# STATISTICS by hedongchenchen

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```									Statistics 312           30 Factorial Designs                            1

Factorial experiment

The values of more than one independent variable (factor) are varied. A
two-factor analysis of variance with more than one observation
per treatment (more than one replication) allows for
examination of interaction effects in addition to "main" effects
in a two-factor experiment.

Examples
1.   Auto mileage based on seven brands and three octanes of gasoline.
2.   Crop yields from five different seed varieties with four fertilizers.
3.   Run times for three different search algorithms with two different
addressing schemes in Monte Carlo simulations.
4.   Breaking strengths of an alloy subjected to ten different annealing
temperatures and three different cooling times.
5.   Smoothness of silicon wafers after being treated for three times,
with two lubricant solutions, at two temperatures.

When there are a levels of factor A crossed with b levels of factor B,
then there are ab treatments and the experiment is called an a x b
factorial design.

MAIN EFFECT The average change in the response produced by
changing the level of a factor.

INTERACTION Occurs when the effect of one factor depends on the
level(s) of the other(s).
Statistics 312   30 Factorial Designs   2
Statistics 312                   30 Factorial Designs                   3

General Linear Model: Roughnss versus Time, Solution

Factor      Type Levels Values
Time       fixed      2 40 60
Solution   fixed      2 1 2

Analysis of Variance for Roughnss, using Adjusted SS for Tests

Time              1     2.5000       2.5000    2.5000   21.90   0.000
Solution          1     0.8410       0.8410    0.8410    7.37   0.010
Time*Solution     1     0.3240       0.3240    0.3240    2.84   0.101
Error            36     4.1100       4.1100    0.1142
Total            39     7.7750
Statistics 312                  30 Factorial Designs                               4
Tukey 95.0% Simultaneous Confidence Intervals
Response Variable Roughnss
All Pairwise Comparisons among Levels of Time

Time = 40 subtracted from:

Time     Lower    Center      Upper   -------+---------+---------+---------
60      0.2833    0.5000     0.7167    (-----------------*-----------------)
-------+---------+---------+---------
0.36      0.48      0.60

Tukey 95.0% Simultaneous Confidence Intervals
Response Variable Roughnss
All Pairwise Comparisons among Levels of Solution

Solution = 1 subtracted from:

Solution     Lower    Center       Upper   ----+---------+---------+---------+--
2          -0.5067   -0.2900    -0.07330   (--------------*-------------)
----+---------+---------+---------+--
-0.45     -0.30     -0.15      0.00
Statistics 312   30 Factorial Designs   5
Statistics 312                30 Factorial Designs                         6

General Linear Model: Time versus Material, Design

Factor      Type Levels Values
Material   fixed      3 Carbide Steel     Titanium

Analysis of Variance for Time, using Adjusted SS for Tests

Material            2   1194.00    1194.00       597.00    27.24   0.001
Design              1     70.08      70.08        70.08     3.20   0.124
Material*Design     2    944.67     944.67       472.33    21.55   0.002
Error               6    131.50     131.50        21.92
Total              11   2340.25

Tukey 95.0% Simultaneous Confidence Intervals
Response Variable Time
All Pairwise Comparisons among Levels of Material*Design

Material = Carbide

Material*Design     Lower    Center      Upper
Carbide Split      -27.14     -8.50      10.14
Steel    Split     -54.64    -36.00     -17.36
Titanium Split     -10.64      8.00      26.64

Material*Design   ------+---------+---------+---------+
Carbide Split             (-----*----)
Steel    Split    (-----*----)
Titanium Split                 (----*-----)
------+---------+---------+---------+
-35         0        35        70
Statistics 312                30 Factorial Designs         7
Material = Carbide
Design   = Split subtracted from:

Material*Design     Lower    Center     Upper
Steel    Split     -46.14    -27.50    -8.863
Titanium Split      -2.14     16.50    35.137

Material*Design   ------+---------+---------+---------+
Steel    Split       (----*----)
Titanium Split                   (-----*----)
------+---------+---------+---------+
-35         0        35        70

Material = Steel

Material*Design     Lower    Center     Upper
Steel    Split     -43.14    -24.50    -5.863
Titanium Split       0.86     19.50    38.137

Material*Design   ------+---------+---------+---------+
Steel    Split        (----*----)
Titanium Split                    (-----*----)
------+---------+---------+---------+
-35         0        35        70

Material = Steel
Design   = Split subtracted from:

Material*Design      Lower   Center     Upper
Titanium Split      25.363    44.00     62.64

Material*Design   ------+---------+---------+---------+
Titanium Split                           (-----*----)
------+---------+---------+---------+
-35         0        35        70

Material = Titanium

Material*Design     Lower    Center     Upper
Titanium Split    -0.1372     18.50     37.14

Material*Design   ------+---------+---------+---------+
Titanium Split                    (----*-----)
------+---------+---------+---------+
-35         0        35        70
Statistics 312         30 Factorial Designs   8