Analysis of Variance (ANOVA)

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Analysis of Variance (ANOVA)

• EPP 245
• Statistical Analysis of
• Laboratory Data

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The Basic Idea

• The analysis of variance is a way of testing
  whether observed differences between groups are
  too large to be explained by chance variation
• One-way ANOVA is used when there are k 2
  groups for one factor, and no other quantitative
  variable or classification factor.

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A B C
9 10 12
7 9 14
7 8 14
9 9 12
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Data Grand Mean Row Deviations from grand
mean Cell Deviations from row mean
Are the row deviations from the grand mean too
big to be accounted for by the cell deviations
from the row means?
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Data
A B C
9 10 12
7 9 14
7 8 14
9 9 12
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Cell Means
A B C
8 9 13
8 9 13
8 9 13
8 9 13
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Deviations from Cell Means
A B C
1 1 -1
-1 0 1
-1 -1 1
1 0 -1
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Red cell folate data Description 22 rows
and 2 columns. data on red cell folate
levels in patients receiving three different
methods of ventilation during anesthesia. Format
folate a numeric vector. Folate
concentration (?g/l). ventilation a factor
with levels 'N2OO2,24h' 50 nitrous oxide
and 50 oxygen, continuously for 24 hours
'N2OO2,op' 50 nitrous oxide and 50
oxygen, only during operation
'O2,24h' no nitrous oxide, but 35-50 oxygen for
24 hours.
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insheet using redcell.csv summarize
folate tabulate ventilation tabulate ventilation,
summarize (folate) graph box folate, over
(ventilation) graph export folate1.wmf oneway
folate ventilation describe ventilation encode
ventilation, generate(dv) Describe dv
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. summarize folate Variable Obs
Mean Std. Dev. Min
Max ---------------------------------------------
------------------------ folate 22
283.2273 51.28439 206 392 .
tabulate ventilation ventilation Freq.
Percent Cum. ----------------------------
------------------- N2OO2,24h 8
36.36 36.36 N2OO2,op 9
40.91 77.27 O2,24h 5
22.73 100.00 -------------------------------
---------------- Total 22
100.00 . tabulate ventilation, summarize
(folate) Summary of
folate ventilation Mean Std. Dev.
Freq. ------------------------------------------
------ N2OO2,24h 316.625 58.717088
8 N2OO2,op 256.44444 37.121797
9 O2,24h 278 33.756481
5 ------------------------------------------
------ Total 283.22727 51.284391
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. oneway folate ventilation
Analysis of Variance Source
SS df MS F Prob gt
F ------------------------------------------------
------------------------ Between groups
15515.7664 2 7757.88321 3.71
0.0436 Within groups 39716.0972 19
2090.32091 ---------------------------------------
--------------------------------- Total
55231.8636 21 2630.08874 Bartlett's
test for equal variances chi2(2) 2.0951
Probgtchi2 0.351
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. describe ventilation storage
display value variable name type format
label variable label ---------------------
--------------------------------------------------
-------- ventilation str10 10s . encode
ventilation, generate(dv) . describe dv
storage display value variable name
type format label variable
label --------------------------------------------
----------------------------------- dv
long 10.0g dv . anova folate
dv Number of obs
22 R-squared 0.2809
Root MSE 45.72 Adj
R-squared 0.2052 Source
Partial SS df MS F Prob gt
F ---------------------------------
------------------------------
Model 15515.7664 2 7757.88321 3.71
0.0436
dv 15515.7664 2 7757.88321
3.71 0.0436
Residual 39716.0972 19 2090.32091
--------------------------------
-------------------------------
Total 55231.8636 21 2630.08874
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Two- and Multi-way ANOVA

• If there is more than one factor, the sum of
  squares can be decomposed according to each
  factor, and possibly according to interactions
• One can also have factors and quantitative
  variables in the same model (cf. analysis of
  covariance)
• All have similar interpretations

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Heart rates after enalaprilat Description
36 rows and 3 columns. data for nine
patients with congestive heart failure before
and shortly after administration of
enalaprilat, in a balanced two-way
layout. Format hr a numeric vector. Heart
rate in beats per minute. subj a factor
with levels '1' to '9'. time a factor with
levels '0' (before), '30', '60', and '120'
(minutes after administration).
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. drop _all . insheet using heart.rate.csv (4
vars, 36 obs) . anova hr subj time
Number of obs 36
R-squared 0.9685
Root MSE 3.5165 Adj R-squared
0.9540 Source Partial SS
df MS F Prob gt F
------------------------------------------------
--------------- Model
9117.52778 11 828.866162 67.03
0.0000
subj 8966.55556 8 1120.81944
90.64 0.0000 time
150.972222 3 50.3240741 4.07
0.0180
Residual 296.777778 24 12.3657407
-----------------------------------------
---------------------- Total
9414.30556 35 268.980159
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. anova hr subj
Number of obs 36 R-squared
0.9524 Root MSE
4.07226 Adj R-squared 0.9383
Source Partial SS df MS
F Prob gt F -------------------
--------------------------------------------
Model 8966.55556 8
1120.81944 67.59 0.0000
subj 8966.55556
8 1120.81944 67.59 0.0000
Residual
447.75 27 16.5833333
-------------------------------------------------
-------------- Total
9414.30556 35 268.980159 . predict
hrhat (option xb assumed fitted values) .
generate hrres hr - hrhat . graph box hrres,
over (time) . graph export hrresxtime.wmf
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. anova hr subj time
Number of obs 36 R-squared
0.9685 Root MSE
3.5165 Adj R-squared 0.9540
Source Partial SS df MS
F Prob gt F -------------------
--------------------------------------------
Model 9117.52778 11
828.866162 67.03 0.0000
subj 8966.55556
8 1120.81944 90.64 0.0000
time 150.972222 3 50.3240741
4.07 0.0180
Residual 296.777778 24
12.3657407 ---------------------
------------------------------------------
Total 9414.30556 35 268.980159
. rvfplot . graph export hrrvf.wmf . rvpplot
subj . graph export hrrvpsubj.wmf . rvpplot
time . graph export hrrvptime.wmf
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