Other experimental designs

Randomized Block designRepeated Measures designs

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.

Experimental Designs

The objective of Experimental design is to reduce the magnitude of random error resulting in more powerful tests

to detect experimental effects

The Completely Randomizes Design

Treats1 2 3 … t

Experimental units randomly assigned to treatments

Randomized Block Design

Blocks

All treats appear once in each block

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 11.9B 11.4 12.9 12.7C 9.9 12.3 11.4D 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 meani = the effect of the ith treatment

j = the effect of the jth Blockij = random error

The Anova Table for a randomized Block Experiment

Source S.S. d.f. M.S. F p-valueTreat 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 .000437418.8 1 437418.8 6280.442 .0004594.683 5 918.937 13.194 .0005969.350 9 663.261 9.523 .0003134.150 45 69.648451117.0 60

13698.183 59

SourceCorrected ModelInterceptDIETBLOCKErrorTotalCorrected Total

Type IIISum ofSquares df

MeanSquare 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 .000437418.8 1 437418.8 6280.442 .0005969.350 9 663.261 9.523 .000904.033 2 452.017 6.490 .003

2680.017 1 2680.017 38.480 .0001010.633 2 505.317 7.255 .0023134.150 45 69.648451117.0 60

13698.183 59

SourceCorrected ModelInterceptBLOCKSOURCELEVELSOURCE * LEVELErrorTotalCorrected Total

Type IIISum of

Squares dfMean

Square F Sig.

R Squared = .771 (Adjusted R Squared = .700)a.

Obtain the ANOVA table

The ANOVA table for the Completely Randomized DesignSource df Sum of Squares

Treatments t - 1 SSTr

Error t(n – 1) SSError

Total tn - 1 SSTotal

Source df Sum of SquaresBlocks 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)

• 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 .000437418.8 1 437418.8 6280.442 .0004594.683 5 918.937 13.194 .0005969.350 9 663.261 9.523 .0003134.150 45 69.648451117.0 60

13698.183 59

SourceCorrected ModelInterceptDIETBLOCKErrorTotalCorrected Total

Type IIISum ofSquares df

MeanSquare 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 .000437418.8 1 437418.8 6280.442 .0005969.350 9 663.261 9.523 .000904.033 2 452.017 6.490 .003

2680.017 1 2680.017 38.480 .0001010.633 2 505.317 7.255 .0023134.150 45 69.648451117.0 60

13698.183 59

SourceCorrected ModelInterceptBLOCKSOURCELEVELSOURCE * LEVELErrorTotalCorrected 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 DesignWe 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 .00036282.267 2.588 14021.994 1301.662 .00036282.267 3.000 12094.089 1301.662 .00036282.267 1.000 36282.267 1301.662 .000

390.233 42 9.291390.233 36.225 10.772390.233 42.000 9.291390.233 14.000 27.874

Sphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-bound

SourceTIME

Error(TIME)

Type IIISum ofSquares df

MeanSquare 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-valueDrug 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 0.0001Error2

668.031 63 10.604

Tables of means

Drug Day 0 Day 1 Day 2 Day 7 OverallControl 118.63 77.88 60.50 55.75 78.19A 103.25 68.25 52.00 51.50 68.75B 103.38 69.38 54.13 51.50 69.59Overall 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

Enzy

me

Leve

l

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-valueAnxiety 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 0.0237Error1

82.85 8 10.3125B 991.5 3 330.5 152.05 0

BA 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 0.1477Error2

52.16667 24 2.17361