How are Polynomial contrasts computed

[William Dudley WNDUDLEY]

I am struggling with a repeated measures model in which I have run growth curve analyses AND repeated measures ANOVA.
In both the growth curve model and ANOVA model I obtain significant linear and quadratic terms.
I understand what the terms mean in the growth model but I do not understand how the contrasts are arrived at  in the ANOVA


Can someone enlighten me?

--
William N. Dudley, PhD
Associate Dean for Research
The School of Health and Human Performance Office of Research
The University of North Carolina at Greensboro
126 HHP Building, PO Box 26170
Greensboro, NC 27402-6170
VOICE 336.2562475
FAX [phone]

[Bruce Weaver]

William Dudley WNDUDLEY wrote

I am struggling with a repeated measures model in which I have run growth
curve analyses AND repeated measures ANOVA.
In both the growth curve model and ANOVA model I obtain significant linear
and quadratic terms.
I understand what the terms mean in the growth model but I do not understand
how the contrasts are arrived at  in the ANOVA


Can someone enlighten me?

If you're asking where the sums of squares for the linear and quadratic contrasts come from, they are linear combinations of the means, with coefficients taken from a table of Orthogonal Polynomial Coefficients, like this one:

   http://www.gseis.ucla.edu/courses/help/op.html

But I'm not sure if this is what you are driving at.

[Mike]

On Thursday, July 15, 2010 9:53 AM, Bruce Weaver wrote:

> William Dudley WNDUDLEY wrote:
>> I am struggling with a repeated measures model in which I have run growth
>> curve analyses AND repeated measures ANOVA.
>> In both the growth curve model and ANOVA model I obtain significant linear
>> and quadratic terms.
>> I understand what the terms mean in the growth model but I do not
>> understand
>> how the contrasts are arrived at  in the ANOVA
>>
>> Can someone enlighten me?
>
> If you're asking where the sums of squares for the linear and quadratic
> contrasts come from, they are linear combinations of the means, with
> coefficients taken from a table of Orthogonal Polynomial Coefficients, like
> this one:
>
>   http://www.gseis.ucla.edu/courses/help/op.html
>
> But I'm not sure if this is what you are driving at.

The original question is a bit ambiguous because one has to infer
that the question is about orthogonal polynomial analysis.  Background
to selection of weights and their interpretation is given in some detail
in Maxwell & Delaney's (2004) Designing Experiments and Analyzing
Data (2nd ed) Chapter 4 "Trend Analysis" but many forego the background
and make use of standard tables that contain coefficients that are to be
used for different set sizes of means (e.g., Hays' Statistics text 2nd ed,
bypasses the rationale and goes directly to a worked example).  I assumed
that the original poster was familiar with this background but was asking
specifically about how SPSS decides which weights to use, that is:

(a) SPSS uses traditional weights like that at the UCL:A website and standard
textbooks (e.g., Maxwell & Delaney)
or
(b) some SPSS specific (idiosyncratic?) weight determination that differs
from the standard sources.

I suspect that one of the SPSS statisticians know the answer.

-Mike Palij
New York University
[hidden email]

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[Anthony Babinec]

Not sure whether this is your question but: In GLM,

on the Options Button, check the Transform box.

 

The contrast coefficients are typically orthonormalized:

 

sum to 0

sum of squares is 1

crossproduct sum is 0.

 

Typically, you would find the Anova table and look for

the single-degree-of-freedom polynomial terms.

 

Tony Babinec

[hidden email]

 

[Bruce Weaver]

Mike Palij wrote

On Thursday, July 15, 2010 9:53 AM, Bruce Weaver wrote:
> William Dudley WNDUDLEY wrote:
>> I am struggling with a repeated measures model in which I have run growth
>> curve analyses AND repeated measures ANOVA.
>> In both the growth curve model and ANOVA model I obtain significant linear
>> and quadratic terms.
>> I understand what the terms mean in the growth model but I do not
>> understand
>> how the contrasts are arrived at  in the ANOVA
>>
>> Can someone enlighten me?
>
> If you're asking where the sums of squares for the linear and quadratic
> contrasts come from, they are linear combinations of the means, with
> coefficients taken from a table of Orthogonal Polynomial Coefficients, like
> this one:
>
>   http://www.gseis.ucla.edu/courses/help/op.html
>
> But I'm not sure if this is what you are driving at.

The original question is a bit ambiguous because one has to infer
that the question is about orthogonal polynomial analysis.  Background
to selection of weights and their interpretation is given in some detail
in Maxwell & Delaney's (2004) Designing Experiments and Analyzing
Data (2nd ed) Chapter 4 "Trend Analysis" but many forego the background
and make use of standard tables that contain coefficients that are to be
used for different set sizes of means (e.g., Hays' Statistics text 2nd ed,
bypasses the rationale and goes directly to a worked example).  I assumed
that the original poster was familiar with this background but was asking
specifically about how SPSS decides which weights to use, that is:

(a) SPSS uses traditional weights like that at the UCL:A website and standard
textbooks (e.g., Maxwell & Delaney)
or
(b) some SPSS specific (idiosyncratic?) weight determination that differs
from the standard sources.

I suspect that one of the SPSS statisticians know the answer.

-Mike Palij
New York University
[e-mail]

From the Help files:

Polynomial contrasts. This is the default for within-subjects factors. The first degree of freedom contains the linear effect across the levels of the factor, the second contains the quadratic effect, and so on. In a balanced design, polynomial contrasts are orthogonal. By default, the levels are assumed to be equally spaced; you can specify unequal spacing by entering a metric consisting of one integer for each level of the factor in parentheses after the keyword POLYNOMIAL. (All metrics specified cannot be equal; thus (1,1,...,1) is not valid.) For example,  
  /WSFACTOR=D 3 POLYNOMIAL(1,2,4). 
  Suppose that factor D has three levels. The specified contrast indicates that the three levels of D are actually in the proportion 1:2:4. The default metric is always (1,2,...,k), where k levels are involved. Only the relative differences between the terms of the metric matter (1,2,4) is the same metric as (2,3,5) or (20,30,50) because, in each instance, the difference between the second and third numbers is twice the difference between the first and second.

--- End of Help File excerpt ---

I didn't know about the option for specifying something other than equal spacing between levels.  Is that something (relatively) new?  

[Mike]

Just a couple of notes:

(1)  Unequal spacing for within-subject orthogonal polynomial analysis
is an old feature, being available in the MANOVA procedure as specified
in the "Big White SPSS Manual" (1990) -- see page 370.  Sure beats
doing it by hand.

(2)  Othogonal polynomial analysis can also be used in between-subjects
designs and is an option in the ONEWAY procedure.  This might be
used, say, in an experimental study where different groups received
different amounts of a drug treatment (e.g., 0mg, 10mg, 20mg, 40mg).
The Old White manual provides info on doing polynomial analysis
in this situation but (a) it is unclear how to specific unequal spacing,
and (b) these designs can be unbalanced (i.e., nonconstant sample
sizes) and along with unequal variances (and apparently unequal
spacing) makes the analysis nonorthogonal as well as introducing
other difficulties.  Maxwell & Delaney deal with unequal sample sizes
in their Chap 6, pp267-269.

-Mike Palij
New York University
[hidden email]



----- Original Message -----
From: "Bruce Weaver" <[hidden email]>
To: <[hidden email]>
Sent: Thursday, July 15, 2010 10:32 AM
Subject: Re: How are Polynomial contrasts computed

> Mike Palij wrote:
>>
>> On Thursday, July 15, 2010 9:53 AM, Bruce Weaver wrote:
>>> William Dudley WNDUDLEY wrote:
>>>> I am struggling with a repeated measures model in which I have run
>>>> growth
>>>> curve analyses AND repeated measures ANOVA.
>>>> In both the growth curve model and ANOVA model I obtain significant
>>>> linear
>>>> and quadratic terms.
>>>> I understand what the terms mean in the growth model but I do not
>>>> understand
>>>> how the contrasts are arrived at  in the ANOVA
>>>>
>>>> Can someone enlighten me?
>>>
>>> If you're asking where the sums of squares for the linear and quadratic
>>> contrasts come from, they are linear combinations of the means, with
>>> coefficients taken from a table of Orthogonal Polynomial Coefficients,
>>> like
>>> this one:
>>>
>>>   http://www.gseis.ucla.edu/courses/help/op.html
>>>
>>> But I'm not sure if this is what you are driving at.
>>
>> The original question is a bit ambiguous because one has to infer
>> that the question is about orthogonal polynomial analysis.  Background
>> to selection of weights and their interpretation is given in some detail
>> in Maxwell & Delaney's (2004) Designing Experiments and Analyzing
>> Data (2nd ed) Chapter 4 "Trend Analysis" but many forego the background
>> and make use of standard tables that contain coefficients that are to be
>> used for different set sizes of means (e.g., Hays' Statistics text 2nd ed,
>> bypasses the rationale and goes directly to a worked example).  I assumed
>> that the original poster was familiar with this background but was asking
>> specifically about how SPSS decides which weights to use, that is:
>>
>> (a) SPSS uses traditional weights like that at the UCL:A website and
>> standard
>> textbooks (e.g., Maxwell & Delaney)
>> or
>> (b) some SPSS specific (idiosyncratic?) weight determination that differs
>> from the standard sources.
>>
>> I suspect that one of the SPSS statisticians know the answer.
>>
>> -Mike Palij
>> New York University
>> [hidden email]
>>
>>
>
> From the Help files:
>
> Polynomial contrasts. This is the default for within-subjects factors. The
> first degree of freedom contains the linear effect across the levels of the
> factor, the second contains the quadratic effect, and so on. In a balanced
> design, polynomial contrasts are orthogonal. By default, the levels are
> assumed to be equally spaced; you can specify unequal spacing by entering a
> metric consisting of one integer for each level of the factor in parentheses
> after the keyword POLYNOMIAL. (All metrics specified cannot be equal; thus
> (1,1,...,1) is not valid.) For example,
>  /WSFACTOR=D 3 POLYNOMIAL(1,2,4).
>  Suppose that factor D has three levels. The specified contrast indicates
> that the three levels of D are actually in the proportion 1:2:4. The default
> metric is always (1,2,...,k), where k levels are involved. Only the relative
> differences between the terms of the metric matter (1,2,4) is the same
> metric as (2,3,5) or (20,30,50) because, in each instance, the difference
> between the second and third numbers is twice the difference between the
> first and second.
>
> --- End of Help File excerpt ---
>
> I didn't know about the option for specifying something other than equal
> spacing between levels.  Is that something (relatively) new?
>
>
>
> -----
> --
> Bruce Weaver
> [hidden email]
> http://sites.google.com/a/lakeheadu.ca/bweaver/
>
> "When all else fails, RTFM."
>
> NOTE: My Hotmail account is not monitored regularly.
> To send me an e-mail, please use the address shown above.
>
> --
> View this message in context: http://spssx-discussion.1045642.n5.nabble.com/How-are-Polynomial-contrasts-computed-tp1221431p1223540.html
> Sent from the SPSSX Discussion mailing list archive at Nabble.com.
>
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