Completely randomized design - University of Saskatchewan
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 yij i ij(CR ) The ANOVA table for the Randomized Block Design Source df Sum of Squares Blocks
n-1 SSBlocks Treatments t-1 SSTr Error (t 1) (n 1) SSError Total tn - 1 SSTotal yij i j ij( RB ) Comments ( CR ) The error term, ij , for the Completely Randomized Design models variability in the reponse, y, between experimental units The error term, ij( RB,) for the Completely Block Design models variability in the reponse, y, between experimental units in the ( CR )
same block (hopefully the is considerably smaller than ij . The ability to detect treatment differences depends on the magnitude of the random error term Example Weight gain, diet, source of protein, level of protein (Completely randomized design) Randomized Block Design Block 1 107 (1) 96 (2) 112 (3) 83 (4) 87 (5) 90 (6) Block 6
112 (3) 83 (4) 93 (5) 81 (6) The Anova Table for Diet Experiment Source Block Diet ERROR S.S 5992.4167 4572.8833 3147.2833 d.f. 9 5 45 M.S. F 665.82407 9.52
914.57667 13.076659 69.93963 p-value 0.00000 0.00000 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:
93 (5) 81 (6) 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 10.1 11.4 12.1 C A B
D 13.4 12.9 12.2 12.3 D B A C 12.7 12.9 11.4 11.9 B D C A Table Chemical A B C D Blocks (Bolt Samples) 1 2 3
10.1 12.2 11.9 11.4 12.9 12.7 9.9 12.3 11.4 12.1 13.4 12.9 The Model for a randomized Block Experiment yij i j ij 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 Treat
S.S. SST d.f. t-1 M.S. MST F p-value 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 Block Diet ERROR S.S 5992.4167 4572.8833 3147.2833 d.f. 9 5 45 M.S. F 665.82407 9.52 914.57667 13.076659 69.93963 p-value 0.00000
0.00000 The Anova Table forTextile Experiment SOURCE Blocks Chem ERROR SUM OF SQUARES 7.17167 5.20000 0.53500 D.F. 2 3 6 MEAN SQUARE 3.5858 1.7333 0.0892 F 40.21 19.44 TAIL PROB. 0.0003 0.0017 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 Block Diet ERROR Source Block Source Level SL ERROR S.S 5992.4167 4572.8833 3147.2833 S.S 5992.4167 882.23333 2680.0167 1010.6333 3147.2833
d.f. 9 5 45 d.f. 9 2 1 2 45 M.S. F 665.82407 9.52 914.57667 13.076659 69.93963 M.S. 665.82407 441.11667 2680.0167 505.31667 69.93963 F 9.52 6.31 38.32 7.23 p-value
0.00000 0.00000 p-value 0.00000 0.00380 0.00000 0.00190 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.
Obtain the ANOVA table Tests of Between-Subjects Effects Dependent Variable: WTGAIN Type III Sum of Source Squares Corrected Model 10564.033 a Intercept 437418.8 DIET 4594.683 BLOCK 5969.350 Error 3134.150 Total 451117.0 Corrected Total 13698.183 df 14 1 5 9 45 60 59 Mean Square 754.574
437418.8 918.937 663.261 69.648 F 10.834 6280.442 13.194 9.523 Sig. .000 .000 .000 .000 a. R Squared = .771 (Adjusted R Squared = .700) 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) Obtain the ANOVA table Tests of Between-Subjects Effects Dependent Variable: WTGAIN Type III
Sum of Source Squares Corrected Model 10564.033 a Intercept 437418.8 BLOCK 5969.350 SOURCE 904.033 LEVEL 2680.017 SOURCE * LEVEL 1010.633 Error 3134.150 Total 451117.0 Corrected Total 13698.183 df 14 1 9 2 1 2 45 60 59 Mean
Square 754.574 437418.8 663.261 452.017 2680.017 505.317 69.648 a. R Squared = .771 (Adjusted R Squared = .700) F 10.834 6280.442 9.523 6.490 38.480 7.255 Sig. .000 .000 .000 .003 .000 .002 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. Table: The enzyme levels -immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgery Subject 1 2 3 4 5 6 7 8
Day 0 Day 1 Day 2 Day 7 108 63 45 42 112 75 56 52 114 75 51 46 129 87 69 69 115 71 52 54 122 80 68 68 105 71 52 54 117 77 54 61
Subject 9 10 11 12 13 14 15 Day 0 Day 1 Day 2 Day 7 106 65 49 49 110 70 46 47 120 85 60 62 118 78 51 56 110 65 46 47 132 92
73 63 127 90 73 68 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 Subject Day ERROR SS 4221.100 36282.267 390.233
df MS 14 301.507 3 12094.089 42 9.291 F 32.45 1301.66 p-value 0.0000 0.0000 The Subject Source of variability is modelling the variability between subjects The ERROR Source of variability is modelling the variability within subjects The repeated measures are in columns Analysis Using SPSS - the data file 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 Source TIME Error(TIME) Type III Sum of Squares Sphericity As sumed 36282.267 Greenhouse-Geis ser 36282.267 Huynh-Feldt 36282.267 Lower-bound 36282.267 Sphericity As sumed 390.233 Greenhouse-Geis ser 390.233 Huynh-Feldt 390.233 Lower-bound 390.233
df 3 2.588 3.000 1.000 42 36.225 42.000 14.000 Mean Square 12094.089 14021.994 12094.089 36282.267 9.291 10.772 9.291 27.874 F 1301.662 1301.662 1301.662 1301.662 Sig. .000 .000 .000 .000
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) 0 122 112 129 115
45 78 67 74 54 59 43 76 64 7 31 50 72 43 66 54 38 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 Source Drug 1 Error Time Time x Drug 2 Error SS 1745.396 df 2 MS 872.698 10287.844 47067.031 357.688 21 3 6 489.897
15689.010 59.615 668.031 63 10.604 F 1.78 p-value 0.1929 1479.58 5.62 0.0000 0.0001 There are two sources of Error in a repeated measures design: The between subject error Error1 and the within subject error Error2 Tables of means Drug Control A B Overall
Day 0 118.63 103.25 103.38 108.42 Day 1 77.88 68.25 69.38 71.83 Day 2 60.50 52.00 54.13 55.54 Day 7 55.75 51.50 51.50 52.92 Overall 78.19 68.75 69.59 72.18 120
Time Profiles of Enzyme Levels 100 Control Enzyme Level Drug A Drug B 80 60 40 0 1 2 3 Day 4 5 6 7 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 subject 1 2 3 18 19 14 14 12 10 12 8 6
BDAC CADB DCBA ABCD BADC CDAB DCBA 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. ABCD BCDA CDAB DABC 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: 1 2 Drivers 3 4 5 1 5.83 D 4.80 P 7.43 F 6.60 R
11.24 C 2 6.22 P 7.56 D 11.29 C 9.54 F 6.34 R Week 3 7.67 F 10.34 C 7.01 R 11.11 D 11.30 P 4 9.43 C 5.82 R
10.48 D 10.84 P 12.58 F 5 6.57 R 9.86 F 9.27 P 15.05 C 16.04 D The Model for a Latin Experiment yij k k i j ij k i = 1,2,, t j = 1,2,, t k = 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 treatment No interaction th i = the effect of the i row between rows, columns and
j = the effect of the jth column treatments ij(k) = random error 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 p-value F MSTr /MSE Source Treat S.S. SSTr d.f. t-1 M.S. MSTr Rows SSRow
t-1 MSRow MSRow /MSE Cols SSCol t-1 MSCol Error SSE (t-1)(t-2) MSE Total SST t2 - 1 MSCol /MSE The Anova Table for Example 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 Source 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: 1 2 Initial Weight Category 3 4 5 6 1 62.1 A 86.2 B 63.9 C 68.9 D 73.8 E 101.8
F Appetite Category 2 3 4 84.3 61.5 66.3 B C D 91.9 69.2 64.5 F D C 71.1 69.6 90.4 D E F 77.2 97.3 72.1 A F E 73.3 78.6 101.9
C A B 83.8 110.6 87.9 E B A 5 73.0 E 80.8 A 100.7 B 81.7 C 111.5 F 93.5 D 6 104.7 F 83.9 E 93.2 A 114.7 B
95.3 D 103.8 C The Anova Table for Example S.S. d.f. M.S. F p-value Inwt 1767.0836 5 353.41673 111.1 0.0000 App 2195.4331 5
2 470.74086 147.99 0.0000 Error 63.61999 20 3.181 8210.0499 35 Source Total Graeco-Latin Square Designs Mutually orthogonal Squares Definition A 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 G B C D E F G A C D E F G A B D E F G A B C E F
G A B C D F G A B C D E G 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 rio d 1 1 2 3 C o w s 4 5 6
The Model for a Greaco-Latin Experiment yij kl k l i j ij kl i = 1,2,, t j = 1,2,, t k = 1,2,, t l = 1,2,, t yij(kl) = the observation in ith row and the jth column receiving the kth Latin treatment and the lth Greek treatment = overall mean k = the effect of the kth Latin treatment l = the effect of the lth Greek treatment i = the effect of the ith row j = the effect of the jth column ij(k) = random error No interaction between rows, columns, Latin treatments and Greek treatments 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 Latin S.S. SSLa d.f. t-1 M.S. MSLa p-value F MSLa /MSE Greek SSGr t-1 MSGr MSGr /MSE Rows
SSRow t-1 MSRow MSRow /MSE Cols SSCol t-1 MSCol Error SSE (t-1)(t-3) MSE Total SST t2 - 1 MSCol /MSE The Anova Table for Example Source
S.S. d.f. Protein Lysine Cow Period Error Total 160242.82 6 6 6 6 24 48 30718.24 2124.24 5831.96 15544.41 214461.67 M.S. 26707.1361 5119.70748 354.04082 971.9932 647.68367
F p-value 41.23 7.9 0.55 1.5 0.0000 0.0001 0.7676 0.2204
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