Two-way Analysis of Variance

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Two-way Analysis of Variance

• Two-way ANOVA is a type of study design with one
  numerical outcome variable and two categorical
  explanatory variables.
• Example In a completely randomised design we
  may wish to compare outcome by age, gender or
  disease severity. Subjects are grouped by one
  such factor and then randomly assigned one
  treatment.
• Technical term for such a group is block and the
  study design is also called randomised block
  design

2
Randomised Block Design

• Blocks are formed on the basis of expected
  homogeneity of response in each block (or group).
• The purpose is to reduce variation in response
  within each block (or group) due to biological
  differences between individual subjects on
  account of age, sex or severity of disease.

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Randomised Block Design - 3

• Randomised block design is a more robust design
  than the simple randomised design.
• The investigator can take into account
  simultaneously the effects of two factors on an
  outcome of interest.
• Additionally, the investigator can test for
  interaction, if any, between the two factors.

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Steps in Planning a Randomised Block Design

1. Subjects are randomly selected to constitute a
   random sample.
2. Subjects likely to have similar response
   (homogeneity) are put together to form a block.
3. To each member in a block intervention is
   assigned such that each subject receives one
   treatment.
4. Comparisons of treatment outcomes are made within
   each block

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Analysis in Two-way ANOVA - 1

• The variance (total sum of squares) is first
  partitioned into WITHIN and BETWEEN sum of
  squares. Sum of Squares BETWEEN is next
  partitioned by intervention, blocking and
  interaction

SS TOTAL
SS BETWEEN
SS WITHIN
SS INTERVENTION
SS BLOCKING
SS INTERACTION
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Two-way ANOVA
method. And an interaction between gender and
teaching method is being sought. Analysis of
Two-way ANOVA is demonstrated in the slides that
follow. The study is about a n experiment
involving a teaching method in which professional
actors were brought in to play the role of
patients in a medical school. The test scores of
male and female students who were taught either
by the conventional method of lectures, seminars
and tutorials and the role-play method were
recorded. The hypotheses being tested
are Role-play method is superior to conventional
way of teaching. Female students in general have
better test scores than male students. Role-play
method makes a better impact on students of a
particular gender. Thus, there are two factors
gender and teaching method. And an interaction
between teaching method and gender is being
sought.
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Analysis in Two-way ANOVA - 2

• Each Sum of Squares (SS) is divided by its degree
  of freedom (df) to get the Mean Sum of Squares
  (MS).
• The F statistic is computed for each of the three
  ratios as
• MS INTERVENTION MS WITHIN
• MS BLOCK MS WITHIN
• MS INTERVENTION MS WITHIN

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Analysis of Two-way ANOVA - 3

• Analysis of Variance for score
• Source DF SS MS F
  P
• sex 1 2839 2839 22.75
  0.000
• Tchmthd 1 1782 1782 14.28
  0.001
• Error 29 3619 125
• Total 31 8240

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Analysis of Two-way ANOVA - 4

• Individual 95 CI
• Sex Mean -------------------------
  -------------
• 0 58.5
  (------------)
• 1 39.6 (-------------)
• -------------------------
  -------------
• 40.0 48.0
  56.0 64.0
• Individual 95 CI
• Tchmthd Mean -------------------------
  -------------
• 0 56.5
  (--------------)
• 1 41.6 (---------------)
• -------------------------
  -------------
• 42.0 49.0
  56.0 63.0

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Analysis of Tw0-way ANOVA - 5
Analysis of Variance for SCORE Source
DF SS MS F
P SEX 1 2839
2839 22.64 0.000 TCHMTHD 1
1782 1782 14.21 0.001 INTERACTN
1 108 108 0.86
0.361 Error 28 3511
125 Total 31 8240
Interaction is not significant P 0.361
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Analysis of Two-way ANOVA - 6
Individual 95 CI SEX Mean
-------------------------------------- 0
58.5
(------------) 1 39.6
(-------------)
--------------------------------------
40.0 48.0 56.0
64.0 Individual 95
CI TCHMTHD Mean ----------------------
---------------- 0 56.5
(--------------) 1
41.6 (---------------)
--------------------------------------
42.0 49.0 56.0
63.0
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Analysis of Two-way ANOVA by the regression
method (reference coding)
The regression equation is SCORE 65.9 - 18.8
SEX - 14.9 TCHMTHD Predictor Coef
SE Coef T P Constant
65.913 3.420 19.27 0.000 SEX
-18.838 3.950 -4.77
0.000 TCHMTHD -14.925 3.950
-3.78 0.001 S 11.17 R-Sq 56.1
R-Sq(adj) 53.1 Analysis of Variance Source
DF SS MS F
P Regression 2 4620.9
2310.4 18.51 0.000 Residual Error 29
3619.0 124.8 Total 31
8239.8
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Analysis of Two-way ANOVA by the regression
method (effect coding)
The regression equation is SCORE 49.0 - 9.42
EFCT-Sex - 7.46 EFCT-Tchmthd - 1.84
Interaction Predictor Coef SE
Coef T P Constant 49.031
1.980 24.77 0.000 EFCT-Sex
-9.419 1.980 -4.76 0.000 EFCT-Tch
-7.463 1.980 -3.77
0.001 Interact -1.838 1.980
-0.93 0.361 S 11.20 R-Sq 57.4
R-Sq(adj) 52.8
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Reference Coding and Effect Coding - 1

• In both methods, for k explanatory variables k-1
  dummy variables are created.
• In reference coding the value 1 is assigned to
  the group of interest and 0 to all others (e.g.
  Female 1 Male 0).
• In effect coding the value -1 is assigned to
  control group 1 to the group of interest (e.g.
  new treatment), and 0 to all others (e.g. Female
  1 Male (control group) -1 Role Play 1
  conventional teaching (control) -1).

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Reference Coding and Effect Coding - 2

• In reference coding the ß coefficients of the
  regression equation provide estimates of the
  differences in means from the control (reference)
  group for various treatment groups.
• In effect coding the ß coefficients provide the
  differences from the overall mean response for
  each treatment group.