1 **EXAMPLE
1******************************************************;
2 *** Example of
unbalanced CRD with
experimental error ***;
3 *** From
Snedecor & Cochran, 1967
(pg 278)
***;
4
*****************************************************************;
5 OPTIONS PS=256
LS=78 NOCENTER NODATE
PAGENO=1;
6 dm
'log;clear;output;clear';
7 DATA MICE
onebyone; INFILE CARDS
MISSOVER;
8
INPUT STRAIN $ 1-4 DAYS NUMBER;
9
TITLE1 'EXST7015: DAYS TO
DEATH FOR MICE INOCULATED WITH 3 TYPHOID STRAINS';
10
TITLE2 'UNEQUAL NUMBER OF MICE IN
EACH TREATMENT';
11 do
i = 1 to number by 1; output onebyone; end;
12
output mice;
13 CARDS;
NOTE:
The data set WORK.MICE has 31 observations and 4 variables.
NOTE:
The data set WORK.ONEBYONE has 224 observations and 4 variables.
NOTE:
DATA statement used:
real time
0.07 seconds
cpu time
0.07 seconds
13 !
RUN;
45 ;
46 PROC PRINT
data=mice; TITLE3 'RAW DATA
LISTING'; RUN;
NOTE:
There were 31 observations read from the data set WORK.MICE.
NOTE:
The PROCEDURE PRINT printed page 1.
NOTE:
PROCEDURE PRINT used:
real time
0.04 seconds
cpu time
0.04 seconds
EXST7015:
DAYS TO DEATH FOR MICE INOCULATED WITH 3 TYPHOID STRAINS
1
UNEQUAL
NUMBER OF MICE IN EACH TREATMENT
RAW
DATA LISTING
Obs STRAIN DAYS
NUMBER i
1 9D
2 6
7
2 9D
3 4
5
3 9D
4 9
10
4 9D
5 8
9
5 9D
6 3
4
6 9D
7 1
2
7 11C
2 1
2
8 11C
3
3
4
9 11C
4 3
4
10 11C
5 6
7
11 11C
6 6
7
12 11C
7 14
15
13 11C
8 11
12
14 11C
9 4
5
15 11C
10 6
7
16 11C
11 2
3
17 11C
12 3
4
18 11C
13 1
2
19 DSC1
2 3
4
20 DSC1
3 5
6
21 DSC1
4 5
6
22 DSC1
5 8
9
23 DSC1
6 19
20
24 DSC1
7 23
24
25 DSC1
8 22
23
26 DSC1
9 14
15
27 DSC1
10 14
15
28 DSC1
11 7
8
29 DSC1
12 8
9
30 DSC1
13 4
5
31 DSC1
14 1
2
47
PROC MIXED DATA=OnebyOne cl covtest;
CLASSES STRAIN;
48
TITLE3 'ANALYSIS OF VARIANCE';
49
MODEL DAYS = STRAIN / htype=3
DDFM=Satterthwaite outp=ResidDataP;
50
repeated / group=strain;
51
run;
NOTE:
Convergence criteria met.
NOTE:
The data set WORK.RESIDDATAP has 224 observations and 11 variables.
NOTE:
The PROCEDURE MIXED printed page 2.
NOTE:
PROCEDURE MIXED used:
real time
0.14 seconds
cpu
time
0.13 seconds
EXST7015:
DAYS TO DEATH FOR MICE INOCULATED WITH 3 TYPHOID
STRAINS 2
UNEQUAL
NUMBER OF MICE IN EACH TREATMENT
ANALYSIS
OF VARIANCE
The
Mixed Procedure
Model Information
Data
Set
WORK.ONEBYONE
Dependent
Variable
DAYS
Covariance
Structure Variance
Components
Group
Effect
STRAIN
Estimation
Method
REML
Residual
Variance Method None
Fixed
Effects SE Method Model-Based
Degrees
of Freedom Method Satterthwaite
Class Level Information
Class
Levels
Values
STRAIN
3
11C 9D DSC1
Dimensions
Covariance
Parameters
3
Columns
in
X
4
Columns
in
Z
0
Subjects
224
Max
Obs Per
Subject
1
Observations
Used
224
Observations
Not
Used
0
Total
Observations
224
Iteration History
Iteration
Evaluations -2 Res Log
Like Criterion
0
1 1027.49256195
1
1
1012.92789611 0.00000000
Convergence criteria met.
Covariance
Parameter Estimates
Standard Z
Cov
Parm Group
Estimate Error
Value Pr Z Alpha
Lower Upper
Residual
STRAIN 11C 5.8633
1.0795 5.43
<.0001
0.05 4.2127 8.7221
Residual
STRAIN 9D 1.8989
0.4903 3.87
<.0001
0.05 1.2126 3.3928
Residual
STRAIN DSC1 6.6327
0.8164 8.12
<.0001
0.05 5.2839 8.5761
Fit Statistics
-2
Res Log
Likelihood 1012.9
AIC (smaller
is better) 1018.9
AICC
(smaller is better) 1019.0
BIC
(smaller is better) 1029.2
Null
Model Likelihood Ratio Test
DF
Chi-Square Pr > ChiSq
2
14.56 0.0007
Type 3 Tests of Fixed Effects
Num Den
Effect
DF
DF F Value Pr > F
STRAIN
2
114 70.10 <.0001
53
PROC UNIVARIATE DATA=ResidDataP PLOT
NORMAL; VAR resid; RUN;
NOTE:
The PROCEDURE UNIVARIATE printed page 3.
NOTE:
PROCEDURE UNIVARIATE used:
real time
0.02 seconds
cpu
time
0.02 seconds
53
!
QUIT;
EXST7015:
DAYS TO DEATH FOR MICE INOCULATED WITH 3 TYPHOID
STRAINS 3
UNEQUAL
NUMBER OF MICE IN EACH TREATMENT
ANALYSIS
OF VARIANCE
The
UNIVARIATE Procedure
Variable:
Resid
Moments
N
224 Sum
Weights
224
Mean
0 Sum
Observations
0
Std
Deviation 2.39433175
Variance
5.73282454
Skewness
0.06043085
Kurtosis
-0.0153611
Uncorrected
SS 1278.41987 Corrected
SS 1278.41987
Coeff
Variation
. Std Error Mean
0.15997802
Basic Statistical Measures
Location
Variability
Mean
0.00000 Std
Deviation
2.39433
Median
-0.03226
Variance
5.73282
Mode
-0.79699
Range
12.00000
Interquartile Range 3.00000
Tests for Location: Mu0=0
Test
-Statistic- -----p Value------
Student's
t t
0
Pr > |t| 1.0000
Sign
M -3
Pr >= |M| 0.7384
Signed
Rank S
-212
Pr >= |S| 0.8277
Tests for Normality
Test
--Statistic--- -----p Value------
Shapiro-Wilk
W
0.988718 Pr < W
0.0758
Kolmogorov-Smirnov
D
0.077823 Pr > D <0.0100
Cramer-von
Mises W-Sq
0.148439
Pr > W-Sq 0.0246
Anderson-Darling
A-Sq
0.857552 Pr > A-Sq 0.0276
Quantiles
(Definition 5)
Quantile
Estimate
100%
Max 6.2030075
99%
5.2030075
95%
4.2030075
90%
3.2030075
75%
Q3 1.2030075
50%
Median -0.0322581
25%
Q1 -1.7969925
10%
-2.7969925
5%
-4.3666667
1%
-5.7969925
0%
Min -5.7969925
Extreme Observations
------Lowest-----
-----Highest-----
Value
Obs
Value Obs
-5.79699
94
5.20301 221
-5.79699
93
5.20301 222
-5.79699
92
5.20301 223
-5.36667
32
5.63333 91
-4.79699
99
6.20301 224
Stem
Leaf
# Boxplot
6
2
1 0
5
6
1 |
5
2222
4 |
4
666
3 |
4
22222222
8 |
3
66
2 |
3
02222222
8 |
2
666666
6 |
2
00022222222222222
17 |
1
6666
4 |
1 0000000022222222222222
22 +-----+
0
66666666666
11 | |
0 2222222222222222222222
22 | + |
-0 44444444444444000000000
23 *-----*
-0 88888888888888888888888
23 | |
-1 4444440000
10
| |
-1 8888888888888888888
19 +-----+
-2
444444000000
12 |
-2
88888888
8 |
-3
444
3 |
-3
88888
5 |
-4 444
3
|
-4
88888
5 |
-5
4
1 |
-5
888
3 |
----+----+----+----+---
Normal Probability Plot
6.25+
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0.25+
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| ***
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-5.75+***
+----+----+----+----+----+----+----+----+----+----+
-2 -1
0
+1 +2
55
PROC GLM DATA=MICE; CLASSES STRAIN;
FREQ NUMBER;
56
TITLE3 'ANALYSIS OF VARIANCE WITH
TESTS OF HOMOGENIETY';
57
MODEL DAYS = STRAIN;
58
MEANS STRAIN / HOVTEST=BARTLETT
HOVTEST=BF HOVTEST=LEVENE(TYPE=ABS)
59
HOVTEST=LEVENE(TYPE=SQUARE) HOVTEST=OBRIEN WELCH;
60
OUTPUT OUT=MICERES PREDICTED=P
RESIDUAL=E; RUN;
60
!
QUIT;
NOTE:
The data set WORK.MICERES has 31 observations and 6 variables.
NOTE:
The PROCEDURE GLM printed pages 4-7.
NOTE:
PROCEDURE GLM used:
real time
0.11 seconds
cpu
time
0.11 seconds
EXST7015:
DAYS TO DEATH FOR MICE INOCULATED WITH 3 TYPHOID
STRAINS 4
UNEQUAL
NUMBER OF MICE IN EACH TREATMENT
ANALYSIS
OF VARIANCE WITH TESTS OF HOMOGENIETY
The
GLM Procedure
Class Level Information
Class
Levels Values
STRAIN
3 11C 9D DSC1
Number
of observations 224
The
GLM Procedure
Dependent
Variable: DAYS
Frequency:
NUMBER
Sum of
Source
DF Squares
Mean Square F Value
Pr > F
Model
2 359.794413
179.897207 31.10
<.0001
Error
221 1278.419872
5.784705
Corrected
Total
223 1638.214286
R-Square
Coeff Var Root
MSE DAYS Mean
0.219626
33.58801
2.405141 7.160714
Source
DF Type I SS
Mean Square F Value Pr > F
STRAIN
2 359.7944135
179.8972067 31.10
<.0001
Source
DF Type III SS Mean
Square F Value
Pr > F
STRAIN
2 359.7944135
179.8972067 31.10
<.0001
EXST7015:
DAYS TO DEATH FOR MICE INOCULATED WITH 3 TYPHOID
STRAINS 6
UNEQUAL
NUMBER OF MICE IN EACH TREATMENT
ANALYSIS
OF VARIANCE WITH TESTS OF HOMOGENIETY
The
GLM Procedure
Levene's Test for Homogeneity of DAYS
Variance
ANOVA of Squared Deviations from Group
Means
Sum of Mean
Source
DF
Squares Square F
Value
Pr > F
STRAIN
2
566.4
283.2 4.56
0.0115
Error
221
13722.0 62.0903
O'Brien's Test for Homogeneity of DAYS
Variance
ANOVA of O'Brien's Spread Variable, W
= 0.5
Sum of Mean
Source
DF
Squares Square F
Value
Pr > F
STRAIN
2
564.1
282.0 4.40
0.0133
Error
221
14152.2 64.0370
Brown
and Forsythe's Test for Homogeneity of
DAYS Variance
ANOVA of Absolute Deviations from Group
Medians
Sum of Mean
Source
DF
Squares Square F
Value
Pr > F
STRAIN
2
23.0777
11.5388 4.99
0.0076
Error
221
511.1 2.3128
Bartlett's
Test for Homogeneity of DAYS Variance
Source
DF
Chi-Square Pr > ChiSq
STRAIN
2
14.4363
0.0007
Welch's ANOVA for DAYS
Source
DF
F Value Pr > F
STRAIN
2.0000 69.64
<.0001
Error
99.2945
EXST7015:
DAYS TO DEATH FOR MICE INOCULATED WITH 3 TYPHOID
STRAINS 7
UNEQUAL
NUMBER OF MICE IN EACH TREATMENT
ANALYSIS
OF VARIANCE WITH TESTS OF HOMOGENIETY
The
GLM Procedure
Level
of
-------------DAYS------------
STRAIN
N
Mean Std Dev
11C
60
7.36666667 2.42142042
9D
31
4.03225806 1.37801478
DSC1
133
7.79699248 2.57540641
61
OPTIONS PS=35 LS=80;
62
63
PROC CHART DATA=MICE;
64
TITLE3 'Histogram of mouse
mortality with typhoid strain';
65
VBAR STRAIN / SUMVAR=DAYS TYPE=MEAN
FREQ=NUMBER; RUN;
NOTE:
The PROCEDURE CHART printed page 8.
NOTE:
PROCEDURE CHART used:
real time
0.01 seconds
cpu
time
0.01 seconds
EXST7015:
DAYS TO DEATH FOR MICE INOCULATED WITH 3 TYPHOID
STRAINS
8
UNEQUAL
NUMBER OF MICE IN EACH TREATMENT
Histogram
of mouse mortality with typhoid strain
DAYS
Mean
8
+
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*****
6
+
*****
*****
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*****
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*****
4
+
*****
***** *****
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***** *****
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***** *****
2
+
*****
***** *****
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--------------------------------------------
11C
9D DSC1
STRAIN
66
OPTIONS PS=256 LS=80;
67
68
PROC NPar1Way DATA=MICE; FREQ NUMBER;
CLASSES STRAIN;
69
TITLE3 'Nonparametric ANALYSIS OF
VARIANCE';
70
*
MODEL DAYS = STRAIN / htype=3 DDFM=Satterthwaite;
71
run;
NOTE:
There were 31 observations read from the data set WORK.MICE.
NOTE:
The PROCEDURE NPAR1WAY printed pages 9-20.
NOTE:
PROCEDURE NPAR1WAY used:
real time
0.05 seconds
cpu
time
0.05 seconds
EXST7015:
DAYS TO DEATH FOR MICE INOCULATED WITH 3 TYPHOID
STRAINS
9
UNEQUAL
NUMBER OF MICE IN EACH TREATMENT
Nonparametric
ANALYSIS OF VARIANCE
The
NPAR1WAY Procedure
Analysis
of Variance for Variable DAYS
Classified by Variable STRAIN
STRAIN
N
Mean
--------------------------------------
9D
31 4.032258
11C
60 7.366667
DSC1
133
7.796992
Source
DF
Sum of Squares Mean Square F
Value
Pr > F
--------------------------------------------------------------------
Among
2
359.794413 179.897207
31.0988
<.0001
Within
221
1278.419872 5.784705
Average scores were used for
ties.
EXST7015:
DAYS TO DEATH FOR MICE INOCULATED WITH 3 TYPHOID
STRAINS
10
UNEQUAL
NUMBER OF MICE IN EACH TREATMENT
Nonparametric
ANALYSIS OF VARIANCE
The
NPAR1WAY Procedure
Wilcoxon Scores (Rank Sums) for
Variable DAYS
Classified by Variable
STRAIN
Sum of
Expected Std
Dev Mean
STRAIN
N
Scores Under
H0 Under
H0 Score
----------------------------------------------------------------------
9D
31
1117.0 3487.50
332.739483 36.032258
11C
60
7082.0 6750.00
426.719398 118.033333
DSC1
133
17001.0 14962.50
473.250796 127.827068
Average scores were used for
ties.
Kruskal-Wallis Test
Chi-Square
51.7109
DF
2
Pr
> Chi-Square <.0001
Median Scores (Number of Points Above
Median) for Variable DAYS
Classified by Variable
STRAIN
Sum of
Expected Std
Dev Mean
STRAIN
N
Scores Under
H0 Under
H0 Score
----------------------------------------------------------------------
9D
31
0.394737
15.50
2.370647 0.012733
11C
60
32.526316
30.00
3.040220 0.542105
DSC1
133
79.078947
66.50
3.371739 0.594579
Average scores were used for
ties.
Median
One-Way Analysis
Chi-Square
41.1408
DF
2
Pr
> Chi-Square <.0001
Van der Waerden Scores (Normal) for
Variable DAYS
Classified by Variable
STRAIN
Sum of
Expected Std
Dev Mean
STRAIN
N
Scores Under
H0 Under
H0 Score
----------------------------------------------------------------------
9D
31
-35.335930
0.0 5.033882
-1.139869
11C
60
4.805868
0.0
6.455667 0.080098
DSC1
133
30.530063
0.0
7.159622 0.229549
Average scores were used for
ties.
Van
der Waerden One-Way Analysis
Chi-Square
50.2485
DF
2
Pr
> Chi-Square <.0001
Savage Scores (Exponential) for
Variable DAYS
Classified by Variable
STRAIN
Sum of
Expected Std
Dev Mean
STRAIN
N
Scores Under H0
Under
H0 Score
----------------------------------------------------------------------
9D
31
-25.261616
0.0 5.066124
-0.814891
11C
60
0.066414
0.0
6.497015 0.001107
DSC1
133
25.195202
0.0
7.205478 0.189438
Average scores were used for
ties.
Savage
One-Way Analysis
Chi-Square
26.3902
DF
2
Pr
> Chi-Square <.0001
Kolmogorov-Smirnov Test for Variable DAYS
Classified by Variable STRAIN
EDF at Deviation from Mean
STRAIN
N
Maximum at Maximum
---------------------------------------------------
9D
31
0.870968 3.333122
11C
60
0.216667 -0.431100
DSC1
133
0.157895 -1.319633
Total
224
0.272321
Maximum Deviation Occurred at Observation 22
Value of DAYS at Maximum = 5.0
Kolmogorov-Smirnov
Statistics (Asymptotic)
KS
0.241248
KSa 3.610676
Cramer-von
Mises Test for Variable DAYS
Classified by Variable STRAIN
Summed Deviation
STRAIN
N from Mean
----------------------------------------
9D
31
4.428843
11C
60
0.095553
DSC1
133
0.749578
Cramer-von
Mises Statistics (Asymptotic)
CM
0.023545
CMa 5.273974
Modified: August 16, 2004
James P. Geaghan