1.1. Convergence tasks and measurements

Convergence to an interlocutor within a conversation or to a model talker during shadowing or other exposure is often measured by the change in difference between the two speakers. In shadowing tasks, subjects are exposed to recordings of a model talker, which they repeat after, and the comparison is made of their speech before and after exposure. Conversational interactions present additional complications in defining reference points for the speakers, because both of the interlocutors are potentially changing.

In shadowing tasks, recordings are often naturally produced (e.g., Pardo, Urmanche, Wilman, & Wiener, 2017; Babel, 2012), but are sometimes manipulated to create extreme values in a particular measure (e.g., Nielsen, 2011; Yu et al., 2013), which ensures that the reference values which subjects are exposed to will be relatively far from all of the subjects’ starting values and differ in the same direction, eliminating possible effects of variation in absolute starting distance or the direction of the difference. However, it is possible that presenting reference values outside the range of normal human performance could undermine the ecological validity of such studies.

In interactional studies, it is difficult to reliably expose subjects to consistent extreme productions; while confederate interlocutors can be trained to produce certain behavioral patterns such as face rubbing and foot shaking (Chartrand & Bargh, 1999) and scripting can control syntactic and lexical choices (e.g., Branigan, Pickering, & Cleland, 2000), confederates cannot precisely control the phonetic details of their speech. Gijssels, Casasanto, Jasmin, Hagoort, and Casasanto (2016) resolved this issue of phonetic control in conversation by using a virtual interlocutor whose speech was entirely controlled and varied only in F0, the variable of interest. Felker, Tronsco-Ruiz, Ernestus, and Broersma (2018) offered a different resolution that allowed control of the productions while still presenting natural speech, by using a ventriloquist setup to present pre-recorded speech as if it were being produced live. While such a setup might allow acoustic manipulations in conversational interactions, the vast majority of current interactional studies of convergence do not control the interlocutors’ speech, so biases created by variation in starting distance can create large problems in analysis of by-speaker patterns.

Even when stimuli are not manipulated, the choice of model talker or the particular assigned interlocutor can influence how distant subjects are from these other speakers. Some studies use multiple model talkers (e.g., Pardo et al., 2017; Babel, McGuire, Walters, & Nicholls, 2014), and often find differences depending on the talker. However, many studies have a single model talker (e.g., Babel, 2010; Dias & Rosenblum, 2016; Mitterer & Ernestus, 2008) or have each participant converse with only a single interlocutor (e.g., Pardo, 2006; Abel & Babel, 2017), which could obscure possible differences in convergent behavior due to particular interlocutors or model talkers or the distance between them and the subjects. Studies always have multiple subjects, which reduces some of the potential noise due to variation in individual starting distance.

1.2. Modelling convergence

In shadowing tasks, subjects’ productions are measured before exposure to the model talker (S_b) and after exposure (S_R), and compared to the recordings of the model talker (R). Convergence in this system would be any change from S_b to S_R that makes S_R more R-like. In interactional tasks, it is similarly possible to compare productions from before the conversation or in conversations with other partners to productions within the shared conversation. When multiple interlocutors are available, subjects’ performance can be approximated with a linear combination of consistency (self-correlation), measured in ${\beta }_{S_b}$ , and convergence, measured in β_R, as well as noise, as given in (1).¹

However, many studies do not model both the effects of the subject and the model talker, instead looking at the change in similarity of the subject and the model talker. Some studies measure similarity subjectively with AXB designs (e.g., Goldinger, 1998; Pardo, Gibbons, Suppes, & Krauss, 2012); these tasks yield holistic judgements about whether the subject’s productions before or after exposure are more similar to the recordings of the model talker. Other studies measure similarity in particular acoustic characteristics as the absolute difference between the subject and the model talker (e.g., Babel, 2012; Pardo, Jordan, Mallari, Scanlon, & Lewandowski, 2013), with convergence as change in that difference (difference-in-difference, DID), as in (2).

DID is sometimes used to quantify convergence in conversational tasks, looking at the change from the two speakers’ starting distance and ending distance (e.g., Pardo, Cajori Jay, & Krauss, 2010), or their distance earlier and later in the conversation (e.g., Levitan & Hirschberg, 2011; Abel & Babel, 2017). When comparing interlocutors’ productions within a shared conversation, conversations may appear to be convergent due to participants independently being similarly influenced by the task, e.g., speaking more quickly as they become familiar with the task or their interlocutor; some studies compare distance between interlocutors to distance between individuals who were not interacting with each other, to control for such effects (e.g., Sanker, 2015; Oben & Brône, 2016), though many do not. In AXB or similar perceptual testing of distance, the X reference point for the interlocutor is sometimes taken from the middle of the conversation, compared to the other speaker’s productions before and after the conversation or before and during the conversation (e.g., Pardo, 2006). Many conversational convergence studies compare synchronous variations in both participants (e.g., Levitan & Hirschberg, 2011; Schweitzer & Lewandowski, 2013), either in addition to or instead of examining overall convergence across the conversation.

Some studies, rather than measuring change in distance between two interlocutors, look at correlations between partners’ productions or compare speakers’ productions in different conditions. Comparing participants’ productions under two manipulation conditions is a particularly common method in syntactic priming paradigms, e.g., comparing participants’ use of dative indirect objects and double accusative constructions when they had been exposed to descriptions using one or the other (Bock, 1986; Branigan et al., 2000). It is also used in some phonetic studies, e.g., comparing a condition of exposure to lengthened VOTs to a condition with shortened VOTs (Nielsen, 2011). Some work also compares correlations between conversational partners to correlations between speakers who were not conversing with each other (e.g., Gregory & Webster, 1996), or models predictors of each speaker’s productions, including the interlocutor’s productions as a predictor (Cohen Priva & Sanker, 2018; Schweitzer & Lewandowski, 2013), which both examine how conversational partners’ speech patterns are related to each other, rather than measuring distance per se.

1.3. Possible issues with DID

We argue that the DID approach has biases that make it unreliable for investigations of individual differences, except when the reference value is outside the range of normal productions. DID can still capture convergence broadly when aggregated across participants, but it introduces a degree of noise that could obscure convergence, particularly when compared across groups of participants.

The first issue is that DID does not distinguish between different trajectories producing the final distance; distance due to lack of convergence is treated the same way as distance due to speakers over-converging. For example, if S_b = 5 and R = 4, then S_R = 5 and S_R = 3 would be equally non-convergent (DID = 0), though the former reflects a lack of change and the latter reflects over-convergence.

The second issue is that eliminating the term for speakers’ consistency ( ${\beta }_{S_b}$ above) and the error term means that DID will underestimate convergence when the speaker and the interlocutor or model talker had a small initial distance, because individual variability from a nearly shared starting point is more likely to appear divergent than variability when the speaker and interlocutor or model talker had a larger starting distance. When the starting distance is small, there is little room for convergence, so noise from random variability is more likely to overshadow actual convergent shifts. This bias may produce the appearance of individual differences in convergence, even when the actual variation is simply reflecting differences in starting distance.

Some work includes a comparison of distance between subjects and their actual interlocutors or model talkers versus distance between ‘pseudo-pairs’ of subjects and speakers or model talkers they did not interact with (e.g., Levitan & Hirschberg, 2011; Miller, Sanchez, & Rosenblum, 2014; Sanker, 2015). Such methods are generally aimed at reducing measurement of convergence in shifts that are actually due to task-based effects. They additionally may reduce possible artifacts due to how convergence is measured, correlating with individuals’ starting distance from the interlocutor or starting distance from the population mean; such artifacts would be present both for real pairs and the pseudo-pairs, and be factored out.

The final issue is that difference-in-difference is susceptible to effects of regression to the mean, because it does not have an error term to control for noise. Effects of novelty and repetition may produce non-representative measured baselines even beyond effects of noise. In comparisons of pre-task and post-task productions of a word list (e.g., Pardo et al., 2010), participants may produce atypical utterances during initial pre-task productions based on lack of familiarity with the task or the particular words. While comparison of speech from early and late in the conversation (e.g., Levitan & Hirschberg, 2011; Abel & Babel, 2017) avoids some of these risks, the data contains additional noise due to less strict control over the items produced and the environments in which they were said. If a speaker’s baseline is not estimated correctly and an extreme variant is taken as that speaker’s baseline, reversion to less extreme values in subsequent performance after exposure or during an interaction could be interpreted as convergence with the interlocutor or model talker, whose baseline value is likely to be less extreme. Such an effect would make speakers whose baseline performance was measured as being closer to the mean seem less convergent than speakers whose baseline performance was further than the mean.

Differences in starting distances between subjects and interlocutors or model talkers and distance between subjects and the population mean may account for some of the variation in convergence across measures. In the same task, there can be differences in overall convergence in different measures (e.g., Babel, 2012; Sanker, 2015) and also differences in effects of conditioning factors (e.g., Bilous & Krauss, 1988; Pardo et al., 2017). Differences in production variability by measure might result in apparent differences in convergence, and obscure potential parallels across measures. Moreover, such measure-specific patterns mean that the results of a study might seem to be very different depending on which measure is used.

In this paper, we use data from four interactional studies of convergence (from Cohen Priva & Sanker, 2018) and simulated data, defined to lack any individual differences in convergence, to demonstrate limitations of the DID approach, and offer an alternative analytical method, linear combination, that does not suffer from these limitations. Though existing theories of convergence remain broadly consistent with the data and our results do not motivate a paradigm shift in analyzing the mechanism of convergence, the methodological issues that we present have implications for the role of individual differences in convergence, as we demonstrate that much of the appearance of individual differences is likely an artifact of how convergence is measured.

2.1. Underlying studies

We use data from the four convergence studies used by Cohen Priva and Sanker (2018). All studies are based on the Switchboard corpus (Godfrey & Holliman, 1997), a large collection of telephone conversations. In this corpus, each speaker was randomly paired with another speaker and given a conversation topic, participating in several such conversations. Most conversations involved interlocutors who participated in at least one other conversation, making it possible to compare their performance across multiple interactions. However, a speaker did not interact with the same interlocutor more than once. This provides a large corpus of natural speech for many speakers conversing with several different partners. The conversation sides are recorded separately, facilitating reliable measurements for each speaker.

Each conversation is annotated for clarity. To ensure reliable acoustic measurements (F0 median and variance), calls with high levels of background noise, echoing, or other issues were omitted. This left 464 speakers used for acoustic measures. For the other measures (uh:um ratio and speech rate), no conversations were omitted, comprising 518 speakers. Conversations averaged 6:20 minutes. The word-level annotations produced at MS State (Harkins, Feinstein, Lindsey, Martin, & Winter, 2003) were used to measure word duration.

We follow Cohen Priva et al. (2017) and Cohen Priva and Sanker (2018) in treating each subject’s performance as the value to be predicted (S_R), the average performance of that subject in other conversations as their baseline (S_b), and the average performance of their interlocutor in other conversations as the reference value (R). We did not use the interlocutor’s performance in the same conversation as the reference value because (a) it could be affected by the subject’s performance and (b) the subject and interlocutor could co-vary without converging, due to factors influencing both of them, such as the conversation topic, or the amicability of their interaction. We do not expect this choice to affect the arguments made in this paper; our goal is to compare different methods of measuring convergence, rather than to compare effects of using different reference values within a given method.

We use the four measures presented by Cohen Priva and Sanker (2018):

• F0 median F0 is the measure of the frequency of wave cycles produced by the vibration of the vocal folds. The measurement of frequency was converted into the mel scale, which provides a better approximation of human perception than Hz (Stevens, Volkmann, & Newman, 1937).

• F0 range F0 range was measured as the log of the ratio of the 75th percentile to 25th percentile of F0 measurements in mels.

• Speech rate Speech rate in a conversation was measured as the mean log speech rate of individual utterances. Following Cohen Priva et al. (2017), point-wise speech rate was measured as the actual utterance duration (including pauses) divided by the expected utterance duration. Expected utterance duration was calculated as the sum of the predicted durations of words in the utterance, each calculated as the predicted value of a linear regression using the median duration of that word in the entire corpus, the length of the utterance, and the distance from the end of the utterance. Unlike F0 measurements, speech rate was calculated based on hand-corrected values.

• uh:um ratio This measure was calculated as the log odds of uh versus um, two frequently-used filled pauses in English. The use of one or the other has been attributed to processing factors (e.g., Clark & Fox Tree, 2002), but is also influenced by other factors such as gender (Acton, 2011). Log odds were calculated as the predicted values plus the residuals of a logistic regression between the number of uh uses and um uses in each conversation side, which could be evaluated even when a subject never used one or the other.

2.2. Convergence models

We compare two approaches to modelling convergence: (1) difference in difference (DID) and (2) mixed-effects linear regression using the subject’s and interlocutor’s baselines as predictors of each subject’s productions. All the measures were standardized.

• DID follows the calculation in (2). Convergence is measured as the difference between two values: (a) The absolute difference between a subject’s performance without the exposure (in other conversations, in this case) and their interlocutor’s baseline, and (b) the absolute difference between the subject’s performance during the shared conversation and the interlocutor’s baseline. Positive numbers indicate convergence: The two speakers have more similar productions during the shared conversation than they do when not speaking to each other. Negative values indicate divergence: Speakers are more distinct from each other during their shared conversation than they were when not speaking to each other. These DID measures can then be used as an input in more complex models: grouped by subject to yield each subject’s tendency to converge, or compared across conditions to detect whether those conditions influence degree of convergence.

• While DID is generally not examined with regression models, we use these models to more closely parallel the alternative linear combination model that we propose, which can only be done with regression. The limitations of DID are based on comparing distance without reference to the raw values for each speaker, rather than being based on how those differences are modelled, so the regression structure should not change the behavior of DID results.

• When multiple subjects and interlocutors are present, as in Cohen Priva et al. (2017), (2) would take the form of (3), given in lme4 syntax. DID stands for the difference-in-difference. The random intercept (1 | subject) can capture some subjects’ tendency to have higher or lower difference-in-difference values, the random intercept (1 | interlocutor) can capture some interlocutors eliciting higher or lower difference-in-difference values, and the random intercept (1 | conversation) can capture particular conversations having higher or lower difference-in-difference values.

  (3)	DID ~ 1 +
  	(1 | subject) + (1 | interlocutor) + (1 | conversation)

  The intercept would be significantly positive if overall convergence is detected. The random intercepts for subject, interlocutor, and conversation would model individual differences in convergent behaviors by individual, by interlocutor, or by conversation.

• Linear combination follows the calculation in (1). Convergence is measured in a mixed-effects linear regression in which subjects’ performance is regarded as a linear combination of their baselines and their interlocutors’ baselines. Random intercepts are minimally used for subject, interlocutor, and conversation. This yields a formula of the form (4), given in lme4 syntax, in which subject.value represents the speakers’ performance in conversation, subject.baseline represents the subject’s performance outside the conversation, and interlocutor.baseline represents the interlocutor’s performance outside the conversation. The random intercepts (1 | subject), (1 | interlocutor), and (1 | conversation) account for additional per-subject, per-interlocutor, and per-conversation variability, respectively.

  (4)	subject.value ~ 1 + subject.baseline + interlocutor.baseline + (1 | subject) + (1 | interlocutor) + (1 | conversation)

  The intercept in this case is expected to be zero if the predictors are standardized, as they are in the studies presented here. The subjects’ baseline models consistency, the extent to which subjects’ speech patterns are consistent across conversations, and the interlocutors’ baseline would model convergence, the extent to which subjects are affected by their interlocutors’ performance.

  The by-subject intercept is expected to explain little variance, as the subject’s baseline is explicitly provided as a fixed predictor, and is included only for completeness. The by-interlocutor intercept is somewhat more necessary, as it can capture variance not due to convergence, such as modifying one’s speech when speaking with figures of authority or older people. The by-conversation intercept could capture interactional effects that are not convergence per se, which influence both speakers’ performance in the same direction, rather than toward one another.

  Additional fixed and random effects could be added to address particular research questions. For instance, adding a by-subject random slope for the interlocutor’s baseline could be used to model individual variation in convergence, yielding a formula of the form (5). The random slope would be non-zero if there is variance among speakers with respect to the reliance on the interlocutors’ baseline, thereby capturing individual differences in convergence. The formula explicitly requests that the variance-covariance matrix between the random intercept and the random slope is not evaluated because the random intercept is expected to be zero (so that the model would converge). This is done by replacing the single pipe | with a double pipe || in the expression specifying the random effects structure per subject.²

  (5)	subject.value ~ 1 + subject.baseline + interlocutor.baseline + (1 + interlocutor.baseline || subject) + (1 | interlocutor) + (1 | conversation)

  Similarly, if two convergence conditions are compared, a fixed effect per condition could be added, and the particular effect of the condition on convergence would be the coefficient for the interaction term between the condition and the interlocutor’s baseline, as in (6).

  (6)	subject.value ~ 1 + subject.baseline + interlocutor.baseline * condition + (1 | subject) + (1 | interlocutor) + (1 | conversation)

  Crucially, measuring convergence would always involve a manipulation of the coefficient of the interlocutor’s baseline in the mixed effects model. The use of this method is currently rather limited, but has been established as effective for measuring convergence. Cohen Priva et al. (2017) use this method, adding demographic fixed predictors, for convergence in speech rate. Cohen Priva and Sanker (2018) use the same method, with by-subject random slopes for the interlocutor’s baseline to measure individual differences in convergence, and find convergence in each of the four datasets (also used here). Cohen Priva and Sanker (n.d.) add two additional datasets, for which the linear combination method also proves powerful enough to capture convergence. Schweitzer and Lewandowski (2013) and Schweitzer and Walsh (2016) use a similar model, though they omit the subject baseline. Omitting the subject baseline as a fixed effect makes the per-subject random intercept serve as the speaker’s baseline. This should be very similar to using the explicit baseline, but the range of baseline values would be assumed to be normally-distributed, which may not be the case, as in the case in bimodal distributions (e.g., F0 values, see Figure 2).

3.1. Study 1: Proximity leading to divergence in DID models

3.1.3. Results and discussion

All models and data are available in the supplementary materials, and are summarized below.

In all of the four measures, the absolute distance between the subject’s baseline performance and the interlocutor’s baseline performance was positively correlated with higher DID values, as shown in (9). That is, the DID models were more likely to find convergence for subjects whose baselines were further from their interlocutors’ baselines. This is consistent with our predictions, and is not expected to hold in other methods of measuring convergence.

1. (9)

1. Study 1: Regression results for the absolute distance between the subject’s baseline performance and the interlocutor’s baseline performance across four measures, for DID models.

1.  
1.  

1. 	β	SE	df	t	p
   F0 median	0.16	0.02	3600	8.2	<0.0001
   F0 variance	0.54	0.02	3368	29.9	<0.0001
   Speech rate	0.45	0.02	3604	29.1	<0.0001
   uh:um ratio	0.43	0.02	4071	25.8	<0.0001

The estimate for the intercepts was negative in all measures, as shown in (10). This indicates that for subjects whose baselines differed very little from their interlocutors, the DID value was negative, not just a smaller positive value, and would thus appear to be divergence.

1. (10)

1. Study 1: Regression results for the DID models’ intercepts, across four measures.

1.  
1.  

1. 	β	SE	df	t	p
   F0 median	–0.17	0.03	1430	–6.3	<0.0001
   F0 variance	–0.60	0.03	1117	–22.7	<0.0001
   Speech rate	–0.51	0.02	991	–21.8	<0.0001
   uh:um ratio	–0.48	0.03	1301	–19.3	<0.0001

In contrast, the linear combination models found no significant effect for the interaction between absolute distance and the interlocutors’ baseline in any of the measures, as shown in (11). These results suggest that there is no special status for the initial distance between the interlocutor and the subject in influencing how much they actually converge, and the significant interactions found in the DID models were indeed only artifacts of how convergence was defined. Figure 1 illustrates the differences between the coefficients for the absolute distance between the interlocutors’ baseline and the speakers’ baseline. The DID model coefficients are large for all measures, while the coefficients for linear combination models are all negligible.

Figure 1

A comparison between the coefficients of DID and linear combination models for the four measures in Study 1. Each point is the estimate for the measure in that model, the thick lines are one standard error in each direction, and the thin lines are two standard errors in each direction. The two types of models are distinguished by color and shape. A dashed line marks zero; the linear combination model coefficients are all close to zero, having no effect, while the DID model coefficients are much larger.

1. (11)

1. Study 1: Regression results for the interaction between the interlocutor’s baseline performance and the distance between the subject’s baseline and the interlocutor’s baseline across four measures, for linear combination models.

1.  
1.  

1. 	β	SE	df	t	p
   F0 median	–0.0061	0.007	2882	–0.9	0.39
   F0 variance	–0.0108	0.013	1215	–0.8	0.42
   Speech rate	0.0018	0.008	780	0.2	0.82
   uh:um ratio	–0.0138	0.011	1988	–1.2	0.21

3.2. Study 2: Extreme baseline values appearing as convergence in DID models

3.2.3. Results and discussion

All models and data are available in the supplementary materials, and summarized below.

In three of the four measures (excluding F0 median), the absolute distance between the subject’s baseline performance and the mean of the distribution was positively associated with higher DID values, as shown in (14). This means that these models were more likely to find convergence for subjects whose initial values were more extreme, consistent with our predictions.

1. (14)

1. Study 2: Regression results for the absolute distance between the subject’s baseline performance and the mean of the distribution for DID models, across four measures.

1.  
1.  

1. 	β	SE	df	t	p
   F0 median	0.034	0.04	3683	0.8	0.4
   F0 variance	0.187	0.03	495	6.8	<0.0001
   Speech rate	0.147	0.02	4645	5.9	<0.0001
   uh:um ratio	0.119	0.03	4663	4.5	<0.0001

The distance between the subjects’ and interlocutors’ baselines (the focus of Study 1) was still positively correlated with higher DID values, as shown in (15), which suggests that these are two distinct effects. As before, the estimate for the intercepts was negative in all measures, signifying that DID was predicted to be negative for small differences in baselines (16).

1. (15)

1. Study 2: Regression results for the absolute distance between the subject’s baseline and the interlocutor’s baseline performance for DID models, across four measures, based on the revised formula in (12).

1.  
1.  

1. 	β	SE	df	t	p
   F0 median	0.16	0.02	3562	7.8	<0.0001
   F0 variance	0.48	0.02	3068	23.9	<0.0001
   Speech rate	0.40	0.02	2844	22.3	<0.0001
   uh:um ratio	0.39	0.02	3584	21.0	<0.0001

1. (16)

1. Study 2: Regression results for the DID models’ intercepts, across four measures, based on the revised formula in (12).

1.  
1.  

1. 	β	SE	df	t	p
   F0 median	–0.20	0.04	2992	–4.7	<0.0001
   F0 variance	–0.68	0.03	452	–23.7	<0.0001
   Speech rate	–0.56	0.02	1557	–22.6	<0.0001
   uh:um ratio	–0.53	0.03	1996	–19.5	<0.0001

There was no robust effect of distance from the median as a predictor of convergence for F0 median, in contrast to the DID models for the other three measures. This lack of effect is likely a result of F0 median having two distinct modes, as shown in Figure 2. Male and female speakers have little overlap in their F0 median values, so the mean of the joint male and female distribution is not a meaningful reference point. Instead, we may expect regression to the mean to be reflected by shifts of male and female speakers towards the respective means of the distribution of their own group. To test this hypothesis, we retrained the models, replacing absolute distance from the mean with absolute distance from the nearest mode (the nearest peak of the distribution). Indeed, in this post-hoc model, all four measures exhibited a significant correlation between high DID values and distance from the nearest mode (17).

1. (17)

1. Study 2: Regression results for the absolute distance between the subject’s baseline performance and the nearest mode of the distribution for DID models, across four measures.

1.  
1.  

1. 	β	SE	df	t	p
   F0 median	0.166	0.06	3674	2.6	0.00938
   F0 variance	0.151	0.04	3660	3.9	0.00012
   Speech rate	0.205	0.04	4720	5.1	<0.0001
   uh:um ratio	0.094	0.03	4667	3.2	0.00120

Figure 2

A density plot of F0 values, averaged for each subject. The values are split by sex on the left panel, and collapsed on the right panel.

In contrast, the linear combination models found no significant effect for the interaction between absolute distance from the mean and the interlocutors’ baseline; that is, there was no interaction between convergence and subjects’ baseline distance from the mean, as shown in (18). This suggests that there is no actual effect of the distance between subjects’ baseline and the mean on convergence. As in Study 1, the initial distance between the subject and the interlocutor was not significant (19). Figure 3 illustrates the differences between the coefficients in the DID and linear combination models. The DID model coefficients are visibly much larger than the linear combination model coefficients, which are consistently close to zero.

Figure 3

A comparison between the coefficients of DID and linear combination models for the four measures in Study 2 (before the post-hoc adjustment). Each point is the estimate for the measure in that model, the thick lines are one standard error in each direction, and the thin lines are two standard errors in each direction. The two types of models are distinguished by color and shape. A dashed line marks zero; the linear combination model coefficients are all close to zero, having no effect, while the DID model coefficients are much larger.

1. (18)

1. Study 2: Regression results for the interaction between the absolute distance between the speaker’s baseline and the mean, and the interlocutor’s baseline performance, across four measures, for linear combination models.

1.  
1.  

1. 	β	SE	df	t	p
   F0 median	–0.0085	0.01	266	–0.8	0.41
   F0 variance	0.0078	0.02	371	0.4	0.72
   Speech rate	0.0122	0.01	286	0.8	0.40
   uh:um ratio	0.0113	0.02	375	0.6	0.52

1. (19)

1. Study 2: Regression results for the interaction between the interlocutor’s baseline, and the absolute distance between the speaker’s baseline and the interlocutor’s baseline, across four measures, for linear combination models, based on the revised formula in (13).

1.  
1.  

1. 	β	SE	df	t	p
   F0 median	–0.0045	0.007	3312	–0.6	0.54
   F0 variance	–0.0123	0.014	1333	–0.9	0.38
   Speech rate	–0.0014	0.009	1118	–0.2	0.88
   uh:um ratio	–0.0157	0.011	2093	–1.4	0.17

These results suggest that the DID approach is prone to two distinct types of artifacts: Extreme baseline values for a subject can result in the overestimation of convergence, and small initial difference between a subject and interlocutor can result in the overestimation of divergence. Study 3 shows how these effects can lead to the overestimation of individual differences in convergence among subjects.

3.3. Study 3: Individual differences

3.3.3. Results

The results for the two models are summarized in (20). For DID models, there was a positive correlation between mean DID and mean performance for all measures; that is, the subject’s distance from the nearest mode was related to measured convergence. For the linear combination models, there was a much smaller relationship between per-individual slopes and mean performance, which did not consistently reach significance. Figure 4 shows the correlation between per-subject mean DID estimates and mean subject performance, for the DID models. Figure 5 shows the relationship between subjects’ mean performance and their convergence slopes for the linear combination models; no significant trend is found.

Figure 4

The relationships between proximity to a mode and DID values. The solid line shows the relationship between the variable using local polynomial regression (loess). The dashed line is the linear relationship. The X-axis is the absolute distance between the subject and the nearest mode. The Y-axis is the DID value. Each panel shows the relationship for a different measure. The results show a clear relationship between proximity to a mode and high DID, though it is much weaker for F0 median than for other characteristics.

Figure 5

The relationships between proximity to a mode and individual differences in convergence, as measured using a random slope for interlocutors’ baseline in a linear combination mixed effects model. The solid line shows the relationship between the variable using local polynomial regression (loess). The dashed line is the linear relationship. The X-axis is the absolute distance between the subject and the nearest mode. The Y-axis is the standardized per-subject convergence slope value. Each panel shows the relationship for a different measure. The results do not show a clear relationship between proximity to the median and convergence.

1. (20)

1. Comparison of the correlations between absolute distance from the nearest mode and the individual differences in the DID and mixed effects slope methods. In every measure, the relationship between the individual differences measure and absolute distance from the nearby mode was higher for DID than for the linear combination method.

1.  
1.  

1. 

These results demonstrate major differences between the two methods of measuring convergence. The DID models produced a consistently significant correlation between the subject’s distance from the nearest mode and the convergence measured for that subject, across all measures. The random slopes measured in the linear combination models did not produce significant correlations between individual convergence and distance from a mode in most measures. The key difference between this method and DID is that it does not assign value to the initial distance between the subject and the interlocutor, and leaves room for the estimation of noise. This allowance for noise and decreased reliance on starting distance make the linear combination method better suited to measuring convergence for particular individuals, without inflating convergence for subjects whose baselines were far from their interlocutors’ and understimating convergence for subjects with values close to a mode. The one measure in which the linear combination method found a statistically significant correlation between a subject’s convergence and baseline distance from the closest mode was speech rate, but as Figure 5 shows, that trend was minimal, was not present in the loess trend (as opposed to DID trends for the same data), and did not remain when p-values were corrected for multiple comparisons.

3.4. Study 4: Sampling in a simulated convergence dataset

3.4.3. Results and discussion

For the ten thousand samples, the median correlation between each subject’s before and after values was 0.802.

In DID models, the absolute distance of the subjects’ distance from the mean resulted in statistically significant (p < 0.05) positive coefficients 24.7% of the time, which is greater than what is predicted by chance. The coefficient for the absolute distance between the speakers and their interlocutors was statistically significant and positive 70.8% of the time.

In the linear combination models, in contrast, the absolute distance of the subject from the mean resulted in statistically significant positive coefficients 2.4% of the time, and the coefficient for the absolute distance between the speakers and their interlocutors was statistically significant and positive 2.5% of the time. Both of the results for the linear combination models are within what would be expected by chance. Figure 6 shows the density plots of the coefficients in DID and linear combinations. It is evident that the distribution of t-values in the linear combination models is centered around zero, and though for a small fraction of the models a positive correlation is found, negative correlations are similarly likely to be found. In contrast, in the DID models the coefficients are systematically biased to be positive, making spurious positive values more likely to be found than chance alone would predict.

Figure 6

Density plots of t-values for the two coefficients in the 10,000 samples taken in Study 4, for both the DID models and in the linear combination models. The dashed line marks 0, where the non-existent effect is supposed to be. The dotted line marks 2, which is roughly the point at which the t-value woulds appear to be significant. The distributions of the linear combination models are centered around zero, while the DID t-values are shifted such that false positive correlations are more likely to be found.

The results clearly indicate that even in the complete absence of underlying relationships between convergence and the participant’s distance from the mode or the starting distance between the participant and the interlocutor, such effects often spuriously emerge in DID models. When studies then look for individual differences in convergence as measured this way, these effects are likely to be interpreted as evidence for differences in individual tendencies in convergence, even though the true variation across individuals is just in their measured baselines. Because distance from the mode correlates with convergence, speakers whose mean performance is exceptional will seem to converge more than other speakers do, while speakers whose mean performance is close to their interlocutors will seem less convergent or even divergent.

We argue that the spurious effects in the DID model are due to mishandling the noise that exists between measurements of an individual. This predicts that spurious effects would be more likely in more noisy measurements than in less noisy ones. We therefore replicated the results presented above with varying amounts of noise, between 0.1 and 2 standard deviations, as sampled from a uniform distribution (that parameter was fixed at 0.5, as discussed above). Indeed, less noise translates to lower t-values for both variables in DID models, and more noise translates to higher t-values for both variables. The values seem to peak at high degrees of noise, but that happens after self-consistency drops below Pearson r = 0.5, which is not typical of phonetic variables (all the variables in our dataset have higher consistency). In contrast, the t-values for the two corresponding variables in linear combination models are not affected by the noise in the sample. Figure 7 presents these results.

Figure 7

The relationship between speaker self-consistency, as measured by noise SD, and the t-values of the DID and linear combination coefficients in Study 4. The x-axis is the degree of noise, in standard deviations (0.5 in the results reported above). The y-axis (left) is the t-value of the coefficient. The solid lines represent the trend of the t-values of the two variables, one in each panel. The Pearson r values associated with each noise value are provided by a black dashed line (marked on the right y-axis). All lines were smoothed by the gam method, using a 4-dimensional spline function for greater smoothing. It is evident that finding convergence is not predicted by the amount of noise present in the data when using linear combination models, but amount of noise does affect DID measures of convergence.

As with the rest of the analysis used in this paper, the code for the sampling procedures is available in the supplementary materials.