Anh Khoa

Ngo Ho

Franc ¸ois Yvon
email: francois.yvon@limsi.fr

Optimizing Word Alignments with Better Subword Tokenization

Word alignments identify translational correspondences between words in a parallel sentence pair and are used, for example, to train statistical machine translation, learn bilingual dictionaries or to perform quality estimation. Subword tokenization has become a standard preprocessing step for a large number of applications, notably for state-of-the-art open vocabulary machine translation systems. In this paper, we thoroughly study how this preprocessing step interacts with the word alignment task and propose several tokenization strategies to obtain well-segmented parallel corpora. Using these new techniques, we were able to improve baseline word-based alignment models for six language pairs.

Introduction

Word alignment is a basic task in multilingual Natural Language Processing (NLP) and is used, for instance, to learn bilingual dictionaries, to train statistical machine translation (SMT) systems [START_REF] Koehn | Statistical Machine Translation[END_REF], to filter out noise from translation memories [START_REF] Pham | Fixing translation divergences in parallel corpora for neural MT[END_REF] or in quality estimation applications [START_REF] Specia | Quality Estimation for Machine Translation[END_REF]. Word alignment can also serve to explain MT decisions [START_REF] Stahlberg | An operation sequence model for explainable neural translation[END_REF]. Given pairs associating a sentence in a source language and a translation in a target language, word alignment aims to identify translational equivalences at the level of individual word tokens and has been initially approached with generative probabilistic models learning alignment in an unsupervised manner [START_REF] Och | A systematic comparison of various statistical alignment models[END_REF][START_REF] Tiedemann | Bitext Alignment. Synthesis Lectures on Human Language Technologies[END_REF].

With rapid advances in neural based NLP, word alignment has recently regained some traction [START_REF] Legrand | Neural network-based word alignment through score aggregation[END_REF] and improvements of the state of the art for multiple language pairs have been reported thanks to neuralized generative models [START_REF] Alkhouli | Biasing attention-based recurrent neural networks using external alignment information[END_REF][START_REF] Alkhouli | On the alignment problem in multi-head neural machine translation[END_REF][START_REF] Ngo-Ho | Neural Baselines for Word Alignments[END_REF], pre-trained multilingual embeddings (Jalili [START_REF] Sabet | SimAlign: High quality word alignments without parallel training data using static and contextualized embeddings[END_REF][START_REF] Nagata | A supervised word alignment method based on crosslanguage span prediction using multilingual BERT[END_REF][START_REF] Dou | Word alignment by fine-tuning embeddings on parallel corpora[END_REF] or more powerful architectures based on the Transformer translation model of [START_REF] Vaswani | Attention is all you need[END_REF], as reported for instance by [START_REF] Garg | Jointly learning to align and translate with transformer models[END_REF]; [START_REF] Chen | Accurate word alignment induction from neural machine translation[END_REF] and [START_REF] Chen | Mask-align: Self-supervised neural word alignment[END_REF].

In addition to using neural architectures, these new models differ from past approaches in that they compute alignments based on a decomposition into subword units [START_REF] Sennrich | Neural machine translation of rare words with subword units[END_REF][START_REF] Kudo | Subword regularization: Improving neural network translation models with multiple subword candidates[END_REF], which makes it possible to easily accommodate open-ended vocabularies and mitigate issues related to the alignment of unknown words, which has always been a challenge for discrete models. Another interesting property of subword units in the context of word alignment is that (a) they ease the generation of many-to-one / one-to-many links, which are difficult to handle in standard asymmetric models such as IBM-1 and IBM-4 [START_REF] Liu | Generalized agreement for bidirectional word alignment[END_REF][START_REF] Tomeh | Maximum-entropy word alignment and posterior-based phrase extraction for machine translation[END_REF][START_REF] Wang | Yet another symmetrical and real-time word alignment method: Hierarchical sub-sentential alignment using f-measure[END_REF]; (b) they also enable to actively manipulate the lengths of the source and target sentences so as to make them more even, arguably a facilitating factor for alignment and translation models [START_REF] Deguchi | Bilingual subword segmentation for neural machine translation[END_REF].

In this work, we take a closer look at the interaction between alignment and subword tokenization and try to address the following research questions: how much of the reported improvements in alignment performance can be linked to subword splitting? which issue(s) of basic alignment models do they mitigate? is it possible to design more active segmentation strategies that would target the alignment problem for specific language pairs? Our conclusions rests on the analysis of a systematic study of word alignment for 6 language pairs from multiple language families. We notably show that subword tokenization also help discrete alignment models. We also study techniques aimed at optimizing tokenization, which enable us to further improve the alignment accuracy and mitigate the problems cause by rare / unaligned words.

This paper is organized as follows: in § 2 we review the pitfalls of generative word alignment models, and analyse in § 3 how their performance vary with changing subword tokenizations. These analyses help to understand why such preprocessing actually improves word based models. Our main proposals are sketched in § 4, where we show how to optimize subword tokenization for better alignments. In § 5, we then briefly review related work, before concluding in § 6.

Pitfalls and limitations of word alignments models

In this section, we experiment with well-known word alignment packages (Fastalign [START_REF] Dyer | A simple, fast, and effective reparameterization of ibm model 2[END_REF], Giza++ [START_REF] Och | A systematic comparison of various statistical alignment models[END_REF], Eflomal [START_REF] Östling | Efficient word alignment with Markov Chain Monte Carlo[END_REF] as well as Simalign (Jalili Sabet et al., 2020)1 ), outlining difficult issues for word alignment models such as the prediction of null links, of many-to-one links, as well as the alignment of rare words. Detailed analyses are in [START_REF] Ngo Ho | Generative Probabilistic Alignment Models for Words and Subwords : a Systematic Exploration of the Limits and Potentials of Neural Parametrizations[END_REF]. Asymmetric alignment models associate each source word with exactly one target word; such alignments are denoted as English → Foreign, when English is the source language. As a preamble, we start with our data condition.

Datasets

Our experiments consider multiple language pairs all having English on one side. Our training sets for French and German are made of sentences from Europarl [START_REF] Koehn | Europarl: A Parallel Corpus for Statistical Machine Translation[END_REF]. For Romanian, we use both the NAACL 2003 corpus [START_REF] Mihalcea | An evaluation exercise for word alignment[END_REF] and the SETIMES corpus used in WMT'16 MT evaluation. For Czech, the parallel data from News Commentary V11 [START_REF] Tiedemann | Parallel data, tools and interfaces in opus[END_REF] is considered, while we use the preprocessed parallel data for Vietnamese in IWSLT'15 [START_REF] Luong | Stanford neural machine translation systems for spoken language domain[END_REF] and the Japanese data from the KFTT [START_REF] Neubig | The Kyoto free translation task[END_REF].

Our evaluations use standard test sets whenever applicable: for French and Romanian, we use data from the 2003 word alignment challenge [START_REF] Mihalcea | An evaluation exercise for word alignment[END_REF]; the German test data is Europarl; 2 for Czech we use the corpus designed by [START_REF] Mareček | Czech-English manual word alignment[END_REF]; the Japanese test data is from the KFTT and the test corpus for Vietnamese is generated from the EVBCorpus. 3 As is custom when evaluating unsupervised alignments, we append the test set to the training corpus at training time, meaning that there is no unknown word in the reference alignments.

Basic statistics for these corpora are in Table 1. 4 English-French and English-German training data (≥ 1.5M) are much larger than the rest (from 122K to under 400K) and we take them as representative of a "large data" condition. Unsurprisingly, the vocabulary sizes of the German, Romanian and Czech corpora are substantially greater than the corresponding English, which contains a smaller number of inflected variants. The opposite pattern is observed for Japanese and Vietnamese, two synthetic languages with less inflectional variability than English.

Evaluation protocol

We use the alignment error rate (AER) [START_REF] Och | Minimum error rate training in statistical machine translation[END_REF], F-score (F1), precision and recall as measures of performance. AER is based on a comparison of predicted alignment links (A) with a human reference including sure (S) and possible (P) links, and is defined as an average of the recall and precision taking into account the sets P and S. AER is defined as:

AER = 1 - |A ∩ S| + |A ∩ P | |A| + |S| (1)
where A is the set of predicted alignments. Note that the English-Romanian, English-Japanese and English-Vietnamese reference data only contain "sure" links, meaning that for these languages pairs, AER and F-measure are deterministically related.

Main observations

Detailed analyses of automatic word alignments, fully documented in (Ngo Ho, 2021), show that:

• Unaligned words are poorly predicted: we collect correctly/incorrectly unaligned words on the source side for the asymmetrical models. For English→ Czech, there are too few English words aligning with Czech words for IBM-1 whereas IBM-4 produces too many unaligned English words (Figure 1).

• Many-to-one/one-to-many links are also poorly predicted, even with symmetrization. 5 This can be seen in Figure 2.

• Larger length differences between parallel sentences yield more errors, as shown in Figure 3. This again hints at the tendency of discrete word models to generate one-to-one alignments.

3 Studying the interaction between alignment and segmentation

Implementation

In this section, we restrict our analysis to Fastalign and Eflomal and study how their performance vary when the subword vocabulary changes. We perform the alignment between subword units generated by Byte-Pair-Encoding [START_REF] Sennrich | Neural machine translation of rare words with subword units[END_REF] and the unigram method of [START_REF] Kudo | Subword regularization: Improving neural network translation models with multiple subword candidates[END_REF], both implemented with the SentencePiece package [START_REF] Kudo | SentencePiece: A simple and language independent subword tokenizer and detokenizer for neural text processing[END_REF]. All parameters of these models are set to their default values. We independently segment sentences in each language with varying vocabulary sizes V ∈ {2K, 4K, 8K, 16K, 32K, 48K}. For Japanese, we do not use the vocabulary size of 2K because it is smaller than the characterbased vocabulary size. For English-Vietnamese, experiments for English vocabulary size of 48K and Vietnamese vocabulary size larger than 32K were not performed. This is because they would imply larger vocabularies than their word-based counterparts. When using the sampling strategy of SentencePiece, we use α = 0.1.

Our results and analyses are however based on word-level alignments. Subword-level alignments are thus converted into word-level alignments as follows: a link between a source and a target word exists if there is at least one alignment link between their any of their subwords.

(32K-32K) (8K-8K) (4K-32K) (16K, 16K) (16K-8K) (16K-2K) (16K-32K) (32K-16K) (8K-8K) (8K-16K) (4K-4K) (4K-
(45K-16K) (48K-32K) (4K-48K) (16K-16K) (39K-16K) (32K-4K) (16K-32K) (48K-16K) (8K-8K) (8K-32K) (16K-2K) (4K-8K)
(16K-32K) (32K-16K) (4K-32K) (32K-16K) (16K-48K) (8K-48K) (8K-32K) (48K-16K) (8K-32K) (8K-32K) (2K-8K) (2K-
(45K-48K) (32K-32K) (32K-32K) (48K-32K) (32K-48K) (48K-16K) (32K-32K) (48K-16K) (16K-8K) (30K-16K) (16K-8K) (2K-16K)
Table 2: AER scores of subword-based models and word-based models. We only report the best result obtained by subword-based models, and the corresponding vocabulary sizes.

Main results

In order to observe how the alignment accuracy varies with the size of the subword vocabulary, we plot precision and recall as a function of the target vocabulary size for each source vocabulary size. As can be seen in Figure 4, having short units (top-left zones) on both sides yields a better recall but a much worse precision. The opposite trend is found in bottom-right zones where we approach word-based models. Note that however with a proper choice of unit size, BPE-based models are able to outperform their word-based counterparts, with a gain of about 2 AER points. This improvement is not clear for unigram-based models (see

Complementary analyses

Unaligned words and alignment types

Figure 5 displays unaligned word patterns generated by several BPE-based models for English-German. Choosing small inventories on the target side yields more fragmented sequences and a reduced number of non-aligned words in the source, as is expected for asymmetrical models. Significantly increasing both recall and precision proves difficult, and we only observe small improvements with respect to the word-based baselines: for instance, with Fastalign, the best BPE-model (4K-32K) removes 40 incorrectly unaligned words and finds 10 correctly unaligned words. Compared with HMM or IBM-4, we also notice that BPE-based models are less prone to over-generate null links. Similar trends were observed for the other language pairs/directions.

We now study how the number of links for each alignment type changes with the vocabulary size (Figure 6). The most noticeable observation is that shorter BPE units (e.g., 2K-2K) generate less one-to-one links and accordingly more of the other alignment types, especially one-to-many and many-to-many links. In other words, tokens that decompose into a sequence of shorter units in the source side have more chance to align with several target tokens. However, this does not result in an increased number of correct one-to-many/many-to-many links. Similar trends were observed for the other language pairs/directions.

Aligning rare words

Using subwords affects the overall distribution of units and helps mitigate issues with rare tokens.

To measure this effect, we collect rare source words (a word is rare if it occurs once in our training data) and plot their F-scores as a function of target and source vocabulary sizes (see Figure 7).

Recall that German has a very large word-based vocabulary size (Table 1). Accordingly, for the German-English direction, we can see a large gain (about +8 points) in F-score when using a reduced German vocabulary size of 32K.

Improving alignment by voting

As a final experiment, we combine multiple BPE-based alignments using a simple voting procedure.This method is parameterized by the required level of agreement (the percentage of models agreeing on an alignment link). Figure 8 shows that considering the BPE models described above and using an agreement level of 70% improves the F-score by almost 2 points for German→English and Japanese→English. Similar results are obtained for the other language pairs, showing that considering multiple segmentations in alignment can be helpful.

Optimizing subword tokenization

In this section, we build on the intuition that pairs of sentences which differ in length are difficult to align [START_REF] Deguchi | Bilingual subword segmentation for neural machine translation[END_REF], suggesting that subword splitting should be used to make the Table 3: Average F-score (over language pairs and directions) for global methods of controlling sequence length difference for Fastalign and Eflomal. We also report the best vocabulary pair found in the vocabulary pair matrix (BVP-M).

improvement for the greedy search, which outperforms the word-based models for Eflomal for English-French, English-German, English-Japanese and English-Vietnamese. The methods presented above consider ways to optimize the length difference at the corpus level, using one subword vocabulary that is used across the board. We study here four local methods that aim to reduce the length differences separately for each sentence pair before training the alignment procedure. With the exception of the first method, they all rely on the unigram algorithm, and use a fixed, predefined, vocabulary size for both languages:

• the first (SP-M) simply picks, among all the considered segmentations of each sentence, the one that minimizes the length difference. When there is more than one minimal segmentation, we select the one for which total source and target lengths is smallest.

• the second8 (SM1-1VP) relies on the idea of [START_REF] Deguchi | Bilingual subword segmentation for neural machine translation[END_REF]: (a) we collect the 10 most likely segmentations for each language using the unigram algorithm; (b) we select the highest probability candidate on both sides, and consider the longer of the two as the anchor segmentation; (c) we pair this segmentation with the in the other language, that is closest in length and maximally likely. We also consider the case SM5-1VP where we include the top five highest probability in the last step for the training data.

• SSM5-1VP extends the previous idea with more candidates: we sample 10 segmentations using the unigram algorithm for each language, then select the 5 pairs of segmentations that have the smallest length difference, and use it as the training data for the word alignment.

• a last idea (SSM5-GS) uses the same strategy as SSM5-1VP, using the "optimal" pair of vocabulary sizes computed by the greedy search algorithm (Algorithm 1).

Subword segmentation is introduced in the context of neural translation in [START_REF] Sennrich | Neural machine translation of rare words with subword units[END_REF], using a reimplementation 9 of the Byte Pair Encoding algorithm of [START_REF] Gage | A new algorithm for data compression[END_REF]. BPE is a greedy, bottom up algorithm that recursively aggregates frequent bigrams into new symbols, and is thoroughly analyzed in [START_REF] Gallé | Investigating the effectiveness of BPE: The power of shorter sequences[END_REF]. The main alternative is SentencePiece introduced in [START_REF] Kudo | Subword regularization: Improving neural network translation models with multiple subword candidates[END_REF][START_REF] Kudo | SentencePiece: A simple and language independent subword tokenizer and detokenizer for neural text processing[END_REF], which implements a form of variable-length probabilistic unigram model, which can be traced back to [START_REF] Deligne | Language modeling by variable length sequences: Theoretical formulation and evaluation of multigrams[END_REF].

With BPE/unigram subword tokenization becoming a standard for many applications, several studies have started to investigate more closely the impact on these preprocessing decisions on the final performance. The implementation of SentencePiece 10 reports a large number of MT experiments aimed to compare BPE and unigram in multiple conditions, concluding that both yield comparable BLEU scores across the board when used with a fixed tokenization in words.

The shortcomings of BPE/unigram segmentations have been the subject of several studies, reporting comparisons with (a) linguistic segmentations [START_REF] Huck | Target-side word segmentation strategies for neural machine translation[END_REF][START_REF] Ataman | Linguistically motivated vocabulary reduction for neural machine translation from turkish to english[END_REF][START_REF] Banerjee | Meaningless yet meaningful: Morphology grounded subwordlevel NMT[END_REF][START_REF] Weller-Di Marco | Modeling word formation in English-German neural machine translation[END_REF] 2019) conduct a systematic exploration considering a large numbers of vocabulary sizes to better understand its impact on NMT performance, comparing several NMT architectures such as shallow/deep-transformer, tiny/shallow/deep-LSTM. [START_REF] Bostrom | Byte pair encoding is suboptimal for language model pretraining[END_REF] evaluate the impact of tokenization on language model pre-training. They conclude that tokenization encodes a surprising amount of inductive bias and that LM-based tokenization produces subword units that qualitatively align with morphology much better than those produced by BPE, suggesting that the latter is better than the former for pretrained models.

The work of [START_REF] Deguchi | Bilingual subword segmentation for neural machine translation[END_REF] is our main inspiration, and explore ways to optimize the subword segmentation, using, as we do, sampling techniques and length-based heuristics to chose the most appropriate target for each source, and observing gains in translation performance.

Conclusion and outlook

In this work, we have studied the interaction between word alignment and word segmentation based on two algorithms (BPE and unigram) and multiple word aligners. Using smaller units notably mitigate issues with rare/unknown words; shorter units also help to retrieve more correct links for non-canonical (one-to-many, many-to-one) alignment links. Based on these observations, we have thoroughly analyzed the variation of alignment scores with respect to vocabulary sizes, showing that the word-based segmentation was less than optimal. We have finally explored various ways to actively optimize the subword tokenization; promising results in this direction have been obtained with the unigram algorithm, owing to its ability to generate multiple high-probability segmentations. We have notably found that adjusting length differences in source and target was a reasonable heuristic to progress towards better joint tokenizations, even though (a) the relationship between length difference and alignment quality was not as clear as one may have wished; (b) inconsistencies have been observed between unigram and BPE. In the future, we will continue to explore inexpensive ways to identify promising joint segmentations and improve the alignment between subword units.

Figure 1 :

 1 Figure 1: Number of correctly/incorrectly unaligned English and Czech words for English→Czech (left) and Czech→English (right).

Figure 3 :

 3 Figure3: F-score (red) and number of correct one-to-one alignments (blue) as a function of a length difference for the direction English-French, computed by Fastalign. The numbers in black are the corresponding number of sentences.

Figure 4 :

 4 Figure 4: Precision and recall of BPE-based alignments for English→Romanian and English→Vietnamese, computed by Fastalign. The darker the cell, the greater the score.

Figure 5 :

 5 Figure 5: Number of correctly/incorrectly unaligned English (left) and German (right) words generated by Fastalign for respectively English→German and German→English. VP-M denotes the vocabulary pair for which the average length difference between source and target sentences is smallest; BVP-M denotes the vocabulary pair yielding the best AER.

Figure 6 :

 6 Figure 6: Alignment errors for BPE-based, word-based asymmetrical (Word asym.) and symmetrical alignments (Word sym.) computed by Fastalign for English→German.

Algorithm 1

 1 Finding the vocabulary pair minimizing the average length differences Require: α: Source side vocabulary size; β: Target side vocabulary size ε: search space radius (default = 2000); ρ: step size (default = 100); Ensure: 1000 ≤ α, β ≤ 50000 while ε ≥ 100 do for ν ∈ {α -ε, α, α + ε}, µ ∈ {β -ε, β, β + ε} do if f (ν, µ) < f (α, β) then α = ν; β = µ; = 2000 end if end for if α and β remain the same then ε = ε -ρ end if end while 4.2 Local methods for controlling the length difference

  and (b) alternative preprocessing schemes such as character-based models (eg. in Sennrich (2017); Sajjad et al. (2017); Cherry et al. (2018)). Ding et al. (

Table 1 :

 1 Basic statistics for the training data and test data

	Corpus		Training data				Test data
		# sent.	word vocab.	char. vocab. # sent.	# words	# non-null
		pairs	Eng.	For.	Eng.	For.	pairs	Eng.	For.	links
	En-Fr	∼1.7M ∼106K ∼112K 111	115	447	7 020	7 761	17 438
	En-Ge	∼1.5M	∼96K ∼311K 218	235	509	10 413 9 945	10 533
	En-Ro	∼250K ∼74K ∼115K 124	131	246	5 455	5 315	5 991
	En-Cz	∼182K ∼62K ∼147K 246	157	2 501 59 724 52 881	67 423
	En-Ja	∼377K ∼156K ∼126K ∼2K ∼5K 1 235 30 822 34 403	33 377
	En-Vi	∼122K ∼42K	∼19K	133	171	3 447 70 049 94 753	81 748

  Table 2).

	English→Romanian		English→Vietnamese
	Precision	Recall	Precision	Recall
	48K 32K 16K 8K 4K 2K Source	48K 32K 16K 8K 4K 2K Source	32K 16K 4K 2K 8K Source	32K 16K 4K 2K 8K Source
	2K 4K 8K 16K 32K 48K Target	2K 4K 8K 16K 32K 48K Target	2K	4K Target 8K 16K	2K	4K Target 8K 16K

A method of generating alignment links based on the matrix of embedding similarities without parallel data. The options are to use mBert(Devlin et al.,

2019) or the multilingual version of Fasttext are used to generate multilingual embeddings from monolingual data. In our experiments, we use the setting: mBert + Argmax. 2 http://www-i6.informatik.rwth-aachen.de/goldAlignment/ 3 https://code.google.com/archive/p/evbcorpus/ 4 We only use training sentences of length lower than 50.

We heuristically merge two alignments with opposite directions to produce a symmetric alignment, by using the grow-diag-final (GDF) heuristic proposed in[START_REF] Koehn | Europarl: A Parallel Corpus for Statistical Machine Translation[END_REF].

The step size ρ remains the same for the whole procedure.

f (α, β) returns the average sequence length difference obtained with vocabularies of size α and β.

This method and next only apply to unigram, which, contrarily to BPE, is based on a sound probabilistic model.

Acknowledgements

This work has been made possible thanks to the Saclay-IA computing platform. 9 https://github.com/rsennrich/subword-nmt 10 https://github.com/google/sentencepiece/blob/master/doc/experiments.md

. The red curve plots the F-score for each level of agreement for German→English and Japanese→English. For both directions, voting improves the F-score of about 2 pts with a 70% level of agreement.

length of parallel sentences more even. We study global and local ways to achieve this goal.

Global methods for controlling length differences

We first consider two ways to find the vocabulary pair minimizing the average length difference:

• the first one (denoted VP-M) simply picks the vocabulary pair that minimizes this value in the matrix of all vocabulary pairs;

• this solution can be improved by using the following greedy search procedure (VP-GS): we compute the average sequence length difference for a vocabulary pair based on a pre-defined search space radius. If we find a new vocabulary pair producing a smaller average than the current pair, we continue to explore the neighbors of this new pair. We reduce the search space radius ε in the case that no new pair is found. 6 Details are in algorithm 1. 7 We collect the average F-score, length difference and English vocabulary size for all language pairs and directions (see Table 3). For BPE-based models, minimizing length difference between the source and target sentence outperforms word-based models with a gain of at least 1 point in F-score. This performance is close to the best results found from the matrix of vocabulary pair. Unigram-based models fail to match such performance, but we still observe an We always consider one single pair of segmentations for the test data: we chose the highest probability pair for SM5-1VP and one pair producing the smallest length difference for SSM5-*.

For BPE-based models (Figure 9), SP-M only outperforms the word-based model for English-French and English-Vietnamese, and fails to achieve better F-scores than the two global methods. The performance of unigram-based method (assuming vocabularies of sizes 16K-16K) is displayed in Figure 10. They all outperform the baseline (a fixed 16K-16K model) and also the word-based models for French, Japanese and Vietnamese. It also seems that including several segmentation samples for each sentence pair in the training data (as in SSM5-1VP ) also helps to improve the performance, resulting in a simple scheme based only on length differences, that consistently outperforms all other unigram-based methods. These results open perspectives for further improving these models, especially for German, Czech and Romanian, for which the 16K-16K setting might be suboptimal. The last method (SSM5-GS) does not succeed in improving SSM5-1VP. Similar observations hold for Eflomal, albeit with better baselines.


Emilie Sandrin

Alexandre J S Morin

Claude Fernet

Nicolas Gillet
email: nicolas.gillet@univ-tours.fr

Complementary variable-and person-centered approaches to the dimensionality of burnout among fire station workers

Keywords: Burnout, latent profiles, recognition, job satisfaction, firefighters, bifactor models

come   L'archive ouverte pluridisciplinaire

Introduction

There is a general recognition [START_REF] Maslach | Job burnout[END_REF]) that a comprehensive assessment of burnout should tap into the three components of emotional exhaustion, cynicism, and reduced professional efficacy. These components are seen as independent [START_REF] Maslach | Job burnout[END_REF], and prior meta-analyses have supported that each of these three components shared unique associations with covariates [START_REF] Alarcon | A meta-analysis of burnout with job demands, resources, and attitudes[END_REF]. However, research has suggested that employees could also experience burnout as a single global construct [START_REF] Schaufeli | The burnout companion to study and research: A critical analysis[END_REF]. This global representation is supported by the high correlations reported among ratings of the three burnout components [START_REF] Maslach | Job burnout[END_REF], and by the presence of stronger associations with covariates when burnout is defined as a global dimension [START_REF] Shirom | A comparison of the construct validity of two burnout measures in two groups of professionals[END_REF].

Unfortunately, these prior investigations have not relied on an operationalization of burnout allowing them to differentiate employees' global (G) and specific (S) levels of burnout, raising a series of potentially critical questions regarding: (a) whether the burnout facets of emotional exhaustion, cynicism, and reduced professional efficacy really retain meaningful specificity beyond the assessment of the overarching burnout construct; and (b) whether this overarching construct exists as a G-entity including specificities mapped by the facets, or whether these facets reflect distinct correlated dimensions without such a common core [START_REF] Morin | Complementary variable-and person-centered approaches to the dimensionality of psychometric constructs: Application to psychological wellbeing at work[END_REF]. Indeed, recent research relying on a bifactor approach has shown that burnout ratings could be simultaneously represented by a G-level across all facets and by S-levels of emotional exhaustion, cynicism, and reduced professional efficacy left unexplained by this G-level (e.g., [START_REF] Hawrot | Factor structure of the Maslach Burnout Inventory-Educators Survey in a Polish-speaking sample[END_REF][START_REF] Isoard-Gautheur | Development and evaluation of the psychometric properties of a new measure of athlete burnout: The Athlete Burnout Scale[END_REF].

Despite the interest of this improved bifactor representation, the criterion-related validity of the resulting G/S components of burnout remains insufficiently explored. Fortunately, alternative variable-(e.g., [START_REF] Isoard-Gautheur | Development and evaluation of the psychometric properties of a new measure of athlete burnout: The Athlete Burnout Scale[END_REF] and person- [START_REF] Leiter | Latent burnout profiles: A new approach to understanding the burnout experience[END_REF] centered approaches exist to support a more thorough investigation of these important questions. The present research illustrates these approaches while contributing to research on burnout by: (a) relying on the variable-centered bifactor modeling framework to account for the multidimensionality of this construct (Studies 1 and 2); (b) relying on person-centered analyses to assess the nature of employees' burnout profiles while taking into account the G/S components (Study 2); and (c) documenting the criterion-related validity of the G/S dimensions (Study 1) and profiles (Study 2) by examining their associations with covariates (organizational, supervisor, and colleagues recognition as well as job satisfaction). [START_REF] Hawrot | Factor structure of the Maslach Burnout Inventory-Educators Survey in a Polish-speaking sample[END_REF] found support for a bifactor model of burnout including one G-factor and three S-factors (emotional exhaustion, cynicism, and reduced professional efficacy). However, their results revealed low reliability for the three S-factors. [START_REF] Mészáros | The bifactor model of the Maslach Burnout Inventory-Human Services Survey (MBI-HSS)[END_REF] and [START_REF] Szigeti | Burnout and depressive symptoms in teachers: Factor structure and construct validity of the Maslach Burnout Inventory-Educators among elementary and secondary school teachers in Hungary[END_REF] reported similar conclusions, showing that the burnout Gfactor and the reduced professional efficacy S-factor were the main sources of variation in item scores. Researchers have also suggested that a two-factor model including only emotional exhaustion and cynicism might be more appropriate, based on methodological and conceptual rationales, challenging the relevance of the reduced professional efficacy component [START_REF] Kalliath | A test of the Maslach Burnout Inventory in three samples of healthcare professionals[END_REF]. Additional studies are needed to confirm that enough specificity exists at the subscale level once G-levels of burnout are considered, and to document the role of reduced professional efficacy as independent, or not, from G-levels of burnout.

Co-Existing Global and Specific Burnout Components

Study 1

Study 1 was designed to investigate the multidimensionality of a measure of burnout. To document the value of considering employees' S-levels of emotional exhaustion, cynicism, and reduced professional efficacy beyond their G-levels of burnout, we also consider the criterionrelated validity of this solution by examining relations between these components and a series of covariates (organizational, supervisor, and colleagues recognition as well as job satisfaction).

Recognition refers to the way the organization, supervisors, and colleagues show their appreciation and acknowledge the efforts and successes of employees [START_REF] Fall | Recognition at work: Validation of a measuring scale in the context of companies[END_REF]. It can be shown by the spontaneous acknowledgment of work well done, by expressing consideration towards the staff, and by words of encouragement in difficult situations [START_REF] Sandrin | Effects of workaholism on volunteer firefighters' performance: A moderated mediation model including supervisor recognition and emotional exhaustion[END_REF]. We focus on recognition from different sources (i.e., organization, supervisor, and colleagues) based on the acknowledgement that providing professional recognition to workers represents a costeffective way to maintain and improve employees' well-being and performance (e.g., [START_REF] Fernet | The effects of work motivation on employee exhaustion and commitment: An extension of the JD-R model[END_REF]. According to the job demands-resources model, recognition at work can be conceptualized as an important resource for employees, suggesting that it should help workers to regulate their short-term fatigue and recover more optimally from work, and thus to avoid burnout [START_REF] Bakker | Job Demands-Resources theory and self-regulation: New explanations and remedies for job burnout[END_REF]. Prior research has demonstrated negative associations between recognition (or similar constructs) and indicators of ill-being at work, including burnout [START_REF] Caesens | The impact of work engagement and workaholism on well-being: The role of work-related social support[END_REF][START_REF] Caesens | A person-centred perspective on social support in the workplace[END_REF][START_REF] Sandrin | Effects of workaholism on volunteer firefighters' performance: A moderated mediation model including supervisor recognition and emotional exhaustion[END_REF]. For these reasons, we expected organizational, supervisor, and colleagues support to be associated with lower G-levels of burnout and with lower S-levels of emotional exhaustion, cynicism, and reduced professional efficacy. Job satisfaction was also considered given its documented association with work performance and organizational retention (e.g., [START_REF] Bowling | Is the job satisfaction-job performance relationship spurious[END_REF]. Research reveals negative associations between job satisfaction and burnout [START_REF] Huyghebaert | Examining the effect of affective commitment to the supervisor on nurses' psychological health as a function of internal locus of control[END_REF], so that highly satisfied employees can often be described as fulfilled and happy. We thus expected job satisfaction to be associated with lower G-levels of burnout and S-levels of emotional exhaustion, cynicism, and reduced professional efficacy.

Method

Participants and Procedure

This study relies on a sample of 247 administrative and technical employees (71 men and 176 women) working in various fire stations located in France. Most participants worked full time (89.1%). Respondents were aged between 24 and 63 years (M = 43.3, SD = 8.5), had an average tenure in their organization of 13.7 years (SD = 7.7), and had an average tenure in their position of 8.6 years (SD = 6.2). This study follows the principles of the Declaration of Helsinki. All participants received an e-mail explaining the study purpose and inviting them to complete an online questionnaire. Once online, participants were first asked to complete a consent form emphasizing the anonymous and voluntary nature of their participation.

Measures

Burnout. Participants completed the 15-item version of the Maslach Burnout Inventory-Student Survey (MBI-SS; [START_REF] Portoghese | Measuring burnout among university students: Factorial validity, Burnout Dimensionality and Profiles 29 invariance, and latent profiles of the Italian version of the Maslach Burnout Inventory Student Survey (MBI-SS)[END_REF] adapted to the work context. This instrument assesses emotional exhaustion (5 items; e.g., "I feel used up at the end of a day at work"; α = .90), cynicism (4 items; e.g., "I doubt the significance of my work"; α = .71), and reduced professional efficacy (6 items; e.g., "At work, I feel confident that I am effective in getting things done"; α = .74; all of these items were reversed coded prior to the analyses).

All items were rated on a seven-point response scale ranging from 0 (never) to 6 (always).

Recognition. Organizational (5 items; e.g., "My organization offers me chances of promotion"; α = .79), supervisor (5 items; e.g., "My supervisor encourages me when I am confronted with a difficult task"; α = .91), and colleagues (4 items; e.g., "My colleagues congratulate me or thanks me for good work"; α = .92) recognition were measured on a fivepoint scale ("Strongly Disagree" to "Strongly Agree") using [START_REF] Fall | Recognition at work: Validation of a measuring scale in the context of companies[END_REF] scale.

Job satisfaction. Job satisfaction was assessed with one-item [START_REF] Shimazu | Workaholism vs. work engagement[END_REF]i.e., "Are you satisfied with your job?") rated on a 1 (dissatisfied) to 4 (satisfied) scale.

Analyses

Analyses were conducted using the Maximum Likelihood Robust (MLR) estimation in Mplus 8 [START_REF] Muthén | Mplus user's guide[END_REF]. Missing responses at the item level (0% to 30.4%; M = 5.3%) were handled using Full Information Maximum Likelihood (FIML; [START_REF] Enders | Applied missing data analysis[END_REF].

Preliminary analyses were conducted to verify the psychometric properties of all measures.

Burnout models were estimated using bifactor-confirmatory factor analyses (bifactor-CFA; [START_REF] Morin | Complementary variable-and person-centered approaches to the dimensionality of psychometric constructs: Application to psychological wellbeing at work[END_REF]. This decision is based on evidence supporting the superiority of bifactor models for burnout measures (e.g., [START_REF] Hawrot | Factor structure of the Maslach Burnout Inventory-Educators Survey in a Polish-speaking sample[END_REF][START_REF] Isoard-Gautheur | Development and evaluation of the psychometric properties of a new measure of athlete burnout: The Athlete Burnout Scale[END_REF].

Independent CFA factors reflecting the recognition measure and the single-item job satisfaction measure were then added to the bifactor solution to estimate the criterion-related validity of the burnout factors using latent correlations. Model adequacy was checked using goodness-of-fit indices, where values > .90 and .95 on the Tucker-Lewis index (TLI) and the comparative fit index (CFI), and values lower than .08 and .06 on the root mean square error of approximation (RMSEA) were respectively taken to reflect acceptable and excellent levels of fit (e.g., [START_REF] Marsh | Goodness of fit in structural equation models[END_REF].

Results and Discussion

Preliminary Analyses: Burnout

The a priori bifactor-CFA resulted in a satisfactory level of fit (χ² = 121.21, df = 75, p ≤ .01, CFI = .96; TLI = .94; RMSEA = .05), and an improvement relative to the CFA (χ² = 160.36, df = 87, p ≤ .01, CFI = .93; TLI = .94; RMSEA = .06). For the CFA, the results revealed reliable factors that were well-defined by a majority of their indicators (see Table 1). However, they also revealed a high correlation between the cynicism and emotional exhaustion factor (r = .51), but an almost null correlation between the reduced professional efficacy factor and the emotional exhaustion (r = -.02) and cynicism (r = -.03) factors. These correlations argue against the need to incorporate a G-factor, as least in relation to ratings of reduced professional efficacy. Supporting this idea, the bifactor-CFA results revealed a burnout G-factor that was well-defined by all of the emotional exhaustion items, moderately well-defined by the cynicism items, but weakly defined by the reduced professional efficacy items, suggesting that these items might tap into a different construct.

We thus estimated an alternative bifactor-CFA encompassing cynicism and emotional exhaustion but including a distinct correlated CFA factor reflecting reduced professional efficacy. This alternative solution resulted in an equally satisfactory level of fit (χ² = 123.32, df = 78, p ≤ .01, CFI = .96; TLI = .94; RMSEA = .05), and in reliable burnout G-and professional efficacy CFA factors, a moderately reliable cynicism S-factor, and a weakly defined emotional exhaustion S-factor. This alternative bifactor solution was thus retained.

This decision was also anchored in a similar examination of the results obtained in Study 2, reported in the online supplements, which converge on a similar solution. The measurement invariance of this solution across studies was also supported. Parameter estimates (reported in the online supplements) from the most invariant solution suggests that when both studies are considered, more specificity remains in the emotional exhaustion S-factor than in the cynicism S-factor, supporting our decision to retain both S-factors in both studies.

Preliminary Analyses: Recognition

The results also support the adequacy of the model underlying the recognition measure (χ² = 126.80, df = 74, p ≤ .01, CFI = .97; TLI = .96; RMSEA = .05), as well as that of the combined models encompassing burnout (the alternative bifactor-CFA solution) and recognition (χ² = 498.16, df = 350, p ≤ .01, CFI = .95; TLI = .94; RMSEA = .04; RMSEA CI = .03 to .05). The parameter estimates from this solution are reported in Table S1 in the online supplements and revealed well-defined factors for all constructs.

Correlations (Criterion-Related Validity)

Latent correlations are reported in Table 2. As hypothesized and in line with past studies [START_REF] Huyghebaert | Examining the effect of affective commitment to the supervisor on nurses' psychological health as a function of internal locus of control[END_REF][START_REF] Sandrin | Effects of workaholism on volunteer firefighters' performance: A moderated mediation model including supervisor recognition and emotional exhaustion[END_REF], the three sources of recognition and job satisfaction were negatively related to the burnout G-factor. Organizational, supervisor, and colleagues recognition were associated with lower levels of reduced professional efficacy (i.e., higher levels of professional efficacy). Colleagues recognition was associated with lower S-levels of cynicism. These results thus support associations involving both the G-and the S-factors of burnout. In the next study, we relied on latent profile analyses (LPA) to examine the combined effects of all four components of burnout in a more holistic manner.

Study 2

This study documents the burnout profiles that characterize employees. These profiles are estimated while jointly considering G-levels of burnout and S-levels of emotional exhaustion and cynicism, as well as reduced professional efficacy as a distinct factor. To ascertain the construct validity of these profiles [START_REF] Morin | Mixture modeling for lifespan developmental research[END_REF], we more specifically consider their criterion-related validity. First, we adopt a multivariate perspective to identify the unique associations between each source of recognition (relative to all others) and the likelihood of profile. Second, we conduct mean-level comparisons of job satisfaction across profiles.

Burnout Profiles

Person-centered research has started to look at how burnout components combine within employees (e.g., [START_REF] Berjot | Burnout risk profiles among French psychologists[END_REF][START_REF] Portoghese | Measuring burnout among university students: Factorial validity, Burnout Dimensionality and Profiles 29 invariance, and latent profiles of the Italian version of the Maslach Burnout Inventory Student Survey (MBI-SS)[END_REF]. However, many previous studies have relied on a combination of burnout components and other variables as profile indicators (work engagement: [START_REF] Mäkikangas | The energy and identification continua of burnout and work engagement: Developmental profiles over eight years[END_REF]; psychological distress and satisfaction with life: [START_REF] Laverdière | Psychological health profiles of Canadian psychotherapists: A wake up call on psychotherapists' mental health[END_REF], making it impossible to isolate the effects of burnout. Among studies focusing solely on burnout, [START_REF] Berjot | Burnout risk profiles among French psychologists[END_REF] identified four profiles based on the emotional exhaustion, cynicism, and reduced professional efficacy components [START_REF] Maslach | Job burnout[END_REF].

However, this study is limited by its reliance on a sample of psychologists, so that additional studies are needed to generalize these findings to other occupations. Likewise, [START_REF] Leiter | Latent burnout profiles: A new approach to understanding the burnout experience[END_REF] identified five burnout profiles among two samples of health-care workers.

More generally, the nature, number, and range of burnout indicators considered in prior studies are quite large (professional inefficacy, cynicism, physical and emotional exhaustion, guilt, etc.). These studies have also relied on a variety of samples (psychologists, teachers, mixed samples, etc.), methods (cross-sectional and longitudinal), and covariates. This variety makes it hard to achieve a clear integration of results. Yet, despite these differences, the results seem to converge on the following profiles: (a) High Burnout Risk; (b) Moderately High Burnout Risk; (c) Moderately Low Burnout Risk; and (d) Low Burnout Risk. However, person-centered evidence emerges from an accumulation of research, making it possible to distinguish core profiles emerging systematically from those who occur only in some contexts, and from those reflecting sample-specific idiosyncrasies.

In addition, all of these studies have relied on indicators ignoring the dual G/S nature of burnout. This limitation is important. When applying person-centered analyses to indicators known to present a G/S structure, relying on indicators unable to properly disaggregate these components results in the erroneous estimation of profiles characterized by matching levels across indicators, thereby reflecting only G-levels of burnout and ignoring specificities located at the subscale level (Morin et al., 2016a[START_REF] Morin | Complementary variable-and person-centered approaches to the dimensionality of psychometric constructs: Application to psychological wellbeing at work[END_REF]. The present study addresses this limitation by documenting burnout profiles defined based on G/S levels of burnout.

Lacking prior guidance from person-centered research in which the multidimensional G/S nature of employees' burnout ratings were properly disaggregated, we leave as an open question the structure and number of profiles that will best reflect burnout profiles.

Nevertheless, in alignment with the consistency of the aforementioned findings, it seems reasonable to assume that some profiles will display a High Burnout Risk (high G-levels of burnout across indicators), Moderate Burnout Risk (average levels of burnout across indicators), and Low Burnout Risk (low G-levels of burnout across indicators) configuration.

Conversely, in accordance with a subset of shape-differentiated profiles obtained in burnout research [START_REF] Berjot | Burnout risk profiles among French psychologists[END_REF][START_REF] Leiter | Latent burnout profiles: A new approach to understanding the burnout experience[END_REF] and with the added value of the approach adopted in the present study to achieve a better disaggregation of G/S levels of burnout (Morin et al., 2016a[START_REF] Morin | Complementary variable-and person-centered approaches to the dimensionality of psychometric constructs: Application to psychological wellbeing at work[END_REF], it also seems reasonable to expect the identification of additional profiles characterized by more differentiated configurations.

Construct Validation

A critical step in the assessment of the construct validity of profiles, especially when relying on a predominantly inductive approach such as the one used here to identify the profiles [START_REF] Morin | Person-centered methodologies in the organizational sciences: Introduction to the feature topic[END_REF] is to document their theoretical and practical implications via the examination of associations between profile membership and theoretically-relevant covariates [START_REF] Morin | Mixture modeling for lifespan developmental research[END_REF]. In Study 2, we extend the results obtained in Study 1 by a consideration of the same set of covariates. This matching consideration is important given that person-centered analyses are generally described as providing the flip side of the coin from variable-centered analyses [START_REF] Morin | Person-centered methodologies in the organizational sciences: Introduction to the feature topic[END_REF]. More precisely, the fact that isolated facets of burnout (e.g., exhaustion S-factor) shared no relation, on their own, with this set of covariates in Study 1 does not exclude the possibility that these facets might play a role in profile definition and, in doing so, in driving associations between profile membership and these covariates.

Despite the importance of recognition at work, no person-centered research has examined the associations between organizational, supervisor, and colleagues recognition and burnout profiles.

However, findings from Study 1 and prior research [START_REF] Fernet | The effects of work motivation on employee exhaustion and commitment: An extension of the JD-R model[END_REF][START_REF] Sandrin | Effects of workaholism on volunteer firefighters' performance: A moderated mediation model including supervisor recognition and emotional exhaustion[END_REF] suggest that lower levels of organizational, supervisor, and colleagues recognition should be associated with a higher probability of membership into profiles presenting higher G-levels of burnout (e.g., High Burnout Risk). Likewise, although we expect well-differentiated associations between the burnout profiles and job satisfaction, the lack of previous personcentered studies relying on a proper disaggregation of the G/S components of burnout, or even considering the associations between job satisfaction and these profiles, precludes the formulation of precise hypotheses. However, Study 1 and prior variable-centered studies have reported negative relations between burnout components and job satisfaction [START_REF] Huyghebaert | Examining the effect of affective commitment to the supervisor on nurses' psychological health as a function of internal locus of control[END_REF]. As a result, we hypothesize that profiles presenting higher G-levels of burnout (e.g., High Burnout Risk) should be characterized by lower levels of job satisfaction relative to profiles presenting lower G-levels of burnout (e.g., Low Burnout Risk).

Method

This study relies on a sample of 654 firefighters (598 men and 56 women) working in

Analyses

As in Study 1, all analyses were realized using Mplus 8 [START_REF] Muthén | Mplus user's guide[END_REF], MLR estimation, and FIML to handle missing data (0% to 2.3%; M = 0.54%).

Preliminary Analyses

As in Study 1, preliminary factor analyses were conducted to verify the psychometric properties of all measures (supporting the conclusions from Study 1) and their measurement invariance across samples used in Studies 1 and 2. Due to the complexity of the measurement invariance models underlying all constructs, these preliminary analyses were conducted separately for the burnout and recognition measures. Factor scores were saved from the most invariant measurement models and used as inputs for the main analyses (for a discussion of the advantages of factor scores in LPA, see Morin et al., 2016a, b). Conclusions from these analyses matched those from Study 1 (see the online supplements for the results). Latent correlations are reported in Table 2, and converge on results similar to those from Study 1, with the main exception that the emotional exhaustion S-factor was negatively related to all covariates in Study 2, possibly due to its higher level of specificity in Study 2.

Model Estimation and Selection

LPA were estimated using 5000 random sets of start values, 1000 iterations, and 200 final optimizations [START_REF] Hipp | Local solutions in the estimation of growth mixture models[END_REF]. We examined solutions including one to eight profiles in which the means and variances of the burnout factors were freely estimated in all profiles [START_REF] Peugh | Modeling unobserved heterogeneity using latent profile analysis: A Monte Carlo simulation[END_REF]. To select the optimal number of profiles, we considered theoretical conformity, meaning, and statistical adequacy of the solutions [START_REF] Morin | Mixture modeling for lifespan developmental research[END_REF]. Statistical indices are also available to guide this decision. Thus, lower values on the Bayesian Information Criterion (BIC), sample-size Adjusted BIC (ABIC), Akaïke Information Criterion (AIC), and Consistent AIC (CAIC) indicate a better fitting model. In addition, a statistically significant pvalue associated with the adjusted [START_REF] Lo | Testing the number of components in a normal mixture[END_REF] Likelihood Ratio Test (aLMR) and the Bootstrap Likelihood Ratio Test (BLRT) supports the added value of a solution when contrasted with a solution including one less profile. Statistical studies support the accuracy of the CAIC, BIC, ABIC, and BLRT, but not that of the AIC and aLMR (e.g., [START_REF] Diallo | The impact of total and partial inclusion or exclusion of active and inactive time invariant covariates in growth mixture models[END_REF][START_REF] Peugh | Modeling unobserved heterogeneity using latent profile analysis: A Monte Carlo simulation[END_REF]. We thus only report these indicators (AIC and aLMR) for complete disclosure, not to guide model selection. All indicators remain influenced by sample size and so often keep on suggesting adding profiles without converging on a specific solution [START_REF] Marsh | Latent profile analysis of academic self-concept dimensions: Synergy of person-and variable-centered approaches to the internal/external frame of reference model[END_REF]. In this situation, the point at which the decrease in the value of these indicators reaches a plateau, on a graphical elbow plot, suggests the optimal solution [START_REF] Morin | General growth mixture analysis of adolescents' developmental trajectories of anxiety: The impact of untested invariance assumptions on substantive interpretations[END_REF]. Finally, the entropy (from 0 to 1) indicates the precision with which the cases are classified into the various profiles, but should not be used to guide model selection.

Criterion-Related Validity

Associations between recognition and profile membership were assessed via multinomial logistic regressions. The three sources of recognition were directly (i.e., the one-step approach) integrated into the LPA solution to examine the unique associations between each source of recognition (relative to all others) and the likelihood of profile membership via a multinomial logistic regression link function. Two models were contrasted [START_REF] Diallo | The impact of total and partial inclusion or exclusion of active and inactive time invariant covariates in growth mixture models[END_REF]. First, the associations between recognition and profile membership were fixed to be zero (null effect models). Second, these associations were freely estimated. However, prior to the integration of recognition, preliminary models were estimated, following the same sequence, to assess the need to incorporate demographic controls: Sex, age, tenure as a firefighter, tenure in the current position, education level, employment schedule, marital situation, and children. Alternative models were contrasted using the information criteria (AIC, CAIC, BIC, and ABIC; [START_REF] Morin | Multiple-group analysis of similarity in latent profile solutions[END_REF]. Finally, job satisfaction levels were contrasted across pairs of profiles using a model-based approach [START_REF] Lanza | Latent class analysis with distal outcomes: A flexible model-based approach[END_REF] implemented via the Auxiliary (DCON) function [START_REF] Asparouhov | Auxiliary variables in mixture modeling: Three-step approaches using Mplus[END_REF] allowing for the comparison of latent profiles (i.e., corrected for classification error) without allowing job satisfaction to change the nature of the profiles [START_REF] Marsh | Latent profile analysis of academic self-concept dimensions: Synergy of person-and variable-centered approaches to the internal/external frame of reference model[END_REF].

Results and Discussion

Latent Profile Analyses (LPA)

The fit of the LPA solutions is reported in Table 4, and the elbow plot is reported in Figure S1 of the online supplements. The CAIC reached its lowest point at six profiles, whereas the BIC, ABIC, and BLRT failed to converge on a solution. The elbow plot shows a plateau in the decrease of the various information criteria somewhere between four and six profiles.

Solutions including four, five, and six latent profiles were thus examined. This examination revealed that they were statistically proper and that, up to five profiles, each increase resulted in the addition of a meaningful profile to the solution. In contrast, adding a sixth profile resulted in the arbitrary division of an existing profile into smaller ones (≤ 2%) with a similar shape. Thus, the five-profile solution was retained. This model is illustrated in Figure 1. The results reveal a high level of classification accuracy of participants into their most likely Burnout Dimensionality and Profiles 13 profiles (see Table S5 in the online supplementary materials), ranging from 83.4% to 94.1% across profiles, consistent with the high entropy value associated with this solution (.76).

Profile Description

Profile 1 characterized employees with very low G-levels of burnout coupled with moderately low S-levels of cynicism, low S-levels of emotional exhaustion, and low levels of reduced professional efficacy. This Very Low Burnout Risk profile corresponded to 3.9% of the employees. Profile 2 characterized employees presenting average G-levels of burnout coupled with high S-levels of cynicism, moderately low S-levels of emotional exhaustion, and low levels of reduced professional efficacy. This Mentally Distanced profile corresponded to 25.6% of the employees. Profile 3 characterized employees with low G-levels of burnout coupled with low levels of reduced professional efficacy, moderately low S-levels of cynicism, and average S-levels of emotional exhaustion. This Low Burnout Risk profile corresponded to 13.4% of the employees. Profile 4 characterized employees presenting high G-levels of burnout coupled with average S-levels of emotional exhaustion and cynicism, and average levels of reduced professional efficacy. This High Burnout Risk profile corresponded to 42.4% of the employees. Finally, Profile 5 characterized employees presenting moderately high G-levels of burnout coupled with high levels of reduced professional efficacy, average Slevels of emotional exhaustion, and low S-levels of cynicism. This Moderately High Burnout Risk profile corresponded to 14.8% of the employees.

Recognition and Profile Membership

As shown in Table 4, the analyses involving the demographic controls supported the null effect model, suggesting a lack of associations between demographic variables and profile membership. These variables were thus excluded from further analyses. In contrast, the results supported the presence of associations between recognition and profile membership (see Table 5). These associations seemed to be primarily driven by colleagues recognition, followed by supervisor recognition, with organizational recognition showing no evidence of sharing associations with profile membership. In accordance with our expectations [START_REF] Fernet | The effects of work motivation on employee exhaustion and commitment: An extension of the JD-R model[END_REF][START_REF] Sandrin | Effects of workaholism on volunteer firefighters' performance: A moderated mediation model including supervisor recognition and emotional exhaustion[END_REF], colleagues recognition was associated with a higher likelihood of membership in all profiles relative to Profile 5 (Moderately High Burnout Risk), as well as into Profile 1 (Very Low Burnout Risk) relative to Profiles 2 (Mentally Distanced), 3 (Low Burnout Risk), and 4 (High Burnout Risk). In addition, supervisor recognition was associated with a higher likelihood of membership into Profile 3 (Low Burnout Risk) relative to Profiles 2 (Mentally Distanced), 4 (High Burnout Risk), and 5 (Moderately High Burnout Risk).

Job Satisfaction and Profile Membership

Our results revealed that levels of job satisfaction varied significantly across profiles. More precisely, levels of job satisfaction (on a 1 to 4 rating scale), together with their 95% confidence intervals (CI), observed in each profile were as follow: Profile 1 (Very Low Burnout Risk): M = 2.97 (CI = 2.75 to 3.19); Profile 2 (Mentally Distanced): M = 2.88 (CI = 2.79 to 2.97); Profile 3 (Low Burnout Risk): M = 2.93 (CI = 2.81 to 3.05); Profile 4 (High Burnout Risk): M = 2.65 (CI = 2.58 to 2.73); and Profile 5 (Moderately High Burnout Risk): M = 2.33 (CI = 2.17 to 2.48). In terms of statistical significance, these results revealed that the levels of job satisfaction were the highest in Profiles 1 (Very Low Burnout Risk), 2 (Mentally Distanced), and 3 (Low Burnout Risk), which did not differ from one another, followed by Profile 4 (High Burnout Risk), and finally by Profile 5 (Moderately High Burnout Risk) which displayed the lowest levels of job satisfaction of all profiles. In line with Study 1, these results confirm that job satisfaction was negatively related to G-burnout [START_REF] Huyghebaert | Examining the effect of affective commitment to the supervisor on nurses' psychological health as a function of internal locus of control[END_REF].

General Discussion

This research adopted a dual variable-and person-centered approach, advocated by [START_REF] Morin | Complementary variable-and person-centered approaches to the dimensionality of psychometric constructs: Application to psychological wellbeing at work[END_REF], to analyze the value of jointly considering G/S dimensions of the burnout construct. Through the application of this framework, we were able to achieve an improved representation of the structure of employees' burnout measurement and profiles.

Multidimensionnality of Burnout

In Studies 1 and 2, results supported a bifactor representation of burnout across two independent samples of employees. This solution revealed co-existing factors representing Glevels of burnout and S-factors reflecting levels of cynicism and emotional exhaustion left unexplained by this G-factor. The burnout G-factor and the S-cynicism factor were both welldefined by most of their indicators, supporting the idea that ratings of cynicism contributed to the assessment of G-levels of burnout while retaining specificity unrelated to these G-levels.

In contrast, the emotional exhaustion S-factor was more weakly defined, suggesting that ratings of emotional exhaustion mainly served to define G-levels of burnout without retaining a lot of unicity. These results reinforce the idea that cynicism and emotional exhaustion are the core dimensions of burnout [START_REF] Schaufeli | The burnout companion to study and research: A critical analysis[END_REF], while ratings of cynicism also incorporate additional information [START_REF] Isoard-Gautheur | Development and evaluation of the psychometric properties of a new measure of athlete burnout: The Athlete Burnout Scale[END_REF].

Finally, this solution revealed that reduced professional efficacy was best reflected as an independent factor, rather than as contributing to the assessment of participants' G-levels of burnout. This observation is consistent with tentative empirical evidence showing that ratings of reduced professional efficacy did not seem to share very strong associations with G-levels of burnout [START_REF] Hawrot | Factor structure of the Maslach Burnout Inventory-Educators Survey in a Polish-speaking sample[END_REF][START_REF] Szigeti | Burnout and depressive symptoms in teachers: Factor structure and construct validity of the Maslach Burnout Inventory-Educators among elementary and secondary school teachers in Hungary[END_REF]. It also supports prior theoretical arguments calling into question the relevance of this so-called third dimension of burnout [START_REF] Kalliath | A test of the Maslach Burnout Inventory in three samples of healthcare professionals[END_REF]. Nevertheless, as suggested by Cherniss (1983), professional efficacy may also represent a protective personal resource against burnout. Indeed, employees who feel confident in having the skills necessary to successfully perform their tasks are more likely to display low levels of burnout [START_REF] Skaalvik | Teacher self-efficacy and teacher burnout: A study of relations[END_REF]. Alternatively, reduced professional efficacy may also be an outcome of burnout, as employees experiencing burnout (i.e., a prolonged state of resource depletion) are less likely to restore their personal resources, leading to lower levels of professional efficacy [START_REF] Kim | Teacher self-efficacy and burnout: Determining the directions of prediction through an autoregressive cross-lagged panel model[END_REF]. Further studies are needed to clarify these issues and to achieve a more nuanced understanding of the links between reduced professional efficacy and the G/S facets of burnout.

In terms of criterion-related validity (a component of the construct validation process), Study 1 revealed that the relations between organizational, supervisor, and colleagues recognition and the burnout components mainly involved employees' G-levels of burnout and reduced professional efficacy. These results support prior research showing that perceived organizational, supervisor, and colleagues recognition were negatively related to employees' G-levels of burnout and to reduced professional efficacy [START_REF] Fernet | The effects of work motivation on employee exhaustion and commitment: An extension of the JD-R model[END_REF][START_REF] Sandrin | Effects of workaholism on volunteer firefighters' performance: A moderated mediation model including supervisor recognition and emotional exhaustion[END_REF]. However, higher levels of colleagues recognition were associated with lower S-levels of cynicism, suggesting that additional associations involving the burnout S-factors also exist.

These results suggest that it is unwarranted to assume that the associations between perceived organizational, supervisor, and colleagues recognition are similar acros all components of burnout. These results match those from previous studies showing that the role of different sources of recognition differs as a function of the variables considered [START_REF] Caesens | A person-centred perspective on social support in the workplace[END_REF].

Having a similar work experience, it might be easier or more natural for a colleague to appreciate the value and meaning of the effort invested and to share enthusiasm about the work to be done. Indeed, colleagues may have a vested interest in supporting the individual's ability to stay and get well by facilitating psychological detachment from their work (S-levels of cynicism net of G-levels of burnout) as this dimension is known to be strongly related to citizenship behaviors and group performance [START_REF] Hunter | Give me a better break: Choosing workday break activities to maximize resource recovery[END_REF]. Assuming equal levels of recognition from the organization, supervisor, and colleagues, colleagues recognition may be more valued as a resource than organizational and supervisor recognition because of the greater frequency to which employees interact with their colleagues. Sharing a comparable work experience, colleagues may have a greater propensity to judge not only the quantity, but also the quality of the task to be performed. Because of the importance of instrumental and emotional recognition, colleagues may also be more inclined to offer recognition, and it might be easier for them to do so, to employees with whom they interact on a daily basis.

Results also supported our expectations [START_REF] Huyghebaert | Examining the effect of affective commitment to the supervisor on nurses' psychological health as a function of internal locus of control[END_REF] by revealing a negative association between job satisfaction and G-levels of burnout. However, they also went against our expectations [START_REF] Kalliath | A test of the Maslach Burnout Inventory in three samples of healthcare professionals[END_REF][START_REF] Leiter | Latent burnout profiles: A new approach to understanding the burnout experience[END_REF] in failing to reveal significant associations between job satisfaction and S-levels of emotional exhaustion and cynicism, as wel as with reduced professional efficacy. Although these results call into question the need to consider these components once employees' G-levels of burnout are considered, we more thoroughly verified this assertion via the adoption of a person-centered approach in Study 2.

Burnout Profiles

The results showed that five profiles best summarized the burnout configurations observed in Study 2. These profiles generally met our expectations and matched results from personcentered studies [START_REF] Berjot | Burnout risk profiles among French psychologists[END_REF][START_REF] Leiter | Latent burnout profiles: A new approach to understanding the burnout experience[END_REF] while allowing us to separately consider the role played by G/S burnout components, a distinction which was ignored in these previous studies. This similarity highlights the robustness of our results and the utility of interventions focused on employees' profiles. Likewise, the results also reinforce the value of jointly considering the G-and S-(cynicism and emotional exhaustion) facets of burnout, coupled with an independent reduced professional efficacy dimension. The similarity across studies relying on different approaches, measures, and operationalizations (i.e., traditional versus G/S) supports the idea that these profiles tap into meaningful psychological processes, just as differences and specificities in results support the unicity of the burnout construct.

Recognition and Burnout Profiles

The present results revealeld strong associations between supervisor and colleagues recognition and profile membership, lending additional support to previous research showing the beneficial role of these two job resources for employees [START_REF] Fernet | The effects of work motivation on employee exhaustion and commitment: An extension of the JD-R model[END_REF][START_REF] Sandrin | Effects of workaholism on volunteer firefighters' performance: A moderated mediation model including supervisor recognition and emotional exhaustion[END_REF]. In alignment with Study 1, supervisor recognition was associated with a higher likelihood of membership into the Low Burnout Risk profile relative to the Mentally Distanced, High Burnout Risk, and Moderately High Burnout Risk profiles. Also aligned with Study 1, perceived colleagues recognition was first associated with a lower likelihood of membership into one of the least desirable profile (Moderately High Burnout Risk) relative to all other profiles. Colleagues recognition was also associated with a higher likelihood of membership into the most desirable Very Low Burnout Risk profile relative to the Mentally Distanced, Low Burnout Risk, and High Burnout Risk profiles, supporting previous reports of negative associations between perceived colleagues recognition and burnout [START_REF] Peterson | Reflecting peer-support groups in the prevention of stress and burnout[END_REF]. These findings reinforce the role played by colleagues recognition noted in Study 1.

Many theoretical arguments have been proposed to account for the benefits of colleagues and supervisor recognition. Thus, some have noted that employees involved in high-quality relationships with their colleagues and supervisors were more likely to be trusted by their peers and managers and thus allowed a greater level of autonomy [START_REF] Caesens | A person-centred perspective on social support in the workplace[END_REF].

Likewise, positive relationships between colleagues and supervisors have also been proposed to elicit positive affective responses during social interactions, to reduce stress-evoking uncertainties or conflicts in the work environment through the availability of informative communication flows, and to generate more positive self-perceptions [START_REF] Fall | Recognition at work: Validation of a measuring scale in the context of companies[END_REF].

Conversely, and contrasting with the results obtained in Study 1 in which this source of work recognition displayed strong associations with employees' G-levels of burnout when considered in isolation, perceptions of organizational recognition showed no evidence of associations with profile membership. As already suggested, such results support the idea that perceived organizational, supervisor, and colleagues recognition may be differently related to burnout components [START_REF] Caesens | The impact of work engagement and workaholism on well-being: The role of work-related social support[END_REF][START_REF] Caesens | A person-centred perspective on social support in the workplace[END_REF][START_REF] Fall | Recognition at work: Validation of a measuring scale in the context of companies[END_REF]. In fact, these results go even further in suggesting the benefits of combining high levels of supervisor and colleagues recognition to ensure that G-levels of burnout remain low. However, they also suggest that perceived organizational recognition may not be as effective at decreasing employees' likelihood of membership into the most problematic burnout profiles, despite circumstantial evidence of variable-centered associations shared with G-levels of burnout when considered on their own. This is interesting given that the bulk of prior research has positioned perceived organizational recognition as a positive driver of work-related functioning in a the more, the better perspective (e.g., [START_REF] Caesens | The impact of work engagement and workaholism on well-being: The role of work-related social support[END_REF][START_REF] Fall | Recognition at work: Validation of a measuring scale in the context of companies[END_REF]. Our findings might suggest a more nuanced view of perceived organizational recognition in line with prior variable-centered results revealing curvilinear relations between perceived organizational support and employees' functioning (e.g., [START_REF] Harris | Is more always better? An examination of the nonlinear effects of perceived organizational support on individual outcomes[END_REF]. More specifically, these studies demonstrated that the most adaptive functioning tended to be associated with moderate to moderately high levels of perceived organizational support.

It is possible that moderate levels of perceived organizational recognition coupled with high levels of perceived colleagues and supervisor recognition might decrease the likelihood of membership into the profiles characterized by the highest G-levels of burnout. In addition, high levels of perceived colleagues and supervisor recognition might be the prerogative of employees able to benefit from perceived organizational recognition. These hypotheses are in line with recent findings showing that high levels of perceived supervisor support may coexist with low levels of perceived organizational support, that employees may display a profile characterized by high levels of perceived organizational, supervisor, and colleagues support, and that this latter profile may be associated with the most positive functioning [START_REF] Caesens | A person-centred perspective on social support in the workplace[END_REF]. This interpretation also helps to make sense of the apparent discrepancy in results obtained across Studies 1 and 2 in relation to the role played by organizational recognition. These results encourage scholars to further examine the distinct and complementary role of perceived organizational, supervisor, and colleagues recognition.

Job Satisfaction and Profile Membership

Our results finally revealed differentiated associations between the burnout profiles and job satisfaction. Employees who had average to very low G-levels of burnout (Very Low Burnout Risk, Low Burnout Risk, and Mentally Distanced) displayed higher levels of job satisfaction than Moderately High Burnout Risk and High Burnout Risk employees. These findings are consistent with our hypotheses and with prior results revealing that workers presenting high G-levels of burnout tend to experience tiredness and reduced functional capacity, which in turn seem associated with a higher likelihood of maladaptive functioning [START_REF] Alarcon | A meta-analysis of burnout with job demands, resources, and attitudes[END_REF].

However, not all profiles differed on job satisfaction in a way that matched our expectations [START_REF] Huyghebaert | Examining the effect of affective commitment to the supervisor on nurses' psychological health as a function of internal locus of control[END_REF]. For instance, the levels of job satisfaction observed in the Very Low Burnout Risk and Low Burnout Risk profiles were impossible to differentiate from those observed in the Mentally Distanced profile. This observation suggests that high S-levels of cynicism may not necessarily be harmful on their own. [START_REF] Leiter | Latent burnout profiles: A new approach to understanding the burnout experience[END_REF] also showed that their Disengaged profile (characterized by high levels of cynicism) was not associated with the lowest levels of job satisfaction. It is important to keep in mind that the interpretation of a S-factor taken from a bifactor model differs from that of a first-order factor.

In a bifactor model, S-factors reflect levels of imbalance in employees' cynicism relative to their G-levels of burnout across all dimensions. Thus, this S-factor reflects a feeling of cynicism that is not backed by a matching feeling of emotional exhaustion. This S-factor might be taken to reflect a type of detachment toward work, as an adaptive way to preserve a sense of purpose, once disaggregated from employees' G-levels of burnout. In line with the protective role of S-cynicism observed in the present research, numerous studies have shown that psychological detachment (i.e., the ability to stop thinking about work-related matters during off-job time; [START_REF] Sonnentag | Recovery from job stress: The stressor-detachment model as an integrative framework[END_REF] is associated with positive functioning. The inability to stop thinking about work has the effect of maintaining employees' psychological systems in a state of activation, thus prohibiting recovery. In contrast, psychological detachment may help to turn off these systems, thus supporting recovery from accumulated work-related energy expenditure (see [START_REF] Sonnentag | Recovery from job stress: The stressor-detachment model as an integrative framework[END_REF], for a review).

Similarly, the lowest levels of job satisfaction observed in the Moderately High profile characterized by high levels of reduced professional efficacy and low S-levels of cynicism suggest that professional efficacy and S-cynicism may possibly play a protective role against low levels of job satisfaction. Supporting this interpretation, the results also showed that levels of job satisfaction were higher in the High Burnout Risk profile than in the Moderately High Burnout Risk profile, which presented a configuration opposite to that of the Mentally Distanced profile in displaying low levels of cynicism coupled with high levels of reduced professional efficacy. Clearly, the cynicism/detachment-professional inefficacy combination does seem to play a particularly important role when the job satisfaction of employees with moderate to high G-levels of burnout is considered. Overall, our results confirm the utility of accounting for G/S facets of burnout coupled with the independent reduced professional efficacy factor when contrasting burnout profiles in relation to covariates. Additional studies considering a broader range of positive (e.g., performance, organizational citizenship behaviors) and negative (e.g., absenteeism, counterproductive behaviors) indicators of work functioning are needed to better understand the mechanisms underlying these relations.

Practical Implications

From an intervention perspective, our findings demonstrate that managers should be attentive to employees exposed to, or rather perceiving being exposed to, low levels of recognition at work, and even more importantly low levels of supervisor recognition. Indeed, our results showed that these workers were more likely to experience higher G-levels of burnout and lower levels of job satisfaction. Consequently, changes designed to increase supervisor recognition sustainably might decrease burnout in the long run. Among possible ways to achieve this objective, supervisors might promote a supportive culture, for instance, by promoting fairness in the application of policies [START_REF] Caesens | A person-centred perspective on social support in the workplace[END_REF]. Informal mentoring and social events might also help to build a stronger support climate among employees [START_REF] Newman | The effects of perceived organizational support, perceived supervisor support and intra-organizational network resources on turnover intentions[END_REF]. The endpoint of these strategies is to create a workplace characterized by supportive and positive interactions among colleagues, subordinates, and supervisors.

In terms of research implications, this research represents a first step toward the identification of the most commonly observed burnout configurations. However, more research is needed to systematically understand the work-related characteristics at play in the emergence of these profiles, as well as the additional occupational groups among which those profiles might emerge more frequently [START_REF] Morin | Multiple-group analysis of similarity in latent profile solutions[END_REF]. For instance, previous research has revealed variations in burnout as a function of work settings, job design, or emotional demands (Huyghebaert et al., 2018b;[START_REF] Sandrin | Effects of workaholism on volunteer firefighters' performance: A moderated mediation model including supervisor recognition and emotional exhaustion[END_REF], making it important to verify whether the identified profiles will generalize across work contexts.

Moreover, the present findings add to mounting research evidence supporting the value of adopting a bifactor representation of burnout ratings [START_REF] Hawrot | Factor structure of the Maslach Burnout Inventory-Educators Survey in a Polish-speaking sample[END_REF][START_REF] Isoard-Gautheur | Development and evaluation of the psychometric properties of a new measure of athlete burnout: The Athlete Burnout Scale[END_REF]. They showcase the importance of adequately accounting for the dual G/S nature of burnout, separated from a distinct factor reflecting reduced professional efficacy. Indeed, researchers and practitioners relying on more classical operationalizations of burnout (e.g., scale scores, CFA) ignore the globality/specificity accounted for in a bifactor model. This is likely to mistakenly suggest that the cynicism and emotional exhaustion facets of burnout are reasonably distinct constructs without a common core and yet displaying comparable associations with covariates (Morin et al., 2016a[START_REF] Morin | Complementary variable-and person-centered approaches to the dimensionality of psychometric constructs: Application to psychological wellbeing at work[END_REF]. This erroneous conclusion would stem from the unmodelled role of the G-factor, and obscure the equally important role played by S-burnout components. Ignoring this duality will thus result in a biased, and far more limited, view of the complex reality of the burnout construct.

To avoid burnout estimates reflecting a confusing mixture of G/S variance likely to be tainted by multicollinearity, researchers and practitioners are thus invited to consider adopting a bifactor representation of burnout ratings. This recommendation strongly advocates in favor of a latent variable approach to research. Yet, this recommendation is easy to transpose to the professional context where practitioners and organizations may still want to manually score burnout questionnaires. In these contexts, the present results suggest the need to develop automated scoring procedures relying on calculations similar to those involved in the generation of the factor scores used in the present research (Morin et al., 2016a[START_REF] Morin | Complementary variable-and person-centered approaches to the dimensionality of psychometric constructs: Application to psychological wellbeing at work[END_REF]). Yet, this consideration reinforces the importance for future research to rely on more representative samples prior to the development of any practically-useful scoring procedure.

Limitations and Future Directions

The present research has some limitations. First, this research capitalized on self-report measures, which may have been influenced by self-reported biases and social desirability. Furthermore, it relied on an adapted version of a questionnaire initially developed for students to assess burnout (the MBI-SS) rather than tools directly created for the assessment of work burnout. However, it would be interesting for future studies to investigate the replicability of our results using alternative measures of burnout. Likewise, upcoming studies should incorporate more objective indicators of organizational and individual functioning (e.g., absenteeism), as well as ratings obtained from multiple informants (e.g., supervisors' ratings of performance). Second, this research involved two samples of administrative and technical employees, and firefighters. Other variable-and person-centered studies are still needed to confirm the generalizability of the results demonstrated here and their associations with a broader range of covariates across a variety of countries, cultures, and occupations (e.g., teachers, sales employees, managers) [START_REF] Morin | Multiple-group analysis of similarity in latent profile solutions[END_REF].

Third, although we considered associations between burnout dimensions and profiles and a variety of covariates (i.e., organizational, supervisor, and colleagues recognition, as well as job satisfaction), our cross-sectional design did not allow us to assess the directionality of these associations, nor to consider spurious associations, reversed causality, or reciprocal influence. Consequently, additional longitudinal research would gain from studying the direction of the relations between predictors, outcomes, and burnout. Finally, it would be worthwhile for future studies to consider a greater variety of work-related (e.g., leadership behaviors) or individual (e.g., perfectionism, job crafting) dimensions that may increase or decrease the likelihood of membership in burnout profiles. of the coefficient; OR: Odds ratio; coefficients and OR reflect the relations between recognition and the likelihood of membership into the first listed profile relative to the second one; organizational, supervisor, and colleagues recognition are factor scores with a standard deviation of 1 and a mean of 0; Profiles 1 (Very Low Burnout Risk), 2 (Mentally Distanced), 3 (Low Burnout Risk), 4 (High Burnout Risk), and 5 (Moderately High Burnout Risk).

various

  French fire stations. Most participants worked full time (87.5%). Respondents were aged between 21 and 63 years (M = 41.1, SD = 8.3), had been working as firefighters for an average of 21.5 years (SD = 8.4), and had an average tenure in their position of 5.7 years (SD = 4.9). Procedures and measures were identical to those used in Study 1. Scale score reliability estimates were as follow: (a) emotional exhaustion (α = .89); (b) cynicism (α = .70); (c) reduced professional efficacy (α = .67); (d) organizational recognition (α = .79); (e) supervisor recognition (α = .93); and (f) colleagues recognition (α = .90).

Figure 1 .

 1 Figure 1. Final 5-Profile Solution

  05; ** p < .01; 1 : indicators are factor scores with a standard deviation of 1 and a mean of 0; G = Global factor (bifactor); S = Specific factor (bifactor).

Figure S1 Elbow

 S1 Figure S1

Table 1

 1 Parameter Estimates from the Preliminary Measurement Models(Burnout, Study 1)

		CFA				Bifactor-CFA			Alternative Bifactor -CFA
		PE	Cynicism	Exhaustion		G-factor	S-PE	S-Cynicism	S-Exhaustion	δ G-factor	PE	S-Cynicism	S-Exhaustion	δ
	Items	λ	λ	λ	δ	λ	λ	λ	λ	λ	λ	λ	λ

** -.23** .10** -.33** .49** .47** .19** -Note. * p < .

Table 4

 4 Latent Profile Analyses (Study 2)

	Model	LL	#fp	Scaling	AIC	CAIC	BIC	ABIC	Entropy	aLMR	BLRT
	Unconditional Latent Profile Analyses
	1 Profile	-3381.41	8	1.08	6778.83	6822.69	6814.69	6789.29	Na	Na	Na
	2 Profiles	-3295.57

Table 5

 5 Multivariate Associations between Recognition and Profile Membership (Study 2) Note. * p < .05; ** p < .01; SE: Standard error

		Profile 1 vs 5	Profile 2 vs 5		Profile 3 vs 5		Profile 4 vs 5		Profile 1 vs 4
	Source of Recognition	Coef. (SE)	OR	Coef. (SE)	OR	Coef. (SE)	OR	Coef. (SE)	OR	Coef. (SE)	OR
	Organizational Recognition	0.36 (.34)	1.43	0.36 (.24)	1.43	0.14 (.29)	1.14	0.06 (.25)	1.06	0.30 (.28)	1.35
	Supervisor Recognition	0.52 (.32)	1.69	0.24 (.21)	1.27	0.89 (.28)**	2.44	0.18 (.21)	1.19	0.35 (.29)	1.42
	Colleagues Recognition	1.26 (.33)**	3.53	0.64 (.18)**	1.90	0.55 (.22)*	1.73	0.41 (.15)**	1.50	0.86 (.31)**	2.35
		Profile 2 vs 4	Profile 3 vs 4		Profile 1 vs 3		Profile 2 vs 3		Profile 1 vs 2
	Source of Recognition	Coef. (SE)	OR	Coef. (SE)	OR	Coef. (SE)	OR	Coef. (SE)	OR	Coef. (SE)	OR
	Organizational Recognition	0.30 (.19)	1.35	0.08 (.23)	1.08	0.23 (.33)	1.25	0.23 (.23)	1.25	0.00 (.29)	1.00
	Supervisor Recognition	0.06 (.18)	1.06	0.71 (.24)**	2.04	-0.37 (.35)	0.69	-0.65 (.25)**	0.52	0.29 (.30)	1.33
	Colleagues Recognition	0.23 (.16)	1.26	0.14 (.19)	1.15	0.72 (.34)**	2.05	0.09 (.20)	1.10	0.62 (.31)*	1.86

Online Supplemental Materials for:

Complementary Variable-and Person-Centered Approaches to the Dimensionality of Burnout Among Fire Station Workers Table S1 Standardized Factor Loadings (λ), Uniquenesses (δ)

Preliminary Measurement Models: Estimation

Preliminary analyses conducted in Study 2 matched the analyses conducted in Study 1 (as described in the main manuscript). In addition, tests of measurement invariance across the samples used in both studies were also realized. More precisely, we assessed [START_REF] Millsap | Table S5 Classification Accuracy: Average Probability of Membership into Each Latent Profile (Column) as a Function of the Most Likely Profile Membership (Row) Profile 1 Profile 2 Profile 3 Profile 4 Profile 5 Sample[END_REF]: (1) configural invariance; (2) weak invariance (loadings); (3) strong invariance (loadings and intercepts); (4) strict invariance (loadings, intercepts, and uniquenesses); (5) invariance of the latent variance-covariance matrix (loadings, intercepts, uniquenesses, and latent variances and covariances); and (6) latent means invariance (loadings, intercepts, uniquenesses, latent variances and covariances, and latent means).

Factor scores were saved from the most invariant of those models. Like the chi-square, chi-square difference tests present a known sensitivity to sample size and minor model misspecifications so that recent studies suggest complementing this information with changes in CFIs and RMSEAs [START_REF] Chen | Sensitivity of goodness of fit indexes to lack of measurement[END_REF][START_REF] Cheung | Evaluating goodness-of fit indexes for testing measurement invariance[END_REF] in the context of tests of measurement invariance. A ∆CFI of .010 or less, a ∆TLI of .010 or less, and a ∆RMSEA of .015 or less between a more restricted model and the previous one support the invariance hypothesis.

Preliminary Measurement Models: Results

The goodness-of-fit results from all models are reported in Table S2. These results clearly support the adequacy of the alternative bifactor-CFA models underlying the burnout measure and it superiority relative to the CFA solution. This solution was thus retained for tests of measurement invariance. The results from these tests, also reported in Table S2, supported the configural and weak invariance of the model. However, neither strong, nor strict, invariance was supported, leading us to consider models of partial invariance in which equality constraints had to be relaxed on the intercepts and uniquenesses of only two items. These models of partial invariance were supported by the data, as well as the next model of invariance of the latent variances and covariances. However, the last model Parameter estimates from this final model are reported in Table S3 andrevealed