Randomized Block Design(Kirk, chapter 7)

BUSI 6480Lecture 6

A blocking strategy rather than independent samples is what makes the difference. Blocking can obtain a more precise test for examining differences in the factor level means.

A block is a factor in which we are not primarily interested.

Difference between Randomized Block Design and Completely Randomized Design

Nuisance Variation can be Accounted for in Blocks

A block may contain p matched subjects A block may contain a single subject that is

observed p times (alternatively called repeated measures design).

A design with repeated measurements in which the order of administration of the treatment levels is the same for all subjects is a subject-by-trials design.

Experimental Design Model RB-p

Yij = + j + i + ij

Where j is the main treatment effect and i is the block effect.

Y Y Y Y Y Y Y Y Y Yij j i ij j i . . ( . . . ) ( . . . ) ( . . . . )Grand Mean Treatment Effect Block Effect Residual Effect

The term MSWG for a CRD model is replaced by the term MSRES for a RB model.

No nesting of subject within treatment.

Relative Efficiency of Randomized Block Design

RE = MSWG (from CRD) / MSRES (from RB)

MSWG can be computed from a RB model by using the following formula.

MSWG = ((n-1)MSBL +n(p-1)MSRES)/(np-1)

Larger RE Implies Smaller Sample Size Is Necessary In RB Model.

The nj number of subjects in each treatment level of a CRD necessary to match the efficiency of a randomized block design is

nj = RE * n (where n is from RB model)

Partial Omega Squared and Partial Intraclass Correlation

Y A B L2

IY A B L

2

2 2| . | .,

The “.” denotes the association between the dependent variable, Y, and the treatment A with the effects of blocks ignored.

A similar equation holds for Y|BL.A where alpha is replaced by .

Computational Formulas for Omega Squared and

Y A B l2 A

A

p 1 F 1

p 1 F 1 np| .

(( )( )

(( )( )

IY A B LA

A

F 1

n 1 F| . ( )

Computations for the Treatment Variability and Block Variability

i2

i2

i

pp 1

npM SA M SR E S

nn 1

npM SB L M SR E S

j

p

p

1

1

/ ( )( )

/ ( )( )

2

2

2

M SR E S

1

nM SA M SR E S

1

pM SB L M SR E S

( )( )

( )( )

Fixed Effects Estimates Random Effects Estimates

Notice similarities!

Between Subjects and Within Subjects Design

Randomized Block Design

Treatment means test

H0: 1 = 2 = K (treatments)

Ha: at least two means in H0 are different

Block means test

H0: μ1 = μ2 = μB (Blocks)

Ha: at least two means in H0 are different

The personnel director at Blackburn Industries is investigating dental claims submitted by married employees having at least one child. Of interest is whether the average annual dollar amounts of dental work claimed by the husband, by the wife, and per child are the same. Data were collected by randomly selected 15 families and recording these three dollar amounts (total claims for the year by the husband, by the wife, and per child).

RB Design Example: Dental Claims by Husband, Wife, Per Child Equal?

The Randomized Block Design

Family Husband Wife Per Child

1 x x x (1st block)

2 x x x (2nd block)

15 x x x (15th block)

H0: µH = µW = µC

Ha: not all three means are equal

Main Treatment Test – Hypothesis Same as in CRD

Can Dental Claim data be analyzed using CRD?

No, since dependency exists between husband, wife, and child, CRD cannot be used.

Note that the error degrees of freedom is smaller for a RB design than for a CRD.

The df for the factor are 3 - 1 = 2, for blocks are 16 - 1 = 15, and for total are 48 - 1 = 47, leaving 47-2-15 = 30 df for error.

Note that df error = (df factor)*(df blocks)

Df Error for RB

Source df SS MS F

Factor 2 80,880.29 40,440.14 40440.14/3264.81 = 12.39

Blocks 15 73,314 4,887.60 4887.60/3264 = 1.50

Error 30 97,944.38 3,264.81

Total 47 252,138.67

In a completely randomized design, the Blocks SS and the SSE would be combined into the SSE.

One F test is for the factor and the other is for the blocks.

RB Design Splits Error Sum of Squares in CRD

To test H0: H = W = C, we use F1 = 12.39. Since 12.39 > F.05, 2, 30 = 3.32, we reject H0 and conclude that the three average claim amounts are not equal.

By observing the sample means, we notice that the claims per child are considerably higher than those for the husband and wife.

As a final note, the block (family) effect is not significant here, since F2 = 1.50 < F.05, 15, 30 = 2.01.

Decision for RB Design Example

Example: Completely Randomized Design with p = 3 Treatments

R u n n e r T r e a tm e n t (L iq u id ) A s s ig n e d

1 B 2 A 3 B 4 C 5 C 6 A 7 B 8 C 9 A 1 0 A 1 1 C 1 2 A 1 3 B 1 4 C 1 5 B

Recall that a completely randomized design to compare p treatments is one in which the treatments are randomly assigned to the experimental units.

Treatments (Liquids)

1

2

3

4

5

CAB

A C B

B C A

A B C

A C B

A randomized block design to compare 3 treatments involving 5 blocks, each containing 3 relatively homogeneous experimental units. The 3 treatments are randomly assigned to the experimental units within each block, with one experimental unit assigned per treatment.

Changing CRD into a RB Design

Blocks (Runners)

The Regression Model for CRD Example

y = 0 + 1 x1 + 2 x2 +

Where: not if 0

Bliquid if 1

not if 0

A liquid if 121

xx

Interpretation of dummy variable coefficients:

A = 0 + 1

B = 0 + 2

C = 0

effectsBlock

66554433

effects Treatment

22110 xxxxxxy

The Regression Model for RB Design

not if 0

1runner if 1

not if 0

Bliquid if 1

not if 0

A liquid if 1321

xxx

not if 0

4runner if 1

not if 0

3runner if 1

not if 0

2runner if 1654

xxx

Data on Page 260 of Kirk can be typed in Excel and used for an RB Design.Use both SAS and SPSS and compare.

HW6: Analyze Data on Page 260 of Kirk

HW6: Put Data on Kirk Page 260 In Long Format & Use Random Factor for Block

In SPSS, data must be in the following format. (SAS can put data in this format for SPSS).

Put variables in the Univarate GLM dialog box.

Click on Model in SPSS and Click on Custom. Do not

include an intercept term in the model.

HW6: Select only Main Effects

Be sure to click “Add” in SPSS Dialog Box.

HW6: Create Plot of Treatment Versus Subjects

Click on the Post Hoc Option and Selecting the Main Treatment and Checking Tukey box. Multiple comparison procedures can only be performed on fixed effect factors.

HW6: Perform the Tukey test

HW6: In SAS, First Import Data from Excel.

DM "Log;Clear;OUT;Clear;" ; options pageno=min nodate formdlim='-';

title 'Randomized block analysis'; PROC IMPORT OUT= mydata DATAFILE= “D:\RB Data KirkP260.xls” DBMS=EXCEL2000 REPLACE; RANGE="Sheet1$"; GETNAMES=Yes; RUN;

HW6: Data Need To Be Put in Column Format for Proc Glm.

Data mydata; set mydata; s = _N_;

Data Treat1; set mydata; resp = a1; TreatLevel = 1;

Data Treat2; set mydata; resp = a2; TreatLevel = 2;

Data Treat3;set mydata;resp = a3;TreatLevel = 3;

Data Treat4;set mydata;resp = a4;TreatLevel = 4;

Data myAnovaData;Set Treat1 Treat2 Treat3 Treat4;

HW6: Note that Colum s and Level are Created

Obs s resp Level

1 1 3 1 2 2 2 1 3 3 2 1 4 4 3 1 5 5 1 1 6 6 3 1 7 7 4 1 8 8 6 1 9 1 4 2 10 2 4 2 11 3 3 2 12 4 3 2 13 5 2 2

HW6: To Export Column Format Data use the Proc Export Command.

This exported data set can be used in SPSS.

proc export data=myAnovaData outfile='D:\MyAnovaDatainColformat.xls' dbms=Excel97 replace;

Run;

Don’t forget to end your SAS program with

“quit;”

Use Proc GLM for RB ANOVA

proc glm data = myAnovaData; class TreatLevel s; model resp = TreatLevel s; random s; means TreatLevel / Tukey; output out=rb4out2 predicted=p rstudent=r ; run;

Get Tukey Comparisons, and out file with predicted and residuals.

In the SAS Output, Expected Mean Squares Are Displayed

Randomized block analysis

The GLM Procedure

Source Type III Expected Mean Square

TreatLevel Var(Error) + Q(TreatLevel)

s Var(Error) + 4 Var(s)

Get Plot of Residual and Predicted Values and Interpret.

symbol1 v=circle;

proc gplot data=rb4out2; plot r*p; run;

Use Original Data Set to Run Multivariate ANOVA (MANOVA)

proc glm data=mydata; model a1 a2 a3 a4 = / nouni; repeated s; run;

This procedure will also give Huynh/Feldt (H-F) test and Geisser-Greenhouse (G-G) test as discussed on pages 279-281.

Test for a difference among Treatment levels.

Get Power With Proc GlmPower Procedure

proc glmpower data = myAnovaData;

class TreatLevel s; model resp = TreatLevel s; Power Alpha = .01 stddev = 1.185 ntotal = 32 power = .; run;

The means of the data set must be in the input file for each level combination. For a RB model, each observation is a mean for a level combination since there is only one observation per cell.

HW6: Using the data on Page 260, determine the efficiency of the RB design versus the CR design.

How many subjects would be required in a completely randomized design to match the efficiency of the randomized block design?

Efficiency of RB Design

For the RB design with the data on Page 260, compute and interpret the value of

Y A B L2

IY B L Aand| . | .

What is the noncentrality parameter for finding the power of the test for the main treatment? Use alpha = .05 and use the Charts in the back of the textbook.

Omega Square and Intraclass Correlation

Different Ways of Running Experimental Designs in SAS