[R-sig-ME] Interpreting lmer() interactions with Helmert contrasts

Becky Gilbert beckyannegilbert at gmail.com
Wed Aug 26 16:51:07 CEST 2015

Hi Steven,

Thanks very much for the clear and helpful explanation.  Thanks also for
highlighting the difference between the two cases.

In case you (or anyone else) is interested, neither factor is a nuisance
variable.  I'm interested in the effects of both Time and Word Type, and
especially their interaction.  We were predicting:

1. RTs for the two trained Word Types (related and unrelated) would
increase at Time 1 (post-training) vs Time -1 (pre-training), but RTs for
the untrained Word Type would be the same for both levels of Time
2. RTs for the unrelated Word Type might show a larger effect of time (i.e.
increase at Time 1) compared to the related Word Type (this is the main
research question)

This goes back to the original question I posted to the list - I tried to
use Helmert contrasts for Word Type to get separate interaction terms
relating to each prediction above, but couldn't work out how to get LRTs
for the two interactions terms (because removing the WordType:Time term
removes the interactions for both WordType contrasts at the same time).
I've since tried explicitly coding the two WordType contrasts as separate
variables (rather than using contrasts() on a single variable) so that I
can remove the interactions with each contrast separately and then assess
their contributions to the model.  However I'm not sure whether this
solution is correct/ideal.


On 24 August 2015 at 19:22, Steven McKinney <smckinney at bccrc.ca> wrote:

> Hi Becky,
> For a model containing A + B + A:B we have two situations
> Case 1)  Interest in A, but the need to have B in the model (B's parameter
> is a nuisance parameter in the model - B needs to be in the model do adjust
> for an important factor so that the model behaves properly, but B is not
> the factor we are interested in testing).  This is what I saw as the
> relevant situation in your case (you seemed to want to test Time, while
> adjusting for WordType).
> Case 2)  Interest in the relevance of both A and B  (discussed in the
> dialog to which you linked below)
> The discussion you link to below provides this hierarchy of models
> l.full = lmer(response ~ A + B + A:B + (1 + A | sub) + (1 | item), data,
> family="binomial")
> l.AB = lmer(response ~ A + B + (1 + A | sub) + (1 | item), data,
> family="binomial")
> l.A = lmer(response ~ A + (1 + A | sub) + (1 | item), data,
> family="binomial")
> l.B = lmer(response ~ B + (1 | sub) + (1 | item), data, family="binomial")
> but omits
> l.reduced = lmer(response ~ (1 | sub) + (1 | item), data,
> family="binomial")
> i.e  the model with neither A nor B.
> Case 1)  If we are interested in A, the omnibus test is
> anova (l.B, l.full)
> If this test is significant, and the contribution of A to the model is of
> a size of scientific relevance, then you can declare A as a significant
> model component and begin to investigate the functional form of that
> contribution.
> The next step would be to test the interaction.  If that is significant
> and of relevant scientific size, then A is important, but its contribution
> differs for different levels of B.  If the interaction is not significant,
> and the sample size was large enough to detect differences of importance,
> then the interaction term can be dropped and the main effects model best
> summarizes the association.
> Case 2) If we are interested in both A and B, the omnibus test for their
> joint relevance is
> anova( l.reduced, l.full ).  This test was not discussed in the link you
> provided.
> If this test yields a non-significant p-value, then A and B are not
> contributing to improving the model fit and their usefulness is
> questionable if the data set size was large enough to detect effect sizes
> of scientific relevance.
> If the p-value is small, then of course we need to assess whether the
> improved model fit is telling us anything of scientific value.  (All
> p-values get small when data set sizes get large - so then the question is
> the relevance of the degree of association.)
> The problem with the discussion you link to is that two test results were
> posited to assess the relevance of both A and B
> "if anova(l.full, l.A) is significant, B has an effect (main effect or
> interaction).
> if anova(l.full, l.B) is significant, A has an effect (main effect or
> interaction)."
> so there's two tests and no discussion of adjustment for multiple
> comparisons.  The omnibus test anova( l.reduced, l.full ) tests both A and
> B simultaneously in one test at the stated type I error rate.  If that test
> is significant, and the effect sizes of A and B in the model are more than
> just trivially small differences of no scientific or biological or medical
> relevance, then you can start assessing the nature of their joint
> contribution to the model.  The first thing to look at would then be the
> interaction term.  If that is significant, and of a relevant size, you are
> done.  Both A and B are important, but the contribution of A depends on the
> level of B.  If the interaction is not significant, then you can look at A
> and B individually and see which is contributing to the model fit at a
> level of scientific or biological or medical relevance.
> Steven McKinney, Ph.D.
> Statistician
> Molecular Oncology and Breast Cancer Program
> British Columbia Cancer Research Centre
> email: smckinney +at+ bccrc +dot+ ca
> tel: 604-675-8000 x7561
> Molecular Oncology
> 675 West 10th Ave, Floor 4, Room 4.122
> Vancouver B.C.
> V5Z 1L3
> Canada
> ________________________________
> From: Becky Gilbert <beckyannegilbert at gmail.com>
> Sent: August 24, 2015 4:54 AM
> To: Steven McKinney
> Cc: Ken Beath; Dan McCloy; r-sig-mixed-models at r-project.org
> Subject: Re: [R-sig-ME] Interpreting lmer() interactions with Helmert
> contrasts
> Thanks very much everyone for the responses.
> @Dan: Thank you for the recommendation about my factor contrast
> coefficients.  I hadn't given much thought to the sign/level association,
> but now that you point it out, it seems obvious that I should do it the way
> you describe.  Here are the model coefficients with recoded contrasts:
> > contrasts(rtData$Time)
>     [,1]
> -1 -0.5  # pre-test
> 1   0.5  # post-test
> > contrasts(rtData$WordType)
>         [,1]        [,2]
> 0 -0.6666667  0.0  # untrained
> 1  0.3333333  0.5  # trained-related
> 2  0.3333333 -0.5  # trained-unrelated
>                               Estimate     Std. Error    t value
> (Intercept)               2.8765116  0.0177527  162.03
> WordType1            -0.0111628  0.0110852   -1.01
> WordType2            -0.0007306  0.0071519   -0.10
> Time1                    0.0268310  0.0195248    1.37
> WordType1:Time1   0.0301627  0.0115349    2.61
> WordType2:Time1  -0.0089123  0.0141624   -0.63
> My interpretations of the interaction coefficients are:
> 1) log RT increases (i.e. RTs slow down) for the two trained (vs
> untrained) Word Types at post-test (Time = 1)
> 2) log RT decreases (i.e. RTs speed up) for the trained-related (vs
> trained-unrelated) Word Type at post-test (Time = 1)..
> However, this doesn't really answer my original question about how to
> assess (and report) the contribution of these two interactions to the model
> fit.  Obviously the t statistic is larger for the Time1:WordType1 compared
> to the Time1:WordType2 interaction coefficients, but that only tells me
> their relative contributions - I would need to know degrees of freedom to
> get p-values, which I understand is not straightforward.  Also, I've read
> that the t statistics for coefficients that are output by summary() for an
> lmer model are sequential tests and thus not the appropriate/desired
> statistics for assessing the contribution of factors (someone please
> correct me if I'm wrong!).  Hence the reason for using LRT to assess this.
> This still leaves me with the problem of not being able to test the
> interactions between Time and the two contrasts for WordType - I can test
> the whole WordType factor and Time:WordType interaction via LRTs, but not
> each contrast within WordType.
> @Steven: thanks for your explanation re interpreting main effects in the
> presence of an interaction, and of the Chi-square LRTs for assessing the
> contribution of factors/terms.
> However I'm confused by this:
> An omnibus test for the statistical significance of a variable of interest
> (say variable A), when that variable is in a model involving an interaction
> with another variable (say variable B) will test the interaction term A:B
> and the main effect A.  The full model has A + B + A:B and the reduced
> model has only B.  Thus a proper omnibus test for the usefulness of A in
> the model will involve the interaction A:B and the main effect A.  This
> test really should be done before testing A:B for proper multiple
> comparisons control.
> Is this what you're saying?
> 1. test A: (A + B + A:B) vs (B)
> 2. test B: (A + B + A:B) vs (A)
> then, if either of the above are significant:
> 3. test A:B: (A + B + A:B) vs (A + B)
> Which I think is the procedure described here:
> https://mailman.ucsd.edu/pipermail/ling-r-lang-l/2011-October/000305.html
> Assuming this is what you meant, will this procedure always get you to
> step 3 (assessing the interaction) in the case of a significant interaction
> without main effects (as in a cross-over interaction).  Sorry if I've
> completely misunderstood!
> Becky
> ____________________________________________
> Dr Becky Gilbert (nee Prince)
> http://www.york.ac.uk/psychology/staff/postgrads/becky.gilbert/
> http://www.researchgate.net/profile/Becky_Gilbert2
> http://twitter.com/BeckyAGilbert
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