Positive interaction with negative effect?

[John Smith III]

Frame is a var with 3 categories: neutral (baseline), N, and C. Effect of C on outcome Qa (measured 1-7) is hypothesized and found to be negative (with a bunch of fixed effects):

reg Qa ib3.frame ib2.oper i.order i.country2, cluster( ResponseId )

             |               Robust
          Qa | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       frame |
          C  |  -.3631873   .0385492    -9.42   0.000    -.4387688   -.2876058
          N  |  -.1823204   .0369146    -4.94   0.000     -.254697   -.1099438

Alignment is another var with 3 categories: independent (baseline), co-partisan, opposite-party. We interact it with frame, in order to examine how the interaction changes C’s effect on Qa. This post is about how to interpret this interaction.

reg Qa ib3.frame##i.alignment ib2.oper i.order i.country2, cluster( ResponseId )

                         |               Robust
                      Qa | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------------------+----------------------------------------------------------------
                   frame |
                      C  |   -.450276   .0725978    -6.20   0.000    -.5926267   -.3079253
                      N  |  -.2098803   .0717897    -2.92   0.003    -.3506466    -.069114
                         |
               alignment |
     Co-partisan source  |   .0471406    .065567     0.72   0.472     -.081424    .1757052
  Opposite party source  |   -.224524   .0666856    -3.37   0.001    -.3552821   -.0937659
                         |
         frame#alignment |
   C#Co-partisan source  |  -.0078699   .1034016    -0.08   0.939    -.2106212    .1948813
C#Opposite party source  |   .2276461   .1024118     2.22   0.026     .0268355    .4284566
   N#Co-partisan source  |  -.0797576    .100597    -0.79   0.428    -.2770096    .1174944
N#Opposite party source  |    .109464   .1014582     1.08   0.281    -.0894767    .3084046

Observe that C##opposite-party interaction term is positive, whereas C’s conditional term is negative, and C’s unconditional effect too was negative earlier. Here are the marginal effects from the interaction:

margins alignment, dydx(frame)
                       |            Delta-method
                       |      dy/dx   std. err.      t    P>|t|     [95% conf. interval]
-----------------------+----------------------------------------------------------------
1.frame                |
             alignment |
   Independent source  |   -.450276   .0725978    -6.20   0.000    -.5926267   -.3079253
   Co-partisan source  |  -.4581459   .0721426    -6.35   0.000    -.5996041   -.3166877
Opposite party source  |  -.2226299   .0741549    -3.00   0.003    -.3680339   -.0772258

Observe that for frame 1 (which is C) the discrete change from the base level (-.22) is smaller in magnitude under opposite party source, compared to the independent source condition (-.45). As seen below, this difference is also statistically significant (careful about the signs here: this time the "contrast" is displayed, which is positive, since it is in the direction of offsetting the otherwise negative effect of C).

margins alignment, dydx(frame) pwcompare

                                             |   Contrast Delta-method         Unadjusted
                                             |      dy/dx   std. err.     [95% conf. interval]
---------------------------------------------+------------------------------------------------
1.frame                                      |
                                   alignment |
   Co-partisan source vs Independent source  |  -.0078699   .1034016     -.2106212    .1948813
Opposite party source vs Independent source  |   .2276461   .1024118      .0268355    .4284566
Opposite party source vs Co-partisan source  |    .235516   .1036335        .03231     .438722

Now the question:
- Researcher A argues that opposite-party interaction mitigates the (negative) effect of C on Qa
- Researcher B argues that opposite-party interaction enhances the (negative) effect of C on Qa,
- Researcher C argues that the (negative) effect of C on Qa does not vary across categories of alignment.
- Researcher D argues that in light of these quantities, no argument can be made in ordinary language, because words could mean anything.
Which interpretation is correct?

Note: This post is a summary of a question asked elsewhere in this forum, which received replies with a lot of text, and somewhat got buried under a philosophy of science discussion. Seeking for more practical advice. Thank you in advance.

[Joseph Coveney]

. . . the question:
- Researcher A argues that opposite-party interaction mitigates the (negative) effect of C on Qa
- Researcher B argues that opposite-party interaction enhances the (negative) effect of C on Qa,
- Researcher C argues that the (negative) effect of C on Qa does not vary across categories of alignment.
- Researcher D argues that in light of these quantities, no argument can be made in ordinary language, because words could mean anything.
Which interpretation is correct?

Note: This post is a summary of a question asked elsewhere in this forum. . . Seeking for more practical advice.

I've attached a profile plot (common in the context of old-style factorial ANOVA), with the baseline Political Messaging Frame condition of Neutral at the origin, the contentious C condition in the midpoint and the N condition at the righthand side for contrast; likewise, I've plotted the baseline Alignment Source of Independent in black, the contentious Opposing-party Source in red and the Copartisan Source in blue. Code is given below; values are taken from the output listing of margins shown by your colleague in the sister thread.

The problem (as your colleague notes but I believe erroneously dismisses*) is that there is a baseline difference in the Opposing-party Source treatment group from the other two Alignment Sources. This renders the marginal change from baseline (Neutral) condition of Political Messaging Frame to the C condition incomparable between the Opposing-party Source treatment group and the other two Alignment Sources.

Your colleague in the other thread argues that 4 cannot be a floor effect, but that neglects the fact that it's adjusted for other covariates in the model. It could (and it looks as if indeed it is) a floor effect, i.e., elevated from a score of 1 to that of 4 by covariate adjustment in the margins command. Moreover, you don't really have much dynamic range in the response, going from a mean of 4.00 ± 0.06 to 4.50 ± 0.05 in a score range of 1 to 7.

So, my interpretation would be:

(i) you very likely have a floor effect in this ordered-categorical questionnaire item's responses, which confounds any interpretation of ostensibly nonadditive effects in linear contrasts in the context of interaction in a linear regression model, and

(ii) despite even that, you cannot really make a statement about the relative change from baseline between the Opposing and Independent Sources as a consequence of C Political Messaging Frame treatment without addressing the counterfactual of what that change would have been had the Opposing-party Source started out at the same response values under the baseline (Neutral) Political Messaging Frame treatment condition.

version 19

clear *

input byte(a b) double(out sem)

0 1 3.995489   .0571731
0 2 4.235885   .0544534
0 0 4.445765   .0475833
1 1 4.037345   .0571095
1 2 4.205853   .0542435
1 0 4.495491   .0463339
2 1 3.998388   .0587389
2 2 4.120602   .0525447
2 0 4.221018   .0479495
end

label define A 0 Independent 1 Copartisan 2 Opposing
label values a A

label define B 0 Neutral 1 C 2 N
label values b B

set scheme s2color
graph twoway ///
    line out b if a == 0, sort lcolor(black) || ///
    line out b if a == 1, sort lcolor(blue) || ///
    line out b if a == 2, sort lcolor(red) ///
        xtitle(Frame) xlabel(0 1 2, valuelabel) ///
        ytitle(Mean Qa) ylabel( , angle(horizontal) nogrid) ///
        text(4.2 0.35 "Independent") ///
            text(4.4 0.4 "Copartisan") ///
            text(4.1 0.375 "Opposing") ///
        legend(off) 

exit

* "So team member 3 is troubled by the fact that under C treatment, there is no outcome difference between sources . . . I explain that this indifference is irrelevant unless coupled with the observation that outcomes would actually differ across sources without C treatment (i.e. neutral conditions), and C reduces the difference by affecting respondents exposed to opposite-party source less than respondents in other source conditions. I explain that the opposite-party source already has a baseline negative association with Qa under neutral frame, and this is offset by the further negative effects of switching from neutral to C frame treatment, precisely because C effect is stronger under conditions other than the opposite-party." [Emphasis added. I don't really understand the meaning of that last part, from "and this is offset . . ." onward.]

[John Smith III]

Thank you. I'm in need of some clarification.
1) You say the other post "argues that 4 cannot be a floor effect, but that neglects the fact that it's adjusted for other covariates in the model." Are we talking about the fixed-effect variables as the covariates in the model? Below I report the margins obtained when we reproduce the regression without these covariates.

reg Qa ib3.frame##i.alignment

margins ib3.frame, over(ib0.alignment)

                               |            Delta-method
                               |     Margin   std. err.      t    P>|t|     [95% conf. interval]
-------------------------------+----------------------------------------------------------------
               alignment#frame |
         Independent source#C  |     4.0057    .059042    67.84   0.000     3.889968    4.121433
         Independent source#N  |   4.226688   .0586611    72.05   0.000     4.111702    4.341674
   Independent source#neutral  |   4.444888   .0584734    76.02   0.000      4.33027    4.559506
         Co-partisan source#C  |   4.011285   .0609585    65.80   0.000     3.891795    4.130774
         Co-partisan source#N  |   4.231834   .0608529    69.54   0.000     4.112551    4.351116
   Co-partisan source#neutral  |   4.495481   .0593082    75.80   0.000     4.379226    4.611735
      Opposite party source#C  |   3.975475   .0581951    68.31   0.000     3.861402    4.089547
      Opposite party source#N  |   4.146858   .0569258    72.85   0.000     4.035274    4.258443
Opposite party source#neutral  |   4.216741    .056563    74.55   0.000     4.105868    4.327615

So essentially the same findings, but I'm not sure if this is what you meant. Let me also share the distribution of responses for the outcome Qa.

                Qa |      Freq.     Percent        Cum.
-------------------+-----------------------------------
 1 Strongly oppose |      2,481       17.96       17.96
                 2 |        982        7.11       25.07
                 3 |      1,489       10.78       35.84
                 4 |      2,514       18.20       54.04
                 5 |      2,268       16.42       70.45
                 6 |      1,566       11.33       81.79
7 Strongly support |      2,516       18.21      100.00

In fact there are a lot of responses below 4. Is there any song reason to believe that there is a floor effect at 4 going on with the predicted means, other than the simply observing that they are all around that number? What leads us to say that this is a floor, rather than simply saying that all the effects are small, so they don't push the conditional means far from the overall sample mean of 4.15?

2) Are you saying that, because there is a baseline difference between source conditions in the neutral frame treatment, we cannot report what source interaction does to frame effects by, for example, displaying discrete change from the base level? (This would also mean a disagreement with Clyde Schechter's remark in the other post, am I correct?)

[Joseph Coveney]

Are you saying that, because there is a baseline difference between source conditions in the neutral frame treatment, we cannot report what source interaction does to frame effects by, for example, displaying discrete change from the base level?

Your request was about interpretation and not reporting, and so the question of interpretation was what I was responding to.

But, yes, I think that the difference in baseline response between Source Alignments—especially where the response’s (Qa) range of values is so constrained—affects the ability to interpret a given difference in discrete change from baseline as evidence of mitigation (or enhancement) of C on the response.

Because the baseline response differs in the Opposing from that of the Independent and Copartisan, in order to support an interpretation that the difference in discrete change between it and them as evidence of altering the effect of C, you will need to address the counterfactual. For example, to interpret the reduction in discrete change as a diminution then you would need to present an argument that, had Opposing’s baseline response been similar to those for Independent and Copartisan, then the profile plot would have looked something like the second graph below (Counterfactual 1). Alternatively, to interpret that the reduction in discrete change that you see in the first graph below (Observed) does not reflect a floor effect by presenting an argument that a profile plot something like the third graph below (Counterfactual 2) will have obtained if your scientific hypothesis is false.

Without such argument or evidence, your team might be relegated to just reporting the value the of the difference in discrete change without the ability to say what it represents, a diminution, an enhancement or nothing at all.

What leads us to say that this is a floor, rather than simply saying that all the effects are small, so they don't push the conditional means far from the overall sample mean of 4.15?

This is related to Counterfactual 2 above. If the effects are small such that they can’t push the observed means beyond that narrow dynamic range (4–4½), then what you’re observing is effectively a floor (and ceiling) effect, especially when your measure’s response range is a very truncated portion of the real line. You’ve got a limited dependent variable (LDV)—a situation that is inherently nonlinear—and you’re fitting a linear regression model to that response and trying to interpret linear contrasts in absolute terms (“displaying discrete change from the base level”) as evidence of nonadditivity (interaction). I suspect that there’s literature on the problems encountered with analogous attempts at interpreting attenuation and enhancement of effects in linear probability models (LPMs) that might be worth looking into.

(This would also mean a disagreement with Clyde Schechter's remark in the other post, am I correct?)

I don’t know which of Clyde’s remarks you’re referring to and so I cannot answer you. I don’t recall ever disagreeing with him.

Observed reflects the revised margins output in #3 above. Do-file for these graphs is attached.

[Alper Yagci]

Hi, this message was erroneously posted.

[John Smith III]

Because the baseline response differs in the Opposing from that of the Independent and Copartisan, in order to support an interpretation that the difference in discrete change between it and them as evidence of altering the effect of C, you will need to address the counterfactual. For example, to interpret the reduction in discrete change as a diminution then you would need to present an argument that, had Opposing’s baseline response been similar to those for Independent and Copartisan, then the profile plot would have looked something like the second graph below (Counterfactual 1). Alternatively, to interpret that the reduction in discrete change that you see in the first graph below (Observed) does not reflect a floor effect by presenting an argument that a profile plot something like the third graph below (Counterfactual 2) will have obtained if your scientific hypothesis is false.

Without such argument or evidence, your team might be relegated to just reporting the value the of the difference in discrete change without the ability to say what it represents, a diminution, an enhancement or nothing at all.

Thank you. I am not sure I fully understand and I’m curious. With your general point about the non-observed counterfactuals, do you mean either the 1st or 2nd of the following?

1. You are warning that, with another research design or other kind of data, some researcher could have legitimately infer that C effect on Qa is mitigated by the presence of a particular frame, but in our case we can’t make that interpretation due to a limitation of our design or the particular distribution of our data.
2. Your point instead is a restatement of the fundamental problem of causality, which, at the extreme, would advise against humans using causal language so it is categorically not possible to infer that C effect on Qa is mitigated by the presence of a particular frame.

(Relatedly, as Clyde Schechter’s remark #7 in the other post, I was referring to his conclusion that “frame C reduces Qa by a smaller amount in the presence of opposite alignment than it does in independent or co-partisan alignment.” Though he reluctantly arrived at that language.)

what you’re observing is effectively a floor (and ceiling) effect, especially when your measure’s response range is a very truncated portion of the real line. You’ve got a limited dependent variable (LDV)

I’d also like to ask more about your usage of the term “floor effect” bec it puzzles me. I am familiar (from applied social science literature) with the floor effect as a measurement-generated process that results in a lot of observations residing at a single observed value at the bottom of the measurement scale, with latent values that would have been observed to have greater variation if only the measurement device allowed. I can’t see how that applies here. I am conducting a social science experiment where I measure a change in respondents’ reported beliefs about the world in response to a momentary exposure to short textual stimulus, so any significant effects are expected and found to be very small, as is typically the case with vignette experiments in social science. Consequently, the treatment is associated with a response variation within a small dynamic range, and then the lower end of this range gets to be called not only a floor but also a product of floor effect? In the meantime, a third of the sample lives way below this floor. How is this a LDV? I’m quite puzzled, but I’d like to learn more. Can you elaborate? I guess my question here is, suppose for a moment a scenario where there is no floor effect but the same data is observed as in here (because that’s possible, correct?). How could we tell if we are not already in that scenario?

[Joseph Coveney]

I am not sure I fully understand and I’m curious.

Ordered-categorical response variables are LDVs and their inherent nonlinearity is what ordered-logistic and ordered-probit regression modeling is intended to accommodate. I realize that when there are at least seven or so categories researchers still often employ methods for use with continuous variables. Your questionnaire item has seven ordered categories and you’ve fitted a linear regression model.

But you do have pile up at both ends of the range—look at the profile of your marginal tabulation of Qa in #3 above—and so you’ll need to be sensitive to the possibility that one or more of your conditional means is in nonlinear territory.

If you want to make claims that results of a linear contrast of coefficients of such a linear model fitted to ordered-categorical responses can be interpreted as evidence of interaction (nonadditivity), then you’ll need to support the assertion that the involved conditional means are still within the quasilinear portion of the curve.

If the Opposing mean under the baseline Neutral condition had been close to that for Independent, then its discrete change from base level would have been comparable to the latter’s. But it wasn’t close. And baseline differences come into play when interpreting differences in change from baseline, especially in nonlinear systems.

If you had had a negative-control treatment group whose mean starts out close to the Opposing mean under baseline conditions and that would prove that your study could have shown a no-effect—that is, show that your conditional mean response was capable of going well below 4 (that your experiment could have shown a treatment having both the same start location as Opposing and the same slope as that of Independent/Copartisan)—then you would have been able to claim that your experimental setup can disprove your scientific hypothesis if it is false. Unfortunately, you don’t have such a negative-control treatment group, and the experiment lacks direct evidence that if your scientific hypothesis is false, then your experimental setup will have shown that to be the case.

In the absence of such evidence, you’ll need to make some kind of argument that the baseline difference is inconsequential. About the only thing I can suggest is to take a pseudomatched-controls approach in a secondary analysis: select a subset of respondents such that their Opposing mean at baseline matches that of the overall Independent or Copartisan baseline mean, and track how they do under C. And for parity, conversely select a subset of respondents such that their Independent or Copartisan baseline mean matches that of the overall Opposing baseline mean and track how their scores change from Neutral to C. Or some variation on this matching theme to try to get at one or both of those counterfactuals mentioned above.

I wanted to avoid getting further hung up on the semantics of floor and ceiling effects, and so I’ve avoided using the terms here. Instead I referred to the central part of the curve as the “quasilinear portion”, and intended to mean that portion predictor-response relationship that is sufficiently far away from those regions where the slope begins to taper off (the latter of which I referred to as “nonlinear territory” in lieu of “approaches regions where floor and ceiling effects begin to come into play”).

Likewise, I didn’t want to pursue matters of what other researchers could have legitimately inferred with another design or other data, expected magnitude of effects in the social sciences, and alternative-universe scenarios. The point that I was trying to make is far more down to Earth, and I can’t helpfully speak to any of those anyway.

And humans using causal language doesn’t bother me: I trust that your i.order variable represents some kind of randomized sequence of presentation of the three levels of each of the two factors to the participants in a crossover or partial crossover experimental study design, perhaps even balanced by permuted Latin squares.

version 19

clear *

input byte sco int cou
1 2481
2 982 
3 1489
4 2514 
5 2268 
6 1566
7 2516
end

set scheme s2color

histogram sco [fweight=cou], discrete fraction xlabel(1(1)7) barwidth(0.1) ///
    ylabel( , format(%04.2f) angle(horizontal) nogrid)

exit

[Alper Yagci]

Thank you Joseph Coveney. Things sound much clearer now. I'll try the pseudo-matching (and one other thing I have in mind). I'll be back with results and maybe questions if you are still available for consulting.

[John Smith III]

Ordered-categorical response variables are LDVs and their inherent nonlinearity is what ordered-logistic and ordered-probit regression modeling is intended to accommodate. I realize that when there are at least seven or so categories researchers still often employ methods for use with continuous variables. Your questionnaire item has seven ordered categories and you’ve fitted a linear regression model.

But you do have pile up at both ends of the range—look at the profile of your marginal tabulation of Qa in #3 above—and so you’ll need to be sensitive to the possibility that one or more of your conditional means is in nonlinear territory.

If you want to make claims that results of a linear contrast of coefficients of such a linear model fitted to ordered-categorical responses can be interpreted as evidence of interaction (nonadditivity), then you’ll need to support the assertion that the involved conditional means are still within the quasilinear portion of the curve.

If the Opposing mean under the baseline Neutral condition had been close to that for Independent, then its discrete change from base level would have been comparable to the latter’s. But it wasn’t close. And baseline differences come into play when interpreting differences in change from baseline, especially in nonlinear systems.

If you had had a negative-control treatment group whose mean starts out close to the Opposing mean under baseline conditions and that would prove that your study could have shown a no-effect—that is, show that your conditional mean response was capable of going well below 4 (that your experiment could have shown a treatment having both the same start location as Opposing and the same slope as that of Independent/Copartisan)—then you would have been able to claim that your experimental setup can disprove your scientific hypothesis if it is false. Unfortunately, you don’t have such a negative-control treatment group, and the experiment lacks direct evidence that if your scientific hypothesis is false, then your experimental setup will have shown that to be the case.

In the absence of such evidence, you’ll need to make some kind of argument that the baseline difference is inconsequential. About the only thing I can suggest is to take a pseudomatched-controls approach in a secondary analysis: select a subset of respondents such that their Opposing mean at baseline matches that of the overall Independent or Copartisan baseline mean, and track how they do under C. And for parity, conversely select a subset of respondents such that their Independent or Copartisan baseline mean matches that of the overall Opposing baseline mean and track how their scores change from Neutral to C. Or some variation on this matching theme to try to get at one or both of those counterfactuals mentioned above.

I wanted to avoid getting further hung up on the semantics of floor and ceiling effects, and so I’ve avoided using the terms here. Instead I referred to the central part of the curve as the “quasilinear portion”, and intended to mean that portion predictor-response relationship that is sufficiently far away from those regions where the slope begins to taper off (the latter of which I referred to as “nonlinear territory” in lieu of “approaches regions where floor and ceiling effects begin to come into play”).

Likewise, I didn’t want to pursue matters of what other researchers could have legitimately inferred with another design or other data, expected magnitude of effects in the social sciences, and alternative-universe scenarios. The point that I was trying to make is far more down to Earth, and I can’t helpfully speak to any of those anyway.

And humans using causal language doesn’t bother me: I trust that your i.order variable represents some kind of randomized sequence of presentation of the three levels of each of the two factors to the participants in a crossover or partial crossover experimental study design, perhaps even balanced by permuted Latin squares.

version 19

clear *

input byte sco int cou
1 2481
2 982
3 1489
4 2514
5 2268
6 1566
7 2516
end

set scheme s2color

histogram sco [fweight=cou], discrete fraction xlabel(1(1)7) barwidth(0.1) ///
ylabel( , format(%04.2f) angle(horizontal) nogrid)

exit

Thank you Joseph Coveney. One clarification & question. In the experiment, neither frame nor alignment are observational, they are both experimentally manipulated. They are both hypothesized and found to have (negative) unconditional effects of their own on outcome Qb. The opposite-party source is expected to have a negative effect on the outcome due to a theorized negative partisanship process: people will rate Qb lower if the source is opposite to their alignment. Due to the same theoretical process, the effect of C frame is expected to be mitigated in the presence of opposite-party source, because negative partisanship reduces how much an opposite source can influence the respondents' outcomes by employing a frame in their messaging.

The findings (quantities) are similar to those obtained on Qa and discussed above. Now let's say we pursue the pseudo-controls approach for Qb, and find a subset of respondents such that their Opposing mean at baseline (neutral frame) matches that of the overall Independent or Copartisan baseline mean. I fear that these will be people who are atypically immune to the negative partisanship process (because they didn't rate Qb lower despite it was Opposite-party). Hence these will be people also atypically immune to the very reason why C effect is expected to be mitigated by the presence of Opposite-party source. In other words, if we do some kind of matching on Opposing means for Qb, and find that C effect is then no different under different alignment conditions, we have theoretical reasons to doubt whether this subset-finding can be generalized to our overall sample. If this is the case, the current experiment setup about Qb does not allow us to get at the counterfactuals you are pointing to, correct?

[Joseph Coveney]

. . . we pursue the pseudo-controls approach for Qb, and find a subset of respondents such that their Opposing mean at baseline (neutral frame) matches that of the overall Independent or Copartisan baseline mean. I fear that these will be people who are atypically immune to the negative partisanship process (because they didn't rate Qb lower despite it was Opposite-party). . . . If this is the case, the current experiment setup about Qb does not allow us to get at the counterfactuals you are pointing to, correct?

No, your fears are unfounded.

The scores of those Opposing participants that average in the neighborhood of 4½ (i.e., those of the Independents) under the Neutral condition would have been 5, 6, 7, 7+ in the absence of the effect of Opposing treatment condition. So, those scores don't reflect any kind of "immunity" to the Opposing treatment effect, they just started out higher (unseen) and were brought down.

Keep in mind that this is a secondary analysis with reduced sample size—you don't need to attain "statistical significance" here; you just need to see whether the profile plot for those Opposing participants goes in the same direction that your scientific hypothesis predicts, that is, the slope from Neutral to C is reduced relative to that for the Independent treatment group..

I realize that this matching approach smacks of selecting (or conditioning) on intermediate outcomes, which can be problematic, and so this exercise should be viewed as a kind of cross-check in support of your argument that the results of the primary analysis reflect what you claim they do.

Another approach for a secondary analysis in support of your primary analysis is to fit an ordered-probit regression model and examine the profile plot of the linear predictor, something like

oprobit Qa i.frame##i.alignment i.(oper order country2), vce(cluster ResponseId)

// or even

meoprobit Qa i.frame##i.alignment i.(oper order country2) || ResponseId:

margins frame#alignment, predict(xb)
marginsplot , xdimension(frame) noci

and see whether the attenuation of the Neutral-to-C slope still obtains for Opposing relative to the other two Alignment treatment conditions. Here, you could argue that the generalized linear model handles the concern about floor effects.

[Joseph Coveney]

Now let's say we pursue the pseudo-controls approach for Qb, and find a subset of respondents such that their Opposing mean at baseline (neutral frame) matches that of the overall Independent or Copartisan baseline mean.

If you're having difficulty with the matching, approach, then an alternative is to take the upper 50% of both Opposing and Independent treatment groups and plot their individual means from Neutral to C. If the hypothesized pattern holds, then there is no longer any reason to object on floor effects / nonlinear region grounds, because even the Opposing mean under the C frame condition will be well above the mean that has already been shown (with the whole dataset) to be feasible to attain with this measure.

Likewise, take the bottom half of both Opposing and Independent groups and plot their individual means from Neutral to C. It is likely that the mean for the former under the C frame condition will lie below 4, perhaps well below. If the same pattern (attenuation of the former's slope compared to the latter) holds, then this, too, buttresses the argument that the pattern predicted by the scientific hypothesis holds independently of baseline value.

Again, these are secondary analyses that seek to explore robustness and to support an argument of the interpretability of the original findings. They are with reduced sample size and you would be looking for whether patterns consistent with your scientific hypothesis prevail, and not necessarily statistical significance of any particular regression coefficient.

By the way, the suggestion in the previous post regarding ordered-probit regression was to fit the model to the entire dataset, and not to any matching subset.