Stats 845 Lecture 14n
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Transcript of Stats 845 Lecture 14n
Other experimental designs
Randomized Block design
Latin Square design
Repeated Measures design
The Randomized Block Design
• Suppose a researcher is interested in how several treatments affect a continuous response variable (Y).
• The treatments may be the levels of a single factor or they may be the combinations of levels of several factors.
• Suppose we have available to us a total of N = nt experimental units to which we are going to apply the different treatments.
The Completely Randomized (CR) design randomly divides the experimental units into t groups of size n and randomly assigns a treatment to each group.
The Randomized Block Design
• divides the group of experimental units into n homogeneous groups of size t.
• These homogeneous groups are called blocks.
• The treatments are then randomly assigned to the experimental units in each block - one treatment to a unit in each block.
The ANOVA table for the Completely Randomized Design
Source df Sum of Squares
Treatments t - 1 SSTr
Error t(n – 1) SSError
Total tn - 1 SSTotal
Source df Sum of Squares
Blocks n - 1 SSBlocks
Treatments t - 1 SSTr
Error (t – 1) (n – 1) SSError
Total tn - 1 SSTotal
The ANOVA table for the Randomized Block Design
( )CRij i ijy
( )RBij i j ijy
Comments
The ability to detect treatment differences depends on the magnitude of the random error term
( )CRij
( )RBij
The error term, , for the Completely Randomized Design models variability in the reponse, y, between experimental units
The error term, , for the Completely Block Design models variability in the reponse, y, between experimental units in the same block (hopefully the is considerably smaller than .( )CR
ij
Example – Weight gain, diet, source of protein, level of protein(Completely randomized design)
Block Block 1 107 96 112 83 87 90 6 128 89 104 85 84 89 (1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
2 102 72 100 82 70 94 7 56 70 72 64 62 63 (1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
3 102 76 102 85 95 86 8 97 91 92 80 72 82 (1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
4 93 70 93 63 71 63 9 80 63 87 82 81 63 (1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
5 111 79 101 72 75 81 10 103 102 112 83 93 81 (1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
Randomized Block Design
The Anova Table for Diet Experiment
Source S.S d.f. M.S. F p-valueBlock 5992.4167 9 665.82407 9.52 0.00000Diet 4572.8833 5 914.57667 13.076659 0.00000
ERROR 3147.2833 45 69.93963
Example 1:
• Suppose we are interested in how weight gain (Y) in rats is affected by Source of protein (Beef, Cereal, and Pork) and by Level of Protein (High or Low).
• There are a total of t = 32 = 6 treatment combinations of the two factors (Beef -High Protein, Cereal-High Protein, Pork-High Protein, Beef -Low Protein, Cereal-Low Protein, and Pork-Low Protein) .
• Suppose we have available to us a total of N = 60 experimental rats to which we are going to apply the different diets based on the t = 6 treatment combinations.
• Prior to the experimentation the rats were divided into n = 10 homogeneous groups of size 6.
• The grouping was based on factors that had previously been ignored (Example - Initial weight size, appetite size etc.)
• Within each of the 10 blocks a rat is randomly assigned a treatment combination (diet).
• The weight gain after a fixed period is measured for each of the test animals and is tabulated on the next slide:
Block Block 1 107 96 112 83 87 90 6 128 89 104 85 84 89 (1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
2 102 72 100 82 70 94 7 56 70 72 64 62 63 (1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
3 102 76 102 85 95 86 8 97 91 92 80 72 82 (1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
4 93 70 93 63 71 63 9 80 63 87 82 81 63 (1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
5 111 79 101 72 75 81 10 103 102 112 83 93 81 (1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
Randomized Block Design
Example 2:
• The following experiment is interested in comparing the effect four different chemicals (A, B, C and D) in producing water resistance (y) in textiles.
• A strip of material, randomly selected from each bolt, is cut into four pieces (samples) the pieces are randomly assigned to receive one of the four chemical treatments.
• This process is replicated three times producing a Randomized Block (RB) design.
• Moisture resistance (y) were measured for each of the samples. (Low readings indicate low moisture penetration).
• The data is given in the diagram and table on the next slide.
Diagram: Blocks (Bolt Samples)
9.9 C 13.4 D 12.7 B 10.1 A 12.9 B 12.9 D 11.4 B 12.2 A 11.4 C 12.1 D 12.3 C 11.9 A
Table
Blocks (Bolt Samples)
Chemical 1 2 3
A 10.1 12.2 11.9
B 11.4 12.9 12.7
C 9.9 12.3 11.4
D 12.1 13.4 12.9
The Model for a randomized Block Experiment
ijjiijy
i = 1,2,…, t j = 1,2,…, b
yij = the observation in the jth block receiving the ith treatment
= overall mean
i = the effect of the ith treatment
j = the effect of the jth Block
ij = random error
The Anova Table for a randomized Block Experiment
Source S.S. d.f. M.S. F p-value
Treat SST t-1 MST MST /MSE
Block SSB n-1 MSB MSB /MSE
Error SSE (t-1)(b-1) MSE
• A randomized block experiment is assumed to be a two-factor experiment.
• The factors are blocks and treatments.
• The is one observation per cell. It is assumed that there is no interaction between blocks and treatments.
• The degrees of freedom for the interaction is used to estimate error.
The Anova Table for Diet Experiment
Source S.S d.f. M.S. F p-valueBlock 5992.4167 9 665.82407 9.52 0.00000Diet 4572.8833 5 914.57667 13.076659 0.00000
ERROR 3147.2833 45 69.93963
The Anova Table forTextile Experiment
SOURCE SUM OF SQUARES D.F. MEAN SQUARE F TAIL PROB.Blocks 7.17167 2 3.5858 40.21 0.0003Chem 5.20000 3 1.7333 19.44 0.0017
ERROR 0.53500 6 0.0892
• If the treatments are defined in terms of two or more factors, the treatment Sum of Squares can be split (partitioned) into:
– Main Effects
– Interactions
The Anova Table for Diet Experiment terms for the main effects and interactions between Level of Protein and Source of Protein
Source S.S d.f. M.S. F p-valueBlock 5992.4167 9 665.82407 9.52 0.00000Diet 4572.8833 5 914.57667 13.076659 0.00000
ERROR 3147.2833 45 69.93963
Source S.S d.f. M.S. F p-valueBlock 5992.4167 9 665.82407 9.52 0.00000
Source 882.23333 2 441.11667 6.31 0.00380Level 2680.0167 1 2680.0167 38.32 0.00000
SL 1010.6333 2 505.31667 7.23 0.00190ERROR 3147.2833 45 69.93963
Using SPSS to analyze a randomized Block Design
• Treat the experiment as a two-factor experiment– Blocks– Treatments
• Omit the interaction from the analysis. It will be treated as the Error term.
The data in an SPSS file
Variables are in columns
Select General Linear Model->Univariate
Select the dependent variable, the Block factor, the Treatment factor.
Select Model.
Select a Custom model.
Put in the model only the main effects.
Tests of Between-Subjects Effects
Dependent Variable: WTGAIN
10564.033a 14 754.574 10.834 .000
437418.8 1 437418.8 6280.442 .000
4594.683 5 918.937 13.194 .000
5969.350 9 663.261 9.523 .000
3134.150 45 69.648
451117.0 60
13698.183 59
SourceCorrected Model
Intercept
DIET
BLOCK
Error
Total
Corrected Total
Type IIISum of
Squares dfMean
Square F Sig.
R Squared = .771 (Adjusted R Squared = .700)a.
Obtain the ANOVA table
If I want to break apart the Diet SS into components representing Source of Protein (2 df), Level of Protein (1 df), and Source Level interaction (2 df) - follow the subsequent steps
Replace the Diet factor by the Source and level factors (The two factors that define diet)
Specify the model. There is no interaction between Blocks and the diet factors (Source and Level)
Tests of Between-Subjects Effects
Dependent Variable: WTGAIN
10564.033a 14 754.574 10.834 .000
437418.8 1 437418.8 6280.442 .000
5969.350 9 663.261 9.523 .000
904.033 2 452.017 6.490 .003
2680.017 1 2680.017 38.480 .000
1010.633 2 505.317 7.255 .002
3134.150 45 69.648
451117.0 60
13698.183 59
SourceCorrected Model
Intercept
BLOCK
SOURCE
LEVEL
SOURCE * LEVEL
Error
Total
Corrected Total
Type IIISum of
Squares dfMean
Square F Sig.
R Squared = .771 (Adjusted R Squared = .700)a.
Obtain the ANOVA table
Repeated Measures Designs
In a Repeated Measures Design
We have experimental units that• may be grouped according to one or several
factors (the grouping factors)Then on each experimental unit we have• not a single measurement but a group of
measurements (the repeated measures)• The repeated measures may be taken at
combinations of levels of one or several factors (The repeated measures factors)
Example In the following study the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery.
The enzyme was measured
• immediately after surgery (Day 0),
• one day (Day 1),
• two days (Day 2) and
• one week (Day 7) after surgery
for n = 15 cardiac surgical patients.
The data is given in the table below.
Subject Day 0 Day 1 Day 2 Day 7 Subject Day 0 Day 1 Day 2 Day 7 1 108 63 45 42 9 106 65 49 49 2 112 75 56 52 10 110 70 46 47 3 114 75 51 46 11 120 85 60 62 4 129 87 69 69 12 118 78 51 56 5 115 71 52 54 13 110 65 46 47 6 122 80 68 68 14 132 92 73 63 7 105 71 52 54 15 127 90 73 68 8 117 77 54 61
Table: The enzyme levels -immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7)
after surgery
• The subjects are not grouped (single group).
• There is one repeated measures factor -Time – with levels– Day 0, – Day 1, – Day 2, – Day 7
• This design is the same as a randomized block design with – Blocks = subjects
The Anova Table for Enzyme Experiment
Source SS df MS F p-valueSubject 4221.100 14 301.507 32.45 0.0000Day 36282.267 3 12094.089 1301.66 0.0000ERROR 390.233 42 9.291
The Subject Source of variability is modelling the variability between subjects
The ERROR Source of variability is modelling the variability within subjects
Analysis Using SPSS- the data file
The repeated measures are in columns
Select General Linear model -> Repeated Measures
Specify the repeated measures factors and the number of levels
Specify the variables that represent the levels of the repeated measures factor
There is no Between subject factor in this example
The ANOVA table
Tests of Within-Subjects Effects
Measure: MEASURE_1
36282.267 3 12094.089 1301.662 .000
36282.267 2.588 14021.994 1301.662 .000
36282.267 3.000 12094.089 1301.662 .000
36282.267 1.000 36282.267 1301.662 .000
390.233 42 9.291
390.233 36.225 10.772
390.233 42.000 9.291
390.233 14.000 27.874
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
SourceTIME
Error(TIME)
Type IIISum of
Squares dfMean
Square F Sig.
Example :
(Repeated Measures Design - Grouping Factor)
• In the following study, similar to example 3, the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery.
• In addition the experimenter was interested in how two drug treatments (A and B) would also effect the level of the enzyme.
• The 24 patients were randomly divided into three groups of n= 8 patients.
• The first group of patients were left untreated as a control group while
• the second and third group were given drug treatments A and B respectively.
• Again the enzyme was measured immediately after surgery (Day 0), one day (Day 1), two days (Day 2) and one week (Day 7) after surgery for each of the cardiac surgical patients in the study.
Table: The enzyme levels - immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7)
after surgery for three treatment groups (control, Drug A, Drug B)
Group Control Drug A Drug B Day Day Day
0 1 2 7 0 1 2 7 0 1 2 7 122 87 68 58 93 56 36 37 86 46 30 31 112 75 55 48 78 51 33 34 100 67 50 50 129 80 66 64 109 73 58 49 122 97 80 72 115 71 54 52 104 75 57 60 101 58 45 43 126 89 70 71 108 71 57 65 112 78 67 66 118 81 62 60 116 76 58 58 106 74 54 54 115 73 56 49 108 64 54 47 90 59 43 38 112 67 53 44 110 80 63 62 110 76 64 58
• The subjects are grouped by treatment– control, – Drug A, – Drug B
• There is one repeated measures factor -Time – with levels– Day 0, – Day 1, – Day 2, – Day 7
The Anova Table
There are two sources of Error in a repeated measures design:
The between subject error – Error1 and
the within subject error – Error2
Source SS df MS F p-value
Drug 1745.396 2 872.698 1.78 0.1929
Error1
10287.844 21 489.897Time 47067.031 3 15689.010 1479.58 0.0000Time x Drug 357.688 6 59.615 5.62 0.0001
Error2
668.031 63 10.604
Tables of means
Drug Day 0 Day 1 Day 2 Day 7 Overall
Control 118.63 77.88 60.50 55.75 78.19
A 103.25 68.25 52.00 51.50 68.75
B 103.38 69.38 54.13 51.50 69.59
Overall 108.42 71.83 55.54 52.92 72.18
Time Profiles of Enzyme Levels
40
60
80
100
120
0 1 2 3 4 5 6 7Day
Enz
yme
Lev
el
Control
Drug A
Drug B
Example : Repeated Measures Design - Two Grouping Factors
• In the following example , the researcher was interested in how the levels of Anxiety (high and low) and Tension (none and high) affected error rates in performing a specified task.
• In addition the researcher was interested in how the error rates also changed over time.
• Four groups of three subjects diagnosed in the four Anxiety-Tension categories were asked to perform the task at four different times patients in the study.
The number of errors committed at each instance is tabulated below.
Anxiety Low High
Tension None High None High
subject subject subject subject 1 2 3 1 2 3 1 2 3 1 2 3
18 19 14 16 12 18 16 18 16 19 16 16 14 12 10 12 8 10 10 8 12 16 14 12 12 8 6 10 6 5 8 4 6 10 10 8 6 4 2 4 2 1 4 1 2 8 9 8
The Anova Table
Source SS df MS F p-value
Anxiety 10.08333 1 10.08333 0.98 0.3517Tension 8.33333 1 8.33333 0.81 0.3949
AT 80.08333 1 80.08333 7.77 0.0237
Error1
82.85 8 10.3125
B 991.5 3 330.5 152.05 0BA 8.41667 3 2.80556 1.29 0.3003BT 12.16667 3 4.05556 1.87 0.1624
BAT 12.75 3 4.25 1.96 0.1477
Error2
52.16667 24 2.17361
Latin Square Designs
Latin Square Designs Selected Latin Squares
3 x 3 4 x 4A B C A B C D A B C D A B C D A B C DB C A B A D C B C D A B D A C B A D CC A B C D B A C D A B C A D B C D A B
D C A B D A B C D C B A D C B A
5 x 5 6 x 6A B C D E A B C D E FB A E C D B F D C A EC D A E B C D E F B AD E B A C D A F E C BE C D B A E C A B F D
F E B A D C
A Latin Square
Definition
• A Latin square is a square array of objects (letters A, B, C, …) such that each object appears once and only once in each row and each column. Example - 4 x 4 Latin Square.
A B C DB C D AC D A BD A B C
In a Latin square You have three factors:
• Treatments (t) (letters A, B, C, …)
• Rows (t)
• Columns (t)
The number of treatments = the number of rows = the number of colums = t.The row-column treatments are represented by cells in a t x t array.The treatments are assigned to row-column combinations using a Latin-square arrangement
Example
A courier company is interested in deciding between five brands (D,P,F,C and R) of car for its next purchase of fleet cars.
• The brands are all comparable in purchase price. • The company wants to carry out a study that will
enable them to compare the brands with respect to operating costs.
• For this purpose they select five drivers (Rows). • In addition the study will be carried out over a
five week period (Columns = weeks).
• Each week a driver is assigned to a car using randomization and a Latin Square Design.
• The average cost per mile is recorded at the end of each week and is tabulated below:
Week 1 2 3 4 5 1 5.83 6.22 7.67 9.43 6.57 D P F C R 2 4.80 7.56 10.34 5.82 9.86 P D C R F
Drivers 3 7.43 11.29 7.01 10.48 9.27 F C R D P 4 6.60 9.54 11.11 10.84 15.05 R F D P C 5 11.24 6.34 11.30 12.58 16.04 C R P F D
The Model for a Latin Experiment
kijjikkijy
i = 1,2,…, t j = 1,2,…, t
yij(k) = the observation in ith row and the jth column receiving the kth treatment
= overall mean
k = the effect of the ith treatmenti = the effect of the ith row
ij(k) = random error
k = 1,2,…, t
j = the effect of the jth column
No interaction between rows, columns and treatments
• A Latin Square experiment is assumed to be a three-factor experiment.
• The factors are rows, columns and treatments.
• It is assumed that there is no interaction between rows, columns and treatments.
• The degrees of freedom for the interactions is used to estimate error.
The Anova Table for a Latin Square Experiment
Source S.S. d.f. M.S. F p-value
Treat SSTr t-1 MSTr MSTr /MSE
Rows SSRow t-1 MSRow MSRow /MSE
Cols SSCol t-1 MSCol MSCol /MSE
Error SSE (t-1)(t-2) MSE
Total SST t2 - 1
The Anova Table for Example
Source S.S. d.f. M.S. F p-value
Week 51.17887 4 12.79472 16.06 0.0001
Driver 69.44663 4 17.36166 21.79 0.0000
Car 70.90402 4 17.72601 22.24 0.0000
Error 9.56315 12 0.79693
Total 201.09267 24
Example
In this Experiment the we are again interested in how weight gain (Y) in rats is affected by Source of protein (Beef, Cereal, and Pork) and by Level of Protein (High or Low).
There are a total of t = 3 X 2 = 6 treatment combinations of the two factors.
• Beef -High Protein• Cereal-High Protein• Pork-High Protein• Beef -Low Protein• Cereal-Low Protein and • Pork-Low Protein
In this example we will consider using a Latin Square design
Six Initial Weight categories are identified for the test animals in addition to Six Appetite categories.
• A test animal is then selected from each of the 6 X 6 = 36 combinations of Initial Weight and Appetite categories.
• A Latin square is then used to assign the 6 diets to the 36 test animals in the study.
In the latin square the letter
• A represents the high protein-cereal diet• B represents the high protein-pork diet• C represents the low protein-beef Diet• D represents the low protein-cereal diet• E represents the low protein-pork diet and • F represents the high protein-beef diet.
The weight gain after a fixed period is measured for each of the test animals and is tabulated below:
Appetite Category 1 2 3 4 5 6 1 62.1 84.3 61.5 66.3 73.0 104.7 A B C D E F 2 86.2 91.9 69.2 64.5 80.8 83.9 B F D C A E
Initial 3 63.9 71.1 69.6 90.4 100.7 93.2 Weight C D E F B A
Category 4 68.9 77.2 97.3 72.1 81.7 114.7 D A F E C B 5 73.8 73.3 78.6 101.9 111.5 95.3 E C A B F D 6 101.8 83.8 110.6 87.9 93.5 103.8 F E B A D C
The Anova Table for Example
Source S.S. d.f. M.S. F p-value
Inwt 1767.0836 5 353.41673 111.1 0.0000
App 2195.4331 5 439.08662 138.03 0.0000
Diet 4183.9132 5 836.78263 263.06 0.0000
Error 63.61999 20 3.181
Total 8210.0499 35
Diet SS partioned into main effects for Source and Level of Protein
Source S.S. d.f. M.S. F p-value
Inwt 1767.0836 5 353.41673 111.1 0.0000
App 2195.4331 5 439.08662 138.03 0.0000
Source 631.22173 2 315.61087 99.22 0.0000
Level 2611.2097 1 2611.2097 820.88 0.0000
SL 941.48172 2 470.74086 147.99 0.0000
Error 63.61999 20 3.181
Total 8210.0499 35
Graeco-Latin Square Designs
Mutually orthogonal Squares
DefinitionA Greaco-Latin square consists of two latin squares (one using the letters A, B, C, … the other using greek letters , , , …) such that when the two latin square are supper imposed on each other the letters of one square appear once and only once with the letters of the other square. The two Latin squares are called mutually orthogonal.Example: a 7 x 7 Greaco-Latin Square
A B C D E F GB C D E F G A
C D E F G A BD E F G A B CE F G A B C DF G A B C D EG A B C D E F
Note:
At most (t –1) t x t Latin squares L1, L2, …, Lt-1 such that any pair are mutually orthogonal.
It is possible that there exists a set of six 7 x 7 mutually orthogonal Latin squares L1, L2, L3, L4, L5, L6 .
The Greaco-Latin Square Design - An Example
A researcher is interested in determining the effect of two factors
• the percentage of Lysine in the diet and • percentage of Protein in the diet
have on Milk Production in cows.
Previous similar experiments suggest that interaction between the two factors is negligible.
For this reason it is decided to use a Greaco-Latin square design to experimentally determine the two effects of the two factors (Lysine and Protein).
Seven levels of each factor is selected• 0.0(A), 0.1(B), 0.2(C), 0.3(D), 0.4(E), 0.5(F), and
0.6(G)% for Lysine and • 2(a), 4(b), 6(c), 8(d), 10(e), 12(f) and 14(g)% for
Protein ). • Seven animals (cows) are selected at random for
the experiment which is to be carried out over seven three-month periods.
A Greaco-Latin Square is the used to assign the 7 X 7 combinations of levels of the two factors (Lysine and Protein) to a period and a cow. The data is tabulated on below:
P e r i o d 1 2 3 4 5 6 7
1 3 0 4 4 3 6 3 5 0 5 0 4 4 1 7 5 1 9 4 3 2 ( A ( B ( C ( D ( E ( F ( G 2 3 8 1 5 0 5 4 2 5 5 6 4 4 9 4 3 5 0 4 1 3 B ( C ( D ( E ( F ( G ( A 3 4 3 2 5 6 6 4 7 9 3 5 7 4 6 1 3 4 0 5 0 2 ( C ( D ( E ( F ( G ( A ( B
C o w s 4 4 4 2 3 7 2 5 3 6 3 6 6 4 9 5 4 2 5 5 0 7 ( D ( E ( F ( G ( A ( B ( C 5 4 9 6 4 4 9 4 9 3 3 4 5 5 0 9 4 8 1 3 8 0
( E ( F ( G ( A ( B ( C ( D
6 5 3 4 4 2 1 4 5 2 4 2 7 3 4 6 4 7 8 3 9 7
( F ( G ( A ( B ( C ( D ( E
7 5 4 3 3 8 6 4 3 5 4 8 5 4 0 6 5 5 4 4 1 0 ( G ( A ( B ( C ( D ( E ( F
The Model for a Greaco-Latin Experiment
klijjilkklijy
i = 1,2,…, t j = 1,2,…, t
yij(kl) = the observation in ith row and the jth column receiving the kth Latin treatment and the lth Greek treatment
k = 1,2,…, t l = 1,2,…, t
= overall mean
k = the effect of the kth Latin treatment
i = the effect of the ith row
ij(k) = random error
j = the effect of the jth column
No interaction between rows, columns, Latin treatments and Greek treatments
l = the effect of the lth Greek treatment
• A Greaco-Latin Square experiment is assumed to be a four-factor experiment.
• The factors are rows, columns, Latin treatments and Greek treatments.
• It is assumed that there is no interaction between rows, columns, Latin treatments and Greek treatments.
• The degrees of freedom for the interactions is used to estimate error.
The Anova Table for a Greaco-Latin Square Experiment
Source S.S. d.f. M.S. F p-value
Latin SSLa t-1 MSLa MSLa /MSE
Greek SSGr t-1 MSGr MSGr /MSE
Rows SSRow t-1 MSRow MSRow /MSE
Cols SSCol t-1 MSCol MSCol /MSE
Error SSE (t-1)(t-3) MSE
Total SST t2 - 1
The Anova Table for Example
Source S.S. d.f. M.S. F p-value
Protein 160242.82 6 26707.1361 41.23 0.0000
Lysine 30718.24 6 5119.70748 7.9 0.0001
Cow 2124.24 6 354.04082 0.55 0.7676
Period 5831.96 6 971.9932 1.5 0.2204
Error 15544.41 24 647.68367
Total 214461.67 48