Date:         Wed, 12 Feb 1997 11:37:49 -0600
Reply-To:     "Nichols, David" <nichols@SPSS.COM>
Sender:       "SPSSX(r) Discussion" <SPSSX-L@UGA.CC.UGA.EDU>
From:         "Nichols, David" <nichols@SPSS.COM>
Subject:      Re: Adjusted verses Estimated means in ANCOVA

>----------
>From:  David Kurzman[SMTP:DAVEK@VAX2.CONCORDIA.CA]
>Sent:  Wednesday, February 12, 1997 12:18 AM
>To:    Multiple recipients of list SPSSX-L
>Subject:       Adjusted verses Estimated means in ANCOVA
>
>Hi there,
>
>When running repeated measures with covariates, I asked for the predicted
>means.
>
>Since the (varying) covariates were significant across time, I wanted to
>present in a table the observed means along with the means
>that are obtained when controlling for the covariate.  The problem is that
>the
>output of predicted (adj vs. est means) are
>uninterpretable.  Am i thinking about this in the wrong way?
>
>Thanks Help is always Aprreciated!
>
>Dave Kurzman
>
> Combined Observed Means for TRAIN
> Variable .. B1L1TRIL
>         TRAIN
>          TEXT             WGT.    61.54167
>                    UNWGT.           61.54167
>      MULTIMED        WGT.    61.68182
>                    UNWGT.            61.82500
>
>
> Adjusted and Estimated Means
> Variable .. T2
>       Factor          Code                  Obs. Mean     Adj. Mean     Est.
>Mean    Raw Resid.   Std. Resid.
>
>
>  TRAIN           TEXT
>   TYPE2           CONSISTE                 -4.29543    -307.11953
>-257.20714     252.91172      37.45693
>   TYPE2           VARIABLE                 -2.66158      85.77838
>250.25013    -252.91172     -37.45693
>
>  TRAIN           MULTIMED
>   TYPE2           CONSISTE                 -3.95285     -94.03544
>299.54121    -303.49406     -44.94831
>   TYPE2           VARIABLE                 -5.45493     298.86247
>-258.36665     252.91172      37.45693
>
>************************************************************
>* David Kurzman                                            *
>* Centre for Research in Human Development                 *
>* Concordia University                                     *
>* Montreal, Quebec, Canada                                 *
>************************************************************
>************************************************************
>* Department of Psychology                                 *
>* Ottawa General Hospital                                  *
>* Ottawa, Ontario, Canada                                  *
>************************************************************
>************************************************************
>* 915 Elmsmere Rd. #504                                    *
>* Gloucester, Ontario   K1J 8H8                            *
>* 613-744-6409                                             *
>* HTTP://VAX2.CONCORDIA.CA/~DAVEK  <- Comments Appreciated *
>************************************************************
>
>When you use the WSFACTORS subcommand in MANOVA (which is what is done when
>you run repeated measures designs), the procedure creates a set of
>transformed
>variables and another set of transformed covariates, and the PMEANS output is
>in terms of these transformed variables. The first of these is named T1 by
>default (unless a RENAME subcommand is used), and it is a normalized sum of
>the dependent variables. It's values thus generally fall above the range
>expected for adjusted means for the original variables, leading to the kind
>of
>confusion noted here. The other transformed variables (T2, T3, etc., by
>default) are normalized contrasts among the original dependent variables, and
>thus their values generally fall below the range expected for adjusted means
>of the original variables. The values produced by PMEANS are valid, but are
>in
>the metric of these transformed variables, not the original variables.
>
>In order to express the PMEANS results in the original variables metric, a
>back-transformation is required. The transformed variables have been created
>by multiplying the matrix of original variable values by a matrix we'll call
>T. In order to back-transform the means of variables in this new metric, we
>need to multiply them by the inverse of T. Since T has a special form (it's
>an orthonormal matrix), it happens that it's inverse is also it's transpose,
>so in order to back-transform these means, all we need do is to multiply the
>PMEANS values by T', the transpose of T. T' is actually printed for us in
>the MANOVA output if we request it via the PRINT subcommand
>(PRINT=TRANSFORM).
>This is labeled "Orthonormalized Transformation Matrix (Transposed)" on the
>output. Actually, all we need to use is the upper left hand square block of
>the printed matrix, of order corresponding to the number of dependent
>variables in our analysis (if there are K dependent variables, we use the
>upper left hand KxK piece--if there are multiple measures, this same piece
>is applied to each set of variables comprising a measure).
>
>Here's an example of what the upper left hand block of a transposed
>transformation matrix looks like for three dependent variables with
>polynomial contrasts:
>
>                    T1         T2         T3
>
> VAR1             .577      -.707       .408
> VAR2             .577       .000      -.816
> VAR3             .577       .707       .408
>
>VAR1-VAR3 are the original dependent variables and T1-T3 are the transformed
>variables. This matrix can be read as T1 = .577*VAR1 + .577*VAR2 + .577*VAR3,
>T2 = -.707*VAR1 + .707*VAR3, and T3 = .408*VAR1 - .816*VAR2 + .408*VAR3. It
>can also be read the other way, as VAR1 = .577*T1 - .707*T2 + .408*T3,
>VAR2 = .577*T1 - .816*T3, and VAR3 = .577*T1 + .707*T2 + .408*T3. The last
>three equations can be used to back-transform the PMEANS results. Or, one can
>set up a matrix equation, with each cell of the between subjects design as a
>column in a second matrix, and premultiply this matrix of adjusted means (the
>rows correspond to T1, T2 and T3) by the above transposed transformation
>matrix.
>
>David Nichols
>Senior Support Statistician
>SPSS Inc.
>nichols@spss.com
>
>
>