MINITAB RELEASE 7.2 *** IBM mainframe running VM/CMS
 (C) Copyright 1989 Minitab, Inc. - ALL RIGHTS RESERVED
 U.S. FEDERAL GOVERNMENT USERS SEE HELP FGU
 FEB. 18, 1992 *** University of Illinois - Chicago
 
 MTB > help ancova
 ANCOVA  model
 
 Subcommands:  COVARIATES    RESIDUALS    TEST
               FITS          MEANS
 
 This command does analysis of variance and covariance for
 orthogonal designs (e.g., latin squares, crossover and
 proportional designs) with fixed effects.  Factors may be
 crossed or nested.  You can store residuals and fitted values,
 print cell and marginal means (adjusted for the covariates), and
 specify your own tests. You may list several response variables
 on one command.
 
 You may specify up to 50 response variables and 50 covariates.
 Models with many terms can take a long time to compute.  One
 rule of thumb is that, in a completely crossed model with all
 interaction terms, each additional factor triples the
 computation time.
 
 
 How to Enter Data for ANCOVA
 
 There should be one row of the worksheet for each observation,
 containing the values of each response variable, the level of
 each factor, and the values of each covariate.  Level numbers
 must be integers from -9999 to +9999 or missing (*).  They need
 not be consecutive nor in any special order.  The patterned data
 capability of SET can be helpful in entering the level numbers
 of a factor.
 
 
 How to Specify the Model in ANCOVA
 
 The model on the ANCOVA line is exactly the same as in the ANOVA
 command.  Covariates are listed in a subcommand.
 
 The example below has two covariates, X1 and X2, and two crossed
 factors, A and B.
 
    ANCOVA Y=A B A*B
      COVARIATES X1 X2.
 
1
 MTB > help anova
 ANOVA  model
 
 Subcommands:  RANDOM    FITS    MEANS      RESTRICT
               EMS       TEST    RESIDUALS
 
 This command performs analysis of variance for multi-way
 balanced designs, i.e., each cell must have the same number of
 observations, and one-way analysis of variance for unbalanced
 designs.
 
 ANOVA calculates all exact F-tests, prints expected mean
 squares, and estimates variance components.  You may specify
 your own tests, store residuals and fitted values, and print
 cell and marginal means.  You can list several response
 variables on one command.
 
 
 How to Specify the Model in ANOVA
 
 ANOVA uses a simplified version of a model as it appears in many
 textbooks.  Here are some examples:
 
    Three factors crossed:      ANOVA Y=A B C A*B A*C B*C A*B*C
    Three factors nested:       ANOVA Y=A B(A) C(A B)
    Crossed and nested design:  ANOVA Y=A B(A) C A*C B*C(A)
 
 List the terms you want in your model after the equal sign.
 Interactions are indicated with asterisks.  For example, A*B is
 the interaction between factors A and B.  Nested factors are
 indicated with parentheses.  For example, B(A) is B nested
 within A and C(A B) is C nested within A and B.
 
 Several rules apply only to ANOVA (and ANCOVA).  You may omit
 the quotes around variable names.  Because of this, variable
 names used in ANOVA must start with a letter and contain only
 letters and numbers.  Alternatively, you can use C1, C2, etc. to
 denote data columns.  You can use special symbols in a variable
 name, but then you must enclose the name in single quotes, as on
 other Minitab commands.  You may not put any extra text on the
 ANOVA line, except after the symbol #.
 
 
 Two symbols allow you to abbreviate a model.  A vertical bar (or
 exclamation point) indicates crossed factors, and a minus sign
 removes terms.  For example:
 
    ANOVA Y=A B|C E                   is equivalent to
    ANOVA Y=A B C B*C E
 
    ANOVA Y=A|B|C - A*B               is equivalent to
    ANOVA Y=A B C A*C B*C A*B*C
 
    ANCOVA Y=A|B(A)|C                 is equivalent to
    ANCOVA Y=A B(A) C A*C B*C(A)
 
 In general, all crossings are done for factors separated by
 bars, unless the cross results in an illegal term.  For example,
 in the third example, the potential term A*B(A) is illegal and
 Minitab automatically omits it.  Note, if a factor is nested
 then you must indicate this when using the bar, as in the fourth
 example with the term B(A).
1
 You can fit reduced models, for example:  Y = A B C A*B is a
 three-factor model with just one two-way interaction.  Models,
 however, must be hierarchical.  For example, if the term A*B*C
 is in the model, then the terms A B C A*B A*C B*C must also be
 in the model.  And if B(A) is in the model, then A must be also.
 
 Several response variables can be included with one model.  For
 example, ANOVA Y1 Y2 Y3 = A B will do three separate analyses,
 one for Y1, one for Y2 and one for Y3.
 
 Minitab checks to see if your model is valid and gives an error
 message if it is not.  Minitab also checks to see if your data
 set is balanced, i.e. has an equal number of observations per
 cell.  Note:  balanced data are not required for one factor
 models.
 
 Models with many terms can take a long time to compute.  One
 rule of thumb is that, in a completely crossed model with all
 interaction terms, each additional factor triples the
 computation time.
 
 
 A Simple Example:
 
 We fit a two-way crossed model using the potato rot data from
 the Minitab Handbook.  Potato rot was measured in potatoes
 stored at two different temperatures (factor 1) and at three
 levels of oxygen (factor 2).
 
    MTB > ANOVA ROT = TEMP|OXYGEN;
    SUBC>  MEANS TEMP|OXYGEN;
    SUBC>  FITS C5;
    SUBC>  RESIDUALS C6.
 
  MTB > help ancova mean
  MEANS  termlist
 
  Termlist contains a list of terms in the model, main effects
  and/or interactions.  List them just as you did for the model on
  ANCOVA.  Minitab will print a table of adjusted means
  corresponding to each term.
 
  If two MEANS subcommands are specified, the second overrides the
  first.
 1
  MTB > help ancova test
  TEST  termlist / errorterm
 
  All tests that ANCOVA does automatically are of the form (MS
  term)/(MSE).  If all your factors are fixed, this is probably
  what you want.  However, if some of your factors are random or
  if you have certain designs, e.g., a split plot, these tests may
  not be appropriate.  The TEST subcommand allows you to specify
  your own tests.
 
  Termlist is a list of terms from the model.  List them exactly
  as you did on the ANCOVA line.  Errorterm is a term in the model
  to be used as the denominator for the F-test.  Or alternatively,
  errorterm can be a linear combination of terms in the model and
  the reserved word, ERROR, can be used to represent MSE.  Minitab
  will then calculate the approximate F-ratio and p-values for
  you.  See the ANOVA TEST subcommand for more details.
 
  MTB > help anova test
  TEST  termlist / errorterm
 
  This subcommand is useful when you want to do F-tests that
  Minitab does not do automatically, e.g., to synthesize F-tests.
  Termlist is a list of one or more terms in the model for which
  you want to construct F-tests.  List these just as you did on
  the model.  Errorterm is a term in the model to be used as the
  denominator in the F-tests.  Alternatively, errorterm can be a
  linear combination of terms from the model and the reserved
  word, ERROR, can be used to represent MSE.  Minitab will
  calculate the F-ratios and p-values for you.  For example:
 
     TEST A / A*B + A*C + B*C - 2*A*B*C - 2*ERROR
 
  The program constructs the synthetic denominator MS for the
  F-test.  This denominator MS is the linear combination of the
  MS's provided with the TEST subcommand, and should have the same
  expectation as the numerator MS under the hypothesis being
  tested.  The synthetic denominator has approximate degrees of
  freedom calculated by the formula:
 
                             ( MS(denom.) )**2
         DF(denom.) = --------------------------------
                         SUM( (Ai*MS(i))**2/DF(i) )
 
  where MS(denom.) = SUM( Ai*MS(i) ), and the MS(i) are the MS's
  for terms in the model with degrees of freedom DF(i).
 
 19-Feb-92