[R-sig-ME] Random effects in multinomial regression in R?
Souheyla GHEBGHOUB
@ouhey|@@ghebghoub @end|ng |rom gm@||@com
Tue Mar 26 14:09:00 CET 2019
Thank you very much Rene for this feedback.
I will surely consider it.
Best,
Souheyla
On Sun, 24 Mar 2019, 13:39 René, <bimonosom using gmail.com> wrote:
> Ok. I try a short one :)
> Also going back to your first mail:
>
> This model is what you want:
>
> score~Time*Group+(1+Time|subject)+(1+Time*Group | words)
>
> No skew in here ;) (but maybe model identification issues depending on the
> number of observations you got.) In other words, (random) by-word
> variations and (random) by-participant variations are taken into account,
> and the "average across stimuli" problem articulated here (Judd, Westfall,
> & Kenny, 2012) is appropriately tackled. As you do not explicitly reference
> to this paper, I just assume that you mean this kind of "skew" (alpha
> inflation due to stimulus aggregation), because I am currently not aware of
> other theoretically or statistically relevant 'skews' there could be.
>
> Best, René
>
> Ps: Why not using brms... :) You get parameter likelihood estimates and
> Bayesian credible intervals for your effect estimates? The model code and
> output-format are basically the same. And you don't need to report p-values
> :))
> Pps: For making the above formula work, you need to recode your data
> matrix to:
>
> Subject word Group Time Score
> 1 1 control Pre 1
> 1 1 control Post 1
> 1 2 control Pre 1
> 1 2 control Post 1
> 1 3 control Pre 0
> 1 3 control Post 0
> 1 4 control Pre 0
> 1 4 control Post 0
>
> Am So., 24. März 2019 um 09:36 Uhr schrieb Souheyla GHEBGHOUB <
> souheyla.ghebghoub using gmail.com>:
>
>> Dear René,
>>
>> When I do Time (PrePost), score (response) will return to be dichotomous
>> (0, 1) and I will be using logistic regression (glmer) using lme4 package
>> instead of brms.
>>
>> So my previous question was about glmer model and whether its results
>> reflect a prepost comparison based on word to word analysis of each
>> participant instead of their sums; the latter will skew results.
>> I appreciate your constant help,
>> Thank you
>> Souheyla
>>
>> On Sat, 23 Mar 2019, 22:01 René, <bimonosom using gmail.com> wrote:
>>
>>> Souheyla, it's a solvable problem :)
>>> But does the model analyse at the level of every word or at the level
>>> of sum of words?
>>> it's both.
>>> Yes, the model tries to predict general tendencies (means/proportions,
>>> conditioned on factors). Observation sums are always involved.
>>> The second answer is, if there is no factor conditioning, then that's
>>> it. With factors, its measuring by item-differences if you want: And it
>>> seems you want: Let 'word type' interact with 'treatment' to predict gain
>>> proportions (0's and 1's for each participant). The relation between word
>>> type and treatment is your domain, but your statistical issues imply the
>>> following brms model: It measures differences between single
>>> questions/words (pre-post on average/proportion on log-scale). If not, the
>>> answer is still 'yes': Hierarchical Bayesian logistic regression can do
>>> what you want :)). I assume, in terms of mixed-effect models (brm is not
>>> mixed-effects least-mean squares, it's hierarchical Bayes) I would go with:
>>>
>>> mod5<-brm(score~Group*Time*Stims+(1|subject)+(1+Time*Stims| words),
>>> family=bernoulli(),...)
>>>
>>> What you will get are Bayesian likelihood estimates for each
>>> mean/proportion (and difference) across participants, for each Group by
>>> every word. After estimation, you can apply traditional statistical
>>> thinking like comparing mean estimates by word, or in general. These (both)
>>> can be accessed via 'emmeans', or 'emmip', functions and you also can
>>> compute Bayes Factors for a change vs no-change hypotheses (or others) for
>>> each question/word (using the hypothesis() function, or prior_samples()
>>> and posterior_samples() ), if you follow your own rules ;)
>>> Best, René
>>>
>>> Am Sa., 23. März 2019 um 18:58 Uhr schrieb Souheyla GHEBGHOUB <
>>> souheyla.ghebghoub using gmail.com>:
>>>
>>>> Dear René and any other member interested in this discussion,
>>>>
>>>> I appreciate the long feedback I received from you. But I can tell I
>>>> could not well convey my concern.
>>>>
>>>> The aim of my analysis is to predict the odds of correct/incorrect
>>>> responses based on some predictors: group (Treatment1/Treatment2/Control);
>>>> Time (pretest/posttest), verbal frequency, concreteness vs.
>>>> abstractness...etc. I have hypotheses such as, Time*Group interaction will
>>>> show a significant effect as I expect students in Treatment Group 1 will
>>>> outperform at the level of Posttest.
>>>> I have a problem of predicting Response with Time as an Independent
>>>> variable which could be summarised in this example of one participant:
>>>>
>>>> Subject word Group Pretest Posttest
>>>> 1 1 control 0 1
>>>> 1 2 control 0 1
>>>> 1 3 control 1 1
>>>> 1 4 control 1 1
>>>> 1 5 control 1 0
>>>> 1 6 control 0 0
>>>> 1 7 control 0 0
>>>> 1 8 control 0 0
>>>> =3 =4
>>>>
>>>> Would the model compare the mean score of pretest against mean score of
>>>> posttest? If yes, it will not truly reflect the data, because its not about
>>>> the sum,
>>>> its about every word on its own. So it should not be about 4 in
>>>> posttest against 3 in pretest means 1 gain.
>>>> In here, there are 2 gains in word 1,2 and one decline in word 5. But
>>>> does the model analyse at the level of every word or at the level of sum of
>>>> words?
>>>> That’s the main problem I have not solved since 3 months now.
>>>> Best,
>>>> SOUHEYLA
>>>>
>>>> On Sat, 23 Mar 2019 at 12:46, René <bimonosom using gmail.com> wrote:
>>>>
>>>>> Hey, things clear up. Thanks for the picture. I want to challenge your
>>>>> concerns:
>>>>> First, you see that observing a main effect of PrePost (or "Time") in
>>>>> mod3 can only mean two things: 1) performance goes up (positive effect); or
>>>>> 2) performance goes down (negative effect), in general. Saying something
>>>>> like this, is what the model is defined for. The conclusion will be this
>>>>> "general increase" or "general decrease" or "no evidence here". If you have
>>>>> a different question from that (i.e. on an item level), then you should
>>>>> specify it in more detail (see below).
>>>>> There are three issues in-between the lines of your questions:
>>>>> 1. is it a statistical concern you have?
>>>>> 2. is it an actual theoretical question you have?
>>>>> 3. is it a matter of making a non-result to a result?
>>>>>
>>>>> 1. The statistical view:
>>>>> Counter-question: who would ever assume that a 1 score for a word in a
>>>>> pre-test will remain constant until eternity. And why should it? The answer
>>>>> is: Nobody, and this is the reason statistics (probability theory) exists
>>>>> :). So the first simple answer to your question is you do not need to test
>>>>> whether observed gains from pre-to-post are 'genuine' (only from 0 to 1,
>>>>> without decline cases) because 'nature' guarantees that there will be
>>>>> decline somewhere. That's "randomness" :) But the question is, what's
>>>>> stronger, gain or decline? See... and there is no problem in it: a general
>>>>> main effect (e.g. an overall gain) still is an overall gain, even if some
>>>>> cases decline.
>>>>>
>>>>> 2. Theoretical view:
>>>>> If, however, such special item dynamics are theoretized in advance,
>>>>> then simply test it :) For instance, the assumption whether the
>>>>> treatment leads to gain on abstract words, but to decline on concrete
>>>>> words, then should find into the model by coding the factor AbsCon
>>>>> (abstract vs. concrete words) as fixed effect (assuming a within
>>>>> participants manipulation).
>>>>> mod4<-brm(score~PrePost*Group*AbsCon+(1+PrePost*AbsCon| subject) +
>>>>> (1+group|words),...)
>>>>> (Note: the 1+XX|subject just means random intercept for subject (1+)
>>>>> plus slope for XX on subject; and writing (1+group|words) is the same
>>>>> as writing (group|words), but you can estimate the slope without the
>>>>> (word) intercept by writing (0+group|words))
>>>>>
>>>>> Without having such a theoretical account testing for can also be done
>>>>> via:
>>>>> mod4<-brm(score~PrePost*Group*words+(1+PrePost*words| subject),...)
>>>>> But you will hardly be able to interpret the interactions in this
>>>>> model because words alone has 28 levels.
>>>>>
>>>>> 3. But, your example seems to suggest a special case... (i.e. an
>>>>> actual Null-Effect):
>>>>> "If a participant has got 5 correct words out of 28 in both pretest
>>>>> and posttest"
>>>>> then there would be no general effect of PrePost in the model above
>>>>> (generalizing to all participants now). And searching for "deeper" model
>>>>> checks looks like rescuing all effects you can get (post hoc). But of
>>>>> course, making an argument like there still is an effect, namely for
>>>>> 5 specific words, which is not observable because there is also a detriment
>>>>> for other 5 words is possible, but requires a solid theory which explicitly
>>>>> predicts this interaction, and an experiment which was explicitly designed
>>>>> to test this interaction. (point 2)
>>>>>
>>>>> So... if it is point 2 you got... Then go ahead :) test it in a
>>>>> meaningful way. Otherwise, simply treat this "effect by words" interaction
>>>>> as random slope (1+group|words), or btw. (1+PrePost*Group|words) is also
>>>>> possible..., which is basically 'statistically' integrating the idea that
>>>>> the treatment*time effects vary (randomly) between stimuli. And doing this
>>>>> in the random effects has the notion of "generalizing" estimation error in
>>>>> the population, and should be preferred to implementing those in the fixed
>>>>> effects, if the 28 words can be seen as a random (non-special) stimulus
>>>>> sample. If this is not the case, then consider coding the "special" thing
>>>>> about the words as fixed effects (e.g. if you want to use the same design
>>>>> again, for testing something, while controlling for stimulus specifics).
>>>>>
>>>>> Best, René
>>>>>
>>>>> Am Sa., 23. März 2019 um 11:59 Uhr schrieb Souheyla GHEBGHOUB <
>>>>> souheyla.ghebghoub using gmail.com>:
>>>>>
>>>>>> Dear René,
>>>>>>
>>>>>> Thank you for the feedback. Actually, my model was originally like
>>>>>> you suggested now (except for slopes I had PrePost without 1 in both words
>>>>>> and subjects. I called PrePost as "Time". I will read more about the
>>>>>> 1+prepost form you mentioned.
>>>>>>
>>>>>> The reason why I gave up this model and looked for something else is
>>>>>> the fact that each subject has 28 words tested twice (pre&post) and I was
>>>>>> not too sure whether such model will take into consideration differences
>>>>>> between pre & post at the level of every word of each participant (which is
>>>>>> what I want) instead of merely comparing every participant's pre mean sores
>>>>>> of 28 words against his post mean score (which is what i should avoid), here
>>>>>> is a short example as to why:
>>>>>>
>>>>>> If a participant has got 5 correct words out of 28 in both pretest
>>>>>> and posttest, there will be multiple interpretations: e.g. They could
>>>>>> refer to the same words (i.e. 0 gain), or they could be totally new words
>>>>>> (i.e. 5 gains) ...etc , hence I am not sure whether such model of pretest
>>>>>> vs posttest will compare each subject score of each word from pretest to
>>>>>> posttest then base its analysis on these score changes instead of comparing
>>>>>> the sum scores between pre and post and which likely skew results.
>>>>>>
>>>>>> I posted about this in stackexchange 3 months ago and was told that
>>>>>> it does compare word to word for every participant, but I am still not
>>>>>> confident enough to use it because all the accurateness of the results and
>>>>>> discussion chapters of my PhD thesis will be based on this decision.
>>>>>>
>>>>>> I look forward to receive feedback from you and any member reading
>>>>>> this,
>>>>>> Souheyla
>>>>>> University of York
>>>>>>
>>>>>> On Sat, 23 Mar 2019, 10:01 René, <bimonosom using gmail.com> wrote:
>>>>>>
>>>>>>> Hi Souheyla,
>>>>>>>
>>>>>>> Well, I guess in your case it is simply more elegant to leave the
>>>>>>> measured predictor out of the fixed effects, because there is also another
>>>>>>> implied question (i.e. about the strength of change between pre and post).
>>>>>>>
>>>>>>> So, another possibility to re-define your model (as logistic
>>>>>>> regression) allowing for better interpretations:
>>>>>>> mod3<-brm(score~PrePost*Group+(1+PrePost | subject)+(1+group |
>>>>>>> words),...)
>>>>>>>
>>>>>>> score = 0 or 1 for a given testitem
>>>>>>> PrePost = Pre vs. Post (basically just an indicator of the
>>>>>>> measurement time point)
>>>>>>> Thus, the PrePost main effect will tell, whether there is a change
>>>>>>> from pre to post (e.g. a gaint), and you can also tell how strong it is (in
>>>>>>> odds ratios).
>>>>>>> And if PrePost interacts with Group, then the change (e.g. a gain)
>>>>>>> is moderated by the treatment, which seems to be your main question.
>>>>>>>
>>>>>>> Now in this model, you can also have by-subject random slopes for
>>>>>>> PrePost of course (because the fixed effect of PrePost is present for every
>>>>>>> subject).
>>>>>>>
>>>>>>> Best, René
>>>>>>>
>>>>>>>
>>>>>>> Am Sa., 23. März 2019 um 10:12 Uhr schrieb Souheyla GHEBGHOUB <
>>>>>>> souheyla.ghebghoub using gmail.com>:
>>>>>>>
>>>>>>>> I read that in multinomial regression, all independent variables
>>>>>>>> should be
>>>>>>>> variables that we manipulate. Can I still have pretest as IV without
>>>>>>>> skewing my results?
>>>>>>>>
>>>>>>>> Best,
>>>>>>>> Souheyla
>>>>>>>>
>>>>>>>> On Fri, 22 Mar 2019, 23:31 Souheyla GHEBGHOUB, <
>>>>>>>> souheyla.ghebghoub using gmail.com>
>>>>>>>> wrote:
>>>>>>>>
>>>>>>>> > Thank you both. I will look into this and see :)
>>>>>>>> >
>>>>>>>> > Best,
>>>>>>>> > Souheyla
>>>>>>>> >
>>>>>>>> > On Fri, 22 Mar 2019, 22:02 Uanhoro, James, <
>>>>>>>> uanhoro.1 using buckeyemail.osu.edu>
>>>>>>>> > wrote:
>>>>>>>> >
>>>>>>>> >> In standard regression models, the assumption is predictor
>>>>>>>> variables are
>>>>>>>> >> measured without error. Test scores will have measurement error,
>>>>>>>> hence
>>>>>>>> >> Doran's comment when test scores are used as covariates. See:
>>>>>>>> Hausman, J.
>>>>>>>> >> (2001). Mismeasured Variables in Econometric Analysis: Problems
>>>>>>>> from the
>>>>>>>> >> Right and Problems from the Left. *Journal of Economic
>>>>>>>> Perspectives*,
>>>>>>>> >> *15*(4), 57–67. https://doi.org/10.1257/jep.15.4.57
>>>>>>>> >> I will note that many practitioners ignore this issue, and it is
>>>>>>>> quite
>>>>>>>> >> common to use predictors measured with error. Consider the
>>>>>>>> number of times
>>>>>>>> >> people use polychotomized income measures, or SES measures as
>>>>>>>> predictors,
>>>>>>>> >> or some other "construct".
>>>>>>>> >> On Mar 22 2019, at 5:39 pm, Souheyla GHEBGHOUB <
>>>>>>>> >> souheyla.ghebghoub using gmail.com> wrote:
>>>>>>>> >>
>>>>>>>> >> Dear Doran,
>>>>>>>> >>
>>>>>>>> >> Could you explain more this point to me, please?
>>>>>>>> >>
>>>>>>>> >> Thank you,
>>>>>>>> >> Souheyla
>>>>>>>> >>
>>>>>>>> >> On Fri, 22 Mar 2019, 21:19 Doran, Harold, <HDoran using air.org>
>>>>>>>> wrote:
>>>>>>>> >>
>>>>>>>> >> Yes, but conditioning on the pre-test means you are using a
>>>>>>>> variable
>>>>>>>> >> measured with error and the estimates you obtain and now
>>>>>>>> inconsistent, and
>>>>>>>> >> that¹s a pretty big sin.
>>>>>>>> >>
>>>>>>>> >> On 3/22/19, 3:49 PM, "Souheyla GHEBGHOUB" <
>>>>>>>> souheyla.ghebghoub using gmail.com>
>>>>>>>> >> wrote:
>>>>>>>> >>
>>>>>>>> >> Dear René,
>>>>>>>> >>
>>>>>>>> >> Thank you for your feedback to me. You are right, dropping the
>>>>>>>> pretest
>>>>>>>> >> from
>>>>>>>> >> covariate if I predict change definitely makes sense to me! But
>>>>>>>> the fact
>>>>>>>> >> that i need to control for the starting levels of participants
>>>>>>>> makes it
>>>>>>>> >> obligatory for me to chose the second way, which is predicting
>>>>>>>> posttest
>>>>>>>> >> instead of change to have pretest scores controlled for.
>>>>>>>> >>
>>>>>>>> >> You also chose (1+group | word) , which is new to me. Does it
>>>>>>>> intend to
>>>>>>>> >> assume the effect of group to vary across words, which is
>>>>>>>> something
>>>>>>>> >> applicable to my data, right?
>>>>>>>> >> I will discuss all this with my supervisor, and may reply here
>>>>>>>> again in
>>>>>>>> >> few
>>>>>>>> >> days if you do not mind.
>>>>>>>> >> Thank you very much
>>>>>>>> >> Souheyla
>>>>>>>> >> University of York
>>>>>>>> >>
>>>>>>>> >>
>>>>>>>> >> On Fri, 22 Mar 2019 at 13:42, René <bimonosom using gmail.com> wrote:
>>>>>>>> >>
>>>>>>>> >> Hi Souheyla,
>>>>>>>> >>
>>>>>>>> >> it seems to me that you will run into problems with your coding
>>>>>>>> of
>>>>>>>> >> change
>>>>>>>> >> (gain, no gain and decline) because the 'change' is by
>>>>>>>> >> definition/calculation depending on the predictor pretest.
>>>>>>>> >> See, according to your coding scheme:
>>>>>>>> >> Change = decline can only occur if pretest=1 (not by pretest=0).
>>>>>>>> >> Change = gain can only occur if pretest = 0 (not by pretest=1)
>>>>>>>> >> Change = No Gain can occur if pretest= 1 or 0
>>>>>>>> >> In other words:
>>>>>>>> >> If pretest = 1 then the possible outcomes can be decline or no
>>>>>>>> gain
>>>>>>>> >> If pretest = 0 then the possible outcomes can be gain or no gain
>>>>>>>> >>
>>>>>>>> >> And if the model result shows you then that the pre-test is
>>>>>>>> >> significantly
>>>>>>>> >> related to p(change-outcome), I guess there is no surprise in
>>>>>>>> it, is it?
>>>>>>>> >>
>>>>>>>> >> So the first solution to this would be simply kicking the
>>>>>>>> pre-test
>>>>>>>> >> predictor out of the model completely, and predict:
>>>>>>>> >> mod1 <- brm(Change ~ Group + (1|Subject) + (1+Group|Word),...)
>>>>>>>> >> (Btw.: actually the first Hierarchical Bayes Model question I
>>>>>>>> see on the
>>>>>>>> >> mixed-effects mailing list :))
>>>>>>>> >>
>>>>>>>> >> Attempt for a further clarification on which random slopes would
>>>>>>>> reflect
>>>>>>>> >> the model's design:
>>>>>>>> >> If you have a within-subjects design, by-subject random slopes
>>>>>>>> are
>>>>>>>> >> possible for the within-subject variable (e.g. if there are two
>>>>>>>> sets of
>>>>>>>> >> words/lists [e.g. abstract vs. concrete words] for each
>>>>>>>> participant, and
>>>>>>>> >> you test whether there is a performance-difference between these
>>>>>>>> >> word-lists, then you can implement by-subject random slopes for
>>>>>>>> words,
>>>>>>>> >> because each participant has seen both sets.) If each
>>>>>>>> participant has
>>>>>>>> >> seen
>>>>>>>> >> only one list (i.e. between subjects design) by subject random
>>>>>>>> slopes
>>>>>>>> >> for
>>>>>>>> >> words are not appropriate, because there is no 'slope' by
>>>>>>>> participant
>>>>>>>> >> (i.e.
>>>>>>>> >> by definition, having a slope requires at least two
>>>>>>>> observations...).
>>>>>>>> >> This
>>>>>>>> >> is always a good rule of thumb without thinking about it too
>>>>>>>> heavily :)
>>>>>>>> >> Ans as you see: you can define a random slope for words:
>>>>>>>> >> (1+Group|Word),
>>>>>>>> >> because each word has been presented in each group (i.e. there
>>>>>>>> can be a
>>>>>>>> >> slope for each word). And intuitively speaking the
>>>>>>>> Treatment-effect can
>>>>>>>> >> vary depending on the stimuli you use, and the slope makes
>>>>>>>> sense. (You
>>>>>>>> >> also
>>>>>>>> >> see in this example that the treatment effect can also vary by
>>>>>>>> subjects,
>>>>>>>> >> but in fact, this subject effect variation IS EQUAL to the
>>>>>>>> effect you
>>>>>>>> >> want
>>>>>>>> >> to test, and having by subject group random slopes would
>>>>>>>> eliminate the
>>>>>>>> >> fixed effect...)
>>>>>>>> >>
>>>>>>>> >> Anyway, there is a second possibility to define your model,
>>>>>>>> depending on
>>>>>>>> >> how you want to interpret it. In the previous model you can say
>>>>>>>> >> something
>>>>>>>> >> about the type-of-change likelihoods depending on the treatment
>>>>>>>> group.
>>>>>>>> >> But
>>>>>>>> >> you could implement the model as binomial as well (i.e. logistic
>>>>>>>> >> regression)
>>>>>>>> >>
>>>>>>>> >> mod2 <- brm(posttest ~ pretest*Group + (1|Subject) +
>>>>>>>> (1+Group|Word),...)
>>>>>>>> >>
>>>>>>>> >> And what you would expect here would be an interaction between
>>>>>>>> pre-test
>>>>>>>> >> and Group. For instance; if pretest=0 & treatment 1 then
>>>>>>>> posttest larger
>>>>>>>> >> than with pretest=0 & treatment 2; but not when pretest=1
>>>>>>>> (because this
>>>>>>>> >> is
>>>>>>>> >> a plausible no gain situation). And so on...
>>>>>>>> >> (And in this model there are no also no further random slopes
>>>>>>>> hidden in
>>>>>>>> >> your design :))
>>>>>>>> >> Hope this helps.
>>>>>>>> >>
>>>>>>>> >> Best, René
>>>>>>>> >>
>>>>>>>> >>
>>>>>>>> >> Am Do., 21. März 2019 um 14:01 Uhr schrieb Souheyla GHEBGHOUB <
>>>>>>>> >> souheyla.ghebghoub using gmail.com>:
>>>>>>>> >>
>>>>>>>> >> Dear Philip,
>>>>>>>> >>
>>>>>>>> >> I understand , here is the structure of my data in case it could
>>>>>>>> help.
>>>>>>>> >>
>>>>>>>> >> I have 3 groups of participants (control, treatment1,
>>>>>>>> treatment2). Each
>>>>>>>> >> group was tested twice, once before treatment (pretest) and once
>>>>>>>> after
>>>>>>>> >> treatment (posttest).
>>>>>>>> >> In each test, they were tested on knowledge of 28 words, scores
>>>>>>>> are
>>>>>>>> >> dichotomous (0 = unknown , 1 = known). Tests are the same.
>>>>>>>> >>
>>>>>>>> >> I calculated change from pretest to posttest :
>>>>>>>> >> if pretest 0 and posttest 0 = no gain
>>>>>>>> >> if pretest 1 and posttest 1 = no gain
>>>>>>>> >> if pretest 0 and posttest 1 = gain
>>>>>>>> >> if pretest 1 and posttest 0 = decline
>>>>>>>> >> So I ended up with a dependent variable called Change with 3
>>>>>>>> levels
>>>>>>>> >> (no_gain, gain, decline) and I tried to predict it using Group
>>>>>>>> and
>>>>>>>> >> Pretest
>>>>>>>> >> as covariates using multinomial logit model. mod0 <- brm(Change ~
>>>>>>>> >> Pretest
>>>>>>>> >> +
>>>>>>>> >> Group) I would like to add random effects for subjects but don't
>>>>>>>> know
>>>>>>>> >> what's the best form when Time factor is absent.
>>>>>>>> >>
>>>>>>>> >> I hope other statisticians who read this could help
>>>>>>>> >> Thank you
>>>>>>>> >> Souheyla
>>>>>>>> >>
>>>>>>>> >> [[alternative HTML version deleted]]
>>>>>>>> >>
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