Twoway Analysis of Variance

© 1998 by Dr. Thomas W. MacFarland -- All Rights Reserved


************
two_anov.doc
************
Background:  Analysis of Variance (ANOVA) methodology is 
             quite effective in determining if two or more 
             group means differ due to chance, or if observed
             differences are indeed the result of true 
             difference between phenomena.  As useful as 
             it may be to determine singular differences
             between multiple groups:

             -- ANOVA analysis is not limited only to 
                studies involving one single variable.  

             -- On the contrary, ANOVA can be used to examine 
                differences with two or more factors (i.e.,
                independent variables) at the same time.

             A common use of ANOVA methodology is to use a 
             Twoway ANOVA statistical test to determine 
             differences (and possible interactions) when 
             variables have two or more categories.  When 
             Twoway ANOVA is used, it is possible to determine 
             if:

             1.  Is there a difference because of variables 
                 acting independently of each other?
     
             2.  Is there a difference because of joint 
                 effects (i.e., interaction)?

             Twoway ANOVA designs can become quite complex, not 
             only to effect but also to interpret.  Yet, this 
             highly useful methodology should not be avoided 
             merely because it is not "user friendly."   On the 
             contrary, Twoway ANOVA should be used perhaps more 
             than it it is, due to the advantage of greater 
             utilization of resources while modeling real-world 
             scenarios.

             Twoway ANOVA designs are often presented in a 
             manner similar to other factorial analyses, such 
             as the Chi-square analysis.  Like the chi-square 
             analysis, Twoway ANOVA uses a factorial organization 
             with data placed in cells.  The information within 
             each cell provides the necessary data for later 
             analysis.  

             A graphic representation of a factorial design is   
             presented below in Figure 1.  When reviewing this 
             representation, be sure to recall that interval or 
             ratio data are used with a Twoway ANOVA design, as 
             opposed to the use of nominal data with a chi-square 
             analysis, which is also organized along the format 
             of a factorial design.
                                

                                          Variable A

                               Category 1            Category 2
                           ______________________________________
                           |                  |                 |
                           |                  |                 |  
                Category 1 |  n1, n2, ...     |  n1, n2, ...    |
                           |                  |                 |
                           |                  |                 |  
             Variable B    |------------------|-----------------|
                           |                  |                 |
                           |                  |                 |
                Category 2 |  n1, n2, ...     |  n1, n2, ...    |
                           |                  |                 |
                           |__________________|_________________|

                                           Figure 1  

                           Comparative Study of Two Variables, 
                           (Variable A and Variable B), with Two 
                           Categories (Category 1 and Category 2)
                           for each Variable


             Thus, when using a Twoway ANOVA, be sure to remember 
             that it is possible to examine three separate 
             hypotheses:

             1.  if means for Variable A are equal to the 
                 population

             2.  if means for Variable B are equal to the 
                 population

             3.  if there is interaction between Variable A 
                 and Variable B

             As such, Twoway ANOVA are often used to help 
             explain "real-world" scenarios, where interaction
             is often found.  This more complex design is 
             different from simplistic designs that can only 
             explain scenarios designed for simplistic modeling.  
             The decision to use a Twoway ANOVA is the decision 
             to see if complex issues can be understood, and 
             possibly acted upon. 


Scenario:    This study examines if there are differences in 
             final examination test scores (the dependent 
             variable) for students in a university senior-
             level software engineering course on two separate 
             factors:

             -- The first factor addresses the method
                of instruction, with:

                -- The first group of students was taught 
                   by traditional lecture (Method Code 
                   = 1).

                -- The second group of students was taught 
                   by Computer Based Training (Method Code 
                   = 2).

                -- The third group of students was taught 
                   by the use of instructional videotapes
                   (Method Code = 3).

                -- The fourth group of students was  
                   enrolled through independent study
                   (Method Code = 4).

             -- The second factor addresses each student's
                possible prior graduation from a community 
                college:

                -- Some students in the senior-level course
                   had previously attended and graduated 
                   from a community college (Grd_CC Code 
                   = 1).

                -- Other students in the senior-level course
                   did not graduate from a community college
                   (Grd_CC Code = 2).

                   This coding scheme (Grd_CC Code = 2) is
                   discrete and therefore includes students who 
                   may have attended a community college but 
                   did not complete the full curriculum needed 
                   to receive an associate's degree. 

             Students were all from a university senior-
             level software engineering course who were
             assigned, through random selection, to
             placement into one of four groups:  instruction
             by traditional lecture, instruction by CBT
             (Computer Based Training), instruction by
             the use of instructional videotapes, and
             independent study.  The teacher worked with
             the registrat's office to obtain information
             about prior graduation from a community 
             college. 

             The teacher was confident that final examination 
             scores represented interval data (i.e., the data 
             are parametric, with the difference between "89" 
             and "90" equal to the difference between "75" 
             and "76").  The teacher also wanted to learn
             more about the effects of teaching method, 
             prior graduation from a community college, and
             the possible effect of interaction between these
             two factors on final examination scores.  As
             such, Twoway ANOVA (Analysis of Variance) was 
             correctly judged to be the appropriate test for 
             this analysis.

             Data from this study are summarized in Table 1. 


             Table 1

             Final Examination Test Scores in a Senior-Level
             Software Engineering Course by Instructional
             Method (Traditional Lecture, Computer Based
             Training, Instructional Videotape, and Independent 
             Study) and by Prior Graduation from a Community 
             College 
             ====================================================    
                            Instructional
                            Method
                            =============
                            1 = Lecture   CC Graduate
                            2 = CBT       ===========
                            3 = Video     1 = Yes
             Student Number 4 = IDS       2 = No      Final Score
             ----------------------------------------------------    

                   01             1         1            089
                   02             1         1            081
                   03             1         2            073
                   04             1         1            084
                   05             1         2            070
                   06             1         2            056
                   07             1         1            070
                   08             1         2            081
                   09             1         2            078
                   10             1         1            069
                   11             1         1            089
                   12             1         2            088
                   13             1         2            045
                   14             1         2            083
                   15             1         1            095
                   16             1         2            077
                   17             1         1            069
                   18             1         1            080
                   19             2         2            093
                   20             2         1            086
                   21             2         1            089
                   22             2         2            095
                   23             2         2            089
                   24             2         1            088
                   25             2         1            098
                   26             2         1            089
                   27             2         2            094
                   28             2         1            095
                   29             2         2            095
                   30             2         2            098
                   31             2         2            087
                   32             2         2            085
                   33             2         1            098
                   34             2         1            093
                   35             2         2            087
                   36             2         1            095
                   37             2         1            093
                   38             2         2            093
                   39             3         2            095
                   40             3         1            096
                   41             3         2            083
                   42             3         2            089
                   43             3         1            088
                   44             3         1            087
                   45             3         1            094
                   46             3         2            097
                   47             3         1            095
                   48             3         2            093
                   49             3         2            085
                   50             3         2            095
                   51             3         1            092
                   52             3         2            082
                   53             3         1            086
                   54             3         1            087
                   55             3         2            089
                   56             3         2            097
                   57             3         1            100
                   58             3         2            093
                   59             3         1            096
                   60             4         2            084
                   61             4         1            085
                   62             4         2            073
                   63             4         1            092
                   64             4         2            057
                   65             4         1            063
                   66             4         1            069
                   67             4         2            073
                   68             4         2            091
                   69             4         1            065
                   70             4         1            074
                   71             4         2            071
                   72             4         2            068
                   73             4         2            062
                   74             4         1            056
                   75             4         1            085
             ----------------------------------------------------    
             
             Note.  Notice how the N (i.e., number of subjects or
                    group members) for each instructional group
                    does not have to be equal.


Ho:          Null Hypothesis:  There is no difference between
             instructional method (instruction by traditional 
             lecture, instruction by Computer Based Training, 
             instruction by the use of instructional videotapes, 
             and independent study) and graduation status from
             a community college (either graduated from a 
             community college or did not graduate from a
             community college) regarding final examination test 
             scores of students enrolled in a university senior-
             level software engineering course (p <= .05).


Files:       1.  two_anov.doc

             2.  two_anov.dat

             3.  two_anov.r01

             4.  two_anov.o01

             5.  two_anov.con

             6.  two_anov.lis


Command:     At the Unix prompt (%), key:

             %spss -m < two_anov.r01 > two_anov.o01


************
two_anov.dat
************
                   01             1         1            089
                   02             1         1            081
                   03             1         2            073
                   04             1         1            084
                   05             1         2            070
                   06             1         2            056
                   07             1         1            070
                   08             1         2            081
                   09             1         2            078
                   10             1         1            069
                   11             1         1            089
                   12             1         2            088
                   13             1         2            045
                   14             1         2            083
                   15             1         1            095
                   16             1         2            077
                   17             1         1            069
                   18             1         1            080
                   19             2         2            093
                   20             2         1            086
                   21             2         1            089
                   22             2         2            095
                   23             2         2            089
                   24             2         1            088
                   25             2         1            098
                   26             2         1            089
                   27             2         2            094
                   28             2         1            095
                   29             2         2            095
                   30             2         2            098
                   31             2         2            087
                   32             2         2            085
                   33             2         1            098
                   34             2         1            093
                   35             2         2            087
                   36             2         1            095
                   37             2         1            093
                   38             2         2            093
                   39             3         2            095
                   40             3         1            096
                   41             3         2            083
                   42             3         2            089
                   43             3         1            088
                   44             3         1            087
                   45             3         1            094
                   46             3         2            097
                   47             3         1            095
                   48             3         2            093
                   49             3         2            085
                   50             3         2            095
                   51             3         1            092
                   52             3         2            082
                   53             3         1            086
                   54             3         1            087
                   55             3         2            089
                   56             3         2            097
                   57             3         1            100
                   58             3         2            093
                   59             3         1            096
                   60             4         2            084
                   61             4         1            085
                   62             4         2            073
                   63             4         1            092
                   64             4         2            057
                   65             4         1            063
                   66             4         1            069
                   67             4         2            073
                   68             4         2            091
                   69             4         1            065
                   70             4         1            074
                   71             4         2            071
                   72             4         2            068
                   73             4         2            062
                   74             4         1            056
                   75             4         1            085


************
two_anov.r01
************
SET WIDTH      = 80
SET LENGTH     = NONE
SET CASE       = UPLOW
SET HEADER     = NO
TITLE          = Twoway Analysis of Variance (TWOWAY ANOVA)
COMMENT        = This file examines if there are differences
                 in final examination test scores (the 
                 dependent variable) for students in a 
                 university senior-level software engineering 
                 course on two separate factors:

                 -- The first factor addresses the method
                    of instruction, with:

                    -- the first group of students was taught 
                       by traditional lecture (Method Code 
                       = 1).

                    -- the second group of students was taught 
                       by Computer Based Training (Method Code 
                       = 2).

                    -- the third group of students was taught 
                       by the use of instructional videotapes
                       (Method Code = 3).

                    -- the fourth group of students was  
                       enrolled through independent study
                       (Method Code = 4).

                 -- The second factor addresses each student's
                    possible prior graduation from a community 
                    college:

                    -- Some students in the senior-level course
                       had previously attended and graduated 
                       from a community college (Grd_CC Code 
                       = 1).

                    -- Other students in the senior-level course
                       did not graduate from a community college
                       (Grd_CC Code = 2), which includes students
                       who may have attended a community college
                       but did not complete the full curriculum
                       needed to receive an associate's degree. 


                 Students were all from a university senior-
                 level software engineering course who were
                 assigned, through random selection, to
                 placement into one of four groups:  instruction
                 by traditional lecture, instruction by CBT
                 (Computer Based Training), instruction by
                 the use of instructional videotapes, and
                 independent study.  The teacher worked with
                 the registrat's office to obtain information
                 about prior graduation from a community 
                 college. 

                 The teacher was confident that final examination 
                 scores represented interval data (i.e., the data 
                 are parametric, with the difference between "89" 
                 and "90" equal to the difference between "75" 
                 and "76").  The teacher also wanted to learn
                 more about the effects of teaching method, 
                 prior graduation from a community college, and
                 the possible effect of interaction between these
                 two factors on final examination scores.  As
                 such, Twoway ANOVA (Analysis of Variance) was 
                 correctly judged to be the appropriate test for 
                 this analysis.
DATA LIST FILE = 'two_anov.dat' FIXED
     / Stu_Code  20-21
       Method       35
       Grd_CC       45
       Score     58-60 

Variable Labels
       Stu_Code   "Student Code"
     / Method     "Method:  Lecture, CBT, Video, IDS"
     / Grd_CC     "Graduated from Community College: Y or N"
     / Score      "Final Examination Score"

Value Labels
       Method     1 'Lecture: Traditional Lecture'
                  2 'CBT: Computer-Based Training'
                  3 'Video: Instructional Videotape'
                  4 'IDS: Independent Study'

     / Grd_CC     1 'Grd_Yes:  Graduated from a CC'
                  2 'Grd_No :  Did NOT Graduate from a CC'

ANOVA Score BY Method(1,4) Grd_CC (1,2)
     / STATISTICS   = ALL
     / FORMAT       = LABELS
COMMENT             = Please note in this analysis how I need to
                      identify which methods (1, 2, 3, and 4)
                      and community college status (1 and 2) to 
                      analyze.  


************
two_anov.o01
************
   1  SET WIDTH      = 80
   2  SET LENGTH     = NONE
   3  SET CASE       = UPLOW
   4  SET HEADER     = NO
   5  TITLE          = Twoway Analysis of Variance (TWOWAY ANOVA)
   6  COMMENT        = This file examines if there are differences
   7                   in final examination test scores (the
   8                   dependent variable) for students in a
   9                   university senior-level software engineering
  10                   course on two separate factors:
  11
  12                   -- The first factor addresses the method
  13                      of instruction, with:
  14
  15                      -- the first group of students was taught
  16                         by traditional lecture (Method Code
  17                         = 1).
  18
  19                      -- the second group of students was taught
  20                         by Computer Based Training (Method Code
  21                         = 2).
  22
  23                      -- the third group of students was taught
  24                         by the use of instructional videotapes
  25                         (Method Code = 3).
  26
  27                      -- the fourth group of students was 
  28                         enrolled through independent study
  29                         (Method Code = 4).
  30
  31                   -- The second factor addresses each student's
  32                      possible prior graduation from a community
  33                      college:
  34
  35                      -- Some students in the senior-level course
  36                         had previously attended and graduated
  37                         from a community college (Grd_CC Code
  38                         = 1).
  39
  40                      -- Other students in the senior-level course
  41                         did not graduate from a community college
  42                         (Grd_CC Code = 2), which includes students
  43                         who may have attended a community college
  44                         but did not complete the full curriculum
  45                         needed to receive an associate's degree.
  46
  47
  48                   Students were all from a university senior-
  49                   level software engineering course who were
  50                   assigned, through random selection, to
  51                   placement into one of four groups:  instruction
  52                   by traditional lecture, instruction by CBT
  53                   (Computer Based Training), instruction by
  54                   the use of instructional videotapes, and
  55                   independent study.  The teacher worked with
  56                   the registrat's office to obtain information
  57                   about prior graduation from a community
  58                   college.
  59
  60                   The teacher was confident that final examination
  61                   scores represented interval data (i.e., the data
  62                   are parametric, with the difference between "89"
  63                   and "90" equal to the difference between "75"
  64                   and "76").  The teacher also wanted to learn
  65                   more about the effects of teaching method,
  66                   prior graduation from a community college, and
  67                   the possible effect of interaction between these
  68                   two factors on final examination scores.  As
  69                   such, Twoway ANOVA (Analysis of Variance) was
  70                   correctly judged to be the appropriate test for
  71                   this analysis.
  72  DATA LIST FILE = 'two_anov.dat' FIXED
  73       / Stu_Code  20-21
  74         Method       35
  75         Grd_CC       45
  76         Score     58-60
  77

This command will read 1 records from two_anov.dat

Variable   Rec   Start     End         Format

STU_CODE     1      20      21         F2.0
METHOD       1      35      35         F1.0
GRD_CC       1      45      45         F1.0
SCORE        1      58      60         F3.0

  78  Variable Labels
  79         Stu_Code   "Student Code"
  80       / Method     "Method:  Lecture, CBT, Video, IDS"
  81       / Grd_CC     "Graduated from Community College: Y or N"
  82       / Score      "Final Examination Score"
  83
  84  Value Labels
  85         Method     1 'Lecture: Traditional Lecture'
  86                    2 'CBT: Computer-Based Training'
  87                    3 'Video: Instructional Videotape'
  88                    4 'IDS: Independent Study'
  89
  90       / Grd_CC     1 'Grd_Yes:  Graduated from a CC'
  91                    2 'Grd_No :  Did NOT Graduate from a CC'
  92
  93  ANOVA Score BY Method(1,4) Grd_CC (1,2)
  94       / STATISTICS   = ALL
  95       / FORMAT       = LABELS
  96  COMMENT             = Please note in this analysis how I need to
  97                        identify which methods (1, 2, 3, and 4)
  98                        and community college status (1 and 2) to
  99                        analyze.
 100



            * * *  A N A L Y S I S   O F   V A R I A N C E  * * *

                 SCORE    Final Examination Score
            by   METHOD   Method:  Lecture, CBT, Video, IDS
                 GRD_CC   Graduated from Community College: Y or N

                 UNIQUE sums of squares
                 All effects entered simultaneously


                              Sum of                 Mean             Sig
Source of Variation          Squares     DF        Square       F    of F

Main Effects                5521.491      4      1380.373    18.308  .000
   METHOD                   5379.834      3      1793.278    23.784  .000
   GRD_CC                    160.120      1       160.120     2.124  .150

2-Way Interactions           177.981      3        59.327      .787  .505
   METHOD   GRD_CC           177.981      3        59.327      .787  .505

Explained                   5704.155      7       814.879    10.808  .000

Residual                    5051.632     67        75.397

Total                      10755.787     74       145.348


75 cases were processed.
0 cases (.0 pct) were missing.


************
two_anov.con
************
Outcome:     The SPSS Analysis of Variance output file for a 
             Twoway ANOVA is rather complex and at first it
             may appear somewhat difficult to interpret.  

             Look at the following edited section of the output 
             file to see just what you need to examine to 
             determine if the variables differ from sample 
             means and if there is any interaction between the
             variables:

                                                       Sig
             Source of Variation                 F    of F

             Main Effects                     
             METHOD                           23.784  .000
             GRD_CC                            2.124  .150

             2-Way Interactions                              
             METHOD   GRD_CC                    .787  .505

             Although you could compare the calculated F 
             statistics to criterion F statistics, it is
             usually easier and just as informative to use
             the probability (i.e., Significance of F) values
             to determine if differences exist.  In this
             example:

             METHOD (Instructional Method)

             -- The calculated Method p value is .000.

             -- The delcared Method p value is .05.

             The calculated p value is less than the declared p 
             value and there is, accordingly, a difference in 
             final examination scores based on instructional
             method.

             GRD_CC (Graduated from a Community College)

             -- The calculated Method p value is .150.

             -- The delcared Method p value is .05.

             The calculated p value exceeds the declared p 
             value value and there is, accordingly, no 
             difference in final examination scores based on 
             prior graduation from a community college.

             METHOD by GRD_CC (2-Way Interaction)

             -- The calculated Method p value is .505.

             -- The delcared Method p value is .05.

             The calculated p value exceeds the declared p 
             value value and there is, accordingly, no 
             interaction between instructional method and 
             prior graduation from a community college.


************
two_anov.lis
************
% minitab

 MTB > outfile 'two_anov.lis'
 Collecting Minitab session in file: two_anov.lis
 MTB > # MINITAB Addendum to 'two_anov.dat'
 MTB > #
 MTB > read 'two_anov.dat' c1 c2 c3 c4
 Entering data from file: two_anov.dat
      75 rows read.
 MTB > print c1 c2 c3 c4
 
 
  ROW    C1   C2   C3     C4
 
    1     1    1    1     89
    2     2    1    1     81
    3     3    1    2     73
    4     4    1    1     84
    5     5    1    2     70
    6     6    1    2     56
    7     7    1    1     70
    8     8    1    2     81
    9     9    1    2     78
   10    10    1    1     69
   11    11    1    1     89
   12    12    1    2     88
   13    13    1    2     45
   14    14    1    2     83
   15    15    1    1     95
   16    16    1    2     77
   17    17    1    1     69
   18    18    1    1     80
   19    19    2    2     93
   20    20    2    1     86
   21    21    2    1     89
 Continue? y
   22    22    2    2     95
   23    23    2    2     89
   24    24    2    1     88
   25    25    2    1     98
   26    26    2    1     89
   27    27    2    2     94
   28    28    2    1     95
   29    29    2    2     95
   30    30    2    2     98
   31    31    2    2     87
   32    32    2    2     85
   33    33    2    1     98
   34    34    2    1     93
   35    35    2    2     87
   36    36    2    1     95
   37    37    2    1     93
   38    38    2    2     93
   39    39    3    2     95
   40    40    3    1     96
   41    41    3    2     83
   42    42    3    2     89
   43    43    3    1     88
   44    44    3    1     87
 Continue? y
   45    45    3    1     94
   46    46    3    2     97
   47    47    3    1     95
   48    48    3    2     93
   49    49    3    2     85
   50    50    3    2     95
   51    51    3    1     92
   52    52    3    2     82
   53    53    3    1     86
   54    54    3    1     87
   55    55    3    2     89
   56    56    3    2     97
   57    57    3    1    100
   58    58    3    2     93
   59    59    3    1     96
   60    60    4    2     84
   61    61    4    1     85
   62    62    4    2     73
   63    63    4    1     92
   64    64    4    2     57
   65    65    4    1     63
   66    66    4    1     69
   67    67    4    2     73
 Continue? y
   68    68    4    2     91
   69    69    4    1     65
   70    70    4    1     74
   71    71    4    2     71
   72    72    4    2     68
   73    73    4    2     62
   74    74    4    1     56
   75    75    4    1     85
 
 MTB > # Before I attempt a TWOWAY ANOVA on this data set,
 MTB > # I first need to determine if the design is balanced
 MTB > # or unbalanced.
 MTB > #
 MTB > # Look carefully at the various groups and you will see
 MTB > # that the number of students in each teaching method
 MTB > # group is not consistent.  Because the numbers are
 MTB > # not consistent, this design is unbalanced.  The same
 MTB > # issue applies to the number of students with a prior
 MTB > # community college associate's degree.
 MTB > #
 MTB > # I will use MINITAB's help command to determine the
 MTB > # proper command for a TWOWAY ANOVA on an unbalanced
 MTB > # design.
 MTB > help
 * You are using MINITAB Statistical Software, Standard Version *
 
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          20 Enhanced Statistical Process Control
          21 Graphical Options for Control Charts
          22 Analysis of Means
          23 Design and Analysis of Experiments
  
 MTB > help commands 8
   Analysis of Variance
  
   AOVONEWAY.....does one way analysis of variance, with each
                 group in separate columns
   ONEWAYAOV.....does one way analysis of variance, with the
                 response in one column, subscripts in another
   TWOWAYAOV.....does balanced two way analysis of variance
   ANOVA.........does univariate and multivariate analysis of
                 variance with balanced designs
   ANCOVA........analyzes orthogonal designs (including latin
                 squares and crossover designs) with crossed
                 and nested factors and additive covariates
   GLM...........does univariate and multivariate analysis
                 of variance with balanced and unbalanced
                 designs, analysis of covariance, and regression
   NESTED........experimental command that analyzes fully nested
                 (hierarchical) designs
   INDICATOR.....creates indicator or dummy variables
  
 MTB > # And I see that glm is used when the design is unbalanced.
 MTB > #
 MTB > # FYI ... the two options here are:
 MTB > #
 MTB > # -- for a balanced design        anova c4 = c2 | c3
 MTB > # -- for an unbalanced design     glm   c4 = c2 | c3
 MTB > # 
 MTB > # I recommend that you stay away from the standard command of
 MTB > # twoway c4 c2 c3 when dealing with a balanced design.
 MTB > #
 MTB > # If you use this command, you will need to do manual
 MTB > # calculations to obtain F values.
 MTB > #
 MTB > # I will now use glm on this unbalanced design.
 MTB > #
 MTB > glm c4 = c2 | c3
 
 Factor   Levels Values
 C2            4     1     2     3     4
 C3            2     1     2
 
 
 Analysis of Variance for C4      
 
 Source     DF     Seq SS     Adj SS     Adj MS       F      P
 C2          3    5372.33    5379.83    1793.28   23.78  0.000
 C3          1     153.84     160.12     160.12    2.12  0.150
 C2*C3       3     177.98     177.98      59.33    0.79  0.505
 Error      67    5051.63    5051.63      75.40
 Total      74   10755.79  
 
 
 Unusual Observations for C4      
 
 Obs.       C4       Fit Stdev.Fit  Residual   St.Resid
  13    45.000    72.333     2.894   -27.333     -3.34R 
  63    92.000    73.625     3.070    18.375      2.26R 
  68    91.000    72.375     3.070    18.625      2.29R 
  74    56.000    73.625     3.070   -17.625     -2.17R 
 
 Continue? y
 R denotes an obs. with a large st. resid.
 
 MTB > # And you will notice that the significance of F (or
 MTB > # the p values in MINITAB's printout) are the same
 MTB > # as what you previously saw in the SPSS printout:
 MTB > #
 MTB > # -- Method p = .000
 MTB > # -- Grd_CC p = .150
 MTB > # -- 2-Way Interaction (Method * Grd_CC) p = .505
 MTB > #
 MTB > # Let me demonstrate the use of the twoway command on 
 MTB > # this unbalanced design, just to show how the analysis
 MTB > # will not continue.
 MTB > # 
 MTB > anova c4 = c2 | c3
 * ERROR * Unequal cell counts.
 
 MTB > stop


--------------------------
Disclaimer:  All care was used to prepare the information in this 
tutorial.  Even so, the author does not and cannot guarantee the 
accuracy of this information.  The author disclaims any and all 
injury that may come about from the use of this tutorial.  As 
always, students and all others should check with their advisor(s) 
and/or other appropriate professionals for any and all assistance 
on research design, analysis, selected levels of significance, and 
interpretation of output file(s).

The author is entitled to exclusive distribution of this tutorial. 
Readers have permission to print this tutorial for individual use, 
provided that the copyright statement appears and that there is no 
redistribution of this tutorial without permission.

Prepared 980316
Revised  980914
end-of-file 'two_anov.ssi'

Please send comments or suggestions to Dr. Thomas W. MacFarland

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