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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

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.

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.

Steps in Planning a Randomised Block Design

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

Analysis in Two-way ANOVA - 1

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

SS TOTAL

SS BETWEEN

SS WITHIN

SS INTERVENTION

SS BLOCKING

SS INTERACTION

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.

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

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

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

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

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

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

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%

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).

Reference Coding and Effect Coding - 2

- In reference coding the β 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 β coefficients provide the differences from the overall mean response for each treatment group.

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