

13

14

15

16

17

18

19

20

21

22

23

24

83.6

76.1

64.2

69.2

77.4

61.0

64.2

35.5

46.9

37.0

43.8

30.0

84.6

55.4

58.4

86.7

63.3

81.3

50.3

30.8

43.1

47.8

62.4

43.0

90.6

92.3

75.4

60.8

76.6

73.8

32.1

36.3

40.8

41.0

60.8

56.9

Anova Table Treating Factors (Plant, Machine) as crossed

Anova Table: Two factor experiment B(machine) nested in A (plant)

Sheet1

SourceSum of SquaresdfMean SquareF

Plant18174.761193055436058.253731018552.81950628360.00000

Machine(Plant)2856.303672222220142.81518361111.24514882060.26171

Error5505.469466666748114.6972805556

Sheet2

Sheet3

GraphPaper Strength

Chart1

97.266666666750.233333333386.266666666748.8666666667

77.033333333356.274.634.2

84.651.96643.6

88.066666666762.433333333372.233333333341.9333333333

86.357.866666666772.433333333355.6666666667

76.333333333346.272.033333333343.3

Plant 1

Plant 2

Plant 3

Plant 5

Machine

Sheet1

Plant12

machine123456789101112

98.759.284.172.383.560.633.644.858.963.963.748.1

93.187.886.3110.389.384.848.257.351.662.354.650.6

10084.183.481.686.183.668.966.545.261.155.339.9

Plant34

machine131415161718192021222324

83.676.164.269.277.46164.235.546.93743.830

84.655.458.486.763.381.350.330.843.147.862.443

90.692.375.460.876.673.832.136.340.84160.856.9

197.266666666777.033333333384.688.066666666786.376.333333333350.233333333356.251.962.433333333357.866666666746.2

486.266666666774.66672.233333333372.433333333372.033333333348.866666666734.243.641.933333333355.666666666743.3

1234

197508649

277567534

385526644

488627242

586587256

676467243

Sheet1

Plant 1

Plant 2

Plant 3

Plant 5

Machine

Sheet2

Sheet3