**Replication Model**
We specify two hypotheses, one for each dependent variable. For product quality, we expect an interaction between the between subjects factor and the within subjects factor. Specifically, we expect that senders will rate their quality threshold for sending referrals as similarly high both when there is no incentive and also when there is a small incentive. For receivers, however, we expect that their perceived quality threshold for a sender to send a referral will decrease when an incentive is added. Specifically, we expect a flat line for sender quality threshold and a negative line for receiver quality threshold.
For acceptability of sending a referral, we expect main effects of each of the between and within subjects factors, but no interaction effect. Specifically, we expect that senders will view the act of referring a product as less acceptable than receivers, but for both groups we expect sensitivity to incentives such that adding an incentive increases how acceptable the act of sending a referral is. That is, we expect positive slopes for each condition, with the line for senders being significantly lower than the line for receivers.
Analysis plan: responses to each DV should be nested within participants, with a regression term for the between subjects factor and a test of the within subjects factor and their interaction. We used the stata xtmixed command, which simulates OLS when reml is used as an option:
xtmixed acceptable condition##time || id:, reml
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log restricted-likelihood = -5914.9516
Iteration 1: log restricted-likelihood = -5914.9515
Computing standard errors:
Mixed-effects REML regression Number of obs = 3000
Group variable: id Number of groups = 1500
Obs per group: min = 2
avg = 2.0
max = 2
Wald chi2(3) = 74.55
Log restricted-likelihood = -5914.9515 Prob > chi2 = 0.0000
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acceptable | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------------+----------------------------------------------------------------
condition |
sender | -.4547611 .1019563 -4.46 0.000 -.6545918 -.2549303
1.time | .275076 .0665149 4.14 0.000 .1447092 .4054428
|
condition#time |
sender#1 | .0835938 .0887786 0.94 0.346 -.0904091 .2575968
|
_cons | 4.379939 .0763879 57.34 0.000 4.230222 4.529657
--------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Identity |
sd(_cons) | 1.543999 .0378135 1.471636 1.619919
-----------------------------+------------------------------------------------
sd(Residual) | 1.206471 .0220417 1.164034 1.250454
------------------------------------------------------------------------------
LR test vs. linear regression: chibar2(01) = 729.47 Prob >= chibar2 = 0.0000
**Alternate Analysis**
As the dependent variables are single-item Likert items we will use maximum likelihood estimation with robust standard errors (Li, 2016) using the following code:
xtmixed acceptable condition##time || id:, vce(robust)
**References**
Li, C. H. (2016). The performance of ML, DWLS, and ULS estimation with robust corrections in structural equation models with ordinal variables. *Psychological methods*, 21(3), 369.
