Valentina Aristodemo

Beatrice Giustolisi
email: beatrice.giustolisi@unimib.it

Giorgia Zorzi

Doriane Gras

Charlotte Hauser

Rita Sala

Jordina Sánchez Amat

Caterina Donati

Carlo Cecchetto

Jordan S Martin

Gesche Westphal-Fitch

Tecumseh Fitch

On the nature of role shift

Keywords: Visual artificial grammar learning, mildly context-sensitive grammars, sequencing, deafness, auditory scaffolding hypothesis

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Introduction

Humans excel at learning abstract patterns without explicit teaching, often with only minutes of exposure to a set of sequences that follow a target pattern or rule. This holds true from infancy to adulthood, and for a wide variety of different stimuli across multiple modalities (e.g. [START_REF] Saffran | Statistical learning by 8-month-old infants[END_REF][START_REF] Marcus | Rule learning by seven-month-old infants[END_REF][START_REF] Gomez | Variability and detection of invariant structure[END_REF][START_REF] Saffran | Dog is a dog is a dog: infant rule learning is not specific to language[END_REF]).

However, based on deficits in sequential processing seen in some deaf participants, Conway and colleagues proposed an influential hypothesis [START_REF] Conway | The importance of sound for cognitive sequencing abilities: The auditory scaffolding hypothesis[END_REF], the "auditory scaffolding hypothesis", which states that the development of general cognitive abilities related to representing temporal and sequential patterns is directly sustained by hearing experience.

The theoretical basis for this intriguing idea is that temporal and sequential changes are the foundations of sound stimuli and thus play a more crucial role in auditory than visual cognition. The socio-clinical and political implications of this hypothesis are not trivial, because sequencing and timing behavior are basic building blocks for many higher cognitive functions (Lashley, 1951). If the auditory scaffolding hypothesis is indeed correct, swift action should be taken to ensure each deaf newborn has his/her sense of hearing technologically restored as soon as possible after birth, independently of the language policy adopted.

As evidence supporting the auditory scaffolding hypothesis, a group of deaf children with cochlear implants showed no learning effect in an implicit sequence learning task [START_REF] Conway | Implicit sequence learning in deaf children with cochlear implants[END_REF]. Specifically, this group of deaf children with cochlear implants, all born in hearing families, and a control group of hearing children matched in age took part in an implicit sequence learning test with visual stimuli. Participants were asked to memorize and reproduce sequences of colored squares presented one by one on a computer screen. Without informing the participants, the task was divided into two phases, with an initial exposure phase followed by a testing phase. During the exposure phase, color sequences followed specific constraints, i.e., transition probabilities between colors were fixed. During the testing phase, half of the sequences followed the constraints of the exposure phase (familiar stimuli), and half of the sequences did not (unfamiliar stimuli). Implicit sequence learning was assessed comparing accuracy in reproducing familiar stimuli with accuracy in reproducing unfamiliar stimuli, with the difference in accuracy between familiar and unfamiliar patterns used to indicate whether implicit learning occurred. While the control hearing group showed a learning effect, the deaf children did not, providing support for the auditory scaffolding hypothesis.

However, the validity of the auditory scaffolding hypothesis has been called into question by several recent studies investigating sequence learning abilities in deaf children and adults (Hall, Eigsti, Bortfeld and Lillo-Martin, 2017;[START_REF] Giustolisi | Visual statistical learning with stimuli presented sequentially across space and time in deaf and hearing adults[END_REF]von Koss Torkildsen, Arciuli, Haukedal andWie, 2018 and[START_REF] Terhune-Cotter | Visual sequence repetition learning is not impaired in signing DHH children[END_REF][START_REF] Terhune-Cotter | Visual sequence repetition learning is not impaired in signing DHH children[END_REF]. From a theoretical perspective, Hall et al. (2017) contested the validity of drawing inferences on the effects of auditory deprivation from a population of deaf children of hearing parents, who were exposed to sound via cochlear implants. In fact, in this population, the period of auditory deprivation mostly overlaps in time with a period of language deprivation. To tease apart the influence of language deprivation versus auditory deprivation on sequencing skill development, Hall et al. evaluated the auditory scaffolding hypothesis in a third group of participants: deaf children without any delay in language exposure, i.e., deaf children of deaf parents, exposed to natural language (a sign language) from birth.

Moreover, Hall and colleagues raised some concerns about the experimental paradigm used in [START_REF] Conway | Implicit sequence learning in deaf children with cochlear implants[END_REF]. As previously mentioned, Conway and colleagues' testing phase consisted of showing the children a series of sequences one at time and asking them to reproduce each sequence.

The underlying hypothesis was that children should reproduce the sequences of the same type as those presented during familiarization better than unfamiliar sequences. But Hall and colleagues point out that children with high working memory span should show no learning effects, simply because their performance would be at ceiling for both types of sequences (i.e., they can correctly remember familiar sequences as well as non-familiar ones), meaning that learning effects would be detectable only in those children who failed to remember the unfamiliar presented sequences.

Hall and colleagues' empirical results directly challenged the auditory scaffolding hypothesis. First, they failed to replicate the results of [START_REF] Conway | Implicit sequence learning in deaf children with cochlear implants[END_REF]: neither hearing children, deaf signing children of deaf parents, nor deaf children of hearing parents, showed significant evidence of learning using Conway's implicit sequential learning task. However, using a serial reaction time task (Nissen and Bullemer, 1987), all the three groups of children showed learning effects. In the serial reaction time task, participants had to provide different responses depending on the position of a target item.

Unbeknown to participants, item position is determined by fixed transitional probabilities between possible locations. Learning these fixed transitional probabilities leads to reduced reaction times.

Interestingly, the fact that deaf children of hearing parents showed learning effects also argues against a possible language scaffolding hypothesis, i.e., that the development of implicit learning skills may depend on the temporal, hierarchical, and inherently social structure of language (Hall et al., 2017).

Also [START_REF] Terhune-Cotter | Visual sequence repetition learning is not impaired in signing DHH children[END_REF] assessed the validity of the auditory scaffolding hypothesis with a sequence learning task roughly similar to that used in [START_REF] Conway | Implicit sequence learning in deaf children with cochlear implants[END_REF]. Participants were asked to repeat sequences of either colored (easily nameable) or monochromatic (less easily nameable) shapes placed in four possible screen locations. Sequences might build upon the preceding one (repeating sequences) or not (random sequences). The comparison between repeating and random sequences was used as an index of sequence learning. In this case, participants were a group of deaf signing children and a control group of hearing non-signing children. Both groups showed sequence learning, and no significant difference was found between the deaf children and the control group.

Von Koss Torkildsen et al. ( 2018) assessed implicit learning of embedded triplets of unfamiliar alien figures (see Arciuli andSimpson 2011, 2012) in a group of prelingually deaf children with cochlear implants and a control group of hearing peers. The exposure phase was composed of a continuous stream of stimuli with embedded triplets, with participants engaged in a cover task (i.e. press a button upon the repetition of the same alien twice). In the testing phase, participants had to choose between two triplets, one that was familiar (i.e. already presented in the exposure phase) and one that was novel (i.e. never presented in the exposure phase but composed of the same aliens). Results showed that the performance of deaf children with cochlear implants was comparable to that of hearing children. In addition, in the deaf group the correlation between sequence learning performance and age of implantation or speech perception level was not significant. The author's discussion focused on the difference between the stimuli used in their task (pictures of aliens) versus those used in [START_REF] Conway | Implicit sequence learning in deaf children with cochlear implants[END_REF]) (colored squares). Von Koss Torkildsen et al. (2018) suspect that differences in verbal rehearsal strategies between deaf children with cochlear implant and hearing children might play a key role in determining the results of [START_REF] Conway | Implicit sequence learning in deaf children with cochlear implants[END_REF] (it is likely that participants verbalized colored squares, whereas it is unlikely that they could verbalize unfamiliar alien figures).

The proposition that the human ability to learn abstract rules in the visual modality without explicit teaching is independent from hearing experience is also supported by results from [START_REF] Giustolisi | Visual statistical learning with stimuli presented sequentially across space and time in deaf and hearing adults[END_REF]. To assess sequential learning in deaf adults with a lifelong lack of hearing experience but early exposure to a sign language, they used a version of the triplet paradigm (see Siegelman, Bogaerts and Frost, 2017) in which the stimuli were composed of abstract black shapes.

Contrary to the auditory scaffolding hypothesis, the vast majority of participants performed above chance level, showing a clear learning ability for sequence regularities without explicit teaching. The goal of the present work is to focus on tree structures with nested and crossed dependencies, which, compared to previous works, is a more abstract level through which sequences can be coded. This level of abstraction is required to account for the processes that we find in one of the more complex types of sequence that we use every day, that is language. Our aim was to evaluate whether a lack of hearing experience really hinders learning of sequential patterns at a higher level of complexity, as expected following the auditory scaffolding hypothesis. We know from previous research that human implicit learning abilities are not restricted to simple regularities but extend to more complex patterns generated by different types of artificial grammars [START_REF] Fitch | Computational constraints on syntactic processing in a nonhuman primate[END_REF][START_REF] Uddén | Implicit acquisition of grammars with crossed and nested non-adjacent dependencies: investigating the push-down stack model[END_REF]Westphal-Fitch et al. 2018). Using formal language theory, grammars can be ranked at different level of complexity on the so-called extended Chomsky hierarchy [START_REF] Jäger | Formal language theory: refining the Chomsky hierarchy[END_REF]. The simplest level in the hierarchy is that of regular grammars (e.g. sequential transition probabilities), with the next level of complexity given by context-free grammars, which also allow the processing of nested dependencies. Natural language syntax requires computations at a still higher level of the extended hierarchy, that of mildly contextsensitive grammars, which allow the processing of both nested and crossed dependencies. Center embedded relative clauses (Figure A) are an example of nested dependencies, whereas crossed dependencies (Figure 1 B) can be found in languages like Swiss German (the example in Figure 1 B is from [START_REF] Shieber | Evidence against the context-freeness of natural language[END_REF]. Context-free and context-sensitive grammars are collectively termed "supraregular" grammars. [START_REF] Shieber | Evidence against the context-freeness of natural language[END_REF], is in Swiss German and its meaning is "that we let the children help Hans paint the house".

The current study builds on previous work by Westphal -Fitch and collaborators (2018), who used a classic artificial grammar learning paradigm to assess visual pattern-processing skills in a group of 20 university students. Participants were exposed for several minutes to visual sequences made up of abstract tile patterns, generated by grammars at different formal levels. Participants were instructed simply to observe the sequences, without being involved in any cover task. Then, in a successive testing phase, they were asked to judge if novel strings were similar to that seen in the exposure phase or not. The artificial grammars that were used spanned all levels relevant for human language, i.e. included a regular grammar (AB N A), a context-free grammar (Mirror grammar) and a mildly contextsensitive grammar (Copy grammar). The AB N A grammar generates strings beginning and ending with an A element, with a variable number of B elements in the middle (e.g. A BB A; A BBBBB A), with a simple long-distance dependency between the two As. The Mirror grammar generates strings in which the second half-sequence is a reversal of the first (e.g. AAB BAA; ABAA AABA), leading to nested dependencies between its elements. The Copy grammar generates strings in which the second half simply reduplicates the first (e.g. AAB AAB; ABAA ABAA), creating crossed dependencies.

Participants were able to learn all three grammars: they correctly categorized as "similar" strings following the same pattern as exposure stimuli (including extensions, i.e. strings of a length not used in the exposure phase, thus showing generalization), and as "dissimilar" ungrammatical foils with incomplete dependencies. Successful learning was observed both at the group level, and individually for almost all participants.

In this study, we tested the auditory scaffolding hypothesis using the paradigm of Westphal-Fitch et al.. Specifically, we assessed the ability of deaf adults to acquire artificial grammars presented visually that differed in complexity. We investigated participants' ability both to recognize familiar stimuli, and to generalize the acquired rule to strings of different lengths than the lengths presented in the exposure phase. Moreover, we tested a control group of hearing participants. Our primary aim was to determine if hearing and deaf participants differ in the strategies they use to perform the task, but also to provide a possibility to replicate Westphal-Fitch et al.'s findings in a different group of participants.

Method

Participants

One group of deaf people and one group of hearing people participated. Deaf participants were 15 Italian Sign Language (LIS) signers (Mage = 33 years; SD = 14 years; range = 18 -62; 7 females, 8 males) recruited from the members of four Italian Deaf Institutes (***omitted to ensure blind review***). They were all born deaf and none of them had any associated disability or further sensory deficits. Seven out of fifteen deaf participants (47%) were native signers, exposed to LIS from birth, while eight out of fifteen (53%) were first exposed to LIS during childhood or adolescence. At testing, all participants were fluent LIS signers and used LIS as their main everyday means of communication.

They also used Italian with different degrees of proficiency. The mean number of years of education was 13.7 years (SD = 2.6). One additional deaf participant was excluded as they scored below the normal range in the Raven's test following Basso, Capitani and Laiacona's norms (1987).

Hearing participants were 15 hearing Italian speakers (7 females, 8 males) with no knowledge of LIS or any other sign language, recruited from ***omitted to ensure blind review*** through online social media and flyers. They were matched with the deaf participants in age (Mage = 34 years; SD = 15 years; range = 18 -59; t(28) = -0.16, p = 0.87) and level of education (Myears_edu = 13.5 years; SD = 2.4; t(28) = -0.07, p = .95). Overall cognitive abilities of the two groups of participants were assessed using Raven's Coloured Progressive Matrices [START_REF] Raven | Guide to Using the Coloured Matrices, Sets a, a b[END_REF]. Raw scores were corrected following Basso, Capitani and Laiacona's norms (1987). Corrected scores mean for the deaf participants was 31.67 (SD = 4.76), for the hearing participants was 33.33 (SD = 3.11), with no significant difference between groups (t(28) = -1.14, p = 0.27, d = -0.42 95% CI [-1.14 , 0.31]). Visuo-spatial working memory span was assessed with the Corsi-block tapping task [START_REF] Corsi | Human memory and the medial temporal region of the brain[END_REF]. The task was administered using the nine square blocks positioned on a plastic board. The mean span for the deaf participants was 5.67 (SD = 0.90) and for the hearing participants 5.53 (SD = 0.99); this difference was not significant (t(28) = 0.38, p = .70, d = 0.14 95% CI [-0.58 , 0.86]). All participants had normal or corrected-to-normal vision. No participant had previous experience with experimental psychology investigations. Participants gave their written informed consent prior to taking part to the experiment and received 20€ reimbursement for their participation. The study was approved by the ethics committee of the xxx (omitted to ensure blind review) and was carried out in accordance with the code of ethics of the World Medical Association (Declaration of Helsinki).

Materials and Procedure

The visual grammar learning approach built on previous work, using short strings of complexcolored tiles to probe pattern perception abilities. Previous research using such stimuli included Please refer to Westphal-Fitch et al. (2018) for a detailed explanation of materials and procedure.

Here, a brief summary is provided. The experimental stimuli are available xxx (omitted to ensure blind review. Reviewers can access experimental stimuli here: https://osf.io/25det/?view_only=32a02f113cec4445babe8757ddfea9ea Materials.zip).

Grammars and stimuli

The "warm-up" grammar was a regular grammar (AB) N that generated sequences of (AB) elements.

The testing grammars were three different grammars with long distance dependencies, located on different levels of the extended Chomsky hierarchy: a regular grammar (AB N A), a context sensitive grammar (Mirror grammar) and a mildly context-sensitive grammar (Copy grammar). The sequences generated by the grammars were composed of colorful abstract decorated small squares (tiles) sized 20 x 20 pixels, clearly belonging to two different categories. Sequence's structure was marked by black rectangles (black tiles) of 16 x 20 pixels (two for AB N A sequences, between A and B elements, and one for Mirror and Copy sequences, in the middle). Each tile was sequentially presented against a black background one at a time, one new tile every 166ms. New tiles were presented adjacent to the location of the previous tile, and tiles remained on the screen until the entire sequence was completed. Thus, the sequences appeared on the screen as if typed on a typewriter.

Then the whole sequence disappeared, and the screen remained blank until the participant responded.

Procedure

All participants received instructions in written Italian, deaf participants were additionally provided with instructions in LIS (video instructions) so that all participants could receive instruction in their preferred language. After completing the warm-up task (using the (AB) N grammar), each participant was tested on the three target grammars in a randomized order. For each grammar, the procedure was divided in two phases: exposure and subsequent testing. During each phase, all participants saw the same sequences, but in a different randomized order. During exposure (duration approx. 2 minutes), participants saw 30 grammatical sequences with N = 2, 3 and 5 1 (Figure 2 shows example of exposure sequences with N=5). During testing, participants saw 87 individual strings: 36 grammatical (N = 2 and N = 3, and extensions to N = 4 and N = 6) and 51 ungrammatical (N = 2, 3, 4 and 6). Ungrammatical "foil" strings included sequences with a missing element, and sequences with the correct number of elements, but incorrect category membership (see Westphal-Fitch et al. (2018), table 1, for more detailed information about the stimuli and appendix 1 for examples of grammatical and ungrammatical sequences). The participants' task in the test phase was to indicate whether the sequence followed the same schema as those seen during the exposure phase or not by pressing a yes/no key on a keyboard. Response time was not limited and no feedback was given. To ensure that participants understood the task and were paying attention, the experimental session was preceded by a training session during which participants were exposed to the warm-up regular grammar, (AB) N . Success on this grammar (accuracy > chance level, i.e. accuracy > 12/15, Exact binomial test, p = .02) was a prerequisite for taking part in the experimental session and it was achieved by all participants.

Results and analysis

We first considered the acquisition of the grammars by analyzing the percentage of "yes" responses (i.e., same schema) for grammatical and ungrammatical stimuli at the group level and individually.

Responses to grammatical and ungrammatical stimuli were compared using Wilcoxon signed-rank test. We analyzed the performance of the two groups separately, and of participants individually.

Then, accuracy data were analyzed using a multilevel Bayesian modeling framework developed in Westphal-Fitch et al. (2018). We assessed whether participants performed the task by effectively using the target grammar. Finally, we directly compared the performances of deaf and hearing participants. Data and code to perform the Bayesian multilevel analysis are available in the second author's GitHub repository (***Link omitted to ensure blind review*** Reviewers can access data and code here: https://osf.io/25det/?view_only=32a02f113cec4445babe8757ddfea9ea AGL_bayesfiles.zip).

Overall group performance

The first analysis dealt with the participants' ability to accept grammatical strings and reject ungrammatical foils. Deaf participants correctly responded "same" to 81% (SD = 19) of AB N A grammatical strings, and incorrectly responded "same" 17% (SD = 19) of AB N A ungrammatical foil strings (W = 120, p < 0.001, Cohen's d = 2.52, d' = 1.85). Considering the Mirror grammar, they responded "same" to 82% (SD = 18) of grammatical and 35% (SD = 22) of ungrammatical strings (W = 120, p < 0.001, Cohen's d = 1.73, d' = 1.31). As for the Copy grammar, "same" responses were given to 82% (SD = 18) of grammatical and 34% (SD = 20) of ungrammatical strings (W = 120, p <0.001, Cohen's d = 1.80, d' = 1.32).

Turning to hearing participants, in the AB N A grammar they responded "same" to 94% (SD = 9) of grammatical and 5% (SD = 7) of ungrammatical strings (W = 120, p <0.001, Cohen's d = 6.10, d' = 3.22). In the Mirror grammar, "same" responses were given to 84% (SD = 15) of grammatical and

Individual above-chance performance

The performance of participants of the two groups was also analyzed at an individual level, considering success in each grammar separately for varying N (see Table 1). Success was determined by comparing the number of correct answers by participants with a criterion calculated using a binomial test. For new grammatical sequences and ungrammatical foils with incomplete dependencies, "success" corresponded to at least 20/30 trials correct (exact binomial test, p = 0.049).

(27%)

Table 1 Individual participant performance on novel grammatical stimuli and ungrammatical foils with missing dependencies. Raw number and percentage of participants who reach the criterion value in the three grammars shown separately for the N conditions.

As Table shows, almost all participants' performance was above chance level with novel strings of N=2 and N=3. With N=4 strings, performance above chance level was observed in about half of the deaf participants, in 10/15 hearing participants in the two supra-regular grammars, and almost all hearing participants (14/15) in the regular grammar. All hearing participants and the majority of deaf participants (11/15) performed above chance level with N=6 stimuli in the regular grammar, whereas performance with N=6 stimuli in the supra-regular grammars was poor for both groups, with almost no deaf participant performing above chance level, and only 3 to 4 hearing participants doing so.

We also considered individual performance on ungrammatical stimuli with a single incorrect tile.

"Success" for these ungrammatical sequences of N=2 or 3 was counted as at least 12/15 correct "different" answers (p=0.02), and for sequences of N=4 as at least 9/10 correct answers (p=0.01).

Results are reported in

Bayesian multilevel analysis

Our previous analyses show that both hearing and deaf participants perform above-chance in most conditions, indicating that participants were sensitive to some structural properties of the grammatical stimuli. However, these results do not necessarily demonstrate induction of the intended AB N A, Mirror, or Copy target grammars, as individuals can potentially utilize a variety of alternate rules, including "short cuts" at different levels of grammatical complexity [START_REF] O'donnell | Using mathematical models of language experimentally[END_REF][START_REF] Fitch | Artificial Grammar Learning Meets Formal Language Theory: An Overview[END_REF]. If such alternative strategies exhibit correlated dependencies with the target grammar across specific subsets of experimental stimuli, statistically significant performance can be attained with an alternate rule. For example, a participant simply selecting all stimuli ending in A elements (a regular rule) would perform similarly to a participant using the supra-regular Mirror grammar for those stimuli that satisfy both rules (e.g. ABBA, ABBBBA, ABABBABA). To address this issue, we used a Bayesian multilevel model developed by Westphal-Fitch and colleagues (2018) to estimate the probability of each participants' performance being consistent with the induction of the target grammars versus a plausible set of alternate grammars. This was accomplished by coding each trial's response as being consistent (1) or inconsistent (0) with each possible grammar, and then comparing the estimated probabilities or odds of grammar-consistent responses across the entire set of responses and stimuli.

The alternative grammars are listed in the supplementary materials. We implemented these multilevel models within a Bayesian framework using Markov Chain Monte Carlo (MCMC) estimation, allowing hypothesis testing at both the individual and group level, based upon posterior probability distributions.

Here we applied this framework to determine (i) whether deaf and hearing participants' responses were more consistent with induction of the target grammars as compared to the alternate grammars, as well (ii) whether deaf and hearing participants differed in their overall performance and responses toward key sequence properties. Using Bayesian logistic regression models with random subject-level effects, we first estimated and compared the probability of the average deaf and hearing participant performing consistently with each of the target and alternate grammars. While absolute probabilities greater than chance (p > 0.50) indicate that performance is non-randomly associated with application of the test grammar, higher probabilities indicate a great chance that participants were in fact applying a specific grammar rule consistently across trials (e.g. p = 0.90 suggests that responses are on average consistent with a grammar in 90% trials, while p = 0.6 indicates non-random performance but consistent application of the rule in only 60% trials). Given that true and alternate grammars are both satisfied for particular subsets of experimental stimuli, and some participants may not apply the same rules consistently across all trials, it is expected that some alternate grammars may have above-chance probabilities of consistent responses. To address this issue, we further assessed whether deaf and hearing participants performed more consistently with the target grammar as compared to alternate grammars. Log odds ratios (LogOR) were used for these comparisons, so that LogOR = 0 indicated equivalent odds of performing consistent with a target and alternate grammar, while LogOR > 0 indicated greater support for the target grammar. Individual random intercepts were further used to assess patterns of individual variation unexplained by the average differences among hearing and deaf participants.

All analyses were done using the brms package [START_REF] Bürkner | Advanced Bayesian Multilevel Modeling with the R Package brms[END_REF], which facilitates Bayesian modeling using the Stan statistical programming language [START_REF] Carpenter | Stan: A probabilistic programming language[END_REF] in the R statistical environment (R Core Team, 2013). Weakly regularizing priors-Normal(0,2) for fixed effects, Half -Cauchy(0,2)

for random effects, and LKJ( 2) for correlation coefficients-were set on all model parameters to reduce the risk of false positives and facilitate more robust inference given our relatively small sample size [START_REF] Mcelreath | Statistical rethinking: A Bayesian course with examples in R and Stan[END_REF]. Following recent suggestions for more informative reporting of statistical models [START_REF] Mcshane | Abandon statistical significance[END_REF], multiple estimates are provided to summarize the posterior distributions of model parameters. In particular, we report the median posterior effect size (i.e. p for median probabilities, β for median logistic regression coefficients, and LogOR for median log odds ratios), the median absolute deviation (MAD) as a robust measure of statistical uncertainty around median estimates, the 90% credible interval (CI), and the posterior probability of a positive effect (i.e., 𝑝𝑝 + ).

When applicable, we also report Cohen's d to provide standardized effect sizes for comparison within and among studies. Note that, in contrast to classical p-values indicating the probability of observing data given a null hypothesis 𝑝(data|H 0 ), the reported 𝑝𝑝 + directly quantify the probability of a positive effect given the observed data 𝑝(H + |data). Values close to 1 thus indicate support for a positive effect, while values close to 0 indicate support for a negative effect. We interpreted effects with 90% CI excluding the null value (i.e. p = 0.5, β = 0, LogOR = 0) as providing clear support for a directional effect, which reflects a posterior probability of at least 0.95 in the expected direction (or alternatively ≤ 0.05 probability for an effect in the opposite direction).

To determine our a priori power for univariate comparisons of average grammar acquisition between deaf and hearing participants, we conducted a simulation-based power analysis [START_REF] Johnson | Power analysis for generalized linear mixed models in ecology and evolution[END_REF]. Specifically, we generated 200 random datasets assuming a small effect size for the mean difference between groups (Cohen's d = 0.3) on the transformed scale of our logistic regression model. Fifteen participant responses were simulated across 87 trials each for a single grammar, with modest between-individual residual variance (σ^2 = 0.25) in mean probabilities. We then analyzed these datasets with Bayesian regression models, following the approach taken for our empirical analyses. Power was assessed by the proportion of simulated datasets for which the regression models were able to recover the simulated group difference with a posterior probability of >95% (or, in other words, where the 90% credible intervals for the slope estimate did not overlap with 0).

Induction of the target grammar

On average, both hearing and deaf participants performance was consistent with induction of the target grammars (Figure 5 A), as evidenced by their overall high probabilities of responding appropriately to the target grammars on any random trial for AB

Comparison of stimulus properties

Despite both hearing and deaf participants providing strong evidence for induction of the target grammar, the overall greater performance of hearing participants for the AB N A and Copy grammars suggested that deaf participants may have responded differently to specific stimulus properties, such as whether a stimulus required recognition or generalization from the training stimuli (N = 2 and 3 vs. N = 4 and 6) or whether the stimulus was similar or dissimilar to the rule used for generating training stimuli. We therefore further explored whether interactions were present on average between trial-level stimulus properties (recognition vs. generalization, similarity vs. dissimilarity) and group (hearing or deaf status) for the AB N A and Copy grammars. Individual-level random slopes were further estimated for both stimulus properties to account for heterogeneity among participants.

Hearing participants were found to perform slightly worse on trials with similar as compared to dissimilar stimuli for the Copy target grammar (β = -0. grammars of varying complexity, up to the mildly context-sensitive level on the extended Chomsky hierarchy, provides clear evidence against the auditory scaffolding hypothesis, especially considering that our deaf participants were all born deaf, and all but one never used a cochlear implant.

All deaf participants used LIS as their preferred means of communication at the time of testing, but only half of them were native signers, namely they acquired LIS from birth in their family by interacting with other deaf signers. The small sample size did not allow us to directly compare the group of native with that of non-native signers. However, we saw no clear evidence that one group performed better than the other, as we had no evidence for specific participants performing worse than the others. To some this is in line with results from Hall and collaborators (2018), which argued against a possible language scaffolding hypothesis for sequencing abilities. They found that implicit sequence learning capabilities develop both in deaf children with a delay in auditory and language exposure (deaf children of hearing parents with hearing restored thanks to cochlear implants), and in deaf children with no hearing exposure and no delay in language exposure (deaf children with a deaf parent). Nevertheless, further studies should focus on this aspect, taking into account not only age of exposure to language, but also each participant linguistic competence, an aspect that was not considered in the present group of participants and that we acknowledge as a limit of the present study.

Another difficulty with the auditory scaffolding hypothesis is the presence of complex sequential patterns in sign languages. Even if it is true that sign languages rely more heavily on simultaneous information compared to spoken languages, this does not imply the absence of sequential processes in sign languages [START_REF] Sandler | Phonological Representation of the Sign: Linearity and Nonlinearity in American Sign Language[END_REF], particularly in the syntactic domain. Because sign languages have been primarily developed by individuals with auditory deprivation, who by that hypothesis should have limitations in processing sequential sequences, the presence of such patterns cannot be easily explained. This is true also in the context of our experiment, as LIS, the language of our deaf participants, has been shown to contain complex syntactic dependencies. A sentence that exemplifies this is the LIS counterpart of a textbook example that illustrates the presence of recursive structure in natural languages, namely a sentence like the following: "GIANNIa COUSIN POSSa COUSIN POSS-rotated-b We also compared the performance of the deaf participants with that of a group of age-and education matched hearing participants to observe if the two groups used similar strategies to perform the task.

Moreover, this allowed us to replicate the results obtained by Westphal-Fitch et al. (2018) with a group of hearing adults with different characteristics. Westphal-Fitch and colleagues' participants were all university students, whereas the present group of hearing participants was a heterogeneous mix of people with different educational backgrounds, and most did not attend university. In essence, the results were replicated: this second group of participants also learned the three intended grammars and generalize the rules to lengths not encountered during the exposure phase, with only some difficulties in generalizing to N=6 in supra-regular grammars.

Regarding the deaf-hearing comparison, both groups showed evidence of correctly identifying and applying the intended grammar, but the hearing group outperformed the deaf group in certain respects. This was especially true for the regular AB N A grammar and for the supra-regular Copy grammar, and was driven by a difference in the willingness to reject ungrammatical strings, with hearing participants more likely to do so. Nonetheless, as discussed above, both groups mastered the grammars; therefore, this difference in performance should not be attributable to superior rule of participants could learn rules at different levels of the formal language hierarchy, thus providing new evidence against the auditory scaffolding hypothesis. A slight decrease in the performance of deaf participants compared to hearing participants was not attributable to differing rule extraction capabilities, but may result from interference during stimulus encoding.

  The studies described so far assessed implicit learning of simple patterns involving fixed transition probabilities across stimuli. They investigate the ability of grouping together continuous items, which is one specific level of abstraction through which sequences can be coded, but not the only one. Following the taxonomy of sequence knowledge proposed by Deahene et al. (2015), at least four additional different systems are needed to represent sequencing abilities, each system based on a different degree of abstraction. Considering thus the multifaced nature of sequence knowledge, and the beforementioned socio-clinical implications of the auditory scaffolding hypothesis, it is in our opinion crucial an evaluation at different levels of sequence processing.

Figure 1 A

 1 Figure 1 A) Example of nested dependencies in natural language syntax. B) Example of crossed dependencies in natural language syntax. The sentence, taken from[START_REF] Shieber | Evidence against the context-freeness of natural language[END_REF], is in Swiss German and its meaning is "that we let the children

  both humans and animals (Stobbe et al., 2012), and the current study was closely based on an earlier publication using the same stimuli with hearing participants (Westphal-Fitch et al., 2018). Because this approach uses purely visual stimuli, but otherwise matches considerable previous work using auditory or written stimuli (Fitch & Friedrici, 2012; Reber,1967; Saffran, 2002), it offers an ideal test-bed for investigating pattern learning abilities in deal participants.

Figure 2

 2 Figure 2 Examples of grammatical sequences for all three grammars with N = 5.

Figure 3

 3 Figure 3 Percentage of "yes" responses ( = same schema) for novel grammatical N = 4 sequences (white) and ungrammatical sequences with incomplete dependencies (gray) in the three target grammars. Upper panel: deaf participants, lower panel: hearing participants.

Figure 4

 4 Figure 4 Percentage of "yes" responses ( = same schema) for novel grammatical N = 6 sequences (white) and ungrammatical sequences with incomplete dependencies (gray) in the three target grammars. Upper panel: deaf participants, lower panel: hearing participants.

For

  grammatical and ungrammatical sequences with incomplete dependencies of N = 4 the criterion was at least 15/20 correct (p = 0.02) and considering N = 6 at least 10/12 correct (p = 0

Figure 5

 5 Figure 5 Average and individual-level probabilities of grammar-consistent responses. Posterior probabilities are summarized by the median estimate (dot) and 90% credible interval (lines) for hearing (black) and deaf (grey) participants. (A) The estimated probability of responding consistently with the target (AB N A, Mirror, Copy) and alternate grammars for an average hearing and deaf participant on a random trial. (B) Individual-level probabilities of responding consistently with the target grammar on a random trial.

Figure 6

 6 Figure 6 Set diagram of the sentence 'the cousin of the cousin of the cousin of Gianni is Luca' in LIS (glosses, the video example is available xxx *** omitted to ensure blind review*** Reviewers can access the file here: https://osf.io/25det/?view_only=32a02f113cec4445babe8757ddfea9ea SupportingInformation-ex fig.6.mp4)

Table 2 .

 2 Most of the deaf and hearing participants performed above chance level in the regular grammar, whereas 40 to 50 % of deaf participants and 53 to 73% of hearing participants performed well in the two supra-regular grammars,

		Deaf participants (total = 15)	Hearing participants (total = 15)
	Condition	AB N A	Mirror	Copy	AB N A	Mirror	Copy
	N = 2 and N = 3	10	6	6	12	9	11
		(67%)	(40%)	(40%)	(80%)	(60%)	(73%)
	N = 4	10	7	7	14	8	11
		(67%)	(47%)	(47%)	(93%)	(53%)	(73%)

Table 2 Individual

 2

participant performance on ungrammatical stimuli with an incorrect tile. Raw number and percentage of participants who reached the criterion value in the three grammars separately for each N condition.

Acknowledgements

We sincerely thank our participants for their patience.

21% (SD = 19) of ungrammatical strings (W = 120, p < 0.001, Cohen's d = 2.05, d' = 1.76). As for the Copy grammar, hearing participants responded "same" to 82% (SD = 18) of grammatical and 18% (SD = 19) of ungrammatical strings (W = 120, p <0.001, Cohen's d = 2.15, d' = 1.81). To summarize, both groups succeeded at learning each of the three grammars: both the deaf and the hearing group correctly accepted novel grammatical sequences and rejected ungrammatical foils.

We then focused on participants' ability to generalize to stimuli of novel lengths. In the exposure phase, participants saw sequences of N = 2, 3 or 5. In the testing phase, we included sequences of N = 4 (generalization to an intermediate length) and sequences of N = 6 (generalization to longer strings).

Discussion

We evaluated the performance of 15 deaf and 15 hearing adults in a visual artificial grammar learning task, assessing implicit learning of three different grammars ranging in computational complexity from a regular grammar to a mildly context-sensitive grammar. Learning was assessed through participants' ability to accept grammatical strings and to reject ungrammatical foils. Test stimuli included sequences of the same length as those presented in the exposure phase and sequences of a different length, thus testing for rule generalization. We assessed visual artificial grammar learning in deaf participants in order to provide new data relevant to the evaluation of the auditory scaffolding hypothesis, which states that the ability to represent sequential patterns requires developmental support by hearing, predicting that congenital deafness should hinder the development of sequential processing in deaf individuals [START_REF] Conway | The importance of sound for cognitive sequencing abilities: The auditory scaffolding hypothesis[END_REF]. Specifically, we assessed sequencing abilities by focusing on a higher level of abstraction of sequential knowledge compared to the previous studies assessing the auditory scaffolding hypothesis. Our results show that, as a group, deaf participants learned all three grammars: they were able to accept novel grammatical sequences and reject ungrammatical foils. Bayesian analyses indicate that they did so by inducing each specific target grammar, not adopting alternative strategies.

Individual performance analysis indicated that almost all deaf participants (14/15 participants) performed above chance level with novel strings of length N=2 and N=3 (the length of sequences of the exposure phase), demonstrating a clear ability to recognize the given grammatical patterns without explicit teaching. Furthermore, half of the participants generalized the rule to a different length N=4, showing that they could effectively generalize the learned rules. Although the vast majority of deaf participants could generalize to N=6 in the regular grammar, this was not the case for the supraregular grammars, where performance was at chance level for almost all deaf participants. It is important to stress that this failure to generalize to N=6 by deaf participants does not represent a general inability to generalize, since they did so with N=4 generalization. Rather, this failure may indicate that N=6 sequences were too long to be tracked by the deaf population for reasons that we discuss below. All in all, the ability shown by the present group of deaf participants to acquire learning abilities in the hearing group compared to the deaf group, because deaf participants never performed worse on generalization relative to recognition. This confirms that deaf participants' difficulties were not triggered by generalization sequences, which provide the most informative evidence for rule extraction.

As generalization sequences elicited a similar performance as recognition sequences, we hypothesize that the deaf-hearing difference observed did not involve rule extraction, but rather some more general difficulty in encoding the incoming sequence in the deaf population. As previously pointed out by Von Koss Torkildsen et al. ( 2018), verbal rehearsal strategies may have a relevant impact on sequence learning tasks performance. Sequence tracking may be more difficult for the deaf population due to visual stimulus interference with their verbal coding strategies. Hearing participants may have implemented some form of verbal (vocal) recoding to track the incoming sequence, e.g. based on tile color (A tiles were gray/purple, whereas B tiles were red/green), or on the tiles' internal shape (A tiles were composed of rounded shapes, B tiles by angular shapes). Such a recoding strategy for the AB N A grammar might run "roundangledangled -angled -round." Deaf participants attempting to implement such verbal encoding would suffer from interference, since verbal recoding of the experimental stimuli would need to use the same visual channel as their signed language. This hypothesis could be explored by future research, for example, comparing hearing and deaf participants using non-visual (e.g. tactile) stimuli, or focusing on hearing participants and seeing if their performance will drop more if they perform the visual AGL task together with a verbal rehearsal task or with a task not tapping into verbal rehearsal. Another explanation that builds on verbal coding is that the sequence of recoded items would be a list of words for hearing participants and a list of signs for the hearing participants. As the sign span is known to be lower than the word span (including in LIS, cf. [START_REF] Geraci | How grammar can cope with limited short-term memory: Simultaneity and seriality in sign languages[END_REF], deaf participants would be at a disadvantage, especially for longer lists.

In summary, the present work investigated visual artificial grammar learning abilities in deaf and hearing adults to test the auditory scaffolding hypothesis considering sequences with the same degree of abstraction as that required to process natural languages. We showed that both groups Notes 1 For the Mirror and the Copy grammar, N refers to the number of long distance dependencies, whereas for the ABNA grammar, N corresponds to the number of B elements. For example, AABBAA is a sequence with N = 3 generated by the Mirror grammar, AABABB is sequence with N = 3 generated by the Copy grammar and ABBBA is a sequence with N = 3 generated by the ABNA grammar.

2 Following standard practice in the sign language literature, we indicate signs by small caps.

Subscripts are used to indicate that two signs are articulated in the same position in the signing space; poss is a possessive pronoun, pe is a demonstrative pronoun and poss-rotated is a special version of the poss sign that extends from the contralateral space to the ipsilateral space to spatially indicate the possession relation. Cf. [START_REF] Mantovan | Syntax: 14[END_REF] for more information on possessive structure in LIS.


London Romero Prieto

Andrea Verhulst

Valérie Delaunay

Michel Guillot

Under-five mortality during the COVID-19 outbreak: evidence from four demographic surveillance systems in low-income countries

Introduction and background

The under-five mortality rate halved during the last 30 years [START_REF] Liu | Global, regional, and national causes of under-5 mortality in 2000-15: an updated systematic analysis with implications for the Sustainable Development Goals[END_REF], but the progress could be jeopardized by the epidemic of COVID-19. Although, data from high-income countries suggest that young populations are less susceptible to the infection, the percentage of under-five deaths due to COVID-19 is very low, and overall mortality rates did not increase during the lockdown [START_REF] Davies | Age-dependent effects in the transmission and control of COVID-19 epidemics[END_REF][START_REF] Bhopal | Children's mortality from COVID-19 compared with all-deaths and other relevant causes of death: epidemiological information for decision-making by parents, teachers, clinicians and policymakers[END_REF][START_REF] Bhopal | Children and young people remain at low risk of COVID-19 mortality[END_REF][START_REF] Odd | Child mortality in England during the COVID-19 pandemic[END_REF]; simulation models anticipate a large increase in maternal and under-five mortality in low-and middle-income countries, due to the indirect effects of the epidemic of COVID-19 [START_REF] Roberton | Early estimates of the indirect effects of the COVID-19 pandemic on maternal and child mortality in low-income and middleincome countries: a modelling study[END_REF][START_REF] Mao | Achieving global mortality reduction targets and universal health coverage: The impact of COVID-19[END_REF][START_REF] Osendarp | The COVID-19 crisis will exacerbate maternal and child undernutrition and child mortality in low-and middle-income countries[END_REF][START_REF] Hillis | Global minimum estimates of children affected by COVID-19-associated orphanhood and deaths of caregivers: a modelling study[END_REF]. Indirect effects are informed by assumptions that build on the experience of the Ebola outbreak of 2014 in West Africa and the epidemic of severe acute respiratory syndrome of 2003 in Taiwan.

The indirect effects may be due to the interruption of health services, the postponement of diagnosis and treatments, the discontinuity of health interventions and vaccination programs, undernutrition, economic hardship, and lack of social support due to the mortality of parents. Some evidence indeed seems to point at a reduction in the utilization of maternal, neonatal, and child health services; and a reduction in birth related emergencies and the number of births in hospitals [START_REF] Ahmed | The effect of COVID-19 on maternal newborn and child health (MNCH) services in Bangladesh, Nigeria and South Africa: call for a contextualised pandemic response in LMICs[END_REF][START_REF] Ashish | Effect of the COVID-19 pandemic response on intrapartum care, stillbirth, and neonatal mortality outcomes in Nepal: a prospective observational study[END_REF][START_REF] Mor | Impact of the COVID-19 pandemic on excess perinatal mortality and morbidity in Israel[END_REF]. These reductions have been related to travel restrictions and the fear of transmission in hospitals. However, there is still little evidence demonstrating the negative impact of COVID-19 on under-five mortality in the context of less-and middle-income countries.

This paper investigates the effect of the epidemic of COVID-19 on under-five mortality in four Health and Demographic Surveillance Systems (HDDS): Siaya-HDSS in Kenya [START_REF] Odhiambo | Profile: the KEMRI/CDC Health and Demographic Surveillance System -Western Kenya[END_REF], Basse-HDSS in the Gambia1 , Niakhar-HDSS in Senegal [START_REF] Delaunay | Profile: The Niakhar Health and Demographic Surveillance System[END_REF], and Matlab-HDSS in Bangladesh [START_REF] Alam | Health and Demographic Surveillance System (HDSS) in Matlab, Bangladesh[END_REF]. Inasmuch as the number of deaths due to COVID-19 is dependent on testing coverage, variations in case definitions, or the performance of health facility-based surveillance, the effect of COVID-19 on under-five mortality is estimated to be indirect, through a measure of excess mortality [START_REF] Leon | COVID-19: a need for real-time monitoring of weekly excess deaths[END_REF]. We calculate the excess of mortality as the difference between the probabilities of dying in a calendar month affected by the pandemic and a counterfactual estimate that only depends on the secular trend in mortality of each site before the outbreak of COVID-19. We approach to this aim by forecasting the life tables of under-five mortality and quantifying the effects of the epidemic of COVID-19 on the most relevant indicators of mortality from zero to five years (e.g., the neonatal mortality, the post-neonatal mortality, the infant mortality rate, the child mortality rate, and the under-five mortality rate). Preliminary results did not identify any evidence of excess mortality that can be attributed to the COVID-19 outbreak, and these findings would contradict the predictions of earlier simulation studies.

Data

This paper uses data collected in four sites of demographic surveillance: Siaya-HDSS in Kenya (2008-2020), Basse-HDSS in the Gambia (2008-2021), Niakhar-HDSS in Senegal (1997-2020), and Matlab-HDSS in Bangladesh . The Siaya-HDSS has been managed through a collaboration between the Kenya Medical Research Institute (KEMRI) and the Centers for Disease Control (CDC). The total population of the site is approximately 260,000 residents living in 393 villages and the population below 5 years is estimated to be 31,000 individuals. The Basse-HDSS has been founded by the Medical Research Council and, as of 2019, covers a population of approximately 190,000 residents living in 219 settlements. The population below the age of 5 years is estimated to be 30,000 individuals. The Niakhar-HDSS was established by the Institut de Recherche pour le Développement of Senegal and the area of surveillance has an approximated population of 45,000 residents living in 30 villages. The Matlab-HDSS has been in operation since 1966 and is administered by the International Centre for Diarrhoeal Disease Research. The Matlab HDSS covers a population of approximately 235,000 residents. Individual records of these sources include the exact date of all life events (i.e., births, deaths, and migration). Information has been collected prospectively, visiting each household three or four times per year. This frequency ensures a short time of retrospection that improves the quality of the reported dates. Compared to other data sources for estimating mortality, HDSS have the advantage that events and exposure are calculated from the same source. In principle, the relatively short periods of retrospection may also contain reporting bias. Synthetic cohort life tables were constructed for estimating under-five mortality by detailed age groups. First, mortality rates were estimated dividing the number of deaths 𝑑 ! " by the number of person-years lived in the same interval of age and time 𝐿 ! " . The following exact-age cut-offs were used: 7, 14, 21, and 28 days; 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 18, and 21 months; and 2, 3, 4, and 5 years. Second, cumulative probabilities of dying 𝑞(𝑥), were calculated from age-specific mortality rates under the assumption of constant mortality hazards within the age interval, which is a tenable assumption considering that age intervals are small. Inasmuch as migration is also a source of attrition in these populations and out-migrants can return to the areas of demographic surveillance after some time abroad, mortality and migration rates were calculated in a simultaneous process with two exits (i.e., mortality and out-migration) and two possible entries (i.e., birth and in-migration). In this regard, permanent out-migrants stop contributing events and exposure after leaving the surveillance site (i.e., right censoring); and returning-migrants do not add exposure while they were living outside of the surveillance area (i.e., interval censoring). Therefore, the number of deaths only include those occurring within the borders of demographic surveillance, as well as the mortality experience of the in-migrant population that moved into the surveillance area sometime after birth.

Methods

Excess under-five mortality during the COVID-19 epidemic

Excess mortality was estimated as the difference between the probabilities of dying observed in one calendar month affected by the pandemic and a counterfactual that is depending on the secular trends in mortality of each site before the COVID-19 outbreak. Compared to other approaches that can be used for estimating excess mortality, forecasting the probabilities of dying (or the mortality rates) have some clear advantages. First, before the pandemic there was a rapid decline in the under-five mortality of these populations, thus the simple average of the last five years tends is likely to overestimate the expected mortality. Second, compared to the number of deaths, mortality rates and probabilities of dying are less sensitive to changes in the exposure. This is important because data collection was interrupted during lockdown periods and not all households are immediately revisited whenever fieldwork resumed. This attrition -or temporary loss of follow-up-, will depress the observed number of deaths, but mortality rates and probabilities of dying will not necessarily be affected because these metrics also account for the changes in the population at risk of dying.

Forecasting the under-five mortality

Following Lee-Carter's approach [START_REF] Lee | Modeling and forecasting US mortality[END_REF], life tables of under-five mortality were forecasted from the model of mortality represented by Equation 1. In this case, the natural logarithm of the cumulative probability of dying at the exact age 𝑥, and time 𝑡 (in months), indicated by 𝑞(𝑥, 𝑡), is a linear combination of the age-specific coefficients {𝑎 # , 𝑏 # , 𝑐 # , 𝑑 # }, and the value of three time-specific parameters {𝑘 $ (𝑡), 𝑘 % (𝑡), 𝑘 & (𝑡)}, indicating three independent mortality indices. From this perspective, the forecasted value of the cumulative probability of dying 𝑞(𝑥, 𝑡 + 𝑠), will result from a new set of time-specific parameters {𝑘 $ (𝑡 + 𝑠), 𝑘 % (𝑡 + 𝑠), 𝑘 & (𝑡 + 𝑠)}, keeping constant the age-specific coefficients of the model {𝑎 # , 𝑏 # , 𝑐 # , 𝑑 # }.

ln[𝑞(𝑥, 𝑡)] = 𝑎

# + 𝑏 # • 𝑘 $ (𝑡) + 𝑐 # • 𝑘 % (𝑡) + 𝑑 # • 𝑘 & (𝑡) + 𝑒 #,( (1)
Life tables of under-five mortality were forecasted after estimating the time parameters and the age coefficients of the model. For each HDSS, all values of the mortality model were estimated in two steps. First, the constant term 𝑎 # was estimated as the simple mean of the dependent variable ln[𝑞(𝑥)]. A total of 22 values of 𝑎 # were estimated, considering the number of cut-off points of age. Second, the other three sets of age-specific coefficients {𝑏 # , 𝑐 # , 𝑑 # } and the three sets of time-specific parameters were estimated by the Singular Value Decomposition (SVD) of a matrix conformed by the deviations from the mean {ln[𝑞(𝑥, 𝑡)] -𝑎 # }, at different ages (in rows) and periods (in columns). The values of {𝑏 # , 𝑐 # , 𝑑 # } correspond to the first three columns of the left eigenvectors, while the values of {𝑘 $ (𝑡), 𝑘 % (𝑡), 𝑘 & (𝑡)} are given by the first three rows of the right eigenvectors multiplied by the first three eigenvalues of the SVD.

Considering the properties of the SVD, if all 22 orthonormal eigenvectors were included in the model, the error term in Equation 1 -represented by 𝑒 #,( -would be equal to zero at all ages and periods of time. Hence, the selection of three indices of mortality and three sets of age-specific coefficients rather than one -as typically implemented-, is to maximize the fitting of the model, while allowing a reasonable increase in the complexity of the time dynamic. Finally, after the SVD, the estimated coefficients and the parameters were rescaled to make the sum of the agespecific coefficients {𝑏 # , 𝑐 # , 𝑑 # } equal to one, across ages; and the mean of the mortality indices {𝑘 $ (𝑡), 𝑘 % (𝑡), 𝑘 & (𝑡)} equal to zero, over time.

Time series analysis

Following a standard notation and 𝐿 representing the lag operator (i.e., 𝑘(𝑡 -𝑠) = 𝐿 ) • 𝑘(𝑡)) [START_REF] Enders | Applied Econometric Time Series (Second Edition)[END_REF], Equation 2 is assuming that the mortality index 𝑘 $ (𝑡) is stationary in its first difference. Hence, the current value of 𝑘 $ is equal to the value of the previous month 𝑘 $ (𝑡 -1), plus a drift 𝜇 $ , plus a compounded mortality shock that is following a particular autoregressive structure, which is the main difference compared to the Lee-Carter model to forecast mortality. From our perspective, any mortality shock 𝜖 $ (𝑡), will prevail for two additional months -at a proportion given by the coefficients 𝜃 $$ and 𝜃 $% ; and any mortality shock will have an echo effect -at a proportion 𝜑 $$ , on the mortality index of same month but the following year. The aim of this effect is to model the interannual variations of the under-five mortality.

𝑘 $ (𝑡) = 𝑘 $ (𝑡 -1) + 𝜇 $ + [1 + 𝜃 $$ • 𝐿 + 𝜃 $% • 𝐿 % ] • [1 + 𝜑 $$ • 𝐿 $% ] • 𝜖 $ (𝑡) (2)
𝑘 % (𝑡) = 𝜇 % + [1 + 𝜃 %$ • 𝐿 + 𝜃 %% • 𝐿 % + 𝜃 %& • 𝐿 & ] • 𝜖 % (𝑡) (3)
𝑘 & (𝑡) = 𝜇 & + [1 + 𝜃 &$ • 𝐿 + 𝜃 &% • 𝐿 % ] • 𝜖 & (𝑡) (4)
Compared to the first index of mortality, the second and third indices follow a simpler structure, as shown in Equations 3 and 4. Both indices are stationary in levels (i.e., unconditional to the previous value) and the compounded mortality shocks do not assume any seasonal effect. The mortality shock of the second mortality index will prevail for three additional months, while the shocks of the third index will have an impact for the next two months.

Estimation, confidence intervals, and smoothing

The coefficients of Equations 2, 3, and 4 -representing the time dynamic of the under-five mortality of each site, were estimated by Maximum Likelihood, assuming that mortality shocks are independent and follow a normal distribution with a constant variance over time. Since the mortality indices {𝑘 $ (𝑡), 𝑘 % (𝑡), 𝑘 & (𝑡)} are unrelated -due to the orthonormal property of the SVD, Equations 2, 3, and 4 were estimated and forecasted independently. Only the life tables pre-dating January 2020 were used for estimation. Given the estimated values of the model and using Equation 1, life tables of under-five mortality were forecasted for the following 24 months, starting in January 2020.

Confidence intervals (CI) were computed by resampling the under-five population 500 times with replacement (bootstrapping) and estimating the model and the forecast for each resample. The aim of the bootstrapped CI is to detect significant differences between the forecasted and the observed mortality in 2020 and part of 2021, that would account for the excess of under-five mortality during the COVID-19 epidemic. The graphical analysis includes smoothed trends and confidence bounds that were calculated from penalized B-Splines [START_REF] Eilers | Flexible smoothing with B-splines and penalties[END_REF].

Preliminary results

Preliminary results correspond to Siaya-HDSS. The Figure 1 shows the estimated values of the neonatal mortality (i.e., the probability of dying within the first 28 days of life, 𝑞(28𝑑)) by calendar months, from January 2008 and December 2020. For convenience, all values are plotted on the 15th of the month. While the panel A shows a full time series, the panel B is a close-up to the year 2020 -mostly affected by the epidemic of COVID-19. In addition to the observed values (in black), Figure 1 shows forecasted values to the years 2020 and 2021, depending entirely on the observations before the COVID-19 outbreak. Forecasted values (in red) assume that mortality shocks 𝜖(𝑡) are equal to zero, then the neonatal mortality converges to a long-term foreseeable trajectory. The Figure 1 shows two different approaches to the CIs: first, we indicate the future uncertainty of these mortality shocks (in green); second, we calculate those CIs related to the size of the under-five population and the precision to measure the neonatal mortality (in pink and blue). In the second approach, we are simply resampling the under-five population with replacement and estimating and forecasting under the same assumptions (bootstrapping). Finally, the excess of mortality will result to compare the observed values in 2020 with the foreseeable trajectory of the same mortality indicator.

As shown in Figure 1 (panel B), the neonatal mortality in Siaya-HDSS had a slightly decline during the second half of 2019 and the observed value of January 2020 was very low. As a result, the observed mortality was below the counterfactual, indicating a negative excess of neonatal mortality for the beginning of 2020. During the following months, some of the observed values were above the expected value; and the Figure 1 is suggesting an excess of neonatal mortality for the most part of 2020. However, the CIs of observed and forecasted values overlap, indicating that the excess of neonatal mortality in 2020 are not necessarily significant -from the statistical point of view.

Figure 2 shows similar estimates, but analyzing the under-five mortality rate (i.e., the probability of dying within the first 60 months of life, 𝑞(60𝑚)). The panel A describes an important decline of this indicator from 2008 to 2020. Compared to the neonatal mortality, the CIs of the under-five mortality are not too much wide, basically, because 𝑞(60𝑚) is greater then 𝑞(28𝑑). Hence, more power (precision) can be reached using the same number of observations. According to Figure 2, the under-five mortality has a distinctive seasonal pattern in Siaya-HDSS, peaking in the midst of the year. Panel B shows that the pattern did not repeat in 2020 and 𝑞(60𝑚) declined during the first part of the year. Although the under-five mortality increased during the second semester of 2020, the observed values were below the counterfactual, indicating a negative excess of under-five mortality. Inasmuch as the CIs of observed a forecasted values overlap, the (negative) excess of mortality during 2020 was not significant.

Figure 1 :

 1 Figure 1: The neonatal mortality-q(28d)A. The full period

Figure 2 :

 2 Figure 2: The under-five mortality rate -q(60m)A. The full period

This site would be part of the analysis if the PIs of Basse-HDSS accept to collaborate in this project.