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+                                                  *
        |                                                +*+
        |                                             **+*
        |                                           **+
        |                                       ****+
        |                                       *++
        |                                     ***
        |                                    **
        |                                 ***
        |                                **
        |                              ***
        |                            ***
    0.25+                         ****
        |                       ***
        |                    ****
        |                   **
        |                ****
        |              ***
        |            **+
        |           *+
        |         ***
        |       +**
        |     ***
        |   +*
   -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 +                               *****
  |       *****                   *****
  |       *****                   *****
  |       *****                   *****
6 +       *****                   *****
  |       *****                   *****
  |       *****                   *****
  |       *****                   *****
4 +       *****       *****       *****
  |       *****       *****       *****
  |       *****       *****       *****
  |       *****       *****       *****
2 +       *****       *****       *****
  |       *****       *****       *****
  |       *****       *****       *****
  |       *****       *****       *****
  --------------------------------------------
           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