Completely Randomized Design with a Single Factor - Download as PowerPoint

Document Sample
Completely Randomized Design with a Single Factor - Download as PowerPoint Powered By Docstoc
					Analysis of Variance for
Standard Designs

          Chapter 15
Completely Randomized
Design with a Single Factor
Table 15.3: A Completely
Randomized Design

Treatment                        Mean
1           y11   y12    y1n1       y1.
2           y21   y22    y2 n2      y2.


t           yt1   yt 2   ytnt       yt .
   The Model for a Completely
   Randomized Design
            yij     i   ij
                           i  1, ..., t ; j  1, ..., ni

where

  yij :   Observation on j th experimental unit
          receiving treatment i
       : Overall treatment mean, an unknown
          constant
   The Model for a Completely
   Randomized Design, cont.

  i     : An effect due to treatment i, an unknown
           constant
   ij: A random error associated with the response
        from the j th experimental unit receiving
        treatment i. We require that the ij s have a
        normal distribution with mean 0 and common
        variance   . In addition, the errors must be
                   2

independent.
Sums of Squares
Total Sum of Squares (TSS)


            TSS    yij  y.. 
                                        2

                        ij


   Partition of TSS

        y        y..    ni  yi.  y..     yij  yi. 
                        2                    2                     2
             ij
       ij                    i                    ij
    Sums of Squares, cont.
Between-treatment sum of squares (SST)



          SST  n  yi.  y.. 
                                   2

                       i



Sum of squares for error (SSE)



          SSE=  yij  yi. 
                                   2

                  ij
        Table 15.4: Analysis of Variance Table
        for a Completely Randomized Design


Source       SS    df         MS                 F

Treatments   SST   t 1   MST=SST/  t  1    MST/MSE
Error        SSE   N t   MSE=SSE/  N  t 
Total        TSS   N 1
 Unbiased Estimates

When                        is true, both MST
and MSE are unbiased estimates of      , the
variance of the experimental error. That is,
under Ho, both have a mean value in
repeated sampling, called the expected
mean squares, equal to        .
Expected Mean Squares

Under   Ha   ,


   E  MSE       2




   E  MST      nT
                  2
  Advantages of the Completely
  Randomized Design

1. The design is extremely easy to construct.
2. The design is easy to analyze even though
   the sample sizes might not be the same for
   each treatment.
3. The design can be used for any number of
   treatments.
   Disadvantages of the Completely
   Randomized Design

1. Although the completely randomized design can be
   used for any number of treatments, it is best suited
   for situations in which there are relatively few
   treatments.
2. The experimental units to which treatments are
   applied must be as homogeneous as possible. Any
   extraneous sources of variability will tend to inflate
   the error term, making it more difficult to detect
   differences among the treatment means.
       Randomized Complete Block
       Design
        Location
1         2        3           4
P1       P2       P3      P4
                                   Confounded factors:
P1       P2       P3      P4
                                   location & type of paint
P1       P2       P3      P4
P1       P2       P3      P4

Table 15.6: Random
assignment of 4 paints to 16
sections of roadway
       Randomized Complete Block
       Design, cont.
Table 15.7: Randomized              Location
complete block assignment of
4 paints to 16 sections of     1     2     3    4
roadway                        P2   P2    P1   P1
                               P1   P4    P3   P2
                               P3   P1    P4   P4
                               P4   P3    P2   P3
  Definition

A randomized complete block design is an
experimental design for comparing t treatments
in b blocks. The blocks consist of t
homogeneous experimental units. Treatments
are randomly assigned to experimental units
within a block, with each treatment appearing
exactly once in every block.
  Advantages of the Randomized
  Complete Block Design

1. The design is useful for comparing t
   treatment means in the presence of a single
   extraneous source of variability.
2. The statistical analysis is simple.
3. The design is easy to construct.
4. It can be used to accommodate any number
   of treatments in any number of blocks.
  Disadvantages of the Randomized
  Complete Block Design

1. Because the experimental units within a block must
   be homogeneous, the design is best suited for a
   relatively small number of treatments.
2. This design controls for only one extraneous source
   of variability (due to blocks). Additional extraneous
   sources of variability tend to increase the error term,
   making it more difficult to detect treatment
   differences.
3. The effect of each treatment on the response must
   be approximately the same from block to block.
Figure 15.1: Treatment Means in
a Randomized Block Design
  Plot of Treatment Mean by Treatment
 100
                  3
  90              4           3
  80              1           4

  70   3          2           1
       4                      2
  60
       1
  50
       2
  40

       1           2         3
               Treatment
 The Hypotheses for Testing
 Treatment Mean Differences
The null hypothesis is no difference among
treatment means versus the research
hypothesis treatment means differ.
     Total Sum of Squares


Partition of TSS:
    Table 15.11: Analysis of Variance
    Table for a Randomized Complete
    Block Design

Source   SS   df    MS                  F
  Expected Mean Squares
When                          is true, both MST
and MSE are unbiased estimates of        ,
the variance of the experimental error.
  Relative Efficiency
 RE(RCB, CR): the relative efficiency
of the randomized complete block
design compared to a completely
randomized design
 Did blocking increase our precision
for comparing treatment means in a
given experiment?
       Latin Square Design
                Secretary                        Secretary
 Problem    1      2   3    4     Problem    1      2   3    4
 I         A      A    C    A     I         A      B    C    D
 II        B      D    A    D     II        B      C    D    A
 III       D      B    D    B     III       C      D    A    B
 IV        C      C    B    C     IV        D      A    B    C
Table 15.14: A Randomized       Table 15.15: A Latin Square
Complete Block Design for the   Design for the Spreadsheet
Spreadsheet Study               Study
  Advantages of the Latin
  Square Design

1. The design is particularly appropriate for
   comparing t treatment means in the
   presence of two sources of extraneous
   variation.
2. The analysis is quite simple.
  Disadvantages of the Latin
  Square Design

1. Although a Latin square can be constructed for any
   value of t, it is best suited for comparing t
   treatments when
2. Any additional extraneous sources of variability tend
   to inflate the error term, making it more difficult to
   detect differences among the treatment means.
3. The effect of each treatment on the response must
   be approximately the same across rows and
   columns.
  Definition 15.2

A t x t Latin square design contains t rows
and t columns. The t treatments are randomly
assigned to experimental units within the rows
and columns so that each treatment appears in
every row and in every column.
  Test for Treatment Effects

We can test specific hypotheses concerning
the parameters in our model. In particular, we
may wish to test for differences among the t
treatment means.
    Table 15.17: Analysis of Variance
    Table for a t x t Latin Square Design

Source   SS   df     MS               F
  Relative Efficiency

RE(LS, CR): the relative efficiency of the
Latin square design compared to a completely
randomized design
Did accounting for row/column sources of
variability increase our precision in estimating
the treatment means?
  Factorial Treatment Structure in a
  Completely Randomized Design


A factorial experiment is an experiment in
which the response y is observed at all factor-
level combinations of the independent variables.
Figure 15.6a: Illustration of the
Absence of Interaction in a 2 x 2
Factorial Experiment
 Mean response




                                     Level 1, factor B
                                     Level 2, factor B




                 Level 1   Level 2


         Factors A and B do not interact
Figure 15.6b,c: Illustration of the
Presence of Interaction in a 2 x 2
Factorial Experiment
Mean response



                                    Level 2, factor B


                                    Level 1, factor B

                Level 1   Level 2
                                                        Factors A and
                                                        B interact
Mean response




                                    Level 1, factor B
                                    Level 2, factor B

                Level 1   Level 2
Table 15.25: Expected Values for
a 2 x 2 Factorial Experiment


                     Factor B
Factor A   Level 1          Level 2
Table 15.26: Expected Values for a
2 x 2 Factorial Experiment, with
Replications


                     Factor B
Factor A   Level 1          Level 2
  Definition 15.4

Two factors A and B are said to interact if the
difference in mean responses for two levels of
one factor is not constant across levels of the
second factor.
  Profile Plot

     See Figure 15.6
 Used to amplify the notion of
interaction: when no interaction is
present, the difference in the mean
response between two levels of one
factor is the same for levels of the other
factor.
    Table 15.27: AOV Table for a
    Completely Randomized Two-Factor
    Factorial Experiment

Source   SS   df   MS             F
        Illustration of Significant,
        Orderly Interaction
Figure 15.8:
Profile plot in              100                                      Level 3, factor B
which interactions                   90
are present, but
                     Mean response
                                     80
interactions are
orderly                              70
                                                                      Level 2, factor B
                                     60
                                     50
                                                                      Level 1, factor B
                                     40

                                          Level 1 Level 2 Level 3 Level 4
                                                       Factor A
        Illustration of Significant,
        Disorderly Interaction
Figure 15.9:
Profile plot in              100
which interactions                   90
are present, and
                     Mean response
                                     80                               Level 2, factor B
interactions are
disorderly                           70                               Level 1, factor B
                                     60
                                     50
                                     40                               Level 3, factor B

                                          Level 1 Level 2 Level 3 Level 4
                                                       Factor A
Factorial Treatment Structure in a
Randomized Complete Block Design
  Estimation of Treatment Differences
  and Comparisons of Treatment
  Means
100(1-)% Confidence Interval for the
Difference in Treatment Means




where s is the square root of MSE in the AOV table
and t/2 can be obtained from Table 2 in the Appendix
for a = /2 and the degrees of freedom for MSE.
Multiple Comparison
Procedures

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:37
posted:6/29/2012
language:English
pages:43