Linear discriminative learning as a computational model implements a discriminative view of learning. In contrast to deep learning models that have multiple hidden layers based on non-linear functions, LDL networks are very simple two-layer networks and are linguistically transparent and interpretable. In LDL, the mental lexicon consists of five high-dimensional numeric matrices, each of which represents a different subsystem: the visual matrix, retina; the auditory matrix, cochlea; the semantic matrix; the speech matrix, speaking; and the spelling matrix, typing. For the current implementation, the semantic and the speech matrix are most important.

With regard to the mappings between vectors, linear mappings are implemented. These mappings are estimated using the linear algebra of multivariate regression. Thus, each mapping is defined by a matrix A that transforms the row vectors in a matrix X into the row vectors of a matrix Y, i.e., Y = XA. Then, A = X′Y, where X′ is the generalised inverse of X. We will return to the mapping of matrices in Section “Comprehension and Production,” and refer the interested reader to Baayen et al. (2019b) for an introduction to the mathematical details, as well as to Milin et al. (2017) for a detailed discussion on the restrictions and possibilities of linear mappings.

Another important feature of LDL is its notion of lexomes, i.e., basic semantic units corresponding to words or morphological functions. As outlined in Chuang et al. (2020), lexomes fall into two groups: content lexomes, and inflectional and derivational lexomes. Content lexomes can be morphologically simple or complex forms, i.e., cat and cats. Inflectional lexomes represent inflectional functions, e.g., number, tense, and aspect. Derivational lexomes represent derivational functions, e.g., morphological categories such as -NESS, -LESS, or UN-. Each lexome is paired with a vector of the aforementioned five subsystems. That is, for the semantic matrix, each lexome is paired with a semantic vector, making each lexome a pointer to a semantic vector on the one hand (Milin et al., 2017), and a location in a high-dimensional space on the other hand. For monomorphemic words, the semantic vector is identical to the semantic vector of the corresponding lexome. That is, the semantic vector of the word cat, cat→, is identical to the vector of the lexome CAT. For complex words, the semantic vector is the sum of its corresponding lexome vectors. That is, the semantic vector of the word cats, cats→, is the sum of the semantic vectors of the lexomes CAT and PLURAL, cat→+plural→. The implementation of LDL and the matrices necessary for the present paper are introduced in the subsequent sections. Please refer to https://osf.io/zy7ar/?view_only=ef43a5caf6444270a56074027d7d6482 for the full documentation of the data set, the implementation in R (R Core Team, 2020), and the R script.

The present paper follows the implementational basics outlined in Section “Introduction to LDL.” However, as we are interested in /s/ durations in pseudowords (and not in real words), there are a number of complications. The most important complication arises from the widely shared belief that pseudowords do not have meaning. So how can we map form and meaning with forms that have no meaning? In a recent study (Chuang et al., 2020) it was shown that the assumption that pseudowords are bare of meaning is most probably wrong. Due to their formal similarity with existing words, pseudowords resonate with the lexicon. As a result, they may in fact carry meaning. The authors demonstrate that quantitative measures gauging the semantic neighbourhoods of pseudowords predict reaction times of lexical decision and acoustic durations. The present study is inspired by these results and implements a similar architecture. To model resonance of pseudowords with the lexicon, both real words and pseudowords must be included in the networks. The following sections will detail the combined LDL implementation of real words and pseudowords.

The pseudowords and their phonetic realisations that this paper is based on are taken from the study of word-final /s/ production by Schmitz et al. (2020). In their study, participants were given pictures of “alien creatures” and their respective names (which were the target pseudowords), a short explanation of a situation, and a question relevant to the situation which was to be answered aloud. For each participant, pairings of pictures and pseudowords were randomised. That is, each pseudoword was represented by different pictures across participants. By button-press, a question was given to elicit an answer with the pertinent type of S while the context slowly faded out. The fading out of the question forced participants not to rely on the reading-aloud of the given context. In total, 24 pairs of pseudowords were used in that study. Each pseudoword form can act as singular or plural noun, e.g., glaits is either realised as singular, i.e., a glaits, or as plural, i.e., two glaits. Additionally, some pseudowords show a number of different realisations by the participants in the experiment, e.g., prups is sometimes produced as /pɹʌps/, and sometimes it is produced as /pɹups/. Thus, not 48 (i.e., 2 × 24) but 78 different phonological forms are included in the pseudoword set. Supplementary Table 1 gives an overview of all pseudowords and their phonological forms.

The second set of words contains real words and their phonetic realisations. Following Chuang et al. (2020) we extracted these words from the MALD corpus (Tucker et al., 2019a). While the MALD corpus contains 26,793 real words, only a subset of 8,285 words is used for a number of reasons. First, some 7,577 words in the corpus contain multiple affixes. As it was unclear how to handle such words, these were excluded. Second, only words for which we have semantic vectors could be used, leading to the exclusion of further 6,828 words. Third, only words with transcriptions available in the CELEX corpus (Baayen et al., 1995) were retained, i.e., there was no transcription available for 818 words. Fourth, 3,285 words showed ambiguities regarding their morphology, e.g., walks as a third person singular verb versus the plural of a noun. As huge numbers of words lead to extensive computation times, we decided to exclude such cases as well. The final set of real words contains 6,165 simple and 2,120 complex word forms.

As introduced in Section “The C Matrix: Form Vectors,” cue matrices are coded in binary form, giving information on which triphones are part of which word. For the current implementation, two such cue matrices are created using the WpmWithLdl package’s (Baayen et al., 2019a) make_cue_matrix function. First C_rw, the real word cue matrix, is created for the set of real words. Then, a second cue matrix, C_pw, is created for the set of pseudowords. However, C_pw is a lot smaller than C_rw as there are only 78 phonological forms for pseudowords, but more than 8,000 for real words. Thus, the C_rw is of dimension 8285×7610, while C_pw is of dimension 78×78. We will come back to this issue of differing dimensions in the next section.

To introduce semantics, i.e., semantic vectors, for the present set of real words, a pre-built semantic matrix A from Baayen et al. (2019b) was used. These authors derived semantic vectors based on the TASA corpus (Ivens and Koslin, 1991). For this, words were parsed into their lexomes, i.e., inflected words were represented by their stem and sense-disambiguated labels for their respective inflectional functions. Ambiguous forms, e.g., walks, were disambiguated using part of speech tagging (Schmid, 1999). Derived words were assigned a lexome for their stem and a lexome for derivational function. Then, following Baayen et al., 2016 and Milin et al. (2017), Naïve Discriminative Learning (Baayen et al., 2011; Sering et al., 2019) was used to build semantic vectors. The Rescorla-Wagner update rule (Rescorla and Wagner, 1972; Wagner and Rescorla, 1972; Rescorla, 1988) was applied incrementally to the sentences of the TASA corpus. That is, for each sentence the algorithm was given the task to predict the lexomes in that sentence from all lexomes of that sentence. This resulted in a 23562×23562 weight matrix A. This matrix lists all lexomes as rows and columns. Thus, for a given lexome at row i, the association strengths of this lexome with all other lexomes as given as columns is contained. In this state of the A matrix, lexomes predict themselves. Thus, the diagonal of the A matrix is set to zero (see Baayen et al., 2019b, for a discussion on this procedure). Lastly, columns which mostly contain zeros, i.e., no information, and show small variances (σ < 3.4 × 10⁻⁸) are removed. The resulting A matrix is of dimension 23,562 × 5030. Following the method outlined in Section “The S Matrix: Semantic Vectors,” a semantic matrix for real words S_rw can be constructed based on A. That is, the semantic vector s→ in S_rw for a simplex word is identical to its corresponding lexome, while the semantic vector s→ in S_rw for a complex word is the sum of its corresponding lexomes. That is, the semantic vector of apple is apple→, while the semantic vector of apples is the sum of the vectors of the lexomes APPLE and PLURAL, i.e., apples→=apple→+plural→. As a set of real words is used, S_rw contains only semantic vectors for this set of real words (instead of, e.g., all word forms of the TASA corpus). The final real word semantic matrix S_rw is of dimension 8285 × 5487.

While this procedure is rather straightforward, the creation of a pseudoword semantic matrix S_pw is not. Due to the nature of pseudowords, their lexomes are not contained within any corpus or our A matrix, for that matter. Instead, one can estimate a pseudoword’s semantic content by utilising the semantic and phonological information on real words, i.e., their C and S matrix (Chuang et al., 2020). That is, the same transformation matrix F that is used for mapping real word cues onto predicted real word meanings (see Section “Comprehension and Production”) can be used to map pseudoword cues onto their estimated semantics. That is, one must first solve

to obtain F. Then, one can make use of the pseudoword cue matrix C_pw, and estimate pseudoword semantics, as

with S_pw denoting the originally estimated semantic matrix for pseudowords. In this semantic matrix, pseudowords of identical segmental makeup show identical semantics as semantics are calculated only based on triphone occurrence, i.e., the semantics of pleeps_singular is identical to the semantics of pleeps_plural. To differentiate between singular and plural pseudowords, the semantic vector of the PLURAL lexome is added to all plural pseudowords in the S matrix. Similarly, the semantic vectors of ALIEN and CREATURE are added to all pseudoword semantic vectors as participants in the original production experiment were told that pseudowords describe alien creatures. As explained in Section “Model B: LDL Measures and Affix Specification,” the pairing of the pictures with pseudowords representing the alien creature was randomised during the experiment by Schmitz et al. (2020). A pertinent pseudoword thus only contains the semantics of “alien creature” as a constant part of its own semantics, while other factors such as appearance, e.g., colour, shape, or number of eyes, differ across participants. We can assume that in the course of the experiment, participants gradually came to realize that the looks of these alien creatures, i.e., colour, shape, etc., are not relevant to their label names. Thus participants were just aware of the fact that these are all alien creatures, without paying much attention to their individual features. Please see the aforementioned complementary material for a detailed implementation.

Pseudoword comprehension and production are not computed and evaluated in isolation but in combination with real words, simulating a real person’s lexicon in a pseudoword comprehension and production situation, respectively. For this, we created a cue matrix C_comb based on a combined set of words, containing all aforementioned real words and pseudowords. In total, 8440 word forms are part of this set of words. A combined semantic matrix S_comb is created by attaching S_pw to S_rw, and reordering its rows to reflect the same order of words as found in C_comb.

Then, using the functions of the WpmWithLdl package (Baayen et al., 2019a) in R, a comprehension model is trained and checked for accuracy. That is, taking form vectors as input for the prediction of semantic vectors of output, S^comb=CcombF is solved. Comprehension is successfully modelled for a word i if its predicted semantic vector s^i is most highly correlated with its targeted semantic vector s_i. This is true for 74.41% of cases (i.e., 6,165 word forms) in our comprehension model. In total, 25.59% of cases (i.e., 2,120 word forms) are incorrectly predicted, with 1,912 simple and 208 complex word forms. None of the incorrectly predicted word forms is a pseudoword.

Similarly, a production model is trained and checked for accuracy using functions of the aforementioned R package. Thus, semantic vectors are provided as input to predict form vectors as output, i.e., to solve T^comb=ScombG. Production is successfully modelled for a word i if its predicted triphones are those triphones present in its targeted cue vector in the correct sequence (possible sequences of triphones will be referred to below as “paths”). This is true for 97.3% of cases (i.e., 8,061 word forms) in our production model. In total, 2.7% of cases (i.e., 224 word forms) are incorrectly predicted, with 98 simple and 126 complex word forms. None of the incorrectly predicted word forms is a pseudoword.

In order to explore the potential of different measures emerging from the network to predict phonetic duration, we extracted a whole range of measures, based on the measures introduced by the WpmWithLdl package (Baayen et al., 2019a) and by Chuang et al. (2020). Please see the Supplementary Material for exploratory analyses of individual measures.

In the following, we first describe the semantic measures before we turn to the phonetic measures.

L1NORM and L2NORM: The L1NORM is the sum of the absolute values of vector elements of a given word’s predicted semantic vector s^, i.e., its city-block distance. The L2NORM is the square root of the sum of the squared values of a given word’s predicted vector s^, i.e., its Euclidian distance. For both variables, higher values imply more strong links to many other lexomes. Thus, both measures may be interpreted as semantic activation diversity.

DENSITY: For DENSITY, the correlation values of a word’s predicted semantic vector s^ and its eight nearest neighbours’ semantic vectors s_n1⋯s_n8 are taken into consideration. The mean of these eight correlation values describes DENSITY, with higher values indicating a denser semantic neighbourhood.

ALC: The Average Lexical Correlation, ALC, is the mean value of all correlation values of a pseudoword’s estimated semantic vector as contained in S_pw with each of the real word semantic vectors as contained in S_rw. Higher ALC values indicate that a pseudoword’s semantics are part of a denser semantic neighbourhood. Thus, ALC may be interpreted as a measure of semantic activation diversity for pseudowords.

EDNN: This variable describes the Euclidian Distance between a pseudoword’s estimated semantic vector s and its Nearest semantic real word or pseudoword Neighbour. Thus, higher values indicate a larger distance to the nearest semantic neighbour. EDNN may be regarded as a measure of semantic neighbourhood density.

NNC: The Nearest Neighbour Correlation is computed by taking a pseudoword’s estimated semantic vector as given in S_pw and checking it for the highest correlation value against all real word semantic vectors as given in S_rw. This highest correlation value is taken as NNC value. Thus, higher values indicate that a pseudoword is semantically close to a real word. Additionally, one can tell which real word a pseudoword’s semantics are closest to. This measure may be interpreted as a measure of similarity between nonce and real words, indicating the co-activation of a real word when confronted with a pseudoword.

SUPPORT: This measure describes the amount of support the word-final triphone (i.e., fs#, ks#, ps#, ts#) obtains for each pseudoword. The value of SUPPORT is extracted from T^. Higher values of this variable indicate a higher semantic support for the word-final triphone which includes the segment of interest, i.e., word-final S.

PATH_COUNTS: PATH_COUNTS describes the number of paths, i.e., possible sequences of triphones, detected for the production of a word by the production model. PATH_COUNTS may be interpreted as a measure of phonological activation diversity, as higher values indicate the existence of multiple candidates (and thus paths) in production.

PATH_SUM: PATH_SUM describes the summed support of paths for a predicted form. PATH_SUM may be interpreted as a measure of phonological certainty, with higher values indicating a higher certainty in the candidate form.

PATH_ENTROPIES: PATH_ENTROPIES contains the Shannon entropy values which are calculated over the path supports of the predicted form in T^. Thus, PATH_ENTROPIES may be interpreted as a measure of phonological uncertainty, with higher values indicating a higher level of disorder, i.e., uncertainty.

ALDC: The Average Levenshtein Distance of all Candidate productions, ALDC, is the mean of all Levenshtein distances of a word and its candidate forms. That is, for a word with only one candidate form, the Levenshtein distance between that word and its candidate form is its ALDC. For words with multiple candidates, the mean of the individual Levenshtein distances between candidates and targeted form constitutes the ALDC. Thus, higher values indicate that a word’s candidate forms are very different from the intended pronunciation. ALDC may be interpreted as a measure of phonological neighbourhood density as it takes into account real word neighbourhoods for pseudowords, i.e., large values indicate sparse real word neighbourhoods.

Besides the aforementioned variables extracted and computed from the LDL implementation itself (see Section “Measures”), the following covariates, adopted from previous analyses of word-final S (e.g., Plag et al., 2017; Tomaschek et al., 2019; Schmitz et al., 2020), are included in the analysis. The main reason for this is to allow us to compare the performance of these predictors with the performance of LDL predictors. LDL measures often correlate with traditional measures (such as lexical frequencies, transitional probabilities, or neighborhood densities), but the traditional measures have no clear correlating mechanisms in learning or processing.

There are, however, also covariates that do not tap into lexical properties, but that control for other influences, such as speech rate, the speaker, gender, the order of stimuli in an experiment, etc. These will be referred to as “non-lexical covariates” and they will also be included in our regression models.

AFFIX: This binary variable indicates whether a word contains an affix, i.e., whether the pertinent pseudoword is a singular or plural form. It takes the value NM for pseudowords without affix, and PL for pseudowords with affix.

SPEAKINGRATE: Analysing durational data, speech rate is a self-evident variable to consider. As speech rate is no inherent part of any LDL measure, we calculated speaking rate as the number of syllables in an utterance divided by the duration of the utterance (e.g., Tomaschek et al., 2019; Schmitz et al., 2020). This was done automatically using a script in Praat (de Jong and Wempe, 2008; Boersma and Weenink, 2019).

BASEDURLOG: Base duration was taken as a more local measure of speech rate (e.g., Plag et al., 2017, 2020; Schmitz et al., 2020). Here, the term “base” refers to the string of segments preceding the word-final S, for both non-morphemic and morphemic pseudowords. Base duration was then log-transformed to achieve a closer to normal distribution.

PAUSEBIN: To account for final-lengthening effects, stretches of silence between the offset of the word-final S and the onset of the following word were measured. Silence of 50 ms and above was considered as pause (Lee and Oh, 1999; Krivokapić, 2007). In order to make sure that closures of following plosives were not mistaken for pauses, their average closure duration (see Yao, 2007) was subtracted of the pertinent measured silence. Following the results by Schmitz et al. (2020), pause information was included as binary variable with the values PAUSE / NO PAUSE.

DISC: As some pseudowords were produced with multiple pronunciations, their transcription was incorporated as a categorical variable. This variable is called DISC after the DISC keyboard phonetic alphabet (Burnage, 1988).

BIPHONEPROBSUMBIN: The summed biphone probability for each pseudoword and its phonological variants is included as the binary variable BIPHONEPROBSUMBIN. It was calculated using the Phonotactic Probability Calculator (Vitevitch and Luce, 2004). The rationale for this variable is that more probable biphones should lead to shorter durations (e.g., Schmitz et al., 2020).

LIST & SLIDENUMBER: To account for priming effects, the list number (1–12) and the point of occurrence during the original experiment by Schmitz et al. (2020) are included.

PREC: To account for potential effects of the consonant preceding the word-final S (Umeda, 1977), it is included as PREC variable (similar to e.g., Tomaschek et al., 2019).

BIPHONEPROB: The probability of the final biphones /fs/, /ks/, /ps/ and /ts/ in monomorphemic words is included as covariate to account for potential effects of phonotactics (see Schmitz et al., 2020, for a detailed explanation).

FOLTYPE: As the segment following the word-final S is no part of the individual pseudoword, it is also not considered in LDL measures. Thus, the covariate FOLTYPE is introduced (similar to e.g., Tomaschek et al., 2019), coding the following segment by its segmental class (i.e., approximant APP for listen, fricative F for find, nasal N for know, plosive P for cook, and vowel V for eat), to account for potential effects of the following word (Klatt, 1976; Umeda, 1977).

SPEAKER, GENDER, AGE, LOCATION and MONOMULTILINGUAL: SPEAKER ID was included to account for general inter-speaker differences in production. GENDER, AGE, and LOCATION, i.e., the place in which the pertinent participant spent the bigger part of their life, were included as well. Additionally, participants who were early bilinguals were categorised as multilingual, while all other participants were categorised as monolingual in MONOMULTILINGUAL.

REAL: Some of the pseudowords in Schmitz et al.’s data set have an orthographically different, but phonologically identical real word counterpart. We introduced the variable REAL to control for this potential confound. This variable is TRUE for pseudowords with such a real word counterpart, and FALSE for those without. We considered the following real words as counterparts as given in Schmitz et al. (2020): glits corresponds to glitz, glaiks corresponds to Gleicks, glifs corresponds to glyphs, and pleets corresponds to pleats.

All of the following analyses make use of the following non-lexical covariates: BASEDURLOG, SPEAKINGRATE, SLIDENUMBER, and PAUSEBIN as variables concerning speech rate and continuity, PREC and FOLTYPE accounting for coarticulatory effects, LIST taking into consideration potential priming effects, MONOMULTILINGUAL, GENDER, LOCATION, AGE, and SPEAKER to account for speaker-individual differences, and REAL to include potential effects of real word counterparts.

We devised three kinds of model: First, a baseline model with the traditional predictor variables (plus the non-lexical covariates). Second, a model with LDL predictors that also includes AFFIX as a covariate (plus the non-lexical covariates). Third, a model that contains only the LDL predictors (plus the non-lexical covariates).

The three kinds of model will allow us to answer our research questions. Recall that our ultimate goal is to understand how systematic durational differences emerge between words of different, but homophonous morphological categories. Traditional lexical variables are predictive but cannot explain how morphology can make its way into durational differences. But these models can show that such differences exist by looking at the effect of the variable AFFIX. This is our baseline model. As an alternative we implement a model that uses LDL measures. If these measures are predictive, they offer an explanation of the morphologically-induced phonetic differences: they emerge as a by-product of the association of form and meaning in the mental lexicon, and this association is the outcome of discriminative learning. By having a model that also includes AFFIX as an additional predictor, we can see whether the LDL measures completely capture the morphological effect, or whether there is a residue of morphological information that is predictive of duration but is still not captured by the LDL measures.

This model is meant to resemble those in previous studies on word-final S duration (e.g., Plag et al., 2017; Schmitz et al., 2020). Thus, we make use of similar variables: AFFIX, BIPHONEPROBSUMBIN, and BIPHONEPROB, as well as those control variables included in all analyses of this paper. None of these covariates showed high correlation coefficients. Hence, no cautionary measures regarding collinearity were taken before an initial full model was constructed. The model selection process proceeded as explained in section “Model B: LDL Measures and Affix Specification.” That is, non-significant variables were excluded in a controlled step-wise fashion.

Then, variance inflation factors were checked. The covariates BIPHONEPROB and PREC showed high VIF values (i.e., 46.53 and 46.88, respectively), indicating potential overfitting of the model (e.g., Zuur et al., 2010; Fox and Weisberg, 2019). Consequently, PREC was removed from the model as it showed the highest VIF value, following the procedure described by Zuur et al. (2010). Re-fitting the model without PREC and re-checking the new variance inflation factor values revealed only non-problematic values.

Finally, the resulting model’s residuals were trimmed (e.g., Baayen and Milin, 2010). Data points with residuals larger than 2.5 standard deviations were removed, ensuring a satisfactory distribution of residuals. This procedure led to a loss of 4 data points, i.e., 0.61% of all data points. An overview of all variables used in the initial model is given in Supplementary Table 2.

This model makes use of all LDL measures as well as of the AFFIX variable. Additionally, the non-lexical covariates are included. One issue to address when considering a model with such a multitude of variables is collinearity (e.g., Baayen, 2008; Tomaschek et al., 2018). To avoid collinearity related problems later on, all variables were tested for correlation using the languageR package (Baayen and Shafaei-Bajestan, 2019). This correlation check resulted in eight correlation coefficients indicating a high degree of correlation, for which we assume the threshold to be |rho| ≥ 0.5. The pairs of correlated covariates as well as their correlation coefficients are given in Table 2.

Due to the high number of correlated variables, we opted for a principal component analysis (PCA; e.g., Venables and Ripley, 2002; Baayen, 2008; Tomaschek et al., 2018) to address collinearity issues. In a PCA, the dimensionality of the data is reduced by transforming the included variables into principal components. These transformations result in linear combinations of the predictors that are orthogonal to each other. Thus, the resulting principal components are not correlated.

The PCA was carried out using the PCAmix function of the PCAmixdata package (Chavent et al., 2017) in R, allowing the simultaneous integration of continuous and discrete variables. All variables given in Table 2 were included in the computation of the principal component analysis, which yields nine principal components. The next step of the PCA is to determine how many of these principal components are meaningful and thus should be retained for further use. For this decision, we followed several rules of thumb (e.g., O’Rourke et al., 2005; Baayen, 2008). First, any component that displays an Eigenvalue greater than 1 accounts for a greater amount of variance than had been contributed by one variable. Such a component is therefore potentially meaningful. Second, one should retain enough components so that the cumulative percent of variance explained is equal to some minimal value. Following other implementations of principal component analyses, we aim at a value of 80% (e.g., O’Rourke et al., 2005). Third, only interpretable components are to be retained. That is, each component is made up of loadings, i.e., parts of the variables included in the PCA’s computation represented by correlation coefficient values. If none of these variables is strongly represented in a component, the interpretability of that component is extremely low, rendering the component of small interest for further analyses. Following these three criteria, we find that the first three of the principal components show an Eigenvalue of one or higher. Also, the first three components account for 84% of variance. Considering the third criterion, all three components are strongly correlated with input variables. We therefore retain components 1 to 3 for further analysis, all of which show an Eigenvalue greater than 1, account for more than eighty percent of variance, and contain strong representations of variables in their loadings.² But what do these principal components mean? The highest loadings of the principal components, i.e., the correlation of the original variables to the pertinent component, are given in Table 3.

COMPONENT1 is most strongly positively correlated with PATH_COUNTS, PATH_ENTROPIES, and ALDC, while it is most strongly negatively correlated with PATH_SUM and SUPPORT. For PATH_COUNTS, higher values indicate the existence of multiple candidates (and thus paths) in production. It thus functions as an indicator of phonological uncertainty. Values of PATH_ENTROPIES relate to the level of uncertainty concerning the path supports of the predicted candidate form, with higher values indicating a higher level of uncertainty. For ALDC, higher values mean that a word’s candidate forms are very different from the intended pronunciation, indicating uncertainty in production. PATH_SUM describes the summed support of paths for a predicted form, with higher values indicating a higher certainty in the candidate form. Higher values for SUPPORT suggest more certainty in the choice of the word-final triphone. COMPONENT1 can thus be described as a dimension that represents phonological or articulatory certainty.

COMPONENT2 is most strongly correlated with L1NORM, L2NORM, NNC, and AFFIX. L1NORM and L2NORM both imply more strong links to many other lexomes with higher values indicating a higher semantic activation diversity. Higher values of NNC suggest a close real word neighbour, which leads to higher levels of co-activation of that real word when confronted with the pseudoword, also leading to higher semantic activation diversity. As for AFFIX, COMPONENT2 is positively correlated with the presence of non-morphemic S data points.

COMPONENT3 is similar to COMPONENT2 as it is also strongly correlated with L1NORM, L2NORM, and AFFIX. Again, for L1NORM and L2NORM higher values indicate higher semantic activation diversity. AFFIX is positively correlated for plural S data points. We will come back to the interpretation of this correlation in Section “Model B: LDL Measures and AFFIX Specification.”

In a next step, models were fitted using linear mixed-effects regression in R (R Core Team, 2020) using RStudio (RStudio Team, 2021) and as implemented by lme4 (Bates et al., 2015), lmerTest (Kuznetsova et al., 2017), and LMERConvenienceFunctions (Tremblay and Ransijn, 2020) to analyse the data on non-morphemic and plural S duration. The dependent variable, duration of S, was log-transformed following standard procedures to reduce the potentially harmful effect of skewed distributions in linear regression models (e.g., Winter, 2019). The name of this variable is SDURLOG.

Following the standard backward step-wise selection process for model selection (e.g., Baayen, 2008), a first model containing all remaining variables is created. That is, COMPONENT1, COMPONENT2, COMPONENT3, DENSITY, ALC, EDNN, BASEDURLOG, SPEAKINGRATE, PAUSEBIN, FOLTYPE, PREC, and REAL were included as fixed effects. The remaining variables, GENDER, LOCATION, MONOMULTILINGUAL, AGE, LIST, and SPEAKER, are included as random intercepts.

This full model was then continuously reduced through step-wise exclusion of non-significant variables. That is, a variable was considered as significant if it passed all of three tests. First, its F-value in the pertinent model had to yield a value below −2 or above 2. Second, the AIC value, i.e., the Akaike information criterion value, of the model including the variable had to be lower than the AIC value of a comparable model without the pertinent variable. Third, the results of log-likelihood tests comparing the model with to a model without the pertinent variable had to yield a p-value below the 0.05 threshold, thus indicating a significant improvement of model fit. This process was verified using the step function of R, which resulted in an identical model.

Then, variance inflation factors (VIFs) were computed. Predictors showing variance inflation factor values equal or greater than 3 are to be excluded due to the high risk of introducing multicollinearity and thus overfitting of the model (e.g., Zuur et al., 2010). For the present model, all variance inflation factor values are below 3.

Finally, the resulting model needed trimming of its residuals (e.g., Baayen and Milin, 2010). Data points with residuals larger than 2.5 standard deviations were removed to ensure a more satisfactory residual distribution. This procedure resulted in a loss of six data points (0.92%). An overview of all variables used in the initial model and their distribution is given in Supplementary Table 2.

This model uses all LDL measures but does not incorporate the AFFIX covariate. As in the previous model, there was a high number of highly correlated variables (see Table 2 with the exception of the correlation of AFFIX and NNC, as AFFIX is not included in this analysis). We therefore again computed a principal component analysis, following the procedure outlined in Section “Model B: LDL Measures and Affix Specification.” Following the first two criteria, we find that two principal components are to be retained. However, considering the third criterion, we find that the two components are not readily interpretable as they show relatively high positive or negative correlations with all or almost all variables, without indicating a clearly discernible dimension underlying the patterns of correlations. We therefore turned to another procedure to reduce collinearity issues.

For each set of variables with a correlation of |rho| > 0.5, models containing only the pertinent variable and a random intercept for subject are fitted and compared. Using log-likelihood tests for model comparison, the variable contained in a significantly better fit model is retained while those variables highly correlated with it are no longer used. In case of a non-significant difference, the variable of the model with the lower AIC value is retained. This procedure leads to the exclusion of L2NORM, PATH_COUNTS, PATH_ENTROPIES, and PATH_SUM.

Linear mixed-effects regression models were fitted according to the procedure given in Section “Model B: LDL Measures and Affix Specification.” That is, an initial full model was fitted with the following variables: L1NORM, ALDC, SUPPORT, DENSITY, ALC, EDNN, NNC, BASEDURLOG, SPEAKINGRATE, PAUSEBIN, FOLTYPE, PREC and REAL. As for random effects, random intercepts for GENDER, LOCATION, MONOMULTILINGUAL, AGE, LIST, and SPEAKER were included.

This full model was then continuously reduced through step-wise exclusion of non-significant variables, following the aforementioned criteria. Then, variance inflation factors were computed, resulting only in non-problematic values (e.g., Zuur et al., 2010). Finally, the resulting model needed trimming of its residuals (e.g., Baayen and Milin, 2010). That is, data points with residuals larger than 2.5 standard deviations were removed, ensuring a more satisfactory residual distribution. This procedure led to a loss of 8 data points, i.e., 1.2% of all data points. An overview of all variables used in the initial model and their distribution is given in Supplementary Table 2.

The final model of traditional measures includes main effects of the following variables: type of S (AFFIX), speaking rate (SPEAKINGRATE), log-transformed base duration (BASEDURLOG), pause (PAUSEBIN), the summed biphone probability (BIPHONEPROBSUMBIN), and following segmental type (FOLTYPE). As for random effects, random intercepts for SPEAKER and random slopes for AFFIX are included. The p-values of the analysis of variance of the final model are given in Table 4.

The marginal R-squared value of the model is 0.43, i.e., fixed effects explain 43% of variation in the data. Taking random effects into account as well, the conditional R-squared value is 0.62. That is, the model explains 62% of data variation in total (see Nakagawa et al., 2017, for details on marginal and conditional R-squared computation). Both R-squared values were computed using the MuMIn package (Barton, 2020). The R-squared values are similar to the values found by Schmitz et al. (2020) on their complete data set.

The estimates of the final model and their p-values are given in Table 5. The reference levels for the categorical predictors are: for AFFIX it is NM, for PAUSEBIN it is no-pause, for BIPHONEPROBSUMBIN it is high, and for FOLTYPE it is APP.

The predictor strength of individual covariates was checked by taking the final model as template. For each predictor variable, a model was fitted lacking the particular variable. This resulted in seven models, each lacking a different predictor. Then, R-squared values were computed for these models and finally compared. The variable leading to the highest decrease in R-squared value as compared to the final model is thus the variable showing the highest predictor strength. The results of this comparison are reflected in the hierarchy given in (1). The decrease in R-squared is greatest when removing BASEDURLOG, followed by PAUSEBIN, and so forth. The resulting order is identical to the one found by Schmitz et al. (2020) for the complete data set.

In the final model including LDL measures as well as the AFFIX covariate as parts of the individual components resulting from the principal component analysis, and fitted according to the procedure described in Section “Model B: LDL Measures and Affix Specification,” we find main effects of the first principal component (COMPONENT1), the third principal component (COMPONENT3), DENSITY, ALC, base duration (BASEDURLOG), following pause (PAUSEBIN), following segmental type (FOLTYPE), and preceding consonant (PREC). Regarding random effects, only a SPEAKER-specific random intercept turns out to significantly improve model fit. The p-values of the analysis of variance of the final model are given in Table 6.

The marginal R-squared value of the final model is 0.42, thus fixed effects explain 42% of the variation in our data. The conditional R-squared value of the final model is 0.60, that is fixed and random effects taken together explain 60% of variation.

The estimates of the final model and their p-values are given in Table 7. The reference levels for the categorical predictors are: for PAUSEBIN it is no-pause, for FOLTYPE it is APP, and for PREC it is f.

Similar to Section “Model B: LDL Measures and AFFIX Specification,” the predictor strength of individual covariates was checked by taking the final model as template. For each predictor variable, a model was fitted lacking the pertinent variable. This resulted in seven models, each missing a different covariate. Then, marginal R-squared values were computed and compared. The model showing the lowest of these values in turn missed the covariate with the highest predictor strength. The result of this procedure is reflected in the hierarchy in (2). The decrease in R-squared is greatest when removing BASEDURLOG, followed by PAUSEBIN, and so forth. In sum, variables containing measures obtained by our LDL analysis appear to be meaningful predictors of S duration.

Figure 3 shows the effect on S duration of the numerical variables included in the model. The estimated values of the dependent variable SDURLOG, i.e., S duration, and BASEDURLOG, i.e., base duration, are back-transformed into seconds. For COMPONENT1, higher values lead to longer S durations, while for COMPONENT3 (panel A), higher values lead to shorter S durations (panel B). Higher values of DENSITY (panel C) and ALC (panel D) come with shorter S durations. Longer bases come with longer S durations (panel E).

The partial effects of the categorical variables included in the final model are illustrated in Figure 4. Pauses lead to longer S durations (panel A), which is most likely a case of phrase-final lengthening (e.g., Cooper and Danly, 1981). There is also an effect of the following segment type, with S being shorter when followed by a vowel (panel B). This difference is significant for all consonant types being compared against vowels with the exception of fricatives. However, as there is only a small number of fricative cases in our data, this non-significant difference is potentially not meaningful. Lastly, there is an effect of preceding consonant on S duration (panel D). S duration is significantly longer if preceded by a voiceless labiodental fricative /f/ or a voiceless velar stop /k/ as compared to cases where S is preceded by a voiceless alveolar stop /t/. All other comparisons are non-significant.

Let us turn to the variables of interest, i.e., those derived from our LDL network. COMPONENT1 acts as a general measure of phonological certainty. High values of COMPONENT1 come with high values of PATH_COUNTS, PATH_ENTROPIES, and ALDC, indicating a high level of phonological uncertainty. At the other end of the COMPONENT1 dimension, high values of PATH_SUM and SUPPORT indicate a high level of phonological certainty. Higher uncertainty appears to lead to longer S durations, while higher certainty appears to lead to shorter S durations.

Recall from Section “Model B: LDL Measures and Affix Specification” that COMPONENT3 relates to semantic activation diversity and to the presence of the plural suffix. Higher values of COMPONENT3 indicate a higher level of semantic activation diversity. Higher levels of activation diversity then lead to shorter S durations (see panel B of Figure 3). High values of COMPONENT3 are positively correlated with the presence of plural S. It appears that the presence of plural makes words semantically more similar to each other as they share this meaning component. Hence it is to be expected that plural words live in a space of greater semantic activation diversity. COMPONENT3 is not only a measure of semantic activation diversity, but also indicates that plural pseudowords show a tendency of having a higher degree of semantic activation diversity as compared to monomorphemic pseudowords in general. DENSITY and ALC also tap into the semantics of pseudowords. That is, similar to COMPONENT3, higher values indicate higher levels of semantic activation diversity. These higher levels then lead to shorter S durations.

The final model of LDL measures only is fitted with main effects of the following variables: L1NORM, ALC, NNC, log-transformed base duration (BASEDURLOG), pause (PAUSEBIN), following segmental type (FOLTYPE), and preceding consonant (PREC). The SPEAKER variable is included as random intercept. The p-values of the analysis of variance of the final model are given in Table 8.

With a marginal R-squared value of 0.41, the fixed effects of this model explain 41% of variation within the data. The conditional R-squared value of the model is 0.61, that is the complete model accounts for 61% of variation.

The coefficients of the final model and their p-values are given in Table 9. The reference levels for the categorical covariates are: for PAUSEBIN it is no-pause; for FOLTYPE it is APP, and for PREC it is f.

As for both other final models, the predictor strength of the individual predictors was checked. Models with one of the predictor variables were constructed based on the complete final model. Then, marginal R-squared values were computed for each of these six models. A comparison of R-squared values then revealed the hierarchy of predictor strength given in (3). That is, the decrease in R-squared is greatest when removing BASEDURLOG, followed by PAUSEBIN, and so forth.

Base duration and speaking rate show identical effects as compared to the model fitted in Section “Model B: LDL Measures and AFFIX Specification,” i.e., longer base durations come with longer S durations, while higher speaking rates lead to shorter S durations. As for categorical variables, pauses again come with longer S durations, and S is shorter if followed by a vowel. There is also an effect of the preceding consonant, with S duration being significantly longer if preceded by a voiceless labiodental fricative /f/ or a voiceless velar stop /k/ as compared to cases where S is preceded by a voiceless alveolar stop /t/. These results are generally in line with those by the analysis in the previous section.

Taking a closer look at the variables of interest, we find that higher values of L1NORM, and ALC, i.e., higher semantic activation diversity, lead to shorter S durations. As in model B, higher levels of semantic activation diversity come with shorter S durations. For NNC, we find that S duration is longer if a pseudoword is semantically similar to a real word. The effects of L1NORM, ALC, and NNC are illustrated in Figure 5.

Previous studies (Zimmermann, 2016; Seyfarth et al., 2017; Tomaschek et al., 2019; Plag et al., 2020, 2017; Schmitz et al., 2020) reported that there are significant differences in the acoustic duration between different types of word-final S in English. Such durational differences challenge established feed-forward theories of morphology-phonology interaction (e.g., Chomsky and Halle, 1968; Kiparsky, 1982) as well as theories of psycholinguistics (e.g., Levelt et al., 1999; Roelofs and Ferreira, 2019; Turk and Shattuck-Hufnagel, 2020). The present study investigated whether measures derived on the basis of a discriminative learning theory are predictive of S durations in nonce words. In particular, we implemented LDL networks that model the production of a word based on its relation to the rest of the lexicon.

We explored the predictive possibilities of LDL measures by fitting three different models: a) a model based on the traditional predictors as used in previous studies (Plag et al., 2017; Tomaschek et al., 2019; Schmitz et al., 2020); b) a model with LDL measures and a variable AFFIX specifying the presence or absence of an affix; and c) a model with LDL measures but without a variable specifying the presence or absence of an affix. Both models with LDL measures show that such measures are predictive of S durations. This result is the most important of our study. While traditional variables such as lexical frequencies, bigram frequencies, transitional probabilities or neighbourhood densities measure important lexical properties, it is unclear why they would manifest themselves in a particular morphological effect in speech production. In LDL such effects can emerge through the mapping of form and meaning in a clearly defined process of discriminative learning.

All regression models showed a similar hierarchy of predictor strength for the variables included in the models. For the traditional model A, AFFIX is the third strongest predictor of S duration and for model B this spot is taken by COMPONENT3, while there is no comparable variable included in model C. Comparing the variance explained by the fixed effects of the different models, we find that the traditional model accounts for most variation, i.e., 43%, while the LDL model including the AFFIX variable accounts for 42%, and the LDL model without the AFFIX variable accounts for 41% of variation. Thus, in terms of marginal R-squared values, all three models are close to each other. To check whether these differences in marginal R-squared values are of significance, the three models were refitted to the untrimmed data set and then compared with an analysis of variance. The results suggest that there is no significant difference between the traditional model and the LDL model including the AFFIX variable. However, the LDL model without the AFFIX variable shows a significantly worse fit (p < 0.01). This seems to indicate that the LDL measures do not capture the full amount of the variance that is captured by the variable AFFIX. This means that there is still something about the morphological function that translates into duration and that is not properly modelled by the associative measurements of the learning network. The same problem holds, incidentally, for the traditional model (model A), in which the usual lexical measures (such as lexical frequencies, neighbourhood densities, etc.) and phonetic covariates (such as pauses, speech rate, etc.) are also not able to cover all durational variance. The morphological residue in both types of analysis remains a conundrum that calls for more sophisticated approaches in future research.

The LDL measures included in our final models are either concerned with semantic activation diversity (COMPONENT3, ALC, and DENSITY in model B; L1NORM, and ALC in model C), semantic similarity (NNC in model C) or with phonological certainty (COMPONENT1 in model B).

Higher degrees of semantic activation diversity come with shorter S durations. This effect is similar to the one which was reported by Tucker et al. (2019b) in a study on stem vowels, and Tomaschek et al. (2019) in their NDL study on S duration. A higher degree of activation diversity makes it “more difficult to discriminate the targeted outcome from its competitors” (Tomaschek et al., 2019:27). As for production, a prolongation of the acoustic signal is dysfunctional if the prolongation maintains or increases the discrimination problem instead of contributing to resolving it (Tomaschek et al., 2019).

In the model without AFFIX as predictor variable, NNC (i.e., a pseudoword’s semantic similarity to its closest semantic real word neighbour) emerges as significant (see model C). Why so? As reported in Table 2, the AFFIX variable and NNC are strongly negatively correlated (rho = −0.89). Post-hoc analysis shows that plural S has significantly lower NNC values as compared to non-morphemic S (Wilcoxon test, p < 0.001). It therefore appears that NNC takes over the role of differentiating between plural and non-morphemic S in model C.

As for phonological certainty, we find that higher phonological certainty leads to shorter S durations, while higher phonological uncertainty leads to longer S durations. Shorter durations in contexts of high phonological certainty may be related to effects of frequency, i.e., highly frequent forms are produced with higher certainty and are thus shorter.

The results of the present study may bring up further questions. First, are the predictive measures found for word-final S duration in pseudowords also predictive for word-final S duration in real words? Tomaschek et al.’s (2019) NDL implementation suggests that it is, but LDL networks still need to be implemented. It would be especially interesting to model those data sets that have yielded seemingly contradictory effects. Second, taking into account that the specification of AFFIX in the modelling process leads to a significantly better model fit, one may ask what the underlying reasons for this significant effect are. This then automatically leads to another question: Is it possible to catch the effect of the AFFIX specification in terms of (new) LDL measures?

To summarize, this paper was the first to investigate durational differences between different types of word-final S (non-morphemic vs. plural S) in pseudowords by means of an LDL implementation, measures, and resulting statistical analyses. The findings yielded important evidence on the question of how such durational difference come to be, i.e., they can be predicted based on their pseudoword’s relations to the lexicon. We demonstrated that durational differences emerge from the pseudoword’s resonance with the lexicon by way of differing degrees of semantic activation diversity and phonological uncertainty. These manifestations of the relations to other words in the lexicon in turn are the result of discriminative learning.